Keywords:

dit l'adage: il n'y a que les montagnes qui ne se (re)rencontrent jamais.

Finally, I would like to acknowledge my family and friends, who supported me during this time. My mum and dad, without whom everything would have been impossible. You are truly the best and I could write pages and pages for you. To my friend Monica, Flavia, Francesca and Giulia, for your everlasting friendship and for supporting me for more than 20 years. To Matteo, Gianluca, Guglielmo. Thanks. A special thank goes to Andrea, for

I would need way more than few lines to thank all the people who contributed to the completion of this thesis and who have been present during these three years.

First of all, a huge thanks to Laurent Ménard, my thesis director. I would like to thank you for the time spent working together during these last three years, for the determination that you transmitted to me, for the continuous attention to details. It is a huge awareness that I will always bring with me. I truly believe that the success of a thesis lies in the passion and willingness to continue that it leaves on you. And if I feel this way, is thank to you.

I would also like to thank Marc-Antoine Verdier, for its kindness and its help, and Yves Charon, for all the advices that he gave me, always encouraging my research. Thank you for your help, for having opened my mind. You enriched me both on a professional and personal point of view.

The results described in this thesis were accomplished thanks to the help and collaboration of Aurélie Desbrée, Ti↵any Beaumont, David Broggio, Aurélie Forbes, Jad Farah and Luis Ammour. Thank you all for giving this thesis the possibility to be successfully completed, I am grateful.

Surely, a huge thanks goes to all the members of IMNC, especially to Philippe Lanièce, Laurent Pinot, Cédric Esnault, Françoise Lefebvre, Marie-Alix Duval, Nathalie Arlaud and Christiane Robin, who helped me several times and with whom I had the pleasure to collaborate with. And of course to Alexandre Liège, who helped me so many times, I will always remember your contagious laugh.

And a thanks to all the PhD student of the lab I had the chance to meet. I am so grateful for your friendship and I will always bring with me all the moments spent together. A huge thanks to the sweet and kind Hussein, it so hard to find people as genuine as you are. To Tim, a friend full of surprises, I miss our talks. To Lèo, the most organized person ever. The time we spent together in Lyon is vivid in my mind. To Emilie, you are a precious girl. And Luis, for all the music evenings spent together. Lastly, I would like to thank the two people who made these three year unique. Fanny, I need no words to explain us and the relationship we created. It is something so strong and intense, we entered so deeply in our thoughts during all the time spent together. And Enzo, for this last year spent together, everyday. You have been a incredible support for me, at work and outside. For all the time spent together à la BNF, or climbing, or only speaking and chilling. Thank you both for everything. Et comme le Although the understanding of the mechanisms of action of ionizing radiation was far from being complete, their clinical usefulness as a means of treating cancer was immediately attempted: the first advanced breast cancer [START_REF] Bernier | Radiation oncology: a century of achievements[END_REF] and a malignant skin tumor [2] were treated just before the end of the century. Since that time, advances in radiation physics, chemistry and biology have made radiation therapy a recognized medical discipline in which many professionals from many disciplines work together to develop an increasingly more accurate, more e cient and less harmful way of treating diseases.

More generally, while advances in science and technology have led to develop and strengthen all therapeutic approaches to cancer (surgery, radiotherapy and chemotherapy), the global cancer burden is expected to almost double by 2030 and will remain the leading cause of death in both developed and less developed countries [START_REF] Bray | Global cancer statistics 2018: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries[END_REF]. New therapeutic approaches are therefore needed to increase cancer control while always diminishing side e↵ects on normal tissues. These should be more and more targeted and patient-specific.

In that context, Molecular Radiotherapy (MRT), also known as in vivo targeted radiotherapy or targeted radionuclide therapy, is a rapidly expanding and e↵ective approach to the management of cancer and benign diseases. The ability to selectively deliver controlled doses to cells is given by the use of a radiopharmaceutical, the combination of an unstable radionuclide and a molecule, known as vector. This vector specifically binds to target cells while the radionuclide induces treatment e↵ects through its decay. The objective of MRT is to impart lethal damage to target cells, while maintaining possible side-e↵ects to normal tissues at tolerable levels. Due to the heterogeneity of the radionuclide biodistribution between individuals, since each patient has di↵erent radiopharmaceutical uptakes and clearance half-times, a proper personalized dosimetry should be a prerequisite to MRT [START_REF] Eanm | Internal Dosimetry Task Force Report on: Treatment Planning For Molecular Radiotherapy: Potential And Prospects[END_REF]. Indeed, di↵erent dose delivered to patients will result in di↵erent treatment responses, both in terms of e↵ects and toxicity [START_REF] Stabin | State of the Art in Nuclear Medicine Dose Assessment[END_REF][START_REF] Strigari | The evidence base for the use of internal dosimetry in the clinical practice of molecular radiotherapy[END_REF][START_REF] Flux | From fixed activities to personalized treatments in radionuclide therapy: lost in translation?[END_REF].

Introduction

Dosimetry in MRT requires the accurate in vivo estimation of the distribution and pharmacokinetics of the radiopharmaceutical over time and methods for converting the number of decays within a particular volume to the absorbed dose in that volume. In practice, measurements could be performed using planar imaging or Single-Photon Emission Computed Tomography (SPECT), usually in combination with a CT scan. Based on a pre-treatment dose assessment, these estimates should first optimize individual treatment planning. The maximum activity that can be administered to each specific patient could be defined according to the objectives of the treatment and the tolerance doses to organs at risk. Furthermore, post-treatment assessment of the absorbed dose delivered to targeted and non-targeted tissues is also essential to refine the treatment planning. Indeed, it allows correlations to be established with the dose planned in the pre-treatment study as well as with the clinical response in terms of disease control and toxicity on organs at risk. While decades of developments have led treatment planning based on dosimetric calculations and post-therapy dosimetric verification to become part of the clinical routine when performing external radiotherapy treatments, this is still not the case for MRT [START_REF] Strigari | The evidence base for the use of internal dosimetry in the clinical practice of molecular radiotherapy[END_REF][START_REF] Chiesa | The conflict between treatment optimization and registration of radiopharmaceuticals with fixed activity posology in oncological nuclear medicine therapy[END_REF]. This deficiency is mainly related to several challenges. The first and most important is the accurate quantification of activity over time in target regions [START_REF] Flux | The Impact of PET and SPECT on Dosimetry for Targeted Radionuclide Therapy[END_REF], considered as the main source of error in radionuclide therapy dosimetry [START_REF] Dewaraja | MIRD Pamphlet No. 23: Quantitative SPECT for Patient-Specific 3-Dimensional Dosimetry in Internal Radionuclide Therapy[END_REF]. The current limitations are related to the fact that clinical SPECT cameras are optimized for diagnostic imaging (< 200 keV) and therefore not suited to the detection of gamma emitters with an energy greater than 200 keV ( 131 I , 90 Y and 177 Lu), generally used during treatments [START_REF] Ljungberg | 3-D Image-Based Dosimetry in Radionuclide Therapy[END_REF]. Even using high-energy collimators, their limited spatial resolution, which ranges from 10 mm to 20 mm at a distance of 10 cm, hinders the quantification of small heterogenous distributions of activity, which can strongly influence the accuracy of dosimetry at a macroscopic scale [START_REF] Lassmann | The Relevance of Dosimetry in Precision Medicine[END_REF]. In addition, dosimetry also poses logistical and practical challenges, since images need to be acquired at di↵erent times after treatment and automatically require more steps than a single diagnostic nuclear medicine examination [START_REF] Flux | From fixed activities to personalized treatments in radionuclide therapy: lost in translation?[END_REF]. Nowadays, the ordinary data collection must be limited to a few points in time, which leads to a lack of accuracy in the definition of the time-activity curve, adding large uncertainties on kinetics and on the calculation of the absorbed dose [START_REF] Strigari | The evidence base for the use of internal dosimetry in the clinical practice of molecular radiotherapy[END_REF].

The objective of my thesis was precisely to propose new instrumental and methodological approaches aiming to strengthen the control of the dose released to patients during molecular radiotherapy. This is achieved by reducing the uncertainties associated to activity quantification (and therefore to the absorbed dose calculation) through the use of a compact and highly optimized imaging system. This work was carried out within the Instrumentation et Imagerie Radio-Isotopique Clinique et préclinique (IIRIC) team of the Imagerie et Modélisation en Neurobiologie et Cancérologie (IMNC) laboratory based in Orsay, France. It consisted in the development and optimization of a miniaturized, high-resolution mobile gamma camera specifically designed to improve Introduction the individual quantitative assessment of the heterogeneous distribution and biokinetics of the radiotracer before and after treatment administration. The study was focused on the treatment of benign and malign thyroid disease with 131 I . Thanks to its selective tropism for thyroid tissue, the use of 131 I has led to high e ciency of treatments with low toxicity for healthy tissues [START_REF] Stabin | Uncertainties in Internal Dose Calculations for Radiopharmaceuticals[END_REF]. However, the importance of the dosimetric approach in the treatment of thyroid disease is still emphasized to optimize the administered activity according to the desired clinical outcome (euthyroidism or hypothyroidism) or to avoid undesired side e↵ects and long-term consequences [START_REF] Stokkel | EANM procedure guidelines for therapy of benign thyroid disease[END_REF][START_REF] Lassmann | EANM Dosimetry Committee series on standard operational procedures for pre-therapeutic dosimetry I: blood and bone marrow dosimetry in di↵erentiated thyroid cancer therapy[END_REF]. Moreover, the simplicity of the treatment method and its widespread use make it an ideal clinical application to provide abundant data to evaluate the dose-response correlation with our new methods, which could lead, in the long term, to a confident patient-specific treatment approach [START_REF] Lassmann | Dosimetry and thyroid cancer: the individual dosage of radioiodine[END_REF]. This thesis manuscript, which summarizes all my work, is divided into four chapters. The first presents the foundations and role of radiation therapy, with an emphasis on molecular radiotherapy. We will begin by describing the physical and radiobiological properties of radionuclides, followed by an overview of those most commonly used in current clinical practice. Then, the role of dosimetry, as well as a detailed description of the methods and variables involved in its accurate definition, will be discussed. As briefly introduced above, despite its obvious interest, the use of dosimetry in MRT is far from being established in daily clinical practice. The challenges necessary for the dissemination of its use will be presented. They will introduce the interest of going further in instrumental developments to access a better estimate of absorbed dose, as will be discussed at the end of the chapter.

The opening of the second chapter will describe the objectives of this work as an instrumental point of view to overcome the challenges and constraints imposed by dose quantification in MRT. The basic components of our mobile camera based on the indirect detection of high-energy gamma radiation by coupling a parallel hole collimator and an inorganic scintillator to a photodetector with silicon photomultipliers will be detailed. From the presentation of the fundamentals of the interaction between gamma rays and matter, the role of the collimator in the overall performance of the camera will be first highlighted as well as its basic features. The limitations of conventional analytical methods for accurately describing the behavior of collimators when high energy gamma rays are imaged will be addressed to demonstrate the need to use Monte Carlo simulation to optimize their geometry. The collimator section is followed by the basic description and definition of the role of the inorganic scintillator in gamma ray detection. The results of a bibliographic study that led to the choice of the material best suited to our application in terms of energy and spatial properties will be presented. In addition, the influence of the optical coating and coupling will be highlighted. The chapter will continue with a short overview of the existing photodetection systems. Particular attention will be paid to silicon photomultipliers. Their operation principle, intrinsic characteristics and advantages and drawbacks fo gamma imaging will be discussed. The Introduction chapter will conclude with a description of the components and characteristics of the readout electronics and of the imaging reconstruction algorithms used during this work.

The third chapter is the core of the characterization and optimization study of our mobile camera. It begins with the experimental characterization of the scintillatorphotodetector assembly in terms of energy resolution, spatial resolution, distortion and uniformity of response. Di↵erent samples of scintillator with di↵erent thicknesses and optical coatings have been investigated. The optimization of the collimator geometry by Monte Carlo simulation will then be presented. It was carried out by simulating several collimator geometries and by finely characterizing the impact of penetration and scatter on the collimator intrinsic properties. These optimizations led to the choice of the scintillator and collimator to develop the first fully operational 5⇥5 cm 2 field of view camera prototype. Its design and intrinsic properties will be described at the end of the chapter.

Lastly, the fourth chapter will present the characterization of the overall performance of the camera with 131 I sources in order to achieve its complete calibration, which is mandatory for activity quantification. Then, the detector's ability to quantify homogenous or heterogenous distributions of activity will be investigated in a realistic clinical context using 3D thyroid phantoms. At the end of the chapter, we will comment on the results obtained by highlighting both the strengths and limitations of the quantification process used.

In conclusion, after a summary of the main results obtained, some perspectives that could guide future improvements of the detector for its implementation in a clinical environment will be also discussed.

Chapter 1

The Role of dosimetry in Molecular Radiotherapy a Transport of energy by electromagnetic waves or atomic particles is generally referred to as radiation. Radiation can be classified into non-ionizing radiation (as visible light, infrared photons, microwaves and radio waves) and ionizing radiation (as X rays, gamma rays, energetic neutrons, electrons, protons and heavier particles) according to their ability to ionize matter. Ionization is the process by which a particle carry enough energy so that, during an interaction with an atom, it can remove tightly bound electrons from its orbits, introducing a potentially damaging charged particle into the environment of the irradiated medium, thereby enabling its use in medicine for therapeutic procedures. The dosimetry of ionizing radiation is the discipline whose objective is to determine the energy that is deposited by ionizing radiation within the media they interact with, also known as absorbed dose (its SI unit, the Gray, is defined as 1 joule of energy absorbed per kilogram of medium). Dosimetry is of particular interest because it allows to estimate, in advance and retrospectively, the extent of the induced e↵ects on tissues. In the use of ionizing radiation for medical purposes, such as oncological radiation therapy or radio-diagnostics, the interaction of ionizing radiation with organisms is, unlike in other situations, a deliberate objective. In this type of application, especially in radiotherapy, dosimetry requires the highest reasonably achievable accuracy because, in this case, it is necessary to minimize the radiation exposure to healthy biological organs, while maximizing those of the organs for which the radiological treatment is intended.

In the following sections the basis of radiation therapy, and more precisely of molecular radiotherapy, will be presented. Those most commonly used radionuclides in clinical practice will be introduced together with their physical and radiobiological properties. Then we will highlight the role and the methods of dosimetry in MRT followed by the current limitations that prevent its use in clinical practice. At the end, we will discuss the instrumental developments that might allow a more accurate evaluation of the absorbed dose. The past few decades have experienced an extensive progress towards the use of ionizing radiations as a therapeutic modality and this trend increases over time. Thus, among the 18.1 million new cancer cases worldwide, as estimated by the International Agency for Research on Cancer (IARC) in 2018 [START_REF]and 9.6 million cancer deaths in 2018 -The International Agency for Research on Cancer (IARC)[END_REF], about half of patients now receive a radiation therapy in one of its form during the course of their illness, either alone or in combination with other treatment modalities such as surgery or chemotherapy [START_REF] Sjögreen Gleisner | Variations in the practice of molecular radiotherapy and implementation of dosimetry: results from a European survey[END_REF].

Combined modalities treatments are nowadays a flagship for the management of a variety of human tumors, allowing to exploit both their locoregional and systemic treatment's capabilities. Surgery is a key step in cancer control since the beginning of the 19th century. It is used to treat bulky tumors that grow as solid masses of tissues. The challenge of locoregional surgery treatments is to precisely define the margins of the target mass in order to remove it, while preserving as much as possible the healthy surrounding tissues, making them unsuitable for the treatment of small and disseminated tumors and metastasis. When locoregional methods are ine↵ective, more emphasis is put on systemic treatment such as chemotherapy. Chemotherapy is the treatment of disease by the systemic administration of a chemical drug that di↵uses in the whole body throughout the blood stream. It may be curative in cases of lymphomas, testicular tumors and tumors in the pediatric group, but unlike surgery, which targets specific areas, chemotherapy works throughout the body a↵ecting some fast-growing healthy cells, like those of the skin, hair, intestines, and bone marrow, thus increasing the occurrence of side e↵ects. Moreover, while it can be e↵ective in the control of metastatic disease in the short term, relapses usually occur due to cancer cell ability to become resistant to the chemotherapeutic agent, leaving the treatment of such metastatic disease as the largest unmet medical need in cancer therapy. The radiation therapies available nowadays are both locoregional, as with the use of external beams or of sealed radiation sources, and systemic, exploiting the use of radiolabeled vectors referred to as therapeutic radiopharmaceuticals. The latter modality, known as Molecular Radionuclide Therapy (MRT), is a unique treatment technique lying between chemotherapy and external beam radiotherapy, which allows local radiation-induced cytotoxic e↵ects, opening the door both to the treatment of hardly accessible and small tumors and to disseminated micro-metastasis [START_REF] Carlsson | Tumour therapy with radionuclides: assessment of progress and problems[END_REF]. Because of the cytotoxic e↵ect of radiation, the main challenge in MRT treatments is to deliver controlled radiation distribution to a target mass while sparing the surrounding healthy tissues.

Although advances in science and technology have led to important improvement in all therapeutics procedures, the cancer incidence and mortality are rapidly growing and will continue to be the leading cause of death worldwide, especially because of disseminated micro-methastasis [START_REF] Bray | Global cancer statistics 2018: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries[END_REF]. The increasing attention paid to cancer management has led to a growing need for new therapies to increase the fight against common and rare cancers, which must be increasingly targeted and patient-specific. Thus, the Chapter 1 Radiation therapy role of radiation therapy as a therapeutic armamentarium is nowadays fundamental. Rapid progress in this field continues to be boosted by advances in new radionuclides and therapeutic agents, imaging techniques, computer technologies, radiation treatment machines as well as improved understanding of the radiobiology at the base of radiation therapy mechanism [START_REF] Baskar | Cancer and Radiation Therapy: Current Advances and Future Directions[END_REF]. It is nonetheless important to understand that the choice of the optimal therapy approach, or of their combination, is performed thanks to the collaboration of specialized doctors (among whom oncologists, radiotherapists and surgeons), according to the nature of the cancer (histological type, interested organ, tumor size, infiltration to the nearby organs and lymph node, eventual metastasis presence), the toxicity and morbidity risk of the treatment, always with the objective of maintaining a good compromise between the benefit and the risk related to the patient heath status.

Radiation-induced cell death mechanisms a

The use of radiation in treatment is based on the assumption that, at a microscopic scale, incident rays or charged particles could induce treatment e↵ects by ionization in irradiated tissues and organs, with cell death or fibrosis as possible outcomes. According to the type of particle emitted, ionizing radiation can be divided into directly and indirectly ionizing radiation.

Directly ionizing radiation consists of charged particles, such as electrons, protons, ↵ particles and heavy ions, which deposit their energy in the medium through direct Coulomb interactions with the orbital electrons of atoms and molecules in cells. In general, charged particles cause both excitation and ionization of the atoms in the biological target. While excitation corresponds to the moving of a bound electron to a higher energy level of the orbital state by absorbing energy, ionization occurred when enough energy is given to the electron to remove it from the host atom. The irradiation of cellular material with such electron gives rise to the production of a flux of secondary electrons, unbound and energetic, which will give their energy to the surrounding medium.

Indirectly ionizing radiation consists of neutral particles as X rays and gamma ray which deposit their energy in the absorbing medium through a two-step process. Firstly, they interact with the medium producing a charged particle, which, secondly, acts as a directly ionizing particle, depositing at least part of its kinetic energy through Coulomb interactions with orbital electrons.

X rays and gamma radiation are the two major forms of electromagnetic radiation used in radiotherapy. X rays are produced when high speed electrons collide with a material of high atomic number like tungsten-molybdenum while gamma rays are produced by intra-nuclear disintegration. Photon beams are the main source of treatment in External Beam Radiotherapy (EBRT), as will be described in Sec 1.1.2. Although the collective ionization behavior of large numbers of photons can be predicted with great accuracy, individual photon interactions occur randomly and, passing through any type of matter, a photon may interact one or more times, or never. However in

Radiation therapy

Chapter 1 each interaction (which normally involves a photoelectric event, a Compton event or a pair production event, see Sec. 2.2), secondary particles are produced, usually electrons or another photon of reduced energy which itself can undergo further interactions. The electrons undergo many ionizing events relatively close to the site of their creation and, therefore, contribute mostly to a local absorbed dose. Any secondary photons which may be created carry energy further away from the initial interaction site and, following subsequent electron-producing interactions, are responsible for the dose deposition occurring at sites which are more distant from the original interaction. Instead in nuclear medicine, three types of ionizing radiation play a relevant role in tumor and normal tissue e↵ects: beta radiation, alpha particles and Auger electrons.

particles are fast electrons, which result from the weak decay of a neutron into proton, in combination with the emission of an electron antineutrino. Since beta particles are electrons, once ejected from the host atom, they behave exactly as do the electrons created following the passage of a gamma ray, giving up their energy to other atoms and molecules through a series of collisions. However, in a beta decay the emitted electron is characterized by a continuous energy spectrum which can go from 0 to the maximum energy available for the reaction rather than being emitted with a fixed, discrete energy. For most beta sources, this maximum value ranges from a few tens of keV to hundreds of keV. Moreover, particles follows a very tortuous path through their Coulomb interactions with the atomic electrons (they are deflected at almost each collision).

An ↵-particle is a mono-energetic helium nucleus ( 4 2 He). Thus, it is much heavier than other subatomic particles emitted from decaying radionuclides and nuclear reactions. With a charge of +2, ↵-particles interact through Coulomb forces between their positive charge and the negative charge of orbital electrons within the absorber atoms. Thanks to their mass (4 MeV) compared to the one of atomic electrons (0.5 MeV), they are almost undeviated during the interaction, following a straight path in matter.

Auger electrons are low energy orbital electrons, which arise from the electronic shells of radionuclides when they decay by electron capture (20-2000 eV). In this decay mode, which is mediated by the weak force, an inner atomic electron is absorbed by a nuclear proton, changing it into a neutron in combination with the emission of an electron neutrino. The vacancies in the inner electron shells, usually in the K or L shell, are filled by a downward electron transition from higher energy shells. The energy transitions di↵erence can be released both by the emission of one or more characteristic X-ray photons or by the transfer of this energy to an atomic electron, which is ejected with a characteristic kinetic energy, the Auger electron.

Regardless of the nature of the radiation, the mechanism by which energy is transferred from the primary radiation to the biological target is always through the secondary electrons. Once these secondary electrons are created, they could induced damage to biological targets and trigger a consequent possible cytotoxic e↵ect. Radiation damage to the cell can be caused by the direct or indirect action of radiation on the DNA (DeoxyriboNucleic Acid) molecules (Fig. 1.1). In the direct radiation action, the radiation hits the DNA molecule directly, disrupting the molecular structure [START_REF] Bailey | Physicists in Medicine, éds, Nuclear medicine physics: a handbook for teachers and students[END_REF]. In the indirect radiation action, the radiation hits the water molecules present in cells surrounding environment, with the consequent creation of an unstable and highly reactive molecular species, known as free radicals. Free radicals are characterized by an unpaired electron in the structure, which is very reactive and therefore reacts with DNA molecules to cause a molecular structural damage. The free radicals created can also migrate and either donate or accept an electron from other molecules, damaging the adjacent biological targets. In normally oxygenated mammalian cells, because water constitutes nearly 70% of their composition, the direct e↵ect accounts for about one third of the radiations induced damage, while the chemical interactions involving free radicals accounts for about two thirds of the damage, being thus the main cause of radiation damage [START_REF] Sage | Radiation-induced clustered DNA lesions: Repair and mutagenesis[END_REF].

In both cases, whether the action of ionization radiation is direct or indirect, the DNA molecule is the main target of radiation in cells. Several types of radiation-induced DNA damage have been identified, such as DNA base damage, DNA protein crosslinks, Single-Strand (SSBS) and Double-Strand DNA breaks (DSBS). For a dose of 1-2 Gy, the lesions induced in the DNA of a cell are approximately >1000 for base damages, around 1000 for SSBS and around 40 for DSBS [START_REF] Bailey | Physicists in Medicine, éds, Nuclear medicine physics: a handbook for teachers and students[END_REF]. After undergoing one of these damages, a cell is regarded as having been "killed" by radiation only if it has completely lost its reproductive capability, through mechanisms as apoptosis and necrosis [START_REF] Eriksson | Radiation-induced cell death mechanisms[END_REF]. In addition to the number of DNA lesions generated, multiple enzymatic DNA repair mechanisms exist, which depend on the type of DNA damage [START_REF] Bailey | Physicists in Medicine, éds, Nuclear medicine physics: a handbook for teachers and students[END_REF]. A general view of DNA damages, repair mechanisms and consequences in mammalian cells is shown in Fig. 1.2. The abnormal bases and single-strand breaks (SSBS), are eliminated by the baseexcision repair (BER) mechanism, and double-strand breaks (DSBS) repaired by recombination repair mechanisms (mostly non-homologous end joining, NHEJ, or homologous recombination, HR). Whereas SSB and base damages can be usually correctly repaired, this is not always the case for DSBS. Therefore, DSBs are considered to be the most severe because they are more likely to result in chromosome aberrations and genomic instability which can ultimately lead to lethal cell damage.

Radiation therapy

As noted above, the energy transfer to the absorbing medium is via secondary electrons created by the passage of the primary ionizing particle or ray. A parameter used to measure the linear rate at which radiation is absorbed in the absorbing medium by the secondary particles is the Linear Energy Transfer (LET). It is defined by the Chapter 1

Radiation therapy

International Commission on Radiation Units and Measurements (ICRU) as the ratio dE/dl, where dE is the average energy locally imparted to the medium by a charged particle of specified energy in traversing a distance dl. The LET is usually expressed in kiloelectronvolt per micrometer (keV/µm). X rays and gamma rays are low LET radiations (around 0.2 keV/µm) which mainly produce individual repairable DNA lesions. Beta-decay emitted particles have low LET radiation too, which produce sparse ionizations and excitation within DNA along a contorted track, causing easily repairable individual DNA lesions. Auger electrons have an intermediate LET (4-26 keV/µm) and emit cascades of electrons that produce highly localized ionizations and excitations within DNA, inducing poorly repairable damage. Alpha particles have high LET (50-230 keV/µm, 250 times higher that the one of ) and produce densely localized ionizations and excitations along a linear track. They are thus highly ionizing and cytotoxic agents resulting in locally, multiple damaged sites that are poorly repairable.

For radiobiological studies, the concept of LET is problematic since it relates to an average linear rate of energy deposition but, at the microscopic level (i.e. at dimensions comparable with the cellular targets), the energy deposited per unit length along different parts of a single track may vary dramatically. In particular, as charged particles lose energy in their passage through a medium via the result of collision and ionizing processes, the LET rises steeply to its highest value towards the very end of their range. The change in LET value along the track length is one reason why average LET values correlate poorly with observed (i.e. macroscopic) biological e↵ects. For example, depending on the energy, low LET radiations can have di↵erent biological e↵ects: 29 kVp X ray is more e↵ective in cell death than 200 220 kVp X-rays [START_REF] Hunter | Review of relative biological e↵ectiveness dependence on linear energy transfer for low-LET radiations[END_REF] or Tritium beta ray (5.7 keV) is much more e↵ective than 15 MeV electrons [START_REF] Neshasteh-Riz | Relative Biological E↵ectiveness (RBE) of 131i Radiation Relative to 60co Gamma Rays[END_REF]. For these reasons, the directly measured Relative Biological E↵ectiveness (RBE) is of greater use as an index of the biological e cacies of di↵erent radiation types. The RBE of a radiation is defined as the ratio of the dose required to produce the same biological end point as a reference low LET radiation. The reference radiation are usually 60 Co gamma rays or high energy (250 kVp) X rays. In general, for many relevant cellular e↵ects, as LET increases, the relative biological e↵ectiveness (RBE) of radiation tends to increase as well. In reality significant di↵erences in RBE of particles have been reported with results varying for di↵erent physical parameters of the beam and biological endpoint and tissue of choice [START_REF] Arena | Space radiation e↵ects on plant and mammalian cells[END_REF]. These observations imply that high-LET radiation may not only cause more serious damage to the cell when compared to low-LET, but also activate di↵erent repair mechanisms. These repair mechanisms are the prerequisites of radiation therapy procedures: the DNA repair capacity is generally greater in healthy cells than in cancerous cells [START_REF] Willers | Recombinational DNA Repair in Cancer and Normal Cells: The Challenge of Functional Analysis[END_REF]. However, although normal tissue toxicity is lower, it is non-zero and it shows variability between patients, indicating inter-individual di↵erences in the intrinsic radiosensitivity [START_REF] Borràs-Fresneda | Di↵erences in DNA Repair Capacity, Cell Death and Transcriptional Response after Irradiation between a Radiosensitive and a Radioresistant Cell Line[END_REF]. Thus, all radiotherapy procedures, which are going to be synthetically presented in the following sections, have to be performed with the goal of achieving a high probability of local target irradiation, while sparing the surrounding healthy tissues and critical organs.

Radiation therapy

Chapter 1

External beam radiation therapy a

External Beam Radiotherapy (EBRT), the most frequently used form of radiotherapy [START_REF] Mehta | Radiotherapy: Basic Concepts and Recent Advances[END_REF], refers to the delivery of tightly targeted radiation beams from outside the body. The absolute number of patients that would have an indication for radiotherapy at least once during the course of the disease would increase from 1700000 patients approximately in 2012 to 2000000 in 2025 [START_REF] Borras | How many new cancer patients in Europe will require radiotherapy by 2025? An ESTRO-HERO analysis[END_REF]. Since the introduction of cobalt therapy in the early 1950s, the goal of EBRT was to cure locoregional cancer, that is localized cancers. Externally applied radiation beams can be produced by several approaches: radioactive sources, such as 60 Co, that emit high energy gamma rays (1.17 and 1.33 MeV), high-energy X-rays or photons produced by linear accelerators (LINAC, 4-25 MV) or particle beams-electrons, protons, or heavier ions-accelerated by other types of accelerators (>70 MeV).

In clinical practice, the mostly radiotherapies approaches used photons (for the treatment of solid tumors) and electrons (for treating superficial lesions) produced by medical linear accelerators. A linear accelerator is a device that uses high Radio-Frequency (RF) electromagnetic waves to accelerate charged particles (i.e. electrons) to high energies in a linear path, inside a structure called the accelerator waveguide. If needed, these electrons then collide with a heavy metal target and as a result of these collisions, high-energy X rays are produced from the target. LINAC have collimators and filters to shape the radiation beam thereby producing more conformal irradiation of the target tissues while minimizing normal tissue exposure (Fig. 1.3). They are also equipped with onboard imaging devices that can verify the accuracy of treatment delivery. Multiple companies manufacture the technologies, o↵ering a range of highenergy X-rays ( Radiation therapy radiotherapy includes direct damage to DNA in cells, or indirect damage caused by the X-rays colliding with molecules within the target, from primary and secondary ionisation events. Conventional radiotherapy is given with curative intent in fractions of 1.8-2.0 Gy daily on weekdays up to 35 times. The purpose of fractionation, introduced more that 80 years ago, is to maximize the killing of cancer cells, while minimizing e↵ects on normal tissue in and around the target volume. Hypofractionation refers to giving a lower number of fractions larger than 2.0 Gy. Several large-scale clinical trials have found that giving a higher (>2.0 Gy) dose of radiotherapy per fraction, and a fewer (<35) number of fractions in total, can be as safe as conventional radiotherapy, resulting in a possible change in the standard of care for common cancers, such as breast cancer and prostate cancer [START_REF] Group | The UK Standardisation of Breast Radiotherapy (START) Trial A of radiotherapy hypofractionation for treatment of early breast cancer: a randomised trial[END_REF][START_REF] Dearnaley | Conventional versus hypofractionated high-dose intensity-modulated radiotherapy for prostate cancer: 5-year outcomes of the randomised, non-inferiority, phase 3 CHHiP trial[END_REF]. These changes, together with improvements in engineering and computing and to the integration of imaging information in every phase of the treatment, highly improving the accuracy of treatments with resulting improvements in the outcome. At present, the precise identification of target volumes for treatment planning is mainly based on the integration of radiological/metabolic imaging, such as Computed Tomography (CT) and Magnetic Resonance Imaging (MRI). Tumor localisation immediately before and during treatment delivery by means of image-guided techniques is becoming a part of clinical practice and is a fundamental prerequisite for high-precision RT. These progresses allowed to evolve from conventional LINAC towards highly conformal techniques, such as Intensity Modulated Radio Therapy (IMRT) and Stereotactic Body Radiation Therapy (SBRT).

Intensity modulated RT (IMRT) was introduced in the early 1990s as an enhancement in the delivery of three-dimensional conformal radiation therapy. This procedure has been made possible by the use of computer controlled multileaf collimators and advanced Treatment Planning System (TPS) with optimized algorithms [START_REF] Nahum | Radio)Biological Optimization of External-Beam Radiotherapy[END_REF]. In opposition to standard planning techniques, where the dose distribution can only be modified by changing, for example, the field weight, angle and shape, IMRT allows to implement inverse treatment plannings: the doses and dose/volume constraints for the tumor and the surrounding normal organs are fixed in advance by the oncologist and then the TPS determines the optimal fluence of each field resulting in an adjusted dose distribution with low dose to normal organs situated tightly adjacent to the target volume (Fig. 1.3).

Stereotactic Body RT (SBRT) refers to the precise irradiation of an image-define lesion by delivering one or a few large dose fractions [START_REF] Brown | The Tumor Radiobiology of SRS and SBRT: Are More than the 5 R's Involved?[END_REF]. It is an increasingly popular ways of delivering radiotherapy because its delivery techniques enable the production of very conformal single high dose radiation field with a steep gradient between the target and the surrounding organs at risk, opening the possibility to ignore the classic fractionation paradigm [START_REF] Hellevik | Radiotherapy and the Tumor Stroma: The Importance of Dose and Fractionation[END_REF]. Essential requirements in SBRT are the accuracy of target delineation and the implementation of target motion management made by image guidance, to ensure a perfect match between dose deliver and patient anatomy. At present, there are a variety of systems available for SBRT. Some of them are based on the traditional LINAC gantry while others have changed the design searching for higher accuracy through greater non-coplanar beam arrangements, as for example CyberKnife

(Accuray, Inc., Sunnyvale, CA, USA) and VERO (Mitsubishi Heavy Industries, Ltd., Japan, and BrainLAB AG, Feldkirchen, Germany) systems, which use multiple beams of radiation converging in three dimensions to focus precisely on a small volume [START_REF] Mehta | Radiotherapy: Basic Concepts and Recent Advances[END_REF].

Technological advances in EBRT seem likely to reach a plateau in the near future [START_REF] Garibaldi | Recent advances in radiation oncology[END_REF]. Even if the tendency towards cost-e↵ective treatments is still open, the biggest developments are linked to a better understanding of tumor biology and consequently in the growth of radiobiology-oriented treatment planning optimisation [START_REF] Nahum | Radio)Biological Optimization of External-Beam Radiotherapy[END_REF], the improvement of the biological e ciency and ballistic selectivity of the radiation beam through the use of protons, light ions such as carbons and oxygen or very high energy electrons and the implementation of new irradiation modalities (spatial and temporal fractionation, ultra-high dose, nanoparticles, ...) [START_REF] Lacombe | Particle therapy and nanomedicine: state of art and research perspectives[END_REF]. The main drawback of the treatment with X rays is that the dose deposition slowly decreases with respect to the penetration depth, which makes it di cult to accurately reach tumors deeply located in the body while simultaneously preserving the surrounding tissues. Thus, gantry-based high-energy beam hadrontherapy is vastly growing. Indeed, inversely to X rays, the hadrons (mainly proton and cobalt ions, 70-250 MeV) deposit almost all of their energy in a sharp peak at the end of their path, in the so-called Bragg Peak, allowing to obtain a more conformal treatment, better sparing the healthy tissues located in proximity of Figure 1.4: Scheme of the inverse planning principles. The desired 2-dimensional fluence map is created by irradiating a sequence of di↵erent fields shaped with a multileaf collimator. An appropriate modulation of the fluence results in a target conformal dose distribution [Source [START_REF] Siggel | Concepts for the e cient Monte Carlo-based treatment plan optimization in radiotherapy[END_REF]]

Chapter 1
Radiation therapy the tumor and allowing a higher control of the disease. Moreover, ions deposit large amount of energy in their path; the produced ionization is really dense and thus able to induce direct multiple strand brakes in the DNA, leading to almost non-repairable damage (higher RBE then photons). This e↵ect opens the way to the treatment of tumors which are resistant to X rays as prescribed by standard medical protocols (Central nervous system cancers, eye cancer). Although huge interest is placed on hadrontherapy performances, the cost of the equipment is still prohibitive for using it on a large number of patients as an everyday clinical practice (89 center in the World, 2019, more then 20 under construction [START_REF] Ptcog | Particle Therapy Co-Operative Group: Facilities under Construction[END_REF][START_REF] Ptcog | Particle Therapy Co-Operative Group: Facilities in Operation[END_REF]) Despite the proved e cacy of EBRT, many cancer patients su↵er from small local recurrence and/or metastases, which cannot be treated with external beams, being too small to be accurately delimited or located too deep in the body. These clinical issues, which require alternative therapeutic strategies, stimulate the development of internal radiation therapy [START_REF] Hellevik | Radiotherapy and the Tumor Stroma: The Importance of Dose and Fractionation[END_REF].

Internal radiation therapy a

Nuclear medicine is a branch of medicine that uses radioactive compounds to diagnose, treat and monitor the response of a variety of diseases, including many types of cancers, heart disease, gastrointestinal, endocrine, neurological disorders and other pathologies within the body. It exploits the spontaneous decay of unstable isotopes to lower energy states, releasing energy in the process. This radioactive decay can result in the emission of therapeutically useful ionizing radiation such as photons, electrons, or other decay products. The radioactive material can be placed close to the target tissue through the use of external implants, the brachytherapy procedures, or can be administered to the patient in the form of radiopharmaceuticals, for molecular radiotherapy procedures.

Brachytherapy a

The word brachytherapy, from the ancient Greek words brachios for "short distance", refers to the therapeutic use of sealed high-energy radioactive sources, called implant, placed directly within or close to the target organ, allowing the delivery of high total dose of radiation in the volume of interest with low deposition in the surrounding normal tissues [START_REF] Mehta | Radiotherapy: Basic Concepts and Recent Advances[END_REF]. Since the first use of radium in the treatment of cancer in the late 19th century, brachytherapy is being used in focused and short treatment courses in which the target volume is clinically accessible and delimitable. Brachytherapy implants come in di↵erent forms, including thin wires, tubes (catheters), ribbons, needles, capsules and small seeds [START_REF] Mehta | Radiotherapy: Basic Concepts and Recent Advances[END_REF]. Commonly used radionuclide are 137 Cs and 131 Cs, 125 I, 192 Ir, 60 Co. The delivery device may be inserted into a body cavity such as the uterus (intracavitary brachytherapy), into a lumen like esophagus (intraluminal brachytherapy) or into body tissues such as prostate or breast (interstitial brachytherapy). Depending

1.2. MOLECULAR RADIOTHERAPY Chapter 1
on the type of implant, the radiation source can deliver a single high dose of radiation in a very short period of time (high-dose-rate temporary brachytherapy, HDR) or continuous low doses of radiation over hours, days or permanently (low-dose-rate brachytherapy, LDR) [START_REF] Mehta | Radiotherapy: Basic Concepts and Recent Advances[END_REF]. Brachytherapy dose rates could vary from 0.4-2.0 Gy/h for LDR to more then 12 Gy/h for HDR [START_REF] Skowronek | Current status of brachytherapy in cancer treatment -short overview[END_REF]. Nowadays, brachytherapy is a standard therapy for cervical, uterus and skin cancers and has an important part of the treatment guidelines for other malignancies, like cervix, breast, prostate, and intraocular cancers. While it was initially used only for cancer treatments, BR is now considered to be also helpful in non-malignant diseases as in prevention of vascular restenosis and in keloids treatment [START_REF] Skowronek | Current status of brachytherapy in cancer treatment -short overview[END_REF]. The e ciency of brachytherapy, as compared to the one of EBRT, is attributed to the implants ability of deliver a higher concentrated radiation dose more precisely to tissues, using nuclide that emits low range particles, as it will be described in Section 1.2.1. However, in contrast to EBRT, brachytherapy is highly invasive, requiring insertion of implants under sedation or anesthesia and high radiation exposure of the medical sta↵ during implantation.

Targeted radiation therapy a

Molecular radiotherapy (MRT), also known as "in vivo targeted radiotherapy" or "targeted radionuclide therapy", is a growing alternative in the treatment of tumors and benign diseases because it o↵ers the advantage of target selectivity achievable in EBRT treatments, while being systemic as chemotherapy. Its features are rather similar to those of low-dose-rate brachytherapy, without being invasive because the mechanism of treatment is performed through the use of a radiopharmaceutical that is usually administered intravenously, even though oral, intra-cavity, and intra-arterial administrations are also employed. The main advantage of MRT is the ability to selectively deliver controlled dose distribution to cancer cells that causes minimal toxicity to surrounding healthy tissues, using optimized vector that deliver radiation directly into the tumor cells. Molecular radiotherapy is currently rapidly evolving thanks to the joint development of new tracers and innovative radionuclides, which open the field to a more targeted treatment of cancers.

In the following sections, the therapeutic impact in the field of MRT procedures will be synthetically presented, starting from the definition of the key properties of a radiopharmaceutical till the description of the MRT protocols currently used in clinical practice. An highlight of the future prospects in MRT will also be provided.

Molecular Radiotherapy a

The use of radioactive compounds for diagnosis and therapy has origins dating back to the 1910s, with the tracer principle highlighted by the radiochemist Georg Karl Von Hevesy. This principle is at the basis of nuclear medicine and relies on the evidence that a minute quantity of a radiopharmaceutical introduced into the body Chapter 1

Molecular Radiotherapy

does not raise any pharmacological response but can be used to monitor the patient's physiological function [START_REF] Sawin | Radioiodine and the Treatment of Hyperthyroidism: The Early History *[END_REF][START_REF] Wessels | Overview of dosimetry for systemic targeted radionuclide therapy (STaRT)[END_REF]. Nuclear medicine first became recognized as a potential treatment speciality in 1946 when Sam Seidlin described in the Journal of the American Medical Association the success of radioactive iodine in treating a patient with advanced thyroid cancer [START_REF] Seidlin | Radioactive iodine therapy: Effect on functioning metastases of adenocarcinoma of the thyroid[END_REF]. Since that time, developments in interventional nuclear medicine were driven by advances in a multidisciplinary science that includes physics, chemistry, computing, mathematics, pharmacology and biology.

Molecular radiotherapy consists in the administration of a therapeutic radiopharmaceutical that specifically binds to target cells producing a local radiation-induced cytotoxic e↵ects. A radiopharmaceutical is composed by an unstable radionuclide linked to a molecule, often called vector, which targets organs though mechanisms of localization that could be as easy as normal metabolic di↵usion or as sophisticated as an antigen-antibody reaction. While the vector targets specific biological sites, the therapeutic radionuclide emits radiation through its decay in the form of charged particles (mainly electrons from beta-decay or alpha particles), which induce treatment e↵ects by ionization in tissues by the triggering of radiobiological mechanisms which could lead to cell death. Apart from charged particles, many therapeutic radionuclides have an associated nuclear emissions (alpha and beta particles, gamma rays), which result in a single or double gamma rays emission, opening the possibility to perform the imaging of the distribution of the radiopharmaceutical after its administration. The resulting gamma-rays can be detected with a scintillation cameras, either in planar mode or employing Single-Photon Emission Computed Tomography (SPECT), or Positron Emission Tomography (PET), in order to characterize the biokinetics of the radiopharmaceutical in both target organs and organs-at-risk, thus enabling quantitative imaging and subsequent individual-patient dosimetry, as will be described in Section 1.3.

The choice of a radiopharmaceutical for a specific therapeutic application relies both on the definition of the tumor-seeking agent with suitable biochemical properties (target selectivity and compatibility) and on the choice of the radionuclide to deprive target cells of their reproductive potential. In next section, a description of the physical and radiobiological basis of radionuclide is given. After a brief description of the vector mechanisms of localizations, the mostly used ones in clinical practice will be presented together with some future perspective and developments.

Physical and radiobiological properties of radionuclides a

The wide variety of tumors and radionuclides make evident that the optimization of MRT is highly dependent on the matching between many biological and physical parameters. The choice of the most appropriate radionuclide for a given therapeutic application mainly depends on its chemical and physical properties and on its biological behavior and stability [START_REF] Das | Options to meet the future global demand of radionuclides for radionuclide therapy[END_REF][START_REF] Dash | Targeted Radionuclide Therapy -An Overview[END_REF]. Moreover, the type of particle emission required for a given application depends also on the nature of the disease to be treated, the volume and size of the target, the distribution of target cells within the target volume as well

as the pharmacokinetics of the tracer and the degree of heterogeneity of radionuclide deposition [START_REF] Dash | Targeted Radionuclide Therapy -An Overview[END_REF]. Radionuclide features must be also consistent with the suitable selective vectors which should be minimally reactive with normal tissues and highly specific to a target, that could vary from a patient to another in terms of histology, volume, and anatomic distribution [START_REF] Wessels | Physical and chemical properties of radionuclide therapy[END_REF].

Regarding its chemical properties, a vector can be as simple as an ion or a more complex agent such as monoclonal antibodies. In all cases, the selected radionuclide is expected to produce a stable radionuclide-vector bond with in vivo kinetic inertness towards dissociation, substitution or redox reactions [START_REF] Zweit | Radionuclides and carrier molecules for therapy[END_REF]. It should be available with high purity (radionuclidic, radiochemical, and elemental purity), free from trace metal contamination and with an high specific activity (activity per quantity of a radionuclide) [START_REF] Dash | Targeted Radionuclide Therapy -An Overview[END_REF][START_REF] Zweit | Radionuclides and carrier molecules for therapy[END_REF]. The important physical properties to consider include the radionuclide half-life and the type, energy and range in tissues of the emitted radiation. The radionuclide physical half-life must be compatible with the pharmacokinetics of the targeting molecule so that a considerable fraction of the radiation dose can be delivered after that it has been fixed in the target-organ [START_REF] Wessels | Physical and chemical properties of radionuclide therapy[END_REF]. A too short physical half-life does not give enough time to the targeting process to take place, leading to a huge non-specific irradiation and potential toxicity for healthy tissues. On the other hand, a too long physical half-life could reduce the irradiation of the targets with a consequent increase of the amount of activity to be injected in order to obtain the necessary treatment irradiation, with the result of serious radioprotection issues associated to patient care. For what concern the other physical properties, the emitted charged particle should delivers a high amount of energy to cause a severe degree of DNA damage, but only close to the specific target site. Therefore, according to the specific application, one needs to match the energy, and hence the range of the radiation, to the cellular distribution of the carrier molecule [START_REF] Zweit | Radionuclides and carrier molecules for therapy[END_REF] and to the shape of the target organs.

Beta emitters a

Until now, MRT has been mainly performed using particle-emitting radionuclides [START_REF] Marcu | Global comparison of targeted alpha vs targeted beta therapy for cancer: In vitro, in vivo and clinical trials[END_REF]. The particle continuous energy spectrum ranges from a few tens of keV to hundreds of keV and consequently the their range in tissue can vary from 1 to 12 mm. This relatively long range in tissue covers hundreds of cell diameters (Fig. 1.5), making -emitters suitable for the treatment of medium-large or heterogeneous tumors but unsuitable for small clusters of tumor cells (non-selective energy deposit). Among the -emitting radionuclides employed in MRT, 90 Y is the most energetic, with a particle energy ranging from 0 to 2284 keV (mean energy of 933 keV) and a corresponding maximum range in water of 11.8 mm, even if 90% of its energy is deposited within the first few mm [START_REF]Beta emitters and radiation protection[END_REF]. Most of the radionuclides used in clinical trials, such as 131 I, 90 Y, 177 Lu, 188 Re, 186 Re, 32 P , 89 Sr, 153 Sm or 67 Cu emit particles with a low LET of ⇠0.2 keV/µm, which mainly produce individual repairable DNA lesions (see Sec. 1.1.1). This low LET has complementary consequences on the therapeutical e↵ects of particles. On one hand, the long range of electrons has the disadvantage of causing possible cytotoxic damage in surrounding non-targeted cells, resulting in a larger dose to healthy tissues. On the other hand, this e↵ect, at a cellular level, reduces the need of targeting each single cell in order to achieve an e↵ective treatment in medium-large volumes. However, particles need several hits at the level of the cell nucleus to cause cytotoxic e↵ects and a much higher radioactivity must reach the target tissue to kill it e↵ectively compared to high LET ↵-particles.

Alpha emitters a

↵-particles are highly ionizing and cytotoxic agents with a LET of 50-230 keV/µm, 250 times higher that the one of (see Sec. 1.1.1). The currently available ↵-particle emitting radionuclides, such as 223 Ra, 213 Bi, 212 Bi, 211 At, as well as 212 Pb and 225 Ac, have energies that goes from 5 to 9 MeV [START_REF] Dekempeneer | Targeted alpha therapy using short-lived alpha-particles and the promise of nanobodies as targeting vehicle[END_REF]. Their short range in tissue (50-100 µm), which varies between 5 and 10 cell diameters (Fig. 1.5), results in a significant amount of energy deposited near the point of emission (see Sec. 1.1.1) which induce DNA damages di cult to repair by the cell. These characteristics give ↵-emitters the potential to deliver radiation in a highly localized and toxic manner, making them e↵ective in eradicating small-volume tumors (or small-residual after surgery), isolated cancerous cells or micro-metastatic tumors, with little exposure to surrounding healthy tissues.

↵ and -emitters have been compared in vitro, in vivo and in several clinical trials [START_REF] Marcu | Global comparison of targeted alpha vs targeted beta therapy for cancer: In vitro, in vivo and clinical trials[END_REF]. Henriksen et al. investigated the bone-seeking property and the potential exposure of red marrow by the ↵-particle emitter 223 Ra compared with those of the -emitter 89 Sr. They showed that 223 Ra can, together with its daughter radionuclides ( 219 Rn, 215 Po), deliver an intense and highly localized radiation dose to the bone surfaces with substantially less irradiation to healthy bone marrow compared to standard bone-seeking -emitters [START_REF] Henriksen | Targeting of Osseous Sites with alpha-Emitting 223ra: Comparison with the beta-Emitter 89sr in Mice[END_REF]. Despite its positive features, the use of ↵-emitters in clinic is reduced because of the limited radionuclide availability and the short physical halflife of progenitors, which limits radionuclide shipment from the site of production and causes a high number of disintegrations before radiopharmaceutical targeting [START_REF] Dekempeneer | Targeted alpha therapy using short-lived alpha-particles and the promise of nanobodies as targeting vehicle[END_REF]. Fur-

Molecular Radiotherapy

Chapter 1 thermore, several other issues concerning ↵-particle emitters should be addressed, such as radiolysis of the ↵-vector bound and the unstable chemical bond of daughter isotopes with the targeting vector [START_REF] Dekempeneer | Targeted alpha therapy using short-lived alpha-particles and the promise of nanobodies as targeting vehicle[END_REF]. Radiolysis is the radiation-induced chemical dissociation of molecules and its occurrence when using ↵-particles is high because of the magnitude of energy deposits by ↵-emitters. This cause the radiolabeling of certain vectors with an ↵-particle emitter using high levels of radioactivity while maintaining appropriate biological properties to be challenging [START_REF] Zalutsky | Labeling monoclonal antibodies and F(ab')2 fragments with the a-particle-emitting nuclide astatine-211: Preservation of immunoreactivity and in vivo localizing capacity[END_REF]. Zalutsky et al. investigated the e↵ects of radiation dose on the synthesis of SAB (N-succinimidyl 3-211 At-astatobenzoate), a compound used for labeling antibodies with the ↵-particle emitter 211 At. They showed that for the high activity levels required for MRT, the radiation dose delivered by 211 At ↵-particles can adversely impact the labeling chemistry and stability of the radiopharmaceutical [START_REF] Zalutsky | Labeling monoclonal antibodies and F(ab')2 fragments with the a-particle-emitting nuclide astatine-211: Preservation of immunoreactivity and in vivo localizing capacity[END_REF]. Another issue is related to the bond between the daughter isotopes and the vector molecule and the lack of stable daughters in lot of decay chains involving emissions of ↵-particles. This is due both to the di↵erent chemical properties of daughters compared to the ones of progenitors and to their high recoil energy, which is 1000 times higher than the binding energy of any chemical compound [START_REF] Dekempeneer | Targeted alpha therapy using short-lived alpha-particles and the promise of nanobodies as targeting vehicle[END_REF]. It causes the chemical bonds between the daughter atoms and the vector to break down and thus, the release of daughter isotopes, which are often ↵-emitters that might raise radiation protection problems.

Auger emitters

a For what concern Auger electrons, they produce multiple ionizations with medium LET along their irregular path with a high level of cytotoxicity (see Sec. 1.1.1) [START_REF] Kassis | Radiobiologic Principles in Radionuclide Therapy[END_REF]. Theoretically, Auger-electron emitters present multiple advantageous characteristics for MRT, and more especially for the treatment of single disseminated tumor cells [START_REF] O'donoghue | Targeted radiotherapy using Auger electron emitters[END_REF]. While ↵ and particles deposit their energy at the cellular scale, the subcellular range of Auger electrons require close proximity to the radiosensitive target (DNA) for high radiotherapeutic e↵ectiveness [START_REF] Kassis | Radiobiologic Principles in Radionuclide Therapy[END_REF]. Thus, Auger radiation is of low toxicity when decay occurs outside the cell nucleus, such as in the cytoplasm or outside the cells, and will therefore cause little damage to non-targeted cells. On the other hand, if Auger electrons are delivered close to cell nucleus, they are able to produce a high proportion of double-strand DNA breaks. For this reason, Auger emitters have to be bound to very specific carriers, capable of crossing cell membranes and reaching DNA while targeting a high percentage of individual target cells [START_REF] Buchegger | Auger radiation targeted into DNA: a therapy perspective[END_REF]. Possible challenging methods of targeting include the use of analogues of DNA precursors, such as iodo-deoxyuridine [START_REF] Das | Options to meet the future global demand of radionuclides for radionuclide therapy[END_REF].

The theranostic approach a

With the term theranostics one refer to an integrated approach to diagnosis and therapy using suitable combinations of molecular targeting vectors and radionuclides. In nuclear medicine, theranostics involves combining diagnostic imaging and therapy by using the same molecule or at least very similar molecules, which are either radiolabeled Chapter 1

Molecular Radiotherapy

with the same isotope or with a pharmacology similar one. A theranostics approach to therapy is based on the idea that imaging (and thus treatment planning for personalized dosimetry) before a MRT treatment could be performed by using the same therapeutic pharmaceutical compound, just substituting the "therapeutic" radionuclide by a "diagnostic" one. This radionuclide should ideally represent another isotope of the "therapeutic" radionuclide, allowing the use of chemically identical vector for both diagnosis and therapy [START_REF] Lopci | Matched pairs dosimetry: 124i/131i metaiodobenzylguanidine and 124i/131i and 86y/90y antibodies[END_REF]. Fig. 1.6 summarizes the physical properties of some of the theranostics radionuclide we are going to describe.

The best established example is with 131 I with its isotopes 123 I and 124 I used for SPECT [START_REF] Benjamin | Measurement of iodine-123 thyroid uptake using a gamma camera with LEAP collimator[END_REF] and PET [START_REF] Jentzen | Pre-therapeutic 124i PET(/CT) dosimetry confirms low average absorbed doses per administered 131i activity to the salivary glands in radioiodine therapy of differentiated thyroid cancer[END_REF] imaging respectively. Among the two 131 I isotopes, 124 I is preferable for theranostics because of the better resolution of PET than of SPECT and on its longer half-life, which leads to more data for quantitative imaging after administration [START_REF] Eberlein | Individualized Dosimetry for Theranostics: Necessary, Nice to Have, or Counterproductive?[END_REF]. A recently published report has demonstrated the use of 124 as a substitute for 131 I for pretherapeutic dosimetry confirming, by the use of PET/CT images, the expected absorbed doses needed for thyroid ablation (>300 Gy) and for successful treatment of metastases (>80 Gy) [START_REF] Wierts | Dose-Response Relationship in Di↵erentiated Thyroid Cancer Patients Undergoing Radioiodine Treatment Assessed by Means of 124i PET/CT[END_REF] (see Sec. 1.3.3.3).

In terms of radiometals [START_REF] Notni | Rethinking the role of radiometal isotopes: Towards a future concept for theranostic radiopharmaceuticals[END_REF], yttrium and copper are elements that include radionuclides useful for clinical PET imaging ( 86 Y and 62 Cu-64 Cu) and for targeted radionuclide therapy ( 90 Y and 67 Cu) [START_REF] Anderson | Copper-64 Radiopharmaceuticals for PET Imaging of Cancer: Advances in Preclinical and Clinical Research[END_REF][START_REF] Walrand | Dosimetry of yttrium-labelled radiopharmaceuticals for internal therapy: 86y or 90y imaging?[END_REF][START_REF] Rosch | The Beginning and Development of the Theranostic Approach in Nuclear Medicine, as Exemplified by the Radionuclide Pair 86y and 90y[END_REF]. Moreover, 47 Sc was proposed as a potential therapeutic match to the PET nuclide 44 Sc [START_REF] Muller | Promising Prospects for 44sc-/47sc-Based Theragnostics: Application of 47sc for Radionuclide Tumor Therapy in Mice[END_REF]. Terbium nuclides have also been proposed for PET and SPECT imaging ( 152 Tb and 155 Tb) as well as for a alpha and beta radionuclide therapy ( 149 Tb and 161 Tb) but these nuclides have not been made available for clinical application yet [START_REF] Lehenberger | The lowenergy beta(-) and electron emitter 161tb as an alternative to 177lu for targeted radionuclide therapy[END_REF][START_REF] Muller | A Unique Matched Quadruplet of Terbium Radioisotopes for PET and SPECT and for -and -Radionuclide Therapy: An In Vivo Proof-of-Concept Study with a New Receptor-Targeted Folate Derivative[END_REF].

Alternatively to the one presented above, another "diagnostic" radionuclide from a chemical element di↵erent to the "therapeutic" one may be used. In this case, however, pharmacological studies need to make sure that the analogy in chemistry reflects a similarity in pharmacology for the two di↵erently radiolabelled compounds [START_REF] Rösch | Generator-based PET radiopharmaceuticals for molecular imaging of tumours: on the way to THERANOSTICS[END_REF]. Among several available couples ( 44 Ti/ 44 Sc, 68 Ge/ 68 Ga or 110m Sn/ 110m In), the 90 Y / 111 In is the most commonly used and Ibritumomab tiuxetan (Zevalin) labeled with the two ra-Figure 1.6: Nuclear data of theranostic isotopes for therapy, PET and SPECT imaging.

[Source [START_REF] Muller | Promising Prospects for 44sc-/47sc-Based Theragnostics: Application of 47sc for Radionuclide Tumor Therapy in Mice[END_REF]]

Molecular Radiotherapy

Chapter 1 dioisotope was the first therapeutic radionuclide-antibody conjugate approved by the U.S. Food and Drug Administration (February 2002) [START_REF] Fisher | MIRD Dose Estimate Report No. 20: Radiation Absorbed-Dose Estimates for 111in-and 90y-Ibritumomab Tiuxetan[END_REF]. 111 In-ibritumomab tiuxetan is usually administered intravenously as a tracer to assess labeled antibody biodistribution and dosimetry, and 90 Y -ibritumomab tiuxetan is administered later as the cancer therapy agent.

One have to notice that for most of the presented matching isotopes, there is a significant discrepancy between the half-lives of the corresponding diagnostic and therapeutic isotope, i.e. 13.2 h and 8.04 d for 123 I/ 131 I or 9.74 min and 2.58 d 62 Cu for 67 Cu, respectively. Thus, even having the same biodistribution (or at list similar), the radioisotopes have di↵erent biokinetic which implies that they could have di↵erent cumulated activity over time. Hence, even allowing low-dose imaging, pre-treatment dosimetric study could lead to erroneous treatment dose evaluation. Thus, even if several studies have demonstrated that a theranostic concept can be applied to predicting absorbed doses (most successful example is the pair 124 I/ 131 I), accurate dosimetric studies are ongoing for several matching radionuclides in order to confirm their applicability as theranostic agents [START_REF] Eberlein | Individualized Dosimetry for Theranostics: Necessary, Nice to Have, or Counterproductive?[END_REF].

Radiopharmaceuticals used in clinical practice a

As already stated, therapy using unsealed sources of radioactivity has the advantages of both systemic administration (as in chemotherapy) as well as selective irradiation (like brachytherapy and external beam radiation) of tissues. For a long time, radionuclide therapy was mainly limited to the use of radioiodine for thyrotoxicosis or thyroid cancer and of 32 P for polycythemia vera. However, considerable advances have taken place in the field during the past three decades, thanks to the joint development of new tracers and innovative radionuclides. Nowadays the list of suitable radiopharmaceuticals and of target organs for MRT application has expanded tremendously [START_REF] Dash | Targeted Radionuclide Therapy -An Overview[END_REF], together with the number of possible mechanisms for delivering radionuclide treatments. Radioisotopes can reach the target taking advantage of several processes and usually di↵erent branches of MRT are identified depending on the targeting mechanisms employed. The two principal processes are active transport and receptor binding.

Active transport is a type of selective carrier-mediated transport across membranes. This mechanism use cellular metabolic processes for the accumulation of radiopharmaceuticals and requires energy, usually from ATP, for the transporters to function. Examples of active transport are the one of 131 I -Na in the thyroid gland, and of metaiodobenzylguanidine (mIBG), a molecule that is structurally related to the norepinephrine transporter, whose over-expression occurs in certain neoplasms such as neuroblastoma, and can therefore be inserted into this carrier to perform transport across membrane. Receptor binding refers to the "lock-and-key" binding of a molecule to a specific receptor site. A classic example is the binding interaction of cell surface antigens with target monoclonal antibodies, such as ibritumomab, tositumomab (Radioimmunotherapy, RIT). Other examples of ligands that bind to particular receptor sites include the use of specific cell surface receptors for accumulating radiolabelled Chapter 1 Molecular Radiotherapy peptides, such as TOC (Octreotide), TATE (Octreotate) and PSMA (Peptide Receptor Radionuclide Therapy, PRRT) [START_REF] Sjögreen Gleisner | Variations in the practice of molecular radiotherapy and implementation of dosimetry: results from a European survey[END_REF]. Other specific mechanisms are used in MRT, such as phagocitosys (in radiosynovectomy), the binding of phosphate-type compounds onto the surface of bone (for bone palliation therapy) and radioembolization, which will be better describe in following paragraphs.

According to the physical and physiological characteristics of the target and to the biochemical pathway selected, di↵erent vectors and radionuclides can be used for a variety of therapeutic applications. Thanks to a survey realized in 2015 by the European Association of Nuclear Medicine (EANM) Internal Dosimetry Task Force survey [START_REF] Sjögreen Gleisner | Variations in the practice of molecular radiotherapy and implementation of dosimetry: results from a European survey[END_REF], the most di↵use MRT procedures in Europe have been identified, together with the proportions of performed treatments, and are summarized in table 1.1. The following paragraphs provide a brief overview of these procedures.

Benign and malign thyroid diseases with 131 I -NaI a Among more than 42.500 treatments performed each year in Europe, therapies involving 131 I -NaI are the most common applications, with around the 71% of the total number of treatments performed. Administration of 131 I -NaI for thyroid diseases has been in common use since the 1940s, when the first e↵ective therapy was performed by Saul Hertz [START_REF] Sawin | Radioiodine and the Treatment of Hyperthyroidism: The Early History *[END_REF]. Since then, it is extensively used to ablate thyroid remnant or inoperable advanced di↵erentiated thyroid cancer and to diagnose and treat patients with benign diseases of thyroid, that is hyperthyroidism conditions such as toxic di↵use goitre (Graves' disease), solitary hyperfunctioning thyroid nodule, toxic multinodular goitre and nontoxic goitre [START_REF] Stokkel | EANM procedure guidelines for therapy of benign thyroid disease[END_REF]. Iodide transport into the thyroid takes advantage of an active cellular metabolic process. Its uptake into the gland is naturally mediated by a specific sodium-dependent iodide transporter, the sodium iodide symporter (NIS), located on the membrane of thyroid follicular cells. While normal thyroid tissue reveals heterogenous NIS protein expression, hyperthyroid and cancerous tissues show huge changes in NIS protein expression with a consequent increased/decreased radioiodine uptake [START_REF] Saito | Increased expression of the sodium/iodide symporter in papillary thyroid carcinomas[END_REF], which forms the basis for the e↵ective diagnostic and therapeutic management of thyroid diseases [START_REF] Dohán | The Sodium/Iodide Symporter (NIS): Characterization, Regulation, and Medical Significance[END_REF].

