Doctorat De

Universite De L'

Marie Pierre

Curie

Présentée Par

Maxime Doury

Mme Frédérique Frouin

Directrice De Thèse

M Alain De Cesare

Co-Encadrant De Thèse

M Ayache Bouakaz

Rapporteur Mme

Marie-Claude Grégoire

Examinateur Rapporteur Mme Isabelle Thomassin-Nagara

Examinateur M Michel Claudon

THESE

Quanti cation of tissue perfusion from dynamic contrast-enhanced ultrasound data relies on appropriate modeling of the curve representing the evolution of the contrast-agent concentration inside the studied tissue. Many factors, experimental or physiological, make inter-subject or intra-subject comparison of these perfusion parameters di cult. In this thesis, the reproducibility and the comparison of various quanti cation methods was investigated through preclinical test-retest experiments and through simulations. e investigated methods were: the log-normal model, the one-compartment model using an arterial input function, and the one-compartment model using a reference tissue. e preclinical experiments revealed the di culty to estimate an arterial input function directly from the image, as well as the necessity to locally correct for the time of arrival of the contrast agent in the tissue in order to ensure the model accurately ts the experimental enhancement curves. A regularized linear estimation of the parameters of the one-compartment model using a reference tissue taking advantage of multiple tissue regions was then proposed to obtain homogeneous relative values of the tissue blood ow and tissue blood volume, expressed relatively to the parameter value inside the reference tissue. e improved robustness and reproducibility of the method was demonstrated. e in uence of factors such as the exam duration, the sampling frequency, the number of tissue regions in the analysis, and the noise amplitude were investigated through simulations, and allowed us to formulate recommendations regarding the acquisition and the analysis of contrast-enhanced ultrasound studies.

iii Quanti cation de la perfusion tissulaire en échographie de contraste: vers la comparaison robuste d'examens par Maxime Doury

Résumé

La quanti cation de la perfusion tissulaire à partir de données dynamiques d' échographie de contraste repose sur une modélisation appropriée de la cinétique de la concentration en agent de contraste dans le tissu étudié. De nombreux facteurs, expérimentaux ou physiologiques, rendent la comparaison inter ou intra-individu de ces paramètres de perfusion di cile. Dans cette thèse, la reproductibilité et la comparaison de di érentes méthodes de quanti cation ont été étudiées dans le cadre d'une étude préclinique de test-retest et sur des simulations numériques. Les méthodes étudiées ont été : le modèle log-normal, le modèle compartimental avec fonction d' entrée et le modèle compartimental avec tissu de référence. Les études précliniques ont montré la di culté d' estimation d'une fonction d' entrée artérielle et la nécessité de corriger localement le temps d'arrivée de l'agent de contraste dans le tissu pour que l'approximation des cinétiques expérimentales par le modèle soit de qualité sufsante. Une estimation linéaire sous contrainte des paramètres du modèle compartimental avec tissu de référence tirant pro t de di érentes zones d'intérêt dans l'image a été ensuite proposée pour obtenir à l' échelle régionale et/ou locale des valeurs relatives cohérentes de débit sanguin tissulaire et de volume sanguin tissulaire, exprimées par rapport aux valeurs dans le tissu de référence. Il a été montré que cette approche est la plus robuste et la plus reproductible. L'in uence des facteurs tels que la durée d'acquisition, la fréquence d' échantillonnage, le nombre de régions utilisées et l'amplitude du bruit a été étudiée sur des simulations et a permis de formuler des recommandations pour l'acquisition et le traitement des études en échographie de contraste.

Résumé étendu Introduction

Cette thèse e ectuée au sein du Laboratoire d'Imagerie Biomédicale (LIB) a été nancée par la Fondation pour la Recherche Médicale (FRM). Le projet global consiste à développer un outil de classi cation multi-paramétrique des tissus tumoraux exploitant diverses modalités d'imagerie ultrasonore, i.e. l' échographie quantitative, l' élastosonographie et l' échographie de contraste. Les données serviront au développement d'un modèle réaliste de croissance tumorale ainsi que de la réponse aux traitements anti-tumoraux. Une première étape clé de ce projet sur laquelle portait mon travail de thèse a consisté à estimer de façon précise et reproductible des paramètres de perfusion à partir de données de contraste ultrasonore, ce a n de les utiliser dans un contexte de suivi longitudinal et pour modéliser l' évolution tumorale.

La quanti cation de la perfusion est une tâche di cile, en e et des variations peuvent intervenir entre les examens, que ce soit au niveau expérimental ou physiologique. Ce processus s'avère néanmoins crucial pour étudier la croissance de tumeurs, avec ou sans traitement.

Si l'imagerie de contraste permet d' étudier la perfusion in-vivo, la comparaison quantitative d' examens reste di cile en raison du manque de reproductibilité des acquisitions. De nombreuses méthodes de quanti cation de la perfusion ont été développées pour analyser des données à une échelle globale, ce qui masque les variations spatiales de la perfusion tissulaire, et ne permet pas d' exploiter les relations entre les paramètres locaux. Le but de cette thèse est de rendre l' estimation de paramètres de perfusion robuste aux variations interexamens, a n de rendre possible la comparaison d' examens tout en révélant l'hétérogénéité spatiale de la perfusion. Notre étude se concentre sur la quanti cation en échographie de contraste, cependant les méthodes proposées pourraient-être étudiées pour la quanti cation avec d'autres modalités d'imagerie de contraste.

Le manuscrit est divisé en trois parties. La première partie cherche à établir un état de l'art des méthodes de quanti cation de la perfusion développées pour les di érentes modalités d'imagerie de contraste. La seconde partie étudie la reproductibilité des paramètres obtenus à l'aide de di érentes approches ainsi que les relations qui unissent les di érents paramètres v estimés avec ces approches. En particulier sont comparés une approche semi-quantitative, un modèle à un compartiment alimenté par une fonction d' entrée artérielle, et un modèle à un compartiment utilisant un tissu de référence. En n, dans la troisième partie nous présentons une nouvelle approche d' estimation du modèle à un compartiment utilisant un tissu de référence exploitant l'hétérogénéité spatiale des motifs de perfusion au sein de la tumeur.

Partie I. Quanti cation de la perfusion : état de l'art

Dans cette première partie, composée du Chapitre , nous établissons un état de l'art des méthodes développées pour quanti er la perfusion. Ce Chapitre présente une sélection de méthodes de quanti cation fondatrices, ainsi que leur évolution à travers les di érentes modi cations qui y ont été apportées pour surmonter di érentes limites. Leur usage est décliné pour chaque modalité d'imagerie (TEP, TDM, IRM et échographie).

Les méthodes de quanti cation ont été classées en trois catégories : semi-quantitatives, déconvolution, ou compartimentales. Les approches semi-quantitatives extraient des paramétres caractérisant la cinétique de la concentration en agent de contraste et sont courrament utilisées pour caractériser la perfusion tissulaire, mais les paramètres de ces modèles n' ont pas de lien direct avec la physiologie. Les approches de déconvolution, ainsi que la majorité des approches compartimentales, nécessitent la connaissance de la fonction d' entrée artérielle. La fonction d' entrée artérielle peut-être obtenue par prélèvements sanguins, ou directement dans l'image. Cependant, les prélèvements sanguins sont invasifs, en particulier les prélèvements artériels, et en raison des fortes concentrations en agent de contraste observées dans les artères l' estimation directe de la fonction d' entrée artérielle dans l'image est a ectée par des artefacts, notamment les e ets de saturation et de volume partiel.

Les di cultés rencontrées lors de l' estimation de la fonction d' entrée artérielle ont motivé le développement de méthodes utilisant un tissu de référence, permettant la quanti cation relative de la perfusion. En e et, un tissu de référence peut être choisi dans une région plus grande qu'une artère et bien perfusée, réduisant ainsi les risques de saturation et de volume partiel. Des modèles compartimentaux sont courrament utilisés pour quanti er la perfusion et d'autres mécanismes plus complexes incluant di usion extravasculaire et métabolisme en TEP, en TDM et en IRM. Cette partie s'achève sur le constat que si un certain nombre vi de transferts ont été opérés d'une modalité d'imagerie à une autre, à notre connaissance les modèles compartimentaux, qu'ils utilisent une fonction d' entrée artérielle ou un tissu de référence, n' ont pas été appliqués à la quanti cation de la perfusion en échographie de contraste.

Partie II. Reproductibilité des méthodes de quanti cation existantes en échographie de contraste et les relations entre les di érentes méthodes

Cette partie est composée du Chapitre dans lequel nous avons étudié la reproductibilité des paramètres de perfusion estimés par di érentes méthodes de quanti cation sur des données d' échographie de contraste, et du Chapitre qui complète le Chapitre précédent en établissant les relations théoriques et empiriques entre les paramètres des di érents modèles, qui permettent de justi er pleinement le choix des paramètres sélectionnés dans l' étude de reproductibilité.

L' étude présentée dans le Chapitre a fait l' objet d'une publication dans Physics and Medicine in Biology sous le titre 'Quanti cation of tumor perfusion using dynamic contrastenhanced ultrasound : impact of mathematical modeling '. Nous y présentons une étude de reproductibilité des paramètres de perfusion en échographie de contraste réalisée sur quatre souris porteuses de tumeurs colorectales. Chaque souris a subi quatre examens test-retest d' échographie de contraste avec injection automatique d'un bolus de microbulles suivant un protocole développé au laboratoire. Les acquisitions étaient espacées de quinze minutes a n de garantir l' élimination totale des microbulles avant chaque nouvelle acquisition. La région perfusée des tumeurs (en excluant la partie nécrotique) a ensuite été segmentée en régions, permettant la dé nition de cinétiques. Ce découpage régional a été choisi a n de montrer l'hétérogénéité de la perfusion tumorale tout en garantissant un rapport signal sur bruit su sant permettant la quanti cation. En n, diverses méthodes de quanti cation ont été appliquées à ces cinétiques régionales a n d' estimer des paramètres de perfusion et il a été possible de les comparer.

En particulier, nous avons comparé la reproductibilité des paramètres de perfusion estimés par les méthodes suivantes : (a) une approche basée sur le modèle log-normal (LN) vii estimant des paramètres semi-quantitatifs, notamment l'aire sous la courbe AUC et le taux de remplissage WIR, (b) un modèle à un compartiment utilisant une fonction d' entrée artérielle (AIF) estimant des paramètres absolus, i.e. le volume sanguin tissulaire V , le ux sanguin tissulaire F, et par déduction la constante de transfert k T = F V , (c) un modèle à un compartiment utilisant un tissu de référence (RT), conduisant à l' estimation de paramètres relatifs, le volume sanguin tissulaire relatif rV RT , le ux sanguin tissulaire relatif rF RT , ainsi que la constante de transfert dans le tissu d'intérêt k T , et par déduction la constante de transfert dans le tissu de référence k R = rV ⋅k T rF . Dans cette étude, les trois modèles ont été ajustés aux cinétiques de perfusion expérimentales en minimisant l' erreur au sens des moindres carrés à l'aide d'algorithmes de minimisation non-linéaires. La valeur du paramètre k R a été xée de façon systématique à une valeur estimée à partir d'une modélisation compartimentale avec fonction d' entrée artérielle. En e et ce paramètre n' est pas identi able dans la formulation non-linéaire du modèle.

A n d' étudier l' e et de la normalisation sur la reproductibilité des paramètres de perfusion, nous avons dé ni des paramètres relatifs à partir des paramètres du modèle LN, notés rAUC et rWIR, et des paramètres absolus du modèle monocompartimental avec fonction d' entrée artérielle AIF, notés rV AIF et rF AIF , en les normalisant par leur valeur dans le tissu de référence précédemment décrit. En n, la prise en compte du temps d'arrivée du bolus dans le tissu étudié étant souvent négligée dans les autres modalités d'imagerie, nous avons étudié son impact sur la qualité de modélisation des données de perfusion.

La nécessité de prendre en compte le temps d'arrivée du bolus dans le tissu d'intérêt a été démontrée en étudiant la qualité de la modélisation obtenue avec les di érents modèles compartimentaux avec et sans estimation d'un temps de retard. En e et, lorsqu'aucun temps de retard n' est pris en compte dans les modèles, le nombre de régions présentant une modélisation de mauvaise qualité est égal à pour le modèle AIF et à pour le modèle RT, sur un total de régions étudiées dans l' étude test-retest ( × × ). Ce nombre s' élève à quel que soit le modèle (AIF ou RT) lorsque le temps de retard est pris en compte.