Neuroblastoma and neuroendocrine tumors with 131 I -mIBG a

The labelling of mIBG with 131 I was firstly reported in 1980 and the first therapeutic use for neuroblastoma, the most common intra-abdominal and extracranial solid tumor of childhood, was performed in 1984 [START_REF] Wiel | Radiolabeled Adrenergic Neuron-Blocking Agents: AdrenomeduDary Imaging with[END_REF]. Treatment with 131 I -mIBG are nowadays the first choice for patients with neuroblastoma and advanced neuroendocrine tumors diseases, since the invasiveness and toxicity of this therapy are favorable compared with that of chemotherapy, external beam radiotherapy and immunotherapy [START_REF] Dash | Targeted Radionuclide Therapy -An Overview[END_REF]. Both neuroblastoma and neuroendocrine tumors express the norepinephrine transporter and, thus, show an avid uptake of mIBG by an active transport mechanism [START_REF] Dash | Targeted Radionuclide Therapy -An Overview[END_REF].

Molecular Radiotherapy

Chapter 1

Treatment

Radiopharmaceutical P (%)

Benign thyroid diseases performed with each radionuclide according to the EANM Internal Dosimetry Task Force survey [START_REF] Sjögreen Gleisner | Variations in the practice of molecular radiotherapy and implementation of dosimetry: results from a European survey[END_REF] Chapter 1

Molecular Radiotherapy

Bone pain palliation therapy a Bone metastasis are a common complication in cancer patients and often occur with severe bone pain that significantly a↵ects patient's quality of life. Bone metastases may occur in almost all tumors at di↵erent frequencies; however, prostate, lung, and breast cancers most commonly result in metastasis [START_REF] Dash | Targeted Radionuclide Therapy -An Overview[END_REF]. Radiopharmaceutical treatment of metastatic bone pain has emerged around 70 years ago as an e↵ective modality that provides palliation of pain to multiple areas of the skeleton simultaneously without significant soft-tissue toxicity [START_REF] Cuccurullo | Bone Metastases Radiopharmaceuticals: An Overview[END_REF]. Several beta-emitting agents exist for treatment of cancer metastatic to bone, mainly for patients with osteoblastic bone metastasis from castration-resistant prostate cancer. The most commonly used are 89 SrCl 2 (Metastron R ), which behaves biologically like calcium, and 153 Sm, 186 Re and 188 Re coordinated with diphosphonates, as EDTMP and HEDP, that seek for the bone matrix and accumulate in areas of increased osteoblastic activity [START_REF]Internal Dosimetry Task Force Report on: Treatment Planning For Molecular Radiotherapy: Potential And Prospects[END_REF]. More recently, a lot of attention has been put on the ↵-emitting radionuclide 223 Ra after the results from a first clinical study [START_REF] Nilsson | First Clinical Experience with alpha-Emitting Radium-223 in the Treatment of Skeletal Metastases[END_REF]. Indeed, radium has a natural a nity for metabolically active bone due to its chemical similarity with calcium. In the form of 223 RaCl 2 (Xofigo R ), it has been approved in 2013 for the treatment of patients with symptomatic bone metastases from castration-resistant prostate cancer [START_REF]Internal Dosimetry Task Force Report on: Treatment Planning For Molecular Radiotherapy: Potential And Prospects[END_REF].

Radiosynovectomy a

The use of therapeutic radioisotopes for treatment of arthritis is widespread. Rheumatoid arthritis (RA) is one of the most common autoimmune inflammatory diseases. It is due to destruction of synovial tissues which causes severe pain, disability, and immobility in patients [START_REF] Schneider | Radiosynovectomy in rheumatology, orthopedics, and hemophilia. -Semantic Scholar[END_REF]. Radiosynovectomy is the intra-articular injection of small radioactive particles (0.05 -2 µm) to restore inflamed and damaged synovial membrane in order to improve patient mobility and to maximally decrease soreness and discomfort, with consequent significant improvement in quality of life. In this procedure, a beta-emitting radionuclide in colloidal or particulate form is injected into the articular cavity. They are recognized as foreign bodies by the outermost cellular layer of the synovial membrane and are therefore phagocyted by these cells. The selection of radionuclides for radiosynovectomy of specific joint is critical since the synovial thickness varies substantially. Thus, selection of the radionuclide is primarily based on the size of the joint to be treated. For smaller joints (as finger) lower energy is preferable and the short range beta emitters such as 169 Er are used. The medium range beta emitters such as 175 Yb, 177 Lu, 186 Re are useful for medium joints as shoulder, ankle, wrist and elbow. 90 Y and 32 P colloid, with tissue penetration of 3 to 11 mm, are used for larger joints such as the knee.

Hepatocellular Carcinoma and Hepatic Malignancies with 90 Y-microspheres a Hepatocellular carcinoma (HCC) is a malignant tumor of the liver hepatocyte, which may present either as primary liver cancer or as secondary liver tumors. HCC is

the sixth most common cancer in the world and the third leading cause of cancer-related mortality with nearly half a million deaths per year [START_REF] Bray | Global cancer statistics 2018: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries[END_REF]. Radioembolization is one of the therapies for the treatment of liver malignancies. The trans-arterial radioembolization technique exploits the fact that hepatic lesions have a preferential blood supply from the hepatic artery to deliver the radioactive particles in hepatic end-arterioles, allowing localized delivery of therapeutic doses, while sparing the surrounding liver parenchyma [START_REF] Dash | Targeted Radionuclide Therapy -An Overview[END_REF]. At present, therapy with 90 Y-microspheres is mainly used. There are two commercially available microsphere devices in which 90 Y is incorporated; one with microspheres made of glass (TheraSphere R ) and the other with microspheres made of resin (SIR-Spheres R ).

Neuroendocrine tumors with radiolabeled peptides a

In the last two decades, there was an explosive growth in the development of radiolabeled peptides for both diagnostic an therapeutic applications. Peptides are molecules consisting of two or more aminoacids linked together with peptide bonds. They regulate many physiological processes and are known for having very high a nity with receptors present on the cell surface. There are several types of receptor in a typical cell and each of them recognizes and binds only a certain type of ligand allowing a very high specificity. The receptors are often over-expressed in cancer cells and targeting these receptors using peptides is the basis of diagnosis as well as therapy. Thus, peptides can be used as transport vehicles to guide the radionuclides to tissues, which express a particular receptor. Peptide Receptor Radionuclide Trehapy (PRRT) is mainly used for the treatment of endocrine tumors because most of them over-express receptors that bind to somatostatin, a peptide hormone that regulates the endocrine system. Several somastosatine analogues have been synthetically produced and radiolabelled with di↵erent radionuclides for the development of diagnostic and therapeutic radiopharmaceuticals. DOTA-Octreotide (DOTA-TOC) and DOTA-Octreotate (DOTA-TATE) are the most used somatostatin analogues that binds to somatostatin receptors on neuroendocrine tumors and provided a stable binding with 90 Y ( 90 Y-DOTATOC) and 177 Lu ( 177 Lu-DOTATATE) to deliver a lethal absorbed dose to the tumor.

Radioimmunotherapy a

Radioimmunotherapy (RIT) uses monoclonal antibodies (mAbs) as a vector for transport of radioactivity to cancer cells. Antibodies are agents of our immune system, primarily used to identify and aid in the clearing of foreign pathogens. The radiolabeled antibodies are used to target the antigens over-expressed on tumor cells and under-expressed in normal cells in order to deliver lethal doses of cytotoxic radiation to the tumor. RIT has significantly progressed in the past decade [START_REF] Breedveld | Therapeutic monoclonal antibodies[END_REF] and is now mainly used to treat hematopoietic malignancies such as Non-Hodgkin's Lymphoma (NHL). Currently, two commercial radiolabelled antibodies are available for treatment of follicular NHL, 90 Y-ibritumomab tiuxetan (Zevalin R ) and 131 I -tositumomab (Bexxar R ).

Chapter 1 Molecular Radiotherapy

Zevalin R was the first therapeutic radionuclide-antibody conjugate approved by the U.S. Food and Drug Administration in February 2002 [START_REF] Fisher | MIRD Dose Estimate Report No. 20: Radiation Absorbed-Dose Estimates for 111in-and 90y-Ibritumomab Tiuxetan[END_REF]. RIT is also used to treat metastatic castration-resistant prostate cancer, the second leading cause of cancer death in men in many parts of the world. Indeed, many prostate cancer patients recur with overexpression of the Prostate-Specific Membrane Antigen (PSMA), even after radical prostatectomy and external beam radiotherapy. Di↵erent radiolabeled ligands targeting PSMA have been developed, but the most reported treatment is 177 Lutetium-PSMA-617 radioligand therapy [START_REF] Von Eyben | PSMA diagnostics and treatments of prostate cancer become mature[END_REF]. Despite encouraging results in term of decrease of PSA with minimal toxicity in patients [START_REF] Baum | 177lu-Labeled Prostate-Specific Membrane Antigen Radioligand Therapy of Metastatic Castration-Resistant Prostate Cancer: Safety and E cacy[END_REF], the use of 177 Lu-PSMA-617 has not yet reached the point of being considered as a standard therapeutic option and the role of PSMA in prostate cancer treatment remains the subject of many clinical trials [START_REF] Von Eyben | PSMA diagnostics and treatments of prostate cancer become mature[END_REF].

Future perspectives and further developments a

While the advances in science and technology have led to huge developments in all therapeutics procedures, the cancer incidence is predicted to double by 2030 continuing to be the principal cause of death in both developed and less developed countries [START_REF] Bray | Global cancer statistics 2018: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries[END_REF]. The increasing focus on cancer management has led to a growing need for new therapies to increase cancer control, which should be increasingly targeted and patient-specific. The demand for therapeutic nuclear medicine is expected to grow rapidly due to its e↵ectiveness in treating a wide range of tumours, which cannot be treated by more traditional procedures, such as surgery, external radiotherapy and chemotherapy. Recent local survey (The Internal Dosimetry Users Group has surveyed UK molecular radiotherapy) highlight this trend, with an 82% and a 250% increase in the number of patients treated using MRT and of MRT treatments administered, respectively, between 2007 and 2017 [START_REF] Rojas | Eighty per cent more patients in 10 years of UK molecular radiotherapy: Internal Dosimetry Users Group survey results from 2007 to 2017[END_REF].

Advances in the characterization of new structures associated with the surface of tumor cell, in the understanding of tumor biological over-expression of oncogenes, recombinant antibody technology, synthetic chemistry and other related disciplines have recently led to enormous progress in the development of new targeting agents exploiting di↵erent physiological, biochemical and molecular properties [START_REF] Carlsson | Tumour therapy with radionuclides: assessment of progress and problems[END_REF]. As a consequence, new vectors have been developed and are in early phase clinical trials [START_REF] Carlsson | Tumour therapy with radionuclides: assessment of progress and problems[END_REF]. Various radiolabelled peptides have been investigated in the recent past, as 90Y-DOTA-octreotide [START_REF] Chinol | Receptor-mediated radiotherapy with 90y-DOTA-DPhe1-Tyr3-octreotide: The experience of the european institute of oncology group[END_REF] and 90 Y-DOTA-lanreotide [START_REF] Pandey | 90y-DOTA-Lanreotide: A potential agent for targeted therapy[END_REF], and other are still in progress, as for 177 Lu-DOTAoctreotate [START_REF] Sandblom | Strategies for optimisation of 177Lu-octreotate therapy -exploring local administration and combination therapy regimens[END_REF], all for the treatment of neuroendocrine malignancies. Other peptides are currently under investigation, as 188 Re labeled peptide, some of which have shown encouraging results [START_REF] Lepaureur | Rhenium-188 Labeled Radiopharmaceuticals: Current Clinical Applications in Oncology and Promising Perspectives[END_REF]. Moreover 188 Re shows similar chemical properties of 99m Tc hence representing a new possible theranostic pair [START_REF] Lepaureur | Rhenium-188 Labeled Radiopharmaceuticals: Current Clinical Applications in Oncology and Promising Perspectives[END_REF]. The biggest problem with radiolabeled peptides is that they often su↵er from rapid blood clearance, high uptake in the kidneys, high metabolic instability, and/or low accumulation in tumor tissue [START_REF] Grzmil | An Overview of Targeted Radiotherapy[END_REF].

Moreover, there is growing interest in the use of nanobodies as radionuclide vehicles. Indeed in the last decade, nanotechnology has shown a great potential for the

early detection, accurate diagnosis, and personalized treatment of various diseases [START_REF] Sahoo | The present and future of nanotechnology in human health care[END_REF]. Their size is comparable to biological molecules such as antibodies, and about 100 to 10000 times smaller than human cells and can interact with the biomolecules both on the surface and inside the cells. With the capacity of large specific activity inside each particle, nanoparticles can be very useful for internal radiation therapy through passive targeting and/or active targeting [START_REF] Perinelli | PEGylated polylactide (PLA) and poly (lactic-co-glycolic acid) (PLGA) copolymers for the design of drug delivery systems[END_REF][START_REF] Shukla | Conclusion and Future Prospective of Polymeric Nanoparticles for Cancer Therapy[END_REF]. Other nanoparticles, which have been approved for human use or are currently in clinical trials, include iron oxide, perfluorocarbon, nanotube, quantum dot, micelle, and dendrimer [START_REF] Yeong | Therapeutic radionuclides in nuclear medicine: current and future prospects[END_REF]. Among nanocarriers, liposomes have been found to be more e↵ective in the passive targeting then other nanosized drug carrier material [START_REF] Moosavian | Aptamer-functionalized liposomes for targeted cancer therapy[END_REF]. Their potential as carriers of radionuclides in internal radiotherapy (also thanks to their excellent biodegradability and biocompatibility) is under investigation since almost two decades [START_REF] Emfietzoglou | An Analytic Dosimetry Study for the Use of Radionuclide-Liposome Conjugates in Internal Radiotherapy[END_REF][START_REF] Emfietzoglou | An Analytic Dosimetry Study for the Use of Radionuclide-Liposome Conjugates in Internal Radiotherapy[END_REF][START_REF] Kullberg | Development of EGF-Conjugated Liposomes for Targeted Delivery of Boronated DNA-Binding Agents[END_REF] and pre-clinical studies are still active [START_REF] Fan | 177lu-labeled Payloaded Liposome for SPECT/Optical Dual-modality Imaging and Combined Radio-and Chemotherapy in LLC Mice Model[END_REF]. Furthermore, PSMA-based PET/CT imaging is increasingly being implemented in many countries and PSMA theranostics are emerging as highly promising tools in the treatment of metastatic castration-resistant prostate cancer. Randomized trials are underway to determine whether PSMA used for diagnostics and treatment of prostate cancer can delay progression to death and reduce mortality [START_REF] Von Eyben | PSMA diagnostics and treatments of prostate cancer become mature[END_REF].

Beyond the development of new specific vectors, there is also an expanding list of novel radionuclides with di↵erent emission characteristics, which can potentially be used for clinical therapy. An evaluation of the literature on therapeutic radionuclides identifies at least 100 radionuclides that are useful or likely to have potential for radionuclide therapy [START_REF] Das | Options to meet the future global demand of radionuclides for radionuclide therapy[END_REF]. In particular, there is a strong interest in ↵ particles for the eradication of small volume tumours and small clusters of cancer cells. Besides 223 Ra, many other ↵-particle emitters, as 211 At, 213 Bi and 225 Ac have suitable characteristics for therapeutic applications. Investigations are still ongoing but although still experimental, promising results on the first treatments (two advanced-stage prostate cancer patients) with 225 Ac-PSMA-617 are reported in literature [START_REF] Kratochwil | Targeted alpha therapy of mCRPC: Dosimetry estimate of 213bismuth-PSMA-617[END_REF] and its e↵ectiveness in treating prostate cancer brain metastases is under investigation [START_REF] Sathekge | Treatment of brain metastases of castration-resistant prostate cancer with 225ac-PSMA-617[END_REF]. Because of the limits of alpha therapy, as described in Section 1.2.1.2, lot of in vitro, in vivo and clinical trials are also carried out [START_REF] Marcu | Global comparison of targeted alpha vs targeted beta therapy for cancer: In vitro, in vivo and clinical trials[END_REF].

While factors such as stringent regulatory approvals and high development costs could hinder the growth of the radiopharmaceutical market, it is expected that there will be an increasing demand for therapeutic radionuclides in the near future given the recent high interest in this area. Indeed, the global market for radiopharmaceuticals, driven by diagnostic and therapeutic radiopharmaceuticals, is expected to reach 10 million USD by 2023, up from 5,8 million USD in 2016. According to the World Nuclear Association, about one in 50 people use diagnostic nuclear medicine each year and the frequency of treatments with radioisotopes is about one-tenth of this figure, but the increasing emphasis on cancer management will accelerate the therapeutic market segment [START_REF]Global Radiopharmaceutical Market Research Report -Forecast to[END_REF]. To meet needs and sustain their growth rate, it is necessary to develop and maintain a constant and reliable supply, hence the need to build new production facilities [START_REF] Das | Options to meet the future global demand of radionuclides for radionuclide therapy[END_REF]. Various important radionuclides used for diagnostic and therapeutic purposes (e.g., 99 Mo, 131 I, 90 Y, 153 Sm, 166 Ho, 177 Lu, and 188 Re) are currently produced by Chapter 1 Molecular Radiotherapy nuclear reactors. Although cyclotrons can produce radionuclides and further research can result in alternative production way for several reactor-produced radionuclides, the roles of nuclear reactors is still important [START_REF] Krijger | The necessity of nuclear reactors for targeted radionuclide therapies[END_REF]. Several main reactors in the world are now aging, posing a challenge to the constant supply of medical radionuclides. To prevent medical radionuclide shortages, "a shared global responsibility is needed by all the stakeholders to replace a small number of aging reactors with modern ones" [START_REF] Yeong | Therapeutic radionuclides in nuclear medicine: current and future prospects[END_REF]. For example since one decade, the world supply of 99 Mo, whose decay product 99m Tc is nowadays used in more than 80% of clinical nuclear medicine studies, start to highly decrease [START_REF] Ballinger | 99mo shortage in nuclear medicine: crisis or challenge?[END_REF]. Most of the world's supply of 99 Mo is obtained from only five aging nuclear reactors while others having been decommissioned and not replaced [START_REF] Ballinger | Short-and long-term responses to molybdenum-99 shortages in nuclear medicine[END_REF]. In recent months nuclear medicine departments start to experiencing a complete shortage of 99m Tc, with the consequence need of reschedule and limit most 99m Tc clinical applications (o cial announcement of the division of nuclear medicine in the department of radiology of the university of Iowa Hospitals and Clinics [102], october 2018).

In parallel with the development of increasingly specific radiopharmaceuticals, one of the main challenges in the use of MRT is the individual determination of absorbed doses by Organs-At-Risk (OAR) and target regions and the activity to be administered for each patient based on these dosimetric assessments. These determinations are essential and rely both on existing proofs that di↵erent biological e↵ects (i.e. di↵erent absorbed doses) can be produced from the same amount of administrated activity, since each patient has di↵erent tumor and normal organ radiopharmaceutical uptakes and di↵erent clearance half-times, and on increasing evidence of the correlation between the dose delivered to patients and the therapeutic response, both in terms of e↵ects and toxicity [START_REF] Stabin | State of the Art in Nuclear Medicine Dose Assessment[END_REF][START_REF] Strigari | The evidence base for the use of internal dosimetry in the clinical practice of molecular radiotherapy[END_REF][START_REF] Flux | From fixed activities to personalized treatments in radionuclide therapy: lost in translation?[END_REF]. The role of a theranostic approach to MRT is thus crucial, because several studies have demonstrated that a theranostic concept can be applied to predicting absorbed doses, as discussed in Sec. 1.2.1.4. Moreover, these dosimetry challenges are even more crucial with new treatments with alphas particles, which yield a massive energy local deposition which induces inducing predominantly non-repairable DNA double-strand breaks. Therefore, MRT should be based on individualized dosimetry, with the goal of maximizing the probability of successful therapeutic response while maintaining the risk of toxicity to organs-at-risk at justifiable levels, as is routinely done in EBRT or brachytherapy.

THE ROLE OF DOSIMETRY IN MRT

1.3 The role of dosimetry in molecular radiotherapy: an ongoing discussion a As discussed in the previous sections, many novel radionuclides, molecules and radiopharmaceuticals have been proposed and introduced in recent years, increasing the list of available therapeutic procedures but consequently also the number of organsat-risk [START_REF] Strigari | The evidence base for the use of internal dosimetry in the clinical practice of molecular radiotherapy[END_REF]. For example, during the treatment with 90 Y and 177 Lu radiopeptides, kidney and liver show high uptake and retention; this is also the case with high activity of 131 I -mIBG and 90 Y-ibritumomab tiuxetan [START_REF] Roelf | Long-Term Follow-Up of Renal Function After Peptide Receptor Radiation Therapy with ˆsup 90ˆY[END_REF][START_REF] Imhof | Response, Survival, and Long-Term Toxicity After Therapy With the Radiolabeled Somatostatin Analogue [ 90 Y-DOTA]-TOC in Metastasized Neuroendocrine Cancers[END_REF]. While toxicity may be mitigated for some organs, for example in the kidneys by blocking renal uptake, few similar strategies exist for other organs, particularly for red marrow, which is considered the primary OAR in MRT. This increase in the number of MRT applications has drawn the attention of the scientific community and the institutional organizations to the need of developing a standardized strategy to determine the amount of radiation received by tissues during a treatment.

The study of the biodistribution of radiation resulting from the systemic administration of a therapeutic radiopharmaceutical is referred as "dosimetry" and requires not only a careful understanding of its distribution and pharmacokinetics, but also methods for converting the number of decay within a particular volume to the absorbed dose by that volume, as will be described in the following sections. A standardized dosimetric approach should first lead to optimize the individual treatment planning, in order to maximize the probability of successful treatment response while sparing normal tissues. In this way, the maximum activity that can be administered to each specific patient should be defined according to the treatment objectives and the tolerance doses to organs at risk. In addition, post-treatment verification of the absorbed dose delivered to targeted and non-targeted tissues is also essential to refine treatment planning. Indeed, it allows correlations to be established with the dose predicted from the pre-treatment evaluation as well as with the clinical response observed in terms of disease control and toxicity on organs-at-risk.

In MRT, patient specific dosimetry-based treatments have been much slower to develop that other existing radiotherapy modalities: the absorbed doses necessary for successfully therapies are not known, nor are the tolerance doses for normal tissues. The lack of extended clinical trials entails major limitations in responding to these shortcomings [START_REF] Falzone | Clinical trials in molecular radiotherapy-Tribulations and Triumphs Report of the NCRI CTRad meeting held at the Lift Islington, 8 June 2018[END_REF]. On the other hand, no treatment in external radiotherapy or brachytherapy can currently be considered without individualized and precise treatment planning [START_REF] Lee | Determination of energy windows for the triple energy window scatter correction method in I-131 on a Siemens SYMBIA gamma camera: a GATE simulation study[END_REF]. The main disadvantage in the lack of dosimetry is the failure to consider patient's individuality, posing them under the risk of either being under or over-dosed, with the consequent need of new treatment or unnecessary exposure to radiation, that could lead to unwanted side e↵ects. Individual dosimetry-based treatment planning and verification of the absorbed dose delivered during treatment must therefore be performed in the best interest of patients, as mandated by the Euratom directives. Indeed, the new European Council Directive 2013/59 "laying down basic safety standards for protection Chapter 1

The role of dosimetry in MRT against the dangers arising from exposure to ionising radiation" mandates the use of dosimetry for treatment planning and dose verification [107].

The planning of treatment by individual dosimetry and the verification of the absorbed dose delivered during treatment must therefore obviously be carried out in the best interests of patients, but also because this is what is prescribed by the Euratom directives.

In the article 56, it states that:

For all medical exposure of patients for radiotherapeutic purposes, exposures of target volumes shall be individually planned and their delivery appropriately verified, taking into account that doses to non-target volumes and tissues shall be as low as reasonably achievable and consistent with the intended radiotherapeutic purpose of the exposure. The optimization principle (ALARA, as low as reasonably achievable) of article 56 highlights that absorbed doses to non-target tissues must be kept at a reasonably low level but not to the point as losing the e↵ectiveness of the treatment. According to the definition 81 of the directive, "radiotherapeutic purposes", means "pertaining to radiotherapy, including nuclear medicine for therapeutic purposes". Consequently, molecular radiotherapy has to be based on individualized dosimetry, with the aim of maximizing the probability of successful treatment response while keeping the risk of toxicity for organs at risk at justifiable levels, as is routinely done in the other form of radiotherapy.

Dose assessment in radionuclide therapy a

The dosimetric approaches developed in MRT to determine the amount of radiation received by the target region and the surrounding OARs will be detailed in the following sections, beginning with the MIRD (Medical Internal Radiation Dose) formalism, an essential reference in the field for absorbed dose assessment [START_REF] Dewaraja | MIRD Pamphlet No. 23: Quantitative SPECT for Patient-Specific 3-Dimensional Dosimetry in Internal Radionuclide Therapy[END_REF][START_REF] Kesner | Modern Radiopharmaceutical Dosimetry Should Include Robust Biodistribution Reporting[END_REF].

Basic principles of dose calculation methods a

The Medical Internal Radiation Dose (MIRD) committee was formed in 1965 within the Society of Nuclear Medicine & Molecular Imaging (SNMMI) with the mission of standardizing dosimetry practices according to the same formalism. Since then, through its phamphlets, the committee reduces complex dosimetric analyses to methods that are relatively simple to use, proposing internal dosimetry approaches for many radionuclides in very di↵erent biological environments. In the phamplet No. 21, MIRD committee proposed a schema and a standardizing nomenclature for the estimation of the absorbed dose D, defined as the average energy imparted to a target volume per unit of tissue mass [START_REF] Bolch | MIRD Pamphlet No. 21: A Generalized Schema for Radiopharmaceutical Dosimetry-Standardization of Nomenclature[END_REF]. The dose delivered to target tissue r t within a patient from a radioactive material distributed uniformly within source tissue r s at time t after

The role of dosimetry in MRT Chapter 1 administration is defined as:

D(r t , t) = e A(r s , t) ⇥ S(r t r s ) ( 1 . 1 )
where e A is the time integrated activity, or cumulated activity, in the region r s and S is the mean dose absorbed by the region r t per unit cumulated activity in the region r s .

As can be seen from equation 1.1, the estimation of the absorbed dose D depends both on a time-dependent (biokinetic) factor incorporated in e A, and a time-independent (physical) factors, represented by S, which will be described in the next two paragraphs.

a) Cumulated activity a

The cumulated activity e A represents the number of decays that take place in a certain source region r s during a treatment time t D (Bq•sec or µC • h) and it embodies most of the biological data needed for dose estimation [START_REF] Snyder | Absorbed Dose per Unit Cumulated Activity for Selected Radionuclides and Organs[END_REF]. e

A includes the characteristics of both uptake and retention of activity in the regions of interest and also consideration about the physical half-life of the radionuclide and the biologic half-time of the radiopharmaceutical. The cumulated activity can be calculated as the area under the time-activity curve:

e A(r s ) = Z t D t 0 A(r s , t) dt (1.2)
The estimation of e A can be performed by indirect or direct measurement of the activity A(r s , t) within the source region r s over time.

Indirect measurement of the cumulated activity : compartmental models Compartmental or biokinetic models consist of a set of first-order coupled di↵erential equations, defined for all organs and sub-organ tissues of interest, to predict the timedependent behavior of incorporated radionuclides. Their purpose is to describe the distribution of activity in di↵erent functional compartments within which the activity is distributed uniformly and which are connected to each other by input/output time transfer constants. The distribution of the activity over time can thus be expressed separately for each of the compartments, in the form of time-activity curves, which makes it possible to compute their cumulated activity. An easy three-compartimental model is shown in Figure 1.7(top-left), where the transfer constants between compartiments are represented by K ij (expressed in time 1 ), together with the three time-activity curves representing the biokinetics of each compartiment (Fig. 1.7)(bottom-left).

The International Commission on Radiological Protection (ICRP) has developed and published, during time, the reference biokinetic models for use in radioprotection, mainly for occupational intake. The reports has been updated over time to adopt more physiologically realistic, flexible and complex biokinetic models that has been adapted for a variety of radiation protection and specific therapeutic applications, as the one shown in Fig. 1.7 developed for dose assessments during radioiodine treatments. [START_REF] Leggett | A Physiological Systems Model for Iodine for Use in Radiation Protection[END_REF][START_REF] Melo | Organ Dose Estimates for Hyperthyroid Patients Treated with 131 I: An Update of the Thyrotoxicosis Follow-Up Study[END_REF]. These models have been widely used because the direct determination of the activity for each individual exposed to radiation could be a significant complication for the daily routine of an ordinary nuclear medicine service. However, although these models have generally been developed on a large cohort of patients, they cannot take into account individual variations in the physiological mechanisms (uptake and excretion) that govern the accumulation of radiopharmaceuticals. An dosimetric evaluation should therefore be performed on an individual basis using direct activity measurements to be accurate.

Direct measurement of the cumulated activity The direct estimation of e

A consists of two steps. The first one is the quantification of the activity A(r s , t) within the source region r s at di↵erent times after the injection, with a number of measurement techniques that may include or not the use of imaging. The second step consists in the generation of the time-activity curve and in its integration to obtain the total number of disintegrations occurring in the source organs over time. Non-imaging measurements of A(r s , t) include counting tissue biopsies, blood samples or excreta (i.e. urine and feces collection) and using external counting probes, such as Geiger-Mueller or sodium iodide probes, to measure whole body uptake and sometimes single source organ uptake, in cases where accumulation is extremely localized. More accurate results could be achieved using quantitative imaging performed with planar cameras or SPECT or PET

The role of dosimetry in MRT Chapter 1 tomographic systems, by converting the counts recorded in the images into an absolute value of activity (Bq or mCi).

Once the activity is quantified in the source region at di↵erent post-injection times, its evolution over time is represented by an activity-time curve that has to be integrated to obtain the cumulated activity. More or less elaborate methods are used to perform the integration. Commonly, the points of the curve are fitted using a mathematical model often made by the composition of exponential functions, whose coe cients are determined by an iterative process [START_REF] Motulsky | Fitting Models to Biological Data using Linear and Nonlinear Regression[END_REF]. Although exponentials are most commonly used, other method are also employed, such as the trapezoidal method or Riemann integration [START_REF] Stabin | Uncertainties in Internal Dose Calculations for Radiopharmaceuticals[END_REF]. Practically, the radiopharmaceutical acts in the patient body until its complete physical decay and/or biological excretion. Thus, the time-integration period of A(r s , t) should be set from t 0 = 0 to at least two to three e↵ective half-lives T 1/2,ef f , a combination between the radionuclide physical half-life T 1/2 and the biological half-life T 1/2,j [START_REF] Bailey | Physicists in Medicine, éds, Nuclear medicine physics: a handbook for teachers and students[END_REF]:

1 T 1/2,ef f = 1 T 1/2 + 1 T 1/2,j (1.3)
Regardless of the method used, the uncertainties on the estimation of the cumulative activity are intrinsically related to the temporal sampling of the activity measurement, since the shape of the fitting curve can be strongly influenced by both the number and the timing of the individual measurements. This accurate temporal sampling is also required by the highly variable shape of the time-activity curve, characterized by a phase of fixation and a phase of elimination of the radiopharmaceutical from the source volume, which vary considerably from one patient to another. For example, fast fixation and clearing of the radiopharmaceutical in the volume of interest necessitate frequent and early acquisition, while slow fixation and clearing require more widely dispersed activity measurements [START_REF] Dewaraja | MIRD Pamphlet No. 24: Guidelines for Quantitative 131i SPECT in Dosimetry Applications[END_REF]. In practice, three activity acquisitions should be considered as the minimum data required to determine the pharmacokinetics of the radiotracer [START_REF] Bailey | Physicists in Medicine, éds, Nuclear medicine physics: a handbook for teachers and students[END_REF] and these data should include a measurement of A(r s , t) at t 0 '0 and t D ' 3•T 1/2,ef f in order to avoid an under-over estimation of the activity, as synthetically shown in Fig. 1.8. The dosimetric acquisition protocol must therefore be established after a detailed study of the pharmacokinetics of the radiopharmaceutical, which makes it possible to define the best temporal sampling.

b) Absorbed dose rate per unit cumulated activity a The cumulated activities in the source organ per unit administrated activity should be further combined with the absorbed dose per unit cumulated activity S for the estimation of the absorbed dose. The S value corresponds to the average dose absorbed by a target region r t for a particle emitted from the source region r s . It contains information on the type of radiations emitted, their energies and geometrical aspects (size and shape of the source and target regions and distance between them), but also on the density and the atomic composition of the tissues traversed. The S-factor is Chapter 1

The role of dosimetry in MRT specific to a given radionuclide and can be analytically defined as:

S(r t r s ) = X i (r t r s , E i ) = 1 M (r t ) X E i Y i (r t r s , E i ) (1.4)
where:

• i represents the mean energy of the i th transition per nuclear transformation decay and is defined as the product of the mean energy E i of the i th nuclear transition and the number of i th nuclear transitions per nuclear transformation Y i ;

• (r t r s , E i ) is the specific absorbed fraction. It is defined as the ratio between the absorbed fraction (r t r s , E i ), the fraction of radiation energy E i emitted from the source tissue r s that is absorbed in the target tissue r t , and the target mass M(r t ). Thus, the calculation of S is based on the physical data of the radionuclides used for the treatment, that can be found in numerous databases published by the ICRP, the National Institute of Standards of Technology (NIST), as well as the MIRD committee [START_REF] Wong | MIRD: Radionuclide Data and Decay Schemes[END_REF], the geometry in which particle transport takes place and the method of calculating the energy loss in the target organ due to the interactions of the particles emitted by the source organ. Di↵erent methods for S calculation will be presented in Sec. 1.3.1.3. It has been shown that the determination of absorbed doses based on MIRD formalism [START_REF] Bolch | MIRD Pamphlet No. 21: A Generalized Schema for Radiopharmaceutical Dosimetry-Standardization of Nomenclature[END_REF] relies on the estimation of the cumulated activity in source organs and of S-factors. These estimations provide two possibilities. Based on a pre-treatment activity evaluation, individual treatment planning optimization are enabled. The goal of the pre-therapeutic study (also know as tracer study) is to define the maximal activity that could be administered to each specific patient, according to the treatment The role of dosimetry in MRT Chapter 1 objectives and to the tolerance doses to organs at risk, for example considering early or late kidney damage or bone marrow toxicity. Furthermore, post-treatment verification of the absorbed dose delivered to targeted and non-targeted tissues is also essential to refine the treatment planning. According to these objectives, the absorbed dose calculation must be carried out in the most accurate and individualized possible way. In this context, functional imaging of the patient plays an important role. The access to patient's specific pharmacokinetics provide the physician with the necessary information to optimize patients treatments and therefore giving them the best and durable responses to their therapy.

The role of imaging in dose assessment a

The accurate estimation of radionuclide biodistribution and biokinetics, that is the estimation of e A, is of primary importance for absorbed dose calculation [START_REF] Eberlein | Individualized Dosimetry for Theranostics: Necessary, Nice to Have, or Counterproductive?[END_REF][START_REF] Siegel | MIRD Pamphlet No. 16: Techniques for Quantitative Radiopharmaceutical Biodistribution Data Acquisition and Analysis for Use in Human Radiation Dose Estimates[END_REF] as it is considered the largest contributor to its uncertainties [START_REF] Dewaraja | MIRD Pamphlet No. 24: Guidelines for Quantitative 131i SPECT in Dosimetry Applications[END_REF]. Uncertainties in activity quantification at each time point will propagate into uncertainties in the absorbed dose and automatically in treatment response assessment, as will be discussed in section 1.3.1.4. In therapeutic applications, where toxicity and e cacy are of concern and where there is much less tolerance for inaccuracies, individual imaged-based dosimetry is needed.

According to MIRD schema, quantitative imaging can be performed with planar scintillation camera and with SPECT (and PET). The most commonly employed imaging method for in vivo quantitation of radioactivity uses 180 opposed planar images (known as the conjugate view CV approach) in combination with transmission data through the subject and a system calibration factor for conversion from count rate to activity in each pixel of the reconstructed image. This technique o↵ered an improvement over the simple single-view procedures because the CV mathematical formalism provides correction for source thickness, inhomogeneity and attenuation. CV has been shown to be useful for activity quantification in studies carried out with physical phantoms, simulated images employing Monte Carlo methods, and patient images.

Planar methods have been evaluated by several authors since many decades [START_REF] Eary | Preliminary validation of the opposing view method for quantitative gamma camera imaging -Eary -1989 -Medical Physics[END_REF][START_REF] Norrgren | Accuracy of the Quantification of Organ Activity from Planar Gamma Camera Images | Cancer Biotherapy and Radiopharmaceuticals[END_REF][START_REF] Delpon | Impact of Scatter and Attenuation Corrections for Iodine-131 Two-Dimensional Quantitative Imaging in Patients[END_REF][START_REF] Zimmerman | Multicentre evaluation of accuracy and reproducibility of planar and SPECT image quantification: An IAEA phantom study[END_REF]. All authors demonstrate the capability of activity quantification with errors in the range of 3-20%, mainly depending on the source used ( 99m Tc or 131 I), on its dimension and position. They noted that penetration and organ overlap were perhaps the most important factor impacting the activity quantification, together with partial volume e↵ect if small volumes were considered. Activity quantification with tomographic imaging e.g. Single Photon Emission Computed Tomography (SPECT) is theoretically superior to that with planar imaging, as problems of organ overlap may be overcome and contrast is increased for small regions.

As with planar approach, SPECT quantification also has been evaluated by several authors [START_REF] Zimmerman | Multicentre evaluation of accuracy and reproducibility of planar and SPECT image quantification: An IAEA phantom study[END_REF][START_REF] Jaszczak | SPECT using a specially designed cone beam collimator[END_REF][START_REF] Dewaraja | Accurate Dosimetry in 131i Radionuclide Therapy Using Patient-Specific, 3-Dimensional Methods for SPECT Reconstruction and Absorbed Dose Calculation[END_REF][START_REF] He | Comparison of conventional, model-based quantitative planar, and quantitative SPECT image processing methods for organ activity estimation using In-111 agents -IOPscience[END_REF][START_REF] Pereira | Image Quantification for Radiation Dose Calculations -Limitations and Uncertainties[END_REF][START_REF] Li | Quantitative Imaging for Targeted Radionuclide Therapy Dosimetry -Technical Review[END_REF]. These studies have found results with various levels of accuracy, but it is often di cult to directly compare them because investigators have used di↵erent levels of activity, di↵erent radionuclides, and di↵erent shapes and sizes of sources.

Chapter 1

The role of dosimetry in MRT However, SPECT quantification proved superior results in many cases, thanks to the implementation of correction methods for attenuation, scatter, septal penetration, background, and organ overlap (good accuracy, within 2-6.5% in some cases [START_REF] Zimmerman | Multicentre evaluation of accuracy and reproducibility of planar and SPECT image quantification: An IAEA phantom study[END_REF][START_REF] He | Comparison of conventional, model-based quantitative planar, and quantitative SPECT image processing methods for organ activity estimation using In-111 agents -IOPscience[END_REF][START_REF] Li | Quantitative Imaging for Targeted Radionuclide Therapy Dosimetry -Technical Review[END_REF]). Despite this fact, when dosimetry is performed, planar method is the most used method as it takes less time for acquisition, is less computationally intensive and it can be implemented relatively easily on modern dual-head gamma cameras [START_REF] Li | Quantitative Imaging for Targeted Radionuclide Therapy Dosimetry -Technical Review[END_REF].

Clear is that to perform high quality dosimetry studies, it is essential to carefully characterize the imaging system to establish attenuation and calibration coe cients, suitable window settings for collection of data for scatter correction and factors for objects of di↵erent size, for any radionuclides used.

With the development of advanced techniques to improve image quality and quantification, is nowadays clear that functional imaging is an extremely powerful tool for dosimetric application. However, several limitation hindered its clinical use, which will be highlighted in section 1.3.3.2.

Dose calculation : anatomic models and computational methods a

Once the cumulative activity e A is calculated by integrating the time-activity curve, the S-factor between each source/target organ should also be determined. The estimation of S, analytically described by the equation 1.4, is performed taking into account the physical data of the radioactive isotopes used for the treatment, i.e the emission spectrum of the particles following the decay of the isotope, the geometry in which particle transport takes place, which is described by anatomical phantoms and the method of calculating the energy loss in the target organ due to the interactions of the particles emitted by the source organ with the material, which may be indirect (Dose point kernel or Dose voxel kernel) or direct (Monte-Carlo simulation).

a) Anatomical models a

The S-factor is highly influenced by the computational phantom defining the spatial shape and tissue compositions of both r s and r t [START_REF] Bolch | MIRD Pamphlet No. 21: A Generalized Schema for Radiopharmaceutical Dosimetry-Standardization of Nomenclature[END_REF]. The description of the patient's morphology is one of the main sources of uncertainty in the calculation of S-factors because its anatomy can be represented more or less faithfully, with the consequence of introducing biases into the final calculation of the absorbed dose [START_REF] Divoli | E↵ect of Patient Morphology on Dosimetric Calculations for Internal Irradiation as Assessed by Comparisons of Monte Carlo Versus Conventional Methodologies[END_REF]. Since the first attempt to calculate S, the representation of the patient was made by di↵erent anthropomorphic models: mathematical models, voxelized models and hybrid models.

Anthropomorphic mathematical models The mathematical models are the oldest of the three approaches. They consist in the combinations/intersections of planes, circular and elliptical cylinders, spheres and cones representing the various organs, which are described by mathematical equations. The first of these models, the socalled Fisher-Snyder model, was developed in the 1960s (Fig. 1

.9a). The mass of the

The role of dosimetry in MRT Chapter 1 organs and their compositions and densities were based on data provided by the ICRP. The phantom was used into MIRD pamphlet No. 5 in 1969 [START_REF] Snyder | MIRD Pamphlet No 5 : Estimates of absorbed fractions for monoenergetic photon sources uniformly distributed in various organs of a heterogeneous phantom[END_REF] for the calculation of absorbed fractions in the emission of monoenergetic sources of photons and then it was used in MIRD pamphlet No. 11 to calculate S values for over 110 radionuclides and for selected source-target organs configurations commonly used in nuclear medicine [START_REF] Snyder | Absorbed Dose per Unit Cumulated Activity for Selected Radionuclides and Organs[END_REF]. New anthropomorphic models were then developed to cover a wider variety of situation, ranging from di↵erent pediatric models [START_REF] Cristy | Mathematical phantoms representing children of various ages for use in estimates of internal dose[END_REF][START_REF] Cristy | SPECIFIC ABSORBED FRACTIONS OF ENERGY AT VARIOUS AGES FROM INTERNAL PHOTON SOURCES[END_REF], to adult women and pregnant women in each trimester of their pregnancy [START_REF] Stabin | Mathematical models and specific absorbed fractions of photon energy in the nonpregnant adult female and at the end of each trimester of pregnancy[END_REF]. Although considerably facilitating the calculation of the absorbed dose, the mathematical models have gradually been replaced, due to their lack of realism, by new so-called voxelized models, more representative of the morphology of patients.

Anthropomorphic voxelized models Instead of representing the di↵erent anatomical regions by complex equations defining simple geometric shapes, 3D voxel phantoms are constructed from Computed Tomography (CT) or Magnetic Resonance (MRI) images recorded from real people. These models are digital images (three-dimensional matrices) made up voxels (three-dimensional pixels) of fixed width, height and thickness. The density of the organs are generally matched to the reference values given in ICRP Publication 89 [START_REF] Lemosquet | Voxel anthropomorphic phantoms : review of models used for ionising radiation dosimetry[END_REF][START_REF] Caon | Voxel-based computational models of real human anatomy: a review[END_REF]. More than 30 voxelised models are nowadays available [START_REF] Zaidi | Computational Anthropomorphic Models of the Human Anatomy: The Path to Realistic Monte Carlo Modeling in Radiological Sciences[END_REF], including various models to represent di↵erent ethnic groups [START_REF] Kramer | All about MAX: a male adult voxel phantom for Monte Carlo calculations in radiation protection dosimetry[END_REF][START_REF] Kramer | All about FAX: a Female Adult voXel phantom for Monte Carlo calculation in radiation protection dosimetry[END_REF][START_REF] Kramer | MAX06 and FAX06: update of two adult human phantoms for radiation protection dosimetry[END_REF][START_REF] Xu | VIP-Man: an image-based wholebody adult male model constructed from color photographs of the Visible Human Project for multi-particle Monte Carlo calculations[END_REF][START_REF] Petoussi-Henss | The GSF family of voxel phantoms[END_REF][START_REF] Zhang | CNMAN: a Chinese adult male voxel phantom constructed from color photographs of a visible anatomical data set[END_REF]. The voxelised phantoms are anatomically realistic, but their voxelized nature limits the representation of anatomical variations, motion and thin organs as the skin. [START_REF] Norris | A set of 4d pediatric XCAT reference phantoms for multimodality research[END_REF][START_REF] Ramos | Internal radiation dose and modeling codes in nuclear medicine: a fresh look at old problems[END_REF].

Anthropomorphic hybrid models These limits explains the arrival of so-called hybrid models (Fig. 1.9c), which combine both the advantages of voxelized and mathematical phantoms [START_REF] Segars | Development and application of the new dynamic NURBS-based cardiac-torso (NCAT) phantom[END_REF][START_REF] Hoseinian | Hybrid phantom approach for radiation dosimetry applications -ProQuest[END_REF][START_REF] Hoseinian-Azghadi | Development of a 9-months pregnant hybrid phantom and its internal dosimetry for thyroid agents[END_REF]. Hybrid phantoms are based on segmented human data to create a realistic anthropomorphic anatomy and use more complex mathematical surfaces and polygon meshes (the Nonuniform Rational Basis Spline, NURBS) to define each anatomical object, so that organ variations and movements are easily obtained by rigid and non-rigid transformation. Thus, the development of hybrid models allows more individuality and flexibility, while preserving the reality of voxel models [START_REF] Segars | Development and application of the new dynamic NURBS-based cardiac-torso (NCAT) phantom[END_REF][START_REF] Segars | The MCAT, NCAT, XCAT, and MOBY Computational Human and Mouse Phantoms[END_REF][START_REF] Segars | Population of anatomically variable 4d XCAT adult phantoms for imaging research and optimization[END_REF][START_REF] Cassola | FASH and MASH: female and male adult human phantoms based on polygon mesh surfaces: I. Development of the anatomy[END_REF].

b) Energy deposit calculation a

The methods for calculating the energy deposits are divided into direct and indirect methods. Direct methods are Monte Carlo methods, whereas indirect methods are analytical methods such as Dose Point Kernel (DPK) and Dose Voxel Kernel (DVK).

Direct methods: Monte Carlo simulation Monte Carlo simulation is applied to areas of science in which the various processes and phenomena can be modeled by statistical methods based on the generation of random numbers. In the present case, the di↵erent processes to be modeled concern the transport of particles in matter: the particles (photons or electrons) undergo a certain number of interactions with the material whose probability is systematically established according to the energy of the particle and the density of the medium. The radiation-matter interaction probability laws determine the type of interaction, the energy lost by the particle during the interaction, the mean free path between two successive interactions for this particle as well as the initial energy of the secondary particle potentially created during the interaction. Usually, the anatomical phantom described above can be easily integrated inside the simulation in order to take into accont the interaction of particles with real human materials and geometries. Among the various generic Monte Carlo codes, some, such as MCNP [START_REF] Mckinney | MCNP 6.1.1 New Features Demonstrated[END_REF], GEANT4 [START_REF] Brada | Proton Therapy in Clinical Practice: Current Clinical Evidence[END_REF], EGS4 [START_REF] Kawrakow | The EGSnrc Code System, Monte Carlo Simulation of Electron and photon Transport[END_REF], PENELOPE [START_REF] Salvat | PENELOPE-2011: A Code System for Monte Carlo Simulation of Electron and Photon Transport[END_REF] and FLUKA [START_REF] Battistoni | The FLUKA code: description and benchmarking[END_REF] are used in nuclear medicine. These codes, and especially GEANT4, have the advantage of being used by a large community of physicists. They adapt to di↵erent ranges of energy, various particles and contain many physical models. However, their level of complexity is often high and this is why other Monte Carlo codes, more specific to imaging and nuclear medicine application, have been created in parallel. Some are only used for SPECT simulation, such as SIMIND [START_REF] Ljungberg | 3-D Image-Based Dosimetry in Radionuclide Therapy[END_REF], SimSPECT (derived from MCNP) [START_REF] Yanch | Monte Carlo simulation in SPECT: complete 3d modeling of source, collimator and tomographic data acquisition[END_REF] or MCMATV [START_REF] Smith | A vectorized Monte Carlo code for modeling photon transport in SPECT[END_REF]. Other specific codes are dedicated to PET modeling, such as PETSIM [START_REF] Thompson | PETSIM: Monte Carlo simulation of all sensitivity and resolution parameters of cylindrical positron imaging systems[END_REF] and EIDOLON [START_REF] Zaidi | Implementation of an environment for Monte Carlo simulation of fully 3-D positron tomography on a high-performance parallel platform[END_REF]. Finally, di↵erent codes can be used for a bigger range of application, as SIMSET and

The role of dosimetry in MRT Chapter 1 GATE, based on GEANT4 [START_REF] Jan | GATE: a simulation toolkit for PET and SPECT[END_REF].

Indirect analytical methods The Dose Point Kernel (DPK) calculation was introduced in MIRD Pamphlet 2 [START_REF] Berger | MIRD Pamphlet No 2 : Energy deposition in water by photons from point isotropic sources[END_REF] and consists in studying the absorbed dose distribution around an isotropic point source of photons or electrons in an infinite homogeneous medium. This analytical approach is based on the superposition principle, which establishes that each source can be defined as a set of elemental point sources, and that the energies deposited for each elemental point source add up. The DPK values are presented as tables of absorbed dose values as a function of the distance to the point source and of the material, which have been obtained by experimental measurements [START_REF] Cross | Tables of beta-ray dose distribution in water[END_REF] or by Monte-Carlo simulation using di↵erent software (EGS4 [START_REF] Furhang | Radionuclide photon dose kernels for internal emitter dosimetry[END_REF], MCNP [START_REF] Janicki | A dose point kernel model for a low energy gamma emitting stent in a heterogeneous medium[END_REF], FLUKA code validated by PENELOPE [START_REF] Botta | Calculation of electron and isotopes dose point kernels with FLUKA Monte Carlo code for dosimetry in nuclear medicine therapy[END_REF]). The dosimetric calculation therefore consists of using the DPK values and then convolving them with the spatial distribution of the considered source. This allows a quick estimation of the energy density in a homogeneous environment. However, when it is desired to define an elemental source in an entire voxel rather than a point source, the MIRD pamphlet No. 17 introduces the possibility of performing a quick calculation of the dose absorbed in a homogeneous medium, just like the DPK, but at the voxel scale [START_REF] Bolch | MIRD Pamphlet No. 17: The Dosimetry of Nonuniform Activity Distributions -Radionuclide S Values at the Voxel Level[END_REF]. The Dose Voxel Kernel (DVK) values are estimated by Monte Carlo simulation placing a source voxel at the center of a voxel geometry consisting of a uniform material. The absorbed dose is then calculated from cubic kernels composed of this central source voxel and some surrounding voxels (e.g. a 3x3x3 voxel kernel), convoluted by the spatial distribution of the source. Dosimetric approaches by DPK and DVK are perfectly adapted to homogeneous interaction environments and have the advantage of their rapidity.

The choice of the dosimetric calculation method, direct or indirect, must therefore be determined in advance according to the particular anatomical region considered. For regions presenting interfaces between several media of di↵erent densities, the direct Monte Carlo method is preferable.

In order to facilitate the calculation of the absorbed dose, various dosimetry software has been developed. The first softwares available to the nuclear medicine community is called "MIRDOSE" [START_REF] Stabin | MIRDOSE: Personal Computer Software for Internal Dose Assessment in Nuclear Medicine[END_REF]. The program, based on the Fisher-Snyder anthropomorphic mathematical model, used libraries of radionuclide decay data and specific absorbed fractions to develop S-values for a source organs (chosen by the user) and the target organs desired. MIRDOSE was later updated to a Java-based personal computer code called OLINDA/EXM (Organ Level INternal Dose Assessment/EXponential Modeling) [START_REF] Stabin | OLINDA/EXM: The Second-Generation Personal Computer Software for Internal Dose Assessment in Nuclear Medicine[END_REF], which includes a database of data for 814 radionuclides and 10 reference anthropomorphic mathematical models [START_REF] Cristy | SPECIFIC ABSORBED FRACTIONS OF ENERGY AT VARIOUS AGES FROM INTERNAL PHOTON SOURCES[END_REF].

Other dosimetric software were designed and directly integrated an anthropomorphic models (mathematical, voxelized or hybrid) with a method for calculating the energy deposit (DPK, DVK or direct calculation by Monte Carlo simulation). For example VoxelDose [START_REF] Gardin | Voxeldose: A Computer Program for 3-D Dose Calculation in Therapeutic Nuclear Medicine[END_REF] includes the calculation of the energy deposition at the voxel scale by DVK convolution with the cumulative activity distribution while SIMDOSE [START_REF] Ljungberg | A 3-Dimensional Absorbed Dose Calculation Method Based on Quantitative SPECT for Radionuclide Therapy: Evaluation for 131i Using Monte Carlo Simulation[END_REF], 3D-

Chapter 1

The role of dosimetry in MRT ID [START_REF] Furhang | Radionuclide photon dose kernels for internal emitter dosimetry[END_REF][START_REF] Prideaux | Three-Dimensional Radiobiologic Dosimetry: Application of Radiobiologic Modeling to Patient-Specific 3-Dimensional Imaging-Based Internal Dosimetry[END_REF] and OEDIPE [START_REF] Chiavassa | OEDIPE: A Personalized Dosimetric Tool Associating Voxel-Based Models with MCNPX[END_REF] includes direct Monte Carlo simulation calculations,.

Despite the many dosimetry tools available to the clinician, we will see in section 1.3.3 that personalized dosimetry-based treatments is far from being systematically used in molecular radiotherapy. There are several reasons for this, but one of them is that clinicians and medicals physicists generally have low confidence in the use of radiation dose analyses, which raises the question of uncertainties associated with dose assessment.

Uncertainties in dose calculation a

As previously described, activity quantification require both the quantification of radiopharmaceutical biokinetics from images and the use of dosimetry computational models. This process requires many steps that are far from trivial and which are subject to uncertainties. Whatever the dosimetric approach used, the absorbed dose assessment chain is composed of di↵erent phases and the overall accuracy of the dosimetric results depends on the level of accuracy of each of them. Their reliability also depends on the rigour with which the study is reported [START_REF] Lassmann | Dosimetry and thyroid cancer: the individual dosage of radioiodine[END_REF]. Understanding and quantifying these uncertainties therefore appears to be a requirement for quality assurance of the absorbed dose calculation and the activity to be administered, as well as for quantitative comparisons and critical analysis of the results from di↵erent investigators.

The di culty of associating uncertainties with the estimation of absorbed dose is considered to be one of the main reasons why dosimetry is not yet systematically integrated into clinical routines [START_REF] Flux | Clinical radionuclide therapy dosimetry: the quest for the 'Holy Gray[END_REF]. In this context, the European Association of Nuclear Medicine (EANM) published in September 2018 a practical guide for the analysis of uncertainties in internal radiotherapy [START_REF] Gear | EANM practical guidance on uncertainty analysis for molecular radiotherapy absorbed dose calculations[END_REF]. This document provides a generalized scheme for determining uncertainties in dosimetric calculations; it applies to the MIRD schema the uncertainty propagation law given in the Guide to the Expression of Uncertainty in Measurement (GUM) published by the Joint Committee for Guides in Metrology (JCGM) [START_REF]JCGM 100. Evaluation of measurement data -Guide to the expression of uncertainty in measurement[END_REF][175][176]. The GUM is accepted as the guiding document for the assessment of uncertainties by the International Organisation for Standardisation (ISO) and the European Committee for Standardisation (CEN) [START_REF] Siebert | Uncertainty in radiation dosimetry: basic concepts and methods[END_REF]. In its simplest form, corresponding to the case where the target volume coincide with the source activity volume and the contribution from other target organs can be considered negligible, the absorbed dose uncertainty u(D) can be derived from the MIRD equation 1.1 applying the propagation schema as follow [START_REF] Gear | EANM practical guidance on uncertainty analysis for molecular radiotherapy absorbed dose calculations[END_REF]:

 u(D) D 2 = " u( e A) e A # 2 +  u(S) S 2 + u( e AS) e AS (1.5)
The uncertainty on the cumulated activity e A has to be determined by the sum of both systematic and random uncertainties on the cumulated activity calculation. There

The role of dosimetry in MRT Chapter 1 are a multitude of uncertainties relied to activity quantifications [START_REF] Ljungberg | 3-D Image-Based Dosimetry in Radionuclide Therapy[END_REF][START_REF] Gear | Characterisation of the attenuation properties of 3d-printed tungsten for use in gamma camera collimation[END_REF]. The most important are now discussed. Firstly, the delineation of the region of interest (ROI) on the image plays an important role in activity quantification. It is fundamental because it automatically defines the number of count detected to be used for activity calculation. Bigger ROI that the lesion leads to the integration of background, penetration and scatter events and thus to an higher amount of detected counts. Smaller ROI cause the spill-o↵ of useful event outside the ROI, with the consequent loss of useful events. Then, the uncertainties on the calibration factor could have a huge impact on the determination of the activity and of its uncertainties. The factor depends both on the reconstructed count rate inside the ROI and on the measurement of the activity used (which include the accuracy of the activimeter used and the precision knowledge of a possible the decay of the source). Moreover, once known the activity at a given time, several data points are needed for the estimation of e A. The choice of the time points for the acquisition and the number of data points acquired impact the fitting methods used and consequently its accuracy, as already discussed in Sec. 1.3.1.1. Together with the uncertainties described above, e A uncertainties could also arise from other random sources such as image noise, patient motion, registration and other post-acquisition operations, which could include the image reconstruction procedure [START_REF] Gear | Characterisation of the attenuation properties of 3d-printed tungsten for use in gamma camera collimation[END_REF]. Studies shown that the calibration factor and the partial volume e↵ect are the most important contribution to the overall uncertainty in the dosimetric procedure [START_REF] Stabin | Uncertainties in Internal Dose Calculations for Radiopharmaceuticals[END_REF][START_REF] Dewaraja | Accurate Dosimetry in 131i Radionuclide Therapy Using Patient-Specific, 3-Dimensional Methods for SPECT Reconstruction and Absorbed Dose Calculation[END_REF][START_REF] Gustafsson | Uncertainty propagation for SPECT/CT-based renal dosimetry in 177 Lu peptide receptor radionuclide therapy[END_REF]. Sure is that the quality of data and the intrinsic characteristics of the system used for image acquisition strongly influence both the calibration factor and the partial volume e↵ect. Indeed, intrinsic limitations on energy resolution, degradation of spatial resolution because of collimator septal penetration by high-energy photons, data loss due to scatter and attenuation and the basic statistical variability of any measurement of a radioactive source, could highly influences the image quality and thus the accuracy of quantification.

The uncertainties on the absorbed dose rate per unit cumulated activity S are less that those on the e A value. The S-factor depends mainly on the specific absorbed fraction and the organ mass M, as shown in equ. 1.4. If Monte Carlo radiation transport simulations with standardized organs are used, the absolute uncertainty in the calculated can be reduced to 5% or less, performing a large number of radiation transport simulations [START_REF] Stabin | Uncertainties in Internal Dose Calculations for Radiopharmaceuticals[END_REF]. But this error increases when the individual's characteristics deviate from those of the population median, used as normalized values of organ mass M. Thus, the more realistic the anatomical models used is, the more u(S) could be considered negligible through direct Monte Carlo radiation transport.

Finally, even if e A and S are calculated separately, one from scintigraphic images and the other from MC simulations, a covariance exists between the two parameters because they both depend of the same volume of interest corresponding to the volume of an organ or tumor. In therapeutic applications, if patient-individualized dosimetry is performed with particular attention to the acquisition of accurate data (collection of a su cient number of time points to characterize each absorption or clearance phase, acquisition of a su cient number of counts for each image), analysis (including image Chapter 1

The role of dosimetry in MRT quantification), processing (i.e. curve adjustment) and measurement of individual organ volumes, many of the model-based uncertainties can be reduced and the overall uncertainty in an individual dose estimate can be reduced [START_REF] Stabin | Uncertainties in Internal Dose Calculations for Radiopharmaceuticals[END_REF].

At present, uncertainty calculation in MRT are rare [START_REF] Lassmann | EANM Dosimetry Committee guidance document: good practice of clinical dosimetry reporting[END_REF].None of the commercially available software o↵ers an assessment of dose-related uncertainties. However, the number of centres performing dosimetry increases with the rapid expansion of MRT. In this context, the uncertainty analysis appears as a primary necessity for identifying and reducing errors and quantifying the degree of confidence that can be placed in dose assessment and consequently, increasing its validity. The assessment of uncertainties is as important as that of the absorbed dose, in order to allow for an acceptable interpretation of the data in clinical practice, with the final goal of increasing the dose-response relationships, which could lead to improved treatment procedures [START_REF] Gear | EANM practical guidance on uncertainty analysis for molecular radiotherapy absorbed dose calculations[END_REF]. Innovative approaches such as Bayesian networks developed in the context of artificial intelligence may help to propagate uncertainties in the calculation of the absorbed dose. This flexible and fast technique has already been successfully applied to consider uncertainties in the surveillance of workers exposed to a risk of internal contamination.

Clinical evidence of the relevance of dosimetry in molecular radiotherapy

a There are su cient and growing evidences of the therapeutic benefits of treating diseases with molecular radiotherapy and the need to systematically combine these treatments with personalized dosimetry. This evidence first relies on the existing proofs that di↵erent biological e↵ects, i.e di↵erent absorbed doses, can be produced by fixed activities, because each patient have di↵erent tumor and normal organ radiopharmaceutical uptake and clearance half-times [START_REF] Smith | Clinical trial of di↵erent doses of 131-I in treatment of thyrotoxicosis[END_REF][START_REF] Eary | Importance of PrecTreatment Radiation Absorbed Dose Estimation for Radioimmunotherapy of Non-Hodgkin's Lymphoma[END_REF]. Moreover, although no multicenter clinical studies have so far demonstrated the dose-response relationship for MRTs, many individual studies have shown that dosimetry results are correlated with both patient safety, i.e. toxicity to OARs, and treatment e↵ectiveness. Several of these results have been summarized in a review article published in 2014 by Strigari et al. for several treatments involving radionuclides, such as radioiodine therapy of benign and malignant thyroid diseases, neuroblastoma, neuroendocrine tumors and liver tumors [START_REF] Strigari | The evidence base for the use of internal dosimetry in the clinical practice of molecular radiotherapy[END_REF].

In the context of benign thyroid diseases, di↵erent studies have been published for the treatment of Graves' disease and multinodular goitre using a dosimetric approach. Among them, Peters et al. shown in 1995 that the therapeutical e↵ectiveness of thyroid treatments depends on the radiation dose absorbed by the thyroid [START_REF] Peters | Radioiodine therapy of Graves' hyperthyroidism: standard vs. calculated 131 iodine activity*. Results from a prospective, randomized, multicentre study[END_REF]. In 1996, they reported a study on radioiodine treatment in 205 patients with Graves' hyperthyroidism and concluded that the treatment reduces thyroid volume in a dose-dependent manner [START_REF] Peters | Reduction in thyroid volume after radioiodine therapy of Graves' hyperthyroidism: results of a prospective, randomized, multicentre study[END_REF][START_REF] Peters | Treatment of Graves' Hyperthyroidism with Radioiodine: Results of a Prospective Randomized Study[END_REF]. For the treatment of di↵erentiated thyroid cancer, both Maxon et al. and Flux et al. [186,[START_REF] Flux | A dose-e↵ect correlation for radioiodine ablation in di↵erentiated thyroid cancer[END_REF] showed that a successful thyroid ablation is dependent on the absorbed dose to the tumor remnants. Moreover, there are di↵erent recent re-

The role of dosimetry in MRT Chapter 1 ports describing a treatment-related dose-e↵ect relationships based on pre-therapeutic 124 I imaging [START_REF] Jentzen | Pre-therapeutic 124i PET(/CT) dosimetry confirms low average absorbed doses per administered 131i activity to the salivary glands in radioiodine therapy of differentiated thyroid cancer[END_REF][START_REF] Wierts | Dose-Response Relationship in Di↵erentiated Thyroid Cancer Patients Undergoing Radioiodine Treatment Assessed by Means of 124i PET/CT[END_REF][START_REF] Jentzen | Assessment of Lesion Response in the Initial Radioiodine Treatment of Di↵erentiated Thyroid Cancer Using 124i PET Imaging[END_REF]. In patients with metastatic liver neuroendocrine tumor, a significant absorbed dose-e↵ect relationship for tumor volume and tumor size reduction after treatment was found ( 90 Y-Labeled somatostatin analogs) [START_REF] Pauwels | Practical Dosimetry of Peptide Receptor Radionuclide Therapy with 90y-Labeled Somatostatin Analogs[END_REF]. Dewaraja et al. also shown an absorbed dose-response correlation with a specific threshold of 2 Gy for patients treated with 131 I -tositumomab for relapsed or refractory B-cell lymphoma previously treated with chemotherapy [START_REF] Dewaraja | 131i-Tositumomab Radioimmunotherapy: Initial Tumor Dose-Response Results Using 3-Dimensional Dosimetry Including Radiobiologic Modeling[END_REF]. For radioimmunotherapies with 177 Lu-labeled compound addressing the antigen (CD-37), Blakkisrud et al. succeeded in establishing a correlation between the hematologic toxicity and absorbed dose to the bone marrow [START_REF] Blakkisrud | Red Marrow-Absorbed Dose for Non-Hodgkin Lymphoma Patients Treated with 177 Lu-Lilotomab Satetraxetan, a Novel Anti-CD37 Antibody-Radionuclide Conjugate[END_REF]. Outside these applications, clinical benefits of dosimetry has also been demonstrated for the use of 131 I -labelled antibodies for radioimmunotherapy and in the treatment of hepatocellular carcinoma with the use of microspheres in the treatment of liver cancers [START_REF] Strigari | The evidence base for the use of internal dosimetry in the clinical practice of molecular radiotherapy[END_REF].

These applications cover most MRT applications, as shown in Section 1.2.2, which demonstrates that increased use of dosimetry-based approaches for organs-at-risk and target regions would significantly improve patient outcome. However, the evidence of dose-e↵ect relationships for MRT has been emphasized without a critical evaluation of the dosimetric methodology used. Regardless of the approach used, each step of the dosimetric computation chain, although responding to the principles described in the MIRD pamphlets, is carried out according to di↵erent methods, which are generally chosen and applied by di↵erent clinical centers mainly according to the resources at their disposal. Thus, there is currently no standardized dosimetric protocol, resulting in di↵erent absorbed doses to target tissues and OARs. For example, high variability between centers of absorbed radiation dose has been demonstrated during treatment of B-cell non-Hodgkin lymphoma using Ibritumomab tiuxetan (Zevalin) labeled with 111 In or 90 Y. The values varies by a factor of 24 between the di↵erent dosimetric results obtained for the kidneys, corresponding to a substantially equivalent whole-body absorbed dose between centers [START_REF] Fisher | MIRD Dose Estimate Report No. 20: Radiation Absorbed-Dose Estimates for 111in-and 90y-Ibritumomab Tiuxetan[END_REF]. Moreover, in the case of di↵erentiated thyroid cancer, di↵erent dose limits (300 Gy [START_REF] Maxon | Relation between e↵ective radiation dose and outcome of radioiodine therapy for thyroid cancer[END_REF] and 50 Gy [START_REF] Flux | A dose-e↵ect correlation for radioiodine ablation in di↵erentiated thyroid cancer[END_REF]) were assessed as threshold for successful treatment.

The lack of standardization across centers and consequently the absence of large randomized clinical trials that could demonstrate the e↵ectiveness of dosimetry in optimizing treatment planning and predicting therapeutic outcomes is certainly a major obstacle to its use [START_REF] Kesner | Modern Radiopharmaceutical Dosimetry Should Include Robust Biodistribution Reporting[END_REF]. However, it cannot justify the absence of post-treatment verification of the absorbed doses delivered, which provides patient-specific information without changing patient management [START_REF] Flux | From fixed activities to personalized treatments in radionuclide therapy: lost in translation?[END_REF].

Chapter 1

The role of dosimetry in MRT [START_REF] Sjögreen Gleisner | Variations in the practice of molecular radiotherapy and implementation of dosimetry: results from a European survey[END_REF].

According to this survey, only 36% of all MRT treatments were performed with an absorbed dose individually planned for each patient; in 63% of cases, absorbed dose planning was never carried out, or carried out in a minority of treatments. Post-therapy imaging was performed for 69% of the treatments but only the 26.0% were performed with the purpose of post-therapy dosimetry. The majority of MRT therapies performed in Europe are prescribed using di↵erent fixed-activity protocols mainly tailored to patient weight and body surface area or adjusted on the diagnosis or the disease stage. All the treatments performed with 131 I -mIBG, 177 Lu/ 90 Y-somatostatin analogs, 177 Lu-PSMA and the therapies including radiation synovectomy, 153 Sm-EDTMP, 89 SrCl 2 , and 223 RaCl 2 , 32 P sodium-phosphate, and 90 Y-ibritumomab-tiuxetan are performed with the administration of fixed activity, in few cases adjusted based on diagnosis, stage, or other clinical factors. Treatments of hepatic diseases with 90 Y-microsphere are the ones with the highest number of absorbed-dose based reported prescriptions. This is justified by the high toxicity associated with radioembolization, which can lead to liver failure and possible death of the patient. Section 1.3.3.3 gives an insight of thyroid treatments with 131 I , which is the most commonly used radionuclide in MRT applications, with the largest geographical spread [START_REF] Sjögreen Gleisner | Variations in the practice of molecular radiotherapy and implementation of dosimetry: results from a European survey[END_REF].

The common use of fixed activities for most radionuclides shows that, although there is existing proofs that di↵erent biological e↵ects can be produced by fixed activities [START_REF] Smith | Clinical trial of di↵erent doses of 131-I in treatment of thyrotoxicosis[END_REF][START_REF] Eary | Importance of PrecTreatment Radiation Absorbed Dose Estimation for Radioimmunotherapy of Non-Hodgkin's Lymphoma[END_REF], and that the treatment response, both in terms of e↵ects and toxicity, is correlated to the dose delivered to the patients [START_REF] Stabin | State of the Art in Nuclear Medicine Dose Assessment[END_REF][START_REF] Strigari | The evidence base for the use of internal dosimetry in the clinical practice of molecular radiotherapy[END_REF][START_REF] Salvatori | Radioiodine therapy dosimetry in benign thyroid disease and di↵erentiated thyroid carcinoma[END_REF][START_REF] Garin | High impact of macroaggregated albumin-based tumour dose on response and overall survival in hepatocellular carcinoma patients treated with 90 Y-loaded glass microsphere radioembolization[END_REF], the clinical need of dosimetry is still not yet well established.

Challenges and constraints of dosimetry for MRT a

The implementation of dosimetry-based treatment planning and the verification of the absorbed dose delivered after treatment are hindered by a number of challenges that have to be addressed.

The role of dosimetry in MRT Chapter 1

The first one is the accurate activity quantification over time in target regions, commonly considered as the main source of error in MRT [START_REF] Flux | The Impact of PET and SPECT on Dosimetry for Targeted Radionuclide Therapy[END_REF][START_REF] Dewaraja | MIRD Pamphlet No. 23: Quantitative SPECT for Patient-Specific 3-Dimensional Dosimetry in Internal Radionuclide Therapy[END_REF] (Sec. 1.3.1.4). As thoroughly described in Section 1.3.1.1, the current established method for quantitative imaging is to measure biokinetics on images acquired at a number of points in time after administration of the radiopharmaceutical, using both planar imaging and SPECT technique [START_REF] Jentzen | Pre-therapeutic 124i PET(/CT) dosimetry confirms low average absorbed doses per administered 131i activity to the salivary glands in radioiodine therapy of differentiated thyroid cancer[END_REF][START_REF] Snyder | Absorbed Dose per Unit Cumulated Activity for Selected Radionuclides and Organs[END_REF][START_REF] Dewaraja | Accurate Dosimetry in 131i Radionuclide Therapy Using Patient-Specific, 3-Dimensional Methods for SPECT Reconstruction and Absorbed Dose Calculation[END_REF][START_REF] Jentzen | Assessment of Lesion Response in the Initial Radioiodine Treatment of Di↵erentiated Thyroid Cancer Using 124i PET Imaging[END_REF].

Although SPECT quantification is considered to be a more robust quantitative modality than planar imaging (which is still the most used [START_REF] Li | Quantitative Imaging for Targeted Radionuclide Therapy Dosimetry -Technical Review[END_REF]) many e↵ects complicate accurate SPECT quantification. The current limitations are primarily related to the fact that more e↵ort is needed to obtain calibration data (conversion factors from counts to absolute activity) compared to planar imaging [START_REF] Pereira | Image Quantification for Radiation Dose Calculations -Limitations and Uncertainties[END_REF], and that most clinical cameras are dedicated to diagnostic imaging with 99m Tc (principal photon emission at 140 keV) and thus, not optimized for detecting gamma emitters with high energy and high administrated activity used in therapy. [START_REF] Ljungberg | 3-D Image-Based Dosimetry in Radionuclide Therapy[END_REF][START_REF] Eberlein | Individualized Dosimetry for Theranostics: Necessary, Nice to Have, or Counterproductive?[END_REF][START_REF] Snyder | Absorbed Dose per Unit Cumulated Activity for Selected Radionuclides and Organs[END_REF]. The use of high therapeutic activities may bias quantification due to counts loss, in the absence of adequate methods to correct pile-up and dead-time [START_REF] Ljungberg | 3-D Image-Based Dosimetry in Radionuclide Therapy[END_REF]. For example, the counting rate capability of currently available gamma-cameras (around 400 kCps) is not suited to early imaging, shortly after administration of several gigabecquerels of activity, such as in the treatment of thyroid cancers using 131 I . Dead-time e↵ects are emphasised by the use of radionuclides emitting high energy gamma rays, such as 177 Lu, 131 I and 90 Y, due to the significant number of photons scattered in the patient and in the camera as well as the penetration e↵ects related to a non-optimized collimator design, that could also contribute to the poor image contrast achieved with conventional gamma cameras [START_REF] Dewaraja | MIRD Pamphlet No. 24: Guidelines for Quantitative 131i SPECT in Dosimetry Applications[END_REF]. With the introduction of hybrid SPECT/CT systems (early 2000s), corrections for these e↵ects can be incorporated into iterative reconstruction methods, combined with CT-based attenuation, and are now more or less standard method even if they require substantial computational time as quantification is performed for dozens or hundreds of reconstructed image slices. However, di↵erent studies showed a large error associated with quantifying 131 I activity in small objects (errors on the order of 30% have been suggested [START_REF] Dewaraja | MIRD Pamphlet No. 23: Quantitative SPECT for Patient-Specific 3-Dimensional Dosimetry in Internal Radionuclide Therapy[END_REF]), primarily due to the partial-volume e↵ect, defined as the spread or blurring of regional uptake to surrounding areas due to finite spatial resolution and scattering inside the collimator of the gamma-camera. The typical spatial resolution of SPECT systems ranges between 10 to 20 mm for high-energy gamma rays at 10 cm (364 keV), which introduce large error in evaluations of the absorbed dose distribution in small targets, such as nodes or tumor remnants, with dimensions less than 2-3 times the system spatial resolution, as well as for heterogeneous activity distributions [START_REF] Dewaraja | MIRD Pamphlet No. 23: Quantitative SPECT for Patient-Specific 3-Dimensional Dosimetry in Internal Radionuclide Therapy[END_REF]. This is an essential limitation, because, as we have already discussed in previous chapters, molecular radiotherapy is a localized treatment modality and, as such, the accuracy with which dosimetry is performed is strongly a↵ected by the intrinsic heterogeneity of radiopharmaceutical uptake, both at a macroscopic and microscopic scale [START_REF] Lassmann | The Relevance of Dosimetry in Precision Medicine[END_REF]. Thus, an accurate spatial localization of radiopharmaceutical is absolutely required for accurate dosimetry [START_REF] Lassmann | The Relevance of Dosimetry in Precision Medicine[END_REF][START_REF] Dekempeneer | Targeted alpha therapy using short-lived alpha-particles and the promise of nanobodies as targeting vehicle[END_REF][START_REF] Jentzen | Assessment of Lesion Response in the Initial Radioiodine Treatment of Di↵erentiated Thyroid Cancer Using 124i PET Imaging[END_REF][START_REF] Jeraj | Molecular Imaging to Plan Radiotherapy and Evaluate Its E cacy[END_REF].

There are also logistical and practical challenges involved in dosimetry, since the Chapter 1

The role of dosimetry in MRT accurate estimation of cumulated activity requires appropriate temporal sampling of the images, which must therefore be acquired several times after the injection. This automatically implies more steps than in a single diagnostic nuclear medicine examination [START_REF] Flux | From fixed activities to personalized treatments in radionuclide therapy: lost in translation?[END_REF]. Nowadays, ordinary data collection is therefore generally limited to a few time points, due to the reduced availability of the imaging detectors in a daily routine. Moreover, patients access to imaging facilities could be limited due to radioprotection concerns and to their general physical conditions. All this leads to a lack of accuracy in the estimation of time-activity curve adding large uncertainties on the calculation of the absorbed dose. Additionally, as in any kind of radiotherapy, dosimetry requires the competence of both physicians, medical physicists and technologists [START_REF] Sjögreen Gleisner | Variations in the practice of molecular radiotherapy and implementation of dosimetry: results from a European survey[END_REF] and performing quantitative imaging for dosimetry analyses may represent a significant time and financial burden for a medical facility [START_REF] Pereira | Image Quantification for Radiation Dose Calculations -Limitations and Uncertainties[END_REF]. Moreover, physicians generally have low confidence in the use of radiation dose analyses [START_REF] Stabin | State of the Art in Nuclear Medicine Dose Assessment[END_REF] and the need of an high accuracy in the dosimetric results in clinical practice when a general standardization is still not present, could require new resourcing and training (Sec. 1.3.1.4). Despite the e↵orts to provide the users with a comprehensive overview on how to perform quantitative imaging for dosimetry, the questions of how to best estimate doses (by the easiest and most accurate way) and what is the associated uncertainties still remain open.

Due to all these limitations, quantitative imaging is often not performed and, therefore, the calculation of absorbed therapeutic dose is rarely available and, if so, is generally based on non-standardized and thus, non trustworthy dosimetric methods [START_REF] Ljungberg | 3-D Image-Based Dosimetry in Radionuclide Therapy[END_REF]. Therefore, a sort of conservatism is maintained with a "one-dose-fits-all" therapeutic approach, resulting in a higher or lower than optimized activity being administered to the majority of patients [START_REF] Sjögreen Gleisner | Variations in the practice of molecular radiotherapy and implementation of dosimetry: results from a European survey[END_REF]. However, the development of new treatments, and in particular those based on the use of alpha-emitting radionuclides, which require better control of the local dose deposit, should gradually change this status quo. On the other side of the board, radioiodine treatments of thyroid disease are already showing strong e ciency with low healthy tissue toxicity due to the selective tropism of 131 I for thyroid tissue. The importance of the dosimetric approach in the treatment of thyroid disease nonetheless remained central. Indeed, as stated by Lassmann et all., "because of the simplicity of the treatment method, the lack of complications, and widespread use, radioiodine treatment of thyroid should be able to provide abundant data for evaluating dose-response criteria that could lead to a confident patient-specific treatment approach" [START_REF] Lassmann | Dosimetry and thyroid cancer: the individual dosage of radioiodine[END_REF].

The controversial case of thyroid's therapies a

Iodine-131 has been extensively used since the 1940s to ablate thyroid remnant or inoperable advanced di↵erentiated thyroid cancer and for the diagnosis and treatment of patients with benign diseases of thyroid, i.e. hyperthyroidism such as toxic di↵use goitre (Graves' disease), solitary hyperfunctioning thyroid nodule, toxic multinodular

The role of dosimetry in MRT Chapter 1 goitre and nontoxic goitre [START_REF] Stokkel | EANM procedure guidelines for therapy of benign thyroid disease[END_REF].

According to the EANM Internal Dosimetry Task Force (IDTF) survey, thyroid therapies performed in Europe are mainly prescribed using di↵erent fixed-activity protocols, adjusted on the specific diagnosis or stage of the disease [START_REF] Stokkel | EANM procedure guidelines for therapy of benign thyroid disease[END_REF][START_REF] Bernard | Radioiodine therapy in benign thyroid disorders. Evaluation of French nuclear medicine practices[END_REF] (54% and 84% for benign disease and remnant ablation/cancer therapy, respectively [START_REF] Sjögreen Gleisner | Variations in the practice of molecular radiotherapy and implementation of dosimetry: results from a European survey[END_REF]). For 38% and <20% of responders (for benign disease and remnant ablation/cancer therapy, respectively), administered activities are individually calculated to give a prescribed absorbed dose to target tissue [START_REF] Sjögreen Gleisner | Variations in the practice of molecular radiotherapy and implementation of dosimetry: results from a European survey[END_REF].

For malign diseases, at present, there are no established values for absorbed doses to remnants and metastases that may be used as prescription values [START_REF]Internal Dosimetry Task Force Report on: Treatment Planning For Molecular Radiotherapy: Potential And Prospects[END_REF]. The activity administered after thyroidectomy is maximized to eliminate residuals, optimizing the chances of complete remission and minimize the risk of recurrence. Since residual lesions are often small and di cult to outline using ultrasounds, target volumes cannot be determined precisely, making it di cult to predict the rates of radioiodine fixation, which can range from 0.1% to 30%. This fixation rate will reflect, depending on the clinical situations, either normal thyroid parenchymal residues or thyroid embryonic residues, ectopic thyroid residues, tumor residues or iodine avid cervical ganglion metastases. In some cases of persistent disease after surgery or recurrence, tumor targets may be noniodine avid. Several studies investigated the correlations between the absorbed doses delivered to remnants and metastases and responses, resulting in a wide range of absorbed doses to achieve a successful ablation. Sometime the outcome of the treatment can be successful in patients who receive a low remnant-absorbed dose of only a few Gy. In other cases, the absorbed doses to successfully treat remnants were higher than 49 Gy, 90 Gy and 300 Gy while for metastases, absorbed doses of over 40 Gy and 80 Gy have been reported [START_REF] Wierts | Dose-Response Relationship in Di↵erentiated Thyroid Cancer Patients Undergoing Radioiodine Treatment Assessed by Means of 124i PET/CT[END_REF][START_REF] Flux | A dose-e↵ect correlation for radioiodine ablation in di↵erentiated thyroid cancer[END_REF].

Consequently, the treatments are performed in more than 80% of cases by administration of a fixed activity of 131 I ranging between 1.11 GBq and 3.7 GBq, only depending on aggressiveness and prognostic factors of the disease [START_REF] Sjögreen Gleisner | Variations in the practice of molecular radiotherapy and implementation of dosimetry: results from a European survey[END_REF][START_REF] Luster | Guidelines for radioiodine therapy of di↵erentiated thyroid cancer[END_REF][START_REF] Vija Racaru | Clinical outcomes 1 year after empiric 131i therapy for hyperthyroid disorders: real life experience and predictive factors of functional response[END_REF]. This activity can reach 7.4 GBq in case of metastatic disease.

At present, one of the protocol used for treating advanced stages of thyroid cancer based on the individual calculation of the activity to be administered is to deliver a maximum absorbed dose of 2 Gy to the blood compartment. Indeed, when high activities are given, red marrow absorbed dose may be of concern; 2 Gy is a widely accepted limit for bone marrow toxicity [START_REF] Lassmann | EANM Dosimetry Committee series on standard operational procedures for pre-therapeutic dosimetry I: blood and bone marrow dosimetry in di↵erentiated thyroid cancer therapy[END_REF] and with the evidence of a similarity between radioiodine concentration in blood and in red marrow [START_REF] Lassmann | Blood and Bone Marrow Dosimetry in Radioiodine Therapy of Di↵erentiated Thyroid Cancer After Stimulation with rhTSH[END_REF], only blood need to be monitored [START_REF] Lassmann | EANM Dosimetry Committee series on standard operational procedures for pre-therapeutic dosimetry I: blood and bone marrow dosimetry in di↵erentiated thyroid cancer therapy[END_REF]. Furthermore, high activity could cause side e↵ects in salivary glands [START_REF] Jeong | Salivary Gland Function 5 Years After Radioactive Iodine Ablation in Patients with Di↵erentiated Thyroid Cancer: Direct Comparison of Pre-and Postablation Scintigraphies and Their Relation to Xerostomia Symptoms[END_REF] but at present there is no absorbed-dose constraint reported for them, even if toxicity has been reported for lower absorbed doses than the constraints used in external beam radiotherapy [START_REF] Liu | Iodine kinetics and dosimetry in the salivary glands during repeated courses of radioiodine therapy for thyroid cancer[END_REF]. For blood absorbed dose estimate procedure low activities of 131 I -NaI have to be administered pre-therapeutically; with this method the radioactivity is checked through blood samples and whole-body imaging with a dual-headed gamma camera [START_REF] Lassmann | EANM Dosimetry Committee series on standard operational procedures for pre-therapeutic dosimetry I: blood and bone marrow dosimetry in di↵erentiated thyroid cancer therapy[END_REF]. This method allows only the estimation of the radiation dose that will be delivered to the haematopoietic system lowering its risk of toxicity, but no informations Chapter 1

The role of dosimetry in MRT about the absorbed dose to the tumor are known, so no information about the e↵ectiveness of the treatment are known. Higher activities might be administered without achieving a real higher therapeutic e↵ect.

For about half of the treatments of benign diseases, administered activities are individually calculated. Usually, methods of dosimetric estimation are based on the calculation of the activity to be injected by the use of Marinelli 's formula [START_REF] Bernard | Radioiodine therapy in benign thyroid disorders. Evaluation of French nuclear medicine practices[END_REF]. According to this formula, the two main factors influencing the prescribed activity are the thyroid uptake and the thyroid volume. The uptake is usually obtained by measuring the fixation rate of a diagnostic activity of 131 I or 123 I with a counting probe. Volumes are mainly measured with an ultrasound. Only in few cases a scintigraphy is used for thyroid uptake and/or gland size using 131 I, 123 I [195]. Because of 123 I high cost and thanks to the similarity of volume and charge between iodide and pertechnetate ions, 99m Tc is also use for thyroid uptake and scintigraphy [START_REF] Stokkel | EANM procedure guidelines for therapy of benign thyroid disease[END_REF]. However, part of the responders who a rm to individually calculate the activity to be administred, only adapt it according to the thyroid volume estimated by palpation or ultrasound [START_REF] Sjögreen Gleisner | Variations in the practice of molecular radiotherapy and implementation of dosimetry: results from a European survey[END_REF]. The therapeutic activity are modulated according to these data and the treatments' objective. This activity may be ablative for the destruction of thyroid parenchyma or hyperfixing thyroid nodules with a dose projection deposited in the thyroid ranging from 200 to 400 Gy or may aim to restore normal thyroid function (euthyroidism), with a projected dose less than 200 Gy. These values result in prescribed activities that range from 200 to 800 MBq [START_REF] Stokkel | EANM procedure guidelines for therapy of benign thyroid disease[END_REF]. When fixed activities are used, the mean values are respectively 370 MBq (185-740 MBq), 555 MBq (185-740 MBq) and 555 MBq (296-925 MBq) for Basedow's disease, toxic solitary nodule and toxic multinodular goiter.

Although there have been few changes over time, there is no internationally agreement on the methods to treat thyroid diseases. The use of a dosimetric approaches or fixed activities for thyroid treatment is still controversial. Indeed, due to the high selective uptake of 131 I in the thyroid or in iodine-avid metastasis, selective irradiation can be obtained and, thus, localized treatments are achievable with a typically low absorbed dose in surrounding tissues and organs in the case of benign diseases [START_REF] Stabin | State of the Art in Nuclear Medicine Dose Assessment[END_REF][START_REF] Smith | Clinical trial of di↵erent doses of 131-I in treatment of thyrotoxicosis[END_REF] and with good outcomes for the malign ones [START_REF] Luster | Guidelines for radioiodine therapy of di↵erentiated thyroid cancer[END_REF]. The administration of fixed or empirically calculated activities is considered simple, i.e. it consumes little time and resources [START_REF] Vija Racaru | Clinical outcomes 1 year after empiric 131i therapy for hyperthyroid disorders: real life experience and predictive factors of functional response[END_REF], although it is well known that the biological half-life of iodine varies considerably [START_REF] Kramer | Biological Half-life of Iodine in Adults with Intact Thyroid Function and in Athyreotic Persons[END_REF]. In this context, there is still great interest in defending the dosimetric approach in order to optimize the administered activity according to the desired clinical outcome while avoiding undesired side e↵ects and long-term consequences [START_REF] Stokkel | EANM procedure guidelines for therapy of benign thyroid disease[END_REF][START_REF] Lassmann | EANM Dosimetry Committee series on standard operational procedures for pre-therapeutic dosimetry I: blood and bone marrow dosimetry in di↵erentiated thyroid cancer therapy[END_REF][START_REF] Smith | Clinical trial of di↵erent doses of 131-I in treatment of thyrotoxicosis[END_REF] In the case of malign diseases, a prior accurate individualized dosimetry, based on a diagnostic activity and quantitative imaging, could provide information on the iodine avid or non-iodine avid nature of the targets to be treated and thus, help to optimize the therapeutic activity to be administered. While focusing the main objective of destroying normal and tumor residues after thyroidectomy in the case of first treatment or residual disease and lymphonode metastases in the case of recurrence, the activity

1.4. ABSORBED DOSE CONTROL IN MRT Chapter 1
to be administered could also be adjusted to limit the side e↵ects of irradiation on high-risk organs (salivary and lacrimal glands) according to tolerance doses.

For benign diseases, since the precision of the planning protocols is currently insu cient to ensure a fine tuning of the administered activity according to the desired clinical outcomes, it is frequent to observe maximalist strategies of these activities even if there is a well-documented strong correlation between the amount of 131 I administered and the incidence of hypothyroidism [START_REF] Smith | Clinical trial of di↵erent doses of 131-I in treatment of thyrotoxicosis[END_REF], a condition in which the thyroid gland is not able to produce enough thyroid hormone. In order to achieve the objective of no recurrence of hyperthyroidism, the hypothyroidism has been accepted as an acceptable consequence of iodine treatment [START_REF] Flux | The Impact of PET and SPECT on Dosimetry for Targeted Radionuclide Therapy[END_REF][START_REF] Salvatori | Radioiodine therapy dosimetry in benign thyroid disease and di↵erentiated thyroid carcinoma[END_REF]. In a recent publication, Hyer et al. show that 60 Gy deposited in the thyroid are enough for 30% of patients to regain normal thyroid function without recurrence [START_REF] Hyer | Dosimetry-based treatment for Graves' disease[END_REF]. Thus, an accurate dosimetry based on quantitative imaging, which correlates the absorbed dose after treatment with the observed clinical e↵ects, is essential to maximize the number of patients with euthyroidism while minimizing the risk of hypothyroid, thereby reducing the need for life-long medication. For example, if a nodule fixes all activity, estimating an optimized dose at the nodule should avoid irradiating the normal thyroid parenchyma and increase the chances of achieving normal thyroid function.

In this context and whatever the thyroid disease, the major challenge for individualized treatment planning and post-treatment verification of the absorbed doses delivered is the development of a precise, simple and cost-e↵ective quantitative method to measure thyroid 131 I biokinetics in patients [START_REF] Vija Racaru | Clinical outcomes 1 year after empiric 131i therapy for hyperthyroid disorders: real life experience and predictive factors of functional response[END_REF].

Towards a better control of the absorbed dose in MRT a

The objective of our project is to propose new instrumental and methodological approaches aiming to strengthen the control of the dose released during MRT by reducing the uncertainties associated to absorbed dose calculation. This is based primarily on the development of a high-resolution mobile gamma camera specifically designed to improve the individual quantitative assessment of the heterogeneous distribution and biokinetics of 131 I before and after treatment administration. The device is optimized, in terms of spatial and energy response, sensitivity and counting rate capability, to meet the specific needs of imaging with high-energy gamma rays and high rate of photon fluence, such as in thyroid therapy using 131 I . The compactness and mobility of the camera should o↵er the possibility to improve image resolution by reducing the distance between the detector and the targeted structures (thyroid, tumor remnants, metastatic nodes or salivary glands) and to optimize their detectability by adjusting the angular view. Its availability and the possibility of using it at the patient's bedside (e.g. in the isolation room) should also be compatible with an accurate temporal sampling of the biokinetics of the tracer before (patient-specific treatment planning) and after the therapy administration, which is a key parameter for an accurate absorbed doses quantification.

Chapter 2

A mobile gamma camera for absorbed dose control during molecular radiotherapy a

In this second chapter, we will first discuss the challenges and constraints of dose quantification that led to the instrumental approach chosen for the development of the miniaturized gamma camera dedicated to the control of the absorbed dose during radionuclide therapy. Then, after a brief presentation of the interaction of gamma particles with matter, we will describe the characteristics of the main components of the detector. We will start by presenting the major role of a collimator for planar gamma imaging and in particular the importance of its optimization when using high energy gamma rays. Because our first prototype is based on the absorption and conversion of the energy of incident gamma rays into light, the description of the basic principle of scintillation will be followed by a bibliographic review which led to the choice of the material to be implemented. An overview of the di↵erent photodetection systems available to readout the scintillation light is then presented, focusing on the operating principles and properties of SiPMs used in our camera. Finally, we will conclude by presenting the electronics and the reconstruction algorithms used in order to optimize the spatial response of the camera.

Challenges and constraints of dose quantification in MRT

a As previously mentioned in Chapter 1, the main source of error in dosimetry for molecular radiotherapy is related to the individual pharmacokinetics assessment of the radionuclide uptake [START_REF] Sjögreen Gleisner | Variations in the practice of molecular radiotherapy and implementation of dosimetry: results from a European survey[END_REF]. At present, the established methods include the measurement of the biokinetics with counting or images acquired at one or two points in time after administration of the radiopharmaceutical. The current limitations of planar scintillation camera or SPECT for dosimetry are mainly due to the fact that most cameras are Challenges and constraints of dose quantification in MRT Chapter 2 not optimized, in terms of performances and ergonomics, for detecting high activity of high-energy gamma emitters during treatment [START_REF] Ljungberg | 3-D Image-Based Dosimetry in Radionuclide Therapy[END_REF]. In that context, our main objective is to develop a high-resolution mobile gamma camera specifically designed to improve the quantitative assessment of the heterogeneous distribution and biokinetics of 131 I in thyroid after treatment administration. To our knowledge, no comparable systems have been reported in literature.

The compactness and mobility of the camera should o↵er the possibility to improve image resolution by reducing the distance between the detector and the targeted structures (thyroid, tumor remnants, metastatic nodes or salivary glands) and to optimize their detectability by adjusting the angular view. The availability of the camera and the ability for using it at the patient's bedside (e.g. in the isolation room) should be also compatible with an accurate temporal sampling of the biokinetics of the tracer after the therapy administration, which is a key parameter for the quantification of the absorbed doses [START_REF] Strigari | The evidence base for the use of internal dosimetry in the clinical practice of molecular radiotherapy[END_REF]. This need is reinforced by the fact that in many nuclear medicine departments focused on oncology, the number of multi-purpose wide-field cameras is expected to decrease in favor of PET systems.

Unlike conventional large field-of-view gamma cameras, the device should also be optimized, in terms of spatial and energy responses and counting capability, to meet the specific needs of imaging with high energy gamma rays and high photon fluence rates, such as in thyroid therapy using 131 I . Due to the large amount of radioactivity administrated during the treatment of benign or malignant thyroid disease (Sec 1.3.3.3), the accuracy of the activity quantification is less dependent on the imaging system sensitivity because the counting statistics accumulated in the image over a reasonable examination time (around 10 minutes) will be su ciently important. Therefore, priority was given to the spatial and energy resolution of the camera. The performance objective is to achieve a system spatial resolution, i.e. the capacity of the camera to discriminate two close objects, ranging between 3 and 5 mm FWHM (at a 5 cm distance), much better than the one of conventional gamma cameras equipped with high-energy collimators (around 1.5 cm at 364 keV). This spatial resolution is suited to the maximum path in tissue of electrons emitted by 131 I (mean and maximal ranges of the 131 I beta emission in soft tissue are 0.4 and 2 mm [START_REF] Autret | Monte Carlo modeling of gamma cameras for I-131 imaging in targeted radiotherapy[END_REF]) and should allow to quantify the heterogenous distribution of the radiopharmaceutical in the targeted tissues and therefore, to improve the accuracy of dosimetry at the macroscopic scale by reducing the partial-volume e↵ects. This e↵ect can introduce significant errors in assessing the distribution of the absorbed dose in small targets, such as nodes or tumor remnants (sizes less than 2 to 3 times the spatial resolution of the system), or for heterogeneous uptake of the radiotracer in organs. The energy resolution defines the capability of the detector to discriminate between emitted radiations with di↵erent energies. A high energy resolution allows to discriminate primary unscattered photons, which can be used to form an accurate image of the activity in the object under study, from scattered photons (generated by Compton e↵ect) mainly produced in the surrounding tissues or in the collimator. This signal forms a background noise which can strongly reduce the Chapter 2

Challenges and constraints of dose quantification in MRT image contrast. Our objective is to reach an energy resolution less than 8% in order to minimize the contamination from high-energy scattered events. The counting rate capability of the camera is also an important issue to reduce the e↵ect of dead time on activity estimates. The final prototype of the mobile camera should provide a maximum count rate around 400 kCps in order to open the door to early imaging, even shortly after administration of several GBq of activity [START_REF] Dewaraja | MIRD Pamphlet No. 24: Guidelines for Quantitative 131i SPECT in Dosimetry Applications[END_REF]. Dead-time and partial-volume e↵ects are particularly important in 131 I imaging because of the significant number of high-energy scattered events (coming from 637 and 723 keV photons) that contributes to the camera dead time and poor image contrast observed with conventional gamma cameras, respectively [START_REF] Dewaraja | MIRD Pamphlet No. 24: Guidelines for Quantitative 131i SPECT in Dosimetry Applications[END_REF].

The final mobile gamma camera should have a field of view of 10⇥10 cm 2 suited to the size of the thyroid (even in the case of enlarged thyroids, such as in Graves' disease). However, the proof of concept presented in this work will be realized on a first feasibility prototype with a 5⇥5 cm 2 field of view. A schema of the camera is shown in Fig. 2.2 and its design will be detailed in Sec. 2.3. To achieve the desired objectives in terms of performance and ergonomics, we chose to build up the camera around a parallel-hole high-energy collimator, coupled to a continuous inorganic scintillator, readout by a recent and well-suited technology based on arrays of Silicon Photomultiplier (SiPMs) detectors. The choice of a scintillator-photodetector assembly instead of a direct detection with high-Z semiconductors (CdTe, CdZnTe) is related to the fact that the sensitivity of these sensors is not adapted to the detection of high energy gamma rays [START_REF] Hutton | Advances in clinical molecular imaging instrumentation[END_REF]. Moreover, scintillation cameras o↵er greater performances versatility by adjusting the scintillator and collimator characteristics separatly, while the features of pixelized semiconductor detectors must be fitted to the collimator used (or vice versa) in order to optimize the camera's sensitivity [START_REF] Scheiber | CdTe and CdZnTe detectors in nuclear medicine[END_REF]. Among the photodetectors currently available, the silicon photomultipliers are the most relevant for developing compact imaging systems with high spatial and spectral performance (Sec. 2.6). We also chose to use a continuous scintillator instead of a pixelated scintillator because a monolithic crystal does not impose a strong compromise between spatial resolution, energy resolution and sensitivity. On the other hand, the use of arrays with long and thin crystals, necessary to obtain a high spatial resolution while maintaining good sensitivity, will automatically degrade the energy resolution due to the multiple light reflections within each crystal. However, continuous scintillators require more sophisticated reconstruction methods to accurately calculate the interaction position of gamma photons from the light distribution measurement (Sec. 2.8).

Before presenting the mobile gamma camera developed during this project and its miniaturized components, a description of the relevant information about gamma rays interactions with matter will be done. The fundamentals of these interaction mechanisms are briefly introduced here and will serve as a basis for understanding the behaviors, features and optimization studies of collimator and crystals, which will be presented respectively in Section 2.4 and 2.5.

INTERACTION OF GAMMA RAYS IN MATTER

Chapter 2

Interaction of gamma rays in matter a

Among the various ways known for gamma rays interaction in matter, only three mechanisms have a real significance in radiation measurements: photoelectric absorption, Compton scattering and pair production, which lead to a partial or complete transfer or conversion of the gamma-ray energy to the absorber. A fourth e↵ect, the coherent scattering, is often neglected in gamma-ray interactions discussions because no energy transfer takes place. However, the direction of the photon is changed and thus it will be considered in future considerations because it could impact image contrast. Photoelectric absorption and coherent scattering predominate for low-energy gamma rays (up to several hundred keV), pair production prevails for high-energy gamma rays (above 5-10 MeV), while Compton scattering is the most probable process over the range of energies between them. For pair production to occur, the incoming energy of the gamma must be above a threshold of at least twice the total rest mass energy of an electron (1.022 MeV). Considering the energy range of interest in this work, which extends to the most energetic emission of 131 I (722 keV), no pair-production is possible and therefore attention is focused only on the other three processes.

The photoelectric e↵ect is the process by which a photon undergoes an interaction with an atom in which it is completely absorbed. The interaction is with the atom as a whole (it cannot occur with free electrons) and it takes place only if the incident photon has a energy slightly higher than the binding energy of an orbital electron (usually with the most bound shells, the K-shells). In this case, the electron receives su cient energy to be ejected from its bound shell and is emitted with a kinetic energy E k given by E k =h⌫ -E b , where h⌫ is the energy of the incident photon and E b is the binding energy of the electron. The ejection of this electron leaves the atom in a ionized state, with a vacancy in one of its bound shells, which is quickly filled through the rearrangement of electrons from other shells or capture of a free electron with the consequent emission of characteristics X-rays. Although these X-rays are usually reabsorbed close to the original site through photoelectric absorption involving less bound shells, they can escape from the volume where they are produced. It is what usually happens in the interaction of gamma rays with the high-Z shielding surrounding radiation detectors, generally made of lead (Pb) or tungsten (W). Their characteristics X-rays (Pb K-shells: 74.9 keV, 72.8 keV and 84.9 keV and tungsten K-shells: 59.3 keV, 58.0 keV and 67.2 keV [206]) usually arrive on the detector and are thus visible on the source energy spectrum. When the photoelectric interaction takes place inside the detector and nothing escapes, then the sum of the kinetic energies of the electrons that are created and absorbed is equal the incident gamma-ray energy. This is the reason why photoelectric absorption is the preferred mode of interaction in radiation applications.

The cross-section of photoelectric absorption per atom over all ranges of E and Z can be roughly approximate by: phe ⇠ Z n E 3.5 (2.1)

Chapter 2

Interaction of gamma rays in matter with n ranging from 4 to 5 depending on the photon energy. Therefore, the process is enhanced for absorber materials with high atomic number Z. This dependence of the photoelectric absorption probability on the atomic number of the absorber is a primary reason for the use of high-Z materials (such as lead and tungsten) in gamma-ray shields and collimators and the reason why scintillator with higher Z are often preferable as absorbers in gamma-ray detectors.

The interaction process of Compton scattering (incoherent scattering) is the predominant interaction mechanism for gamma-ray energies typical of radioisotope sources. It occurs between an incident gamma-ray photon and a weakly bound electron of the absorber, when the incident energy of the photon h⌫ is much higher than the binding energy of the electron. The photon transfers only a part of its energy to the electron, which acquires a kinetic energy T e , and is then scattered as a secondary photon with an energy h⌫ 0 and an angle ✓ in respect to its original direction. To satisfy the conservation of momentum and energy, the energy h⌫ 0 of the secondary photon and the kinetic energy of the electron T e are correlated with the photon incident energy h⌫ by means of the following equations:

h⌫ 0 = h⌫ 1 + h⌫ mec 2 (1 cos✓) T e = h⌫ h⌫ 0 (2.2)
where m e c 2 is the ress-mass energy of the electron (0.511 MeV). The cross section of Compton scattering per atom of the absorber is described by the Klein-Nishina formula, which depends linearly on the atomic number Z of the attenuator (i.e. it depends on the number of electrons available as scattering targets) and thus:

Compton ⇠ Z (2.3)
Moreover, Klein-Nishina formula predicts that all angles of scattering could occur. Thus, a continuum of energies can be transferred to the electron, ranging from zero up to the maximum predicted by the Equation 2.2 corresponding to the photon backscattering (✓=180 ), the so-called Compton edge. For incident gamma rays of 364 keV ( 131 I principal peak), the photon back-scattered has an energy of 150 keV and the corresponding Compton edge is at 214 keV.

In addition to Compton scattering, another type of scattering can occur, in which the gamma-ray photon interacts coherently with all the electrons of an absorber atom. This scattering, also known as Rayleigh scattering, neither excites nor ionizes the atom and the gamma-ray photon retains its original energy after the scattering interaction. Thus, the direction of the photon is changed but no energy is transferred. The probability of coherent scattering is higher for low energy photons and the average deflection angle decreases with increasing energy; moreover, it is most prominent in high-Z absorbers.

Interaction of gamma rays in matter

Chapter 2

According to the previous descriptions, a gamma-ray photon from a beam incident on an absorber could be removed from the beam either by absorption or by scattering. The fraction of photons surviving a thickness t is given by :

I I 0 = e µ•t (2.4)
where I 0 is the number of incident photons and µ is the material linear attenuation coe cient, which represents the total probability per unit path length that a photon is removed from the beam. If we define N v , the number of target entities in a unit volume present in an absorber of density ⇢, and tot , the total cross section per unit target, the linear attenuation coe cient µ (cm 1 ) is defined as:

µ = tot • N v = tot N A M ⇢ (2.5)
where N A and M represent the Avogadro constant (mol 1 ) and the molar mass (g.mol 1 ), respectively. Since each of the interaction processes described above removes a gammaray photon from the beam with a specific probability of occurrence per unit path length, the total probability µ tot is simply given by the sum of the probabilities of all interaction mechanisms that take place at the energy of interest. However, the use of the linear attenuation coe cient µ is limited by the fact that it varies with the density of the absorber. Therefore, the linear mass attenuation coe cient µ mass , defined as µ/⇢ (cm 2 .g 1 ), is much more widely used because it gives information about a material with a specific chemical structure, independently of its raw density. [START_REF] Berger | NISTXCOM : Photon Cross Section Database[END_REF]. Figure 2.1 shows the theoretical fraction of photoelectric, Compton and Rayleigh scattering as a function of the Z of the absorber at 364 keV. The e↵ective Z of di↵erent scintillator compounds, which will be described in Section 2.5.3, is highlighted, as well as Z of Pb and W, which are the most commonly used materials in gamma-ray shields and collimators (see Sec.2.4).

Design of the high-resolution mobile gamma camera

a When designing a gamma-ray detector, there is a wide range of possibilities in terms of combining its di↵erent components, which should be matched in order to optimize its performances in relation to the targeted application. Our miniaturized gamma camera has been conceived to meet the specific requirements of activity quantification in order to improve personalized radioiodine treatment of thyroid diseases. As described in Section 2.1, this involves accurately measuring the biodistribution and biokinetics of radiotracers in tumor remnants or thyroid tissues, after administrating a large amount of activity to the patient. From an instrumental point of view, it is therefore necessary that the camera has high spatial and energy resolutions and high counting capability, in order to guarantee good image quality and accurate quantification. It must also be compact and mobile for practical daily clinical use. Looking for these needs, we chose to design the first prototype of the camera (5⇥5 cm 2 field of view) around a high-energy parallel-hole tungsten collimator coupled to a continuous inorganic scintillator. The scintillator is readout by an array of Silicon photomultiplier (SIPM) detectors combined with a dedicated miniaturized multi-channel electronics (Fig. 2.2). The parallel multi-hole collimator is the central element of a gamma camera because it largely determines its overall performances. The main role of the collimator is to absorb gamma rays emitted by the source that do not propagate in the desired direction in order to obtain an image corresponding to the planar projection of the activity present in the object under study, thus strongly influencing system sensitivity and spatial resolution. Its importance is even more pronounced for imaging with 131 I than for common nuclear medicine isotopes such as 99m Tc, because of the higher energy of the photopeak (364 keV) and the multiple gamma-ray emissions that both scatter and penetrate the collimator, significantly degrading spatial resolution and contrast and biasing the quantification, as will be described in Section 2.4. The photodetection system, based on a monolithic inorganic crystal that converts the energy from the incident gammaray into optical-wavelength photons, must be capable to reduce this scattering noise by accurately measuring the energy of each detected photon. The detector should also reconstruct the interaction position of the gamma photon that depends mainly, as also the energy response of the detector, on the amount and distribution of scintillation light on the photodetector. These optical parameters are influenced by the intrinsic nature of the scintillator material, its thickness and its optical coatings, as will be described in Section 2.5. The spatial and energy responses of the camera also strongly depend on the electro-optical features of the SiPM array (gain, photon detection e ciency, correlated noise, ...), which provides an electrical signal proportional to the number of light photons detected, as will be described in Section 2.6. Finally, this electric signal is exploited to compute the two-dimensional spatial position of detected events using reconstruction methods that will be described in details in Section 2.8.

Each of these detection components has an impact on the overall performance of the camera and thus a detailed understanding of each of them is essential. The basis and principles of high-energy parallel-hole collimators, the theoretical notion of scintillation and the role of the inorganic scintillator will be described, as well as the operating principle and main features of the SiPM photodetection system and the readout electronics. At the end, a focus on the image reconstruction methods used for gamma-ray imaging will be given.

The high-energy parallel-hole collimator a

The collimator limits the performance of a nuclear medicine imaging system more than any other single component and thus its optimization is of primary importance: poor design can result only in poor overall performance. The design and basic considerations on parallel-hole collimators will be discussed in this section, together with a brief description of other collimator geometries and existing manufacturing methods. Then, the analytical models commonly used to optimize the design of parallel-hole collimators, particularly useful for low energy (<150 keV) applications, will be presented. Their limitations when high-energy isotopes, such as 131 I , are used will also be discussed to justify the need to implement Monte Carlo simulations in order to accurately model the di↵erent interactions that can occur in the collimator.

Principle and basic designs a

In order to convert the three-dimensional distribution of the activity in an organ or tissue of interest into a realistic two-dimensional image, gamma-ray detectors use a collimator, that is a plate made of high density and high Z material containing a single hole or a closely packed structure of holes. Collimator features are directly dependent on the shape and orientation of the hole, the hole dimensions (length and diameter), the septal thickness and the material. Ideally, a collimator should work as a binary filter to select incident photons: the only ones traveling through a hole could reach the scintillator and be detected while those interacting with its septa should be fully absorbed. In reality, no thickness of septal material is su cient to stop all gamma rays and only a tiny fraction of the gamma rays isotropically emitted by the source could pass through the collimator inside a hole without hitting the septal material, which leads to the so-called geometric behavior of a collimator [START_REF] Moore | Collimator design for single photon emission tomography[END_REF]. In addition, a significant amount of gamma rays could penetrate through one or more septa walls without interacting or could be scattered and deflected to the detector by Compton or coherent interactions in a septa. The collimator penetration and scatter components are generally undesirable because their emission point cannot be correlated with their detection point. A large number of penetrating or scattering photons contribute to the creation of a substantial background noise, which highly degrades the image. Their impact strongly depends on the energy of the gamma-ray emissions used (Fig. 2.

3).

Therefore, a primary consideration in the design of collimators is to ensure that a large amount of gamma rays are absorbed by photoelectric e↵ect, always maintaining the amount of those that penetrate and scatter at negligible levels (see Sec. 2.4.3). To achieve this objective, the first obligation is to manufacture collimators with materials characterized by high density and linear attenuation coe cient, such as lead or tungsten (see Sec. 2.2). The linear attenuation in terms of photoelectric absorption, incoherent and coherent scatter fractions, as well as the corresponding interaction probability P for an arbitrary thickness of 0.5 mm calculated using equation 2.4 are summarized in Table 2.1 for tungsten and lead. All values are calculated by considering the nominal

The high-energy parallel-hole collimator Chapter 2 density of the two raw metals, 11.36 g/cm 3 and 19.3 g/cm 3 for lead and tungsten, respectively [210]. Commercially available collimators for clinical gamma cameras are mainly made of lead because of its lower cost and easier manufacturing (see Sec. 2.4.2) [211]. However, the higher interaction probability of tungsten at 364 keV, and more generally at high energy, makes it very interesting for imaging applications in molecular radiotherapy.

Beyond the amount of penetrated and scatter events, a collimator is characterized by its intrinsic spatial resolution and e ciency. These two quantities depend directly on the collimator geometry and are usually described by simple geometric equations that will be discuss in Section 2.4.3.1. The collimator e ciency is defined as the fraction of gamma-rays emitted isotropically by a point source that actually pass through the collimator. The above definition is unit-less and, in order to be more clinically relevant, the e ciency is often converted to sensitivity (cpm/µCi or cps/MBq). The intrinsic spatial resolution is defined as the full width at half-maximum (FWHM) of the radiation profile from a point or line source of radiation projected by the collimator into the detector and refers to the sharpness or smallest identifiable detail of the gamma-ray image projected on the detector. For a conventional gamma camera, the collimator Chapter 2

The high-energy parallel-hole collimator spatial resolution is combined with the intrinsic resolution of the detector to produce an overall system resolution that is always highly dominated by the collimator contribution.

Since each application has its own specific requirements in terms of e ciency and spatial resolution, a trade-o↵ between all these parameters should be considered individually, as well as the choice of collimator geometry that best suits the specific needs of imaging.

Parallel-hole collimators

a Among all the available basic collimator geometries, parallel-hole collimators are usually employed in 2D and 3D imaging and are considered as the standard collimator in clinical practice [211]. They consist of a structure of parallel holes disposed in a specific array shape (Fig. 2.4). Hexagonal holes arranged in a hexagonal array are generally used, but other hole shapes (i.e. square, circular or triangular) can also be made, but are less common.

The first characteristic of this type of collimator is the one-to-one size relationship between the object emitting gamma rays and the image produced (no magnification). Each point on the projection image only gives information on gamma-rays emitted by a labeled molecule placed somewhere along the line passing through a hole in the collimator so that each point of the source has a unique identifiable point on the image. Lead collimators in commercially available gamma cameras are not categorized according to their geometrical features, but to the maximum gamma ray energy for which their septal thickness is considered su cient to absorb a majority of the incident gamma rays (see Table 2.1). Low-energy collimators are designed and optimized for being used with 99m Tc (140 keV). Medium-energy collimators are usually designed for 67 Ga and thus for energies up to 300 keV [START_REF] Macey | Comparison of Low-and Medium-Energy Co mators for SPECT Imaging with Iodine-123-Labeled Antibodies[END_REF]. Clinical high-energy collimators are instead optimized for 131 I sources. Although designed for a specific range of energies, all collimators may su↵er penetration and scatter when imaging isotopes with higher energy Lead Tungsten ). The corresponding probability of interaction is calculated for a 0.5 cm thick material. secondary emissions, such as 123 I (159 keV and 528 keV, 83.3% and 1.4 % respectively) and 131 I (364 keV, 637 keV and 723 keV, 82%, 7.2% and 1.8% respectively). However, the majority of low-and medium-energy applications use single gamma emitters.

While maintaining the constraint of low septal penetration, several types of low-energy (LE) collimators with di↵erent combinations of spatial resolution and e ciency are proposed to meet the specificities of di↵erent clinical applications. Some of them are optimized to enhance system spatial resolution (low-energy high resolution LEHR or ultra high resolution LEUHR collimators) with typical spatial resolutions ranging from 2 to 3.5 mm at a source distance of 10 cm and a corresponding sensitivity of 100 and 300 cpm/µCi, respectively. Others are optimized for sensitivity above 1000 cpm/µCi (low-energy high sensitivity LEHS), but with a loss of spatial resolution (around 11 mm at 10 cm). Collimators whose characteristics fall between these two extremes are referred to as general purpose or all purpose (low energy general purpose LEGP). Main performance and features of the di↵erent families of low-energy collimators are summarized in Table 2.2. Imaging with high energy gamma rays, such as those emitted by 131 I , is much more complex due to both the higher energy of the main photopeak (364 keV) and the multiple secondary emissions at even higher energy. The 637 and 723 keV photons have low intensity, but can contribute significantly to the image because they have a relatively low attenuation in collimators designed for an energy of 364 keV. Since high-energy (HE) collimators are mainly optimized for diagnostic applications and, therefore, focus on reducing scatter and penetration while maintaining acceptable sensitivity, the geometries chosen for clinical collimators have thick septa (greater than 2 mm) and large diameter holes, which leads to a poor spatial resolution (typical value of 13 to 15 mm at a source distance of 10 cm) (Tab. 2.2). The need to improve spatial resolution for activity quantification requires refining the geometry of the collimator with respect to standard clinical configurations, as will be discussed in Section 2.4.3.

The standard choice of lead as a material must also be questioned, and tungsten should be preferred because of its higher probability of interaction at 364 keV (Tab. o↵ers more flexibility in optimizing the collimator either towards increased sensitivity or spatial resolution. For example, the high linear attenuation coe cient of tungsten makes it possible to thin the septa and thus, improve geometric e ciency for a fixed spatial resolution and a fixed fraction of penetrating photons. More generally, the much higher density of tungsten compared to lead should significantly reduce scatter and septal penetration into the collimator resulting in reduced image artifacts, improved image contrast and thus, better dose quantification [START_REF] Ljungberg | 3-D Image-Based Dosimetry in Radionuclide Therapy[END_REF][START_REF] Dewaraja | Characterization of scatter and penetration using Monte Carlo simulation in 131i imaging[END_REF].

Other collimators geometries a

The trade-o↵ between sensitivity and spatial resolution imposed by the geometry of the parallel-hole collimator (see Sec. 2.4.3.1), as well as the need of imaging organs or lesions smaller or larger than the active area of standard gamma cameras, have drawn attention to the development of new collimators geometries, such as converging fan-beam, converging or diverging cone-beam and pinhole collimators. These di↵erent geometries have been tested for years because they provide di↵erent optimized performances for specific tasks, such as increased sensitivity or increased resolution thanks to the magnification or reduction of images [START_REF] Jaszczak | SPECT using a specially designed cone beam collimator[END_REF][START_REF] Gullberg | Review of convergent beam tomography in single photon emission computed tomography[END_REF][START_REF] Tsui | Evaluation of cardiac cone-beam single photon emission computed tomography using observer performance experiments and receiver operating characteristic analysis[END_REF][START_REF] Beekman | The pinhole: gateway to ultra-high-resolution three-dimensional radionuclide imaging[END_REF].

When the object of interest is much smaller than the detector area, an important performance gain is achieved using converging collimators, which magnify the projection of the object on the detector and, therefore, improve the e↵ective spatial resolution and sensitivity of the system. In fan-beam converging collimators, the holes converge toward a focal line parallel to the axis of rotation. The holes are tilted in the transverse plane and parallel in the axial direction. They have been designed and mainly used to image small organs, such as the brain and heart [START_REF] Jaszczak | Single Photon Emission Computed Tomography Using Multi-Slice Fan Beam Collimators[END_REF][START_REF] Pareto | Iterative reconstruction with correction of the spatially variant fan-beam collimator response in neurotransmission SPET imaging[END_REF][START_REF] Ichihara | Evaluation of SPET quantification of simultaneous emission and transmission imaging of the brain using a multidetector SPET system with the TEW scatter compensation method and fan-beam collimation[END_REF]. Collimators that converge both transaxially and axially, known as cone-beam converging collimators, are now employed, thereby exploiting almost the entire active surface of the detector [START_REF] Jaszczak | Cone beam collimation for single photon emission computed tomography: Analysis, simulation, and image reconstruction using filtered backprojection[END_REF][START_REF] Annis | Tomographic imaging with concentric conical collimator[END_REF]. Pinhole collimators are also commonly employed to obtain high-resolution images of small organs, such as the thyroid gland [START_REF] Beekman | The pinhole: gateway to ultra-high-resolution three-dimensional radionuclide imaging[END_REF][START_REF] Ghanem | Comparison of Pinhole and High-Resolution Parallel-Hole Imaging for Nodular Thyroid Disease[END_REF]. They consist of a single small hole aperture at the tip of a cone made of lead or tungsten. The inverted image of the source is projected through the aperture on the detector with a high magnification, which allows to reach sub-millimeter resolutions. On the other hand, the sensitivity achieved remains rather low (150 cpm/µCi for a SR of 6-7 mm at 10 cm), but if the detector is large enough, it can be improved by combining many pinholes into a multiple-pinhole collimator [211, [START_REF] Beekman | The pinhole: gateway to ultra-high-resolution three-dimensional radionuclide imaging[END_REF][START_REF] Weber | Pinhole SPECT: An approach to in vivo high resolution SPECT imaging in small laboratory animals[END_REF]. Finally, when the object is relatively large, it may be appropriate to use cone-beam diverging collimators in order to enlarge the field of view, but with a consequence loss of spatial resolution and sensitivity. The use of new highresolution detector technologies can help to compensate for this loss of resolution. A schematic view of the di↵erent collimator types and their impact on image are given in Figure 2.5.

In general, the choice of the type of collimator to be used depends mainly on the size ratio between the object under study and the detector field of view, the required spatial resolution and/or sensitivity [211]. Although divergent collimators could be an interesting option to widen the camera's field of view, while maintaining its compactness, they su↵er, like pinhole collimators, from prohibitive defects to quantify activity during radiotherapy. The main one is that sensitivity and spatial resolution are modified by the magnification or compression factor that varies with distance between the source and the collimator, but also depends on the position of the source relative to the main axis of the system. This means that target regions placed at di↵erent depths will be projected onto the same image with di↵erent scaling factor. These information related to the size and depth of the source is important in the quantification process. More generally, the strong dependance of the sensitivity of pinhole collimators on distance Chapter 2

The high-energy parallel-hole collimator (inverse of square distance) almost precludes any possibility of quantification. These collimators also su↵er from image distortions when the source is o↵set from the main axis of the detector. In comparison, and as already mentioned, parallel-hole collimators ensure a magnification of one regardless of the position of the source. Their sensitivity, which will be described in Section 2.4.3.1, is also theoretically independent of the distance from the source.

Manufacturing methods a

The design of the collimator also involves the practical issues related to its manufacturing. There are di↵erent production methods, all of which have specific advantages and disadvantages. The materials generally used are lead (Pb), tungsten (W), gold (Au), uranium (U), and platinum (Pt). Due to their cost, Pb and W are by far the most commonly used.

Parallel, fan-beam and cone-beam collimators are traditionally manufactured using Pb by stamping and stacking lead foils, or by casting molten lead. These are relatively easy and inexpensive techniques, but they have their limitations. As a general rule, the minimum diameter of the holes is 1.2 mm and the minimum septal thickness is 0.15 mm [211]. These specifications are su cient for traditional clinical collimators, but to build very high resolution collimators, smaller holes and septa are required. X-ray lithography combined with metal electroforming is an alternative solution that produces very high-precision collimators with a variety of metals (e.g., Cu, Ni, Pb, Ag, and Au). For example, 0.025 mm thick gold septa can be designed [START_REF] Makarova | Fabrication of antiscatter grids and collimators for X-ray and gamma-ray imaging by lithography and electroforming[END_REF]. However, all traditional manufacturing techniques cannot be use with tungsten. Indeed, this metal is very brittle, di cult to machine and its high melting point compared to lead (3422 C versus 328 C) prohibits casting techniques. Other techniques are therefore necessary for the manufacture of tungsten collimators. Among them, the stacking of photochemically etched tungsten foils to form the collimator pattern was tested [START_REF] Makarova | Fabrication of collimators for gamma-ray imaging[END_REF]. The limitations of this technique include the need for very precise layer-to-layer alignment in order to obtain straight and uniform septa. It is also possible to mill or drill tungsten alloys, containing nickel, iron, and copper, using a diamond drill, or to machine them using electric discharge machining (EDM) [START_REF] Zhong | Laser direct manufacturing of tungsten nickel collimation component[END_REF]. However, these techniques are prohibitively expensive and complex shapes cannot be easily produced [211].

In recent years, metal additive manufacturing, also referred to as metal 3D-printing, is gaining popularity, allowing to produce complex shapes from a 3D computer-aided design file [START_REF] Zhong | Laser direct manufacturing of tungsten nickel collimation component[END_REF][START_REF] Deprez | Rapid additive manufacturing of MR compatible multipinhole collimators with selective laser melting of tungsten powder[END_REF]. 3D printing has been available for nearly 30 years and its applications have grown rapidly. Printers are generally based on three main techniques, namely thermal extrusion, powder deposition and stereolithography. Although they are more commonly used to produce plastic and ceramic parts, additive manufacturing techniques are nowadays also used with metals. Selective laser melting (SLM) using metal powder deposition technology is increasingly used for the manufacture of metal objects, as it represents a more cost-e↵ective solution for customized manufacturing [START_REF] Uhlmann | Investigation on Additive Manufacturing of Tungsten Carbide-cobalt by Selective Laser Melting[END_REF]. A number of commercial companies currently o↵er tugsten SLM (M&I Materials UK, Smit Rontgen, NL, Layerwise, Belgium) [START_REF] Gear | Characterisation of the attenuation properties of 3d-printed tungsten for use in gamma camera collimation[END_REF]. The SLM technique uses a laser as the energy source and the powder as the starting raw material. The technique is based on a layer-by-layer production: a high-power laser scans small sections of a powder layer and selectively fuses the metal powders to build up the desired geometry (Fig. 2.6). After scanning a cross section, a new layer of powder is applied on top and the process is repeated until the object is finished [START_REF] Enneti | E↵ect of process parameters on the Selective Laser Melting (SLM) of tungsten[END_REF]. It has been shown that SLM techniques can produce tungsten parallel-hole collimators with holes as small as 525 µm and a septal thickness of 150 µm [START_REF] Deprez | Preclinical SPECT imaging based on compact collimators and high resolution scintillation detectors[END_REF], which would be impossible with the previously described techniques. However, an in-depth study of the accuracy of SLM technique for the manufacture of tungsten objects showed an average deviation of 20% from the nominal thicknesses requested for septa [START_REF] Gear | Characterisation of the attenuation properties of 3d-printed tungsten for use in gamma camera collimation[END_REF]. This lack of accuracy is due to both the finite size of the laser beam and the grains of tungsten powder [START_REF] Enneti | E↵ect of process parameters on the Selective Laser Melting (SLM) of tungsten[END_REF]. In addition, the densification of the powder depends strongly on the properties of the laser (laser power, hatch spacing, scanning speed) and the scanning strategy used (longitudinal, transverse or rotated scanning) [START_REF] Gear | Characterisation of the attenuation properties of 3d-printed tungsten for use in gamma camera collimation[END_REF][START_REF] Enneti | E↵ect of process parameters on the Selective Laser Melting (SLM) of tungsten[END_REF]. The actual densities reported for SLM vary according to the geometry and complexity of the finished part between 80 to 96 % of the theoretical density of the pure tungsten [START_REF] Gear | Characterisation of the attenuation properties of 3d-printed tungsten for use in gamma camera collimation[END_REF][START_REF] Deprez | Preclinical SPECT imaging based on compact collimators and high resolution scintillation detectors[END_REF]. Even if the minimum e↵ective density achieved with tungsten SLM remains much higher than the lead density, the influence of SLM on the material density, and therefore on its attenuation coe cient, could have a non-negligible consequence when using a high energy source such as 131 I , due to the greater amount of penetration and scatter that could occur.

Collimator geometry optimization a

Being the main factor limiting the performance of planar imaging systems, collimator optimization is an important and common step in the development of gamma-ray detectors. For low-energy gamma-ray imaging, this optimization is always carried out using the analytical expression of the intrinsic spatial resolution and e ciency, which will be briefly described in the following section. The limitation of this model lies on the fact that it completely underestimates both gamma penetration and scatter [START_REF] Moore | Collimator design for single photon emission tomography[END_REF].