Sur les études test-retest, les paramètres absolus du modèle LN se sont montrés peu reproductibles avec des coe cients de variation médians pour les régions de l' ordre de % pour AUC et % pour WIR, suggérant une forte sensibilité des paramètres aux variations viii inter-examens. Les paramètres absolus du modèle AIF se sont révélés plus reproductibles avec des coe cients de variation médians de l' ordre de % pour V et % pour F. Cependant l' étude a démontré la sensibilité de ces paramètres à la fonction d' entrée artérielle utilisée, et les di cultés rencontrées lors de son estimation dans l'image en raison de la petite taille de l'artère dans le champ de vue et des fortes concentrations en agent de contraste observées dans les gros vaisseaux. La normalisation des paramètres des modèles LN et AIF, donnant ainsi naissance aux modèles rLN et rAIF, a permis une réduction signi cative de la variabilité inter-examen. En e et, les paramètres liés au volume sanguin tissulaire, rAUC et rV AIF , ont des coe cients de variation médians de l' ordre de %, tandis que les paramètres liés au ux sanguin tissulaire, rWIR et rF AIF , ont des coe cients de variation médians légèrement supérieurs à %. L' estimation directe de paramètres relatifs à l'aide du modèle RT s' est révélée la plus reproductible de l' étude, avec des coe cients de variation médians légèrement inférieurs à % pour rV RT et à % pour rF RT . Dans le Chapitre , nous avons étudié l'impact de di érents facteurs, pouvant être la source d' erreurs de quanti cation, sur les paramètres de perfusion des modèles rLin et rReg.

Nous nous somme intéressés aux facteurs liés aux paramètres de l'acquisition, comme le niveau de bruit dans les cinétiques, la durée des examens ou la fréquence d' échantillonage, ou aux facteurs dépendant de la stratégie d'analyse, comme le nombre de régions d'intérêt ou les caractéristiques du tissu de référence choisi. Nos études de simulation ont montré que les xii paramètres du modèle rLin sont a ectés par des biais systématiques variant d'une région à l'autre. L'utilisation du modèle rReg permet de réduire les variations régionales du biais d' estimation, en particulier en ce qui concerne les paramètres k T et k R , mais aussi les paramètres rV et rF dans une moindre mesure. L' estimation de l'approche rReg s' est également montrée plus robuste et plus homogène que l' estimation de l'approche rLin en cas d'acquistions plus courtes ou moins bien échantillonnées. En n, l'impact du choix du tissu de référence sur la justesse de l' estimation a été démontré à travers notre étude de simulation, mais une étude plus approfondie reste à mener a n de mieux en appréhender les mécanismes.

Conclusion

Le modèle rReg s' est avéré être prometteur pour quanti er la perfusion, et nous avons démontré son applicabilité en échographie de contraste dans ce manuscrit. Si l' objectif est de quanti er la perfusion dans plusieurs régions d'intérêt (ces régions pouvant être de taille variable), nous recommandons de prendre en compte les relations existant entre les di érents paramètres régionaux pour éviter les incohérences entre régions, tout en rendant l' estimation plus robuste. Le modèle rReg peut-être appliqué à une échelle plus ne, i.e. à l' échelle du pixel ou du macro-pixel, pour mieux étudier l'hétérogénéité fonctionnelle du tissu étudié. Cependant l'absence de correspondance pixel à pixel dans notre étude test-retest nous a poussé à utiliser un découpage régional, permettant ainsi une comparaison des paramètres d'une étude à l'autre.

La majorité des acquisitions d' échographie de contraste se font encore en D, rendant ainsi la comparaison d' examens di cile. Il est en e et impossible de s'assurer que le même plan soit imagé dans deux examens acquis à quelques jours d'intervalle, et c' est encore plus di cile lorsque l' on étudie l' évolution d'un tissu (type tissu tumoral) dont la forme et la taille changent entre les examens (lié à la croissance tumorale ou à la réponse à une thérapie). Nous recommandons donc l'usage de données D lorsque cela est possible, en particulier lorsque l' on cherche à réaliser un suivi longitudinal. Les développements récents en échographie D sont donc prometteurs pour les applications de suivi tumoral. En e et l' échographie de contraste sera alors capable de rattraper les modalités d'imagerie tomographiques, donnant ainsi accès à des informations plus pertinentes sur la forme, la taille, la structure et la fonction xiii des lésions, tout en permettant une imagerie en temps-réel, non-ionisante et peu coûteuse.

L'imagerie de contraste D est systématique en TEP avec une résolution isotrope, elle est plus limité en TDM, et se développe en IRM. La capacité du modèle rReg à être appliqué à ces modalités d'imagerie doit-être étudiée plus en détails, et nécessitera parfois des adaptations pour prendre en compte les caractéristiques du tissu étudié et de l'agent de contraste ou du traceur injecté. En e et, l'adaptation de la méthode d' estimation régularisée à diverses architectures de modèles compartimentaux doit-être réalisée. Par ailleurs, l' étude de la perfusion dans le foie nécessiterait d'autres adaptations du modèle, notamment par la prise en compte d'une entrée portale en plus de l' entrée artérielle. Les structures vasculaires dans le rein se traduisant par plusieurs 'phases ' de perfusion dues à la superposition au niveau macroscopique de di érentes structures vascularisées de façon très di érentes, le modèle doit être adapté pour les prendre en compte. En e et l'utilisation du rein comme tissu de référence, en prenant en compte les di érentes phases, pourrait encore améliorer la qualité de l' estimation.

L'impact du tissu de référence sur les paramètres de perfusion du modèle rReg a été démontré dans le Chapitre , cependant une étude approfondie est nécessaire pour mieux dé nir les caractéristiques du tissu de référence idéal, mais aussi pour comprendre l'impact du choix d'un tissu de référence non-idéal. Inclure plusieurs tissus de référence dans le modèle pourrait rendre l' estimation des paramétres plus robustes aux caractéristiques des tissus de référence.

Pour conclure, le choix d'une méthode de quanti cation de la perfusion reste une tâche di cile qui dépend des données et du but de l' étude. La comparaison d' examens est particulièrement di cile en raison des variations expérimentales et physiologiques qui se produisent entre les examens. Cette thèse a démontré la capacité du modèle relatif à un compartiment à quanti er la perfusion à partir de données d' échographie de contraste de façon reproductible et robuste à l' échelle régionale, révélant ainsi l'hétérogénéité fonctionnelle des tumeurs étudiées et rendant plus robustes les comparaisons intra-examen et inter-examen. Il est absolument nécessaire de prendre en compte les relations entre les paramètres de perfusion des di érents tissus étudiés, puisque nous avons démontré l'intérêt de cette approche sur la justesse, la robustesse et la reproductibilité de l' estimation des paramètres dans le modèle xiv rReg. Il faut noter que les modèles utilisant un tissu de référence se sont montré plus reproductibles et robustes que les modèles utilisant une fonction d' entrée artérielle sur les données que sur lesquelles nous avons travaillé, mais ce point sera peut-être remis en cause si les dicultés rencontrées dans l' estimation de la fonction d' entrée artérielle arrivent à être surmontées. En n notons que sous thérapie, l'usage de tissus de référence pour suivre l' évolution de la perfusion de tumeurs sous thérapie soulève une question quant à l' e et des traitements sur le tissus de référence, notamment en ce qui concerne les traitements anti-angiogéniques.

Dans la continuité de ces travaux, nous souhaitons confronter les paramètres de perfusion estimés par le modèle rReg aux résultats obtenus en histologie pour valider la méthode.

Les paramètres de perfusion du modèle rReg pourraient être utilisés pour réaliser une classi cation des tissus tumoraux en trois classes, selon qu'ils soient nécrotiques, hypoxiques ou proliférants. Les résultats de cette classi cation alimenteront alors un modèle réaliste de croissance tumrale prenant en compte la réponse thérapeutique. -Block diagrams and rst-order di erential equations of compartmental models with (a) one tissue compartment, (b) two tissue compartments, (c) three tissue compartments. C A (t) is the contrast-agent concentration in arterial blood, C V (t) is the tissue vascular concentration, C I (t) is the tissue interstitial concentration, C C (t) is the tissue cellular concentration. K is the unidirectional transfer rate of contrast-agent from blood to tissue vascular space and is related to blood ow and capillary permeability. k is the unidirectional transfer rate of contrast-agent from the tissue vascular space to blood and is de ned as K V D , where V D is the contrast-agent fractional distribution volume. Similarly k and k are the unidirectional transfer rates of contrastagent between tissue vascular space and interstitial space, and k and k are the unidirectional transfer rates of contrast-agent between interstitial space and intracellular space. K is classically expressed in milliliter of blood per minute per milliliter of tissue (mL.min -.mL -), and the other rate constants k , k , . . . , k in fraction of contrast-agent per minute (min -). . . . . . . . . . xxiii -Illustration of the data pre-processing steps. Le : e contours of the tumor and its necrotic core have been overlaid on a contrast enhanced image (in ochre color). e perfused tumor area was divided into radial layers and angular sectors. A reference tissue region (in green color) and a renal cortex region (in blue color) were also delineated.

. Cancer and tumor microenvironment

A tumor is a neoplasm composed of mutated cells undergoing abnormal growth. All tumors are not cancers, in particular benign tumors are not cancers, they are not invasive and usually not life-threatening. Cancer is in fact synonym of malignant tumor. . Cancerous cells are insensitive to the anti-growth signals originating from the surrounding environment.

. Cancerous cells are immune to apoptotic signals, emitted by the environment, and controlling the death of malfunctioning cells.

. e number of replication cancerous cells can achieve is unlimited.

. A er cancerous cells reached a point where further proliferation is limited by the supply in oxygen and nutrients, they switch to an angiogenic state that triggers the construction of a supplying vascular network, allowing rapid cell proliferation despite its chaotic structure.

. When nutrients and space become limiting growth factors, cancerous cells are able to migrate and invade surrounding tissues where nutrients and space are not limiting factors to create new cancerous cell colonies known as metastases.

Later, in , Hanahan and Weinberg [ ] proposed a second generation of cancer hallmarks that include the six acquired traits presented in the previous paragraph, but they add four more hallmarks:

. Cancerous cells are able to modify their metabolism to most e ectively support tumor proliferation.

. Cancerous cells are resistant to immunological destruction by lymphocytes and macrophages.

. Cancerous cells exhibit unstable genomes favoring genetic mutations, that o en result in an accelerated tumor progression.

. Immune cells ghting the proliferation of cancer cells cause tissue in ammation, which can contribute to the acquired traits presented in their rst paper.

.

. CANCER AND TUMOR MICROENVIRONMENT

Treatment of cancer e three major treatments of cancer are surgery, radiotherapy, and chemotherapy.

Surgery aims at removing the whole tumor, however this procedure is highly invasive.

Radiotherapy consists in the irradiation of the tumor by high-energy X-rays, gamma-rays or charged particles, damaging cancerous cells, but also the surrounding tissues. ese two techniques are only applicable to localized solid tumors, and some cancerous cells can remain a er the treatment.

If the cancer has spread throughout the body, chemotherapy is o en used to eliminate remaining and migrating cells a er removal of the primary tumor by means of surgery or radiotherapy. Chemotherapy relies on the injection of a cytotoxic agent, which main purpose is to eradicate the remaining cancerous cells. Cytotoxic agents however, are not speci c of cancerous cells and are globally toxic for the patient, limiting the injected doses. Moreover, since cancerous cells are highly prone to mutations, they can develop a resistance to the cytotoxic agent.

Recently, anti-angiogenic treatments were proposed to disrupt or to limit the development of new vascular structures providing the tumor with the oxygen and nutrients necessary for cell division, therefore limiting further growth of the tumor. ese treatments were shown to normalize the chaotic neovascularization in tumors, allowing a more e cient delivery of cytotoxic therapies inside the tumor. However, since many pathways regulating angiogenesis exist, cancerous cells can bypass the targeted pathway and activate another one, making the tumor resistant to the anti-angiogenic treatment.

Cancer monitoring

. Perfusion imaging

Perfusion imaging is a branch of medical imaging that focuses on the visualization and characterization of tissue vascularization. A tracer or contrast-agent is injected intravascularly, either as a bolus or as an infusion, and the passage of the tracer in the tissue is observed using one of the various imaging modalities presented below. Indeed, despite the various physical phenomena involved during image acquisition in the various contrast-enhanced imaging modalities, analysis of the passage of the contrast agent follows the same general principle. Microbubbles can be disrupted using ultrasound pulses with high mechanical indices [ ].

Nuclear medicine

is phenomenon allows a type of acquisition speci c of contrast-enhanced ultrasound in addition to conventional approaches. Indeed, a er a continuous infusion of microbubbles reached its steady-state, a series of disruptive pulses is sent, then the re lling of the microbubbles in the tissue is observed in real-time using a low mechanical contrast-speci c sequence to ensure minimum destruction of the microbubbles.