Although the literature has provided a wide range of articles focused on the characterization of scatter and penetration in collimator when using 131 I [190, 203, 231], none of them allows this information to be integrated for fast and accurate design of high-energy collimators. The implementation of Monte Carlo simulations is therefore essential to refine the geometry of high-energy parallel-hole collimators, such as the one implemented in our mobile gamma camera. These simulations will be introduced in Section 2.4.3.2, while their detailed description, as well as the results of the optimization study, will be presented in Section 3.2.

Analytical models a

The spatial resolution and e ciency of a parallel, multihole collimator were firstly presented by Anger in 1964 for a point source in air based on simple geometrical considerations [START_REF] Anger | Scintillation Camera with Multichannel Collimators[END_REF]. The collimator e ciency E, defined as the fraction of gamma rays passing through the collimator per gamma-ray emitted by the source, is given by:

E = K 1 L 2 d 4 (d + t) 2 (2.6)
where L, d and t are the collimator length, hole diameter, and septal thickness (Fig.

2.7)

. K is a constant that depends on hole shape and pattern and is equal to p 3/8⇡ for hexagonal holes in an hexagonal array. According to Equation 2.6, the collimator e ciency for a point source in air is independent of the source-to-collimator distance b

The high-energy parallel-hole collimator Chapter 2 when the counting rate is measured over the entire detector area [START_REF] Sorenson | [END_REF].

The collimator resolution R c depends on its thickness, the diameter of its holes and the distance b of the point source from the collimator using the relationship:

R c = d (L + b + c) L (2.7)
where c represents the gap between the collimator and the detector, which should always be as low as possible and which is often not considered in R c expression. This equation shows that the spatial resolution of the collimator degrades linearly with increasing distance from the collimator surface, which means that the structures closest to the collimator will be imaged with the sharpest details. This underlines the importance of placing the camera as close as possible to the structure of interest (see Sec.

2.1). Furthermore, it should be noticed that, for higher values of collimator thickness, the rate of resolution increase with distance is lower. In other words, the slope of the line R c vs. b is less important for thicker collimators, a consideration that has a direct consequence on the imaging of a deep or thick lesions. It also follows from Equations 2.6 and 2.7 that the resolution improves as the ratio (d/L) decreases while the e ciency, which goes approximately as (d/L) 2 (eq. 2.6), will automatically drop significantly. For a given septal thickness, the collimator resolution is therefore improved at the expense of e ciency and vice versa. The implications of this trade-o↵ in the design of collimators are discussed further in Section 3.2.3. Finally, it is also important to note that at short distances, the Equation 2.6 can strongly diverge from the experimental values for a source very close to the collimator surface, because the e ciency then depends strongly on the source location relative to the collimator holes [START_REF] Weinmann | Design of optimal collimation for dedicated molecular breast imaging systems[END_REF]. Thus, extended sources, especially linear sources, are generally used experimentally to minimize these e↵ects by integrating a multitude of source-septa positions.

As already stated, E and R c formulas were obtained by purely geometrical considerations and the penetration and scatter of photons through the walls of the collimator holes were not considered [START_REF] Anger | Scintillation Camera with Multichannel Collimators[END_REF][START_REF] Sorenson | [END_REF]. To take into account the e↵ects of penetration, it was proposed to modify the analytical model by considering that the length of the collimator hole should be reduced at both ends by a factor 1/µ, where µ is the total linear attenuation coe cient of the collimator material at the energy considered [START_REF] Mather | Gamma-ray collimator penetration and scattering e↵ects[END_REF]. The lenght L in Equations 2.6 and 2.7 is therefore replaced by the e↵ective lenght L e , defined as L e =L -2/µ. The contribution of penetration is also included in collimator optimization by setting the transmission factor of a gamma ray through a material of thickness w, defined as the shortest distance that a photon can travel through a single septa of thickness t (Fig. 2.7), at less than 5%. Indeed, experimental studies have shown that it is possible to achieve "acceptable images" when the gamma ray attenuation along the path length w is equal to or greater than 95% [START_REF] Anger | Scintillation Camera with Multichannel Collimators[END_REF]. Thus, when optimizing the collimator, the required septal thickness t is often imposed to be able to stop more than 95% of incident photons using the simple penetration criteria in the Chapter 2

The high-energy parallel-hole collimator form of e µw = 0.05 or, if not set at 5%, the usual criteria is to accept a reasonably low level of septal penetration [START_REF] Moore | Collimator design for single photon emission tomography[END_REF][START_REF] Razavi | Characterization of low, medium and high energy collimators for common isotopes in nuclear medicine: A Monte Carlo study[END_REF]. From this probability of septal penetration and the analytical formulas of S and E, the optimization procedure of the collimator geometry consists in maximizing the e ciency for a given target resolution, as firstly described in 1968 [START_REF] Keller | Optimum dimensions of paralleI-hole, multi-aperture collimators for gamma-ray cameras[END_REF] and used by many collimator designers [START_REF] Formiconi | Optimization of high energy collimator design[END_REF][START_REF] Weinmann | Design of optimal collimation for dedicated molecular breast imaging systems[END_REF]. After setting a desired spatial resolution at a specific depth in Equation 2.7, the optimization can be solved by analytically expressing the e ciency as a function of SR and the penetration criteria, then maximizing it by zeroing its first derivative.

When the first order penetration correction of the collimator length was presented [START_REF] Mather | Gamma-ray collimator penetration and scattering e↵ects[END_REF] and definitely accepted as a necessary correction to Anger's equations, it was clear that it could not work for high energy emitters [START_REF] Gerber | Parallel-Hole, Collimator Design[END_REF]:

"It is seen that for a collimator designed for high-energy gamma rays, the error in the resolution is 10% or greater for small source-to-collimator distances. This error is reduced for lower energy gamma and large source-tocollimator plus collimator-to-detector distances."

The reported error is related both to the underestimation of the real penetration contributions and to the total failure to take scatter into account. The estimation of penetration proposed by the definition of the e↵ective length of the collimator L e is only performed on single septa and cannot allow to obtain an a-priori estimation of the total septal penetration. Moreover, the contribution of penetration and scatter events depends on the size and the distance between the source and the collimator, as will be seen in Section 3.2.3. For example, short distances induce larger penetration because the thickness of material to cross a septa (w, Fig. 2.7) decreases as the angle of incidence of gamma ray with respect to the collimator increases. In 1957, Mather calculated an approximate scatter fraction for a single hole through a slab of material [START_REF] Mather | Gamma-ray collimator penetration and scattering e↵ects[END_REF]. However, this calculation is far from intuitive when considering multi-hole collimators or multiple scattering and even less acceptable for high-energy gamma rays [START_REF] Moore | Collimator design for single photon emission tomography[END_REF]. Thus, since that time, the use of Monte Carlo simulation has been widespread to obtain information on the scatter component.

Monte Carlo simulations a

With the advent of high-speed computers, Monte Carlo (MC) simulation techniques have become powerful and popular tools in various fields of nuclear medicine and physics in general. Monte Carlo techniques are widely used to evaluate the performance of new collimators and scanners in single photon imaging [START_REF] Jaszczak | Cone beam collimation for single photon emission computed tomography: Analysis, simulation, and image reconstruction using filtered backprojection[END_REF][START_REF] Weinmann | Design of optimal collimation for dedicated molecular breast imaging systems[END_REF][START_REF] Lee | Optimization of an ultra-high-resolution parallel-hole collimator for CdTe semiconductor SPECT system[END_REF], to find optimum imaging conditions and parameters [START_REF] Pareto | Iterative reconstruction with correction of the spatially variant fan-beam collimator response in neurotransmission SPET imaging[END_REF][START_REF] Gils | Impact of reconstruction parameters on quantitative I-131 SPECT[END_REF] and to study the e↵ects of attenuation, scatter and penetration on image quality [START_REF] Dewaraja | Characterization of scatter and penetration using Monte Carlo simulation in 131i imaging[END_REF][START_REF] Kumar | Characterization of parallel-hole collimator using Monte Carlo Simulation[END_REF]. The basic method of optimization through MC simulations involves tracking the path of gamma radiation from its point of emission to its absorption. The probabilistic nature of the transport

The high-energy parallel-hole collimator Chapter 2 is simulated by using random numbers. Among the various Monte Carlo simulation codes available (see Sec. 1.3.1.3), we have chosen to use GATE, the Geant4 [START_REF] Agostinelli | Geant4-a simulation toolkit[END_REF] Application for Tomographic Emission. GATE is an opensource software, which incorporates Geant4 libraries in a modular and versatile toolkit specifically dedicated to the simulation of imaging (SPECT, PET, CT, Optical) and radiotherapy [START_REF] Jan | GATE: a simulation toolkit for PET and SPECT[END_REF]. In this work, GATE simulations were used for the design and optimization of the collimator and were carried out in two phases. In the first one, the intrinsic properties of several parallel-hole collimator geometries were studied in depth, in terms of spatial resolution, e ciency and contribution of penetration and scatter. The structure of the simulation will be described below and the analysis of the results will be discussed in Section 3.2. This optimization led to the choice of five di↵erent configurations, with complementary intrinsic features and therefore imaging characteristics (Sec. 3.2.4). Their imaging properties were then investigated in a more realistic clinical context to meet the specific needs of quantifying activity during thyroid treatments, previously discussed in Section 2.1. This second phase will be briefly introduced below and will be further deeply described in Section 3.2.5.

Three physical models are available to simulate electromagnetic processes in GATE : the standard package for processes between 1 keV and 100 TeV, the low-energy package for processes between 250 eV and 100 GeV, and the PENELOPE package for processes between 250 eV and 1 GeV (see Sec. 2.2). The Penelope models, whose physics models have been developed for the PENELOPE code (PENetration and Energy LOss of Positrons and Electrons, 2011), have been used because they have been specifically developed for Monte Carlo simulation with particular attention to low energies. We decided to use it although we principally simulated 131 I (364 keV) because we verified the validity of the code even at lower energy. The default production thresholds (cut-o↵ value) was used (set to 1.0 mm). No optical physics processes are simulated inside the crystal. Its intrinsic properties have been set on the basis of experimental results (see Sec. 3.2.5.2).

Collimator intrinsic properties optimization Eight collimator lengths L (from 3 to 6.5 cm each 0.5 cm), twelve hole diameters d (from 0.5 to 2.25 mm each 0.25 mm) and twelve septa thicknesses t (from 0.75 to 2.125 mm each 0.125 mm) have been considered in order to cover a wide range of configurations that could allow to perform an accurate analysis of the collimator behaviors according to its geometry, as will be presented in Section 3.2.2. The collimator has been simulated as a 10⇥10⇥L cm 3 box of tungsten (⇢ =19.3 g/cm 3 ) with an internal geometric repetition of air holes (hexagonal hole in a hexagonal array). They are coupled to a box volume representing the scintillator. In this first part of the study, the scintillator volume is not sensitive, as will be discussed later. For each collimator, the model simulates a line source of 131 I (diameter of 0.01 mm) placed in air at a source-to-collimator distance b of 5 cm. This value has been chosen as an approximation of the maximal distance of a big solitary hyperfunctioning thyroid nodule (maximal diameter of 4 cm [START_REF] Cantalamessa | Thyroid Nodules in Graves Disease and the Risk of Thyroid Carcinoma[END_REF][START_REF] Erbil | Graves' disease, with and without nodules, and the risk of thyroid carcinoma[END_REF]) and considering an extra-thyroid Chapter 2

The high-energy parallel-hole collimator tissue, i.e. the thickness of the tissue between the surface of the neck skin and the thyroid, of 0.5 cm (average value of adult male extra-thyroid tissue according to the ICRP report 23 [244]). Optimizing the collimator for a distance of 5 cm will guarantee an even better resolution when imaging lesions at shorter distances. The five principal 131 I emissions were modeled (80.2 keV, 284.3 keV, 364 keV, 637 keV and 722.9 keV with a branching ratio BR of 2.6%, 6.2%, 81.6%, 7.1% and 1.8%, respectively). An accurate characterization of the energy and spatial distributions of penetration and scatter events of these di↵erent geometries has been performed and will be presented in Section 3.2.3, as well as the investigation of their impact on collimators intrinsic properties. This characterization has been performed thanks to the use of GATE actors. Actors are GATE tools to collect additional selective information during a simulation. They can provide both general information, such as the number of times a new particle is passing through a volume (FluenceActor), or more specific information, such as the average linear energy transfer of the track (LETActors). For our purpose, PhaceSpace actor was used. It records information about particles entering the volume to which the actor is attached. Two actors were employed, one attached to the collimator box and the other to the crystal box. The name of the particle, energy, position and direction along the three axes, the name of the volume where the particle is produced and the process that produced the particle that pass through these volumes are recorded as output from the simulations [START_REF] Brun | ROOT -An object oriented data analysis framework[END_REF]. In order to study the intrinsic properties of the collimator, no sensitive detector were attached to the crystal and only the information provided by the actors were stored. Thus, the energy associated to each event corresponds to the energy they had when crossing the specific actors and not the energy they released into the crystal. For each configuration, the number of simulated gamma rays was set to produced at least 10000 counts (1% uncertainty) at the output of the collimator with the lowest sensitivity.

Optimization of the overall camera for dosimetry during radioiodine therapy in thyroid diseases The overall imaging capabilities of the detector were estimated by simulating the complete detector. The camera consists of one of five high-energy parallel-hole collimators selected during the first phase. The collimator is directly coupled to a 0.6 cm thick scintillating crystal (CeBr 3 (Ce) with a density of 5.18 g/cm 3 ) (see Sec. 2.5.5) and a 5 cm thick box made of epoxy representing the photodetection system. The camera was placed inside a 1 cm thick lead shielding ( =11.4 g/cm 3 ). A field of view of 10⇥10 cm 2 , corresponding to the final mobile gamma camera, was considered. The radioiodine treatment of hyperthyroidism was modeled using the voxelized human XCAT phantom that allows to simulate a real distribution of 131 I activity, as detailed in Section 3.2.5.1.

All simulations were run on the laboratory cluster. It consists of a head node, a login node and several compute nodes referred to as worker nodes, for a total of 256 cores available (2.60 GHz for each processor). The ROOT output files were analyzed using the Python programming language with its root numpy extension module, which provides an e cient interface between ROOT and NumPy (the fundamental package for scientific computing with Python).

The inorganic scintillator a

The detection of ionizing radiations by the scintillation light produced in certain materials, know as scintillator, is one of the oldest techniques and remains one of the most useful methods available for the detection and spectroscopy of radiation. Scintillators materials present a property known as luminescence. A luminescent material converts the energy of an incident particle into a number of low energy photons in the visible or near visible range, which are therefore easy to be detected using photodetection systems. The most widely applied scintillators include inorganic crystals and organic-based liquids and plastics. The high atomic number and density of inorganic crystals favour their choice for gamma-ray detection, while organic compounds are often preferred for beta spectroscopy and fast neutron detection. As presented in Section 2.3, the miniaturized camera developed in this work convert the gamma energy thank to the use of a continuous inorganic scintillator. The choice of the scintillator used to implement the first camera prototype has been based on a two-step process. First of all, in order to choose the best scintillation materials, which could provide a good compromise between energy and spatial response (Sec. 2.1), a detailed literature review of currently available scintillators has been carried out, as we will present in this section. Then, the fine characterization of the chosen scintillation material, in terms of energy and spatial performances, has been conducted as a function of its thickness and its optical properties, as it will be presented in Section 3.1.3.

Before discussing the bibliographic study and the choice made afterwards, a brief description of the scintillation mechanism in inorganic materials will be presented, limiting discussions to the processes necessary to understand the di↵erences in behavior of the di↵erent types of inorganic scintillators, as well as some of their main properties regarding gamma-ray detection. At the end of the section, some fundamental properties of the scintillator mounting for an e cient light collection will be highlighted.

Basic principles of scintillation a

The scintillation mechanisms in inorganic materials depend on the energy states of the crystal lattice of the specific material considered. As shown in Figure 2.8, the electrons have only discrete energy bands available within a lattice: the lower band, called the valence band, represents the electrons that are essentially bound to the lattice sites, whereas the conduction band represents those that have su cient energy to freely migrate in the crystal. In addition, there is an intermediate band of energies, called the forbidden band, in which electrons can never be found. The energy absorption can result in the excitation of an electron from the valence band to the conduction band, Chapter 2

The inorganic scintillator In pure crystals, these emissions are an ine cient process. Moreover, typical gap widths are such that the resulting photon energy would be too high to lie in the visible range. To enhance the probability of visible photon emission during the described deexcitation process, small amounts of activators are commonly added to the pure crystal, modifying its energy band structure by creating new sites in the lattice forbidden band. As a result of these lower energy states, lower energy gaps are created, from which the electron can de-excite back to the valence band, giving rise to a visible photon, the basis of the scintillation process. The amount of light quanta emitted by the scintillator per unit of energy deposited by ionizing radiation in the medium is identified as its light yield (LY). The de-excitation sites are called luminescence centers or recombination centers. Their energy structure in the crystalline lattice determines the emission spectrum of the specific material.

A charged particle passing through the detection medium will form a large number of electron-hole pairs created by the excitation of electrons from the valence to the conduction band. The positive hole will quickly drift to the location of an activator site and ionizes it, because the ionization energy of the activator will be less than that of a typical lattice site. Meanwhile, the electron is free to migrate through the crystal until it encounters such an ionized activator. At this point, the electron can drop into the activator site, creating a neutral configuration that can have its own set of excited energy states, which are illustrated in Figure 2.8 (horizontal lines within the forbidden gap). If the activator is properly chosen, the radiative transition to the ground state will occur very quickly and with high probability for the emission of a visible photon. Typical half-lives for such excited states are on the order of 50-500 ns. Since the migration time for the electron is much shorter than these half-lives, all excited activator configurations are essentially formed at once, then de-excite with the half-life characteristic of the excited state. The decay time (⌧ ) of these states determines the time characteristics of the emitted scintillation light. Some inorganic scintillators can adequately be characterized by a single decay time or a simple exponential, although more complex time behavior is often observed. Others undesired processes can compete with this radiative de-excitation. For example, the electron can create an excited configuration of the activator site whose transition to the ground state is forbidden.

The inorganic scintillator Chapter 2

Such states then require additional energy to elevate them to a higher state from which de-excitation to the ground state is possible. One possible source of energy is thermal excitation and the resulting delayed emission of light (a dozen to a hundred of µs [START_REF] Douraghy | Evaluation of scintillator afterglow for use in a combined optical and PET imaging tomograph[END_REF]) is called phosphorescence or afterglow.

According to these basic principles, several important optical properties of inorganic scintillators can be highlighted [START_REF] Knoll | Radiation detection and measurement[END_REF]:

• it should convert the kinetic energy of charged particles into detectable light with a high scintillation e ciency, that is a high light yield; The increase in the amount of light produced improves both the spatial and energy resolutions of detectors based on continuous crystals.

• this conversion should be linear -the scintillator LY should be proportional to the deposited energy. The non-linearity of the energy response degrades the energy resolution of the crystal;

• it should be transparent to the wavelength of its own emission, to avoid selfabsorption;

• the decay time ⌧ of the induced luminescence must be short for fast signal pulses to be generated. This means that the scintillation material should convert the largest incident radiation energy fraction to prompt fluorescence, while minimizing non-radiative processes. The scintillation decay time directly impacts the counting capabilities of scintillator-based detectors.

Some mechanical properties may also be of interest, such as crystal stability (variation in light output as a function of temperature and time), environmental or chemical durability or the absence of weak cleavage planes. Other highly desirable properties exist but are more application related; for example crystal radiation hardness, which may be irrelevant for detectors in many imaging applications, is extremely important for detectors in high-energy physics experiments [START_REF] Derenzo | The quest for the ideal inorganic scintillator[END_REF]. Beyond the properties listed above, considered as the basic properties of scintillating materials, other characteristics need to be addressed for the detection of gamma rays with scintillators [START_REF] Derenzo | The quest for the ideal inorganic scintillator[END_REF], and more particularly to meet the specific needs related to our dosimetric application (Sec. 2.1). All these highlighted properties will be taken into account in our literature review to compare the inorganic scintillators currently available.

Inorganic scintillator-based detector properties a

The first requirement for the detection of high-energy gamma rays is to use a scintillation material with high density (⇢) and high atomic number (Z), which will ensure a high stopping power and reasonably good probability of photoelectric absorption (P phe ) (see Sec. 2.2). The advantage is also to reduce the thickness of Chapter 2

The inorganic scintillator the scintillating material, a key point for both compact detectors and high spatial resolution when using continuous scintillators. The scintillator hygroscopicity must also be taken into account. The air-tight housing required by hygroscopic crystals allows them to be isolated from normal humidity, preventing their deterioration, but the glass window covering the scintillator output acts as an optical di↵user, which influences light extraction (amount and distribution) to the photodetector and thus spatial and energy resolutions of the gamma camera (see Sec. 2.5.4).

To optimize light extraction, the refraction index (RF) of the scintillator should be near to the one of the photodetector input window (Sec. 2.5.4). Moreover, since the global properties of a scintillator-based detector depend on the conversion of the light into an electrical signal performed by the photodetection system, the maximum wavelength ( max ) of the scintillator own emission should be compatible with the spectral sensitivity of the photodetector (Sec. 2.6).

The last properties that is usually considered to describe the performance of a scintillator material, and that will be analyse in our literature review, is its energy resolution (ER). The energy resolution is defined as the FWHM of the pulse height distribution spectrum of a monoenergetic source divided by the location of the peak centroid. The width of the peak reflects the fluctuation in the energy response of a given scintillator-photodetector and is essentially limited by the statistical noise arising from the quantum nature of the measured signal itself, which is subject to random variation from one event to another, even though exactly the same amount of energy is deposited in the detector. Thus, a high light yield is mandatory to improve the energy resolution by increasing photon statistics. The energy resolution is also a↵ected by other intrinsic properties of the scintillator material, such as its energy non-proportional response [START_REF] Moszynski | Energy resolution of scintillation detectors[END_REF] or inhomogeneities in the scintillator causing local variations in the light output. In addition, an experimentally determined energy resolution also depends on many other e↵ects, such as the non-uniform reflectivity of the scintillator's surface treatment and optical coating, as will be detailed in Section 2.5.4, the photodetector system it is coupled to or the electronic noise [START_REF] Moszynski | Energy resolution of scintillation detectors[END_REF].

No material meets all these criteria simultaneously and the choice of a particular scintillator is always a compromise between these and other factors, such as the cost or method of crystal growth that can limit the possible shapes and maximum accessible volumes.

Overview and comparative bibliographic study of inorganic scintillators for medical imaging a

The choice of the scintillators to be tested for our specific application was preceded by an in-depth review of a wide collection of available papers on scintillators (articles and technical datasheets). Among all the inorganic scintillators described in literature and used for a large variety of applications, which goes from calorimeters for high energy physics to cosmology applications, we have selected a group of 13 materials as potential candidates for our medical application, although none of them o↵ers the ideal combi-

The inorganic scintillator Chapter 2 nation of all the necessary properties. The chosen materials are listed in Table 2.3 with their main characteristics in terms of density (⇢), e↵ective atomic number (Z ef f ), light yield (LY), energy resolution at 662 keV (ER), decay time (⌧ ), refraction index(RF), wavelength of maximum emission ( max ) and hygroscopicity (Hyg). Together with the properties found in literature, the chemical composition of the scintillators were used to determine their linear attenuation coe cient using the NIST XCOM program [START_REF] Berger | NISTXCOM : Photon Cross Section Database[END_REF], which allowed to calculate the photoelectric absorption probabilities P phe for a 0.5 cm thick scintillator at 356 keV (see Sec. 2.2). If possible, these values were validated comparing them to literature [START_REF] Saint-Gobain | E ciency Calculations for Selected Scintillators[END_REF]. The key characteristics of each crystal are marked in bold. There is considerable variability in published values for a given material, most likely because of di↵erences in impurity levels, variations in samples dimensions and optical quality (which could change from a producer to another) and the photodetector used. For example, di↵erent LY and ER values are measured with LYSO using di↵erent types of PMT (LY=28600 /MeV and ER=8.1% at 662 keV with a Photonis XP5200B PMT [251]; LY=32000 /MeV and ER=8.9% with a Photonis XP20Y0QDA photomultiplier [START_REF] Wanarak | Light yield nonproportionality and energy resolution of Lu1.95y0.05sio5:Ce and Lu2sio5:Ce scintillation crystals[END_REF]). If di↵erent values were found, their mean values were considered. In order to facilitate scintillators comparison, two summary diagrams are also presented. In order to highlight the main di↵erences between the scintillators, the nominal values of the properties are multiplied by scaling factors. The properties of di↵erent scintillation materials are presented in Figure 2.9(up) and have been grouped into di↵erent families, as we will see below, making it easier to visualize the main property trends. In Figure 2.9(down), the materials are shown according to their properties, with emphasis on the individual characteristics of the scintillator.

Scintillation cameras have for many years mainly used alkali-halide scintillators such as thallium doped sodium iodide (NaI(Tl)) crystals (yellow curve in Fig. 2.9), characterized by good light yield but with slow decay time and low density (low stopping power). The main argument for the choice of NaI(Tl) is related to the maturity in its production technology. Very large samples can be grown with high quality allowing the production of the large slabs required for SPECT detector heads. In addition, the mechanical processing of these crystals is relatively easy and well optimized and their moderate hygroscopicity is not a too serious problem to deal with. Therefore, the cost of this scintillator is well controlled and reasonable.

Since the discovery of lanthanum bromide (LaBr 3 (Ce)) in 2001 [START_REF] Van Loef | High-energy-resolution scintillator: Ce3+ activated LaBr3[END_REF], an impressive R&D e↵ort has focused on the rare-earth halide family (blue curves in Fig. 2.9), in which lanthanum cloride (LaCl 3 (Ce)), cerium bromide (CeBr 3 (Ce)) and strontium iodine (SrI 2 (Eu)) are placed. Their choice is motivated by their very high light yield and energy resolution in the domain of energy relevant for medical imaging applications, enhancing scatter rejection capability and thus quality and contrast of images (Table 2.3). Moreover, apart from SrI 2 (Eu), their really fast decay time make them suitable for fast counting. Because of its high performances, which also include a really fast decay time, LaBr 3 (Ce) crystal is usually referred as being a gold standard for precise Chapter 2

The inorganic scintillator energy spectroscopy. An interesting and successful attempt of crystal engineering has been experienced on LaBr 3 (Ce) or which strontium doping has been shown to improve energy resolution by up to 2% at 662 keV [START_REF] Alekhin | Improvement of gamma-ray energy resolution of LaBr3:Ce3+ scintillation detectors by Sr2+ and Ca2+ co-doping[END_REF]. However, it includes the 138 La internal radioactivity typical of La-halide scintillator (around 0.92 counts.s 1 .cm 3 ) and, most of all, its manufacture makes it really expansive. The reduced cost of LaCl 3 (Ce), which shows a really high scintillation linearity and good energy resolution, overcomes the main limit of LaBr 3 (Ce), but its density is half as low [259]. CeBr 3 (Ce), which does not su↵er from the intrinsic 138 La background, also o↵ers a reduced cost compared to LaBr 3 (Ce), a significantly improved probability of photoelectric absorption compared to LaCl 3 (Ce) and its really good spectrometric performances make it suitable for application in medical imaging [START_REF] Lecoq | Development of new scintillators for medical applications[END_REF]. SrI 2 (Eu) has the highest light output of the scintillators studied and a really high energy resolution, which make it an interesting candidate for high energy resolution gamma ray spectroscopy [START_REF] Cherepy | Prospects for High Energy Resolution Gamma Ray Spectroscopy with Europium-Doped Strontium Iodide[END_REF]. An energy resolution lower than the one of LaBr 3 (Ce) at 662 keV is limited by a non negligible self-absorption [START_REF] Giaz | Char-acterization of new scintillators: SrI 2 :Eu, CeBr 3 , GYGAG:Ce and CLYC:Ce[END_REF]. Its very slow decay time makes it unsuitable for applications requiring high counting rate capabilities and currently its manufacture only allows to produce samples of up to 1⇥1 inch. More generally, the size limitation due to current growth methods (maximum diameter of 3 inches), non-competitive prices compared to NaI(Tl) and high hygroscopicity make the rare-earth halide family less attractive for large scanners implementation, but their high spectrometric properties make them highly competitive for small field of view detectors [START_REF] Lecoq | Development of new scintillators for medical applications[END_REF][START_REF] Pani | Excellent pulse height uniformity response of a new LaBr3:Ce scintillation crystal for gamma ray imaging[END_REF].

Sustained attention is paid to oxide scintillators because of their very high density. Among them, both Lutetium based and garnet materials have attracted a lot of interest and have been extensively studied in last decades. Examples include many cerium-doped scintillators, such as LuAP [START_REF] Moszynski | Properties of the new LuAP:Ce scintillator[END_REF], LuYAP [START_REF] Kuntner | Intrinsic energy resolution and light output of the Lu0.7y0.3ap:Ce scintillator[END_REF], LSO, GSO, LGSO [START_REF] Balcerzyk | A study of energy resolution and nonproportionality[END_REF], LYSO, LPS [START_REF] Pidol | High E ciency of Lutetium Silicate Scintillators, Ce-Doped LPS and LYSO Crystals for Medical Applications[END_REF], YAP [START_REF] Kapusta | A high-energy resolution observed from a YAP:Ce scintillator[END_REF], YAG and LuAG [START_REF] Chewpraditkul | Scintillation Properties of LuAG:Ce, YAG:Ce and LYSO:Ce Crystals for Gamma-Ray Detection[END_REF] [START_REF] Lecoq | Development of new scintillators for medical applications[END_REF]. The Lutetium-based family (green curves in Fig. 2.9) includes a wide class of materials that have been extensively studied since the discovery of LSO(Ce) in 1990. Lutetium increases the detection eciencies of high-energy gamma rays due to its high e↵ective atomic number and high density (Table 2.3). Other advantages of Lutetium-based mixtures include the short decay time, a good machining properties and the non-hygroscopicity that allows to produce crystal arrays, which are extensively used in PET detection systems. However, they are non linear and they have a high intrinsic background self-activity, which arises from the beta decay of 176 Lu isotope (2.6% of natural Lu abundance and activity around 263 counts.s 1 .cm 3 for LYSO(Ce) [START_REF] Pidol | High E ciency of Lutetium Silicate Scintillators, Ce-Doped LPS and LYSO Crystals for Medical Applications[END_REF]). This background noise can lead to significant dead time and therefore reduce the counting capabilities of the camera when large volumes of scintillator are used. It also decreases the contrast of single-photon imaging. Among the Lutetium-based family, the cerium doped lutetium silicate (LYSO(Ce)) is the more widely used. Two other materials have been studied, the Lutetium fine silicate (LFS) and a rare-earth silicate single-crystals from Crytur (CRY-019). They display quite comparable behaviors to LYSO(Ce) in terms of probability of photoelectric absorption and light yield, but for a lower cost. LFS shows a higher light yield and

The inorganic scintillator Chapter 2 CRY-019 a slightly higher energy resolution. This di↵erence could be due to a slightly di↵erent energy linearity.

The garnet family (red and orange curves in Fig. 2.9) shows excellent compromises between the halide and lutetium-based scintillators (Table 2.3). From the variety of heavy garnets, Gadolinium Aluminium Gallium Garnet doped with Cerium (GAGG(Ce)) appears to be the most attractive material. The presence of Gd provides a reasonably good stopping power, better than that of halides, but without intrinsic radioactivity. GAGG(Ce) is also much brighter than LYSO(Ce) with a higher associated energy resolution, but its probability of photoelectric absorption remains significantly lower, despite a close density, due to its smaller e↵ective Z. GYGAG(Ce), a transparent ceramic garnets, o↵ers the possibility to achieve high-resolution gamma-ray spectroscopy at low cost [START_REF] Giaz | Char-acterization of new scintillators: SrI 2 :Eu, CeBr 3 , GYGAG:Ce and CLYC:Ce[END_REF] and the potential to be grown in sizes and shapes similar to the most widely used NaI(Tl) [START_REF] Cherepy | Development of Transparent Ceramic Ce-Doped Gadolinium Garnet Gamma Spectrometers[END_REF]. However, both GAGG and GYGAGG have a quite slow decay time, which could impact system dead time making them less suited for high counting rate applications [START_REF] Cherepy | Scintillators with potential to supersede lanthanum bromide[END_REF][START_REF] Cherepy | Strontium and barium iodide high light yield scintillators[END_REF]. To overcome the timing limitation of GAGG(Ce), a new cerium doped Gadolinium Fine Aluminum Gallate (GFAG(Ce)) scintillation crystal has been recently grown [START_REF] Lee | Digital silicon photomultiplier readout of a new fast and bright scintillation crystal (Ce:GFAG)[END_REF]. The GFAG(Ce) crystal, like GAGG(Ce), has high density and high light output, is not hygroscopic and has no intern self-radiation. For above reasons, GFAG(Ce) seems to be really promising in fields of gamma spectroscopy and nuclear medicine. However, the maximum diameter produced is currently 5 cm, which further limits its scope of application.

The main limitation of these three garnets is their maximum emission wavelength, which is not adapted to the loss of sensitivity at longer wavelengths of many photodetectors (bialkali photocathode, silicon photomultiplier with p/n structure, ...) and thus decreases the quantum e ciency of photon-to-electron conversion. The rare-earth silicon oxide scintillator GPS(Ce) adresses this gap (Tab. 2.3). It has similar properties to GAGG(Ce) but with a shorter decay time and a lower maximum emission wavelength. However, monocrystalline growth is still considered di cult [START_REF] Kawamura | Scintillation Characteristics of Ce:$Gd 2si 2o 7$ (Ce 2.5-30 mol%) Single Crystals Prepared by the Floating Zone Method[END_REF][START_REF] Kawamura | Floating Zone Growth and Scintillation Characteristics of Cerium-Doped Gadolinium Pyrosilicate Single Crystals[END_REF]. Lastly, Cerium-doped lanthanum gadolinium pyrosilicate (La-GPS(Ce)) shows to be promising, containing both gadolinium and lanthanum. It is expected to give higher spectrometric performances with the optimization of its production. Being a new material, further investigations are still ongoing (only small arrays are available today).

Light Collection in scintillation detector assembly a

In any scintillation detector, one would like to collect the largest possible fraction of the light emitted isotropically from the track of the ionizing particle, in order to enhance the system energy resolution, while optimizing the spread of the light distribution according to the characteristics of the photodetector (generally by limiting the width of this distribution), in order to promote higher spatial resolution. Light collection conditions a↵ect the energy resolution of a scintillator in two distinct ways. First,

Chapter 2

The inorganic scintillator the statistical broadening of the energy response function will worsen as the number of scintillation photons that contribute to the measured pulse is reduced. In this case, the best resolution can only be achieved by collecting the maximum possible fraction of all photons emitted by the scintillation event. Secondly, the uniformity of light collection will determine the variation in signal pulse height when the position of the radiation interaction changes throughout the scintillator. Perfect uniformity would ensure that all events that deposit the same energy, regardless of where they occur in the scintillator, would result in the same pulse height. In the case of scintillators used in medical imaging, generally with a maximum thickness of a few centimeters, the non-uniformity of light collection could significantly contribute to the overall energy resolution (see Sec. 3.1.3.1). Therefore, the improvement of light collection e ciency (ratio between the number of photons arriving on the photodetector to the total number of photons generated in the scintillator) and light spreading over the photodetection system is and has always been a crucial element in the optimization of scintillation detectors. In practical cases, with the exception of very large scintillators that could su↵er from optical self-absorption, the collection and distribution of light is mainly impacted by the scintillator treatment surfaces and optical coating, the thickness of the crystal and the one of the light guide and by the nature of the optical coupling with the photodetector.

Scintillator optical coating a

Because the light is emitted isotropically, only a small fraction of it can directly reach the scintillator surface coupled to the photodetection system, while the rest of the light goes towards the other surfaces and will behave on the basis of its optical and physical properties. Depending on the applications, these surfaces, which can be polished or slightly rough, are generally wrapped with reflectors (that can be either specular or di↵usive) or with an absorbing material. With reflective coating, the light that would escape through the lateral and upper surfaces of the scintillator is reflected back within the crystal following Fresnel or Lambert's cosinus laws and, after one or more reflections, it could be collected by the photodetector. Specular reflectors act as mirror-like surfaces and transform a thin crystal into a light guide, in which multiple total reflections carry light away from the interaction point, widening the light distribution and consequently, degrading the spatial resolution. For this reason, in most scintillators used for medical imaging, the top surface of the crystal is covered with a white di↵use reflector in order to achieve the highest reflectivity while avoiding the mirror-like e↵ect. White epoxy, Teflon tape, and titanium oxide based white paint are commonly used.

On the other hand, reflected light from the sides can significantly contribute to the width and tails of the light spread distribution for most light source locations, especially near the crystal edges. This lateral reflection becomes even more problematic as the crystal thickness increases and the source point of interaction moves away from the photodetector plane, i.e. for low-energy gamma rays. Under these conditions, it is more di cult to preserve the spatial information on the location of the origin of the gamma

The inorganic scintillator Chapter 2 ray interaction. This causes a deterioration of the spatial performance of the detector, mainly at the edges. Spatial linearity defects at the periphery of the detector cause, in particular, a compression of the useful field of view. For these reasons, scintillation crystal with absorbent edges is generally preferred for applications with small field of view camera. By removing internal reflections at the edges, light spreading and distorsion of the light distribution are reduced, improving the reconstruction of the event position on the whole detector area and thus, the resolution and spatial linearity of the camera. However, the cut of the light distribution by the absorbent edges obviously leads to a decrease in the number of photons collected, whose intensity will depend on the interaction position of the gamma ray in the scintillator. This degrades the local energy resolution and therefore the uniformity of the energy response.

Scintillator-photodetector optical coupling a

The part of the light that reaches the surface of the scintillator coupled to the photodetector array is forced to face a change in the refraction index of the medium. The high index of most inorganic scintillators (typically between 1.8 and 2.3) compared to the lower index of the entrance window of the photodetector (around 1.5 for glass, quartz or resin) can lead to the capture of a significant fraction of the light inside the scintillator by total internal reflection if no optical coupling is made between the two polished surfaces of the scintillator and the photodetector. E cient coupling may be achieved by filling the interface with an optical material, such as grease or thyxotropic gel, with an intermediate refraction index. Photonic crystals have been also proposed to improve the light extraction e ciency [START_REF] Lecoq | Pushing the Limits in Time-of-Flight PET Imaging[END_REF]. A photonic nano-structure corresponds to a dielectric material whose periodic arrangement in the range of the wavelength of the photons allows to overcome the limit of total internal reflection at the interface between the crystal and the photodetector. Recent results shown an enhancement in the scintillation light extraction using nanoprinted photonic crystals by over 40% for a LSO scintillator compared to the grease-coupling [START_REF] Singh | Enhanced Scintillation Light Extraction Using Nanoimprinted Photonic Crystals[END_REF].

When using hygroscopic materials, it is not possible to directly couple the scintillator to the photodetector because crystals are encapsulated in a sealed aluminum housing with a glass entrance window, which plays the role of a thin light guide (typical thickness between 1 and 2 mm). This guide creates one additional optical interface between the crystal and the photodetector window that may decrease light extraction and mainly enlarge the light distribution reaching the photodetector. The influence of the guide on the light spreading depends on the thickness of the scintillator, which is generally the main factor determining the width of the light distribution. An e cient reconstruction of the interaction position of the gamma ray requires that the scintillation light is spread over an optimal number of photodetection units (see Sec. 2.8.2). A too narrow light distribution will lead to positioning errors due to the discrete nature of the photodetector, which may cause its pattern to appear in raw flood images. On the other hand, a too wide distribution will increase the statistical noise on the signal collected by each detection unit, which will degrade the spatial resolution.

Chapter 2

The inorganic scintillator

Scintillators selected for our study

a As introduced at the beginning of this section, we realized a bibliographic study for identifying scintillator materials that could be interesting candidates for our application. Several scintillator materials were investigated. Some of them, as the NaI(Tl) or the LYSO(Ce), because they are the most commonly used scintillators for medical imaging [START_REF] Knoll | Radiation detection and measurement[END_REF]. Others, because they have been widely identified as good scintillating materials due to their excellent properties, as LaBr 3 (Ce), while others have been selected as promising materials, such as GPS(Ce) and LaGPS(Ce).

As a results of this investigation, CeBr 3 (Ce) was chosen to implement the first prototype of the mobile camera, as it best meets the specific needs of our project (Sec. 2.1). The advantages of CeBr 3 (Ce) are its high light yield of 60.000 ph/MeV with a fast decay time (17 ns) suitable for fast counting; an excellent light yield proportionality [START_REF] Lecoq | Development of new scintillators for medical applications[END_REF] and an high energy resolution (4.1% at 662 keV), slightly lower than that of the other halide scintillators and of the gold standard LaBr 3 (Ce) but at a cost twice as low. In addition, it has no intrinsic radioactivity, which reduces the contribution of the background noise to the counting rate and thus, improves the image contrast. It also o↵ers the possibility of being produced in a 5⇥5 cm 2 square shape to match the dimensions of our photodetection system (see Sec. 2.6).

Main disadvantage of CeBr 3 (Ce) is its moderate stopping power for high energy gamma rays compared to the Luthetium-based scintillators and some heavy garnets, such as GAGG(Ce) (probability of photoelectric absorption of 11,6 % at 356 keV). However, as previously mentioned, the sensitivity of the mobile camera is not a key feature for dosimetric imaging after the administration of 131 I , due to the large amount of radioactivity used during the treatment of benign or malignant thyroid diseases. Priority was therefore given to the light output of the scintillator in order to optimize the spatial and energy performance of the camera. Another disadvantage of CeBr 3 (Ce) is its hygroscopicity, which requires it to be sealed in an aluminium housing with a thin glass window. The production of this scintillator is also currently limited to sizes of 7⇥7 cm 2 . However, in view to the future development of a clinical camera with a field of view of 10⇥10 cm 2 , we have verified that it is possible to optically assemble four 5⇥5 cm 2 CeBr 3 (Ce) in the same mechanical housing, at a cost comparable to that of a single 5⇥5 cm 2 LaBr 3 (Ce) sample. Finally, the spectral distribution of CeBr 3 (Ce)'s light emission does not perfectly match the sensitivity of the silicon photomultipliers used in our camera (maximum sensitivity around 450 nm), but the expected loss of sensitivity remains low, particularly compared to what would have been achieved with LaCl 3 (Ce) or GAGG(Ce).

As already described in previous sections, several potential sources of fluctuation a↵ect scintillation light collection, including the optical coating of the scintillator, its thickness and the specific features of the photodetector to which it is coupled [START_REF] Moszynski | Energy resolution of scintillation detectors[END_REF]. In order to study the impact of crystal thickness and optical coating on light extraction and thus, on the intrinsic performance of the camera, four 5⇥5 cm 2 CeBr 3 (Ce) samples provided by Scionix have been tested (see Sec. Together with the scintillator features presented in this section, the global properties of a scintillator-based gamma-ray detector also depend on the intrinsic features of the photodetector used. The properties of the photodetector used in our camera will be presented in Section 2.6. The reconstruction algorithm also plays a fundamental role and can totally or partially restore the positioning errors near the edges of the scintillator, the compression e↵ect of the useful field of view and the non-uniformity of spatial and energy responses (Sec. 2.8.2).

Chapter 2 2.6. THE PHOTODETECTION SYSTEM

The photodetection system a

The photodetector collects the scintillation light and converts it into a measurable electric current, with a charge proportional to the number of incident scintillation photons. Despite major advances in technology in the past 40 years, the basic design of clinical imaging instrumentation has remained relatively unchanged. The most common instruments in clinical nuclear medicine (planar imaging, SPECT and PET system) continue to make extensive use of photomultiplier tube arrays (PMTs) to convert light produced by the interaction of gamma radiations within a scintillation crystal. It is only recently that new technologies and many novel designs (particularly for organspecific imaging) have begun to have an impact on clinical applications [START_REF] Casey | First human images from a next generation SiPM based PET/CT system with improved time and spatial resolution[END_REF]. This recent evolution opens new perspectives for future molecular imaging. An overview of the existing photodetection devices will be presented in this section with a focus on silicon photomultipliers, chosen to build our mobile camera because they allow the development of compact imaging systems with high spatial and spectral performance.

Overview of photodetection devices a

Two photodetector technologies are currently available : the vacuum technology and the solid-state detectors. Among vacuum devices, photomultipliers tubes (PMTs) are now the most widely used. Their advantages and drawbacks for imaging will be briefly described in the following paragraph. The main development in detector design concerned the adoption of solid-state devices. This involves two distinct types of technology: solid-state detectors that provide direct conversion of gamma photon energy into an electronic signal, replacing the conventional combination between scintillators and PMTs and solid-state semiconductors used as photosensors. As already stated in Section 2.1, the choice of a scintillator-photodetector assembly instead of direct detection with high-Z semiconductors (CdTe, CdZnTe) is mainly due to the fact that their sensitivity is not suited for the detection of high energy gamma rays [START_REF] Hutton | Advances in clinical molecular imaging instrumentation[END_REF]. However, a quick overview of their working principle will be given.

Vacuum-based detectors a

The PMTs are basically composed of a photocathode, allowing the conversion of incident photons into photoelectrons, a multiple stages dynodes string, for electrons multiplication, and an anode, from which the final signal can be readout. The e ciency of the photocathode photoelectric conversion is expressed by the quantum e ciency (Q e ), the ratio between the number of photoelectrons released and the number of incident photons on the cathode. The quantum e ciency depends on the wavelength of the incident light. PMTs can be extremely sensitive to ultraviolet, visible and nearinfrared light, depending on the material of the photocathode material. The dynodes string makes it possible to amplify, in an avalanche process, the original photoelectrons created at the photocathode. The multiplied electrons are then collected at the anode,

The photodetection system Chapter 2 generating a measurable electric signal with a gain of about 10 6 , proportional to the number of incident photons on the photocathode. The detection e ciency, representing the probability of detecting an incident photon, is expressed as the product between the photocathode quantum e ciency and the collection e ciency of the dynodes.

Based on this operation mode, multiple PMTs structures have been developed to meet the specific requirements of di↵erent application fields (see Fig. 2.11). For example, standard PMTs (Fig. 2.11a) are mainly used for spectrometric applications, where there is no need to have information about the interaction position of the incident photon on the photocathode. To meet these needs, position-sensitive PMTs have been developed. These phototubes have a particular grid type dynode structure capable of confining the photoelectrons multiplication avalanche around the photon interaction position on the photocathode. The electrons avalanche is then collected on a crossed anode and read out by a network of resistances that allows the centroid position of the light distribution to be reconstructed using charge division from a reduced number of electronic channels. Other available technologies provide similar information, such as MCPMT (MicroChannel PMT), which consists of a photomultiplier tube incorporating a micro-channel plate instead of conventional discrete dynodes or HPMT (Hybrid PMT), which uses a photocathode coupled to a semiconductor detector. The Multi-anode photomultiplier tube (MAPMT) is another photomultiplier structure. It is composed of multiple miniaturized photomultipliers placed side by side in a closepacked arrangement. Extremely compact MAPMTs (5⇥5 cm 2 and 3 cm thickness)

Collimator intrinsic proper/es : Insight Chapter 2

The photodetection system composed of 8⇥8 or 16⇥16 elements are currently available (Flat panel H8500 and H9500, Hamamatsu, Fig. 2.11b). Detection systems based on vacuum technology have been widely used for several decades. Their operational modes are well known, they o↵er large sensitive areas, simplifying the design of optical systems; they also achieve good performance at high counting rates and o↵er picosecond timing resolution with microchannel plate models (10ps). However, their operating supply voltage is high, in the order of kV, they are fragile and bulky and their performance deteriorates in the presence of a magnetic field (incompatible for a combination between SPECT-PET and MRI systems). Most of these drawbacks can be overcome with detectors based on solid-state technologies.

Solid-state detectors

Solid-state detectors with direct conversion Solid-state detectors with direct conversion are not a new technology [START_REF] Scheiber | CdTe and CdZnTe detectors in nuclear medicine[END_REF], but their late introduction for clinical imaging (GE Healthcare, 2016 [START_REF] Healthcare | NM CT 870 CZT | GE Healthcare[END_REF]) is probably due to their relatively high cost. The two most commonly used detection materials are Cadmium Zinc Telluride (CdZnTe or CZT) and Cadmium Telluride (CdTe). Until now, the use of CZT in imaging instruments has been limited to hand-held surgical cameras and small organ-specific imaging systems, where the cost of detector volume was less prohibitive [START_REF] Iniewsky | CZT detector technology for medical imaging[END_REF]. These detectors can directly convert gamma rays into electrons and holes. Indeed, the interaction of a gamma ray with the detector produces an ionization with free electrons/holes, which are then attracted by an applied polarization voltage producing a measurable charge proportional to the absorbed photon energy. They are designed with small elements or pixels with a common electrode so that spatial localization is based on simply identifying the element where the signal originates. The intrinsic resolution of the system is therefore defines by the size of the pixels (range from 1 to 2 mm [START_REF] Hutton | Advances in clinical molecular imaging instrumentation[END_REF][START_REF] Iniewski | CZT Pixel Scaling for Improved Spatial Resolution in Medical Imaging[END_REF]).

There are several advantages to CZT solid-state detectors. At first, they have a very high energy resolution, typically less than 5% at 140 keV, which leads to good energy discrimination [START_REF] Hutton | Advances in clinical molecular imaging instrumentation[END_REF][START_REF] Scheiber | CdTe and CdZnTe detectors in nuclear medicine[END_REF]. However, studies reveal that, even at low energy, a significant tail on the photopeak is produced as a results of charges di↵usion when the signal resulting from photon interaction is shared in neighboring elements. This e↵ect could be reduced by detecting and adding coincident neighboring signals. Nevertheless, the tail contributes significantly to the counts that need to be taken into account in scatter correction, which requires the adoption of complex multi-window methods, mainly if the scatter of multiple higher energy emissions are present, like with radionuclides such as 123 I and 131 I [START_REF] Holstensson | Model-based correction for scatter and tailing e↵ects in simultaneous 99m Tc and 123 I imaging for a CdZnTe cardiac SPECT camera[END_REF]. Instruments based on CZT detectors can also be very compact and for applications where the camera can be placed close to the activity source so as to minimize the influence of the collimator resolution, the system resolution is superior to that of a conventional camera. Another advantage is the use of the full detection area, without the problems associated with inaccurate positioning close to the edges that characterized gamma cameras with scintillators.

The main drawback of high-Z semiconductors for our application is their low stop-

The photodetection system Chapter 2

ping power for high energy gamma rays ( 131 I or for PET applications) [START_REF] Hutton | Advances in clinical molecular imaging instrumentation[END_REF]. Indeed, recent studies show good improvement in the sensitivity of CZT detector but at energies in the order of 99m Tc emissions [START_REF] Goshen | Feasibility study of a novel general purpose CZT-based digital SPECT camera: initial clinical results[END_REF]. In addition, to isolate the electric signal at a single pixel, the thickness of the detector is generally limited to a few millimeters. Finally, when a semiconductor array is used in a gamma camera, the collimator must be designed to fit the size of its pixels (or vice versa) in order to minimize dead zones in the camera. Thus, the overall performance of solid-state gamma cameras, in terms of spatial resolution and e ciency, is less easily adjustable than with the conventional structure combining a scintillator and a collimator.

Solid-state detectors for photodetection This class of detectors is used to convert the light emitted by a scintillator into an electrical signal, e↵ectively replacing PMTs. The recent interest in the application of this technology to molecular imaging is mainly due the fact that, unlike PMTs, they are insensitive to magnetic fields and therefore well suited to hybrid imaging combining PET or SPECT and MRI devices.

In addition, as for CZT, they have the advantage of being well suited to applications that require compact imaging systems with small FoV [START_REF] Moehrs | A detector head design for small-animal PET with silicon photomultipliers (SiPM)[END_REF][START_REF] Carminati | Validation and Performance Assessment of a Preclinical SiPM-Based SPECT/MRI Insert[END_REF].

Semiconductor detectors can be classified into three main categories: photodiodes, Avalanche PhotoDiodes (ADP) and Silicon PhotoMultipliers (SiPMs). All these technologies are based on the PN junction, used in di↵erent conditions and operation modes.

Photodiodes are based on a PN junction or PIN structure, in which an electronhole pair is created by an incident photon with su cient energy. If photon absorption occurs in or near the junction's depletion region, or in its proximity, the carriers are swept from the junction by the built-in electric field. A photocurrent, proportional to the number of incident photons, is then produced by the migration of holes to the anode Collimator intrinsic proper/es : Insight Chapter 2

The photodetection system and electrons to the cathode. Photodiodes have a high quantum e ciency, ranging from 60 to 90% in the visible spectra depending on the wavelength. However, due to their relatively low electronic gain, in the order of unity, they are di cult to use with light pulses of less than hundreds of photons.

Avalanche photodiodes are based on the same operating principle as photodiodes but they operate with higher voltages (100-500 V). An incident photon generates a finite number of electron-hole pairs, creating a current proportional to the number of incident photons. The typical gain of nearly 100 allows the detection of low light levels. This is only valid if the operating voltage is lower than the breakdown voltage (Sec. 2.6.2.2). If the voltage is higher, the APD starts operating in Geiger mode and the device is referred to as Geiger-mode APD (GM-APD), also called single photon avalanche diodes (SPADs). Under these conditions, the electric field is so intense that a single carrier injected in the depletion region triggers a self-sustaining avalanche. The main limitation of the diode operating in Geiger-mode is that the output signal is the same regardless of the number of incident photons. Indeed, the output signal is rather a binary response. On the other hand, the gain is much higher, about 10 6 , which is comparable to the traditional PMT.

In order to partially overcome the limitation of the SPAD response, the diode can be segmented into tiny micro-cells, each operating in Geiger-mode. These micro-cells are connected in parallel to a single common output. The measured signal therefore corresponds to the sum of the charges emitted by the individual microcells activated by the absorption of a photon and is proportional to the number of micro-cells hit by a photon. This structure is called Silicon PhotoMultiplier (SiPM). SiPMs are low-light level detectors. Their photodection e ciency is superior to that of a conventional PMT in the wavelength range from near ultraviolet to near infrared (up to 40 % at 420 nm) and their gain is similarly large (between 10 5 and 10 6 ), but using a much lower operating voltage (<100V). These properties allow them to detect and count single photons arriving on the micro-cells with a very high temporal resolution (<100 ps) [START_REF] Giussani | Imaging in Nuclear Medicine[END_REF][START_REF] Acerbi | Understanding and simulating SiPMs[END_REF]. The other advantages of SiPMs are their high compactness (silicon substrate thickness less than 500 µm), very low energy consumption (<100 µW/mm 2 ), robustness and again, insensitivity to magnetic fields up to 15 T. SiPMs can also be assembled into arrays of di↵erent sizes and shapes to develop detectors for a wide range of applications. SiPMs are already established photodetector in many applications. The following section provides an overview of the basic principles of an SiPM, in order to understand its operating principles and main parameters.

The photodetection system

Chapter 2

The Silicon Photomultiplier array a

SiPMs have been chosen as photodetectors to be integrated in the first prototype of our miniaturized gamma camera because of the advantages highlighted above. Our SiPMs array consists of four Hamamatsu S13361-3050NE-08 monolithic arrays with an overall area of 51.6⇥51.6 mm 2 and a sensitive area of 48⇥48 mm 2 (Fig. 2.13). Each array is composed of 8⇥8 pixels with a 3⇥3 mm 2 e↵ective photosensitive area and a micro-cell size of 50 µm. The through-silicon-via (TSV) technology allows to reduce the dead area between pixels to 0.2 mm. The optical window of the arrays is made of an epoxy resin with a refractive index of 1.55. The four arrays are mounted on a single PCB containing passive elements (resistance, capacitances) for bias as well as four digital thermal sensors, on the backside. The operational principle and main features of SiPMs will be briefly described in the following sections. For each properties, emphasis will be made on the characteristics of the SiPMs array used in our photodetection system. Chapter 2

Operation principles of SiPMs

The photodetection system photon avalanche diode (SPAD) operating beyond the breakdown voltage (V bd ). It also integrates a resistor for passive quenching. All cells are independent and connected to a common readout on a silicon substrate.

In a conventional p-on-n structure, each cell consists of a highly doped n-type silicon substrate (n++), on which is deposited a low n-doping epitaxial layer. The junction is basically obtained by creating a p+ area on the surface of the epitaxial layer. Usually, a second n-type region is created under the n+ area. This allows the breakdown voltage to be set to a desired value. The final structure, composed of the overlapped layers, is shown in Figure 2.14. When a voltage is applied to the cell, the epitaxial layer is associated with a weak electric field and a drift region is created. On the other hand, the thin depletion area (smaller than a µm) that is created at the junction between the n and p+ area is associated with a high electric field (3 to 5•10 5 V/cm). This is called the avalanche region, where the charge multiplication takes place. The uniformity of the electric field within the pixel is guaranteed by a p-doped edge structure, called a guard ring. A layer of silicon oxide (SiO 2 ) is deposited on the surface to electrically decouple the cells and create an optical window to enhance the penetration of photon in the silicon. Finally, the cells are interconnected by a common aluminum strip through individual resistors of a few hundred k⌦, called quenching resistors R q .

An SiPM can be represented by a circuit sequence equivalent to a SPAD (Fig. 2

.15).

A SPAD is represented by the capacitance of the inner depletion layer C d (the diode capacitance in reverse bias), connected in parallel with the series resistor R s of the silicon substrate and the switch S, which represents the trigger of an avalanche. R s includes both the resistance of the neutral region inside silicon, as well as the space charge resistance (R S ⇠1 k⌦). The R s is connected in series to a voltage generator V bd , the breakdown voltage, which is the threshold voltage to trigger the Geiger avalanche process. According to this model, the SPADs is then connected in series with the quenching resistor R q and the external bias (V bias ), i.e. the operating voltage. The evolution of the signal produced by a SPAD can be divided in three phases : the rest phase, the discharge and the recovery phase. In quiescent mode, the switch is open and the capacitance is charged to the operating voltage V bias , with V bias > V bd (Geiger mode). No current flows through the circuit unless a carrier is generated, either by photon absorption or thermal vibration (dark event). When the electron-hole pair is created nearby the p-n junction, the electron is drifted through the high-field region and there is a certain probability, known as turn-on probability, of initiating an avalanche discharge. Each carrier can trigger several new avalanches. Once triggered, the avalanche process is self-sustaining, which means that without quenching, a steady current could flow indefinitely through the device. However, the electron gain is limited in current by the quenching resistor R q . In the discharge phase, the switch is closed, C d discharges through R s and the voltage drops to a value close to V bd . When the the capacitance C d is completely discharged, the avalanche process is shut down by the quenching resistance R q and the equivalent circuit switch opens again (Fig. 2 In the center, the typical SiPM pulse that corresponds to simultaneous firing of several cells (black). In orange, the single cell pulse. The amplitude and the area are proportional to the number of detected photon at the same time. On the right, the pulses pile-up corresponding to a non-simultaneous firing of three micro-cells [Source [START_REF] Ng | Physics of Semiconductor Devices[END_REF]].

times larger than the response of a single-cell, since the N independent current pulses simply add up at the SiPM terminals (Fig. 2.16(center)). Note that the amplitude and surface area of each SiPM pulse, which is the total charge delivered by the detector, are proportional to the number of photons detected (neglecting primary and correlated noise). If the number of incoming photons is larger than the number of micro-cells in the SiPM, saturation occurs and neither the amplitude nor the surface area of the output pulse can provide information on the number of incoming photons. In the case of a non-simultaneous firing of several micro-cells, pulses pile-up may occurs at the SiPM terminals. As an example, Figure 2.16(right) shows three micro-cells firing in rapid sequence. As before, the area of the SiPM signal is still proportional to the number of the detected photons. When SiPMs are coupled to scintillating crystals, the SiPM output signal gives the characteristic light emission profile of the crystal itself assuming that the decay time of the crystal is longer than the recovery time of the SiPMs and that the SiPMs operate in non-saturation. By calculating the integral of the SiPM signal, it is possible to determine the amount of charge produced by the SiPM, which is proportional to the amount of light emitted by the crystal and to the total energy deposited by the incident particle.

SiPMs intrinsic characteristics a

The main intrinsic characteristics of SiPM, such as its breakdown voltage, gain, photon detection e ciency, dynamic range and thermal and correlated noises, are presented in the following paragraphs.

The photodetection system

Chapter 2

Breakdown voltage The breakdown voltage (V bd ) is the minimum (reverse) bias voltage that leads to self-sustaining avalanche multiplication in SPADs. However, output current pulses can only be observed for V bias >V bd . The bias beyond the breakdown voltage is called overvoltage (V ov ). By definition:

V ov = V bias V bd (2.8)
In principle, the higher the overvoltage, the higher the SiPM performance. In reality, as we will see later, since the detection e ciency tends to saturate with V ov , while the noise continues to increase, an upper limit to the optimum SiPM bias voltage exists. The breakdown voltage is defined by the p/n junction characteristics of the particular SiPM technology. Its temperature dependence is also determined by the structural characteristics of the SPADs. Indeed, as the temperature increases, crystal vibrations become more important and the primary carriers have a greater probability of interacting with the crystalline lattice. As a result, the breakdown voltage increases.

Gain

The gain is the number of charges created by a carrier during an avalanche process in a cell. It can be expressed by the relation:

G = Q cell q e = C d V bias V bd q e (2.9)
where Q cell is the maximum charge that can be accumulated in a cells, C d is the microcell capacitance and q e the elementary charge. Since C d does not vary significantly with temperature (within a reasonable range of temperatures), any gain shift with temperature must be attributed to the variation in breakdown voltage with temperature. To compensate for this e↵ect and maintain a constant gain, a higher bias voltage must be applied. Considering that the terminal capacitance (total capacitance of all pixels) of the 3⇥3 mm 2 Hamamatsu SiPMs used in our photodetection system is 320 pF, i.e. a micro-cell capacitance of 89.3 fF, a gain of about 1.6•10 6 can be attained for an overvoltage of 3 V. Such an elevate gain allows to observe light signals of the other of one photo-electron.

Photon detection e ciency The photon detection e ciency (PDE) is the probability that a photon arriving on the SiPM surface is detected producing an output pulse. It can be defined as the ratio between the number of detected photons over the number of incoming photons. The PDE is defined as the product of three factors:

P DE(V ov ) = Q e • F F • P t(V ov ) ( 2 . 1 0 )
where Q e is the quantum e ciency, F F the fill-factor and P t the triggering probability.

The quantum e ciency Q e expresses the probability that a photon striking the SiPM is actually transmitted to the silicon through the dielectric layer on the top of the silicon surface, absorbed in the silicon and finally converted in an electron/hole pair. Q e is a function of the wavelength and angular incidence of incoming photons.

Chapter 2

The photodetection system F F represents the fill-factor, i.e. the geometric e ciency of the micro-cells array. It accounts for the fact that each micro-cell in the SiPM necessarily has a dead area on its periphery to accommodate isolating structures and metal lines for signal routing. The fill factor varies from one SiPM to another depending on the size of the micro-cell and the manufacturer. The SiPMs used in our array have a fill factor of 74%.

The triggering probability P t is the probability that a generated electron/hole pair successfully initiates a self-sustaining avalanche process and thus an output current pulse. P t is a strong function of the overvoltage, and it increases with V ov according to the dependence of electron and hole ionization rates on the induced electric field. P t also depends on the wavelength since the probability of initiating an avalanche depends on the depth inside the SPAD where the electron/hole pair was generated and therefore on the wavelength of the incoming photon. For SiPMs based on the n-on-p structure type, which combines a n+ layer on a p+ substrate, only photons with a wavelength greater than 500 nm (green and red) will be able to interact in the p-doped area. Blue and UV light (380-540 nm) has a shallow absorption in silicon (100-500 nm) and most is absorbed near the cell surface, inside the n-p junction, where the probability of triggering an avalanche is particularly low. Thus, this structure is optimal for the detection of green or red light. Conversely, SiPMs based on a p-on-n structure (p layer on a n+ substrate), such as the one shown in Figure 2.14, have a PDE optimized for the blue wavelength. For our specific application, since the wavelength of photons emitted inside the CeBr 3 scintillator after gamma ray interaction is in the order of 380 nm,

The photodetection system Chapter 2

SiPMs optimized in the blue region was preferred. Specifically, our SiPM has a PDE of 40% at 450 nm and >35% from 400 to 550 nm (Fig. 2.17). The drop in sensitivity in UV region is due to the transmittance of the epoxy resin covering the silicon layer. It can be improved by using silicon resin. The functional dependence of PDE on overvoltage is contained in the P t term, since both Q e and FF are, for a given wavelength, constant factors.