Indeed, the spatial resolution of contrast-enhanced ultrasound images depends on the central frequency of the ultrasound probe, but is usually of about µm. e spatial resolution of nuclear medicine images is generally much coarser, for instance the resolution of PET images is typically about . cm. Additionally, ultrasound images can be acquired at high frame rates, i.e. ranging from Hz to more than kHz using plane wave imaging. ese ne temporal resolutions can however be obtained without sacri cing the image spatial resolution or quality. is is for instance not true in MRI where a compromise must be made between the temporal resolution, the spatial resolution, and the signal to noise ratio.

Contrast-enhanced ultrasound is a non-ionizing imaging modalities, enabling repeated acquisitions without exposing the patient to radiations. is is especially important in pathology monitoring and treatment response monitoring applications. Contrast-enhanced ultrasound is the only real-time perfusion imaging modality available in clinical routine, allowing direct visualization of the passage of microbubbles in the imaging plane on the scanner terminal, whereas tomographic imaging modalities, i.e. SPECT, PET, X-ray CT or MRI, usually require o ine precessing to generate the dynamic perfusion volume. Ultrasound scanners are smaller than other imaging apparatuses, most scanners are actually designed to be moved around the patient and are therefore mounted on wheels. Some miniature scanners, speci cally designed for mobility applications, even include batteries for autonomous usage.

Additionally, ultrasound scanners are much cheaper than other imaging apparatus, and do not require expensive or short-lived consumables.

. . AIMS AND OUTLINE

. Aims and outline

. . Aims

Reliable quanti cation of tumor perfusion is a challenging yet necessary task to establish cancer diagnosis and monitor tumors undergoing therapy. Contrast-enhanced imaging is a great tool to assess perfusion, however quantitative exam comparison remains di cult because of the poor reproducibility of the acquisitions. Moreover, most of the perfusion quanti cation methods are developed to estimate parameters at a global scale, therefore either hiding the local variations in tissue perfusion or not accounting for the relations between the local estimates. is thesis aims at making the estimation of perfusion parameters robust to inter-exam changes, in order to enable exam comparison while revealing the spatial heterogeneity of the tissue vascular function. Various quanti cation approaches will therefore be investigated in this thesis, including semi-quantitative approaches, and compartmental models using either an arterial input function or a reference tissue. Additionally, various estimation methods and their impact on the estimated parameters will be investigated. Our study focused on contrast-enhanced ultrasound data, however the proposed quanti cation methods could be investigated in other perfusion imaging modalities.

. . Outline

e document is divided into three parts. In the rst part we establish a state of the art of the methods for quanti cation of perfusion. In the second part we assess the reproducibility of existing quanti cation methods. And in the third part we propose a new quanti cation method, and then assess its reproducibility and its sensitivity to various factors.

Part

Part I

Quanti cation of perfusion: state of the art

Foreword

In the rst part of this thesis, we reviewed the methodological developments of perfusion quanti cation methods. Various modalities were historically used to assess perfusion, from the most invasive requiring catheterization for blood sampling to the most advanced in-vivo imaging techniques, however they all require injecting a tracer to monitor its concentration throughout the experiment. Quanti cation of perfusion consists in the estimation of parameters characterizing the physiology of the tissue under investigation, in particular regarding the distribution of blood or the exchanges between blood and tissue.

Chapter addresses the three main quanti cation approaches used to characterize perfusion exams, i.e. semi-quantitative, deconvolution, and compartmental. e semi-quantitative approaches are the most intuitive, they derive perfusion parameters directly from the tracer concentration curves, from which physiological parameters can be derived. Deconvolution approaches consider the tissue as a black-box system fed by an arterial input, and estimate the tissue response to an instantaneous injection making few assumptions on the underlying physiology and tracer characteristics. Deconvolutions result in impulse responses with unknown shapes, from which perfusion parameters are usually derived. Compartmental models can be viewed as explicit deconvolution, where the shape of the impulse response in known and parameterized by physiologically relevant parameters.

e methods are presented by imaging modalities, however methodological transfers between modalities was common and was emphasized when a clear continuity was observed.

e evolution of a technique is presented chronologically when possible to reveal the incremental development of the methods.

N.B. All the references cited in this part of the thesis are pooled together and presented at the end of Part I.

Chapter

Quanti cation of perfusion exams:

A review

Generally speaking, quanti cation of perfusion consists in deriving parameters from blood ow measurements, regardless of the method used for the measure. is topic has been on the is chapter fundamentally aims to review the various methods used to quantify perfusion exams acquired using one of the imaging modality brie y presented in Chapter . Quanti cation of perfusion exams is possible assuming the signal intensity is linearly related to the concentration of tracer in the tissue. Verifying this assumption is not included in the scope of this thesis, it will therefore be considered true in the following of this manuscript. Additionally, perfusion imaging modalities yields macroscopic measurements, i.e. at the pixel or

. Semi-quantitative methods

. . Generalities

Semi-quantitative methods are probably among the most intuitive as they extract perfusionrelated parameters directly from raw, interpolated, or noise-ltered enhancement curves.

. . Nuclear medicine

Semi-quantitative parameters have been used to di erentiate stenosed from healthy kidney in scintigraphy exams, i.e. a nuclear medicine technique using a gamma camera, following bolus injection of [ m Tc]DTPA. Miles et al. also adapted the method to account for both arterial and venous perfusion of the liver, using the splenic enhancement as venous input because the portal vein was not present in the image [ ]. Arterial perfusion was calculated using the maximum slope before peak splenic enhancement, while venous perfusion was calculated using the maximum slope a er peak splenic enhancement.

In Blomley et al. proposed the liver subtraction method to quantify liver perfusion [ ]. Instead of estimating the upslope in the late liver enhancement curve, they rst subtracted the splenic enhancement curve multiplied by the ratio of arterial to splenic arterial perfusion, i.e. the ratio of the maximum slope in the early liver enhancement curve to the maximum slope in the spleen. e maximum slope of the corrected enhancement curve is then used to calculate portal perfusion. In this study, the time and value of peak enhancement were obtained using a gamma variate t, allowing a ner estimation of these parameters. Whenever possible the authors used the enhancement curve in the portal vein instead of the splenic vein. Authors demonstrated the use of portal perfusion in one patient with metastatic livers, as well as in four patients with cirrhotic livers. Facing the small number of cases in the previous study, a collaborating group extended the application of the method in cirrhotic livers with more cases [ ]. ey estimated arterial and portal perfusion in a group of twenty patients with viral-induced cirrhosis and in fourteen controls.

While arterial perfusion did not di er between groups, they found a signi cant reduction in portal perfusion among patients compared to controls, and a strong correlation between portal perfusion and the prothrombin ratio which is an indicator of hepatic parenchymal damage.

Inspired by the work of Miles et al with the cardiac output of the patient, a good linearity of relative CBF, and recommend using the relative parameter as a predictor of the reversibility of an ischemic stroke.

. . Magnetic resonance imaging

Following the methodological developments of Gadolinium-DTPA complex (Gd-DTPA) by the baseline signal intensity, the maximum signal intensity, and the time-lapse between the start of the injection and the point of maximum signal intensity. Using these estimates, the de ned the signal intensity ratio as the percentage of increase over the baseline signal intensity, as well as the relative enhancement slope. ey observed an increase of the signal intensity ratios and relative enhancement slope in malignant tumors compared to benign tumors, the latter showing smaller overlap between malignant and benign groups. Enhancement slope enabled malignancy prediction with an accuracy of . % using a cuto value of %/min. ey also reported lower peak intensity ratios and wash-in rates in areas with necrosis or peritumoral edema.

Wilke et al.

[ ] investigated the use of contrast-enhanced magnetic-resonance imaging for the quanti cation of myocardial perfusion in a canine study, and compared the estimated parameters to the blood ow obtained using radiolabelled microspheres. A bolus of Gd-DTPA was injected during Turbo-FLASH acquisition, and a six circular regions of interest were delineated in the myocardium, as well as one region of interest in each ventricular cavity, yielding a total of nine regional time-intensity curves. All the time-intensity curves were tted with a gamma-variate curve to both correct for recirculation, and limit the impact of extravascular di usion, using only the samples occurring before the curves decreases to % of its peak value. e mean transit time of each curve was calculated numerically, and the correlation of its inverse, of the time of peak intensity, as well as the initial slope (de ned as the ratio of the peak intensity to the time of peak intensity) with absolute myocardial blood ow estimates from radiolabelled microspheres were investigated. Good correlations with the ground truth (r ≥ . ) were reported for the three semi-quantitative parameters cited above. Semi-quantitative methods based on indicator dilution theory were nonetheless extensively used to quantify perfusion in contrast-enhanced ultrasound exams.

Bolus injection

Model-free quanti cation methods

From an historical standpoint, Bommer et al assessed for the half wash-out time, with respective correlation coe cients of . , . , and . . In addition, the relative systolic wall thickening was also estimated, and while it correlated well with the coronary artery ow measured by an electromagnetic owmeter placed directly on the artery, no correlation was foud with the ultrasound half wash-out time.

A few years later, Vandenberg et al. [ ] performed a similar canine study with induced ischemia and hyperemia, intending to predict myocardial blood ow from semi-quantitative parameters derived from contrast-enhanced ultrasound time-intensity curves. In addition to peak intensity time and value, they investigated the use of the wash-in rate. Peak intensity value and wash-in rate were found correlated to myocardial blood ow, however correlations CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW were moderate (r = . and . respectively). However, the relative changes of the wash-in rate exhibited a stronger correlation (r = . ) with the relative changes of the myocardial blood ow, i.e. with induced ischemia and hyperemia.

In a clinical study, Ten Cate et al. [ ] estimated the total curve duration, the area under the curve, and the half wash-out time from time-intensity curves obtained in the ventricular septum from end-diastolic images. ey performed multiple regression analysis between the parameters extracted from contrast-enhanced ultrasound data to angiographic parameters, i.e. the percentage of coronary area stenosis, and the minimal lumen area, derived from data acquired using the protocol described in [ ]. Authors reported relations of various natures between the above-mentioned parameters, i.e. linear, inverse, exponential, and logarithmic. In particular, the strongest correlation was found between the area under the curve and the percentage of coronary area stenosis (exponential relation). Additionally, while all correlation were found signi cant, this couple of parameters was the only one with a strong correlation (r ≥ . ). Noting the discrepancies between the pre-clinical and the clinical results, the authors suggest they found their sources in modi cations of the experimental setup, i.e. the injection method or the nature of the ultrasound contrast agent.

In , Bleeker et al. [ ] evaluated the stability, size, and ultrasonic properties of multiple ultrasond contrast agents, and investigated the feasibility of blood ow estimation through in vitro experiments. eir ndings were in favor of the Albunex contrast agent, which were the only microbubbles exhibiting su ciently longlasting stability (i.e. in size and number) when exposed to ultrasound waves. In addition, a linear relation between the concentration of microbubbles and both backscatter coe cient (i.e. re ected power) and attenuation coe cient (i.e. transmitted power) was reported for low concentrations. ey also found strong linear relations between the ultrasonic properties of the contrast agent and the ow estimated using an indicator-dilution theory. ey emphasis the questionability of using grey-levels from ultrasound images to characterize blood ow, and instead recommend using attenuation based on transmission techniques, or backscattering a er proper correction of signal attenuation. In vitro experiments showed strong relations between the ultrasonic properties of Albunex contrast agent and the ow estimated using indicator-dilution theory.

A somehow similar in vitro study was conducted a few years later by Heidenreich et al. the tissue blood ow. e model is based on the estimation of the area under the curve and mean transit time for both input and studied tissues. e ratio of the area under the curve in the studied tissue to the input yields the tissue blood volume, the system mean transit time is approximated as twice the di erence in mean transit times, and the tissue blood ow was classically de ned as the ratio of tissue blood volume to system mean transit time.

In vitro calibration data revealed the log-compressed nature of the data in the experimental system, allowing linearization of the video intensity for low to moderate concentrations, and therefore conversion to volumetric concentration of microbubbles. Excellent agreement was found between measured and estimated ow rates, using measurements at low concentration. Authors report on the di culty to nd a pure blood pool in the ultrasound imaging plane, but also to image both tissue and input with su cient sensitivity and without satura- properties throughout the acquisition of the ultrasound images was outlined, and the impact of hydrostatic pressure and gas saturation level of the solution as reported in the litterature was investigated. e choice of the representation of the time-intensity curves, i.e. video intensity, log-transformed intensity, or concentration from calibration data, and its impact on the estimated parameters were discussed. A possible explanation for undetected changes when using the peak intensity as an indicator of blood ow was proposed, authors related it to signal saturation and illustrate this phenomenon in simple cases where the changes in blood ow are caused by a change in blood volume only or in transit time only. Additionally, electronic issues resulting from signal acquisition, signal processing, and electronic thresholding were discussed. Physiological factors like tissue-dependant hematocrit were also investigated. Finally, methods which showed promising results in preliminary studies at the time of the review (i.e.