Dynamic range

The dynamic range is the range of light intensities for which the output signal of the SiPM is proportional to the input light signal. Since a single microcell touched simultaneously by two photons will be blind to one of them, the dynamics depends exclusively on the number of micro-cells in the SIPM. The optimal operating conditions for SiPMs are those for which the average number of photons hitting the single cell is minimal (less than 2). On the other hand, if the number of incident photons is high, the output signal (number of fired cells) will saturate according to the total number of cells in the device following the relationship:

N f = N cell (1 e N P DE N cell ) ( 2 . 1 1 )
where N f is the main number of fired cells, N cell is the number of micro-cells in the SiPM and N is the number of optical photons incident on the SiPM. If the number of detected photons is small compared to the number of cells (N PDE<<N cell ), the output is linear. This means that the density of cells must be suited to the intensity of light to be detected. The SiPM chosen for our application is composed of 3584 cells. Considering a maximum energy deposited by the gamma particles of 364 keV (main emission of 131 I ) in an CeBr 3 (Ce) with a light yield of 60 photons/keV, a light collection e ciency of 25% and a scintillation light spreading over nine SiPMs, the number of incident photons on a 3⇥3 mm 2 SiPM is around 600. According to the equation 2.11, the corresponding number of fired cells is approximately 200, taking into account a PDE of 35%. Therefore, the number of cells in our SiPMs far exceeds the maximum number of fired cells for the light levels expected in our application.

Dark count rate

The noise in SiPMs is represented by spurious output current pulses produced in the absence of light. This noise can be divided into primary noise and correlated noise. The latter will be discussed in next paragraph. The primary noise includes avalanche pulses triggered both by thermally generated carriers or carriers generated by tunneling in the high-field region. Each pair of carriers, regardless of their production mechanisms, generates an avalanche that produce exactly the same output signal as a photon.

The thermal generation rate typically depends on the quality of the epi-layer. The higher is the defect concentration, the higher is the rate of avalanche pulse generation in dark conditions. Thermal excitation can be significantly reduced by cooling the system, but the most e↵ective is to improve the silicon purity during production. Tunneling is a quantum mechanical phenomenon. Due to the influence of the electric field, electrons Chapter 2

The photodetection system have a certain probability for tunneling through the forbidden band to reach a state in the conduction band with the same energy. Unlike thermal excitation, which is possible without an external bias voltage, the tunneling e↵ect increases with the electric field and cannot be reduced by cooling. The number of dark counts per second induced by these two processes is referred to as the dark count rate (DCR). This rate increases with the number of cells, the volume of the depletion area, the number of impurities in the crystal, the temperature and the overvoltage. Under common thermal conditions around 25 C, the thermal excitation is the dominating e↵ect, while for cryogenic temperatures tunneling generation becomes dominant and the dependence on temperature reduces [START_REF] Acerbi | Understanding and simulating SiPMs[END_REF]. Our SiPMs have a mean DCR of the order of 0.5 MHz for a gain of 1.7•10 6 , a detection threshold of 0.5 photoelectrons and a temperature of 25 C [START_REF] Hamamatsu | MPPC arrays in a chip size package miniaturized through the adoption of TSV structure -S13361-3050 series[END_REF].

Correlated noise

In addition to primary noise, there are two other sources of noise in SiPMs, the afterpulsing (AP) and optical crosstalk (OC). Both AP and OC events originate from an existing current pulse (which can be either a photon event or a dark event) and for this reason they are referred as to correlated noise. The probability of having an AP or OC event strongly depends on the current density in the original avalanche and on the triggering probability P t. Large micro-cells have high gain, resulting in a high number of carriers flowing during the avalanche and therefore high probability of AP and OC. At the first order, it is therefore essential to implement SiPMs with small micro-cells in order to contain the correlated noise [START_REF] Piemonte | Development of an automatic procedure for the characterization of silicon photomultipliers[END_REF]. Afterpulsing is due to the carriers trapped in silicon defects during the avalanche multiplication that are released later during the recharge phase of the SPAD. The net e↵ect is that a new current pulse is observed on the tail of the original current pulse (Fig. 2.18(left)). The AP probability increases with the overvoltage because of the corresponding increase in gain. Optical crosstalk involves visible photons emitted during the avalanche multiplication that are re-absorbed in neighboring cells or even in the inactive region of the same cell, causing additional current pulses. Direct optical crosstalk occurs when an emitted photon reaches the active region of another cell triggering an additional avalanche at about the same time as the original avalanche. This generates pulses with amplitudes corresponding to several fired cells instead of one (Fig. 2.18(right)). Delayed optical crosstalk involves photons that are reabsorbed in the inactive regions of the SiPM. The generated electron (or hole) must then di↵use into the active region of a cell before it can trigger a new avalanche. The correlated pulse therefore has a certain time-delay (in the order of few ns) with respect to the original one (Fig. 2.18(right)). As for afterpulses, the OC probability increases, as a first approximation, in a quadratic way with the overvoltage. Its intensity is also inversely proportional to the distance between cells, which limits the maximum density of cells in an array. Recent improvements have significantly reduced the level of crosstalk and afterpulses compared to previous generations of SiPMs (Fig. 2

.18(down)).

Timing The SPAD timing resolution is defined as the time jitter between the e↵ective time of arrival of the photon at the sensor and the time when the output current pulse is

.18: Afterpulsing in SiPMs (up-left). The same microcells fires again during the recharge phase. The amplitude of the afterpulse is lower than 1 photoelectron since C d does not recharge to V bias when AP occurs. Optical crosstalks in SiPMs (up-right). One can observe a single-cell signal with an amplitude of 1 photoelectron, direct crosstalks with two cells firing at the same time and a delayed crosstalk with a second cell fired after few ns [Source [START_REF] Ng | Physics of Semiconductor Devices[END_REF]]. Pulse waveform comparison of an older Hamamatsu SiPMs array (down-left) with the one used in our application (down-right) showing huge reduction of the optical crosstalk and afterpulses probabilities [START_REF] Hamamatsu | MPPC arrays in a chip size package miniaturized through the adoption of TSV structure -S13361-3050 series[END_REF].

recorded. In SPADs, the area of active pixels and the thickness of the depletion region are very small. This results in a very fast avalanche built-up (of a few tens of ps) and therefore, a fast response to the photoelectrons. For applications that requires a high detection rate (low dead time), the most suitable SiPM has small micro-cells (low C d ) and low quenching resistance in order to reduce the time constant of the recharge phase. New challenges in the fields of time of flight PET, high energy physics, time resolved X-ray detection and spectroscopy are pushing developments to achieve single photon timing resolutions as good as 10 ps [START_REF] Lecoq | Pushing the Limits in Time-of-Flight PET Imaging[END_REF]. When coupled to a scintillation crystal, SiPMs have superior timing performances when compared to the most recent PMT-based systems [START_REF] Hutton | Advances in clinical molecular imaging instrumentation[END_REF]. Recent results show that it is possible to develop a commercial LSObased PET system with a timing resolution of 250 ps [START_REF] Casey | First human images from a next generation SiPM based PET/CT system with improved time and spatial resolution[END_REF][START_REF] Casey | A next generation SiPM based PET/CT system with improved time and spatial resolution[END_REF]. Further improvement Chapter 2

The photodetection system in timing resolution can be expected as the technology develops [START_REF] Lecoq | Pushing the Limits in Time-of-Flight PET Imaging[END_REF].

Advantages and drawbacks of SiPMs for gamma imaging a

As previously mentioned, Silicon Photomultiplier detectors provide high gain, comparable to the PMT based on vacuum technology, they have the advantage to operate at low voltage operation (between 30 and 70 V depending on the technology) and have low power consumption. They also exhibit good photon detection e ciency (20-40%) peaked in the blue or green light region depending on the device. Furthermore, due to their high gain, they do not require complex electronic readout and benefit from a very compact structure. These are all key features when optimizing a miniaturized photodetector to be coupled with a scintillator. Main drawbacks of SiPMs for gamma detection using scintillators concern the temperature stability. Indeed, its performance, including gain, PDE and thermal and correlated noises, are highly temperature dependent because the temperature directly a↵ects the breakdown voltage. However, we know that this influence can be accurately o↵set without cooling system by tuning the bias voltage as a function of the temperature variation, in order to maintain a constant overvoltage [START_REF] Hudin | Characterization and Optimization of silicon photomultipliers for the development of intraoperative beta probes[END_REF] (Sec. 2.6.2.2). Due to the high amount of scintillation light that will be collected by the SiPM, the e↵ect of the dark count rate, which can reach MHz at room temperature (much higher than the kHz DCR of conventional PMTs) could also be easily remove by adjusting the electronic threshold without any loss of detection e ciency for high-energy gamma rays used in molecular radiotherapy. Finally, the correlated noise can deteriorate the energy resolution of a gamma camera based on SIPMs. In fact, the stochastic nature of afterpulsing and optical crosstalk leads to a fluctuation in the gain of a SiPM since the number of micro-cells fired by the original avalanche can be greater than 1 and fluctuates from an event to another. This gain fluctuation, which is equivalent to a multiplication noise (excess noise factor) and increases with the overvoltage, must be minimized by taking into account the compromise with other detection parameters, such as gain and PDE (Sec. 3.3.2.1).

The SiPMs arrays from Hamamatsu (S13361-3050NE-08) introduced at the beginning of this section have been selected for our application because they provide a good compromise between gain, DCR, PDE and crosstalk probability (Fig. 2.19), while o↵ering an extremely small dead area between each pixel (200 µm wide and 400 µm for the two bisectors of the array, where the four individual arrays are joined), which allows to improve scintillation light collection. A study conducted on the S13361-3050 series confirmed the SiPM characteristics given by the manufacturer and shown that these arrays have been improved in all measured characteristics compared to the previous series (S12642-0808) [START_REF] Renschler | Characterization of Hamamatsu 64-channel TSV SiPMs[END_REF]). Table 2.5 summarizes the main features of the SiPMs used for the development of the our mobile camera.

Chapter 2 2.7. THE READOUT ELECTRONICS

The readout electronics a

The processing of SiPMs signals is carried out by a miniaturized reading electronics composed of an analog and a digital part. Analog processing is performed by eight EASIROC (Extended Analogue SI-PM ReadOut Chip) integrated circuits. The signals at the ASICs output are then processed by a combination of an external two-channel Analog-to-Digital converter (ADC) with 12-bit dynamic performance and a sampling rate of 2 Msamples/s, a programmable logic circuit (FPGA, Field-Programmable Gate Array) and a USB interface, which controls the digitization, acquisition and transfer of the data. The electronic modules were designed and produced at LAL (Laboratoire de l'Accélérateur Linéaire, Orsay) for the miniaturized camera based on SiPM arrays, called MAGICS (Miniaturized Gamma Imager for Cancer Surgery) [START_REF] Dinu | SiPM arrays and miniaturized readout electronics for compact gamma camera[END_REF].

EASIROC ASIC a

Each EASIROC ASIC consists of 32 parallel channels (two ASICs for each SiPM array), each with its own front-end electronics dedicated to the SiPMs gain adjustement and readout [START_REF] Dinu | SiPM arrays and miniaturized readout electronics for compact gamma camera[END_REF].

The diagram of a single channel of the ASIC is schematically represented in Figure 2.20. The cathode of the SiPM is connected to the power supply that provides the high voltage V HT . The signal at the anode of the SiPM is converted to a voltage by means of a 50 ⌦ input resistance. In order to finely and individually adjust the supply voltage of each SiPM, the ASIC is equipped with a 8-bit DAC (Digital to Analog Converter) connected to the anode of the SiPM, which delivers a low power input voltage V DAC ranging from 0 to 2.5 V. The voltage through the SiPM is equal to V bias = V HT -V DAC . This allows to improve the gain uniformity of the system, as will be further discussed in Section 2.7.1.2.

The analog signals are sent to two parallel processing channels equipped with a pre-amplifier, one with a low gain (LG) and the other with a high gain (HG), both adjustable over 4 bits to obtained the desired dynamic range between 160 fC and 320 pC [START_REF] Dinu | SiPM arrays and miniaturized readout electronics for compact gamma camera[END_REF]. The amplified signals are then transferred to a circuit called slow shaper, which both integrates and shapes the signal with a time constant adjustable over 8 values (3 bits) from 25 to 175 ns. The amplitude of the signals at the output of the slow shapers is proportional to the integral of the current pulse, and therefore to the charge of the event. The signal of the high gain preamplifier is also sent to a fast shaper circuit equipped with a short time constant (15 ns), which is connected to a discriminator where the signal is compared to a threshold voltage (V th ) common to all 256 channels and adjustable over a 10-bit dynamic range. If an event detected by the SiPMs exceeds the threshold voltage on at least one of the 256 channels, the slow shaper signal is put on "hold", the simultaneous recording of the 256 channels is triggered and the integrated charge of the signal is registered. The delay between the trigger and the hold signal is adjustable over 6 bits (from 0 to 160 ns with a step of 2.2 ns) depending on the shaping time of the signal. The integrated charge signal is digitized by the two-

FPGA and USB a

The front-end module is mounted on four 256-channels motherboards combining an FPGA (Altera Cyclone III), a FTDI FT2232H (USB protocol 2.0 Hi-Speed, 480 Mbit/s) for the communication with a PC and a DC/DC convertor, which delivers the high voltage to the SiPMs arrays with an accuracy of 35 mV. The FPGA allows an easy programming of all ASIC parameters. It organizes the information relative to each event and stores digital data in a bu↵er memory before to be transferred via an USB communication port (USB 2.0 High spec, 480 Mbits/s).

Chapter 2

The readout electronics The overall dimensions of the electronic module (the four Printed Circuit Boards) are 5⇥5⇥4 cm 3 (Fig. 2.21). The four PCBs can be stacked in a compact configuration, perfectly adjusted to the dimensions of SiPMs array. The power consumption of the readout electronics is 4.48 mW/channel and 155 mW/chip. The front side of the first PCB is composed of the four SiPMs arrays of the photodetection system (Sec. 2.6.2). Its back side contains passive elements (resistances, capacitances) for the bias supply of SiPMs, as well as four digital thermal sensors. The thermal sensors are connected to the FPGA in order to monitor the average temperature in real-time and thus, be able to maintain a constant overvoltage on the SiPMs by adjusting the bias voltage with the DC/DC convertor according to the temperature (Sec. 3.3.2.3).

Electronic characteristics a

In order to adapt and optimize the electronics readout performance to the detected signals, its parameters, which will be briefly described in the following paragraphs, have to be characterized and adjusted.

Trigger threshold a

The threshold voltage for triggering the hold of the slow shaper is set by a 10 bits DAC, common to the 32 channels of each ASIC. It can supply a voltage ranging from

The readout electronics Chapter 2

1.1 to 2.4 V with a step of 1.3 mV. A DC voltage of 900 mV is added at the output of the preamplifier in order to be able to register small amplitude signals. This moves the electronic noise pedestal closed to the minimal voltage of the trigger threshold. Thus, the charge distribution of the output events measured by the SiPMs array have to be firstly corrected, channel by channel, by subtracting the electronic pedestal value. In practice, this threshold have to be adjusted according to the type of measurement. During the uniformity characterization of the SiPMs array using a pulsed LED source, the need to detect low-light level signals leads to choice of a threshold value less than half of a photoelectron (Sec. 2.7.1.2). When detecting gamma ray sources coupling the SiPMs array to a scintillator, a higher threshold value can be chosen. Indeed, the interaction of gamma rays in the most common inorganic scintillators generates such a quantity of photons that the amplitude of the signal created at the output of the SiPM is much higher than the level of the dark noise. This allows the trigger value to be easily adjusted to reduce the dark count rate to low frequencies, without reducing the number of photopeak events detected.

Adjustment of the SiPMs bias voltage a

As already introduced, the supply voltage of each SiPM can be finely adjusted thanks to a 8-bits DAC connected to the SiPM anode. This feature is essential since the SiPMs performances presented in the previous section are strongly dependent on the overvoltage. The 8-bit DAC delivers an input voltage V DAC that can be tuned between 0 and 2.5 V with a resolution of less than 10 mV. The maximum standard deviation of V DAC over the 256 channels is of 104 mV for the same DAC input value.

Although the SiPMs array response is expected to be intrinsically almost uniform due to the manufacturer's selection of the SiPM to be assembled in each array, very small di↵erences in the breakdown voltage of each SiPM due to their physical structure (Fig. 2.14) may translate in significant variations in terms of gain and photon detection e ciency. The response uniformity of the array can be further improved by fine-tuning the supply voltage delivered to each SiPM in order to o↵set the di↵erences in breakdown voltage.

The uniformity of the photodetection system was evaluated by illuminating the SiPM array with a pulsed LED source ( =470 nm, T=50 ns). The LED is placed at about 25 cm from the detector. It was covered with a light di↵user in order to ensure an uniform irradiation of the SiPM array. The amount of incident light was adjusted to an intensity corresponding to around 50 photoelectrons. The time constant of the slow shaper and the preamplifier gain were set to 50 ns and 500 fF, respectively. The temperature is maintained at a constant value of 25 C by placing the photodetection module in a climatic chamber (Fig. 3.1). Figure 2.22(left) shows the average charge of the LED spectra on the 256 SiPMs as a function of the applied bias voltage before DAC correction, i.e. by applying the same DAC value to all electronic channels. The response uniformity of the array was improved iteratively by comparing the mean energy of the The readout electronics Chapter 2

Preamplifiers Gain a

The gain of the two parallel preamplifiers is proportional to the ratio between their input capacitances (1.5 and 15 pF for the low and high gain, respectively) and a compensation capacitance on 4 bits, also called feedback capacitance, ranging from 100 fF to 1.5 pF. Therefore, this theoretically makes it possible to have a gain between 1 and 15 for the LG preamplifier and between 10 and 150 for the HG preamplifier. The dynamics of the electronics thus extends by adapting the gain of the preamplifier. It has been defined for each scintillator and each radioactive source (maximum gamma ray energy) in order to benefit for the whole dynamics of the digitizer while avoiding saturation e↵ects.

Dead time a

The analog data digitalization with the 12-bits ADC, their internal transfer to the FPGA and the subsequent transfer to the computer via the USB communication port generate a minimum amount of time during which the acquisition system is unable to recorded any event, known as dead time. This dead time could strongly hamper the quantification ability of the camera in the case of detecting high amount of radioactivity such as that injected into patients during the treatment of thyroid disease (Sec. 1.3.3.3). For each scintillation event in which at least one of the ASIC channel reaches the set threshold value, the system is put on hold and record the charge collected on the 256 channels. Each channel is then coded on 3 bytes leading to a total weight of data of 787 bytes per event, which includes some additional information such as temperature sensor values and event number. The digitalization of the analogue signals (hold and internal transfer) takes around 28 µs. The data transfer through the USB (non-compressible) lasts about 28.9 µs on average (mean transfert rate of 27.2 Mb/s) and up to 45 µs, for a maximum total time of 73 µs per event. This leads to an average acquisition rate of 13.6 10 3 event/s, which is way below the maximum USB 2.0 capabilities. This counting capability is far from the initial objectives set for the clinical prototype of the mobile camera (400kCps, see Sec. 2.1). However, it is largely su cient for the feasibility prototype and for preliminary phantom studies that will use a low amount of radioactivity compared to the values administrated to patients during radioiodine treatment of thyroid disease.

The acquisition software a

The monitoring of the ASIC through the FPGA, real-time data acquisition and recording are performed using a specific software developed with the Insight Toolkit ITK, built through QtCreator using the Visual Studio C++ compiler. The electronics parameters described above (preamplifier gain, shaping time, delay, trigger level, setting of the 8-bit DAC for adjustment of the SiPMs bias voltage, ...) are transmitted to ASIC by defining the input parameters of an internal bu↵er, called slow-control. The slowcontrol is transmitted to ASIC via the FPGA, whose parameter setting are directly controlled by the acquisition software. It also allows the SiPM bias voltage to be adjusted and the temperature to be checked and recorded as a function of time.

The software, developed with the visualization Toolkit VTK, also supports realtime data visualization (Fig. 2.23). In addition to the 2D image of the interaction position of gamma rays in the scintillator, the single spectra corresponding to each SiPM, the overall pulse height spectrum and a map representing hits per pixel are available. An energy selection on the pulse height spectrum can be made to visualize only the image corresponding to the selected events. Acquisition parameters, such as acquisition time, temperatures on the boards and the number of events detected are also displayed. In addition, image display parameters can be directly chosen from the software (reconstruction algorithm, number of pixels in the image) and spatial ROIs can be easily drowned to estimate the number of events in a selected structure. However, the accurate analysis of the data is always performed o✏ine from the raw data using di↵erent programs developed in Python. When all data are displayed in real-time with the reconstruction algorithm corresponding to the iterative adjustment reconstruction method (Sec. 2.8), the acquisition rate is around 3 10 3 event/s.

Image reconstruction algorithms a

Signal processing makes it possible to reconstruct the interaction position of gamma rays in the scintillator, as well as their deposited energy, from the charge distribution measured on the SiPMs array. Many methods for obtaining spatial information from a monolithic scintillator have been presented in the past and the choice of the reconstruction algorithm is important because it can significantly influence the spatial performances of the imaging devices. In following paragraphs, various algorithms will be discussed. First, the barycenter methods, the simplest reconstruction algorithm by definition and implementation, will be presented, as well as its limitations. They will emphasize the importance of taking into account the shape of the light distribution (known as PSF, point spread function) inside the scintillation material, in order to extract accurate spatial information. This opens the door to analytical models based on PSF fitting procedures, among which the analytical model chosen in this work will be presented. These are the most commonly used procedures if an accurate estimate of the gamma interaction position is required, although they are generally time-consuming [START_REF] Li | Nonlinear least-squares modeling of 3d interaction position in a monolithic scintillator block[END_REF].

Classical Barycenter method a

The Barycenter method is the simplest reconstruction algorithm by definition and implementation. The algorithm for calculating the barycentre, also known as the Center of Gravity (CoG), follows the basic principles of image reconstruction introduced by Anger [START_REF] Anger | Scintillation Camera with Multichannel Collimators[END_REF] according to which it is possible to reconstruct the gamma ray interaction point assuming that it is the center of gravity of the collected charge distribution. For a regularly spaced m⇥n array of SiPMs, the interaction position (x, y) of an event can be obtained by means of:

(x, y) = P m i=1 P n i=1 P i,j • (x i , y j ) P m i=1 P n i=1 P i,j (2.12)
where P i,j is the amplitude of the SiPM signal at the position (x i , y j ) corresponding to the pixel of the i th row and j th column of the array. The advantage of this standard method is the short computing time, allowing a large number of events to be analyzed in real time. One of the limitations is that all SiPMs in the array are treated with the same importance, regardless of the position of the incident event. Indeed, the algorithm gives a good estimate of the interaction point in the assumption that the charge distribution is symmetrical around this point and that the influence of the light distribution tails is negligible. In the presence of long tails and light background, typical of the light distribution measured with continuous crystals, linear weighing generates a distortion of the projection that changes the average value of the distribution. In this case, the algorithm reduces the spatial linearity range. The calculation of the centroid is also influenced by the shape of the PSF over the SiPMs array and does not reconstruct the actual position of the scintillation event when applied to an asymmetric Chapter 2

Image reconstruction algorithms distribution, which occurs near the edges of the crystal. In this case, the calculation of the position is influenced by the truncation of the light distribution. To reduce the influence of the light distribution tails and the inhomogeneity of the background noise level, a threshold can be applied on the charge value registered by the SiPMs before calculating the barycenter of the charge distribution.

Other alternative barycenter methods have been developed by implementing a weighting of pixels signals before calculating the event position. For example, Pani et al. proposed to apply a quadratic weighting factor to pixel values (weighted barycenter method) [START_REF] Pani | New position arithmetic for scintillation camera based on floating weight system[END_REF]. This weighting increases the weight of the pixels corresponding to the centre of the charge distribution compared to the tails. This implies a reduction in the width of the charge distribution and a reduction in the contribution of tails. However, if used with thin organic scintillators in which light spreads over a small number of pixels, the application of a quadratic weighting factor involves the exaltation of the anodes in the vicinity of the scintillation event. As a results, this algorithm produces a strongly non-uniform response.

Analytical model fitting methods a

Due to the importance of the PSF shape in the reconstruction approach, another group of algorithms has been developed, which consists in fitting a reference function describing the scintillation light distribution on the charge distribution measured by the photodetector array. Several reference functions of the scintillation light distribution reaching the photodetector have been defined from empirical observations (Gaussian, Cauchy, Scrimber-Baker) [START_REF] Ling | Parametric positioning of a continuous crystal PET detector with depth of interaction decoding[END_REF][START_REF] Fabbri | A new iterative algorithm for pixilated and continuous scintillating crystal[END_REF] or theoretical calculations [START_REF] Li | Nonlinear least-squares modeling of 3d interaction position in a monolithic scintillator block[END_REF]. One of the most commonly used functions in the literature, and the one used during this work, is the Scrimger-Baker distribution function [START_REF] Scrimger | Investigation of Light Distribution from Scintillations in a Gamma Camera Crystal[END_REF]. For an interaction of a gamma ray in a su ciently large scintillator and coupled to a photodetector, the radial intensity of the light distribution, i.e. the light density in photon per unit area I(r) generated by the scintillation event on the photodetection plane in a polar coordinate system, can be described by approximating the definition of the solid angle [START_REF] Eanm | Internal Dosimetry Task Force Report on: Treatment Planning For Molecular Radiotherapy: Potential And Prospects[END_REF] :

I(r) = I 0 ✓ 1 + ⇣ r z 0 ⌘ 2 ◆ 3 2 (2.13)
where I 0 = I(0), z 0 is the distance between the photodetection plane and the interaction position of the gamma ray and r its radial distance. In a Cartesian coordinate system whose origin is at the center of the photodetection plane, the radial distance is given by r 2 = (x i -x 0 ) 2 + (y j -y 0 ) 2 , where (x 0 ,y 0 ) is the interaction position and (x i ,y i ) the coordinates of the pixel corresponding to the i th row and j th column of the array. The fitting process is generally performed using an iterative algorithm based on two methods: maximum-likelihood estimation (MLE) or weighted least-squares (WLS). In the first case, the parameters of the PSF are estimated by maximizing their likelihood. In the

Image reconstruction algorithms

Chapter 2 second one, the fit consists in iteratively finding the best PSFs parameters by minimizing the sum of the squares of residuals (deviations predicted from analytical of measured data). Di↵erent MLE and WLS algorithms exist and have been developed over time for estimating the interaction position of a gamma ray in a scintillator [START_REF] Ling | Parametric positioning of a continuous crystal PET detector with depth of interaction decoding[END_REF][START_REF] Fabbri | A new iterative algorithm for pixilated and continuous scintillating crystal[END_REF][START_REF] Gagnon | Maximum likelihood positioning in the scintillation camera using depth of interaction[END_REF][START_REF] Hunter | Calibration Method for ML Estimation of 3d Interaction Position in a Thick Gamma-Ray Detector[END_REF][START_REF] Lerche | Maximum likelihood based positioning and energy correction for pixelated solid state PET detectors[END_REF]. In both cases, the algorithms are based on iterative adjustments and convergence methods. In this work, the position estimation relies on the Levenberg-Marquardt non-linear least squares method, which was implemented using the scipy.optimize Python library [START_REF] Levenberg | A method for the solution of certain non-linear problems in least squares[END_REF][START_REF] Marquardt | An Algorithm for Least-Squares Estimation of Nonlinear Parameters[END_REF]. The method is based on a maximum neighborhood method, according to which the model is first expanded as a Taylor series. Then, the PSF parameters are calculated at each iteration until the di↵erence between the reference function and the experimental one is lower than a target value (the steepest-descent, or gradient, method) [START_REF] Marquardt | An Algorithm for Least-Squares Estimation of Nonlinear Parameters[END_REF]. This fitting process could be computationally expensive due to the multiple parameters allowed to vary during iterations. However, it does provide very accurate position information, which results in a good linearity response near the crystal edges (Sec. 3.1.3).

Chapter 3 Development and optimization of the mobile gamma camera a

In this Chapter we will present the results of the optimization and characterization of the first prototype of the 5⇥5 cm 2 field of view (FoV) gamma camera. The optimization was carried out using both experimental and theoretical approaches.

The experiments were focused on the choice of the best scintillator-photodetector assembly, in terms of spatial and energy performance. The CeBr 3 crystal has been chosen as the scintillation material to develop the first detection head of the camera. The advantages and drawbacks of this choice were discussed in Sec. 2.5.5. The theoretical studies were first based on the design of the high energy parallel-hole collimator, which largely determines the spatial resolution and sensitivity of the imaging device. It was performed using Monte Carlo simulations, as previously discussed in Sec. 2.4.3.2. The optimization of the collimator geometry was achieved by precisely characterizing the energy and spatial distributions of scattered and penetration events, the main factors degrading both image quality and dose quantification, as well as with their impact on the intrinsic properties of the collimator. This analyse will be described in Section 3.2. The final choice of the collimator was made by modeling the design of the complete camera placed in an anatomically realistic voxelized phantom. The intrinsic performances of the camera, spatial and energy resolutions, were set according to the previous experimental results. These results will be detailed in Section 3.2.5.

This optimization study led to the integration of the first camera prototype, which will be presented in Section 3.3 .

Performance evaluation of the scintillator pho-

todetector assembly for gamma imaging a This section is dedicated to the characterization and optimization of the intrinsic response of the detection head of the camera, in terms of spatial and energy performances. According to the bibliographic study (Sec. 2.5.5), CeBr 3 has been chosen as Scintillator-photodetector evaluation Chapter 3 the scintillator material to be coupled to the SiPM array. As the imaging performance of the gamma camera depends directly on the number of scintillation photons detected and the distribution of the light inside the scintillator, which in turn depend on the geometry of the scintillator and the nature of its optical coatings, four samples CeBr 3 have been tested: two thicknesses, 6 mm and 1 cm, and two optical coatings covering the edges of the crystal, absorbing and reflective (Fig. 2.10). Their intrinsic characteristics were compared to those of a 6 mm thick LaBr 3 sample with absorbing edges, considered to be the gold standard in terms of energy and spatial response [START_REF] Pani | Excellent pulse height uniformity response of a new LaBr3:Ce scintillation crystal for gamma ray imaging[END_REF], as discussed in Section 2.5.3. The geometrical features of the scintillators are summarized in Table 2.4.

Experimental set-up a

The scintillators were coupled to the SiPMs array by using an optical grease (BC-630 Saint Gobain) with a refractive index of 1.46. A 1.5 cm height and 5 mm thick lead shielding was put around the crystal to ensure both alignment with the photodetector and protection against gamma rays arriving on the side. The scintillator-detector assembly coupled to the electronic module were placed inside a climatic chamber (BINDER KB 115E3.1), which allows to maintain a constant temperature with an accuracy of ±0.1 C in a range between -10 and +50 C (Fig. 3.1). All measurements were performed at a temperature of 25 C. The temperature on the SiPMs is monitored by the acquisition software thanks to the four thermals sensors connected to the FPGA (Sec. Chapter 3

Collimated source

Motorized platform

Scintillator-photodetector evaluation 2.7.2). The measurements were performed by applying an optimal overvoltage of 69 V on the SiPMs array. This voltage was optimized for energy resolution performance following an experimental protocol that will be described in more details in Section 3.3.2.1. The preamplifiers gain, delay and trigger threshold of the electronics readout have been adjusted for each configuration in order to optimize their energy resolution, to reduce the dynamic saturation and avoid any dead time due to the dark count rate, as discuss in Section 2.7.1. Others parameters of the readout electronics were set according to the data presented in Section 2.7. The response uniformity correction of the SiPMs array, described in Section 2.7.1.2, was also implemented.

The spatial and energy performance has been evaluated using 1 mm sealed 57 Co and 133 Ba sources (Eckert & Ziegler Strahlen), characterized by a principal -rays emission at 122 keV (85.6%) and 356 keV (62%), respectively. The sources were mounted on a 3-axis motorized platform controlled by a computer, which allows the entire camera to be scanned with a position accuracy of 2 µm (Fig. 3.1). The spectrometric properties were investigated using a 3.3 cm thick tungsten collimator with a single 4 mm diameter hole (Fig. 3.2(A1-A2)). The intrinsic spatial performance were evaluated by using two 0.5 mm diameter single-hole tungsten collimators with a thickness of 2 cm and 3.2 cm for the 57 Co and 133 Ba sources, respectively (Fig. 3.2(B1-B2) and (C1-C2)).

Collimator intrinsic proper/es : Insight

Measurements of the intrinsic imaging performances a

The intrinsic imaging performance was evaluated in terms of energy resolution, intrinsic spatial linearity, spatial resolution and uniformity. The charge distribution of each detected event provided by the SiPMs array is firstly corrected, channel by channel, by subtracting the electronic pedestal value (Sec. 2.7.1.

1). The position of

Scintillator-photodetector evaluation Chapter 3 the interacting gamma-ray in the crystal is then reconstructed with an iterative fitting method based on the Scrimger-Becker analytical model of the light distribution and a Levenberg-Marquardt non-linear least square algorithm (Sec. 2.8.2). The energy deposited by the gamma ray is assumed to be proportional to the sum of the charge intensity in the 256 pixels. Mean values and standard deviations of the performance are measured in the Useful Field of View (UFOV), corresponding to the complete FOV of the camera, in the Central FoV (CFOV, 75% of the UFOV), defined according to NEMA standards [313], and in the complementary region of the CFOV, the CFOV , that has been defined in order to highlight the impact of crystals edges on the intrinsic performances.

Energy response a

For each configuration, the linearity of the energy response was first measured by using four isotopes : 241 Am (59 keV gamma-rays emission), 57 Co (122 keV) and 133 Ba (80 keV and 356 keV). The point sources were collimated with the 4-mm diameter single-hole collimator (Fig. 3.2A) placed in the central area of the detector. A gaussian fit was applied on the full-energy peaks of the energy spectra to determine the photopeak location in channel numbers. Double gaussian fits were used for the 57 Co and 133 Ba spectra in order to account for the influence of the secondary emissions at 136 keV (10.68%) and 383 keV(8.94%), respectively. The energy linearity is evaluated from a linear fit on the photopeak positions as a function of the gamma-ray energy. The slope of the fit is also used as the keV per channel calibration factor for the energy resolution calculation. The energy resolution is evaluated using the sources collimated with the 4-mm diameter single-hole mounted on the 3-axis motorized platform (Fig. 3.1) and moved over the whole FoV of the scintillator performing a 10⇥10 scanning array with 5 mm steps. Only reconstructed events inside a 5 mm diameter circular Region of Interest (ROI) centered on the spot are considered. At each scan position, several thousand of events were recorded (around 3000 events per spot). The FWHM energy resolution and photopeak location are calculated from a double gaussian fit performed on the energy spectra. The energy resolution of the 100 spot are represented in an energy resolution map. In addition to this map, a pulse height response map or intensity map, defined for each position as the relative displacement of the photopeak with respect to the highest position of the photopeak in the field of view, was also computed. The inverse of this intensity map is used to create a 10⇥10 correction matrix to adjust local non-uniformities in the energy response of the camera related to spatial variations in the scintillation light collection. Depending on its interaction position, the energy of each event is multiplied by the corresponding correction factor. This procedure was applied on the data obtained from a flood-field irradiation that will be described in the following section. A corrected total spectrum is thus obtained, from which the overall system energy resolution is calculated.

Chapter 3

Scintillator-photodetector evaluation

Uniformity a

The intrinsic uniformity is a measure of the response of a collimatorless imaging system to a uniform radiation flux. It has been estimated from the flood-field irradiations of the camera with the 133 Ba source placed on the central axis of the detector at 25 cm of distance and images were. The flood flied images were analyzed following the NEMA protocol images treatment [313]. An energy selection of twice the energy resolution is applied around the main peak of 133 Ba . For example, the energy window is 329-383 keV (15% of the peak position) for the 6 mm thick CeBr 3 scintillator with reflective coatings. The correction on the energy response has been also applied (Sec. 3.1.2.1). For each configuration, a 64⇥64 pixels image with an average number of detected events per pixel greater than 8000 is produced and smoothed with a 3⇥3 kernel gaussian filter. Both integral and di↵erential uniformity are calculated. Typical measures of integral uniformity are done by means of the following equation, as described in the NEMA protocol [313]:

IU = 100% ⇥ C max C min C max + C min (3.1)
where C max and C min are respectively the maximum and minimum number of counts per pixel across the whole image. However, this method is considered to be not robust for small FOV gamma cameras when outlying pixel values are present [START_REF] Bhatia | A scheme for assessing the performance characteristics of small field-of-view gamma cameras[END_REF]. In these cases, it is suggested to report the ratio of the standard deviation of counts per pixel to the mean counts, as a measure of the integral uniformity (IU). This definition was finally used as a measure of the integral uniformity (IU). The di↵erential uniformity (DU) is calculated from ten integral uniformity values for each pixel, corresponding to the IU in the 2, 4, 6, 8 and 10 nearest neighbor pixels in each pixel's row and column. These values are collected and used to produce a histogram for di↵erential uniformities. The mean of this distribution is used as a measure of DU [START_REF] Bhatia | A scheme for assessing the performance characteristics of small field-of-view gamma cameras[END_REF].

Intrinsic spatial performances a

The intrinsic spatial performance was estimated with the 57 Co and 133 Ba sources collimated by the 0.5 mm diameter tungsten collimator fixed on the motorized platforms. A 17⇥17 scan of the scintillators with 3 mm steps was performed. In order to have more spatial information near the edges of the crystal, a scan lines is added between the two outermost spots of the main scanning array. An energy selection of twice the energy resolution is applied around the main peaks of 57 Co and 133 Ba . For exemple, the energy window is 107-137 keV and 329-383 keV (15% of the peak position) for the 6mm thick CeBr 3 scintillator with reflective coatings. For each scan, several thousand of events were recorded (at least 1500 events per spot). A two-dimension gaussian fit was then performed on the two-dimension histogram of the reconstructed positions in order to accurately evaluate the x i,rec and y j,rec coordinates of the center of the distribution for each scan position (i, j), as well as the two associated standard deviations of the point spread function x i and y j . In order to get rid of most of the Scintillator-photodetector evaluation Chapter 3 internal radioactive background of the LaBr 3 scintillator and noise coming from natural radioactivity and cosmic rays, a spatial cut was applied, keeping only events localized within a 3 mm diameter ROI around the maximum bin of the histogram.

The horizontal local spatial linearity L x i corresponding to the spot (i, j) with an horizontal mechanical position x i,mec , is calculated using its reconstructed position x i,rec and its adjacent spots' mechanical and reconstructed positions, x i±1,mec and x i±1,rec respectively, according to the following formula :

L x i = 1 2 ( x i,rec x i 1,rec x i,mec x i 1,mec + x i+1,rec x i,rec x i+1,mec x i,mec ) ( 3 . 2 )
A similar formula is applied to obtain the vertical local linearity L y j value. A local linearity equal to 1 illustrates a linear behavior of the reconstruction algorithm to identify the position of the center of the light distribution in the crystal after gamma interaction. Any variation from the unity reflects a local distortion that impacts the spatial resolution. The spatial resolution (SR) along the x direction for each scan position (x i,mec , y j,mec ), SR i,x , is defined as the ratio between the FWHM of the point spread function along the x direction and the horizontal local L x i :

SR i,x = 2.355 ⇥ x i L x i (3.3)
The spatial resolution along the y direction, SR j,y , is obtained in the same way. The intrinsic spatial resolution is calculated as the average value of SR i,x and SR j,y , which is then deconvolved by the diameter of the collimator hole.

For each line (x direction) and column (y direction), a linear fit was performed on the mechanical position -reconstructed position curve obtained with the 13 internal spots, in order to estimate a best-fit curve (Fig. 3.12). Indeed, being at least 7 mm far from the border, these spots are supposed to be free from border e↵ects which could impact their accurate position reconstruction. These best-fits allow to obtain the horizontal and vertical local intrinsic spatial distortion (LISD x and LISD y ), calculated as the horizontal (vertical) distance, in mm, between the spot's reconstructed position x i,rec (y i,rec ) and the best-fit curve. A map of the local intrinsic spatial distortion is also performed in which the direction of the local distortion for each scan position (i, j) is given by a vector (LISD x , LISD y ) and its intensity by the norm of this vector. The maximum value of LISD x and LISD y are identified as the absolute linearity according to NEMA standards [313]. These values are reported in both the UFOV and CFOV.

Investigation of the light distribution a

The light distribution inside the scintillators have been investigated by finely irradiating two SiPMs of the array (one at the center and one close to the edge and the central axis of the detector) with the 0.5 mm collimated 133 Ba source mounted on the 3-axis motorized platform. In order to cover the entire width of a pixel, 33 irradiation spots, 100 µm far from each other, were performed on both pixels. The light distribution has been reconstructed by recording the charge measured by each SiPM of the central line of the photodetector for each irradiation spot and by representing this charge as a function of the distance between the irradiation point and the center position of the considered SiPM.

Results

a

The intrinsic performance of the five crystals investigated following the experimental methods described in the previous section is going to be presented. A final overview of the results is also given in Section 3.1.3.5. To facilitate the following discussions, the five tested scintillators will be identified with the abbreviations summarized in Table 2.4.

Spectrometric performances a

In order to qualitatively compare the relative response of the five samples in terms of light yield, the energy spectra of a single central collimated 133 Ba spot obtained using the same parameters of the electronics readout (preamplifier gain and trigger threshold) are shown in peak position measured with C6R suggests that its scintillation performance, in term of LY and light collection, are close to the one of the absorbing LaBr 3 . When the thickness of the crystal increases from 6 mm to 1 cm, the amount of light collected decreases. For reflective coatings, the small di↵erence observed (9.8%) may be due to both self-absorption of the scintillation light and variations in the quality of scintillator production. On the other hand, the much higher di↵erence observed with absorbing coatings (22%) is obviously due to the increase in light absorption by the edges of the crystal. A slightly visible peak at 59 keV (c) represents the characteristics X-rays of the tungsten collimator. Paying attention on the LaBr 3 spectrum (red line in Fig. 3.4), one can notice the shoulder on the right side of the 122 keV peak, which corresponds to the 136 keV emission that is not resolved by the other scintillators. Moreover, the Compton edge peak at 39.4 keV is also visible. The calibrated 133 Ba spectra (Fig. 3.4(bottom)) show the principal emission peak at 356 keV (62%)(the 383 keV is not resolved, 8.94%) (a), 302 keV (18.34%) (b) and 276 keV (7.16%) (c), the 356 keV Compton edge at 207 keV (d), the backscatter peak at 149 keV (e) and the superimposition of the 133 Ba emission at 81 keV and the characteristics lead X-rays (f). The ER values in the UFOV, CFOV and CFOV are summarized in Table A.1. The spatial maps of the energy resolution and intensity measured with 57 Co and 133 Ba are shown in Figures 3.5 and 3.6, respectively. LaBr 3 shows the best ER at values both 122 keV and 356 keV, with a relative improvement compared to the C6R sample of 10% and 15% at 122 keV and of 8.3% and 6% at 356 keV in the UFOV and CFOV, respectively. The higher degradation of the energy resolution of CeBr 3 at low energy suggests that it is characterized by a higher non-proportional response compared to LaBr 3 , as reported in literature [START_REF] Alekhin | Improvement of gamma-ray energy resolution of LaBr3:Ce3+ scintillation detectors by Sr2+ and Ca2+ co-doping[END_REF]. The C6R sample shows the best spectrometric performance among the four CeBr 3 samples, with an ER of (7.86±0.03)% at 356 keV in the CFOV. In this region, the ER values are improved by 27%, 10% and 33% at 122 keV and by 36%, 1.2% and 40% at 356 keV compared to the C6A, C10R and ). Indeed, it significantly reduces the degradation of the mean ER observed at the periphery of the scintillators with absorbent edges. The relative decrease in energy resolution between CFOV and CFOV is 9% and 5% for the C6R and C10R scintillators at 356 keV, respectively, but more than 25% with absorbing edges. This is correlated with the loss of light measured as the source approaches the absorbent edges, which can vary from -27% to -36% depending on the thickness and the gamma ray energy (Fig. 3.5(right) and 3.6(right)). For the reflective edges, the light collection remains almost constant (decrease of less than 10.5% regardless of gamma ray energy and crystal thickness). As expected, the ER is generally degraded when the thickness of the scintillator increases, due to a less e cient light collection. This is remarkable at 122 keV, because of the higher interaction position of gamma rays within the crystals (10% and 11% reduction in ER at 122 keV for the reflective and Chapter 3

Scintillator-photodetector evaluation absorbing samples, respectively), while at 356 keV no remarkable di↵erences could be highlighted, especially for the reflective samples (1.2% and 6%). For the latter, the higher amount of reflections near the edges when the thicker walls of C10R are considered results in a lower loss of light collection than C6R (Fig. 3.6) and therefore, a slight increased in ER near the edges (8.67% and 8.41% in the CFOV at 356 keV, respectively). We did not found a clear explanation of the huge energy resolution loss close to the edges of the C6A sample when the 133 Ba source was used, clearly visible on its energy resolution maps (Fig. 3.6). This loss is indeed higher than the one visible when the 57 Co source was used (decrease of more than 50% between the CFOV and the CFOV for the 133 Ba source, compared to a loss of 32% for the 57 Co ), which suggests the presence of an e↵ect not related to the intrinsic properties of the crystal. Figure 3.7 shows a representative 133 Ba spectrum in the UFOV, CFOV and CFOV obtained from the flood field irradiation of the C6R scintillator, before and after applying the energy correction matrix. As expected, the energy correction mainly impacts the overall position and width of the full-energy peak near the edges of the crystal (Fig. 3.7(right)) and results in a significant improvement of the overall ER compared to the mean ER in the UFOV from (8.14±0.03)% to (7.68±0.01)%. Without any correction of the energy response uniformity, the overall ER is (11.17±0.05)% in the UFOV. This influence is obviously even more evident when the energy response is less uniform, such as with the two CeBr 3 absorbing samples, which show a relative improvement of the overall ER of 30% and 20% compared to the mean ER of C6A and C10A, respectively.

Light distribution

a Some representative results of the light distribution reconstruction (light PSF) are shown in Figure 3.8 for the two 1 cm thick samples. The Scrimger and Becker light distribution function used by the adjustment reconstruction method is superimposed to the experimental profiles (Sec. 2.8.2). The intensity of the PSF tails of the reflective samples are higher than the one of the absorbing samples, both at the center and at the edges of the crystal, evidence of the increase and scatter of light reaching the photodetector due of reflections. The analytical light distribution function allows the central position of the PSF for both samples to be reconstructed when it is positioned in the central area of the FoV. A good linearity are thus expected. As the interaction approaches the periphery, the absorbent edges cut the PSF while ensuring that the peak is properly positioned. Very close to the edge, the light collected by the secondto-last pixel is the same as that collected by the lateral pixel, indicating large amounts of reflection near the edges. Therefore, the reconstruction of the PSF center is harder for the algorithm, which causes the compression of the image (Figs. 3.9, 3.10 and 3.11). This e↵ect is not or less visible with the absorbent edges and increase with the crystal thickness. Moreover, although good peak positioning can be performed, the algorithm is unable to describe the actual shape of the PSF when reflective edges are involved, both at the center and at the edges of the detector. These results highlight the need to adapt the adjustment function parameter to the di↵erent crystalline coatings both to The results that we are going to discuss have been obtained with the same Scrimger and Becker function for all samples. Indeed, optimization processes could be really time consuming and have not been performed not being the principal purpose of this work. Further development are whatever potentially attractive to increase the intrinsic feature of reflecting samples.

Uniformity a

The flood field images obtained with the 133 Ba source are shown in Figure 3.9. One can first notice that the LaBr 3 is characterized by a degradation of its response uniformity in two regions, probably due to crystal aging (degradation of the hygroscopic scintillator properties by humidity). When the amount of light collected is low (e.g. for the two absorbing samples), macroscopic non-uniformities start to be visible, due to the e↵ect of the trigger threshold on event detection (one can see the shape of the four SiPMs arrays associated with di↵erent ASICs and therefore with di↵erent trigger threshold). The same e↵ect is visible when using a 57 Co source with thick samples, which leads to a low light intensity detected per pixel, as can be qualitatively seen on the 17⇥17 scanning images (Fig. 3.10). The light distribution produced by samples with absorbing edges is also narrower and has lower tails (Fig. 3.8). Under these conditions, the light is spread over a small number of SiPMs and positioning errors lead Chapter 3

Scintillator-photodetector evaluation to the appearance of the photodetector array pattern in the flood images (this e↵ect is obviously more visible with the 6 mm thick scintillator).

The integral and di↵erential uniformities in the UFOV and CFOV are summarized in Table 3.2 for both 57 Co and 133 Ba flood field irradiation. During quality control of commercial gamma cameras, integral uniformity has typical values from 2% to 5% in the CFOV [START_REF] Cherry | Physics in Nuclear Medicine[END_REF]. Although the IU was measured using a slightly di↵erent approach than the one commonly used (Sec. 3.1.2.2), this reference will be used. The IU values measured in the CFOV are less than 5% at 356 keV for the L6A, C6R and C10R samples. As previously observed on the flood images (Fig. 3.9), the absorbing samples exhibit lower uniformity in the CFOV due to the decrease in the collected light and its narrower spatial distribution. The significant degradation of IU in the UFOV is assumed to be related to the spatial compression of the image near the edges, which causes an artificial accumulation of the measured activity. Therefore, the reflective samples C6R and C10R, which are characterized by higher image compression near the edges (Fig. 3.10 and 3.11), exhibit high IU values in the UFOV. Even having absorbing edges, the C6A shows a really high value of IU in the UFOV. This is probably due to an image compression due to a non optimized crystal edges manifacturing. For di↵erential uniformity, a value of less than 3% in the CFOV is usually considered as reference during gamma camera quality controls [316]. The non uniformities previously highlighted for the absorbing samples are translated into DU values higher that 3%. As expected, the lowest DU value in the CFOV is obtained with the C10R, which produces the broadest light distribution with the highest tails. The small di↵erence between IU and DU observed for L6A and the reflecting samples demonstrates that there is no systematic spatial variation across the field of view, which is not the case for the absorbing samples (Fig. 3.9).

Spatial properties a

A summary of the spatial performance of the di↵erent scintillators is presented in Table A.2. The images of the 17⇥17 scans are shown in Figure 3.10 and 3.11 for the 57 Co and 133 Ba sources, respectively. Figures 3.13 to 3.16 represent the intrinsic spatial performance in the form of maps. Each box on the map corresponds to the average of the spatial resolution or local intrinsic spatial distortion measured for the scan positions associated with that region (3⇥3 mm 2 ). Among the 0.6 cm thick samples, LaBr 3 exhibits the best SR ((0.63±0.09) mm at 356 keV in the UFOV), largely due to its higher light output, but also to a narrower light distribution, produced by a finer light guide (Fig.s 3.13 and 3.14). Moreover, the reduced light spread close to the absorbing edges of the crystal improves the image reconstruction, allowing to obtain sub-millimeter absolute linearities in the UFOV (Tab. A.2). For the same reasons, the narrower light distributions produced by the two 0.6 cm thick CeBr 3 compared to those of 1 cm allow to obtain a sub-millimeter intrinsic spatial resolution at 356 keV in the whole field of view ((0.74±0.28) mm in the UFOV for C6R) (Tab. A.2 and Fig. 3.14). These values increase above 1 mm at 122 keV ((1.17±0.38) mm in the UFOV for C6R), due to the lower amount of scintillating light collected on the SiPM array and the shallower gamma interaction in the scintillator, which results in a broadening of the light distribution (Tab. A.2 and Fig. 3.13). The C6R and C6A samples exhibit almost comparable spatial resolution at 356 keV, but C6R shows an increased SR in the CFOV (+13%) due to its higher light collection (Fig.

Chapter 3

Scintillator-photodetector evaluation 3.4), while C6A exhibits a better SR in the CFOV (+5.5%) thanks to better spatial linearity close to the edges (Fig. 3.12). If the spatial resolution is degraded with 1 cm thick scintillators, it remains very high at 356 keV as can be seen qualitatively in Figure 3.11 and very close to one millimeter in the CFOV (Tab. A.2 and Fig. 3.14).

All scintillators have a comparable and very low LISD in the CFOV (Tab. A.2 and Figs. 3.12, 3.15 and 3.16), which underlines the very good ability of the reconstruction algorithm to identify the position (x,y) of the light distribution center independently of the scintillation light spread on the photodetector (Fig. 3.8). As expected, the reflective samples C6R and C10R show a greater deterioration in spatial performance/LISD near the edges compared to the absorbing samples (Tab. A.2 and Figs. 3.12, 3.15 and 3.16), because the light distribution can no longer be modeled by a Scrimger-Becker function (Fig. 3.8). This deformation of the light distribution is even stronger for 1cm samples at low energy due to the cumulative e↵ects of the shallow interaction of the gamma rays and the higher thickness of the crystal, which increase the spread of light and 3.4 summarizes the most important performance presented in the previous paragraphs: the overall system ER, the intrinsic spatial resolution SR and the local intrinsic spatial distortion LISD in the CFOV and UFOV at 356 keV. One can immediately notice that the 6 mm thick CeBr 3 with reflective edges o↵ers the best compromise in terms of ER (7.68±0.01)%, sub-millimetric SR ((0.74±0.28) mm in the UFOV) and very low spatial distortion ((0.15±0.10) mm in the UFOV) thanks to its high light yield and light collection (Fig. A.1). The high spectrometric performances achieved with the 1 cm thick reflecting CeBr 3 makes it also a good potential candidate that could provide a 46% increase in photopeak e ciency compared to the 0.6 mm thick scintillator, thus o↵ering the possibility of shorter acquisition times or an increase in counting statistics (qualitatively visible in Fig. A.1). However, it is first necessary to improve its spatial performance close to the edges, which reduces the e↵ective useful FoV due to compression e↵ects (Figs. 3.11 and 3.14). This could be performed by optimizing the image reconstruction based on fitting methods with more e cient analytical models of light distribution to take into account significant reflections at the edges (Fig. 3.8) or by using more sophisticated reconstruction methods, such as neural networks.

Compared to the C6A sample, the 6 mm thick reflective CeBr 3 has very similar spatial performance, but the reflective surface treatment leads to much better overall and local ER and excellent pulse height uniformity on the whole field of view, resulting in an increase in Compton rejection capability, contrast and counting uniformity of the images. For these reasons and in contrast to the common trend to choose a crystal with absorbing edges if high spatial performance is required (Sec. 2.5.4.1), the 6 mm thick reflective CeBr 3 has been chosen as the scintillator to be integrated in our first 5⇥5cm 2 prototype.

COLLIMATOR OPTIMISATION

Optimization of the high-energy collimator by

Monte Carlo simulation a Monte Carlo simulation is a fundamental tool for modeling the behaviour of a gamma camera and in particular its collimator in order to optimize its performance and improve its quantification capabilities [START_REF] Zaidi | Computational Anthropomorphic Models of the Human Anatomy: The Path to Realistic Monte Carlo Modeling in Radiological Sciences[END_REF][START_REF] Lazaro | Validation of the GATE Monte Carlo simulation platform for modelling a CsI(Tl) scintillation camera dedicated to small-animal imaging[END_REF]. Indeed, the impact of gamma penetration and scatter, the main factors degrading both image quality and dose quantification, are governed by probability distributions laws that are hardly predictable by the simple conventional parallel-hole analytical models commonly used for collimators optimization. In this section, we will focus our attention on the results obtained during the fine optimization of the high-energy parallel-hole tungsten collimator introduced in Section 2.4. At first, the di↵erent families of events participating in the formation of the image will be presented. Then, the intrinsic properties of the collimator are discussed and analyzed as a function of its geometry and of the di↵erent types of events (Sec. 3.2.3). Five collimator configurations with complementary properties in terms of e ciency and spatial resolution were selected following the optimization phase that will be presented in Section 3.2.4. Finally, these configurations were evaluated in a more realistic clinical context simulating thyroid treatment with 131 I in order to choose the final geometry of the collimator (Sec. 3.2.5).

Simulation insight a

In order to estimate the collimators intrinsic properties and to study the spatial distribution of scattered and penetrated events, the geometry of various hexagonal shaped parallel-hole collimators made of tungsten was modeled using GATE Monte-Carlo simulations (v.8.0 [START_REF]GATE: Geant4 Application for Emission Tomography: a simulation toolkit for PET and SPECT, OpenGATE Collaboration[END_REF]). They were coupled to a volume representing the scintillator (Fig. 3.17). In the first part of the simulation study, the scintillator volume is not sensitive. The exhaustive simulations description is given in Section 2.4.3.2.

The design of the collimators was investigated with a 131 I line source placed in air at a source-to-collimator distance b of 5 cm. The line source was tilted of 24 with respect to the X axis of the collimator in order to provide an oversampling of the line profiles with di↵erent source-collimator holes relative positions (Sec. 2.4.3.1). The energy and spatial distributions of the events were accurately identified thanks to the use of GATE PhaceSpace actors. Two actors are used, one to record events arriving at the collimator input surface and the other to record events leaving the collimator and arriving at the scintillator input surface. For each event, energy, position, directions along the three axes and an identification number (Event ID) are registered. The event ID is used to match the information on each event provided by the two actors. The energy of each event is blurred with a gaussian function, whose standard deviation is set in order to obtained a 2.5 % energy resolution at 364 keV. The objective being to study the intrinsic properties of the collimator, we have therefore chosen, at first order, to simulate a camera with very high spectrometric performance. Among all events arriving and passing through the collimator, only those with an energy belonging to a 5 % energy window (2⇥FWHM) around the 364 keV emission of 131 I are selected (355 -373 keV). The direction along the three axes allows events to be separated into two categories:

• non-scattered events, which are events that do not interact within the collimator and thus maintain their direction unaltered

• scattered events, which are events that changed their direction before arriving on the scintillator after interacting in the collimator.

Chapter 3

The positions when crossing the two actors combined with the geometrical structure of each collimator configuration, allows events to be subdivided into two additional categories:

• geometric events, which are events entering the collimator through one of its unit, defined as the hexagonal surface centered on a hole and whose width is equal to that of a hole plus a septa, and emerge from it by the same unit.

• non-geometric events, which are events entering into one unit of the collimator and exiting through another unit, i.e. crossing at least half of a septa.

By combining the direction and the position information, four families of events are finally defined:

1. the non-scattered geometric events (E1) 2. the non-scattered and non-geometric events (E2), also called penetrated events or simply penetration 3. the scattered geometric events (E3a)

4. the scattered non-geometric events (E3b)

A simplified schema illustrating these four families of events is shown in Figure A.2.

Detection criteria for collimator optimization

a Each collimator geometry is evaluated in terms of its spatial resolution (SR), e ciency (E), useful e ciency (uE) and e↵ective septal penetration (eSP). The collimator spatial resolution (SR) was calculated from a double Gaussian fit adjusted on the spatial profile obtained from the sum of vertical profiles (0.04 mm) each composed by the events emitted by 0.04 mm small sources along the line source. The profiles were then translated to account for the angular o↵set. The central Gaussian models the peak events and the second one models the ones present in the tails of the spatial profil. The collimator resolution (FWHM) is obtained from the standard deviation of the central Gaussian. It is related to the analytical expression R c of the collimator resolution (Sec. 2.4.3.1) by the equation [START_REF] Weinmann | Design of optimal collimation for dedicated molecular breast imaging systems[END_REF]:

F W HM = 2 p 2ln2 = p 2ln2 R c (3.4)
Figure 3.19 shows an example of a spatial profile together with the contributions of geometric (E1+E3a) and non-geometric events (E2+E3b). As can be qualitatively noticed, by selecting the events inside a 2 ROI around the spatial peak profile, all the geometric events are included together with a small part of the non-geometric events. This non-geometric contribution, which will be described in more detail in Section 3.2.3.1, does not degrade image quality and even contributes to increased e ciency. Events within this 2 wide ROI can therefore be considered useful for image formation.

The collimator e ciency E is defined as the ratio between the total events detected in the energy window considered and the total 364 keV events emitted by the source (Sec. 2.4). According to this definition, a geometric configuration of the collimator characterized by a high percentage of non-geometric events (E2+E3b) located outside the source image would show an artificially high e ciency. To take into account the "usefulness" of events in image formation, we have therefore introduced the useful e ciency (uE), defined as the ratio of all events detected within a ROI of 2 around the line source image, divided by the total number of 364 keV events emitted by the source.

For the same reason and in order to quantify the real e↵ect of both non-geometric scatter and penetration on image quality, the ratio between all the non-geometric events outside a ROI of 2 around the line source image and the total events detected is calculated and named e↵ective septal penetration (eSP). A summary of the intrinsic properties of the collimators evaluated during the optimization study is given in Table 3.5. The 2D spatial distribution and corresponding profiles of the four families of detected events are shown in Figure 3.20 for a 3.5 cm long collimator with 0.225 cm diameter hole and 0.0875 cm thick septa. Following the definitions given in Section 3.2.1, non-scattered geometric events E1 and non-scattered non-geometric events E2, i.e penetration, are characterized by an unchanged energy registered on the two actors. They are both pure 364 keV events characterized by di↵erent spatial distribution on the image. As shown in Figure 3.20, the non-scattered geometric events E1 are distributed in accordance with the position of the source, they only contribute to the main peak of the profile and thus, represent the events filtered by an ideal parallel-hole collimator. The penetrated events E2 are rather distributed over the entire field of view and contribute mainly to profiles tails. The two families of scattered events (E3a, E3b) are primarily characterized by changes in the direction of incident gamma rays. Being selected inside the 364 keV energy window, they could arise from Rayleigh scattering of 364 keV gamma rays, from Compton scattering of 364 keV photons with small scattering angles or from Compton scattering of 131 I high energy gamma rays (636 keV and 722 keV) (see Sec. 2.2). As shown in Figure 3.20, the geometric scattered events E3a participate to the increase of the main peak, while the scattered non-geometric events E3b are distributed over the entire field of view. As expected, all non-geometric events (E2+E3b) are expected to degrade image quality and contrast. The total image and corresponding spatial profiles, showing the four families of detected events, is presented in Figure 3. 21(top). Two other geometric configurations of the collimator are shown: a 3.5 cm long collimator with 0.225 cm diameter holes and 0.2 cm septa (Fig. 3.21(center)) and a 6.5 cm long collimator with 0.225 cm diameter holes and 0.0875 cm thick septa (Fig. 3.21(bottom)). As can be qualitatively noticed, the relative percentages of geometric, non-geometric, scattered and non-scattered events vary significantly with the geometry of the collimator, impacting the intrinsic properties of the collimator in di↵erent ways that are hardly predictable with analytical models (see Sec. 2.4.3.1). This in-depth comparison is at the heart of the results that will be presented in next Figure 3.22 shows the energy distribution of detected events around the 364 keV photopeak of 131 I for eight geometric configurations of the collimator : two lengths, 3.5 and 6.5 cm, two hole diameters, 0.075 and 0.225 cm, and two thicknesses of septa, 0.875 and 0.2 cm. In each spectra, the contribution of the four families is highlighted. Several trends could be qualitatively extracted from these spectra allowing a better understanding of the upcoming results.

Collimator optimisation by

As expected, the primary spectrum corresponding to the non-scattered geometric events E1 and the penetration spectrum E2 are similar in shape and display a photopeak at 364 keV. However, E1 events remain the main contributors to the photopeak, except when large holes and thin septa are used. Under these conditions, non-geometric contributions (E2+E3b) increase strongly and even become dominant (Fig. 3.22c-g). For fixed d and t, the relative contribution of E1 is always higher in longer collimators (Fig. 3. 22a-e, b-f andd-h). For fixed diameter, it increases when the septa increases, for both long and short collimators. This increase is negligible for small hole diameters for which the non-geometric contribution E2+E3b (and in a complementary way the geometric contribution) is more defined by the large number of septa to be crossed for an equivalent length of the collimator than by the thickness of a single septa (Fig.

3.22a-b and e-f) (see Sec. 3.2.3.2).