) were reported, including various alternative imaging schemes (e.g. acoustic velocity, radiofrequency data, second harmonic imaging).

Model-based quanti cation methods

In , Kaul et al. [ ] proposed a rst-order gamma variate model to quantify blood ow using consecutive contrast-enhanced echocardiography acquisitions in an open-chest canine study with varying blood ow. ey compared their results in the myocardium and in the coronary bed with the transmural myocardial blood ow measured using radiolabelled microspheres, and to direct coronary ow measurements obtained by an electromagnetic owmeter. ey studied the in uence of the injection site by performing their experiments in two groups of eight dogs. In the rst group, dogs were injected a bolus of microbubbles in the circum ex artery, while in the second group they were injected in the le main coronary.

Authors reported a good correlation of both myocardial and coronary blood ow measurements with the parameter of the gamma variate model which in uences the width of the modeled enhancement curve, α, for both groups (mean r = . and . respectively), but a poor correlation of the peak intensity (mean r = . and . respectively). Pooling estimates from the eight dogs did not a ect the correlation with α for the rst group (r = . ), however for the second group, pooling data considerably dropped the correlation (r = . ).

Scattered plots reveal the varying slopes obtained in the di erent dogs, emphasizing the im-pact of the injection site. Indeed, when injecting in the le main coronary the contrast agent is dispersed through the coronary tree, making quanti cation of myocardial blood ow intrinsically relative. Authors suggested that absolute quanti cation of myocardial blood ow may be possible, injecting closer to the branches of the coronary tree, but were well aware that the proposed semi-quantitative parameters are not absolute themselves. linearly related to the wash-in time. ey performed a reproducibility study in ve untreated patients with a total of twelve acquisitions, with varying injected quantity and/or injection duration of contrast-agent from one acquisition to another. While the wash-in time in the metastatic tissue exhibited variations up to %, the wash-in time ratio was more reproducible exhibiting an average deviation of %, with a maximum of %, revealing the e ect of normalization. e ability of the wash-in time ratio to assess treatment e ciency was investigated in a longitudinal study, performed in seven patients undergoing a combination of cytotoxic and antiangiogenic treatments. Major ndings include the ability to discriminate good from bad responders to therapy, the ground truth responder classi cation being assessed by experts using conventional criteria, i.e. number and size of the lesions, blood tests for serum tumor markers, and liver function tests. Indeed, four out of ve good responders exhibited a signi cant rise of the wash-in time ratio a er the rst therapy cycle, revealing the early normalization of the lesion microvascularity, and the mean increase in wash-in time ratio among the good responders at the end of the treatment was %. for the distribution of blood velocities in the tissue. e model assumes a spherical distribution of blood vessel directions with varying blood velocity. is assumption yields an initial linear increase of the concentration until the vessels perpendicular to the imaging plane with the highest velocity are fully re lled. A er the initial linear increase in signal intensity, the fully lled vessels do not contribute to signal increase anymore. is yields a non-linear increase until only the vessels with the lowest velocity remain to be fully lled. Finally, all the vessels present in the imaging plane are fully lled and a plateau intensity is reached.

Infusion injection

In

Authors describe an iterative method to estimate the instant of maximum and mean velocity. ey state the slope of the replenishment curve observed at these times can be used to evaluate the maximum blood velocity in the region of interest, assuming the width of the ultrasound beam is known. e in uence on the width of the blood velocity distribution is visually described. e method was evaluated in a murine study with continuous infusion and intermittent imaging, followed by a bolus injection and intermittent imaging as using a dialysis cartridge with tubular capillaries, and in vivo experiments in the renal cortex of healthy volunteers, separated in two age groups. While it achieved a signi cantly better t than the exponential model using a Wilcoxon signed-rank test, whether the proposed model was using two, three, or four di erent tracts, the di erences in the mean squared error remained extremely small and exhibited non-negligible overlap. Furthermore, while the authors report a signi cant di erence in the initial replenishment slope among the two age groups using the proposed model, based a Mann-Whitney U-test, they do not report on the slope di erences obtained using the exponential model, nor on the apparent concordance of the plateau values obtained using the two models.

. Deconvolution methods

. . Generalities

Deconvolution-based methods are model-free approaches, assuming a linear and stationary system without any assumption on the underlying structures and processes [ ]. Resolving the deconvolution equations necessitates the knowledge of at least two measurements: an input function, and either a residual measurement (i.e. the amount of tracer remaining in the system) or an output measurement (i.e. the amount of tracer leaving the system). In particular, dynamic perfusion imaging grants access to spatially-distributed measurements of the residual tissue function.

e resolution process aims at estimating the impulse-response function of the system, which is theoretically independant on the input, and because no assumption is made on the structure of the system, no assumption is made on the shape of the impulse-response function or on the unit of the measurements. Additionally, the impulse-response function can the second one yields the impulse response of the pulmonary circulatory system, independently of the systemic recirculation of the tracers. e deconvolution was performed using numerically without any curve tting, despite the reported unstability of the solution. In order to limit this instability, the experiments were adapted consequently. Visual checking was used to assess the accuracy of the estimation, and manual correction was performed when necessary. Biases induced by correction of recirculation through mono-exponential interpolation of the wash-out were reported in case of early recirculation.

A popular matrix-based deconvolution technique was proposed by Valentinuzzi and Montaldo Volachec [ ] in . e method relies on the matrix formulation of the convolution process, however the set of derived linear equations is solved successively, therefore avoiding matrix inversion which is known to be an ill-conditioned problem. Authors acknowledge the sensitivity of the technique to noise, and warn about the impact of imperfect noise ltering on the estimation process. e role of the sampling interval is also discussed, the authors reveal that increasing it can only improve estimation accuracy up to a point, but demonstrated that increasing it too much could actually increase the estimation error.

In , a deconvolution method said to be insensitive to noise, recirculation, and im- curves. e impact of noise on the estimation of the impulse response by the direct method was rst demonstrated on simulated data, establishing the necessity of either using an appropriate noise ltering technique, enforcing some predetermined form to the impulse response, or applying some kind of regularization to the estimation process in order to avoid oscillations and negative values. Six estimation methods enforcing either of the above mentioned strategies were then investigated regarding various considerations. e impact of the data starting point was investigated, especially in the output curve which is usually widely spread and delayed. e authors report poor impulse response estimation in case of noisy time-varying data, despite the accuracy of the reconvolved curves. en the impact of the numerical approximation which is a consequence discretization process was investigated, while trapezoidal integration exhibited lower error than rectangular integration, both methods proved extremely sensitive to noise.

Additionally, in

Lassen [ ] warned about the inability to estimate the blood ow unless the shape of the impulse response is accurately known prior to deconvolution. He said this was necessary in order to accurately determin the height of the initial bolus. And while this is not the object of his letter, he also warned about the lack of direct relations between blood ow and the various semi-quantitative parameters proposed in the litterature.

. . Nuclear medicine

Following the development of renography based on gamma-camera acquisitions by Short study. e approach accounts for both the establishment of an equilibrium of the tracer in the extravascular space, spread of the bolus during its passage through the kidney, and the removal of the tracer from blood by the kidney assuming bi-exponential clearance in the cardiac measurements. e impulse response of the kidney vascular network was estimated from the Laplace-derived formula established in the paper, involving the renal time-activity curve, its derivative, its integral and various constants to be estimated. Because scintigraphy is a projection imaging technique, the background activity was subtracted using the regions between the two kidneys to obtain the activity as speci c to the kidney as possible.

is was encouraged by the similarity of the activity curves in this region with those of the nephrectomy sites observed in een kidney-ablated subjects. Authors were able to distinguish healthy kidneys from those a ected by various diseases, indeed all a ected kidneys exhibited longer minimum, mean, and maximum transit times, and a wider transit time distribution. Additionally, they were able to distinguish between the various studied diseases using the same criteria. purposes, the regional cerebral blood ow was also estimated using microspheres measurements, which was considered the ground truth. Fluorescent microspheres with varying colors were injected in the le atrium one minute before every contrast-enhanced computed tomography acquisition, blood was sampled in the femoral artery during two minutes a er the injection. Rabbit brains were then excised a er the last computed tomography acquisition, and cerebral blood ow was estimated using the method presented in [ ]. e authors report good agreement of the measured regional cerebral blood ow as they obtained CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW a slope close to unity in a linear regression analysis, and correlation between measurements was good yet not excellent (r = . ). When repeated studies were performed, i.e. in ve of the six rabbits included in the study due to sudden death of the sixth rabbit, no signi cant di erences were found between the parameters from the various acquisitions, however the computed-tomography estimates of regional blood ow were ten percent more variable than the one estimated using microspheres.

In ment between the reference method and both the commercial package and the presented method estimates of cerebral blood ow (R = . and . respectively), as well as between the two investigated deconvolution methods (R = . ). However, the methods were shown to di er in terms of spatial ltering of the parametric maps, as well as in the detection thresholds of vessels. Indeed, regions with arterial branches are ltered away using the commercial so ware, while they exhibit a higher cerebral blood ow using the improved least squares method. Note that these regions with branches are neither visible in the reference method.

In , Eastwood et al. [ ] used the commercial perfusion quanti cation so ware presented above to quantify brain perfusion in patients with acute stroke. Hoe ner et al. [ ] provided a review of the technique, as well as the various applications, and limitations of this quanti cation so ware. e larger anterior artery was used as an arterial input function, and . . DECONVOLUTION METHODS the superior sagittal sinus was used as a venous out ow function. e deconvolution method requires arterial and venous measurements in order to compare the shape and height of the time-attenuation curve of each pixel with the shape and height of the arterial and venous time-attenuation curves before proceeding to the actual deconvolution, i.e. the estimation of the tissue impulse response. e cerebral blood volume was then derived as the area under the tissue impulse response, the cerebral blood ow as the plateau value of the tissue impulse response, and the mean transit time as the ratio of these two parameters. e authors report signi cant di erences between the healthy and a ected hemispheres in either of the three parameters derived from the deconvolution of signals. ey also observed significant di erences between low-enhancement and normal-enhancement regions for cerebral blood ow and cerebral blood volume. Additionally, inter and intra observer variation were investigated and showed good agreement between and within observers.

In , Cuenod et al. [ ] proposed a deconvolution-based method for the quantication of perfusion in rats with metastasis in the liver using contrast-enhanced computed tomography accounting for both arterial and portal supplies, with bolus injection of iobitridol, a iodinated contrast-agent. Because the arterial and portal components are totally mixed in the sinusoidal capillaries, the system assumes a common impulse response for both components shaped as a Weibul function, however the two inputs are weighted. is technique allows the estimation of the hepatic blood ow, which can be separated in two components, yielding the arterial and portal blood ows, and the tissue blood volume. e method di ers from classical deconvolution because of the strong a priori on the shape of the impulse response, which is de ned parametrically, considerably reducing the number of parameters to estimate but also the exibility of the method. e mean transit time of the contrast agent was also estimated from the impulse response of the system. Additionally, the complementary weights of the two blood supplies allow the estimation of a parameters called the hepatic perfusion index, which is the ratio of the arterial blood ow to the total blood ow. e perfusion of micro and macro metastases were compared to normal liver tissue using the above cited parameters, and signi cant di erences were found in mean transit time, total hepatic blood ow, and portal blood ow for both types of metastases, but also in hepatic perfusion index and the tissue blood volume for macro metastases. e theoreti- e three methods were investigated on synthetic datasets varying the tracer delay in the le hemisphere. e block-circulant approach was the most insensitive to tracer delay while the standard method was the most sensitive, and the delay-corrected method yielded intermediate results. sections, one being the section of interest, the other one containing the feeding arteries. e arterial input function was estimated directly in the imaging plane, either in the carotid or in the vertebral arteries, and the segmentation was performed on pixel-by-pixel maps of semiquantitative parameters, i.e. the full width at half maximum, peak intensity, and time to peak intensity. More precisely, adaptive thresholding was rst performed on the full width at half maximum and peak intensity maps, and only pixels with peak intensities over % of the maximum intensity in the image were retained. e authors de ned the regional cerebral blood volume as the ratio of the area under the curve in the tissue of interest to the area under the arterial input function, as suggested by indicator-dilution theory, but corrected this estimate for both brain tissue density and hematocrit variations from arteries to smaller blood vessels. Fourier deconvolution enforcing noise ltering by means of a Wiener lter was used to estimate the impulse response of the tissue of interest. en the mean transit time was estimated as the ratio of the area under the impulse response to its height. Regional cerebral blood volume and blood ow were estimated in both gray and white matter for every patient. e authors report an overall decrease of both perfusion parameters with age, and suggest this explains the large standard deviations of their estimates.

. . Magnetic resonance imaging

In , Wilke et al. [ ] used a low-order vascular transport operator, presented in [ ],

to model the tissue impulse response using MRI with bolus injection of an intravascular contrast agent, i.e. polylysine-Gd-DTPA. ey compared their perfusion estimates in the myocardium of dogs, with and without induced coronary stenosis, to the perfusion parameters obtained using radiolabelled microspheres. Contrast signal intensities were normalized according to the pre-contrast values, as well as from the signal intensities observed in an oil CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW phantom placed under the chest of the dogs. e vascular transport operator accounts for the pure delay of the contrast agent, but also for the dispersion of the bolus by means of a fourth-order linear di erential operator, resulting from the combination of two attenuated second-order operators in series. e model is therefore parameterized by two parameters only, i.e. the mean transit time of the impulse response, and the relative dispersion which is de ned as the ratio of the standard deviation of the impulse response to its mean transit time.

e model is tted using a fast iterative derivative-free approach presented in [ ], which performs non-linear least-squares tting without making assumptions on the data variance through data weighting. Despite imperfect registration between magnetic resonance estimates and tissue analysis for quanti cation of microspheres, good agreement between the measurements was reported regarding both myocardial blood volume and myocardial blood ow. Myocardial segments that were hypoperfused because of coronary stenosis were accurately identi ed using the proposed method. e weight of the regularization term in the algebraic method, as well as the thresholding value for the SVD approach were determined empirically, and chosen as the values that yielded the most accurate impulse responses and cerebral blood ow estimates on twelve representative simulated cases. Multiple factors and their impact on the regional cerebral blood ow estimates were investigated in this simulation study, i.e. noise, shape of the impulse response, cerebral blood ow, cerebral blood volume, and time-delay. Expectedly, the model-based approach was not able to correctly estimate cerebral blood ow when the shape of the impulse response was not exponential and therefore did not match the single-compartment model assumption. e Fourier method proved sensitive to the cerebral blood ow, indeed the estimation error was negatively correlated with the simulated value, but also to the simulated vascular structure. e regularized algebraic approach proved to overestimate blood ow for a low cerebral blood volume ( %), and to yield more accurate estimates for higher cerebral blood volume ( . %), the authors therefore concluded on the sensitivity of the method to cerebral blood volume. However the authors discuss the possible role of the regularization weight in these variations, as modifying the cerebral blood volume e ectively modi es the signal to noise ratio, which could then di er from the conditions for which the weight was optimized. While the opposite behaviour seems to be observable in the SVD estimates, the authors considered that the bias in the estimation of cerebral blood ow was insensitive to cerebral blood volume. Additionally, when a time delay was added to the simulated curve, the SVD greatly underestimated cerebral blood ow, in particular for large simulated ow values, emphasizing the need for delay correction prior to the estimation process. Model-free approaches were able to estimate cerebral blood ow accurately, however they generally failed to accuratly match the shape of the impulse response function. Globally, all approaches were less precise in case of the authors mainly attribute this delay to the delay between the acquisition of the di erent slices. ey however suggested that appropriate time correction should be applied when quantifying multiple slices using a common arterial input function estimated in a single slice.

However, some cases exhibited time-delays in regions supplied by other arteries, in particular the cerebellum, and in the occipital region which are mainly supplied by the posterior cerebral artery. For all approaches the cerebral blood ow was de ned as the maximum of the impulse response curve, and the cerebral blood volume as its area under the curve.

In healthy patients, the exponential and SVD approaches yielded very similar parametric maps. e regularized algebraic approach yielded very high ow in the grey matter which can be explained by the high cerebral blood volume in this tissue. e Fourier approach yielded poor contrast between the grey and white matters, con rming the sensitivity of the method to the cerebral blood volume and vascular structure. Strong di erences, i.e. almost -fold, were observed between the cerebral blood ows estimated using the exponential and Gaussian models, indeed a better agreement was found between the SVD and exponential approaches. Moreover, the t quality was considerably lower using the Gaussian model, suggesting that the exponential model is a better approximation to model cerebral impulse response. rst acquisition was performed at rest, and the second was performed during adenosineinduced maximum hyperemia. Fermi-based deconvolution was then performed for both exams, using the le ventricle blood pool as an input function. e myocardial blood ow was de ned as the maximum value of the Fermi function, i.e. initial value, and the myocardial blood reserve was obtained as the ratio of the hyperemic blood ow to the basal blood ow. e coronary ow reserve estimated in the le anterior coronary artery using intracoronary Doppler ultrasound was used as a gold standard. Good agreement was found between the reference measurement and the perfusion reserve estimated in the le anterior coronary artery region (r = . ) with a regression slope close to the unit but with high uncertainty ( . ± . ). Moreover, even if the data were not presented, agreement between the perfusion in the le anterior coronary artery region and the perfusion reserve averaged over the entire myocardium was reported. ey also assessed the ability of the method to estimate blood ow reserve from synthetic data, resulting from random solutions of the multiple-pathway perfusion model described in [ ] with simulated uniform random noise. e authors report excellent correlation between the simulated and estimated values (r = . ) with a regression slope close to the unit ( . ± . ). However, accounting for capillary permeability responsible for the leakage of contrast agent in the interstitial space in the simulation process, i.e. varying the permeability surface area product, had a direct impact on the simulated curves, but induced a variation of less than ten percent in the estimates of the myocardial perfusion reserve. e impact of varying upstream bolus dispersion, capillary permeability, blood volume, blood ow, sampling frequency, and injection speed on the estimated parameters was also investigated by simulation. e authors concluded the ability of model-based deconvolution to determine the myocardial perfusion reserve in perfusion magnetic resonance exams.

Inspired by the work of

. . Ultrasound

In as measured by the reference method, despite a seamingly good correlation of the two measurements. However, using the deconvolution-based method an excellent agreement of the estimates is reported, indeed the correlation is high (R = . ) with a mean bias of . %, and a standard deviation of %, demonstrating the feasibility of the method. e absolute parameters tend to be slightly less variable than the semi-quantitative parameters of the method without prior deconvolution, however no statistical tests were reported, and neither were the estimated parameter values, preventing the validation of this quanti cation method by comparison to the in vitro experimental parameters. In vivo, three consecutive acquisitions were performed in ve tumor-bearing mice, using an arterial input function estimated directly from the imaging plane in a feeding vessel that was previously detected using Power Doppler imaging. In terms of reproducibility, the quantitative parameters obtained by deconvolution was found superior to the semi-quantitative parameters derived from the modeled curve. Once again, only the variability of the parameters was reported, the estimated values were not reported, and no statistical analysis was performed. K is the unidirectional transfer rate of contrast-agent from blood to tissue vascular space and is related to blood ow and capillary permeability. k is the unidirectional transfer rate of contrast-agent from the tissue vascular space to blood and is de ned as K V D , where V D is the contrast-agent fractional distribution volume. Similarly k and k are the unidirectional transfer rates of contrast-agent between tissue vascular space and interstitial space, and k and k are the unidirectional transfer rates of contrast-agent between interstitial space and intracellular space. K is classically expressed in milliliter of blood per minute per milliliter of tissue (mL.min -.mL -), and the other rate constants k , k , . . . , k in fraction of contrast-agent per minute (min -). subcompartments: a vascular space fed by an 'input function ', an interstitial space, and a cellular space. e other assumptions are that the quantity of contrast agent inside the system is conserved, and that the quantity of contrast agent leaving a compartment is proportional to the quantity inside it. e exchanges of contrast agent between the compartments are modeled using rst-order di erential equations, establishing the relations between the concentration of contrast agent in each compartment in terms of physiological parameters, i.e. blood ow rates, tissue volumes, transfer constants. But also because in the approach proposed by Renkin [ ], the presence of tracer outside capillaries was neglected despite the steady state reached by the system during the experiment. Oppositely the approach proposed in this study relies on non-steady state measurements, ensuring lower extracapillary concentrations, and allowing accurate permeability estimates.

In

In

C A (t) K 1 C V (t) C I (t) C C (t) k 2 k 3 k 4 k 5 k 6 (c) C A (t) K 1 C V (t) k 2 (a) C A (t) K 1 C V (t) C I (t) k 2 k 3 k 4 (b) dC V (t) dt = K C A (t) -k C V (t) (a) dC V (t) dt = K C A (t) -(k + k )C V (t) + k C I (t) dC I (t) dt = k C V (t) -k C I (t) (b) dC V (t) dt = K C A (t) -(k + k )C V (t) + k C I (t) dC I (t) dt = k C V (t) -(k + k )C I (t) + k C C (t) dC C (t) dt = k C I (t) -k C C (t) (c)
e theoretical developments presented in the three approaches above served as the foundation for compartmental analysis of perfusion exams. e following sections present the adaptation of compartmental models to quantify perfusion in nuclear medicine, x-ray, magnetic resonance, and ultrasound imaging. When comparing approaches, one should consider the characteristics of the tracer (intravascular vs. di using, reversible vs. irreversible), the characteristics of the vasculature in the tissue of interest (single vs. dual input), the estimation method (linear vs. non-linear).

. . Nuclear medicine

Many compartmental analysis developments in the eld of nuclear medicine, and in particular positron emission tomoraphy (PET), were motivated by metabolic studies such as the consumption of glucose. ese metabolic measurements were permited by the development of labelled tracers, and in particular of labelled glucose. Indeed, at the cellular level high consumption of glucose is synonym of fast metabolism. Studying the consumption of glucose can reveal information on the cellular activity in the tissue, which is particularly relevant to study neuronal activity or to detect tumors and assess their malignancy. is is an extension of the method proposed by Gjedde [ ] to account for rapid binding and unbinding of the tracer where the graphical analysis is performed similarly. Arterial blood was sampled throughout the experiment, and the arterial concentration in radiolabeled cocaine complex was used as an input of the model. In terms of two-compartment model, the distribution volume can be derived from the slope of the linear portion of the plot. In terms of three-compartment model, an additional measurement is necessary to determine the ratio of binding to unbinding rate from the ratio of the slopes in a receptor tissue to a non-receptor tissue assuming the plasmatic volume is negligible.

e authors later extended the graphical analysis method to estimate distribution volume ratios [ ], de ned as the ratio of the distribution volume of a tracer in a receptor tissue to a receptor-free tissue. e method allows the estimation of the distribution volume ratio from studies without arterial blood sampling, using only the curves extracted from the image in two regions. e direct distribution volume ratio estimation using the proposed method was compared to the ratio of the distribution volumes in the two tisues obtained using blood sampling data in a PET study with injection of C-labelled raclopride or dthreo-methylphenidate. e method requires the knowledge of the average tissue to blood rate constant, as well as the determination of the initial time at which the slope should be estimated, which can be obtained from baseline experiments requiring blood sampling for calibration.

In constants from blood to tissue and from tissue to blood, as well as its metabolizing rate, by solving the system of equations using a linear least-squares methods. e proposed linear estimation method was compared to the non-linear estimation of the three-compartment model, and good agreement were found between the estimates of the two methods, especially for blood to tissue transfer rate, and for the cerebral metabolic rate for glucose utilization, a composite index which de nition can be found in [ ].

In , Gunn et al. [ ] reviewed the compartmental models used for quanti cation of dynamic PET data, whether assessing perfusion, metabolism or ligand binding. is theoretical study investigates both models using arterial input functions and models using reference tissues, as well as reversible and irreversible tracers, regardless of the number of compartments involved. General solutions for the impulse response are given in the form of sum of weighted exponential functions, which weight and rate constant have to be determined. partment is in equilibrium with the free compartment, then these compartments can be merged, reducing the number of parameters by two. Similarly, the simple one-compartment model, that only considers the exchanges between plasma and tissue through two rate constant parameters, can be used in ow limiting conditions, i.e. when a quick equilibrium is reached between the free compartment and the binding compartment, or when using nonbinding intravascular tracers. e perfusion obtained with the rst approach exhibited a strong correlation with microsphere measurements, which was not true for the second approach. e reproducibility of CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW the rst approach was then investigated in a clinical experiment with healthy patients, and a good inter-and intra-observer agreement was reported.

. . X-ray imaging

. .