The relative contribution of scattered non-geometric events E3b is closely related to the penetration contribution E2 and therefore increases as the length of the collimator, the diameter of the holes or the thickness of the septa decrease (Fig. 3. 22a-e, b-f, d-h andc-g). The scattered non-geometric spectrum is composed of two components : a relatively flat contribution across the photopeak window corresponding to the scattering of 637-and 723-keV photons in the collimator and the 364 keV scatter component which peaks just below the photopeak energy and extend well below (the extension above the energy of the primary 364 keV photon is due to the finite energy resolution of the detector). The high-energy component is dominant when the overall penetration contribution is low, i.e. when the combination of a long collimator, small holes or thick septa reduces the geometric opening of the collimator and stops the majority of scattered non-geometric events from 364 keV photons. According to Figure 3.22, we can also note that the contribution of 284 keV photons to events in the photopeak window is negligible for the considered energy resolution. In short collimators, the E3b contribution is higher than the E2 contribution when small diameter are used, independently of the septa (Fig. 3.22a-b). As will be shown further (Sec. 3.2.3.2), this behavior is only found for really small diameters. In general, for diameters bigger than 0.075 cm, the penetration contribution is always higher than the scatter one for all septa (Fig. 3.22c-d). The scatter geometric component E3a is mainly composed of 364 keV events that scatter within a hole by Rayleigh or Compton interaction (with small scattering angles).

Chapter 3

Collimator optimisation by Monte Carlo simulation

Impact of collimator geometric parameters a) E↵ective septal penetration a

The e↵ective septal penetration eSP has been calculated from the penetration events (E2) and the scattered non-geometric events (E3b), as described in Section 3.2.2. Figure 3.23a shows the total eSP as a function of the septa thickness for all diameters and for three collimator lengths of 3.5 cm, 5 cm, and 6.5 cm. In addition, the relative contributions of the E2 and E3b components to the total number of detected events, renamed as eSP-E2 and eSP-E3b, are plotted in Figure 3.23b and 3.23c, respectively. Finally, Figure 3.23d and 3.23e show the individual relative contribution of E2 and E3b in the total eSP, respectively.

The eSP decreases with increasing septa thickness for all collimator lengths (Fig. 3.23). Obviously, as the thickness of the septa increases, and more generally as the geometric opening of the collimator decreases (small hole diameter and long collimator), the amount of material that a gamma rays finds along its path to the detector increases, thus enhancing both its absorption and probability to scatter. However, as the total probability of attenuation increases faster than the probability of Compton scattering, the relative contribution of scatter increases with respect to penetration when the septa thickness increases (Fig. 3.23d-e). When long collimators are used, Compton reabsorption increase and a kind of balance is achieved between the relative contribution of penetration and scatter (Fig. 3.23d-e).

For a 3.5 cm long collimator with 2 mm diameter holes, the penetration and scatter represent more than 64 % and 15.5 % of the events in the photopeak for 1 mm and 2 mm thick septa, respectively (Fig. 3.23a). More than 90 % are associated with collimator penetration and scatter by the low-energy gamma ray emission at 364 keV (Fig. 3.24). These levels of penetration and di↵usion are much higher than what can be estimated from the attenuation along the path length w, corresponding to the shortest path a gamma ray can cross through a single septa (Sec. 2.4.3.1). For example, the estimated penetration from the calculation of the attenuation along w is 4.7 % for a 3.5 cm long collimator with 2 mm diameter holes and 1 mm thick septa, while the simulated value is more than 44 %. This demonstrates once again that the simplistic description of the penetration used for the analytical models significantly underestimates its contribution for high-energy gamma ray imaging.

For small hole diameters, especially for long collimators, the eSP is no more a↵ected by the increase of the septa thickness and maintains a constant value whose intensity decreases with the length of the collimator (4 % and 2 % for a length of 5 cm and 6.5 cm, respectively). The same plateau is reached for larger diameters, but for thicker septa (Fig. 3.23a). One can see that the slow evolution of eSP towards this plateau observed for thin collimators with small hole diameters is constrained by the scatter component, which reaches a minimum value for much higher thicknesses of septa than the penetration component (minimal value of the penetration contribution of 4.5 %, 1.5 % and the 1.2 % for the three lengths) (Figs. components in Figure 3.24, one can notice that the eSP decrease is more precisely slowed down by the increase in the relative contribution of Compton scattering from the high energy gamma rays at 636 keV and 722 keV, which becomes the main component of scattered non-geometric events, when the geometric opening of the collimator becomes very small, i.e. for small holes and thick septa (Fig. 3.24e). In other configurations, the scattered non-geometric events are largely dominated by the elastic di↵usion of the 364 keV gamma rays, which always decreases with septa thickness (Fig. 3.24c). When eSP is dominated by the high-energy scattering, its variation according to the septa thickness is modified in relation to the decrease in the linear attenuation coe cient between 364 keV and 636 keV (from 2.53 cm 1 to 0.75 cm 1 and from 1.49 cm 1 to 1.2 cm 1 for photoelectric and Compton scattering interactions, respectively) : the relative contribution of scattered non-geometrics events to the total number of detected event increases (Fig. Chapter 3

Collimator optimisation by Monte Carlo simulation ), useful e ciency (uE) and relative contribution of nonscattered geometric events E1, non-scattered non-geometric events E2, scattered geometric events E3a and scattered non-geometric events E3b as a function of the hole diameter d, the septa thickness t and the collimator length L.

L d t E u E E 1 E 2 E 3 a E 3 b (cm) (cm) (cm) ( 10
b) E ciency and useful e ciency a Figure 3.25a shows the collimator e ciency as a function of the septa thickness and the hole diameter and for three representative lengths of the collimator (3.5 cm, 5 cm and 6.5 cm). The simulated values are superimposed on the values calculated from the analytical model of the e ciency (dotted lines) described by Equation 2.6 using the e↵ective length of the collimator L e for a 364 keV energy (Sect. 2.4.3.1). The relative contributions of the non-scattered geometric events (E1) (Fig. 3.25b), non-scattered non-geometric events (E2) (Fig. 3.25c), scattered geometric events (E3a) (Fig. 3.25d) and scattered non-geometric events (E3b) (Fig. 3.25e) in the collimator e ciency are also reported. Finally, the e ciency values and the relative contribution of the four components are also summarized in Table 3.6 for six representative configurations of the collimator geometry.

As predicted by the analytical model, the total e ciency varies with the geometric opening of the collimator and therefore decreases as the thickness of the septa increases, the diameter of the holes decreases or the length of the collimator increases (Fig. 3.25a). The accordance between the simulated data and the analytical model is perfect when penetration and scatter are low, i.e. for long collimator lengths and small hole diameters. However, the analytical model is unable to predict collimator behavior when these components become important, even by introducing the e↵ective length of the collimator to account for penetration (see Sec. 2.4.3.1).

Chapter 3

The penetration component E2 is the main contributor to the artificial increase of the e ciency for short collimators with large holes and small septa. The relative contribution of penetrated events to collimator e ciency is 61.8 %, 45.2 % and 27.7 % for 3.5 cm, 5 cm and 6.5 cm long configuration, respectively (0.225 cm diameter and 0.0875 cm long) (Fig. 3.25c and Table 3.6). This contribution drops below 20 % for all configurations when the septa thickness is set at 0.2 cm. The higher is the length of the collimator, the lower is the impact of penetration and the influence of the hole diameter and septa thickness on the geometric component (and by complementarity, on the penetration component), especially for small diameters. For example, the relative contribution of the geometric component varies by less than 4.5% when the septa thickness ranges from 0.16 to 0.2125 cm for a 6.5 cm long collimator, whatever the hole diameter.

The same behaviors is shown for the scattered non-geometric contribution E3b at small septa-big diameters (Fig. 3.25e). This component always remains below 20 %, but contributes to the di↵erence observed between the simulated data and the analytical model for short collimators when the septa thickness increases (Fig. 3.25a). Finally, the impact of geometric scattered events E3a to the collimator e ciency is very small (always lower than 4 %), but become the dominant scatter component when the geometric opening of the collimator is very small (Fig. 3.25d and Table 3.6). This means that scatter within the same hole becomes the most likely way out of the collimator as the amount of material to be passed increases.

The useful e ciency only considers the component of detected events useful for image formation by removing events outside a 2 region around the reconstructed position of the source that contribute to contrast degradation (see Table 3.5). Under these conditions, a large part of the non-geometric component is not taken into account in the e ciency calculation. This is clearly visible in Figure 3.26, which shows the useful e ciency and the relative contribution of E1, E2, E3a and E3b events as a function of septa thickness and hole diameter for three lengths of the collimator (3.5 cm, 5 cm and 6.5 cm). The geometric non-scattered contribution E1 is always higher than 88 %, 90.5 % and 91.9 %, while the penetration component E2 is always lower than 8 %, 6.4 % and 5.5 % for the three collimator lengths, except for configurations with the smallest septa and biggest diameter with the 3.5 cm long collimator (Fig. 3.26b-c). The scatter contribution, which is now dominated by the scattered geometric component due to the spatial selection, is always lower than 5 % (Fig. 3.26d-e) By removing non-geometric events, the useful e ciency shows a very good agreement with the analytical model for all septa thicknesses and hole diameters, regardless of the collimator length (Fig. 3.26a). As can be seen qualitatively, the 3.5 cm long configurations shows better agreement with the analytical model using the e↵ective length of the collimator L e (doted lines), while for long lengths, the simulated data are close to the analytical model with the actual collimator length L (dashed lines). The corrected analytical model therefore reasonably reproduces the contribution of nongeometric events (scattered and non-scattered), when they involve only the crossing of one or two septa (included inside the 2 ROI), but it fails to model it correctly when penetration occurs over greater distances, as we have seen previously (Fig. 3.26a). As the collimator length increases, the non-geometric component is strongly reduces and the penetration correction of the analytical model is no longer necessary.

c) Spatial Resolution a Figure 3.27 illustrates the influence of the septa thickness on the collimator spatial resolution (Fig. 3.27 left and right, respectively). As expected, smaller septa favour the penetration of events and thus, higher tails in the spatial distribution and lower contrast, mainly for short collimators. However, the FWHM of the profiles is not influenced by the septa thickness as predicted by the analytical model of the intrinsic resolution R c (Eq. 2.7). The increase of the standard deviation observed in Figure 3.28 for short collimators with large holes and thin septa shows that this influence netherless becomes significant for configurations characterized by a huge amount of penetration and scatter (20 % di↵erence between the minimum and maximum SR value for a 3.5 cm thick collimator with 0.225 diameter hole). The simulation also accurately reproduces the linear variation of the spatial resolution with hole diameter and collimator length (Fig. 3.28). The simulated configurations allow to reach spatial resolutions ranging from 0.08 mm to 0.72 mm FWHM.

Influence of the energy resolution a

Figure 3.29 shows the impact of energy resolution on the e↵ective septal penetration. As expected, a poorer energy resolution induces an increase of the e↵ective septal penetration due to the selection of a higher amount of scatter events (Fig. 3.29c). The impact is higher for shorter collimators. Indeed, the relative contribution of scatter in eSP is 31 % and 45 % with an energy resolution of 2.5 % and 39.7 % and 58.7 % with an resolution of 7.5 % for a 3.5 cm and 6.5 cm thick collimator, respectively (hole diameter of 0.175 cm and septa hickness of 0.0875 cm). The spectrum in Figure 3.30 indicates that the degradation of the energy resolution, and thus the widening of the energy window, increases the scatter component coming from the high energy gamma rays (637-and 723-keV photons) more than the 364 keV scatter component. The higher scatter component directly impacts the e ciency of the collimator, which increases up to 15 % for a 3.5 cm collimator length with 0.225 cm diameter holes and 0.075 cm thick septa, when energy resolution varies from 2.5 % to 7.5 % (Fig. 3.31). The e↵ect of energy resolution is less important for a 6.5 long collimator (+3.4 % for the same configuration) and has no influence on the useful e ciency and the spatial resolution.

Choosing the collimator: which compromise for highenergy gamma imaging ?

a The advantage of nuclear medicine is the capability of deriving quantitative information from 2D images. However, these images are often blurred because the finite spatial resolution of the system degrades the contrast of any lesions with a size less than 2 times the system resolution. This consideration suggests the desirability of collimators with better spatial resolution. On the other hand, the statistical noise in the images will a↵ect the precision with which activity may be estimated, suggesting the need to use collimators with good e ciency. Thus, which is the optimal collimator design for quantification? The optimization study, introduced in Section 2. in a multiple-step procedures that will be now presented. Because of the finite number of configurations simulated, interpolation and extrapolation methods have been used in order to extract the collimator configurations that could satisfy the requested properties in terms of spatial resolution, e ciency and scatter and septal penetration (see Sec. 2.1).

Step 1 : Spatial resolution and useful e ciency interpolation Fit of the SR values The slope ↵ of the linear regression on the spatial resolution curves as a function of the hole diameter (Fig. 3.28) has been compared to its expected values derived from the analytical model expression of R c (see Eq. 2.7) with (↵ th = L+b L ) and without penetration correction (↵ thP en = Le+b Le ), i.e. using the actual length L or the e↵ective length L e of the collimator (Table 3.7). As can be noticed, the ↵ values are in very good agreement with the expected values from the geometric analytical model including penetration correction, with a maximal di↵erence ranging between 3 and 1 % when the collimator length moves from 3.5 cm to 6 cm. Therefore, these equations derived from the analytical model were used in the rest of the study to interpolate the spatial resolution of the collimator for various hole diameters and collimator lengths.

As already mentioned, the very good accordance between simulated data and the analytical model relies on the fact that penetration and scatter non-geometric components mainly contributed to the spatial profile tails, while their impact inside the main peak is always low (Figs. 3.20, 3.21 and 3.27). Thus, by using a double gaussian fit ajusted on the spatial profile, the extracted SR is sensitive only to the FWHM of the peak, which is not or only slightly influenced by penetration and scatter. The small amount of penetration in the profile peak, which becomes significant for short collimators, explains why the di↵erences observed with the analytical model are greater for

Fits of the uE values

From the detailed description of the useful e ciency realized in Section 3.2.3.2, it has been shown that this parameter, by removing a large part of the non-geometric component, is more representative of the real e ciency of the collimator, i.e. the one that will contribute to the formation of the source image and thus, to the quantification. The useful e ciency shows a very good agreement with the geometric analytical model (Fig. 3.26). This accordance was quantified through a polynomial fit on the uE curves as a function of the septa thickness using Equation 2.6 for each diameter hole and collimator length:

uE = ⌫ • d 4 (d + t) 2 (3.5)
The mean value of the constant coe cient ⌫ obtained from each hole diameter, together with their standard deviation, are calculated for each collimator length and compared to the theoretical values given by the geometric analytical model of the collimator e↵ciency (eq. 2.6) with (⌫ th = 3 Comparison of the linear variation coe cient between the spatial resolution and the hole diameter obtained from the simulation (↵) and the analytical model with (↵ th = L+b L ) and without penetration correction (↵ thP en = Le+b Le ). The distance b between the source and the collimateur is 5 cm. The standard deviation on ↵ is three orders of magnitude smaller than the ↵ value.

p 3 8⇡ • 1 L 2 ) and without penetration correction (⌫ thP en = p 3 8⇡ • 1 Le 2 ) (Table

Chapter 3

Collimator optimisation by Monte Carlo simulation the thicker one, there is a slightly better match with the model without penetration correction. The general good agreement with the corrected analytical model allows to use this set of equations to interpolate the useful e ciency of the collimator for various septa thicknesses, hole diameters and collimator length.

Step 2: Choice of the maximum level of scatter and penetration a Among all simulated configurations, only those with a very low percentage of e↵ective septal penetration eSP have to be selected for our application. We chose to set the eSP threshold to 7.5 %. As we have already said, this criterion is much more restrictive than the usual 5 % maximum septal penetration accepted when optimizing collimator geometry and calculated from the shortest path a gamma ray can travel through a single septa (Sec. 3.2.3.2). This threshold has been chosen because it allows to select at least one configuration for each collimator length greater than 4 cm.

Step 3 : Extrapolation of the t value corresponding to the eSP threshold a

The minimum septal thickness to reach the 7.5 % eSP threshold was extrapolated from the curves of the e↵ective septa penetration eSP as a function of the septa thickness, as shown in Figure 3.32. An exponential interpolation was made between each simulated eSP value to find the septa value corresponding to an eSP of 7.5 %. At the

L L e ⌫ (⌫) ⌫ th ⌫ thP en ⌫/⌫ th ⌫/⌫ thP en
(cm) (cm) (10 5 cm 2 ) (10 5 cm 2 ) (10 5 cm 2 ) (10 end of this step, we obtained a family of new geometric configurations, characterized by combinations of hole diameter, collimator length and septa thickness allowing to achieve a 7.5 % e↵ective septa penetration. Figure 3.33 summarizes the features of all these configurations. One can note that the 7.5 % eSP value is not reached for the 3 cm and 3.5 cm long collimator configurations, i.e. that the eSP value is always above 7.5 %. For hole diameters smaller than 0.1 cm, the 7.5 % eSP value is rarely reached because of two di↵erent reasons. The first one concerns long collimators (5.5, 6 and 6.5 cm), for which the eSP value remains below 7.5 % over the studied range of septal thicknesses (Figs. 3.23 and 3.32). The second e↵ect is visible with 4.5 cm long collimators and also explains why the 7.5 % eSP threshold is reached for a 4 cm long collimator with 0.125 cm diameter holes but not with smaller diameters. It is due to the increase of Compton scattering of Chapter 3

Collimator optimisation by Monte Carlo simulation high-energy gamma rays (636 and 722 keV) in the collimator observed when the septa thickness increases for short collimators with small holes (Sec. 3.2.3.2). This highenergy scatter component causes the increase or slowly decrease of eSP with the septa thickness (Fig. 3.24) and can prevent the 7.5 % threshold from being reached (4.5 cm long collimator with 0.5 mm hole diameter and 4 cm long collimator with hole diameters less than 1.25 mm) or force the septa thickness to be increase in order to reach it (4.5 cm long collimator with 0.075 cm diameter holes). This last consequence is illustrated in Figure 3.33, which shows that the linear relationship between the minimum septa thickness and the hole diameter is no longer respected and that the value of the septa to reach the 7.5 % eSP threshold is always higher than the value predicted by the linear trend. This linear regression was used to interpolate the minimum septa thickness for various hole diameters and to extrapolate the missing data for small diameters.

Step 4 : Extraction of spatial resolution and useful e ciency a

The geometric configurations defined from the third step were injected inside the theoretical equations of SR and uE derived from the analyticals models in the first step. Figure 3.34(up) shows the useful e ciency and the spatial resolution of the collimator configurations with an e↵ective septal penetration of 7.5 % corresponding to the 8 hole diameters and 6 collimator lengths simulated. Each point corresponds to a specific combination of hole diameter, collimator length and septa thickness. In order to improve clarity, these geometric configurations are gathered by hole diameter using interpolation curves for collimator lengths ranging from 4 to 6.5 cm.

The shape of each SR-uE colored curve can be understood by taking as an example the 0.125 cm hole diameter (interpolated gray curve in Fig. 3.34(up)). This diameter satisfies the requirement of a 7.5 % eSP for 6 configurations characterized by 6 di↵erent lengths and 6 growing septa. The thickness of the septa does not vary linearly with the length of the collimator but rather in inverse proportion (Fig. 3.33). This means that below a certain collimator length, the significant increase in septa thickness required to maintain a septal penetration at 7.5 % results in a decrease in e ciency that is no longer o↵set by the decrease in collimator length. Therefore, for each hole diameter, the e ciency goes through a maximum when the length of the collimator varies from 4 cm (higher value of the spatial resolution) to 6.5 cm (lower value of the spatial resolution). Figure 3.34(down) shows the SR and uE values for 40 extrapolated diameters (between 0.05 and 0.225 cm) obtained from interpolation methods.

Step 5 : Choice of optimal geometric configurations a Among all configurations shown in Figure 3.34(down), five collimators with a spatial resolution ranging from 1 to 5 mm FWHM were selected in order to o↵er di↵erent imaging capabilities (Fig. 3.34(down)). If more than one configuration had the expected SR, the one with the thinnest septa (in order to reduce the collimator artefacts) and the highest useful e ciency was selected. The geometric features and intrinsic performance 3.9: Geometric features and intrinsic performance of the five collimator configurations selected. Both spatial resolution (SR) and Useful E ciency are measured for a source distance of 5 cm.

of the five collimator configurations are summarized in Table 3.9. As can be noticed, the requested SR are attained at the same collimator length of 5.5 cm. This constant value would have been di↵erent if the collimator had been optimized for another distance between the source and the collimator. The collimator ( 1) is characterized by a very high spatial resolution of 1 mm at a distance of 5 cm, thanks to its very small hole diameter, but at the expense of its e ciency, which drops to a value of 0.23 10 5 . On the other side, the collimator (5) was chosen for its spatial resolution of 5 mm. Its larger hole diameter of 0.25 cm allows a forty-fold increase in e ciency (9.42 10 5 ) compared to the configuration [START_REF] Bernier | Radiation oncology: a century of achievements[END_REF].

Collimator optimisation by Monte Carlo simulation Chapter 3

Optimization of the collimator for thyroid treatment a

The five collimator configurations selected at the end of the optimization process were chosen to o↵er complementary performance in terms of spatial resolution and efficiency, while guaranteeing low scatter and septal penetration. The study was carried out in a very simple geometry, where the influence of crystal and of the environment was not considered. In order to choose the best collimator geometry to implement in our first camera prototype, the performance of the detector must necessarily be treated as a whole. To achieve this objective, the camera was fully modeled (see Sec. 2.4.3.2) and we investigated its ability to detect small heterogeneous distributions of activity as a function of the five collimators by placing itself in a realistic clinical context simulating the treatment of thyroid diseases with a voxelized phantom. is fully modeled, including realistic activity distributions and clinical practices in terms of activity injected and acquisition time.

a) XCAT voxelized human phantom The XCAT phantom is an anatomically realistic phantom developed to match the reference human body dimension values and materials given in the ICRP Publication 89 [START_REF] Valentin | ICRP Publication 89: Basic anatomical and physiological data for use in radiological protection: reference values[END_REF][START_REF] Segars | 4d XCAT phantom for multimodality imaging research[END_REF]. Being interested to the investigation of small hetereogenities in the thyroid and seeking for a reduced computational time, the phantom has been divided in three slices with two di↵erent voxels dimensions: two phantoms with 3⇥3⇥3 cm 3 voxels for the head and the body and one phantom with 1⇥1⇥1 cm 3 voxels for the neck (Fig. 3.35). The phantoms are generated in the form of an Interfile which can be easily implemented inside GATE [START_REF] Jan | GATE: a simulation toolkit for PET and SPECT[END_REF] thanks to the use of nested parameterized volume [START_REF]GATE: Geant4 Application for Emission Tomography: a simulation toolkit for PET and SPECT, OpenGATE Collaboration[END_REF]. In the raw image created, voxels belonging to the same body organ are indexed by the same reference number. Together with the image, an attenuation map [START_REF] Valentin | ICRP Publication 89: Basic anatomical and physiological data for use in radiological protection: reference values[END_REF] and an activity map defining a binary relationship between voxel-material and voxel-activity were also created. In this way, the voxelized phantom can act as both an attenuation medium and a radiation source. In order to simulate realistic hyperthyroidism features [START_REF] Cantalamessa | Thyroid Nodules in Graves Disease and the Risk of Thyroid Carcinoma[END_REF][START_REF] Erbil | Graves' disease, with and without nodules, and the risk of thyroid carcinoma[END_REF][START_REF] Silberstein | Comparison of Outcomes After 123i Versus 131i Preablation Imaging Before Radioiodine Ablation in Di↵erentiated Thyroid Carcinoma[END_REF], hyper-fixing thyroid nodules were modeled by two small spherical volumes with 0.6 cm and 1.2 cm diameter completely inserted within the voxelised neck phantom at a depth corresponding to the mean thyroid thickness (Fig. 3.36). The head phantom is vertically rotated by an angle of 15 in order to put the camera in full contact with the neck. The space created between the body and the neck is filled with an adipose tissue-equivalent material (Fig.

3.35).

b) Modeling of the clinical treatment protocol The simulated activity in the thyroid was determined according to the mean value of the activity administrated during hyperthyroidism treatment procedures, set at 330 MBq [START_REF] Melo | Organ Dose Estimates for Hyperthyroid Patients Treated with 131 I: An Update of the Thyrotoxicosis Follow-Up Study[END_REF][START_REF] Bernard | Radioiodine therapy in benign thyroid disorders. Evaluation of French nuclear medicine practices[END_REF], and the mean Collimator optimisation by Monte Carlo simulation Chapter 3 uptake of an hyperfunctioning thyroid 24h after treatment (65%, compared to the 30% characteristic of a normal thyroid [START_REF] Melo | Organ Dose Estimates for Hyperthyroid Patients Treated with 131 I: An Update of the Thyrotoxicosis Follow-Up Study[END_REF]). 24 h have been chosen as being representative of a common uptake measurements time performed in clinical practice [START_REF] Eanm | Internal Dosimetry Task Force Report on: Treatment Planning For Molecular Radiotherapy: Potential And Prospects[END_REF]. The nodules to normal thyroid tissue uptake ratio was set to 6:1 [322] and 10 minutes acquisitions were simulated. A realistic background coming from the radiotracer uptake in surrounding normal organ was modeled based on a iodine biokinetics models [START_REF] Melo | Organ Dose Estimates for Hyperthyroid Patients Treated with 131 I: An Update of the Thyrotoxicosis Follow-Up Study[END_REF]. This model was specifically developed for patients who hyper-fix iodine in the thyroid gland because of a hyperthyroid disease and is a simplified version of a more complex biokinetic model for iodine in normal individuals [START_REF] Leggett | A Physiological Systems Model for Iodine for Use in Radiation Protection[END_REF].

The compartmental model is based on 131 I data from a small, kinetically wellcharacterized subcohort of patients of the Thyrotoxicosis Therapy Follow-up Study (TTFUS), which is a follow-up study of around 36.000 hyperthyroid patients treated from the mid-1940s to the mid-1960s. The iodine biokinetics model allows to derive the normalized activity of 131 I (N D ) over time in all its compartments, which are organs that concentrate iodine after its administration (thyroid, salivary glands, stomach wall, urinary tract/bladder and liver) and spaces in which iodine circulates (iodide pool, thyroxine in extracellular fluids, plasma and extrahepatic organs, contents of stomach, small intestine and colon), thanks to the use of simple equations [START_REF] Melo | Organ Dose Estimates for Hyperthyroid Patients Treated with 131 I: An Update of the Thyrotoxicosis Follow-Up Study[END_REF]. The procedure to obtain these equations is described in detail in the corresponding article [START_REF] Melo | Organ Dose Estimates for Hyperthyroid Patients Treated with 131 I: An Update of the Thyrotoxicosis Follow-Up Study[END_REF]. The functional compartments within which the 131 I activity is uniformly distributed are shown in Fig. 3.37. Compartments are connected to each other by input/output time transfer constants. The intercompartment transfer rates were derived by fitting measured time-activity data for each individual patient in the subcohort selected to the initial compartment model using the SAAM II computer code.

For organs that do not correspond to a specific compartment, the model defined a set of equations that allowed to obtain their N D from the fraction of N D of a compartement (or more than one compartement) that goes in the organ. These fraction values are tabulated for all source organs that fix iodine, whose characteristics are based on the ICRP Publication 89 [START_REF] Lamart | COMPARISON OF INTERNAL DOSIMETRY FACTORS FOR THREE CLASSES OF ADULT COMPUTATIONAL PHANTOMS WITH EM-PHASIS ON I-131 IN THE THYROID[END_REF] reference voxel phantoms for both adult male and female. Thus, knowing fraction parameters and normalized N D over time for each compartment, we were able to calculate the activity to be fixed in each source organ of the XCAT Phantom at 24 h from the 131 I administration. Knowing the number of voxels per organ in the phantom, the number of emission-per-voxels-per-organ were defined and integrated in the activity maps. Figure 3.38 shows the normalized number of disintegrations over time for all compartments of the model that were used to define organ activity. Only the five main compartments are highlighted, which includes the iodide pool (total-body plasma and extracellular fluid space), the thyroid glands, stomach walls and stomach contents. These contributions, except for the thyroid itself, are less than 0.5% of the total activity present 24 h after administration of the radionuclide. This means that the contribution of the background is very low. However, imagining to use image the thyroid after a shorter time range, for example after 6 h from the injection, the contribution of the five main compartments grow to a value of 9%. Considering the mean activity injected during benign thyroid treatment (300 MBq), a thyroid uptake of 0.65% plus the compartiment uptake of 10% and a total e ciency in the order of 10 6 , the final prototype counting rate (400 kCps, see Sec. 2.1) will allow to perform early imaging without having dead time issues.

Analysis protocol a

As already described, 131 I realistic activities were simulated thanks to the use of a 3D XCAT voxelized phantom. The information on the gamma rays emitted by the source phantom and filtered by the collimator were recorded by a GATE PhaseSpace actor placed on the volume defining the crystal. No sensitive detector was used at this stage (see Sec. 2.4.3.2). Each image correspond to around 120•10 9 particles generated (256 cores available, 2.60 GHz for each processor).

In a second step, the output file from the actor, one for each collimator, was used as a source of radiation, i.e gamma rays are generated from the information provided by the actor (energy, emission point and propagation direction). The phantom was removed and the crystal volume was filled with a 6 mm thick CeBr 3 (see Sec. 2.5.5). The interactions of gamma rays within the crystal were recorded. The scintillator intrinsic properties, i.e. its energy and spatial resolutions that depend on the transport of light in the crystal, were not directly modeled. The performance of the scintillatorphotodetector assembly were set according to the experimental results (see Sec. 3.1.3) and integrated in the 2D images by using convolution methods. The images were also processed with a specifically designed filter to eliminate the high frequency artifacts coming from the collimator septa (Sec. 3.2.5.2) [START_REF] García | The HURRA filter: An easy method to eliminate collimator artifacts in high-energy gamma camera images[END_REF]. Finally, the ability of the camera to detect small nodules was quantified by calculating the nodules Contrast (C) and Contrast-to-Noise ratio (CNR) on the thyroid images for each collimator configuration (Sec. 3.2.5.2). a) Energy and spatial gaussian blurring The intrinsic properties of the scintillation material, which will cause fluctuations in the energy and spatial response of the camera, have not been modeled in the simulation. Each detected event is characterized Chapter 3

Collimator optimisation by Monte Carlo simulation by the position and the total energy deposited in the scintillator. When multiple interactions occur, the position corresponds to the centroid of the mean energy deposits. The specific response of the given scintillator-photodetector assembly is then integrated by performing a gaussian blurring on the energy and position of each event. The sigma of the blurring is set according to the spatial and energy performance experimentally measured for the 6 mm thick CeBr 3 (Sec. 3.1.3.5). In particular, the sigma of the spatial blurring corresponds to the intrinsic spatial resolution of 0.75 mm FWHM measured at the photopeak emission of a 133 Ba source (356 keV). The sigma of the energy resolution was extracted for each energy from a fit realized on the energy resolutionenergy curve (see Sec. 3.3.2.2), for which a 8% FWHM energy resolution is achieved at 356 keV. Thus, the final results in terms of Contrast and Contrast-to-Noise ratio will be presented with an energy window of 15% (337-391 keV). Figure 3.39 shows the 256⇥256 pixels images of the thyroid with and without spatial blurring obtained for the collimator with the 3 mm intrinsic spatial resolution.

The pattern of the collimator is all the more visible on the images as the intrinsic spatial resolution of the collimator increases (due to the increase in septa thickness) (Tab. 3.9). b) Frequency filter for collimator artifacts The hexagonal periodic structure of high-energy parallel-hole collimator is most of the time highly visible on planar images due to the thickness of the septa. This pattern is even more pronounced with our camera because of its very high intrinsic spatial resolution. A filter, specifically dedicated to remove these artifacts was implemented during the image analysis [START_REF] García | The HURRA filter: An easy method to eliminate collimator artifacts in high-energy gamma camera images[END_REF]. The filter procedure consists in three steps (Fig. 2. eliminate the high frequencies that correspond to the collimator artifacts by using a mask on the frequency image;

3. calculate the inverse 2D fast Fourier transform to come back to the spatial domain and obtain the filtered 2D image.

This filter allows to drastically improve the visualization of the activity distribution and the definition of the nodules boundary. It was also checked that it had no impact on the measurement of spatial resolution and the number of counts in a given region of interest. Thus, it has a huge impact on activity quantification, for which the delineation of lesions is crucial, mainly for small-sized lesions (Sec. 4.2.3). c) Nodules detectability Figure 3.41 shows the final images of the thyroid simulated for the five collimators after the complete analysis protocol. The ability of each configuration to detect small heterogeneous activity distributions was quantified by calculating the nodule Contrast (C) and the nodule Contrast-to-Noise ratio (CNR):

C = S B B (3.6) CNR = S B (B) (3.7)
where S is the counts inside a circular region of interest (ROI) with a diameter equal to the nodules dimension (Fig. 3.41) and B is the background counts coming from the healthy tissues estimated from the mean value over ten small ROIs drawn on the lobe without nodule and then rescaled to the surface of the nodules ROIs. (S) is the standard deviation of the counts inside the backgrounds ROIs. The higher the C and CNR values, the more the nodule is discriminated from the background noise. A nodule is considered to be detectable when its CNR is higher that 4, corresponding to a p-value lower than 10 4 to be interpreted as the probability of a false positive [START_REF] Waterstram | Nuclear Medicine and PET/CT -E-Book: Technology and Techniques[END_REF].

And the winner is a

Figure 3.42 shows the nodules C and CNR as a function of the overall spatial resolution of the collimator (eq. 3.3). The shape of the contrast curve as a function of the intrinsic resolution of the collimator is dominated by the partial volume e↵ect which results in a decrease in contrast when the resolution degrades. Indeed, the blurr introduced by a limited spatial resolution spreads the signal over several pixels of the image. When considering a ROI defined around nodules, part of the activity emitted by the nodule is also detected outside this ROI (spill-out e↵ect), which results in a reduction of the signal S in Equation A.1. This e↵ect becomes significant when the size of the structures studied is less than 2-3 times the spatial resolution of the system. Therefore, the contrast value calculated for the nodule of 6 mm and 12 mm reaches a maximum for collimators with an intrinsic resolution of 2 mm (overall system spatial resolution of 2.1 mm) and 3 mm (overall system spatial resolution of 3.06 mm), respectively. The slight decrease observed for the collimator (1) with the higher spatial resolution is probably due to the impact of the noise coming from the scattering in surrounding tissue, whose relative contribution becomes more important when the sensitivity of the collimator is extremely low. The maximum value of the contrast does not correspond to the uptake ratio of 6:1 between nodules and healthy tissues because the estimation of the signal coming from healthy tissues in the ROI corresponding to the nodule is overestimated when it is considered equivalent to the signal measured on the lobe without nodule (di↵erences in volumes). Figure 3.42(left) shows that the nodule contrast-to-noise ratio is high regardless of the nodule diameter and the collimator. This contrast-to-noise ratio is mainly dominated by the sensitivity of the collimator, which increases as the inverse of the square of its spatial resolution. Therefore, it increases linearly with spatial resolution until this resolution becomes greater than 2-3 times the size of the nodules. Under these conditions, the partial volume e↵ect becomes significant and the rate of increase of the contrast-to-noise ratio becomes lower. All collimator configurations allow to exceed the detectability threshold of 4. The lowest value, corresponding to the collimator (1), is 6.4.

In order to be able to detect and quantify small uptake heterogeneities (small benign nodules have a diameter from 0.4 to 3 cm [START_REF] Cantalamessa | Thyroid Nodules in Graves Disease and the Risk of Thyroid Carcinoma[END_REF][START_REF] Silberstein | Comparison of Outcomes After 123i Versus 131i Preablation Imaging Before Radioiodine Ablation in Di↵erentiated Thyroid Carcinoma[END_REF] and tumor remnants are often smaller than 1 cm), the choice of the final collimator configuration has been made in order to maximize the detectability of small targets. For this reason, the collimator (2), which o↵ers the best compromise between contrast and signal-to-noise ratio, was chosen to develop the first prototype of the camera, whose design will be presented in the next section.

The first feasibility prototype of the mobile gamma camera

a A first prototype of the mobile camera was developed on the basis of the scintillator characterization and collimator optimization studies presented in Sections 3.1 and 3.2. Coupled with the miniaturized photodetection system, this detection head makes it possible to achieve the desired objectives in terms of spatial and energy resolutions, signal-to-noise ratio and compactness. This section presents the complete characteristics of the camera, as well as the influence of the control parameters (supply voltage and temperature) on its intrinsic performance. The expected sensitivity and spatial resolution of the camera as a function of the source distance to the collimator are also presented.

Integration of the fully operational camera a

A picture of the first 5⇥5 cm 2 Field of View (FoV) complete prototype is shown in Figure 3.43. The overall dimensions of the camera are 12.6⇥8.6⇥8.6 cm 3 and it weights 4.54±0.01 kg (the weight of the collimator only is 3.76±0.01 kg). Figure 3.44 shows pictures of its components: the photodetection system based on the SiPMs array and the miniaturized readout electronics (Fig. 3.44(up-left)), presented in Section 2.6, coupled to a 6 mm thick CeBr 3 crystal with reflective edges (Fig. 3.44(up-right)), described in Section 2.5, and to the 5.5 cm thick tungsten parallel hole collimator with of 8 mm of tungsten. Its base is also expanded to a width of 8⇥8 cm 2 and a thickness of 10 mm (Fig. 3.44(down-right)). This additional material shields the scintillator against gamma rays coming from the front of the camera. The scintillator is coupled to the photodetection system with optical grease (BC-630, Saint Gobain) (Fig. 3.44(upright)). The photodetection system and readout electronics are held in an aluminum housing with external dimensions of 6.5⇥6.5⇥5 cm 3 (Fig. 3.44(down-left)). A 1.5 cm height and 5 mm thick lead shielding around the crystal also ensures both alignment with the SiPMs array and protection against gamma rays arriving on the side. It also prevents the weight of the collimator from resting on the scintillator by maintaining a 100 µm gap between them. An aluminum cap is specifically designed to guarantee stable alignement and coupling between the collimator and the crystal (Fig. 3.43). The camera is connected to an external acquisition box, including the power supply unit, through a 2 m long electrical wires that transfers the digitized signals to the USB communication port of a PC.

Focus on the intrinsic performance a

The intrinsic properties of the collimator for a 5 cm source distance and the intrinsic performance of photodetector/scintillator assembly for a nominal bias and a temperature of 25 C are summarized in Table 3.9 and Tab. 3.4, respectively. Since the camera will be able to be used with di↵erent energy of gamma emitters and in clinical conditions with uncontrolled temperature, we have evaluated the influence of these parameters on the camera's performance. The expected response of the detector as a function of the distance between a pinpoint source and the collimator was also studied by simulation.

First feasibility prototype Chapter 3

Influence of overvoltage a

A set of measurements was performed at 25 C to study the impact of the overvoltage on system energy resolution (ER). Single spots in the center of the FoV with a collimated 57 Co source (4 mm diameter) were performed. The bias voltage of the SiPMs array ranged from 67.5 to 69.5 V. The peak position and the ER were calculated from a double gaussian fit performed on the full-energy peaks of the 57 Co energy spectra (122 keV) and on the secondary peaks at 136 keV.

The variation of peak position and energy resolution as a function of the applied voltage is presented in Figure 3.46. The peak position and energy resolution are very sensitive to voltage variation (Sec. 2.6.2.2). The peak position, proportional to the total gain of the SiPMs, increases by about 5%/mV . The evolution of energy resolution is a little bit more complexe : ER decreases until 69 V, where it reaches a minimum value (12.2±0.03 %) and then slightly increases with the applied voltage. These trends could be explained by the opposite e↵ect of photodetection e ciency (PDE) and correlated noise on energy resolution. The bias voltage increases the PDE and therefore the probability of an incident photon to be detected : the statistical fluctuation on the charge collected decreases and the energy resolution improves. The increase with overvoltage is explained by the increase in the probability of an avalanche being triggered in the SiPM. When this probability tends towards 1, PDE saturates. At the same time, the correlated noise, which causes a fluctuation in the gain of the SiPMs equivalent to a multiplication noise, also increases with the bias voltage. Figure 3.46 shows that the e↵ect of the correlated noise becomes dominant above 69V, which leads to the saturation and then the degradation of the energy resolution. This optimal bias voltage of 69 V was used for the characterization study of the scintillators (Sec. 3.1).

Influence of gamma-ray energy a

The energy linearity and energy resolution as a function of the energy of the incident gamma rays have been studied at 25 C with di↵erent 4 mm diameter collimated sources : 241 Am (60 keV), 57 Co (122 keV) and 133 Ba (80 keV and 356 keV). For the linearity measurements, the SiPMs gain has been kept constant (0.3 pF), while it has been optimized for the measurement of the energy resolution of each specific source. The peak position and the ER were calculated from a single gaussian fit for 241 Am and the 80 keV emission peak of 133 Ba and from a double gaussian fit for 57 Co (122 and 136 keV) and others emission peaks of 133 Ba (356 keV and 383 keV). As shown is Figure 3.47(left), a really good energy linearity is attained (R-squared of 0.998). An energy resolution of (18.0±0.1)%, (15.7±0.6)%, (12.43±0.05)% and (7.9±0.3)% is achieved at 60 keV, 80 keV, 122 keV and 356 keV, respectively (Fig. 3.47(right)). The energy resolution as a function of the gamma ray energy E has been fitted with the following equation (red dotted line in Fig. 3.47(right)):

E E = aE b (3.8)
where a is a parameter that depends on the crystal (non-lineaity, 137.3±0.4), while b is expected to be 0.5 for an ideal scintillator in which the statistical noise dominates the energy resolution [START_REF] Knoll | Radiation detection and measurement[END_REF][START_REF] Moszynski | Energy resolution of scintillation detectors[END_REF]. A value of b = (0.493±0.007) is found.

Influence of temperature

a All the experimental performance presented up to this point have been performed inside a climatic chamber (Fig. 3.1) that allows to maintain a constant temperature (25 C). When using the camera at room temperature, significant variations in performance, especially in energy response, are observed. As this camera is designed for use at the patient's bedside, the temperature dependence of SiPMs gain was evaluated in term of photopeak shift in order to develop a temperature correction method.

The camera was placed inside the climatic chamber and its energy response was evaluated with a 4 mm diameter collimated 133 Ba source by varying the temperature between 25 and 50 C with steps of 1 C. The peak position of the 356 keV emission at 25 C measured with the optimized bias voltage (69 V) was considered as the reference position. For each temperature, two acquisitions have been done: one at 69 V to highlight the peak shift due to temperature (called non corrected peaks) and one at the new bias voltage allowing the photopeak to be held at the position of the reference peak (corrected peak). The photopeak positions as a function of the temperature are shown in Figure 3.48(left). The charge response of SiPMs drops with temperature (4%/ C) due to the increase of the breakdown voltage, which decreases the overvoltage applied to the detector and therefore the gain, photon detection e ciency, crosstalk and after-pulses probabilities of SiPMs (Sec. 2.6.2). This dependance to temperature can be completely o↵set by tuning the bias voltage as a function of the temperature in order to maintain a constant overvoltage, as shown in Figure 3.48(left). Figure 3.48(right) presents the voltage correction allowing to keep the peak position of the 356 keV emission to a constant value over temperature. A fit on this curve allows to extrapolated for any temperature the corrected bias voltage to be used. This temperature compensatedmethod was implemented for phantom measurements that will be presented in Chapter 4.

Chapter 3

First feasibility prototype

Expected sensitivity and spatial resolution as a function of distance a

The sensitivity, spatial resolution and contribution of scatter and penetration were estimated as a function of the distance between a pinpoint source and the collimator by Monte Carlo simulations. The camera was fully modeled, including the collimator, the 6 mm thick CeBr 3 and the photodetection system (as a simple volume made of epoxy). For the collimator, a density of 18 g/cm 3 has been used and not of 19.3 g/cm 3 as the one used during the optimization of the intrinsic properties of the collimator. This because of more precise information given by the 3D printer manufacturer. A sensitive detector has been attached to the crystal to simulate crystal response. A 131 I line source was placed at di↵erent distances from the collimator surface, from 0.5 to 7 cm, with 0.5 cm step. All 131 I emissions were simulated. The line source was tilted of 24 with respect to the x axis of the collimator. An energy window of 15% around the 364 keV peak was applied. The rest of the analysis protocol is the same as the one described in Section 3.2.5.2 (including the Gaussian blurring to simulate the intrinsic spatial and energy response of the scintillator-photodetector assembly).

Di↵erent from what is expected from the theoretical e ciency of a parallel-hole collimator, Figure 3.49(up-left) shows that the sensitivity of our camera slightly linearly decreases with the distance of the source, from a maximum value of 2.57 Cps/MBq at 1 cm to 2.37 Cps/MBq at 7 cm. At a source distance of 5 cm, the simulated overall e ciency of the camera is 2.43 Cps/MBq. We shown that the intrinsic e ciency of the collimator is (14.9±0.4)•10 6 with an energy selection of 15% (Sec. 3.2.3.3). Considering that the full energy peak e ciency of a 6 mm thick CeBr 3 is 15.6% at 364 keV (values obtained from simulations and comparison with the literature [START_REF] Panwar | E ciency of CeBr3 detector: Simulations and measurements using a positron emitter[END_REF]), one will expect an overall e ciency of 2.32 Cps/MBq. The di↵erence of 6.3% is mainly due to the additional contribution of the higher energy emissions of 131 I at 637 and 723 keV, which penetrate the collimator and scatter inside the crystal. This component was not taken into account in the intrinsic e ciency of the collimator since the scintillator was not modeled. The presence of the scintillator also modifies the contribution from high-energy events scattering in the collimator. This is confirmed by the calculation of the contribution of scatter and penetration estimated as the ratio between events detected outside a 2 region centered on the image of the pinpoint source divided by all detected events Figure 3.49(up-right). At a 5 cm distance, this contribution is 9.3% compared to the 7.5% eSP value set in the optimization study (Sec. 3.2.4). As we have just discussed, the di↵erence is first due to the contribution of high energy penetrated events, which scatter in the crystal, but also to geometric events that arrived outside the 2 spatial selection and were now considered in the estimation of eSP (Tab. 3.5). We can notice that the eSP is higher at shorter distances. This e↵ect is due to the increase amount of di↵usion at short distances, as will be shown in Sec. 4.3.1.

The FWHM spatial resolution of the camera increases linearly with the distance (Fig. 3.49(down)). At 5 cm, the spatial resolution is 2.5±0.01 mm, which must be compared to the 2 mm intrinsic spatial resolution of the collimator. The di↵erence observed is obviously due to the convolution with the intrinsic spatial resolution of the Chapter 4

Quantitative planar imaging for dosimetry during thyroid treatment a

In the last chapter of this work the evaluation of the overall performance of the first 5⇥5cm 2 prototype of the mobile camera with 131 I sources will be discussed. First, the studies carried out to calibrate the system for quantification will be presented. This involves measuring the camera response as a function of the source-to-collimator distance using capillary tubes and thin cylindrical sources of di↵erent diameters placed in the air and in a tissue-equivalent medium. Then, a preliminary assessment of the camera's ability to quantify 131 I activities will be reported following the MIRD protocols [START_REF] Siegel | MIRD Pamphlet No. 16: Techniques for Quantitative Radiopharmaceutical Biodistribution Data Acquisition and Analysis for Use in Human Radiation Dose Estimates[END_REF] and using di↵erent 3D-printed thyroid phantoms.

Camera calibration for quantification a

The knowledge of the detector's overall response, including spatial resolution and sensitivity, is a prerequisite for quantitative imaging dedicated to dosimetry-guided radionuclide therapy. First of all, understanding how the system sensitivity changes with distance from the source is absolutely mandatory for activity quantification. The calibration factor represents the system sensitivity value that converts count rate to absolute activity in each pixel of the image. It was shown to be the most important contribution to the overall uncertainty in the dosimetric procedure, besides partial volume correction [START_REF] Siantar | Treatment Planning for Molecular Targeted Radionuclide Therapy[END_REF]. This calibration measurements were performed using di↵erent 131 I sources in liquid solution.

Experimental protocol a

All measurements were made at room temperature. Temperature variations were compensated by using the procedure described in Section 3.3.2.3. The parameters of the photodetection system and readout electronics (bias voltage of the SIPMs array, preamplifiers gain, trigger threshold, ...) were set according to the optimal settings discussed in Sections 2.7.1. The spatial resolution was measured using a glass capillary tube with an internal diameter of 0.86±0.05 mm (Clark Electromedical instruments GC150F-10) filled with 131 I (6.7 MBq). The tube was tilted of 24 with respect to the X axis of the camera in order to provide an oversampling of the line profiles with di↵erent source-to-collimator holes relative positions. The line source was placed on a 3D printed ABS support (0.97 g/cm 3 ), which allowed it to be moved precisely from 1 to 15 cm from the collimator in 2 cm steps. A last acquisition at an additional distance of 0.3 cm (thickness of the support) was also made. The detected events are selected in an energy window of 15% centered on the 131 I principal gamma emission peak (337-391 keV) and reconstructed in 128⇥128 pixels planar images with the Levenberg-Marquard iterative fitting method (Sec. 2.8.2). Images are then processed with a filter specifically designed to remove the high frequency artifacts produced by the high-energy collimator [START_REF] García | The HURRA filter: An easy method to eliminate collimator artifacts in high-energy gamma camera images[END_REF] (see Sec. 3.2.5.2 for more information about this filter). The frequencies corresponding to the structure of the collimator are removed using a spatial filter. An image without collimator artifacts is then obtained by applying the inverse FFT to the filtered spectrum [START_REF] García | The HURRA filter: An easy method to eliminate collimator artifacts in high-energy gamma camera images[END_REF]. The method is illustrated on the line source and the 0.5 cm diameter cylindrical source.

Camera calibration for quantification

Chapter 4

Camera calibration for quantification source angular o↵set. The FWHM spatial resolution is calculated from the profile of the line spread function with a double-gaussian fit in order to account for both the main peak, consisting mainly of geometric events, and the tails of the line spread function, composed of both penetration and scatter events.

System Sensitivity a

The sensitivity was measured using five 3D-printed cylindrical source phantoms (VeroClear, µ=0.116±0.002 cm 1 ) with a diameter of 4, 3, 2, 1 and 0.5 cm and a thickness of 3 mm (Fig. 4.2). The cylindrical sources were filled with 131 I in liquid solution. The source activity was controlled with an activimeter (MEDI 405, accuracy of±3%) and ranged from a minimum of 8.9 MBq for the smallest cylinder (151 MBq/ml) to a maximum 55. [START_REF] Stabin | State of the Art in Nuclear Medicine Dose Assessment[END_REF] MBq for the largest one (14.7 MBq/ml). For each cylinder, 20 minutes acquisitions were performed at 8 di↵erent distances from the collimator surface, from 1 to 15 cm in 2 cm steps thanks to the use of a mechanical ABS support, which also guarantees the centering of the cylinders in relation to the camera. A picture of the set-up is shown in Figure 4.3. Measurements were made with and without intervening attenuation medium, in the form of 5⇥5⇥1 cm 3 bricks of ABS (⇢=0.97 g/cm 3 ). Several lead bricks have been placed around the camera to screen it from background radiation. The image reconstruction protocol is identical to the one used for the spatial resolution measurement and was applied on 128⇥128 pixels images (Fig. 4.1(right)). The sensitivity of the camera (cps/Mbq) is measured as the number of counts detected within ROIs defined on the images of the cylindrical sources divided by the acquisition time and the source activity, corrected for the decay over time. The ROIs definition protocol will be described in the next paragraph.

As already mentioned, 131 I has two high-energy photon emissions at 637 and 723 keV, which can scatter and contaminate the 364 keV photopeak window, resulting in lower image contrast, overestimation of system sensitivity and therefore, quantification bias. In planar imaging, this e↵ect is commonly corrected through the implementation of the Triple Energy Windows (TEW) scatter correction technique [START_REF] Siegel | MIRD Pamphlet No. 16: Techniques for Quantitative Radiopharmaceutical Biodistribution Data Acquisition and Analysis for Use in Human Radiation Dose Estimates[END_REF]. The easiest Camera calibration for quantification Chapter 4 way to implement it is to estimate the fraction of scattered photons in the photopeak window to be subtracted from the total number of counts within it, using two additional windows, one below and one above the photopeak [START_REF] Siegel | MIRD Pamphlet No. 16: Techniques for Quantitative Radiopharmaceutical Biodistribution Data Acquisition and Analysis for Use in Human Radiation Dose Estimates[END_REF][START_REF] Dewaraja | Quantitative 131i SPECT with triple energy window Compton scatter correction[END_REF]. Despite its practicality for clinical implementation, various studies have shown that the TEW scatter correction method could cause a significant overestimation of the scatter contribution inside the photopeak [START_REF] Gils | Impact of reconstruction parameters on quantitative I-131 SPECT[END_REF][START_REF] Robinson | The influence of triple energy window scatter correction on activity quantification for 177lu molecular radiotherapy[END_REF]. Thus, sensitivity was estimated both without and with scatter correction, performed with 7.5% wide energy windows below and above the photopeak (309-336.7 keV and 391.3-418.6 keV).

Regions of interest definition a

Quantitative measurements can be strongly dependent on the method used to define the regions of interest (ROI) around the lesions. Great care should therefore be taken when defining a ROI for system calibration. This results in more robust and reproducible calibration results and therefore lower uncertainties on quantification measurements. In this work, and in general in SPECT imaging [330], the criteria required to define the ROI and to measure the detected counts should involve the selection of all or most of the geometric events and strongly depends on the size of the volume being studied, i.e. whether its dimensions are smaller or larger than the system spatial resolution (FWHM). For large lesions with a simple geometry, as the one usually used for camera calibration, the circular ROI is defined as having a diameter Camera calibration for quantification equal to the FWHM of the source profile [330,[START_REF] Barquero | 131 I activity quantification of gamma camera planar images[END_REF]. For small lesions, this definition could be erroneous, because part of the geometric counts could fall outside the ROI, due to the partial volume e↵ect. In this case, a valid alternative is represented by the full width at tenth of maximum (FWTM) of the central profile of the source, because it is considered as covering almost all the pixels with geometric events [START_REF] Barquero | 131 I activity quantification of gamma camera planar images[END_REF]. There is no clear indication in the litterature of when to use the FWHM or FWTM. Thus, for each cylinder and each distance, the two ROI methods were tested by fixing the diameter as the mean value of the FWHM or FWTM of the vertical and horizontal central profiles of the source image. The FWHM and FWTM values were extrapolated from a simple gaussian fit for the smallest cylinder or from a gaussian-rectangular fits for the others. The images of the smallest and the biggest cylindrical source after definition of the FWHM and FWHM 2D-ROIs are shown in Figure 4.4A.

A third method is used in order to be able to define ROIs encompassing nonstandard-shape lesions [START_REF] Lin | Fully Automatic Region of Interest Selection in Glomerular Filtration Rate Estimation from 99mtc-DTPA Renogram[END_REF]. It consists in hiding the pixels of the image having an intensity value lower than a threshold value. In clinical practice, a preliminary rectangular or elliptical ROI is drawn around the lesion of interest; the maximum count in this ROI is then defined and all pixels, whose values exceeds a fixed percentage of this maximum value, are selected in the ROI. The problem with this definition is that the choice of the percentage is arbitrary and therefore very variable in the literature (30%, 15% or 10% values has been found [START_REF] Lin | Fully Automatic Region of Interest Selection in Glomerular Filtration Rate Estimation from 99mtc-DTPA Renogram[END_REF]). Moreover, the possible presence of hot pixels or clusters of hot pixels in a small part of the image can have a significant impact on the definition of the ROI. For these reasons, we decided to define the threshold value from the image pixels intensity histogram. This histogram is characterized by two distinct intensity distributions: the first one, lower in pixels intensity (low-intensity peak), mainly related to scatter and penetration contributions outside the source image, and the uptake peak, higher in pixel intensity, mainly related to geometric events in the source image. The threshold value is set in a way to remove the low intensity pixels distribution. Figure 4.4(B) shows the pixels intensity histograms for the smallest and largest cylinder at 5 cm distance, in which both the threshold-pixel (red) and the pixel intensity corresponding to the FWHM (green) and FWTM (blue) 2D-ROIs are highlighted. The intensity distribution of the two peaks depends strongly on the size of the source dimension and therefore on its relationship with the camera FOV. Whatever the size of the source, the threshold is set at the minimum value between the low-intensity peak and the uptake peak. The quantification capability of the camera was assessed by using 3D-printed thyroid phantoms with and without nodules in order to model a realistic clinical examination. The phantoms and the quantification protocol, which follows the MIRD recommendations [START_REF] Siegel | MIRD Pamphlet No. 16: Techniques for Quantitative Radiopharmaceutical Biodistribution Data Acquisition and Analysis for Use in Human Radiation Dose Estimates[END_REF], will be presented in this section.

Camera calibration for quantification

The 3D thyroid phantoms a

The 3D-printed thyroid phantoms, developed at the Institute for Radiation Protection and Nuclear Safety (IRSN) (patent n FR 1650 855), were designed using a mathematical model describing the geometry of the thyroid [START_REF] Ulanovsky | Influence of measurement geometry on the estimate of 131(I) activity in the thyroid: Monte Carlo simulation of a detector and a phantom[END_REF] and adapting it to match volumes to ICRP recommendations [START_REF] Valentin | ICRP Publication 89: Basic anatomical and physiological data for use in radiological protection: reference values[END_REF]. A set of age-specific thyroid phantoms (thyroid lobes plus isthmus) for 5, 10, 15 year old children and the adult [START_REF] Beaumont | Development and test of sets of 3d printed age-specific thyroid phantoms for131i measurements[END_REF] have been developed and 3D printed. In this work, we used two phantoms representing the thyroid of a 5 year old (FANTHY5y) and 15 year old child (FANTHY15y) (Fig. 4.5). Their overall volume are 3.2 ml and 11.5 ml, for both lobes and isthmus, corresponding to a width of 2.2 cm and 4.5 cm and a height of 2.5 and 3.8 cm, respectively. The choice of not using the adult phantom (width of about 4.97 cm) is related to the limited field of view of the first prototype of our camera (5⇥5 cm 2 ). Phantoms are made of a tissue-equivalent polymeric resin (VeroClear, µ=0.116±0,002 cm 1 ) chosen because its attenuation coe cient is close to that of adipose tissue (0.106 cm 1 , NIST [210]).

Since the final objective of this work is to improve the quantitative assessment of the heterogeneous distribution of 131 I in the thyroid (Sec. 2.1), a 15 year old thyroid and a resin with a higher attenuation coe cient for the vertebral column (VeroWhite, µ=0.124±0.001 cm 1 ) [START_REF] Beaumont | Development and test of sets of 3d printed age-specific thyroid phantoms for131i measurements[END_REF]. The neck has an overall diameter of 10.8 cm and a height of 8.4 cm.

Experimental protocol a

The thyroid phantoms were filled with 131 I in a colored liquid solution. The activities, controlled with an activimeter (MEDI 405), were 52.06±1.56 MBq and 49.90±1.50 MBq for the FANTY5y and FANTY15y, respectively. For the pathologic phantoms, the lobes (around 7.8 cm 3 for both lobes) were filled with a concentrated 131 I dilution of 2.5 MBq/ml and the nodules with four lobe-to-nodule activity concentration ratios ( 1:2, 1:3, 1:4, 1:6) to simulate di↵erent normal thyroid tissue-to-nodule uptake ratios. The activity within each nodule was measured with the activimeter. Then this value was always compared to the estimated activity, obtained by accurately measuring the activity of the primary dilution (activity in a 1 ml or 5 ml syringe measured with the activimeter before filling the nodules) and by measuring the liquid volume insert inside the nodule. Table 4.1 summarizes the expected and measured activities in the nodules for each ratio. The di↵erences observed are due to inaccurate filling.

The complete phantom was positioned on a support that allows it to be moved vertically and horizontally (±0.5 mm), to ensure the centering of the thyroid (healthy phantoms) or the nodules (pathological phantom) with the camera FoV, and to rotate it using a motorized rotation stage (±0.02 ). The camera was inserted inside a 2 cm thick lead shielding specifically designed for the detector. The complete experimental set-up is shown in Figure 4.7. A distance of 3 mm was set between the neck phantom and the collimator. For each phantom, several planar images with di↵erent angular views were performed : 0 (anterior), 180 (posterior) and 90 (lateral). For each angular view, a 25 minutes acquisition time was set for the healthy phantoms and 15 minutes for the pathological phantom. Considering the average value of the activity administrated to the patient during hyperthyroidism treatments (more than 300 MBq) and the high thyroid uptake (up to 65% of the administrated activity), the same statistics on the image could be attained with acquisition of few minutes (1 to 3 minutes). All measurements were made at room temperature and temperature variations were compensated using the procedure described in Section 3.3.2.3. The image reconstruction and analysis protocols are identical to the one used for the calibration with the cylindrical phantoms, including the TEW scatter correction method. All images are smoothed with a two-sigma gaussian filter for better visualization.

Due to the shape of the target regions, the definition of the ROI that should encom- The first peak correspond to the background noise outside the thyroid, the second to the isthmus and the third to the lobes passed the uptake in the thyroid images without nodules is always performed using the threshold method. Figure 4.8 shows an example of the pixels intensity histograms and the corresponding ROIs contours defined from the threshold value on the images of the healthy thyroid phantoms acquired at 0 . Same procedure were used for phantoms with nodules (Fig. 4.9(right)), except for the 1 cm diameter nodule with the lowest lobe-tonodules activity concentration (1: 2), which could not be analyzed with the threshold method because of an uniform distribution of the pixels intensity between the nodules uptake and the lobe (Fig. 4.9(left)) from the pixels histogram. In this case, a vertical profile on the nodules allowed to extract the FWTM of the activity distribution, which were then used as the ROI radius. It should be noted that the use of di↵erent ROI methods implies the use of di↵erent calibration factors when quantifying activities, as we will see in the following paragraph.

The lateral images were used to estimate the depth and thickness of the thyroid and nodules in the neck (Fig. 4.10), both useful for the activity quantification. The camera was positioned so that the edge of its field of view coincides with the edge of the neck. Depths and thicknesses (both with and without nodules) were estimated by applying the threshold method and extracting the information from the contour automatically Preclinical planar imaging Chapter 4 Figure 4.9: Anterior images of the small nodule when filled with a 1:2 lobe-to-nodule activity concentration ratio (left) and the big nodule filled with a 1:6 activity concentration ratio (right). When the lowest low uptake ratio is used, the ROI around the nodules is defined as the FWTM of the nodule intensity. In the other cases, the threshold method is employed. generated by the threshold, as illustrated in Figure 4.10. The lobe thicknesses of the FANTHY phantoms were estimated as the mean thickness calculated from five (1 pixel width) adjacent horizontal profiles on the lateral images (Fig. 4.10). As visible from these images, the isthmus is well visible and it highly impacts the central profiles of the image. To get rid of it, the five horizontal profiles are drawn just above the outgrowth. Instead, the thickness of the nodules is defined as the width of the corresponding ROI (t j value in Fig. 4.10). The mean value of nodule thickness was estimated only if the threshold method is applicable (concentration ratio greater that 2). Finally, the depth of lobes and nodules is defined as the horizontal distance between the right edge of the image and the centroid of the ROI (d value in Fig. 4.10). The measured values were compared to those corresponding to the actual geometric characteristics of the phantoms. The lateral image of the pathological phantom was also used to estimate the thickness t of the lobe region adjacent to the nodule used for the background noise correction method (Fig. The activity quantification in the healthy thyroid was determined using both single-view planar images and the conjugate view (CV) approach, according to MIRD pamphlet 16 [START_REF] Siegel | MIRD Pamphlet No. 16: Techniques for Quantitative Radiopharmaceutical Biodistribution Data Acquisition and Analysis for Use in Human Radiation Dose Estimates[END_REF]. In the single view method, the source activity A j in the thyroid is individually measured an orthogonal view to estimate the source organ thickness and position. When considering a source regions j with negligible surrounding background activity, the activity A j can be calculated as :

A j = r I A I P e µet f j C (4.1)
where I A and I P are the conjugate-view counting rate and e µetn represents the transmission factor through the overall thickness t n of the neck phantom ((10.81±0.01) cm) with an overall e↵ective linear attenuation coe cient µ e . In this case, the total µ e can be precisely calculated knowing the geometry and the attenuation coe cient of the phantoms materials [START_REF] Beaumont | Development and test of sets of 3d printed age-specific thyroid phantoms for131i measurements[END_REF]. However, in a realistic situation, these values are not directly measurable. One approach could be to estimate the e µetn by using the theoretical attenuation coe cient of adipose tissue and spine at 364 keV (0.106 cm 1 and 0.152 cm 1 respectively [210]) together with standard spine dimensions reported by the ICRP [START_REF] Siegel | MIRD Pamphlet No. 16: Techniques for Quantitative Radiopharmaceutical Biodistribution Data Acquisition and Analysis for Use in Human Radiation Dose Estimates[END_REF]. Theoretical values of µ and t n lead to a e µe value di↵erent from only 0.65% compared to that calculated with the precise phantom geometry/attenuation. In order to perform the activity quantification in a realistic way, the e µetn value calculated with ICRP values was used. C represents the system calibration factor extrapolated from the distance-sensitivity curve without absorbing medium, as described above. The factor f j represents the source self-attenuation correction, estimated from the source region attenuation coe cient µ j and the source thickness t j as :

f j = µ j t j /2 sinh(µ j t j /2) (4.2)
The factor f j involves only the source region characteristics and will not deviate significantly from 1 until µ j or t j become large. For example, considering the mass attenuation coe cient of water and adipose tissue at 364 keV (0.109 cm 1 for water [210]) the f j value deviates by less than the 0.1% from the unity for a thickness t j of 1 cm. For t j =5 cm, the deviations from the unity increases and goes up to 1.25% and 1.15%, for water and adipose tissues, respectively, thus causing a non negligible e↵ect on activity quantification. The thickness of the lobes t j was estimated for the two phantoms from the lateral images of the thyroid (Fig. 4.10) and the calculation of f j was used to quantify the activity.