Magnetic resonance imaging

. . Ultrasound

To our knowledge, no proper compartmental approach had been proposed prior to the method proposed in [ ] and in Chapter of this document. Indeed, compartmental models exhibit simple structures in contrast-enhanced ultrasound because the microbubbles remain strictly in the vascular bed, alleviating the need for additional extravascular compartments.

Recently, a compartmental model was used to characterize contrast-enhanced ultrasound data by Rizzo et al. [ ]. However the method is based on a gamma-variate model of the circulating bubbles with the addition of an irreversible compartment representing a fraction of microbubbles that can get trapped in complex vasculature. is compartment is modeled by the addition of the integral of the tted gamma-variate curve to the model, and the weight of this term must be estimated. e model was tted at the pixel level using a variational Bayes estimator which priors were derived from the non-linear t of the same model at the regional level. e authors reported a better t in half of the pixel using the model with the irreversible compartment compared to a simple gamma-variate model in a cohort of subjects su ering from arthritis. ey designed an automatic algorithm to detemine which model should be used according to the data. e fraction of trapped microbubbles also improved classi cation of patients with rheumatoid and non-rheumatoid arthritis.

. Discussion

From an experimental standpoint, the pioneering studies reviewed in this chapter su er from major limitations which can be explained by the simultaneous development of the imaging technology, contrast agents, and quanti cation approaches.

Many of these studies used preliminary ultrasound contrast agents, known to be unstable proximatively respect this condition when low mechanical index imaging is used, this is why these experiments are o en extremely conclusive, and yield perfusion parameters in good agreement with the ground truth. However, in vivo, in addition to the natural and induced disruption, circulating microbubbles are partially ltrated through the lungs and liver, the mass conservation principle is therefore not respected.

Model-free quanti cation was common in early studies, it is however inherently sensitive to noise, especially as contrast-enhanced ultrasound time-intensity curves are corrupted by a multiplicative noise. erefore quantifying perfusion using parameters dependant on the peak-intensity is extremely unreliable, as it corresponds to the highest noise amplitude.

In pioneering cardiac studies, sampling frequencies were low and did not allow accurate characterization of the high-frequency wash-in resulting from bolus injection, therefore numerous studies focused on contrast-agent wash-out.

Following methodological and computational developements, model-based quanti cation was proposed to t time-intensity curves, and allowed the estimation of many semiquantitative parameters. While physical explanations were found for some explicit models, most studies merely used the models for noise ltering, correction of recirculation and time interpolation. Indeed, semi-quantitative parameters were estimated from tted curves, then blood ow and blood volume estimates were derived from indicator dilution theory.

Whether extracted directly from the time-intensity curve or from the modeled curve, semi-quantitative parameters are inherently sensitive to experimental conditions. For instance, area under the curve, peak-intensity or even wash-in and wash-out rates, are directly related to the injected dose, moreover time parameters such as the wash-in and wash-out times, the time to peak-intensity or the mean transit time of the contrast agent are in uenced by the site, speed and duration of the injection.

In addition, physiological parameters such as heart rate or even blood pressure can a ect the kinetics or the ultrasound response of the microbubbles, and therefore a ect the timeintensity curves. Because semi-quantitative parameters are directly extracted from the timeintensity curve, they su er from physiological inter-exam variations too. is is also true for the replenishment of tissues following disruptive pulses in the case of infusion injection.

Because of the sensitivity of the semi-quantitative parameters to both experimental and CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW physiological conditions, and because indicator-dilution theory is hardly applicable to contrastenhanced ultrasound as discussed above, one should consider the derived parameters as relative indicators of perfusion, but certainly not as absolute parameters.

Deconvolution approaches estimate the response of a tissue to an instantaneous injection of tracer using an arterial input measurement, a tissue measurement, and none or little a priori on the shape of the estimated impulse tissue response. Deconvolution methods based on the matrix formulation rely on matrix inversion, which is known to be an ill-conditioned problem. Iterative algorithms were proposed to estimate the tissue impulse response without inverting the matrix, however they are extremely sensitive to noise as they are prone to cumulative error. Unconstrained or poorly constrained deconvolution can sometimes yield impulse responses with negative values, which is not physiologically possible. Constrained and regularized deconvolution were proposed to estimate positive, decreasing and smooth impulse response functions, but still few assumptions are made on the shape of the tissue response.

e shape of the estimated curves can be extremely heterogeneous depending on the estimation method and constrains, and extracting parameters can be di cult. To reduce the sensitivity of the deconvolution to noise and to limit oscillations of the tissue response, noise ltering techniques can be enforced rather simply in Fourier or singular value decomposition or by direct ltering of the time-intensity curve, but even small ltering errors can yield imprecise estimates.

Compartmental modeling roughly consist in a model-based deconvolution with strong a priori on the shape of the tissue impulse response and a reduced number of physiologically relevant parameters. e small number of degrees of freedom restricts the shape of the input function convolved with the tissue impulse respons to a limited set of possible curves, making the estimation process less sensitive to noise. Linear formulations of these models can alleviate the need for initialization, but also make the estimation process more robust to noise, yielding stable estimates with fewer or sparser samples as discussed in Chapter .

An arterial measurement is required for some semi-quantitative approaches, by almost all deconvolution approaches, and by most compartmental models. Accurate arterial measurement allows the normalization of the tissue measurement and the estimation of perfu-. . DISCUSSION sion parameters that are independant of the injection.

Arterial measurement can be performed through blood sampling, this technique is especially popular in the elds of nuclear medicine and radiology as continuous radioactivity measurements apparatus have been extensively developed and commercialized. Nonetheless, patient needs to be catheterized and this procedure remains invasive.

Image-based arterial measurements are extremely unreliable regardless of the imaging modality. Indeed the aorta is not always present in the eld of view and a large artery feeding the tissue of interest can be hard to identify, especially in two-dimensional data.

Additionally, partial volume e ect, attenuation and saturation artifacts can a ect the measured signal intensity. Indeed, in most modalities a trade-o must be made between space and time resolution, and the small size of the artery combined to the high speed of a bolus in an artery makes the estimation of the arterial input function extremely di cult. In contrast-enhanced ultrasound the noise is multiplicative, which makes the high-intensity arterial measurements highly unreliable.

A healthy reference tissue can be chosen in the image to normalize the perfusion parameters in the tissue interest by their value in the reference region. e reference region should be chosen in a large, homogeneous and well perfused tissue to limit partial volume e et and saturation artifacts in addition to reducing the noise in the reference tissue enhancement curve.

Direct estimation of relative perfusion parameters can also be obtained using compartmental modeling, alleviating the need for an arterial measurement. e relative perfusion quanti cation methods were generally formulated with a single enhancement curve to characterize, even though they were o en applied later to the region or even pixel level.

Whether tting an explicit model to a single tissue curve for noise ltering or estimating Regarding contrast-enhanced ultrasound, which is the core modality addressed in this thesis, semi-quantitative approaches are by far the most commonly used approaches. Indeed, many manufacturers of ultrasound scanners and contrast-agent implemented these techniques in commercial so wares, which explains their popularity in clinical appication.

A few approaches relying on deconvolution were also proposed for characterization of contrastenhanced ultrasound exams, but compartmental modeling remains extremely rare in the CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW ulrasound litterature. is can be explained by the dependance of the majority of these techniques on the ability to perform an accurate arterial measurement. However accurate measurement is not always possible as arterial regions are usually small and exhibit high tracer concentration, making the estimated curve subject to saturation and partial volume e ects. Finding an artery can be especially tricky in two-dimensional data, and even more limiting when attempting to compare two or more exams. Indeed, imaging the exact same plane is extremely di cult even for experienced radiologists, especially in the case of evolving tissues like growing tumors or when monitoring the e ect of a treatment. e impact of the arterial function on the parameters of a one-compartment model for quanti cation of contrast-enhanced ultrasound data will be investigated in Chapter .

When an arterial input function cannot be estimated, or at least not accurately, another tissue present in the image can be used as a reference for comparison purposes. Just like the absolute values, the relative or normalized perfusion parameters can be used to perform relative comparison of the tissues observable in a single exam. However, parameter normalization allows comparison of the same tissue observed in di erent exams, using the reference tissue as a basis for comparison, and assuming the reference tissue did not change between the two exams. e reproducibility of relative approaches will be investigated in the following part of the thesis. [ ] E Klotz and M König. Perfusion measurements of the brain: using dynamic CT for the quantitative assessment of cerebral ischemia in acute stroke. European journal of radiology, ( ): -, June .

Bibliography

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[ ] T J Knopp, W A Dobbs, J F Greenleaf, and J B Bassingthwaighte. [ ] E M Renkin. [ ] C Tei, T Sakamaki, P M Shah, S Meerbaum, K Shimoura, S Kondo, and E Corday. Myocardial contrast echocardiography: a reproducible technique of myocardial opacication for identifying regional perfusion de cits. Circulation, ( ): -, March

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[ ] F J Ten Cate, J K Drury, S Meerbaum, J Noordsy, S Feinstein, P M Shah, and E Corday.

Foreword

In the second part of this thesis, we perform an in-depth investigation of three major perfusion quanti cation techniques. e techniques investigated are a semi-quantitative method using the log-normal distribution, a one-compartment model using an image-based arterial measurement, and a one-compartment model using an image-based reference tissue.

e rst two methods yield absolute parameters while the third directly yields relative parameters. Relative parameters were derived from the absolute parameters, their value inside a tissue of interest are de ned as the ratio of the absolute parameter value in the tissue of interest to the absolute parameter value in the reference tissue. In an attempt to reveal the spatial functional heterogeneity of tumors while ensuring reasonable signal-to-noise ratio in the time-intensity curves, the models were tted to regional time-intensity curves resulting from the cutout of the perfused tumor area.

Barrois et al. [ ] previously proposed an original criterion based on a multiplicative noise

model to re ect the characteristics of noise in ultrasound data. e criterion was not used in this study since it was unable to t the regional data accurately despite using the least-square estimates as initial values. Instead, we optimized a conventional least-squared criterion and use a non-linear estimation method to t all models to data.

e log-normal model is commonly used in the litterature to t perfusion curves, and in particular contrast-enhanced ultrasound data. It is mainly used to lter out the noise of the time-intensity curves, and semi-quantitative parameters are usually derived from the tted curve, e.g. the area under the curve, the peak enhancement, or the wash-in rate. e log-normal model was used to t the global mean time-intensity curve inside the perfused area of tumors [ ]. is study revealed a poor reproducibility of the global semi-quantitative parameters despite the improvement resulting from the use of an automatic injection setup, which was also used in Chapter .

Compartmental modeling performs a 'normalization ' of the studied time-intensity curve by the input of the system through deconvolution, theoretically making parameters independant on the characteristics of the tracer injection, i.e. quantity, duration. Compartmental approaches were primarily developed for nuclear medicine imaging techniques, they were then extended to other perfusion imaging modalities, in particular to contrast-enhanced magnetic resonance imaging. e one-compartment model is the most basic form of compartmental model, and is particularly designed for intravascular tracers like ultrasound contrast agents. However no reference of this model applied to contrast-enhanced ultrasound data was found in the litterature. e method allows the estimation of absolute perfusion parameters, i.e. tissue blood ow, and tissue blood volume. However, as discussed in Chapter , arterial measurements derived from the image are subject to artifacts, and are in fact not always possible.

To alleviate the need for arterial measurements, relative methods making use of a reference tissue instead were developed [ ], allowing direct estimation of relative perfusion parameters. ese relative parameters are closely related to the absolute parameters, except they are normalized according to the reference tissue. e reference tissue can be selected in a large and homogeneous region, free of large blood vessels, reducing the in uence of noise, contrast saturation, and partial volumes.

e reproducibility of the regional parameters estimated using two absolute methods and three relative methods was investigated through test-retest experiments in order to limit the inter-exam changes to their minimum. Additionally, the impact of accounting for the have high CV, whatever the approach, with median values of about % for BV, and % for BF. e positive impact of normalization was established, showing a coherent estimation of relative parameters, with reduced CV (about % for BV and % for BF using the rRTd approach). ese values were signi cantly lower (p < . ) when compared to CV of absolute parameters. e rRTd approach provided the smallest CV and should be preferred for estimating relative perfusion parameters.

. Introduction

Reliable quanti cation of tumor perfusion is a challenging objective in order to establish Prior to imaging, animals were individually placed in an induction chamber, where anesthesia was induced with % iso urane in air with delivery rate of L/min. Anesthesia was maintained with % iso urane in air delivered by a face mask with the same delivery rate.

e temperature of the animal was regulated using a thermostatic heating plate (Minerve, Esternay, France). Each mouse was secured in position with surgical tape so that the oper-CHAPTER . QUANTIFICATION OF TUMOR PERFUSION USING DCE-US: IMPACT OF MATHEMATICAL MODELING ator could not inadvertently reposition it during the procedure.