... with the pathological phantom a

The quantification of activity in the nodules of the pathological phantom is more tricky due to the superposition of activity from the lobes. However, a simple geometric subtraction technique is suggested by the MIRD recommendations [START_REF] Siegel | MIRD Pamphlet No. 16: Techniques for Quantitative Radiopharmaceutical Biodistribution Data Acquisition and Analysis for Use in Human Radiation Dose Estimates[END_REF]. It consists in using the activity calculation procedures already described but adding a multiplicative coe cient to correct this background. Under conditions where the target lesion is completely embedded in an uniform background activity, the fraction F of the counts Chapter 4 Preclinical planar imaging p I A I P that originates only from the target lesion of thickness t j can be calculated using :

F = nh 1 ( I ADJ I A ) • (1 t j t ) i • h 1 ( I ADJ I P ) • (1 t j t ) io 1/2 (4.3)
where I ADJ is the count rate through the phantom from a region adjacent to the nodule ROI with the same area and t is the thickness of this adjacent region. I A , I P were previously defined. t j is the nodule thickness. t j and t are measured from the lateral images (Fig. 4.10). In the case where the source is located near the camera midline or when only the anterior image is available, a simplified expression for this fraction F could be used [START_REF] Siegel | MIRD Pamphlet No. 16: Techniques for Quantitative Radiopharmaceutical Biodistribution Data Acquisition and Analysis for Use in Human Radiation Dose Estimates[END_REF]:

F 0 = h 1 ( I ADJ I A ) • (1 t j t ) i (4.4)
The quantification of the activity in the nodules was performed by using the methods based on the single view planar images and the conjugate view approach described in Section 4.2.3.1. For the single view approach, the background correction was just taken into account by multiplying A j by the fraction F 0 of the geometric mean counts. For the conjugate view approach, the calculation was only modified by multiplying the expression of A j (Eq. A.3) by the fraction F . It is important to note that our nodules are partially inserted inside the thyroids lobes (Fig. 4.6 ). The previous correction will therefore cause an overestimation of the background if no precautions are taken.

The standart deviation of the measured activity A j in the thyroid and nodules was estimated as the ratio of the quadratic addition of each single parameter relative variance in equation A.3. Finally, the accuracy of the quantification achieved with the mobile camera is estimated from the Recovery Coe cient (RC), corresponding to the ratio between the measured activity A j and the known activity in the thyroid phantom and the nodules corrected for its decay over time.

RESULTS

Chapter 4

Results

a

In the first part of this section, the spatial resolution and sensitivity of the mobile camera obtained during system calibration will be discussed. Then, the results of the activity quantification with the healthy and pathological phantoms will be presented. The ability of the camera to detect small radiolabed lesions will be also evaluated in terms of nodule contrast and contrast-to-noise ratio. ). Several explanations can be given to justify this di↵erence between experimental and simulated data. Firstly, the dimension of the capillary glass used for the SR measurement (0.86±0.05 mm), compared to the extremely small one used during the simulation (0.01 mm), impact the spatial resolution value. However, the main reason of the di↵erence is related to the selective melting method chosen for the manufacture of the collimator, which consists in using a high-power laser to fuse grains of tungsten powders in order to obtain the desired three-dimensional shape (Sec. 2.4.2). This implies a certain "porosity" of the material Chapter 4

Results

and therefore a lower e↵ective density than pure tungsten, which will also depend on the printing process [START_REF] Gear | Characterisation of the attenuation properties of 3d-printed tungsten for use in gamma camera collimation[END_REF]. Reported densities for SLM vary depending on the geometry and complexity of the finished piece and can range from 80 to 96% of the theoretical density (19.3 g/cm 3 ) [START_REF] Gear | Characterisation of the attenuation properties of 3d-printed tungsten for use in gamma camera collimation[END_REF][START_REF] Deprez | Preclinical SPECT imaging based on compact collimators and high resolution scintillation detectors[END_REF]. The finite size of the laser beam and tungsten powder grains also implies an inaccuracy in the actual size of the holes and septa obtained, which can modify the characteristics of the collimator compared to those expected (Fig. 3.45). We thus tested, by simulation, di↵erent collimator sizes (diameter and holes, thickness fixed at 5.5 cm) and densities. In these simulations, for what concern the geometry, the complete experimental set-up was considered: the complete camera (as described in Sec. 3.3.1), the glass line source on its ABS support and the environment (lead bricks, table, walls). Other simulation characteristics are the same as described in Sec. 2.4.3.2. Figure 4.11 compares the experimental resolution with the simulated values obtained by considering both a collimator with the expected geometry features (density of 18 g/cm 3 ; diameter and hole dimension as found by the optimization procedure, given in Sec. 3.3.1) and a collimator with a 12% increase in hole size (from 1.15 to 1.3 mm) and a 20% decrease in e↵ective density. The results obtained with the first simulation (red in Fig. 4.11) suggest us that the printed collimator has both a lower density and a di↵erent diameter-septa as the one requested, as suggested by the di↵erent spatial resolution values, the spatial resolution fit slope and consistent with the increase in penetration and scatter shown on the image profiles of the line source (Fig. 4.11(left)). On the contrary, the second configuration simulated allows to obtained a SR of (2.12±0.03) mm, (3.04±0.02) mm and (5.44±0.05) mm at 1, 5 and 15 cm, 4%, 3% and 1% lower that the experimental one, respectively.

Sensitivity Figure 4.12 shows the sensitivity S of the mobile camera as a function of the distance from the collimator without intervening media. The sensitivity was estimated without (Fig. 4.12(up)) and with TEW correction (Fig. 4.12(down)) and for the three ROIs methods described in Section 4.1.1.3 (FWHM, FWTM and threshold method). The sensitivity decreases with the distance for all sources. This trends contrasts with what is expected for parallel hole collimators [START_REF] Siegel | MIRD Pamphlet No. 16: Techniques for Quantitative Radiopharmaceutical Biodistribution Data Acquisition and Analysis for Use in Human Radiation Dose Estimates[END_REF], but it is in agreement with other results found in the literature [START_REF] Barquero | 131 I activity quantification of gamma camera planar images[END_REF]. The small cylinder is the only one with almost constant sensitivity with distance, regardless the ROI method (standard deviation of 0.06 Cps/MBq and of 0.1 Cps/MBq of the sensitivity with the distance when using the FWTM and the threshold method with TEW, respectively). This behavior can be understood by observing the spectra measured with the smallest and largest cylindrical sources at 1 cm and 15 cm from the collimator (Fig. 4.13). Events detected in the whole FoV (Fig. 4.13(left)) and within a circular ROI on the source image with a diameter corresponding to the size of the source (Fig. 4.13(right)). One can notice that scatter contribution in collimator is higher at shorter distances and for larger sources (Fig. 4.13(left)). Since scatter events are distributed over the entire FOV of the camera, a ROI selection removes a large part of this contribution for small sources, but it remains important for large sources especially at short distances (Fig. 4.13(right)). The 0.5 cm diameter source is the only one that shows an independence of the amount of scatter and penetration with distance (Fig. 4.13(left)). Scattered and penetrated events are therefore responsible for the evolution of sensitivity with the distance and size of the source observed on Figure 4.12.

From the sensitivity measured without TEW correction (Fig. 4.12(up)), it can be seen that the use of the FWHM ROI on the 0.5 cm and 1 cm diameter cylinder causes a non negligible sensitivity decrease. It is probably due to the partial volume e↵ect which causes a rejection of an important amount of when fixing a ROI dimension that is too small compared to the image of the cylinder. This leads to the assertion that this ROI method is not suitable for small sources (Sec. 4.1.1.3). When using the FWTM method, the sensitivity of the small cylinders increases significantly to such an extent that it shows a higher sensitivity that the one achieved with large cylinders. This e↵ect is probably due to the relative integration of an higher amount of background pixels compared to bigger sources when using an ROI dimension bigger than the cylinder image. This behavior, causing an artificially higher e ciency for the smaller sources, lead us to state that even this method is not suitable for small lesions.

The use of the threshold method gives sensitivity values for the two small source that fall between those obtained with the other two methods, i.e. between 0.9 and 1.7 cps/Mbq. No significant trend di↵erences can be highlighted between the three ROI methods for large cylinders. We can demonstrate that even in this case, the use of the FWTM lead to a general increase in sensitivity because of the integration of background events (Fig. 4.4A) (this e↵ect is relatively higher for smaller sources). As introduced in Section 4.1.1.3, the FWHM method is preferable for these dimensions and provides sensitivity values similar to the threshold method (maximum di↵erence of 6% at all distances for this sources). resolution, we modeled the influence of the collimator structure on the sensitivity by reducing the density of the material (Fig. 4.14(right)) One can see that by decreasing the density, it is possible to reproduce the shape of sensitivity curves, i.e. an increase at short distances and for large source diameters, even if the absolute values of the sensitivity deviate from the experimental data. As previously seen, this increase in sensitivity at short distances is due to the scattering in the collimator. Therefore, it seems, as we have already observed for the spatial resolution, that the features of the collimator manufactured by 3D printing do not comply with those expected, particularly with regard to the e↵ective density of tungsten, which is lower. This di↵erence has little impact on the sensitivity measured with small sources that are less sensitive to di↵usion as mentioned above (Fig. 4.13).

Because the contribution of scatter is higher for large sources and short distances (Fig. 4.13), the impact of the TEW correction is more significant for these configurations (Fig. 4.12 (down)). The TEW correction applied on large cylinders makes the sensitivity independent of the source size when using the FWTM ROIs. In these configurations, the di↵erence in sensitivity is less than 2% for all distances. When using the threshold method, i.e. when adapting the ROI to the distribution of the image pixels and not on the image profile, the dependence of sensitivity on sources dimensions is greatly reduced for almost all cylinders. Following these considerations and thanks to the possibility of adapting the shape of the ROI to the conformation of the lesion, the sensitivity values obtained with the threshold method were used for the activity quantification, as described in Section 4.2.3. Finally, the figure 4.15 shows the sensitivity measured with the intervening medium and the threshold method, with and without TEW correction. One can note that the decrease in sensitivity with distance is well described by the theoretical attenuation of the attenuation medium (µ=0.111 cm 1 ) (Fig. 4.15(left)). The calibration factors C abs and C used for the quantification with single-view planar images were extrapolated from these distance-sensitivity curves. In the order: 1:2, 1:3, 1:4 and 1:6. In this case, the camera was moved to be centered to the lobe. The image saturation level of the small nodule images (up) is di↵erent from the ones of the big nodule (down).

Nodules detectability

a Figure 4.16 shows the anterior images of the small and big thyroid nodule, when filled with di↵erent activity concentration ratios between the nodule and the thyroid lobes. From this images we calculated the Contrast and Contrast to noise ratio, as define in Sec. 3.2.5.3. The results are reported in Fig. 4.17. As expected, the Contrast growth proportionally to the lobe-to-nodule concentration increase, as it clearly visible from the images in Fig. 4. [START_REF] Lassmann | Dosimetry and thyroid cancer: the individual dosage of radioiodine[END_REF]. For what concern the Contrast-to-noise ratio, all nodules achieved a values higher than 4, thus all are considered to be detectable. However, the small nodule is characterized by a value of 4.7±0.3, thus just above the detectability threshold. As will be better discussed after, the quantification protocol applied on this nodule has been simplified because of the wall that separated the nodule from the lobe, clearly visible in Figure 4.16.

Estimating the thickness and depth of structures of interest

a As introduced previously, we tested the camera's ability to measure the dimensions and positions of structures of interest (lobes and nodules). The thickness of lobes and nodules was measured, as well as their depth in the neck phantom. The results are summarized in Table 4.2 and compared to the actual values known from phantoms geometry.

For both lobes and nodules, the measured thickness is in very good agreement with the expected one (7.8% and 3.8% for the lobes of FANTHY 5y and 15y and 6% and 8% for the BN and SN, respectively, compared to the actual value). Thanks to the nodules spherical shape, their thickness can be also estimated from the images at 0 and 180 . This estimation leads to values of (19.20±1.25) mm and (9.65±0.78) mm at 0 and (22.43±2.25) mm and (13.69±3.02) mm at 180 for the BN and SN, respectively. The 14.4% and 29% degradation observed from the 180 angular view is related to the Chapter 4

Results

system spatial resolution increase with distance (Fig. 4.11). However, thanks to its compactness, the camera can be positioned optimally and always as close as possible to the structure of interest, in order to estimate accurately their dimension. Although these results highlight the capability of the camera to well determine the dimensions of a lesions, one can notice that small di↵erences in thickness values have a really small impact in quantification, as introduced in Sec. 4.2.3.1. Indeed, t j comes into play in the quantification process during the calculation of the factor f j , eq. 4.2 that, for the highest thickness found (Big Nodule, Tab. 4.2), deviates of less than 0.2% from the unity.

The measured depth of the nodules in the neck phantom is (15.22±0.69) mm and (18.20±1.07) mm for the SN and BN, respectively (Tab. 4.2). The actual values are 14.5 mm and 16.5 mm (Tab. 4.2). The small di↵erences could be related to slight misalignment of the camera with respect to the neck (in the order of a millimeter). As introduced in the previous paragraph, the estimation of nodules depth is important because it allows the calibration factor to be extrapolated from the sensitivity-distance curves. However, a small imprecision in the depth estimation such as the one reported (5% and 9% for the SN and the BN compared to the actual geometrical value) leads to variation in the calibration factor of less than 0.4% for both nodules (estimated from the sensitivity curve of the 1 cm and 2 cm diameter cylinder reported in Fig. 4.12). The measured values of the nodules depth have been used for the activity quantification.

One should notice that the robustness of the threshold methods as a tool for measuring the dimension and positioning of a lesion strongly depends on the activity in the structure : a higher nodule-to-lobe activity concentration ratio allows for easier threshold positioning. Thus, for an activity average value of more than 300 MBq as the typical one used during hyperthyroidism treatments (65% of uptake), the method could be considered as a useful tool for activity quantification.

Preliminary assessment of the camera ability to quantify

131 I with homogenous activity distribution a The results of the activity quantification study for the two healthy phantoms are summarized in Table 4.3, with and without TEW correction. Moreover, an histogram of all the results which helps to quickly visualize them is shown in Fig. 4.18.

An almost perfect activity quantification is achieved with FANTHY 15y when using the single view method with TEW correction. The recovery coe cient is almost 100% with the anterior image and 97% with the posterior view. The recovery factor measured with FANTHY 5y is around 96% for both views. The lowest recovery coe cient measured with TEW correction (around 94%) are obtained when using the posterior single view method with attenuation. This e↵ect is probably due to the use of a calibration factor extrapolated from the sensitivity-distance curve with attenuation, which was obtained using an absorbing media with a slightly di↵erent attenuation coe cient compared to the one of the neck phantom (0.111 cm 1 for the attenuations media used

Results

Chapter 4

NO TEW TEW RC (%)

A-SV P-SV A-SV abs P-SV abs CV during the calibration procedure and 0.116 cm 1 for the phantom). The underestimation of attenuation leads to underestimation of activity, which is more important at greater distance where the attenuation is higher. the calibration factor measured without attenuation media and post-correction of the attenuation gets better results because it uses the actual attenuation of the intervening media. The conjugate view protocol, in which both anterior and posterior views are used, allows to obtain a RC value of 96% and 98% for FANTHY 5y and FANTHY15y, respectively.

The RC values measured without TEW correction are on average 9.8% and 15.8% (FANTHY 5y and 15y respectively) higher than the expected. This shows that a greater amount of scatter is present inside the photopeak during phantom acquisition than during calibration with the cylindrical source, due to a larger scattering volume. Although this phantom corresponds to a relatively ideal (but realistic) clinical situation (no superimposition of background activity), the reported results highlight the optimized imaging features of our camera for activity quantification with very simple quantification methods. In light of these results, the quantification of nodule activity will be presented only using the single view method based on a calibration factor measured without attenuation media and the conjugate view method, both with TEW correction.

Results

Chapter 4

RC (%)

Correction 1

(1:2) (1:3) (1:4) (1:6) (1:2) (1:3) (1:4) (1:6)

A-SV P-SV CV

Preclinical planar imaging quantification of heterogenous activity distribution

a Table 4.4 summarizes the quantification of the nodule activity performed on the images obtained with the pathological thyroid phantom. The third and fourth columns of this table contain the activity recovery coe cients without background correction (in the form of a F or F 0 factor, as discussed in Sec. 4.2.3.2). Fig. 4. 19(top) shows an histogram of the obtained results.

It can be noted that due to the partial overlay of nodules with thyroid lobes, the RC is almost always greater than 100%. As expected, this overestimation of the activity in the nodule is all the more important when the lobe-to-nodule activity concentration ratio is low, i.e. when the relative contribution of the activity from the lobe to the image of the nodule increases (109% and 116% with A-SV method for a ratio of 1:2 compared to 98% and 111% for a ratio of 1:6, for the SN and BN nodules, respectively). Moreover, when the lobe-to-nodule activity concentration ratio is low, the partial volume e↵ect strengthens the overestimation of the radiotracer uptake estimate in the nodule (the spill-in of activity from the surrounding thyroid tissue is larger than the spill-out of nodule activity to the neighboring regions). Due to its anatomic position and size in relation to the lobe (Fig. 4.6), the 2 cm diameter nodule shows very low volumes overlay and RC values very close to 100% are achievable even without background correction. When considering a concentration ratio of 1:6, RC values significantly lower than 100% are obtained (Tab. 4.4). In these conditions, the partial volume e↵ect leads to underestimation of the radiotracer uptake estimate in the nodule (the spill-out of nodule activity to the neighboring regions is larger than the spill-in of activity from surrounding thyroid tissue).

For geometric reasons, the e↵ect of activity overlay and the impact of the partial volume e↵ect is even more evident with the small nodule. This leads to an overestimation of the nodule activity of 10% (1:6 ratio) to 16% (1:2 ratio). As shown in Figure 4.10(down-right), an estimation of the lobe thickness superimposed to the nodule can be performed from the lateral image, which allows a background correction as the one described in Section 4.2.3.2 to be implemented. Two corrections are made. The first one using a F factor in which the lobe thickness t from the lateral images and a I ADJ value (estimated from background ROIs drown on the lobe and then rescaled to the nodule ROI) are used. In the second correction, F was estimated in the same way but by dividing the I ADJ value by two, considering, from the anterior image of the phantom, that only about half of the 1 cm diameter nodule is superimposed on the lobe (Fig. 4.9) The results of these corrections are summarized in the fifth and sixth columns of the Table 4.4. It can be seen immediately that the first correction leads to huge underestimation of the nodule activity due to an overestimation of the background. The use of the second correction, although not based on a precise estimate of volume overlay, leads to RC values of the SN activity close to 100% except for the lowest lobe-to-nodule activity concentration ratio. The correction methods were not applied to the activity The very good results achieved during these preliminary assessment of the mobile camera ability to quantify 131 I with homogenous or heterogeneous activity distributions demonstrate that its performance are suited to achieve accurate quantification in a fairly realistic clinical context. First of all, the high spatial resolution of the camera reduces the partial volume e↵ect, since the high recovery coe cients of activity in nodules were Chapter 4

Results

obtained without any correction of its influence. Indeed, the high spatial resolution allows more accurate selection of events inside ROIs by using advanced methods with a negligible impact of spatial volume e↵ect. This advantage is illustrated in Fig. 4.20, which compares the images of the two healthy thyroid phantoms obtained with our mobile camera and a standard gamma camera equipped with a high-energy collimator (Siemens Symbia T2 GC, SR=13.4 mm at 10 cm). The high spatial performance of the camera and its ability to be placed close to the structure of interest also allows an accurate estimation of the position and dimensions of this structure which are then used to improve quantification. This preliminary study also highlights the importance of accurate calibration for quantification. This calibration resulted in promising quantification results even with a single view method and a simple TEW correction. More marginally, we also shown how a simple background correction can help to improve quantification.

However, the achievement of adequate recovery coe cient values cannot be considered as an evidence of a complete reliability of the quantification process. These results show the excellent ability of the camera to be used for activity quantification but furthers investigation, in particular with the use of smaller nodules and larger volume overlay with the lobes could help to better test it in a more realistic clinical situation. Moreover, these results were also facilitated by an intrinsic reduced partial volume effect of the pathological phantom due to the wall that separated the nodule from the lobe, clearly visible in Figure 4.16. For reasons of simplification, we also considered the presence of the absorbing media all over the space between the thyroid and the camera, which is not true considering both the curved shape of the neck and the small air gap. This approximation is expected to have less influence on the activity quantification using the posterior view.

Chapter 5

Conclusions and prospectives a

Molecular radiotherapy is a growing alternative in the treatment of tumors and benign diseases. One of the main challenges of MRT is the accurate individual determination of the dose absorbed by the organs-at-risk and target regions, in accordance with the European Directive 2013/59/Euratom, in order to set the activity to be administered for each patient, as currently done for external radiotherapy and brachytherapy. In molecular radiotherapy, the determination of absorbed doses is based on the MIRD (Medical Internal Radiation Dose) formalism. This established method relies on quantitative measurements of radiopharmaceutical biokinetics in the target region and the organs-at-risk on images acquired at a number of times after administration of the radiopharmaceutical. At present, images are mainly performed using planar scintillation camera or SPECT (Single Emission Computed Tomography). The current limitations of dosimetry protocols are mainly due to the fact that most cameras are not optimized, in terms of spatial resolution, dead time and ergonomics, to detect high activity of high-energy gamma emitters during treatment [START_REF] Ljungberg | 3-D Image-Based Dosimetry in Radionuclide Therapy[END_REF].

To this aim, we have developed a high-resolution mobile gamma camera specifically designed to improve the quantitative assessment of the heterogeneous distribution and biokinetics of 131 I after treatment administration. Beyond The compactness and mobility of the camera should o↵er the possibility to improve image contrast by reducing the distance between the detector and the targeted structures (thyroid, tumor remnants, metastatic nodes or salivary glands) and to optimize their detectability by adjusting the angular view. Moreover, the availability of the camera and the ability for using it at the patient's bedside should be compatible with an accurate temporal sampling of the biokinetics of the tracer. The first prototype of the mobile camera exhibits a 5⇥5 cm 2 field of view and consists of a high-energy parallel-hole collimator, made with 3D printing, coupled to a continuous inorganic scintillator, readout by an array of Silicon Photomultiplier (SiPMs) detectors.

The masterpiece of the camera is its tungsten collimator, which largely determines the spatial resolution and sensitivity of the overall detector device. The geometry of the parallel-hole collimator in the energy range of 131 I was optimized with Monte Carlo simulations, with a special focus on the detailed analysis of the spatial and energy distribution of penetration and scattered components that have a huge impact on both image contrast and quantification. The chosen geometry shows a very low septal penetration, an high intrinsic spatial resolution of 2 mm FWHM at 5 cm and an e ciency of 1.24 10 5 . Several inorganic continuous crystals of CeBr 3 with di↵erent optical coatings have been also deeply investigated during this work. We have shown that a 6 mm thick CeBr 3 with reflective edges o↵ers the best compromise in terms of energy resolution (7.68±0.01 %), sub-millimetric spatial resolution (0.74±0.28 mm) and very low spatial distortion (0.15±0.10 mm) in the UFOV. All detection elements have been integrated to design a fully operational camera and its performance were evaluated as a function of the operating parameters, such as bias voltage ad temperature.

The first feasibility prototype of the camera was calibrated with 131 I phantoms, which is a mandatory prerequisite for quantitative imaging dedicated to dosimetryguided radionuclide therapy. The system calibration leads to an overall spatial resolution of 3.14±0.03 mm and a sensitivity of 1.23±0.01 cps/MBq at a distance of 5 cm. This sensitivity decreases with distance and slightly changes with source size due to the influence high-energy scattering inside the collimator. The calibration procedure also stressed the importance of the method used to define the region of interest around the target structure in order to accurately quantify the accumulated activity. A threshold method, which allows to optimize the ROI according to the specific image intensity distribution, was implemented. In order to increase step by step the complexity of the image quantification protocol, the first preclinical planar studies involved the use of two di↵erent 3D-printed thyroid phantoms homogeneously filled with 131 I . To go further and to model spatial heterogeneities of the activity distribution, we performed quantification on small nodules partially embedded in the thyroid. Activity recovery factors that deviate of around 2% from the unity have been found for all structures, except for the smallest nodules (1 cm diameter) when they are filled with very low lobe-to-nodules uptake ratio (1:2). These preliminary results are very promising for the use of our camera as a tool for accurate quantification of absorbed doses. They first point out that the high spatial resolution of the camera reduces the partial volume e↵ect for small structures and leads to a better definition of ROI. Its imaging capabilities and compactness also provide useful and accurate anatomical information on structures of interest (depth and thickness to within 5% accuracy) that can be used to improve quantification by correcting attenuation, source self absorption or activity superimposition.

The next step toward the goal of using the camera in real clinical practice is to integrate a 10⇥10 cm 2 FoV detector, perfectly suited to the size of the thyroid gland even when enlargement occurs (Graves disease). To maintain or increase the desired objectives in terms of performance and ergonomics, both scintillator and photodetector optimization are planned to be realized. Since crystal growth techniques currently limit the maximum size of crystal to a diameter of 3 inches, the final scintillator will probably consist of several crystal plates assembled in the same mechanical housing. In this case, particular attention will be paid to the choice of optical coupling between the crystals in order to minimize the impact of segmentation on the collection and spatial distribution of scintillation light. Increasing the camera's field of view to a surface area of 10⇥10 cm 2 will also require increasing the sensitive surface of the photodetector array, while maintaining the number of electronic channels and therefore the cost of the device. This will be achieved by using SiPM arrays with 6⇥6 mm 2 elements instead of the current 3⇥3 mm 2 SiPMs.

Although we know that sensitivity is not a key element for imaging during treatment due to the high level of activity administrated, the ultimate objective is also to benefit from the imaging abilities of the camera during treatment planning, which uses tracer activities. In order to improve the versatility of the camera performance, two interchangeable detection heads will therefore be developed : a high-sensitivity configuration for imaging during treatment planning of hyperthyroidism or imaging of metastatic lymph nodes and salivary glands during treatment of thyroid cancers and a low-sensitivity configuration for estimating the absorbed dose in thyroid or tumor remnants after treatment administration. The intrinsic sensitivity of the camera will be also enhance by increasing the thickness of the CeBr 3 scintillator from 6 mm to 10 mm (and potentially 15 mm), which makes it possible to double sensitivity without degrading spatial and energy performance, as shown in our study. However, this increase in thickness, combined with the use of larger SiPM photodetectors, will probably have an impact on the measurement of the scintillation light distribution. This e↵ect should be accurately studied and sophisticated reconstruction algorithms of the interaction point, such as artificial neural networks, should be implemented, in order to maintain the expected intrinsic spatial resolution and low image distortion. Beyond the field of view, it is also necessary to significantly improve the counting capabilities of the photodetection system. Indeed, at the moment, the camera is characterized by an acquisition rate limited to 3 kCps when all data are displayed in real-time on the acquisition software. New communication protocols should be implemented to achieve the objective of a counting capacity of about 400 kHz in order to open the door to early imaging, just after the administration of large therapeutic activities.

To go further, the quantification protocol should also be optimized to cover a wider range of clinical situations, including smaller nodules and greater volume overlap with the lobes. Imaging smaller structures will increase the impact of partial volume e↵ect. The images we obtained during the calibration of the camera (five cylinders with di↵erent diameters) could help to implement the commonly used partial-volume correction (PVC) [START_REF] Dewaraja | MIRD Pamphlet No. 24: Guidelines for Quantitative 131i SPECT in Dosimetry Applications[END_REF], in the form of empiric recovery coe cients calculated from the ratio of the observed count rate measured within a ROI matching the true volume of the phantom with the count rate of all counts originating from the phantom. Although the high spatial resolution of our camera limit spill e↵ects, this correction remains essential for the accurate estimate of the dose delivered in targets with a diameter smaller than 1 cm an, more generally, when the distribution of activity is very heterogenous. Bigger nodule-to-lobe overlapping will decrease nodules detectability and increase the quantification bias. An accurate background correction could be performed by identifying volumes superimposition. These information could be estimated by performing images at several angles. For these purposes the compactness of our camera will be a key features. The definition of the optimal number of angles of incidence to obtain accurate dosimetry should be performed, always trying to minimize the total acquisition time. As well as providing a way to deal with issues of under or overlying background activity, tomographic acquisitions performed with our high resolution camera could help to precisely outline overlapped lesions and to better delineate heterogeneous uptakes of activity in a lesion. Accuracy and robustness of the quantification protocol (including scatter, partial-volume e↵ect and dead time corrections, number of angular views, ...) will be evaluated by using a set of 3D thyroid phantoms with inserts of di↵erent shape and sizes modeling nodules or tumor remnants and will be compared to the performance achieved with a conventional gamma camera equipped with a high-energy collimator.

Even after optimizing the imaging quantification protocol, many other steps need to be undertaken to obtain absorbed dose quantification. Among them, we can mention the realization of multiple images over time to estimate the cumulated activity. These steps could be firstly tested with the use of phantoms, which implies that the biokinetics would only integrate the physical period of iodine. Secondly, the biokinetics including the biological clearance could be considered during the clinical evaluation of the camera. In the case of thyroid cancers, several images of the thyroid remnants will be performed at di↵erent times after the treatment. In the case of hyperthyroidism, images will be done during the patient-specific treatment planning and after treatment administration. One the challenge will be to determine the best methodological way to integrate di↵erent type of activity measurements (planar, SPECT, counting probe, mobile camera) in the absorbed dose evaluation trying to reduce the uncertainties.

As far as quantification is concerned, this work represents only the first step towards the development of a device capable to give access to the absorbed dose during thyroids treatments. Surely, the interest of personalized dosimetry goes well beyond thyroid and is even more essential for more targeted treatments (i.e. alpha emitters). However, the promising results open the door to new developments and optimizations than could help the a rmation of pre and post-treatment dosimetry in clinical practice.

Appendix A Résumé

La radiothérapie moléculaire (RTM), également appelée radiothérapie interne vectorisée, est une approche e cace et en pleine expansion pour la gestion du cancer et des maladies bénignes. La capacité d'administrer sélectivement des doses contrôlées aux cellules est donnée par l'utilisation d'un radio-pharmaceutique, combinaison d'un radionucléide instable et d'une molécule appelée vecteur. Ce vecteur se lie spécifiquement aux cellules cibles par des mécanismes de localisation qui peuvent être aussi simples qu'une di↵usion métabolique ou aussi complexes qu'une réaction antigène-anticorps, tandis que le radionucléide émet des radiations par sa désintégration sous forme de particules chargées (principalement des électrons issus de la désintégration bêta ou des particules alpha), qui induisent des e↵ets thérapeutiques par la ionisation des tissus qui déclenche des mécanismes radiobiologiques pouvant entraîner la mort cellulaire. Le but de la radiothérapie moléculaire est d'infliger des dommages létaux à des cellules cibles tout en limitant les potentiels e↵ets secondaires sur les tissus normaux environnant à des niveaux tolérables. En raison de l'hétérogénéité de la biodistribution des radionucléides entre les individus -chaque patient ayant des absorptions et des clairances (une clairance à mi-temps) radiopharmaceutiques di↵érentes -une dosimétrie personnalisée doit être une condition préalable à un traitement par radiothérapie moléculaire [START_REF] Eanm | Internal Dosimetry Task Force Report on: Treatment Planning For Molecular Radiotherapy: Potential And Prospects[END_REF]. En e↵et, des doses di↵érentes délivrées aux patients entraîneront des réponses di↵érentes au traitement, tant en termes d'e↵ets que de toxicité [START_REF] Stabin | State of the Art in Nuclear Medicine Dose Assessment[END_REF][START_REF] Strigari | The evidence base for the use of internal dosimetry in the clinical practice of molecular radiotherapy[END_REF][START_REF] Flux | From fixed activities to personalized treatments in radionuclide therapy: lost in translation?[END_REF].

La dosimétrie dans les RTM nécessite l'estimation in vivo précise de la distribution et de la pharmacocinétique du produit radiopharmaceutique en fonction du temps, ainsi que des méthodes de conversion du nombre de désintégrations dans un volume donné en la dose absorbée dans ce volume. En clinique, les mesures peuvent être e↵ectuées à l'aide de l'imagerie planaire ou de la tomographie par émission monophotonique (TEMP) généralement en combinaison avec une tomodensitométrie (TDM). En amont, sur la base d'une évaluation de la dose pré-traitement, ces estimations doivent optimiser la planification du traitement individuel. L'activité maximale pouvant être administrée à chaque patient de façon spécifique et peut alors être définie en fonction des objectifs du traitement et des doses de tolérance aux organes à risque. De plus, l'évaluation post-traitement de la dose absorbée délivrée aux tissus ciblés et non Résumé ciblés est également essentielle pour a ner la planification du traitement. Elle permet d'établir des corrélations avec la dose prévue dans l'étude pré-traitement mais aussi avec la réponse clinique en termes de contrôle de la maladie et de toxicité sur les organes à risque.

Bien que des décennies de développement aient permis que la planification de traitements basée sur des calculs dosimétriques et la vérification dosimétrique postthérapeutique soient aujourd'hui une routine clinique lors de traitements de radiothérapie externe, ceci n'est toujours pas le cas pour la radiothérapie moléculaire [START_REF] Strigari | The evidence base for the use of internal dosimetry in the clinical practice of molecular radiotherapy[END_REF][START_REF] Chiesa | The conflict between treatment optimization and registration of radiopharmaceuticals with fixed activity posology in oncological nuclear medicine therapy[END_REF].

Cette di culté implique plusieurs défis. Le premier, et le plus important, est la quantification précise de l'activité dans le temps dans les régions cibles [START_REF] Flux | The Impact of PET and SPECT on Dosimetry for Targeted Radionuclide Therapy[END_REF] considérée comme la principale source d'erreur en dosimétrie par traitement par radionucléides [START_REF] Dewaraja | MIRD Pamphlet No. 23: Quantitative SPECT for Patient-Specific 3-Dimensional Dosimetry in Internal Radionuclide Therapy[END_REF]. Actuellement, les limitations sont dues à la mauvaise détection, par les caméras SPECT, des émetteurs gamma avec une énergie supérieure à 200 keV ( 131 I, 90Y et 177Lu), généralement utilisés lors des traitements [START_REF] Ljungberg | 3-D Image-Based Dosimetry in Radionuclide Therapy[END_REF]. Même en utilisant des collimateurs hautes énergies, leur résolution spatiale limitée qui varie de 10 mm à 20 mm à une distance de 10 cm, empêche la quantification de petites distributions hétérogènes d'activité, ce qui peut fortement influencer la précision de la dosimétrie à l'échelle macroscopique [START_REF] Lassmann | The Relevance of Dosimetry in Precision Medicine[END_REF]. De plus, la dosimétrie pose également des défis logistiques et pratiques, car les images doivent être acquises à di↵érents temps après le traitement et nécessitent de fait plus d'étapes qu'un seul examen diagnostique en médecine nucléaire [START_REF] Flux | From fixed activities to personalized treatments in radionuclide therapy: lost in translation?[END_REF]. Actuellement, l'acquisition de données doit être limitée à quelques points dans le temps, ce qui conduit à un manque de précision dans la définition de la courbe temps-activité, ajoutant de grandes incertitudes sur la cinétique et sur le calcul de la dose absorbée.

L'objectif de ma thèse était de proposer de nouvelles approches instrumentales visant à renforcer le contrôle de la dose délivrée aux patients lors d'un traitement de radiothérapie moléculaire. Ceci est réalisé en réduisant les incertitudes associées à la quantification de l'activité (et donc au calcul de la dose absorbée) grâce à l'utilisation d'un système d'imagerie compact et hautement optimisé. Ce travail a été réalisé au sein de l'équipe Instrumentation et Imagerie Radio-Isotopique Clinique et pré-clinique (IIRIC) du laboratoire Imagerie et Modélisation en Neurobiologie et Cancérologie (IMNC) basé à Orsay, en France. Il consistait à mettre au point et à optimiser une gamma-caméra mobile miniaturisée à haute résolution spécialement conçue pour améliorer l'évaluation quantitative individuelle de la distribution hétérogène et de la bio-cinétique du radiotraceur avant et après administration du traitement. L'étude était axée sur le traitement des maladies bénignes et malignes de la thyroïde à l'aide de l' 131 I . Grâce à son tropisme sélectif pour les tissus thyroïdiens, l'utilisation de l' 131 I a conduit à une grande e cacité des traitements avec une faible toxicité pour les tissus sains [START_REF] Stabin | Uncertainties in Internal Dose Calculations for Radiopharmaceuticals[END_REF]. Cependant, l'importance de l'approche dosimétrique dans le traitement des maladies thyroïdiennes est toujours prononcée afin d'optimiser l'activité administrée en fonction du résultat clinique souhaité (euthyroïdie ou hypothyroïdie) ou pour éviter les e↵ets secondaires indésirables et les conséquences à long terme [START_REF] Stokkel | EANM procedure guidelines for therapy of benign thyroid disease[END_REF][START_REF] Lassmann | EANM Dosimetry Committee series on standard operational procedures for pre-therapeutic dosimetry I: blood and bone marrow dosimetry in di↵erentiated thyroid cancer therapy[END_REF]. La simplicité de la méthode des traitements et son utilisation courante en font une application clinique idéale pour fournir une abondance de données permettant d'évaluer la corrélation dose-réponse A.1. ENJEUX DU PROJET avec nos nouvelles méthodes, ce qui pourrait mener, à long terme, à une approche thérapeutique spécifique au patient [START_REF] Lassmann | Dosimetry and thyroid cancer: the individual dosage of radioiodine[END_REF].

A.1 Enjeux du projet L'objectif final du projet est de développer une gamma-caméra mobile avec un champ de vue de 10x10 cm 2 , adapté à la taille de la thyroïde. La camera est composée d'un collimateur haute énergie à trous parallèles, couplé à un scintillateur inorganique continu, lu par des matrices de photomultiplicateurs en silicium (SiPM).

La compacité et la mobilité de la caméra devrait permettre d'améliorer la résolution de l'image en réduisant la distance entre le détecteur et les structures ciblées (thyroïde, restes de tumeurs, ganglions métastatiques ou glandes salivaires). La possibilité de l'utiliser au chevet du patient (par exemple dans la chambre d'isolement) doit également être compatible avec un échantillonnage temporel précis de la biocinétique du traceur après l'administration du traitement.

Contrairement aux caméras à grand champ de vue utilisée en clinique, le dispositif doit être optimisé, en termes de réponse spatiale et énergétique et de capacité de comptage, pour répondre aux besoins spécifiques de l'imagerie à rayons gamma à haute énergie et à haut taux de fluence photonique, comme dans les traitements de la thyroïde à l'aide de l' 131 I. En raison de la grande quantité de radioactivité administrée pendant le traitement des maladies thyroïdiennes bénignes ou malignes, la précision de la quantification de l'activité dépend moins de la sensibilité du système d'imagerie car la statistique de comptage accumulée sur l'image pendant un temps raisonnable (environ 10 minutes) sera su samment élevée. La priorité a été donc donnée à la résolution spatiale et énergétique de la caméra.

L'objectif de performance est d'atteindre une résolution spatiale du système, c'està-dire la capacité de la caméra à distinguer deux objets proches, comprise entre 3 et 5 mm FWHM (à une distance de 5 cm), bien meilleure que celle des gamma-caméras classiques équipées avec des collimateurs hautes énergies (environ 1.5 cm à 364 keV). Cette résolution spatiale devrait permettre de quantifier la distribution hétérogène du radiopharmaceutique dans les tissus ciblés et donc d'améliorer la précision de la dosimétrie à l'échelle macroscopique en réduisant les e↵ets de volume partiel. La résolution en énergie définit la capacité du détecteur à discriminer les radiations émises à di↵érentes énergies. Une haute résolution énergétique permet de distinguer les photons primaires non diffusés, qui peuvent être utilisés pour former une image précise de l'activité dans l'objet étudié, des photons di↵usés (générés par e↵et Compton) principalement produits dans les tissus ou dans le collimateur. Ce signal forme un bruit de fond qui peut fortement réduire le contraste de l'image. Notre objectif est d'atteindre une résolution énergétique inférieure à 8% afin de minimiser la contamination par les événements hautes énergies di↵usés. La capacité de comptage de la caméra est également un facteur important pour réduire l'e↵et du temps mort sur les estimations de l'activité. Le prototype final de la caméra mobile devrait fournir un taux de comptage maximal d'environ 400 kCps A.2. CONCEPTION ET OPTIMISATION DE LA GAMMA-CAM ÉRA MOBILE afin de permettre la réalisation d'images juste après l'administration de plusieurs GBq d'activité [START_REF] Dewaraja | MIRD Pamphlet No. 24: Guidelines for Quantitative 131i SPECT in Dosimetry Applications[END_REF].

A.2 Conception et optimisation de la gamma-caméra mobile

La preuve de concept présentée dans ce travail a été réalisée sur un premier prototype avec un champ de vue de 5x5 cm 2 . Pour atteindre les objectifs souhaités en termes de performances et d'ergonomie, nous avons choisi de construire la caméra autour d'un collimateur hautes énergies à trous parallèles, couplé à un scintillateur inorganique continu. Le scintillateur est lu par une technologie récente et particulièrement bien adaptée basée sur des matrices des photomultiplicateurs au silicium (SIPM) associés à une électronique multicanaux miniaturisée dédiée. L'optimisation du design de la camera a été focalisée sur le choix de la nature et de la géométrie du scintillateur, du revêtement optique du scintillateur et sur l'optimisation du collimateur par simulation Monte Carlo.

Le collimateur

Le collimateur est l'élément central d'une gamma caméra car il détermine en grande partie les performances globales. Son rôle principal est d'absorber les rayons gamma émis par la source qui ne se propagent pas dans la direction souhaitée afin d'obtenir une image correspondant à la projection planaire de l'activité présente dans l'objet étudié, influençant ainsi fortement la sensibilité et la résolution spatiale du système. Son importance est encore plus marquée pour les images avec de l' 131 I ou encore pour les isotopes de médecine nucléaire courants tels que 99mTc, en raison de l'énergie plus élevée du photopic (364 keV) et des multiples émissions de rayons gamma qui di↵usent et pénètrent dans le collimateur, dégradant considérablement la résolution spatiale, le contraste et donc la quantification.

L'optimisation de la géométrie du collimateur à trous parallèles a été réalisée à l'aide de simulations Monte Carlo basées sur la plateforme de simulation GATE [START_REF] Jan | GATE: a simulation toolkit for PET and SPECT[END_REF]. Les propriétés intrinsèques du collimateurs à trous hexagonaux ont été étudiées avec une source linéaire d' 131 I en termes de résolution spatiale (SR), d'e cacité (E) et de pénétration septale (SP). Une caractérisation précise de l'énergie et de la distribution spatiale des événements de di↵usion et de pénétration a elle aussi été e↵ectuée. Ils sont les principaux facteurs qui dégradent à la fois la qualité de l'image et la quantification de la dose et ils ne peuvent être estimés à partir de modèles analytiques conventionnels à cette énergie (364 keV) [START_REF] Anger | Scintillation Camera with Multichannel Collimators[END_REF]. Parmi toutes les configurations simulées, cinq configurations avec un faible taux de penetration et de di↵usion et une resolution spatiale allant de 1 à 5 mm ont été choisies en fonction des résultats des simulations et grâce a des méthodes d'interpolation. Les propriétés d'imagerie de la camera ont ensuite été A.2. CONCEPTION ET OPTIMISATION DE LA GAMMA-CAM ÉRA MOBILE étudiées par simulation Monte Carlo en fonction des collimateurs dans un contexte clinique plus réaliste simulant le traitement de la thyroïde à l' 131 I en utilisant un fantôme voxélisé. Cette etude est celle qui nous a permis de choisir la configuration finale. L'optimisation du collimateur sera brièvement présentée dans la section A.4.

Le scintillateur

Le scintillateur convertit les rayons gamma qui n'ont pas été absorbés par le collimateur en photons de longueur d'onde optique, qui sont ensuite détectés par le photodétecteurs.

Le choix du scintillateur utilisé pour intégrer le premier prototype de la caméra a été e↵ectué en deux étapes. Tout d'abord une vaste collection de documents disponibles sur les scintillateurs (articles et fiches techniques) a été faite. À la suite de cette enquête, le CeBr 3 (Ce) a été choisi pour mettre en oeuvre le premier prototype de la caméra mobile car il répond le mieux aux besoins spécifiques de notre projet. Les avantages du CeBr 3 (Ce) sont son rendement lumineux élevé (60 ph/keV) avec un temps de décroissance rapide (17 ns);une excellente proportionnalité du rendement lumineux et une résolution énergétique élevée (4.1% à 662 keV). De plus, il n'a pas de radioactivité intrinsèque, ce qui réduit la contribution du bruit de fond au taux de comptage et améliore ainsi le contraste de l'image. Le principal inconvénient du CeBr 3 (Ce) est son pouvoir d'arrêt modéré pour les rayons gamma de haute énergie (6.3%) par rapport aux scintillateurs à base de luthetium (22.1% pour le LYSO(Ce)). Cependant, comme mentionné précédemment, la sensibilité de la caméra mobile n'est pas un élément clé de l'imagerie dosimétrique après l'administration de l' 131 I, en raison de la grande quantité de radioactivité utilisée lors du traitement des maladies thyroïdiennes bénignes ou malignes. La priorité a donc été donnée au rendement lumineux du scintillateur afin d'optimiser la performance spatiale et énergétique de la caméra. Un autre inconvénient de CeBr 3 (Ce) est son hygroscopicité, qui exige qu'il soit scellé dans un boîtier en aluminium avec une fenêtre en verre mince. Cette fenêtre, qui recouvre la sortie du scintillateur, agit comme un di↵useur optique qui influence l'extraction de la lumière vers le photodétecteur et donc la résolution spatiale et énergétique de la gamma-caméra. Ensuite, afin d'étudier l'impact de l'épaisseur des cristaux et du revêtement optique sur l'extraction de la lumière et donc sur les performances intrinsèques de la caméra, quatre CeBr 3 (Ce) de 5x5 cm 2 (Scionix) ont été testés expérimentalement: deux épaisseurs, 0.6 cm et 1 cm, et deux revêtements optiques recouvrant les bords du cristal, absorbant et réfléchissant. Leurs caractéristiques intrinsèques ont été comparées à celles d'un échantillon de LaBr3 3 (Ce) de 6 mm d'épaisseur à bords absorbants. Geiger reliée en série à une résistance de décharge, connectées en parallèle sur un même substrat de silicium. Le photodétecteur utilisé pour la camera est composé de quatre matrices monolithiques Hamamatsu S13361-3050NE-08 couvrant une surface totale de 51.6x51.6 mm 2 et une zone sensible de 48x48 mm 2 . Chaque matrice est composée de 8x8 pixels avec une zone photosensible e↵ective de 3x3 mm 2 , composés de 3600 cellules de 50 µm de dimension latérale, Les performances des SiPMs (e cacité de détection, gain, probabilité de crosstalk et d'afterpulses, bruit d'obscurité) sont directement liées à la surtension aux bornes du détecteur (di↵érence entre la tension d'alimentation et la tension de seuil d'avalanche), mais dépendent également de la température, ce qui impose la mise en oeuvre d'un système de compensation. Les quatre matrices sont montées sur un seul circuit imprimé contenant des éléments passifs (résistance, capacités) pour la polarisation, ainsi que quatre capteurs thermiques numériques, à l'arrière. L'électronique de lecture miniaturisée contient huit ASIC EASIROC à 32 canaux avec une électronique frontale entièrement analogique dédiée au gain trimming et à la lecture des SiPMs [START_REF] Hunter | Review of relative biological e↵ectiveness dependence on linear energy transfer for low-LET radiations[END_REF]. La charge analogique est numérisée en utilisant un CAN externe à deux canaux avec une performance dynamique de 12 bits et un taux d'échantillonnage de 2 Msamples/s. Le front-ed est monté sur trois cartes mères avec 256 canaux combinant un FPGA (Altera Cyclone III), un FTDI FT2232H (protocole USB 2.0 Hi-Speed, 480 Mbit / s) pour la communication avec un PC et les alimentations (y compris le convertisseur DC / DC, qui fournit la haute tension aux SiPM). L'uniformité de la réponse des matrices SiPM est améliorée par un CNA à 8 bits (0-2,5 V) intégré à l'entrée des ASICs, ce qui permet de régler individuellement chaque tension de polarisation des SiPMs. Ce CNA est également utilisé pour compenser avec précision l'influence de la variation de température sur les performances du SiPM, notamment le gain intrinsèque et l'e cacité de la photodétection. Les quatre capteurs thermiques sont connectés au FPGA, qui ajuste en temps réel le convertisseur DC/CC en fonction de la température moyenne afin de maintenir une surtension constante sur les SiPM (précision de 35 mV). Le système de photodétection et l'électronique de lecture sont positionnés dans un boîtier en aluminium de dimensions extérieures 6.5x6.5x5 mm 3 .

Le photodetecteur et l'électronique d'acquisition

A.3 Performances de la tête de la camera

Les quatre cristaux de CeBr 3 et le LaBr 3 présentés dans la section A.2 ont été couplés au système de photodétection et étudiés en termes de réponses spatiales et énergétiques. Pour plus de facilité, nous les identifierons à l'aide d'abréviations : C6A (0.6 cm d'épaisseur, bords absorbant), C6R (0.6 cm, bords réfléchissant), C10A (1 cm, bords absorbant), C10R (1 cm, bords réfléchissant) et L6A (LaBr 3 , 6 mm, bords absorbant).

A.3.1 Performances intrinsèques de la tête de la camera

Les performances intrinsèques de l'ensemble scintillateur-photodétecteur ont été évaluées à l'aide d'une source de 133 Ba (356 keV) collimatées avec un collimateur en tungstène de 4 mm (performances spectrométriques) ou de 0.5 mm (performances spatiales) de diamètre et monté sur une plateforme motorisée. Toutes les mesures ont été e↵ectuées à 25 C, à l'intérieur d'une chambre à température contrôlée. La polarisation des SiPMs a été optimisée pour les performances de résolution énergétique et réglée à 69 V. Pour chaque scintillation détectée, la position d'interaction du rayonnement gamma dans le scintillateur est reconstruite à l'aide d'une méthode d'ajustement itérative basée sur le modèle analytique de Scrimger-Becker de la distribution de la lumière et un algorithme non linéaire des moindres carrés de Levenberg-Marquardt [START_REF] Levenberg | A method for the solution of certain non-linear problems in least squares[END_REF], après l'application d'un seuil en charge sur le contenu des pixels pour diminuer l'influence des queues de la distribution et du bruit électronique. L'énergie déposée par le rayon gamma incident est supposée être proportionnelle à la somme de l'intensité lumineuse dans tous les pixels. Les valeurs moyennes et les écarts-types des propriétés sont mesurés dans le champ de vision utile (UFOV) ; dans le champ de vision centrale (CFOV, 75% de l'UFOV), définie selon les normes NEMA [313] ; et dans la zone complémentaire du CFOV, le CFOV , définie pour souligner les performances intrinsèques des scintillateurs près des bords.

Les performances spectrométriques ont été évaluée à l'aide de la source collimatée (4 mm de diamètre), qui a été déplacée sur tous les champs de vue possibles du scintillateur e↵ectuant un scan 10⇥10 avec un pas de 5 mm. La résolution énergétique (RE) et la position du photopic sont calculés à partir d'un double ajustement gaussien e↵ectué sur les spectres énergétiques. Pour les performances spatiales, la source collimatée (0.5 mm) a été déplacée sur tous les champs de vue possibles du scintillateur en e↵ectuant un scan 17⇥17 avec un pas de 3 mm. La position de la source et la largeur à mi-hauteur (LMH) sont calculées à partir d'un ajustement gaussien sur la projection du pic de mesure le long des axes x et y. La distorsion spatiale est estimée par la di↵érence absolue entre la position reconstruite et la position mécanique de la source. La résolution spatiale (RS) est estimée à partir de la LMH corrigée grâce à la distorsion locale et déconvoluée par le diamètre du collimateur. Pour chaque ligne (direction x) et colonne (direction y), un ajustement linéaire surnommé best fit a été e↵ectué sur une section de courbe tenant compte de 13 positions reconstruites en fonction de 13 positions mécaniques. Ces ajustements permettent d'obtenir la Distorsion Spatiale Intrinsèque Locale horizontale et verticale (DSIL x et DSIL y ), calculée comme la distance horizontale (verticale), en mm, entre la position reconstruite du spot et la courbe du best fit. La valeur maximale du DSIL est identifiée comme la Linéarité Absolue (LA), selon les normes NEMA [313].

A.3.2 Résultats

Afin de comparer qualitativement la réponse relative des cinq cristaux en termes de rendement lumineux, les spectres d'énergie d'un seul point central de 133 la quantité de lumière recueillie lorsque des réflexions se produisent. De plus, lorsque l'épaisseur du cristal passe de 6 mm à 1 cm, la quantité de lumière recueillie diminue. Les performances des di↵érents cristaux à 356 keV sont présentées dans le Tableau A.1. Le LaBr 3 montre le meilleur RE avec une amélioration relative par rapport au C6R de 8.3% et 6% à 356 keV dans l'UFOV et CFOV, respectivement. Le C6R montre la meilleure performance spectrométrique parmi les quatre échantillons de CeBr 3 , avec un RE de (7.86±0.03)% à 356 keV dans le CFOV. Dans cette région, les valeurs de RE sont améliorées du 36%, 1.2% et 40% à 356 keV par rapport aux échantillons C6A, C10R et C10A respectivement. Ce classement est évidemment corrélé à la quantité de lumière captée, mise en évidence par les pics en énergie de la figure A.1. Comme prévu, le RE est généralement dégradé lorsque l'épaisseur du scintillateur augmente, en raison d'une collection de lumière moins e cace. La diminution relative de la résolution énergétique entre la CFOV et CFOV est de 9% et 5% entre les scintillateurs C6R et C10R à 356 keV respectivement, mais supérieure à 25% avec les bords absorbants. Ceci est corrélé avec la perte de lumière mesurée lorsque la source s'approche des bords absorbants, qui peut varier de -27% à -36% selon l'épaisseur.

Un résumé des performances spatiales des di↵érents scintillateurs est présenté dans le tableau A.2. Parmi les échantillons de 0.6 cm d'épaisseur, le LaBr 3 présente le meilleur RS ((0.63±0.09) mm à 356 keV dans l'UFOV), principalement en raison de son rendement lumineux plus élevé mais aussi d'une distribution lumineuse plus étroite, produite par un guide optique plus fin (1 mm, par rapport au 1.5 mm des quatre A.4 Optimisation du collimateur a L'optimisation qui a conduit au choix du collimateur pour la mise en oeuvre de notre caméra a débuté par l'étude de la distribution spatiale et énergétique des événements de pénétration et de dispersion ainsi que de leur influence sur les propriétés intrinsèques d'un collimateur.

A.4. OPTIMISATION DU COLLIMATEUR

Les événements E1 sont distribués en fonction de la position de la source et ils ne contribuent qu'au pic principal du profil. Les événements pénétrés E2 sont plutôt répartis sur l'ensemble du champ de vision et contribuent principalement aux queues de profils. Les événements géométriques di↵usés E3a participent à l'augmentation du pic principal, tandis que les événements non géométriques di↵us E3b sont répartis sur l'ensemble du champ de vue. Comme prévu, tous les événements non géométriques (E2+E3b) devraient dégrader la qualité et le contraste de l'image. Chaque géométrie simulée a été évaluée en termes de résolution spatiale (SR), d'e cacité (E), d'e cacité utile (uE) et de pénétration septale e cace (eSP).

La résolution spatiale du collimateur (SR) a été calculée à partir d'un double ajustement gaussien ajusté sur le profil spatial obtenu à partir de la somme des profils verticaux composés chacun par les événements émis par de petites sources de 0.04 mm le long de la source linéaire. La résolution spatiale du collimateur est obtenue à partir de l'écart-type de la distribution gaussienne centrale ( gauss ). Comme on peut le constater qualitativement à partir de la figure figure A.2(c), en sélectionnant les événements à l'intérieur d'une région d'intérêt (ROI) avec un diameter égal à 2 gauss autour du profil spatial, tous les événements géométriques sont inclus, avec aussi une petite partie des événements non géométriques. Cette contribution non géométrique ne détériore pas la qualité de l'image. Les événements qui s'inscrivent dans ce ROI peuvent donc être considérés comme utiles pour la formation de l'image.

L'e cacité E du collimateur est définie comme le rapport entre tous les événements détectés dans la fenêtre en énergie considérée et tous les événements émis à 364 keV par la source. Pour tenir compte de l'"utilité" des événements dans la formation de l'image, nous avons introduit l'e cacité utile (uE), définie comme le rapport de tous les événements détectés dans une ROI de 2 gauss autour de l'image de la source linéaire, divisé par le nombre total des événements émis par la source.

Pour la même raison et afin de quantifier l'e↵et réel de la di↵usion non géométrique et de la pénétration sur la qualité de l'image, le rapport entre tous les événements non géométriques en dehors d'une ROI de 2 gauss autour de l'image et les événements totaux détectés est calculé et nommé pénétration septale e↵ective (eSP).

A.4.2 Impact des paramètres géométriques du collimateur

Résolution spatiale RS La figure (a) montre la RS en fonction des diamètres et pour tous les epaisseur simulés. Les configurations simulées permettent d'atteindre des RS allant de 0.08 mm à 0.72 mm FWHM. Nous avons remarqué que les simulations reproduisent la variation linéaire de la résolution spatiale avec le diamètre du trou et la longueur du collimateur et que la FWHM des profils n'est pas influencée par l'épaisseur du septa, comme le prévoit le modèle analytique (prévoit par le modèle analytique).

Pénétration septale e↵ective eSP La figure (b) montre l'eSP total en fonction de l'épaisseur des septa pour tous les diamètres et pour deux longueurs de collimateur de 3.5 cm et 6.5 cm. L'eSP diminue avec l'augmentation de l'épaisseur des septa A.7 Imagerie planaire quantitative pour la dosimétrie pendant le traitement de la thyroïde Dans la dernière partie de ce travail, l'évaluation de la performance globale du premier prototype de la caméra mobile avec l' 131 I a été e↵ectuée. Au debut, une calibration du système a été e↵ectuée puis l'évaluation préliminaire de la capacité de la caméra à quantifier l' 131 I a été réalisé conformément aux protocoles MIRD [START_REF] Siegel | MIRD Pamphlet No. 16: Techniques for Quantitative Radiopharmaceutical Biodistribution Data Acquisition and Analysis for Use in Human Radiation Dose Estimates[END_REF], grace à l'utilisation de di↵érents fantômes thyroïdiens imprimés en 3D.

A.7.1 Calibration de la camera

La résolution spatiale et la sensibilité du système en fonction de la distance sourcecollimateur (de 1 cm à 15 cm du collimateur par pas de 2 cm) ont été mesurées à l'aide d'un capillaire et de cinq sources cylindriques de 3 mm d'épaisseur et avec di↵érents diamètres (0.5, 1, 2, 3 et 4 cm) remplies de 131 I. La sensibilité a été mesurée sans et avec milieu di↵usant, sous la forme de plusieurs briques d'ABS (5x5x1 cm 3 , =0.97 g/cm 3 ). L'activité des sources a été contrôlée avec un activimètre (MEDI 405, précision de ±3%).

Elle variait d'un minimum de 8.9 MBq pour le plus petit cylindre à un maximum de 55.5 MBq pour le plus grand. Dans le capillaire il y avait 6.7 MBq. Toutes les mesures ont été e↵ectuées à température ambiante. L'analyse d'image utilisée est la même que celle présentée dans la section A.5. La résolution spatiale a été mesurée comme décrit pour les simulations dans la section A.4.1. La sensibilité de la caméra (cps/MBq) a été mesurée par le nombre de comptes détectés dans des ROI définies sur les images des sources cylindriques, divisé par le temps d'acquisition et l'activité de la source, corrigé par la décroissance dans le temps. Une methode de seuil a été utilisé pour la definition des ROI. La sensibilité a été estimée avec et sans correction de la di↵usion, grâce à la technique du Triple Energy Windows (TEW) [START_REF] Siegel | MIRD Pamphlet No. 16: Techniques for Quantitative Radiopharmaceutical Biodistribution Data Acquisition and Analysis for Use in Human Radiation Dose Estimates[END_REF]. A.8 Méthodes de quantification par imagerie planaire La quantification de l'activité dans les thyroïdes sains a été déterminée à la fois par des images planaires par une approche en vue unique (VU) et par une approche en vue conjuguée (VC), conformément au MIRD 16 [START_REF] Siegel | MIRD Pamphlet No. 16: Techniques for Quantitative Radiopharmaceutical Biodistribution Data Acquisition and Analysis for Use in Human Radiation Dose Estimates[END_REF]. Dans la méthode à vue unique, l'activité de la source A j dans la thyroïde est mesurée individuellement à partir des images planaires A ou P [START_REF] Siegel | MIRD Pamphlet No. 16: Techniques for Quantitative Radiopharmaceutical Biodistribution Data Acquisition and Analysis for Use in Human Radiation Dose Estimates[END_REF]. Le taux de comptage dans la ROI autour de l'image thyroïdienne, c'est-à-dire I A ou I P pour les images A et P respectivement, est divisé par le facteur de calibration du système C abs (cps/MBq) mesuré avec milieu absorbant. Dans une approche alternative, I A ou I P sont divisés par un facteur d'atténuation sous la forme de e µd (µ=0.116 cm 1 ) et par le facteur de calibration C (cps/MBq) mesuré sans milieu absorbant. C abs et C sont extrapolés à partir des courbe de distance-sensibilité obtenue avec les sources cylindriques avec et sans milieu absorbant. Elles correspondent aux valeurs de sensibilité mesurées avec la source cylindrique dont la surface est la plus proche de celle de la ROI autour de l'image thyroïdienne et pour une profondeur d estimée à partir des images latérales. La méthode de la vue conjuguée est le protocole le plus couramment utilisé pour la quantification in vivo de la radioactivité avec des images planaires. Il utilise deux images opposées de 180 et une images orthogonale pour estimer l'épaisseur et la position de la source. Lorsque l'on considère une région source j avec une activité de fond négligeable (comme pour les thyroids sains), l'activité A j peut être calculée à partir de l'equation suivante:

A j = r I A I P e µet f j C (A.3)
où e µetn représente le facteur de transmission à travers l'épaisseur totale t n du fantôme du cou ((10.81±0.01) cm) avec un coe cient d'atténuation linéaire global µ e , calculé avec précision en connaissant la géométrie et le coe cient d'atténuation des fantômes [START_REF] Beaumont | Apport de l'impression 3D pour la réalisation de familles de fantômes d'étalonnage dédiés à la personnalisation de la mesure en dosimétrie interne[END_REF]. C représente le facteur de calibration du système extrapolé à partir de la courbe de distance-sensibilité sans milieu absorbant, comme décrit ci-dessus. Le facteur f j représente la correction d'auto-atténuation de la source, estimée à partir du coe cient d'atténuation de la région source µ j et de l'épaisseur de la source t j (estimée à partir des images latérales) [START_REF] Siegel | MIRD Pamphlet No. 16: Techniques for Quantitative Radiopharmaceutical Biodistribution Data Acquisition and Analysis for Use in Human Radiation Dose Estimates[END_REF]. La quantification de l'activité dans les nodules du fantôme pathologique est plus délicate en raison de leur superposition avec l'activité dans les lobes. Cependant, une technique simple de soustraction géométrique est suggérée par les recommandations du MIRD [START_REF] Siegel | MIRD Pamphlet No. 16: Techniques for Quantitative Radiopharmaceutical Biodistribution Data Acquisition and Analysis for Use in Human Radiation Dose Estimates[END_REF]. Elle consiste à utiliser les procédures de calcul d'activité déjà décrites (eq. A.3) mais en ajoutant un coe cient multiplicatif F pour corriger ce fond. Ce facteur comprend l'épaisseur du nodule t j , l'épaisseur du lobe t et un facteur I ADJ qui représente le taux de comptage à travers le fantôme d'une région adjacente au ROI du nodule avec la même surface. La quantification de l'activité dans les nodules a été réalisée en utilisant les images planaires à vue unique (SV) et l'approche CV. L'écarttype de l'activité mesurée A j dans la thyroïde et les nodules a été estimé comme le rapport de la somme quadratique de la variance relative de chaque paramètre dans l'equation A.3. Enfin, la précision de la quantification obtenue avec la caméra mobile est estimée à partir du Coe cient de Récupération (RC), correspondant au rapport entre l'activité mesurée A j et l'activité mesurée dans le fantôme thyroïdien ou dans les nodules, corrigés pour sa dégradation dans le temps.

A.10 Conclusion

L'objectif de ma thèse était de développer une gamma caméra mobile haute résolution spécialement conçue pour améliorer l'évaluation quantitative de la distribution hétérogène et de la biocinétique de l' 131 I après son administration pour le traitement de maladie de la thyroïde. Le premier prototype de la caméra, avec un champ de vue de 5x5 cm 2 , consiste en un collimateur à trous parallèles à haute énergie, réalisé en impression 3D, couplé à un scintillateur inorganique continu, lu par une technologie récente basée sur des matrices de photomultiplicateurs au silicium (SiPM). Le collimateur en tungstène détermine en grande partie la résolution spatiale et la sensibilité global du système. Son optimisation, réalisée à l'aide de simulations Monte Carlo, a permis l'analyse de la di↵usion et de la penetration des rayons gamma et l'etude de leur influence sur ses propriétés intrinsèques. Ceci a permis de choisir une géométrie finale avec une pénétration très faible, une résolution spatiale intrinsèque élevée (2 mm à 5 cm) et une e cacité de 1.24 10 5 . Plusieurs cristaux continus inorganiques de CeBr 3 avec di↵érents revêtements optiques ont été étudiés. Nous avons montré qu'un CeBr 3 de 6 mm d'épaisseur avec des bords réfléchissants o↵rait le meilleur compromis en termes de résolution énergétique (7.68±0.01), de résolution spatiale sous-millimétrique ((0.74±0.28) mm dans le UFOV) et de très faible distorsion spatiale ((0.15±0.10) mm dans le UFOV).

Le premier prototype de la caméra a été calibré avec de l'131I. La calibration du système conduit à une résolution spatiale globale de (3.14±0.03) mm et à une sensibilité moyenne de (1.23±0.01) cps/MBq, le deux à 5 cm de distance. Nous avons e↵ectué les premières études précliniques avec l'utilisation de di↵érents fantômes thyroïdiens imprimés en 3D, avec et sans nodules, remplis de 131 I. Des résultats très prometteurs ont été atteints (valeurs de RC proches de l'unité), qui mettent en évidence ses performances adaptées à une quantification précise dans un contexte clinique assez réaliste. Ces résultats préliminaires sont très encourageants pour l'utilisation de notre caméra comme un outil de quantification précise des doses absorbées, à la place ou en plus des gamma caméras conventionnelles mais une étude plus approfondie, notamment avec l'utilisation de nodules plus petits et une superposition plus importants avec les lobes, pourrait aider à mieux la tester dans une situation clinique encore plus réaliste. Title : Development of a portable gamma imaging system for absorbed radiation dose control in molecular radiotherapy Keywords : Gamma imaging, nuclear instrumentation, Monte Carlo simulations, targeted radionuclide therapy, dosimetry Abstract : Targeted radionuclide therapy is still a developing area among the different treatment modalities against cancer. However, its range of applications is rapidly expanding thanks to the emergence of new radiopharmaceuticals labeled with beta or alpha emitters (peptides, 223 Ra alpha-therapy, 221 As alphaimmunotherapy, ...) (Ersahin 2011). In that context, the large heterogeneity of absorbed doses and the range of effects observed, both in terms of toxicity and response, demonstrate that individualised patient dosimetry is essential to optimize this therapy (Strigari 2011). In clinical practice, patient-specific dosimetry of tumors and organs-at-risk (liver, kidney, ...) is image-based and rely on the quantification of radiopharmaceutical uptake as a function of time. These images can be obtained from either a pre-therapy tracer study or from a previous therapy procedure. The detection constraints imposed by the treatment protocols are very different from those associated with diagnostic imaging. (Flux 2011[START_REF] Walrand | Dosimetry of yttrium-labelled radiopharmaceuticals for internal therapy: 86y or 90y imaging?[END_REF]. Thus, conventional gamma cameras are not suited for detecting high activity of gamma emitters with energy below 100 keV ( 223Ra) or greater than 300 keV ( 131 I, 90 Y ). Moreover, high activities of the injected tracer typically require isolation of the patient, making the use of standard imaging devices difficult. Finally, the availability of these devices is incompatible with an accurate temporal sampling of the kinetics of the tracer, which is a key parameter for the quantification of the absorbed doses. The objective of my thesis was precisely to propose new instrumental and methodological approaches aiming to strengthen the control of the dose released to patients during molecular radiotherapy. This is achieved by reducing the uncertainties associated to activity quantification (and therefore to the absorbed dose calculation) through the use of a compact and highly optimized imaging system. Specifically, the work consisted in the development and optimization of a miniaturized, high-resolution mobile gamma camera specifically designed to improve the individual quantitative assessment of the heterogeneous distribution and biokinetics of the radiotracer before and after treatment administration. The study was focused on the treatment of benign and malign thyroid disease with 131 I. The first prototype of the mobile camera, with a field of view of 5x5 cm 2 , consists of a high-energy parallelhole collimator, optimized with Monte Carlo simulation and made with 3D printing, coupled to a 6 mm thick continuous CeBr 3 scintillator readout by a recent and well-suited technology based on arrays of Silicon Photomultiplier (SiPMs) detectors. Its intrinsic properties, in term of energy and spatial response, have been tested with collimated point source of 57 Co and 133 Ba. The first feasibility prototype has been then calibrated with a line and five cylindrical sources filled with 131 I. The system calibration leads to an overall spatial resolution of (3.14±0.03) mm at a distance of 5 cm and a sensitivity that decreases with distance and slightly changes with source size. An average sensitivity of (1.23±0.01) cps/MBq has been found at 5 cm. In order to test the quantification capability of the camera, the first preclinical planar studies involved the use of different 3D-printed thyroid phantoms filled with 131 I, with and without nodules. Although corresponding to a relatively ideal, but realistic, clinical situation (no superimposition of background activity), the optimized imaging features of the camera leads to very promising results, with activity recovery factors that deviate of around 2% from the unity.

Universit é Paris-Saclay

Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France

Figure 1 . 1 :

 11 Figure 1.1: (left) Direct and indirect radiation damage to the DNA and (right) schematic illustration of selected types of radiation damage to the DNA.

Figure 1 . 2 :

 12 Figure 1.2: Radiation-induced DNA damage, repair mechanisms and possible consequences in mammalian cells. [Source [24]]

Figure 1 . 3 :

 13 Figure 1.3: Multileaf tungsten collimator whose leaves are positioned to spare normal tissue and direct radiation to the target.

Figure 1 . 5 :

 15 Figure 1.5: Auger electrons, ↵-particle and-particle track range in tissue compared to cell dimension.[START_REF] Pouget | Clinical radioimmunotherapy-the role of radiobiology[END_REF]

[Article 56 ,

 56 European Council Directive 2013/59]

Chapter 1 The

 1 role of dosimetry in MRT Collimator intrinsic proper/es : Insight

Figure 1 . 7 :

 17 Figure 1.7: Example of a simple three-compartimental model (top-left) and the timeactivity curve expressed separately for each of the compartments (bottom-left). Example of a specific complex bio-kinetic model (Hays and Wegner) based on bioassay and external measurements of 131 I in young adult males during the first 3 h after intravenous injection (right) [111].

Figure 1 . 8 :

 18 Figure 1.8: Two possible influences of curve fitting caused by the number and the timing of activity measurements. The solid line gives the real activity versus time; the dotted line represents the exponential curve fitted to the measurements, which are shown as black dots. (left) Early-acquired and (right) late-acquired data could cause a big under estimation of the activity in the source region [Source [21]].

Figure 1 . 9 :

 19 Figure 1.9: Examples of (a) a female anthropomorphic mathematical phantoms [130], (b) the ICRP 110 reference anthropomorphic voxelized male phantom[START_REF] Menzel | ICRP Publication 110. Realistic reference phantoms: an ICRP/ICRU joint e↵ort. A report of adult reference computational phantoms[END_REF] and (c) the RPI adult female (RPI-AF) anthropomorphic hybrid phantom[START_REF] Zhang | RPI-AM and RPI-AF, a pair of mesh-based, size-adjustable adult male and female computational phantoms using ICRP-89 parameters and their calculations for organ doses from monoenergetic photon beams[END_REF]

Figure 2 . 1 : 2 2. 3 .

 2123 Figure 2.1: Fraction of photoelectric, Compton and coherent scattering interaction as a function of the Z of the material. The e↵ective Z of di↵erent pure materials (Pb and W) and scintillator compounds is highlighted.

Design of the mobile gamma camera Chapter 2 Figure 2 . 2 :

 222 Figure 2.2: Design of the first 5⇥5 cm 2 mobile camera prototype.

Chapter 2 2. 4 .

 24 THE HIGH-ENERGY PARALLEL-HOLE COLLIMATOR

Figure 2 . 3 :

 23 Figure 2.3: Simulated response of a high-energy collimator response to a point source of activity with common energies in nuclear medicine imaging : 99m Tc (140 keV), 67 Ga (188 keV) , 131 I (364 keV) and PET isotopes (511 keV). [Source [209]]

Figure 2 . 4 :

 24 Figure 2.4: Geometric schema and photography of a parallel-hole collimator with hexagon holes arranged in an hexagonal array.

Figure 2 . 5 :

 25 Figure 2.5: Schematic view of di↵erent types of collimators used in clinical practice.

2 Figure 2 . 6 :

 226 Figure 2.6: Schematic view of a selective laser melting procedure.

Chapter 2 Figure 2 . 7 :

 227 Figure 2.7: (left) Schematic side view of a parallel-hole collimator. The holes of length L and diameter d are separated by septa of thickness t. (Right) The shortest path a gamma gamma ray can travel through a single septa is a distance w.

2. 5 .

 5 THE INORGANIC SCINTILLATOR Chapter 2

Figure 2 . 8 :

 28 Figure 2.8: Schema of the energy band structure of an activated crystal lattice.

3 )

 3 ( /keV ) (% at 662 keV) (ns)

Figure 2 . 9 :

 29 Figure 2.9: Summary diagrams of the scintillators properties. Nominal values are multiplied by scaling factors K: K LY =1.3, K P phe =4, K max =0.2, K RF =50, K 1/ER = 2.5 and K 1/⌧ =2000.

Figure 2 . 10 :

 210 Figure 2.10: Pictures of the LaBr 3 (Ce) (right), 6 mm thick CeBr 3 (Ce) with reflective edges (center) and 1 clm thick CeBr 3 (Ce) with absorbant edges (left).

  3.1.3) and compared to a LaBr 3 (Ce) sample. Their geometries and main features are summarized in Table 2.4. Pictures of the samples are shown in Figure 2.10.

Figure 2 . 11 :

 211 Figure 2.11: A standard photomultiplier tube (Hamamatsu R6231) (left) and a 8⇥8 flat panel (MAPMTs H9500, Hamamatsu) (right)

Figure 2 .

 2 Figure 2.12: A 8⇥8 CZT solid-state detector (3⇥3 mm 2 pixel, Kromek) (left) and a 8⇥8 SiPMs array (3⇥3 mm 2 pixel, Hamamatsu S13361-3050NE-08 series) (right).

  aA silicon photomultiplier (also called solid-state photomultiplier, SSPM, or multi pixel photon counter, MPPC) consists of a matrix of small-sized sensitive elements called micro-cells (or pixels), all connected in parallel. Each micro-cell is a single-Collimator intrinsic proper/es : Insight

Figure 2 . 13 :

 213 Figure 2.13: SiPMs array used in our mobile camera. It consists of four Hamamatsu S13361-3050NE-08 monolithic arrays.

Figure 2 . 14 : 2 Figure 2 . 15 :

 2142215 Figure 2.14: Cross-sectional layout of a p-on-n single cell type [295].

Figure 2 . 16 :

 216 Figure 2.16: Current produced by a single micro-cell in correspondence with photon absorption (or dark noise) (left). Its amplitude corresponds to 1 photoelectron. The pulse has a very fast rising edge and a slower trailing edge (recovery).In the center, the typical SiPM pulse that corresponds to simultaneous firing of several cells (black). In orange, the single cell pulse. The amplitude and the area are proportional to the number of detected photon at the same time. On the right, the pulses pile-up corresponding to a non-simultaneous firing of three micro-cells [Source[START_REF] Ng | Physics of Semiconductor Devices[END_REF]].

Figure 2 . 17 :

 217 Figure 2.17: Photodetection e ciency as a function of the wavelength of the Hamamatsu S13361-3050NE-08 monolithic SiPM arrays [297] .

Figure 2 . 20 :

 220 Figure 2.20: Simplified diagram of a single channel of the EASIROC chip. An input DAC allows the adjustment of the SiPM bias voltage over 2.5 V. Two voltage-sensitive preamplifiers, with a gain adjustable over a dynamic range from 160 fC to 320 pC, are connected to a fast trigger line consisting of a shaper and a discriminator. Two variable slow shapers and two Track and Hold blocks allow the charge to be measured.

Figure 2 . 21 :

 221 Figure 2.21: Electronic module of the SiPMs array composed of four printed circuit boards.

Figure 2 . 23 :

 223 Figure 2.23: Screenshot of the acquisition software specifically developed for our detector data acquisition.

123 2. 8 .

 1238 IMAGE RECONSTRUCTION ALGORITHMS Chapter 2

Figure 3 . 1 :

 31 Figure 3.1: Experimental set-up including the climatic chamber, the 3-axis motorized platforms, the collimated radioactive sources and the pulsed LED source.

Figure 3 . 2 :

 32 Figure 3.2: Tungsten collimators used in the evaluation of the intrinsic spatial and spectrometric performance of the gamma camera : 4 mm (A1-A2) and 0.5 mm diameter single-hole collimators (B1-B2 and C1-C2).

Figure 3 . 3 :

 33 Figure 3.3: Pulse height spectra of the five samples tested using a 4 mm diameter collimated 133 Ba source.

Figure A. 1 .Figure 3 . 4 :

 134 Figure 3.4: Superimposition of 57 Co (top) and 133 Ba spectra (bottom) measured with the five crystal samples.

Figure 3 .

 3 4 shows the calibrated spectra obtained irradiating the five samples with 57 Co and 133 Ba collimated sources. The 57 Co spectra in Figure 3.4(top) show the principal emission peak at 122 keV (a). The peaks around 80 keV (b) are supposed to be due to both the backscatter peaks (82.6 keV) from the 122 keV emissions and characteristics X-rays emitted by the lead shielding surrounding the crystal ('78 keV).

Figure 3 . 7 :

 37 Figure 3.7: 133 Ba spectra measured in the UFOV (left), CFOV (centre) and CFOV of the C6R scintillator (right) before and after correction of the energy response uniformity.

Figure 3 . 8 :

 38 Figure 3.8: Light distribution within the 1 cm thick crystals for gamma rays interacting in the centre or near the edges of the detector ( 133 Ba ). The Scrimger and Becker light distribution function used for the adjustment reconstruction method is superimposed to the experimental profiles

Figure 3 . 9 :

 39 Figure 3.9: Flood field uniformity 128x128 images obtained by irradiating the head of the camera without collimator with a source of 133 Ba (the correction on the energy response uniformity was applied). Notice that the edges of the C10A are clearly squared compared to the other sample, as visible from the scintillator photo in Sec.2.10.

Figure 3 . 10 :Figure 3 . 11 :Figure 3 . 12 :

 310311312 Figure 3.10: Images of the 17⇥17 scans performed with a 57 Co collimated source (0.5 mm diameter). The spots on the images are spaced 3 mm apart in both directions.

Figure 3 . 13 :Figure 3 . 14 :Figure 3 . 15 :Figure 3 . 16 : 3 Scintillator-photodetector evaluation 3 . 1 . 3 . 5

 31331431531633135 Figure 3.13: Intrinsic spatial resolution maps (left) and intrinsic spatial resolution as a function of the mechanical position (right) for the central row (black) and column (red) of the scan, measured with a 57 Co collimated source (0.5 mm diameter).

Figure 3 . 17 :

 317 Figure 3.17: Schematic view of the Monte Carlo simulations performed to study the properties of tungsten collimators as a function of its length L, hole diameter d and septa thickness t (left). Each collimator was coupled to a volume representing a nonsensitive scintillator (blue volume in the central figure). The five principal 131 I emissions in the form of a line source was simulated (yellow line in the right figure). The green lines represent the trajectories of the gamma rays emitted by the source.

Figure 3 . 18 :

 318 Figure 3.18: Schematic decription of the events identification realized with the GATE actors.

Figure 3 . 19 :

 319 Figure 3.19: Example of the spatial profile of the linear source image together with the contributions of the geometric (E1+E3a) and non-geometric events (E2+E3b).

Chapter 3 Collimator

 3 optimisation by Monte Carlo simulation sections.

3 3Figure 3 . 23 : 3 Figure 3 . 24 :

 33233324 Figure 3.23: E↵ective septal penetration eSP (a) and relative contribution of nonscattered non-geometric events eSP-E2 (b) and scattered non-geometric events eSP-E3b (c) to the total number of detected events as a function of the hole diameter and the septa thickness and for three lengths of the collimator. The relative contribution of the E2 (d) and E3b events (e) in eSP is also presented.

Figure 3 . 25 :

 325 Figure 3.25: Collimator e ciency E (a) and relative contributions of non-scattered geometric events E1 (b), non-scattered non-geometric events E2 (c), scattered geometric events E3a (d) and scattered non-geometric events E3b (e), as a function of the hole diameter and the septa thickness, for three lengths of the collimator (3.5 cm, 5 cm and 6.5 cm). The e ciency calculated from the analytical model is represented by dotted lines.

Figure 3 . 26 :Figure 3 . 27 :

 326327 Figure 3.26: Collimator useful e ciency uE (a) and relative contributions of nonscattered geometric events E1 (b), non-scattered non-geometric events E2 (c), scattered geometric events E3a (d) and scattered non-geometric events E3b (e), as a function of the hole diameter and the septa thickness, for three lengths of the collimator (3.5 cm, 5 cm and 6.5 cm). The e ciency calculated from the analytical model is represented by dashed (L) or dottes lines (L e ).

Figure 3 . 28 :

 328 Figure 3.28: Collimator spatial resolution as a function of the hole diameter and the collimator length. Each point indicates the mean value and the standard deviation of the simulated spatial resolution for all septa thicknesses at a fixed lenght and diameter. A linear regression is adjusted on the simulated data for each collimator length.

3 dFigure 3 . 29 :Figure 3 . 30 :

 3329330 Figure 3.29: E↵ective septal penetration eSP (a) and relative contribution of the nonscattered non-geometric events eSP-E2 (b) and scattered non-geometric events eSP-E3b (c) to the total number of detected events for an energy resolution of 2.5 %, 5 % and 7.5 %. The length of the collimator is 3.5 cm.

Figure 3 . 31 :

 331 Figure 3.31: Collimator e ciency for an energy resolution of 7.5 % and a collimator length of (left) 3.5 cm and (right) 6.5 cm. The e ciency calculated from the analytical model is represented by dottes lines (L e ).

Collimator optimisation by Monte Carlo simulation Chapter 3 low

 3 L values without penetration correction (Tab. 3.7).

Figure 3 . 32 :Figure 3 . 33 :

 332333 Figure 3.32: Determination of the minimum septa thickness to reach the 7.5 % eSP threshold for each hole diameter and collimator lengths of 3.5 cm, 5 cm and 7.5 cm.

Figure 3 . 35 :

 335 Figure 3.35: Simulation set-up used for the choice of the collimator: the complete camera and a 3D XCAT voxelized phantom.

3. 2 . 5 . 1

 251 Simulation set-up a Figure 3.35 shows a picture of the simulation set-up. The camera consists of one of the five selected high-energy parallel-hole collimators, a scintillating crystal (magenta in Fig. 3.35) and a volume representing the photodetection system (blue in Fig. 3.35). More details are given in Section 2.4.3.2. A radioiodine treatment of hyperthyroidism

Figure 3 . 36 :

 336 Figure 3.36: Vertical slice of the neck XCAT phantom showing the two small nodules (0.6 cm and 1.2 cm diameter). The gray scale of the image indicates di↵erent segmented volumes.

Figure 3 . 37 :

 337 Figure 3.37: Schematic representation of the iodine biokinetics model for hyperthyroid patients. Four di↵erent types of lines (heavy solid, light solid, dashed and dotted lines) represent the di↵erent ways in which the intercompartment transfer rates (h 1 ) were obtained [Source [112]].

Figure 3 . 38 :

 338 Figure 3.38: Normalized number of 131 I disintegrations N D in all compartments of the hyperthyroidism biokinetics model in a range of 48h (left). Zoom on the compartments contribution (right).

Figure 3 . 39 :

 339 Figure 3.39: Raw images and images after energy and spatial blurring simulated for the collimator with the 3 mm intrinsic spatiale resolution.

Figure 3 . 40 :

 340 Figure 3.40: Filtering procedure to remove artifacts produced in planar images by parallel-hole high-energy collimators.

Figure 3 . 42 :

 342 Figure 3.42: Nodule Contrast (left) and Contrast-to-noise ratio (right) for the five collimator configurations as a function of the global spatial resolution.

Figure 3 . 43 :

 343 Figure 3.43: First 5⇥5 cm 2 FoV prototype of the mobile camera.

Figure 3 . 44 :

 344 Figure 3.44: View of the SiPMs array and the miniaturized readout electronics (up left and right). Coupling between the scintillator and the photodetection system (down-left). The high-energy parallel-hole collimator made in tungsten with 3D-printing coupled to the scintillator/photodetector assembly (down-right)

Figure 3 . 45 :

 345 Figure 3.45: Hole structure of the 3D-printing tungsten collimator (M&I Materials)

Figure 3 . 46 :

 346 Figure 3.46: Photopeak position (left) and energy resolution (right) at 122 keV as a function of the bias voltage on the SiPMs array.

Figure 3 . 47 :

 347 Figure 3.47: Energy linearity (left) and energy resolution (right) as a function of the energy of the incident gamma rays.

Figure 3 . 48 :

 348 Figure 3.48: Position of the 122 keV emission peak as a function of the temperature before and after voltage correction (left) and correction voltage to be applied as a function of the temperature (right).

Chapter 4 4. 1 . 1 . 1

 4111 System spatial resolution a

  Figure 4.1(left) shows a schematic summary of the filtering protocol applied on the 128⇥28 images of the line source. The collimator SR was measured on the filtered images from the sum of the profiles through the line source image along the X and Y directions after their translation to account for the 5

Figure 4 . 1 :

 41 Figure 4.1:Image analysis protocol used to remove the artifacts produced by the highenergy collimator. The representative frequencies of the image are obtained from its Fast Fourier Transform (FFT). The frequencies corresponding to the structure of the collimator are removed using a spatial filter. An image without collimator artifacts is then obtained by applying the inverse FFT to the filtered spectrum[START_REF] García | The HURRA filter: An easy method to eliminate collimator artifacts in high-energy gamma camera images[END_REF]. The method is illustrated on the line source and the 0.5 cm diameter cylindrical source.

Figure 4 . 2 :

 42 Figure 4.2: Set of five cylinders (4, 3, 2, 1 and 0.5 cm diameter) used for the calibration of the mobile camera.

Figure 4 . 3 :

 43 Figure 4.3: Experimental set-up used to measure the camera sensitivity.

Figure 4 . 4 : 4 4. 2 .

 4442 Figure 4.4: A: Profiles (right) and corresponding FWHM (center) and FHTM (left)2D-ROIs applied on the images of the 0.5 cm (up) and 4 cm (down) diameter cylindrical sources. B: Pixels intensity histograms with the pixel value corresponding to the threshold method (red) and the FWHM (green) and FWTM (orange) ROIs (left). The ROI contours defined from the threshold value (right) for the 0.5 cm (up) and 4 cm (down) diameter cylinders is superimposed to the ROIs obtained with the FWHM and FWTM. The images were acquired at a distance of 1 cm from the collimator.

Figure 4 . 5 :

 45 Figure 4.5: (a) 5 year old (FANTHY5y) and (b) 15 year old (FANTHY15y) children 3D-printed thyroid phantoms (Patent n FR 1650 855) filled with 131 I in liquid solution.

Figure 4 . 7 :

 47 Figure 4.7: Experimental set-up used for quantitative planar imaging with the thyroid phantoms. Sides views of the camera with FANTHY 15y thyroid phantom (top) and the camera with its lead shielding and the pathological phantom (bottom).

Figure 4 . 8 :

 48 Figure 4.8: Pixels intensity histograms (a) and corresponding ROI contours defined using the threshold method for the 5 year old (a) and 15 year old (b) children thyroid phantoms. The first peak correspond to the background noise outside the thyroid, the second to the isthmus and the third to the lobes

  4.10(bottom-right)) (Sec. 4.2.3.2).

4. 2 . 3

 23 Planar imaging quantification methods ..

  .

4. 2

 2 .3.1 ... with the healthy phantoms a

Figure 4 . 10 :

 410 Figure 4.10: Lateral images of the 5y and 15y FANTHY phantoms (top) and the phantom with nodules (1:6 concentration ratio) (bottom). The lateral image allows to measure nodules and lobes thicknesses t j , to extrapolate volumes depth d and to estimate the amounts of tissue superimposition t.

4. 3 . 1

 31 Spatial resolution and sensitivity as a function of the collimator-to-source distance Spatial Resolution Figure 4.11 shows the spatial resolution of the system as a function of the distance. It varies linearly from a minimum value of (2.21±0.03) mm at 1 cm from the collimator to a maximum of (5.47±0.05) mm at 15 cm. A value of (3.14±0.03) mm has been obtained at 5 cm, compared to the 2.5 mm FWHM resolution estimated from Monte Carlo simulations (Sec. 3.3.2.4

Figure 4 . 11 :

 411 Figure 4.11: Experimental and simulated (left) spatial profiles at 5 cm of distance and (right) Spatial resolution as a function of the source distance. Simulation 1 is the one performed using the density and geometry as obtained from the collimator optimization procedure ( =18 g/cm 3 , d = 1.15 and t = 0.07) and Simulation 2 are the one performed by reducing the collimator density of 20% and increasing the hole size of 12%.

Figure 4 . 12 :

 412 Figure 4.12: Sensitivity of the camera as a function of the diameter and distance of the source from the collimator with intervening media. The sensitivity was estimated without (top) and with TEW correction (bottom).

Chapter 4 ResultsFigure 4 . 13 :

 4413 Figure 4.13: Normalized spectra measured with the 0.5 cm and 5 cm diameter cylindrical 131 I sources at 1 cm and 15 cm distance from the collimator. Events detected in the whole FOV (left) and within a circular ROI (threshold) on the source image (right) were considered. The three energy windows used for event selection in the 364 keV photopeak of 131 I and TEW correction are shown.

Figure 4 .

 4 Figure 4.14: (left) Experimental sensitivity with TEW correction and threshold method compared with the (center) simulated sensitivity of our collimator and (right) simulated sensitivity with a density of = 10 g/cm 3 all with TEW correction and threshold method.

Figure 4 . 4 Figure 4 . 15 :

 44415  shows the experimental and simulated sensitivities of the camera as a function of the distance. We can observe that the simulated sensitivity is lower than the experimental data for short distances (less than 3 cm) and large sources and higher in other configurations. In a similar way to what has been done for spatial

Figure 4 . 16 :

 416 Figure 4.16: Images of the thyroid phantom with nodules. (left) Anterior image of the nodules filled with a concentration ratio of 1:4. The camera was centered in relation to the center of the phantom. (top-right) Small nodule and (bottom-right) big nodule when filled with the four ration.In the order: 1:2, 1:3, 1:4 and 1:6. In this case, the camera was moved to be centered to the lobe. The image saturation level of the small nodule images (up) is di↵erent from the ones of the big nodule (down).

Figure 4 . 17 :

 417 Figure 4.17: Nodules (left) contrast and (right) contrast-to-noise ration as a function of the lobe-to-nodule activity concentration ratio.

Figure 4 . 18 :

 418 Figure 4.18: Histogram of the Recovery Factors obtained during the quantification process with the FANTHY thyroid phantoms performed with and without TEW correction.

Figure 4 . 19 :

 419 Figure 4.19: Histogram of the Recovery Factors obtained during the quantification process with the thyroid phantom with nodules. RC values (top) for the two nodules with no background subtraction correction and (bottom) for the small nodules where two different background correction in the form of F and F 0 (eq. 4.2) are applied respectively to the CV and SV.

Figure 4 . 20 :

 420 Figure 4.20: FANTHY 5y and FANTHY 15y images obtained (top) with a clinical Gamma Camera (Siemens Symbia T2 GC) at a distance of 10 cm and (botton) with our miniaturized gamma camera at 3 mm of distance.

a

  Parmi les photodétecteurs actuellement disponibles, les photomultiplicateurs au silicium (SiPMs) sont les plus pertinents pour développer des systèmes d'imagerie compacts à hautes performances spatiales et spectrales. Les SiPMs sont des matrices de cellules élémentaires, composées d'une photodiode à avalanche fonctionnant en mode A.3. PERFORMANCES DE LA T ÊTE DE LA CAMERA

Figure A. 1 : 22 CFOV 8 . 25 Table A. 1 :

 1228251 Figure A.1: Spectres des cinq échantillons testés à l'aide d'une source de 133 Ba équipé d'un collimateur en tungstène de 0.5 mm de diamètre

Figure A. 3 :Figure A. 5 :Figure A. 6 :

 356 Figure A.3: Résolution spatiale en function des diameteres simulés (a), Pénétration septale e↵ective (b), E cacité E (c) et E cacité utile uE (d) pour deux épaisseurs en function des septa.

A. 7 . 2 Figure A. 7 :

 727 Figure A.7: FANTHY5y (a), Phantom avec nodules (b) et set-up des measures avec FANTY15y inséré dans le cou (c).

Figure A. 8 :

 8 Figure A.8: Image antériore de FANTHY5y (a) et du gros nodule avec un rapport de fixation lobenodule de 1:6 (b). Image latérale de FANTHY5y (c) et du phantom pathologique (rapport 1:4) (d).

  et 2 cm de diamètre (Petit Nodule (PN) et Gros Nodule (GN) respectivement). Les fantômes sont constitués d'une résine polymère (VeroClear, µ = 0.116±0.002 cm 1 ) choisie parce que son coe cient d'atténuation est proche de celui du tissu adipeux (0.106 cm 1 , NIST [210]). Les fantômes thyroïdiens ont été insérés dans un fantôme de cou imprimé en 3D développé pour simuler avec précision l'atténuation des tissues, Fig A.7(c). Les fantômes thyroïdiens ont été remplis d' 131 I en solution liquide colorée. Les activités, contrôlées avec un activimètre (MEDI 405), étaient de 52.06±1.56 MBq et de 49.90±1.50 MBq pour FANTY5y et le FANTY15y, respectivement. Pour les fantômes pathologiques, les lobes (environ 7.8 cm 3 pour les deux lobes) ont été remplis avec une dilution de 131 I de 2.5 MBq/ml et les nodules avec quatre rapports de concentration d'activité lobe-nodule (1:2, 1:3, 1:4, 1:6) pour simuler di↵érents rapports d'absorption des nodules. Pour chaque fantôme plusieurs images planaires ont été réalisées avec di↵érentes vues angulaires : 0 (antérieure), 180 (postérieure) et 90 (latérale), avec des temps d'acquisition de 25 minutes et de 15 minutes pour les fantômes sains et le fantôme pathologique respectivement. En raison de la forme des lobes, la définition des ROI sur les images a été e↵ectuée avec la méthode du seuil, Fig. A.8. Les images latérales ont été utilisées pour estimer la profondeur et l'épaisseur de la thyroïde et des nodules dans le cou, les deux utiles pour la quantification de l'activité tel qu'il sera présenté dans la prochaine section, Fig. A.8(c-d). L'image latérale du fantôme pathologique a également été utilisée pour estimer l'épaisseur t de la région du lobe autour du nodule utilisé pour la méthode de correction du bruit de fond, Fig. A.8(d), comme sera décrit dans la section suivante.

Figure A. 9 :

 9 Figure A.9: (a) RS. (b) Spectra de quatre sources cylindriques. (c) E cacité sans TEW e (d) avec TEW.

A. 9 . 1 Figure A. 10 :Figure A. 11 :

 911011 Figure A.10: Images antérieures des deux phantoms sains obtenues avec notre camera optimisée (a) et avec un gamma camera clinique conventionnelle (b). Images antérieures des quatre rapport de fixation lobes-nodules dans le phantom pathologique.

  Titre : D éveloppement d'un imageur gamma ambulatoire pour le contr ôle de la dose en radioth érapie interne vectoris ée Mots cl és : Imagerie gamma, instrumentation nucl éaire, simulation Monte Carlo, th érapie par radio-nucl éide, dosim étrie R ésum é : La th érapie interne par radionucl éides est encore aujourd'hui un domaine peu exploit é parmi les diff érentes modalit és de traitement contre le cancer. Son spectre d'applications est toutefois en pleine évolution gr âce notamment à l'apparition de nouveaux radiopharmaceutiques émetteurs beta ou alpha (peptides, alpha-th érapie223 -Ra, alphaimmunoth érapie 221 -As,...) (Ersahin 2011). Dans ce contexte, la grande h ét érog én éit é des doses d élivr ées et des effets observ és, à la fois en terme de toxicit é et de r éponse, d émontrent qu'une dosim étrie personnalis ée est essentielle pour optimiser le traitement(Strigari 2011). En pratique clinique, la dosim étrie de la tumeur et des organes à risque (foie, rein, ...) repose sur l'image de la biodistribution et de la cin étique pr écise du radiopharmaceutique chez chaque patient. Ces images peuvent être r éalis ées avec un traceur pr é-th érapeutique pour planifier le traitement ou apr ès celui-ci, afin de corr éler directement les effets observ és aux doses d élivr ées de mani ère à optimiser le protocole (activit é maximum à injecter, intervalle entre les injections). Les contraintes de d étection impos ées par les protocoles de traitement sont tr ès diff érentes de celles associ ées à un examen diagnostique(Flux 2011[START_REF] Walrand | Dosimetry of yttrium-labelled radiopharmaceuticals for internal therapy: 86y or 90y imaging?[END_REF]. Les gamma-cam éras conventionnelles ne sont ainsi pas adapt ées à la d étection de fortes activit és de rayonnements gamma d' énergies inf érieures à 100 keV ( 223 -Ra) ou sup érieures à 300 keV ( 131 -I, 90 -Y). D'autres part, les fortes activit és des traceurs inject és exigent g én éralement que le patient reste isol é, ce qui le rend donc plus difficilement accessible par les techniques d'imagerie standard. Enfin, la disponibilit é de ces syst èmes est incompatible avec un échantillonnage temporel pr écis de la cin étique du traceur, qui joue un r ôle tr ès important dans la quantification des doses absorb ées. L'objectif de ma th èse était de proposer de nouvelles approches instrumentales visant à renforcer le contr ôle de la dose d élivr ée aux patients lors d'un traitement de radioth érapie mol éculaire. Ceci est r éalis é en r éduisant les incertitudes associ ées à la quantification de l'activit é (et donc au calcul de la dose absorb ée) gr âce à l'utilisation d'un syst ème d'imagerie compact et hautement optimis é. Il consistait à mettre au point et à optimiser une gamma-cam éra mobile miniaturis ée à haute r ésolution sp écialement conc ¸ue pour am éliorer l' évaluation quantitative individuelle de la distribution h ét érog ène et de la biocin étique du radiotraceur avant et apr ès administration du traitement. L' étude était ax ée sur le traitement des maladies b énignes et malignes de la thyroïde à l'aide de l' 131 I. Le premier prototype de la cam éra, avec un champ de vue de 5x5 cm 2 , consiste en un collimateur à trous parall èles à haute énergie, r éalis é en impression 3D et optimis é par simulations Monte Carlo, coupl é à un scintillateur inorganique continu, lu par une technologie r écente bas ée sur des matrices de photomultiplicateurs au silicium (SiPM). Ses propri ét és intrins èques, en termes d' énergie et de r éponse spatiale, ont ét é test ées avec des sources ponctuelles de57 Co et133 Ba. Le premier prototype de la cam éra a ét é calibr é avec de l' 131 I. La calibration du syst ème conduit à une r ésolution spatiale globale de (3.14±0.03) mm et à une sensibilit é moyenne de (1.23±0.01) cps/MBq, le deux à 5 cm de distance. Nous avons effectu é les premi ères études pr écliniques avec l'utilisation de diff érents fant ômes thyroïdiens imprim és en 3D, avec et sans nodules, remplis de 131 I. Des r ésultats tr ès prometteurs ont ét é atteints (valeurs de RC proches de l'unit é), qui mettent en évidence ses performances adapt ées à une quantification pr écise dans un contexte clinique assez r éaliste.

  Radiopharmaceuticals used in clinical practice . . . . . . . . . . 1.2.3 Future perspectives and further developments . . . . . . . . . . 1.3 The role of dosimetry in MRT . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Dose assessment in radionuclide therapy . . . . . . . . . . . . . 1.3.1.1 Basic principles of dose calculation methods . . . . . . 1.3.1.2 The role of imaging in dose assessment . . . . . . . . . 1.3.1.3 Dose calculation : anatomic models and computational methods . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1.4 Uncertainties in dose calculation . . . . . . . . . . . . 1.3.2 Clinical evidence of the relevance of dosimetry in molecular radiotherapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Current clinical practice : a lack of individualized dosimetry . . 1.3.3.1 Dosimetry in current therapy procedures . . . . . . . . Absorbed dose control in MRT . . . . . . . . . . . . . . . . . . . . . . Performance evaluation of the scintillator photodetector assembly for gamma imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.1.1 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . 128 3.1.2 Measurements of the intrinsic imaging performances . . . . . . . 129 3.1.2.1 Energy response . . . . . . . . . . . . . . . . . . . . . 130 3.1.2.2 Uniformity . . . . . . . . . . . . . . . . . . . . . . . . 131 3.1.2.3 Intrinsic spatial performances . . . . . . . . . . . . . . 131 3.1.2.4 Investigation of the light distribution . . . . . . . . . . 132 3.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3.1.3.1 Spectrometric performances . . . . . . . . . . . . . . . 133 3.1.3.2 Light distribution . . . . . . . . . . . . . . . . . . . . . 139 3.1.3.3 Uniformity . . . . . . . . . . . . . . . . . . . . . . . . 140 3.1.3.4 Spatial properties . . . . . . . . . . . . . . . . . . . . . 142 3.1.3.5 Overview of the results . . . . . . . . . . . . . . . . . . 149 3.2 Collimator optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . 150 The first feasibility prototype . . . . . . . . . . . . . . . . . . . . . . . 185 3.3.1 Integration of the fully operational camera . . . . . . . . . . . . 185 3.3.2 Focus on the intrinsic performance . . . . . . . . . . . . . . . . 187 3.3.2.1 Influence of overvoltage . . . . . . . . . . . . . . . . . 188 3.3.2.2 Influence of gamma-ray energy . . . . . . . . . . . . . 189 3.3.2.3 Influence of temperature . . . . . . . . . . . . . . . . . 189 3.3.2.4 Expected sensitivity and spatial resolution as a function of distance . . . . . . . . . . . . . . . . . . . . . . . . 191 4 Quantitative planar imaging for dosimetry during thyroid treatment 193 4.1 Camera calibration for quantification . . . . . . . . . . . . . . . . . . . 193 4.1.1 Experimental protocol . . . . . . . . . . . . . . . . . . . . . . . 193 At the end of 19 th century, several fundamental discoveries changed diagnostic and therapeutic possibilities in medicine and, particularly, in oncology: in 1895 Wilhelm Roentgen discovered X-rays, in 1896 Henry Becquerel described the phenomenon of radioactivity of uranium and in 1898 Marie and Pierre Curie discovered radium and polonium, establishing the foundations for what is nowadays called Radiation Therapy.

	CONTENTS	CONTENTS
	3 Development and optimization of the mobile gamma camera	127
	3.1 List of abbreviations ADC Analog-to-Digital converter ADP Avalanche PhotoDiodes AL Absolute linearity AP Afterpulsing BER Introduction a

Contents Introduction 1 The Role of dosimetry in Molecular Radiotherapy 1.1 Radiation therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Radiation-induced cell death mechanisms . . . . . . . . . . . . . 1.1.2 External beam radiation therapy . . . . . . . . . . . . . . . . . 1.1.3 Internal radiation therapy . . . . . . . . . . . . . . . . . . . . . 1.1.3.1 Brachytherapy . . . . . . . . . . . . . . . . . . . . . . 1.1.3.2 Targeted radiation therapy . . . . . . . . . . . . . . . 1.2 Molecular Radiotherapy . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Physical and radiobiological properties of radionuclides . . . . . 1.2.1.1 Beta emitters . . . . . . . . . . . . . . . . . . . . . . . 1.2.1.2 Alpha emitters . . . . . . . . . . . . . . . . . . . . . . 1.2.1.3 Auger emitters . . . . . . . . . . . . . . . . . . . . . . 1.2.1.4 The theranostic approach . . . . . . . . . . . . . . . . 1.2.2 1.3.3.2 Challenges and constraints of dosimetry for MRT . . . 1.3.3.3 The controversial case of thyroid's therapies . . . . . . 1.4 3.2.1 Simulation insight . . . . . . . . . . . . . . . . . . . . . . . . . . 150 3.2.2 Detection criteria for collimator optimization . . . . . . . . . . . 152 3.2.3 Analysis of the intrinsic properties . . . . . . . . . . . . . . . . 154 3.2.3.1 Spatial and energy distributions of the detected events 154 3.2.3.2 Impact of collimator geometric parameters . . . . . . . 159 3.2.3.3 Influence of the energy resolution . . . . . . . . . . . . 166 3.2.4 Choosing the collimator: which compromise for high-energy gamma imaging ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 3.2.5 Optimization of the collimator for thyroid treatment . . . . . . 176 3.2.5.1 Simulation set-up . . . . . . . . . . . . . . . . . . . . . 176 3.2.5.2 Analysis protocol . . . . . . . . . . . . . . . . . . . . . 180 3.2.5.3 And the winner is . . . . . . . . . . . . . . . . . . . . 182 3.3 4.1.1.1 System spatial resolution . . . . . . . . . . . . . . . . 194 4.1.1.2 System Sensitivity . . . . . . . . . . . . . . . . . . . . 195 4.1.1.3 Regions of interest definition . . . . . . . . . . . . . . 196

Table 1 .

 1

	131 I -Na	34

1: List of some widely used and clinically established therapeutic radiopharmaceuticals. P (%) represents the percentage of treatments (42.853 treatments in total)

Table 2 .

 2

		140 keV		364 keV		364 keV
		µ	P	µ	P	µ	P
		(cm 1 )	(%)	(cm 1 )	(%)	(cm 1 )	(%)
	Photoelectric	24.68	99.9	2.04	63.9	2.53	71.8
	Compton	1.09	42	0.86	34.95	1.49	52.5
	Coherent	1.34	48.8	0.23	10.86	0.34	15.6
	Total	27.11	99.9	3.13	79.1	4.36	88.7

1: Theoretical linear attenuation coe cients for lead and tungsten at photopeak energies of 99m Tc (140 keV) and 131 I (364 keV) for the photoelectric, Compton and coherent scattering interactions (XCOM database

[210]

Table 2 . 2 :

 22 Geometry and performances of di↵erent parallel hole collimator usually used in clinical practice: Low Energy collimator with High Resolution (LEHR), Ultra High Resolution (LEUHR) and High Sensitivity LEHS), Medium Energy (ME) and High Energy (HE) collimators.

	Chapter 2		The high-energy parallel-hole collimator
		LEHR	LEUHR LEHS ME	HE
	Thickness (cm)	2-3	3-3.5	2-2.5	4-4.5	5.5-6
	Diameter (cm)	1.10-1.15 1.15-1.20 2.5	2.5-3	4
	Septa (cm)	0.15-0.2 0.15-0.2 0.3-0.4 1.1-1.4 1.3-3
	Spatial Resolution (mm) 7.5-8.5	6-7.5	10-12	13-15
	Sensitivity (cpm/µCi)	200-250	100	>1000 200-200 ¡150
	Septal Penetration (%)	1-1.5	<1	1-1.5	1.5-3.5 >3.5

2.1), which

Table 2 . 3 :

 23 Density (⇢), e↵ective atomic number (Z

	ef f ), light yield (LY), energy resolution (ER), decay time (⌧ ), refraction index (RF), wave-
	lenght of maximum emission (

max ) and hygroscopicity (Hyg) of some commercially available inorganic scintillators used for medical imaging.

The photoelectric absorption probabilities (P phe ) is calculated at 364 keV for a 0.5 cm thick scintillator at 356 keV. The key characteristics of each scintillator are marked in bold.

Table 2 . 4 :

 24 Geometries and main characteristics of the five crystals tested. The abbreviation reported in last column will be used during the characterization of these sample in Sec.3.1.3

		Thickness Optical	Light guide	Producer	Abbr.
		(cm)	coating	thickness (mm)
	LaBr 3 (Ce) 0.6	Absorbing 1	Saint-Gobain	L6A
		0.6	Reflective 1.5		C6R
	CeBr 3 (Ce)	0.6 1	Absorbing 1.5 R e fl e c t i v e 1 . 5	Scionix	C6A C 1 0 R
		1	Absorbing 1.5		C10A

Table 3 .

 3

	Chapter 3

1: Mean value and standard deviation of the energy resolution (ER) in the UFOV, CFOV and CFOV and overall ER (after correction of the energy response uniformity) at 122 keV ( 57 Co ) and 356 keV ( 133 Ba ).

Table 3 . 2 :

 32 Integral uniformity (IU) and di↵erential uniformity (DU) in the CFOV and UFOV at 122 keV ( 57 Co ) and 356 keV ( 133 Ba ) after correction of the energy response uniformity.

			133 Ba
		L6A C6R C6A C10R C10A
	IU	UFOV 7.9	11.53 20.71 17.89 10.23
	(%)	CFOV 3.12	2.41	7.72	2.97	8.41
	DU	UFOV 3.11	2.82	5.8	4.96	5.4
	(%)	CFOV 1.82	1.26	3.5	0.97	3.2

Table 3 .

 3

				57 Co
			L6A	C6R	C6A	C10R	C10A
	LISD (mm) AL	UFOV 0.07 ± 0.04 0.16 ± 0.12 0.12±0.28 0.27 ± 0.21 0.17 ± 0.16 CFOV 0.04 ± 0.02 0.05 ± 0.02 0.07±0.03 0.07 ± 0.04 0.06 ± 0.03 CFOV 0.08 ± 0.04 0.23 ± 0.14 0.18±0.30 0.44 ± 0.22 0.20 ± 0.16 UFOV 0.43 2.10 1.53 4.15 1.40
	(mm) CFOV	0.07	0.16	0.09	0.39	0.15
				133 Ba
			L6A	C6R	C6A	C10R	C10A
	SR (mm) LISD (mm) AL	UFOV 0.63 ± 0.09 0.74 ± 0.28 0.72± 0.09 1.53 ± 1.08 1.07 ± 0.26 CFOV 0.54 ± 0.05 0.61 ± 0.04 0.70± 0.04 1.05 ± 0.10 0.93 ± 0.08 CFOV 0.76 ± 0.21 0.91 ± 0.35 0.86± 0.30 2.20 ± 1.34 1.31 ± 0.20 UFOV 0.09 ± 0.06 0.15 ± 0.10 0.12 ± 0.03 0.22 ± 0.11 0.13 ± 0.11 CFOV 0.05 ± 0.03 0.06 ± 0.03 0.05 ± 0.02 0.08 ± 0.05 0.04 ± 0.1 CFOV 0.10 ± 0.06 0.20 ± 0.13 0.14 ± 0.28 0.41 ± 0.13 0.14 ± 0.1 UFOV 0.61 2.02 1.3 3.03 0.87
	(mm) CFOV	0.22	0.3	0.09	0.50	0.06

SR

UFOV 1.00 ± 0.25 1.17 ± 0.38 1.35 ± 0.38 2.59 ± 0.99 1.87 ± 0.53 (mm) CFOV 0.87 ± 0.03 1.00 ± 0.03 1.17 ± 0.05 1.85 ± 0.15 1.58 ± 0.08 CFOV 1.19 ± 0.29 1.42 ± 0.49 1.44 ± 0.40 3.77 ± 1.25 2.43 ± 0.44 3: Intrinsic spatial resolution (SR), local intrinsic spatial distortion (LISD) and absolute linearity (AL) for the five scintillators at 122 keV ( 57 Co ) and 356 keV ( 133 Ba ).

Table 3 .

 3

			133 Ba
		L6A	C6R	C6A	C10R	C10A
	Overall ER	7.21±0.01	7.68±0.01 11.40±0.03 7.84±0.01 12.10±0.01
	SR UFOV (mm) 0.63 ± 0.09 0.74 ± 0.28 0.72± 0.09 1.53 ± 1.08 1.07 ± 0.26 SR CFOV (mm) 0.54 ± 0.05 0.61 ± 0.04 0.70± 0.04 1.05 ± 0.10 0.93 ± 0.08 LISD UFOV (mm) 0.09 ± 0.06 0.15 ± 0.10 0.12 ± 0.03 0.22 ± 0.11 0.13 ± 0.11 AL UFOV (mm) 0.61 2.02 1.3 3.03 0.87

4: Overview of the key performance of the five scintillators at 356 keV.

Table 3 . 5 :

 35 Summary of the definitions of the intrinsic properties of the collimators evaluated during the optimization study. A is the activity of the 131 I source and I the intensity of the radiation emission at 364 keV.

		Monte Carlo simulation		Chapter 3
	SR	E ciency (E)	Useful E ciency (uE)	E↵ective Septal Penetration (eSP)
	2 gauss	(E1+E2+E3) A⇥I	(E1+E2+E3) A⇥I	in 2	(E2+E3b) E1+E2+E3 out 2
	3.2.3 Analysis of the intrinsic properties
	3.2.3.1 Spatial and energy distributions of the detected events
	a

Table 3 .

 3

				6 )	(10 6 )	(%)	(%)	(%)	(%)
		0.075 0.0875	2.19	1.44	64.3	16.9	2.7	16.0
	3.5	0.075 0.225 0.0875 0.2	0.63 170.90	0.49 32.8	76.7 17.3	8.5 61.8	3.9 0.7	10.9 20.3
		0.225	0.2	23.02	17.7	71.8	17.1	3.3	7.9
		0.075 0.0875	0.74	0.64	86.3	5.9	2.8	4.9
	5	0.075 0.225 0.0875 0.2	0.25 42.24	0.22 14.33	88.4 33.3	5.5 45.2	3.6 1.2	2.5 20.2
		0.225	0.2	8.95	7.90	84.9	8.7	3.5	2.8
		0.075 0.0875	0.41	0.37	91.2	4.3	2.3	2.2
	6.5	0.075 0.225 0.0875 0.2	0.14 14.71	0.12 7.99	92.2 54.7	4.0 27.7	3.0 1.9	0.8 15.6
		0.225	0.2	4.86	4.40	88.6	6.8	3.3	1.4

6: E ciency (E

  .8). As already mentioned and explained (Sec. 3.2.3.2), shorter collimators show a better agreement with the model including penetration correction, while for

	L	L e	↵	↵ th	↵ thP en	↵/↵ th	↵/↵ thP en
	(cm)	(cm)
	3.00	2.54	3.07	2.67	2.97	1.15	1.04
	3.50	3.04	2.72	2.43	2.64	1.12	1.03
	4.00	3.54	2.45	2.25	2.41	1.09	1.01
	4.50	4.04	2.25	2.11	2.24	1.07	1.01
	5.00	4.54	2.12	2.00	2.10	1.06	1.01
	5.50	5.04	1.99	1.91	1.99	1.04	1.00
	6.00	5.54	1.88	1.83	1.90	1.03	0.99
	6.50	6.04	1.82	1.77	1.83	1.03	0.99
	Table 3.7:

Table 3 . 8 :

 38 5 cm 2 ) Comparison of the linear variation coe cient ⌫ obtained from the simulation and from the analytical model of the e ciency with and without penetration correction (⌫ th = Le 2 ). The mean and standard deviation values of ⌫ over all hole diameters are reported for each collimator length.

	3.00 2.54	18.24	0.15	13.6	19.0	1.34	0.96
	3.50 3.04	12.50	0.06	10.0	13.2	1.25	0.94
	4.00 3.54	9.11	0.07	7.66	9.77	1.19	0.93
	4.50 4.04	6.96	0.04	6.05	7.50	1.15	0.93
	5.00 4.54	5.39	0.02	4.90	5.94	1.10	0.91
	5.50 5.04	4.37	0.04	4.05	4.82	1.08	0.91
	6.00 5.54	3.57	0.03	3.40	3.99	1.05	0.90
	6.50 6.04	3.02	0.03	2.90	3.36	1.04	0.90
	p 8⇡ • 1 3 L 2 and ⌫ thP en =	p 8⇡ • 1 3

Table 4 . 1 :

 41 Big nodule (BN) and small nodule (SN) activity concentration and overall activity (expected and measured) for di↵erent uptake ratios between normal tissue and nodules. The lobes (7.8 ml) were filled with a 131 I dilution of 2.5 MBq/ml for a total activity of 19.6 MBq. The volume of nodules is 0.42 ml and 3.9 ml for the SN and BN, respectively.

	Chapter 4

Table 4 . 2 :

 42 Measured dimensions and positions of structures of interest (lobes and nodules) from the lateral images. Comparison with the actual values.

			Lobe	Lobe	Big	Small
			(FANTHY 5y) (FANTHY 5y)	nodule	nodule
	Thickness Expected	7.63	18.6	19.5	9.2
	(mm)	Measured	8.23±0.61	19.32±0.48	20.68±0.95	10.05±0.88
	Depth	Expected	11.8	15.4	14.5	16.5
	(mm)	Measured	12.9±0.82	16.8±0.90	15.22±0.69	18.20±1.07

Table 4 . 3 :

 43 Recovery coe cient (RC) measured for the thyroid phantoms activity using two single view (SV) methods with the anterior (A) and posterior (P) images and the conjugate view method (CV). The calibration factor was measured with (SV abs ) and without absorbing media (SV). All methods were investigated with and without TEW correction. The activity inside the phantom was 52.06±1.56 MBq and 49.90±1.50 MBq for the 5y and 15y phantom, respectively.

		NO TEW		TEW
	RC (%)	FANTHY 5y FANTHY 15y FANTHY 5y FANTHY 15y
	A-SV	108.62±1.67	116.08±1.77	96.45±1.50	100.53±1.55
	P-SV	113.57±1.75	117.85±1.81	96.44±1.52	96.88±1.51
	A-SV abs	106.29±1.63	111.03±1.70	95.65±1.43	96.74±1.51
	P-SV abs	112.63±2.01	119.45±2.05	94.12±1.79	94.75±1.73
	CV	110.25±1.72	116.2±1.79	96.23±1.53	98.04±1.54

The single view method based on

Table 4 . 4 :

 44 Recovery coe cient (RC) measured for the nodules activity of the pathological thyroid phantom using the anterior (A) and posterior (P) single view (SV) with absorbing media and the conjugate view method (CV), both with TEW correction. The activity of the small nodule (SN) is also calculated with background correction. The activity inside the nodules and the lobes are summarized in Tab. 4.1.

  RSUFOV 0.63 ± 0.09 0.74 ± 0.28 0.72± 0.09 1.53 ± 1.08 1.07 ± 0.26 (mm) CFOV 0.54 ± 0.05 0.61 ± 0.04 0.70± 0.04 1.05 ± 0.10 0.93 ± 0.08 CFOV 0.76 ± 0.21 0.91 ± 0.35 0.86± 0.30 2.20 ± 1.34 1.31 ± 0.20 DSIL UFOV 0.09 ± 0.06 0.15 ± 0.10 0.12 ± 0.03 0.22 ± 0.11 0.13 ± 0.11 (mm) CFOV 0.05 ± 0.03 0.06 ± 0.03 0.05 ± 0.02 0.08 ± 0.05 0.04 ± 0.1 CFOV 0.10 ± 0.06 0.20 ± 0.13 0.14 ± 0.28 0.41 ± 0.13 0.14 ± 0.1 Table A.2: Résolution spatiale intrinsèque (RS), distorsion spatiale intrinsèque locale (DSIL) et Linéarité Absolue (LA) pour le cinq cristaux à 356 keV ( 133 Ba ). CeBr 3 ) (à celles des quatre CeBr 3 de 1.5 mm). De plus, la di↵usion de la lumière, réduite près de ses bords absorbants, améliore la reconstruction de l'image permettant ainsi d'obtenir des linéarité absolues (LA) submillimétriques dans l'UFOV. Pour les mêmes raisons, les distributions lumineuses plus étroites produites par les deux CeBr 3 de 0.6 cm d'épaisseur par rapport à celles des 1 cm permettent d'obtenir une résolution spatiale intrinsèque submillimétrique à 356 keV dans tout le champ de vision ((0.74±0.28) mm dans l'UFOV avec le C6R). Les échantillons C6R et C6A présentent une résolution spatiale presque comparable à 356 keV, mais C6R montre une RS accrue dans le CFOV (+13%) en raison de sa plus grande collection de lumière, tandis que C6A présente un RS meilleur dans le CFOV (+5.5%) grâce à une meilleure linéarité spatiale près des bords. Cependant, même si la résolution spatiale est dégradée avec les scintillateurs de 1 cm d'épaisseur, elle reste quand même très élevée, très près d'un millimètre dans le CFOV. Tous les scintillateurs ont un DSIL comparable et très faible dans le CFOV, ce qui souligne la très bonne capacité de l'algorithme de reconstruction à identifier la position (x,y) du centre de distribution lumineuse indépendamment de l'étalement de la lumière sur le photodétecteur. Parmi les cristaux CeBr 3 testés, celui de 6 mm d'épaisseur à bords réfléchissants o↵re le meilleur compromis en termes de RE (7.68±0.01)% et RS ((0.74±0.28) mm dans le UFOV), avec une très faible distorsion spatiale ((0.15±0.10) mm dans l'UFOV) grâce à son rendement lumineux et à sa collecte de lumière élevée. Pour ces raisons le C6R a été choisi comme le scintillateur à intégrer à nos prototype 5x5 cm 2 .

			L6A	C6R	C6A	C10R	C10A
	LA	UFOV	0.61	2.02	1.3	3.03	0.87
	(mm) CFOV	0.22	0.3	0.09	0.50	0.06

  Estimation de l'épaisseur et de la profondeur des structures d'intérêt L'épaisseur mesurée est en très bon accord avec celle attendue (2.8% et 3.8% pour les lobes de FANTHY5y et 15y et de 6% et 8% pour le GN et le PN, respectivement, par rapport au lobe réel). La profondeur des nodules dans le fantôme de cou est de (15.22±0.69) mm et de (18.20±1.07) mm pour le PN et le GN, respectivement; les valeurs réelles sont 14.5 mm et 16.5 mm.

Acknowledgments

The photodetection system

The readout electronics light spectrum of each SiPMs and adjusting it through the fine tuning of each 8-bits DACs. After DAC correction, the response uniformity is significantly improve (Fig.

2.22(right)

). The standard deviation of the response over the 256 SiPMs decreases by about 80% for a bias voltage of 68.5 V (final standard deviation of 1.3% at 68.5 V).

Time constant and delay a

The time constant of the slow shaper must be defined in order to optimize the integration of the detected signals. A too short integration time does not allow to completely estimate the charge, while too long a time will cause the integration of primary and correlated noise events, with the consequent decrease of the signal-tonoise ratio. This time constant should be adapted to the duration of the detected signal, which depends both on the temporal characteristics of the signal from the SiPMs array (micro-cells recharge time) and the scintillator decay time (17 ns for CeBr 3 , Tab. 2.3). The recharge time constant (⌧ = C d •R q ) is around 20 ns for the S13361-3050NE-08 array used in our photodetection system (Fig. 2.18). Thus, an integration time constant of 50 ns has been chosen for all measurements with the CeBr 3 scintillators.

Once the time constant has been defined, the time delay between the trigger and the hold signal have to be adjusted so that the slow shaper signal is maintained when it reaches its maximum and for a su cient time to be sure to register a charge value proportional to the one released by the event. The optimal delay has been defined as the one that optimize the energy resolution of the photodetection system.

(5)

d (cm)

Useful Efficiency (10 -5 ) scintillator/photodetector assembly (Sec. 3.4), but lies also on the fact that events emitted along the line source that are detected in each vertical ROI contribute to the profile, which was not the case for the simulation of the intrinsic response of the collimator. phantom with two small spherical inserts was also developed in collaboration with the IRSN in order to model thyroid nodules or tumor remnants. For reasons of simplification, the isthmus has been removed from this pathological thyroid phantom and the two lobes are therefore independent. Figure 4.6 shows the CAD schema and the picture of the phantom. The same tissue-equivalent polymeric resin has been employed as for the healthy thyroid phantoms. The nominal diameters of the nodules are 1 cm and 2 cm diameter, corresponding to an internal volume of 0.53 cm 3 and 4.20 cm 3 , respectively. During the filling of the nodules, we were able to estimate their real volume, which appears to be of 0.42 cm 3 and of 3.9 cm 3 for the small and the big nodule, which correspond to a diameter of 0.92 cm and of 1.95 cm, respectively. These dimensions can been considered to be representative of realistic hyperthyroidism features [START_REF] Cantalamessa | Thyroid Nodules in Graves Disease and the Risk of Thyroid Carcinoma[END_REF][START_REF] Erbil | Graves' disease, with and without nodules, and the risk of thyroid carcinoma[END_REF][START_REF] Silberstein | Comparison of Outcomes After 123i Versus 131i Preablation Imaging Before Radioiodine Ablation in Di↵erentiated Thyroid Carcinoma[END_REF]]. They will be from now identified as Big Nodule (BN) and Small Nodule (SN). Both are partially embedded in the thyroid lobes in order to simulate tissue superimposition. This results in a reduction in the internal volume of the lobes (5.25 cm 3 for the healthy lobe) by 37 % and 13 % for the right and left lobe of the pathological phantom, respectively. A 1.75 mm thick wall of resin separates the nodules from the thyroid lobes (minimum thickness to ensure mechanical stability and tightness).

Preclinical planar imaging

All thyroid phantoms were inserted inside a 3D-printed neck phantom developed to precisely simulate neck attenuation (Fig. 4.7). In order to take into account the possible backscattering of particles, a spine, including the vertebral column (0.152 cm 1 ) and the spinal cord (0.113 cm 1 ), was modeled and inserted in the neck phantom. It was printed using the same tissue-equivalent polymeric resin employed for the thyroid to simulate the neck adipose tissue and the spinal cord (VeroClear, µ=0.116±0.002 cm 1 )

A.4. OPTIMISATION DU COLLIMATEUR

A.4.1 Critères d'optimisation du collimateur