. . Image acquisition

Dynamic contrast-enhanced US sequences were acquired using a L W transducer and a Sequoia US system (Acuson, Siemens, Mountain View, CA, USA) with constant mechanical index ( . ), dynamic range ( dB), and time gain compensation settings. e imaging plane was selected as the largest cross-section of the tumor and the probe was xed to a support in the selected position.

A bolus of µL of SonoVue (Bracco Suisse SA, Geneva, Switzerland) diluted to % was injected at a rate of . mL/min using a controlled injection system to improve acquisition reproducibility [ ]. is diluted concentration was proposed to reduce attenuation artifacts.

Each acquisition consisted of a minute dual-mode recording, including B-Mode and Contrast Pulse Sequencing (CPS) images, using a frame rate of Hz during the rst seconds (including the wash-in phase and the beginning of the wash-out phase), and Hz for the remaining time. Four consecutive (test-retest) studies were acquired for each mouse without any modi cation in the setup. Fi een minute breaks were observed between acquisitions to ensure the destruction of all circulating micro-bubbles.

. . Data pre-processing

Linear echo-power kinetics were extracted from log-compressed video data using a validated home-made so ware calibrated using dose-ranging data [ ]. Both probe and animal motion were assumed negligible for the selected sequences.

Tumors were segmented on the B-mode acquisition and necrotic zones were excluded as . Methods

. . Quanti cation of tumor perfusion

Table . summarizes the main features of the eight methods tested, three absolute and ve relative, for tumor perfusion quanti cation. Some methods require the de nition of an arterial region in order to estimate the AIF, its estimation is presented in section . . . Relative quanti cation methods require the selection of a reference tissue region (labeled with subscript R). is region was segmented in order to de ne a homogeneous area, while being large enough in order to reduce noise in uence on the subsequent analysis (Figure -).

For all methods, quantitative parameters were derived by the minimization of the rootmean-square error between the time-intensity curve inside the tumor, C T (t), and the corresponding tted curve, using an interior-point algorithm (MATLAB, MathWorks, Natick, MA, USA). To make the comparison between models easier, we focused on volume- e kinetics C T (t) is tted according to the equation ( . ):

C T (t) = AUC T √ πσ T (t-∆ T ) exp -[ln (t-∆ T )-µ T ] σ T if t > ∆ T , = otherwise, ( . )
where AUC T is the area under the C T curve, µ T and σ T are the expectation and standard deviation of the distribution C T (τ) AUC T when substituting ln(t -∆ T ) with τ, and ∆ T represents the time shi between the start of the acquisition and the arrival of the contrast agent in the tumor. As the area under the curve AUC T is related to the tissue blood volume [ ] and the wash-in rate WIR T (derived from AUC T , µ T and σ T ) is related to the tissue blood ow [ ], these two parameters were estimated in addition to ∆ T in the remaining analysis.

One-Compartment model with an Arterial Input Function: aAIF and aAIFd

[ ]. e mathematical expression of C T (t) is given by equation ( . ): ) . . METHODS where C A (t) is the kinetics inside the feeding artery (the AIF), V T the tissue blood volume (in %), F T the tissue blood ow (in s -), and d T (in s) the time-delay of the contrast agent from the feeding artery to the tumor. When it is neglected (d T = ), the model is noted aAIF. When it is estimated in addition to V T and F T , the model is noted aAIFd.

C T (t) = F T ∫ t-d T C A (τ) exp - F T V T (t-d T -τ) dτ if t ≥ d T , = otherwise, ( .
To estimate the AIF, a bounding box surrounding arterial vessels was rst de ned ac-

rAUC = AUC T AUC R , rWIR = WIR T WIR R , and D T-R LN = ∆ T -∆ R , ( . )
where (AUC T , WIR T , ∆ T ) and (AUC R , WIR R , ∆ R ) are the absolute LN parameters estimated in the tumor and in the reference tissue respectively using equation ( . ).

Relative One-Compartment model with an Arterial Input Function: rAIF and rAIFd.

is model estimates three parameters: the relative blood volume rV AIF , the relative blood ow rF AIF and the time delay between the arrival of the contrast in the tumor and the reference tissue D T-R AIF according to equation ( . ):

rV AIF = V T V R , rF AIF = F T F R , and D T-R AIF = d T -d R , ( . )
where (V T , F T , d T ) and (V R , F R , d R ) are the perfusion parameters estimated in the tumor and in the reference tissue respectively using the AIF according to equation ( . ). is method is referred to as rAIF when D T-R AIF is set to zero and rAIFd otherwise.

Relative One-Compartment model using the Reference Tissue kinetics: rRT and rRTd [ , ]. is model estimates three parameters: the relative blood volume rV RT , the relative blood ow rF RT and the time delay between the arrival of the contrast in the tumor and the reference tissue D T-R RT , the subscript RT being used for distinguishing this approach from the previous one (section . . ). Assuming that the tumor and the reference region have a common AIF, the kinetics C R (t) and C T (t) can be described by equation ( . ). When replacing C A (t) by its expression as a function of C R (t) in equation ( .), C T (t) can then be described by equation ( ):

C T (t) = rF RT C R t -D T-R RT + (k R -k T ) ∫ t-D T-R RT d R C R (τ) e -k T t-D T-R RT -τ dτ if t ≥ d T , = otherwise, ( .
)
where k R = F R V R and k T = F T V T . e parameter k R was chosen as the mean value of the k R values estimated with the relative AIF approach (rAIFd) and was set to . . e parameters rF RT , k T andD T-R RT were estimated by solving equation ( .). e parameter rV RT was then . . METHODS deduced using equation ( . ):

rV RT = V T V R = F T k T k R F R = rF RT k R k T . ( .
)
e method is referred to as rRT when D T-R RT is set to zero and rRTd otherwise.

. . Data analysis

For each model, the quantitative assessment of the t was achieved using the normalized root mean square error (N RMSE), and the fraction of information that is modeled (FMI), as de ned in [ ]. e N RMSE was de ned by:

N RMSE = nt ∑ nt t= C f it (t) -C (t) max t (C(t)) -min t (C(t)) , ( .
)
where C and C f it are the observed and tted kinetics and nt is the total number of frames.

A good t corresponds to N RMSE close to and FMI close to %. For each sub-region, results for which FMI < % were judged as poor quality ts.

In order to assess the reproducibility of the parameters θ hl of the mouse m l (l from to

) in the sub-region s h (h from to ), coe cients of variation CV hl were estimated using the four test-retest studies, as follows:

CV hl = ∑ k= (θ hl (k) -µ hl ) µ hl
, where . Results signi cantly improved the modeling quality, according to both criteria. Furthermore, the number of cases for which FMI < % was largely reduced when taking into account the time delays. For these reasons, results obtained without time delays (aAIF, rAIF and rRT)

µ hl = k= θ hl (k), (

. . Model comparison through quality of t criteria

were not further reported.

. . Model comparison through coe cients of variation

All mean values and standard deviations of the perfusion parameters are given for each mouse in Appendix, in Table . . high correlation between the ow-based parameters: WIR, rWIR, F, rF AIF , and rF RT . is gure shows also that there is a large range of values for each parameter within one tumor, demonstrating that perfusion parameters inside the di erent sub-regions of the tumor are far from being similar. Finally, it proves that the slopes may be quite di erent from one study to another, and that the use of relative parameters contributes to largely reduce the di erences between the test-retest studies, the estimation of rF AIF being less robust than the estimation of rWIR or rF RT for this speci c example. Table . illustrates the in uence of the AIF choice on the estimation of volume, ow, and time delay parameters. On this speci c exam, two AIF were generated, the rst one (AIF ) with thresholds rPE * = % and ∆T TP * = s, the second one (AIF ) with thresholds rPE * = % and ∆T TP * = . s (as shown in Figure -). e variations were very large for V and F parameters, while they remained moderate for rF AIF and time delays, and very low for rV AIF .

V (%) rV AIF (%) F ( -s -) rF AIF (%) d T (s) D T-R AIF (s) AIF . ± . . ± . . ± . . ± . . ± . . ± . AIF . ± . . ± . . ± . . ± . . ± . . ± .

Table . :

Mean ± standard deviation of the parameters estimated with the aAIFd and rAIFd models, using two di erent sets of cut-o s to generate the AIF functions. parameters (rAUC, rV AIF , and rV RT ) have signi cantly lower CV than absolute volume parameters (AUC and V ). No signi cant di erence was found when comparing WIR, F, and rW IR, but the CV of these parameters are signi cantly higher than those of rF AIF and rF RT .

Numerical values of these mean CV are given for each mouse in Appendix, in Table . .

. Discussion

Using a test-retest study with a controlled bolus injection, it was possible to assess the variability of DCE-US perfusion parameters. To reduce this variability, the interest of estimating relative parameters, which necessitates the de nition of a reference tissue region, was practically demonstrated. Our study also shows the importance of choosing an appropriate method for the estimation of the parameters, because estimation methods have an impact on parameter variability. us the reference tissue approach (rRTd) can be recommended, Two groups of parameters were built (horizontal lines below the parameter names) such that there were no signi cant intra-group di erences while there were statistically signi cant inter-group di erences (marked by * ).

to estimate parameters such as AUC and WIR from explicitly de ned models, e.g. using the aLN model. To reduce the variability of the estimated parameters, Dizeux et al. proposed a controlled injection system [ ]. However, the present study shows that the di erences between two consecutive exams are still not negligible for a regional analysis. Since in PET and DCE-MRI, deconvolution approaches and compartmental models have proved their e ciency to make parameters more robust to inter-exam changes, we decided to test some of these approaches. , (aAIF model), we have shown that the quality of t is worse than the one obtained when using the explicit aLN model. When introducing a time delay parameter, an option that is unfortunately generally overlooked [ ], a much more accurate t of regional tumor kinetics was obtained. Indeed, the quality of t using aLN (depending on four parameters) and this aAIFd model was equivalent in terms of N RMSE.

Our study shows the crucial role of the AIF estimation in the variability of the perfusion parameters (see Table . ). When focusing the eld of view in the main plane of the tumor, it Recirculation is a major problem when dealing with modeling techniques adapted to rst-pass studies. However, its quantitative impact is reduced in DCE-US when compared to other modalities since the destruction of microbubbles, in the lungs in particular, makes the number of microbubbles much smaller in the second pass (and following) than it is in the rst pass. We deliberately did not try to model the recirculation, when we chose the aLN model to t tumor kinetics or when we rst t the AIF using the aLN model. In addition, using simulated data, we showed that the practical impact on parameters estimation when . . CONCLUSION neglecting the recirculation was limited, especially for relative parameters, since the coecients of variation between parameters estimated with and without simulating the recirculation e ect were less than the ones estimated through the test-retest studies (this recirculation e ect was about % on CV values with the rRTd approach).

e use of normalized parameters induced a signi cant reduction of coe cients of variation in our test-retest study. Furthermore, estimating relative volume and ow parameters using equation ( . ), which eliminates the need for an AIF, is more robust than using the AIF directly. It should also be noted that the small D displacements occurring between the four test-retest studies can partly explain the CV, due to the imperfect spatial alignment between sub-regions from one exam to the following one. perfusion parameters was established due to the inherent variations of experimental and physiological conditions for the log-normal modeling and to the di culties in estimating a correct AIF in the image eld of view for the compartmental approach. To reduce this variability, the use of relative values of these regional perfusion parameters was proposed, requiring in all cases the delineation of a reference tissue region. To estimate these relative parameters, the reference tissue model proved to be the most reliable computing approach.

us we recommend the use of this model to estimate reliable relative perfusion parameters.

Appendix is appendix gives the numerical results that were obtained for each mouse (m , m , m , and m ), including mean values and standard deviations of regional parameters (Table . )

and coe cients of variation of these parameters (Table . ).

Absolute parameters

Relative parameters . eory is section gives the closed-form expressions of AUC, PE, T TP, MT T, WIR, WOR, and T D parameters for the Log-Normal model, aLN (Table . ), and for the one-compartment model, aAIFd (Table . ). e equations of the models are given in Section . . of Chapter .

AUC (a.u.) V (%) rAUC (%) rV AIF (%) rV RT (%) m . ± . . ± . . ± . . ± . . ± . m . ± . . ± . . ± . . ± . . ± . m . ± . . ± . ± . ± . ± . m . ± . . ± . . ± . . ± . . ± . WIR (a.u. s -) F (s -) rWIR (%) rF AIF (%) rF RT (%) m . ± . . ± . . ± . . ± . . ± . m . ± . . ± . . ± . . ± . . ± . m . ± . . ± . . ± . . ± . . ± . m . ± . . ± . . ± . . ± . . ± . Delay ∆ (s) d (s) D T-R LN (s) D T-R AIF (s) D T-R RT (s) m . ± . . ± . -. ± . . ± . . ± . m . ± . . ± . . ± . . ± . . ± . m . ± . . ± . . ± . . ± . . ± . m . ± . . ± . -. ± . . ± . . ± .
AUC was de ned as the in nite integral of the function, PE as the value taken by the function where its derivative is null, T TP the time where the function derivative is null, MT T as the expected value of the function normalized by its AUC, the WIR and WOR as the derivative of the function where its second order derivative is null, and T D as the time-delay of the function.

AUC A T TP e µ-σ PE A e σ -µ σ √ π MT T e µ+ σ WIR A σ √ π y σ -e y-µ-y σ , where y = σ +σ √ σ + WOR A σ √ π -z σ e z-µ-z σ , where z = σ -σ √ σ + T D ∆ T
Table . : Closed-form expressions of perfusion parameters using the aLN model, WOR being the absolute value of the maximum negative slope.

AIF

Kδ (t) Krect a (t) C A (t) AUC KV T KV T V T ∫ +∞ C A (τ) dτ T TP a {t P C T (t P -d T ) = V T C A (t P )} PE KF T KV T a -e - aF T V T F T e - F T V T t P ∫ t P C A (τ) e F T V T τ dτ MT T V T F T V T F T + a V T F T + MT T C A W IR ∞ KF T a F T C A (t I ) -V T C T (t I -d T ) {t I dC T dt (t I -d T ) = V T dC A dt (t I ) , dC A dt (t I ) > } WOR KF T V T KF T a -e - aF T V T F T C A (t O ) -V T C T (t O -d T ) {t O dC T dt (t O -d T ) = V T dC A dt (t O ) , dC A dt (t O ) < } T D d T d T d T
Table . : Closed-form expressions of perfusion parameters using a one-compartment model (aAIFd) and assuming three di erent shapes of AIF: impulse function (δ), rectangle function of width a and height a, rect a (t), and general case C A (t). In the rst two cases, K stands for the injected concentration. In the general case, MT T C A stands for the mean transit time of C A (t).

.

. DATA ANALYSIS AIF Kδ (t) Krect a (t) C A (t) rAUC V T V R V T V R V T V R rW IR - F T F R F T C A (t I,T )-V T C T (t I,T -d T ) F R C A (t I,R )-V R C R (t I,R -d R ) {t I,T dC T dt (t I,T -d T ) = V T dC A dt (t I,T ) , dC A dt (t I,T ) > } {t I,R dC R dt (t I,R -d R ) = V R dC A dt (t I,R ) , dC A dt (t I,R ) > } rT D d T -d R d T -d R d T -d R
Table . : Closed-form expressions of the relative perfusion parameters using a relative onecompartment model (rAIFd) and assuming three di erent shapes of AIF: impulse function (δ), rectangle function of width a and height a, rect a (t), and general case C A (t). In the rst two cases, K stands for the injected concentration.

. Data Analysis

Coe cients of determination R θ i ,θ j were estimated from the least-squares linear regression between the regional estimates of parameters θ h i and θ h j , one estimate per sub-region s h ).

ese coe cients were computed independently for each of the DCE-US studies ( mice m l × test-retest studies R k ). A linear regression was also computed between sets of parameters (the sub-regions of the studies were polled together) to assess the consistency of the relationships between parameters.

. Results . ) generate high linear links and thus information redundancy. For that reason we further focused on three derived parameters: AUC, which is related to tissue blood volume V , according to the closed-form expressions given in Table . , WIR, which is mainly related to tissue blood ow F), and the delay parameter ∆. In addition, Table . However, when it comes to the comparison of longitudinal exams, it is crucial to have comparable parameter values. us, relative parameters seem to be the most powerful solution, provided that the reference tissue can be de ned in each exam, and that its characteristics are not modi ed between successive exams. Some studies stated that AUC is related to tissue blood ow, this theoretical and experimental study demonstrates its relation to tissue blood volume instead.

e heat maps shown in

. Conclusion

A comprehensive comparison of the parameters estimated by di erent approaches was proposed, showing high correlations between the volume-based and ow-based parameters respectively estimated.

A erword e analytical expression of the area under the curve reveals its straightforward relation with the fractional blood volume. is semi-quantitative parameter is proportional to the physiological parameter, and can be used for relative comparison of tissues observed in a single exam. Furthermore, using the normalized area under the curve successfully corrects for the inter-exam variations, and therefore makes the comparison more robust.

e relation of the wash-in rate with the tissue blood ow is more complex, and while the normalized parameter is analitically and empirically proportional to the normalized blood ow, the slope of the relation varies from one experiment to another. is explains the weaker improvement observed in terms of reproducibility for the normalized wash-in rate compared to the absolute value.

is study reveals the sensitivity of the arterial measurement to segmentation. Indeed, arterial regions exhibiting both small areas and high signal intensities, small changes in the segmentation can result in large changes in the mean arterial curve. In addition, a lognormal model was tted to the arterial curve for noise-ltering prior to quanti cation. Given the noise level in those high intensity regions, the tted curve is likely biased. Absolute perfusion parameters were strongly a ected by varyiation of the segmented artery region, however relative parameters exhibited a better agreement.

is formulation of the reference tissue model introduces an unidenti able parameter, i.e. the rate constant in the reference tissue. In this study, this parameter was given the mean value obtained with the one-compartment model using the arterial measurement.

And while the estimated parameters are most likely biased depending on the discrepency of the xed parameter value with its actual value, direct estimation of relative perfusion parameters was the most robust approach in our study. In the next part of this thesis, a linear formulation of the reference tissue model is presented to address this issue, along with a regularized estimation scheme to improve parameter reproducibility and comparability. Radiology, ( ): -, February .

[ ] R N Gunn, S R Gunn, and V J Cunningham. Positron emission tomography compartmental models. J Cereb Blood Flow Metab, ( ): -, .

[ ] C Hoe el, S Mulé, L Huwart, Frédérique Frouin, J P Jais, O Helenon, and J-M Correas.

Renal blood ow quanti cation in pigs using contrast-enhanced ultrasound: an ex vivo study. Ultraschall Med, ( ): -, April . Chapter formally details the linear and regularized estimation methods to solve the reference tissue model. It provides a full development of the method proposed in [ ]. In addition, this Chapter aims at evaluating the sensitivity of perfusion parameters to varying data characteristics, and analysis method settings. In particular, the impact of acquisition duration, acquisition frequency, noise amplitude, number of regions used for the regulairzation, as well as the characteristics of the reference tissue were investigated. In order to provide a ground truth, synthetic data based on preclinical experiments was generated using a one-compartment model. is Chapter is an extended version of a paper that we planned to submit to Medical Image Analysis.

N.B. All the references cited in this part of the thesis are pooled together and presented at the end of Part III.

Chapter Regularized Linear Resolution of a

One-Compartment Model to Improve the

Reproducibility of Perfusion Parameters in CEUS . Abstract

Contrast-enhanced ultrasound (CEUS) has been proposed to monitor tumor therapy, in complement to size measurements. Estimating reliable perfusion parameters from CEUS studies is essential in order to propose adapted therapy options according to the parameter values. e variability of these parameters was assessed in an ideal case of consecutive test-retest CEUS studies, in a mouse tumor model. e impact of mathematical modeling on parameter variability was investigated on these data. Four models were compared in tumor sub-regions : the log-normal model (LN), the relative LN model (rLN) where parameters of LN are normalized by the parameters estimated inside a reference tissue (RT) region, a linear resolution of a one-compartment model based on the RT (rLin), a modi ed version of rLin implementing regularization (rLinReg) to ensure coherent results between the di erent sub-regions of the tumor. Results show that LN model had highest coe cients of variation. e positive impact of normalization using RT (rLN) was established, showing reduced coe cients of variation. e rLin approach showed large variations especially for ow parameters. Its regularization version, rLinReg, greatly improved parameter reproducibility while providing coherent results between the sub-regions. In conclusion,the rLinReg approach provided the smallest coe cients of variations and should be preferred for estimating perfusion parameters in CEUS.

. Introduction

Reliable quanti cation of tumor perfusion is a challenging, yet necessary, milestone to reach in order to e ciently monitor tumor growth and treatment e ciency. Contrast-enhanced ultrasound (CEUS) is a non-invasive tool allowing real-time quantitative vascular imaging: for every sampling time and every pixel in the image, the linearized signal intensity is proportional to the concentration of contrast agent for low concentrations.

Recommendations for the quanti cation of CEUS studies rely on explicit modeling of time-intensity curves (TICs), e.g. using a log-normal model [ ]. en, semi-quantitative parameters are usually derived directly from the modeled TIC, e.g. area under the curve (AUC) and wash-in rate (WIR). ese parameters are directly a ected by inter-exam changes occurring either in physiology, e.g. heart rate, blood pressure, or in experimental conditions, e.g. injected quantity, or injection speed [ ]. Controlled injections and compartmental modeling have been proposed to reduce this variability [ ]. To overcome the issues related to the estimation of a correct arterial input function, the use of a reference tissue (RT) region (e.g. [ ]) has been successfully tested [ ]. In the present study, a linear formulation of the one-compartment model is presented and evaluated. is formulation allows the evaluation of an otherwise unidenti able parameter, characterizing the RT region, which value had to be set arbitrarily to . . To prove the interest of this new approach, the coe cients of variation of perfusion parameters estimated at a regional scale were compared using four di erent approaches: ) the log-normal model (LN), ) the relative LN model (rLN), where parameters are normalized by the (LN) parameters estimated inside the RT region, ) a linear resolution of the one-compartment model based on the RT region (rLin), ) a modi ed version of rLin implementing regularization (rLinReg) to ensure a coherent estimation of the ratio between blood ow and blood volume in the RT region when taking into account the di erent sub-regions in the tumor.

. Materials . . Animals

All experiments were conducted in accordance with the institutional guidelines and the recommendations for the care and use of laboratory animals.

ey were based on a murine model of Murine Colon Carcinoma (CT ). Tumor fragments ( -mm ) were implanted days prior to the CEUS acquisitions in the right ank of Balb/C mice. Anesthesia was maintained during the whole acquisition through a face mask delivering % iso urane in air delivered at a L/min rate.

. . Image acquisition

Tumors were imaged in their largest cross-section plane, mice motion was limited using surgical tape securing animal position during and between acquisitions. A controlled injection system was used to inject, at a rate of . mL/min, a µL bolus of SonoVue (Bracco Suisse SA, Geneva, Switzerland) diluted to %. Meanwhile, dynamic contrast-enhanced US sequences were acquired using a L W transducer coupled to a Sequoia US system (Acuson, Siemens, Mountain View, CA, USA) in dual-mode, i.e. anatomical B-Mode along with Contrast Pulse Sequencing (CPS) images. Mechanical index was set to . , dynamic range to dB, and time gain compensation was applied. e frame rate was set to Hz during the rst seconds (including the wash-in phase and the beginning of the wash-out phase), and Hz for the remaining time.

For the four mice in the study, four consecutive (test-retest) data-sets were acquired without any modi cation in the setup. Fi een minute breaks were observed between acquisitions to ensure the disruption of previously injected micro-bubbles. . Methods

. . Data pre-processing

Linear echo-power TICs were calibrated from log-compressed video data using a laboratorymade so ware. Both probe and animal motion were assumed negligible for the selected sequences.

Tumors (herea er labeled with subscript T) were segmented on the B-Mode images and the non-perfused areas were removed for data analysis. In order to preserve the signalto-noise-ratio (SNR) of the TICs while revealing the spatial heterogeneity of the tumor, a regional analysis of the tumor area was performed. e perfused tumor region was divided into N T = sub-regions according to radial layers and angular sectors (Figure -). en mean regional TICs C i T (t), for i = , ...N T were computed. As three of the four quanti cation methods require the selection of a RT region, for each mouse, this RT region (herea er labeled with subscript R) was chosen to be easily identi able on the di erent test-retest studies. A muscular region close to the kidney was generally selected, the renal cortex being excluded from the RT region due to the complexity of perfusion patterns observed inside this structure.

Finally for each sub-region, a time delay parameter, D i , representing the time of arrival . . METHODS of the contrast agent in the considered region, was estimated as follows:

D i = max t d dt (C i T * W * W), ( .
)
where W is an average lter with a xed width empirically set to . seconds. Using this speci c time delay, all regional TICs were registered in time for subsequent analysis.

. . De nition of the four models

Log-Normal model (LN)

is method based on the log-normal distribution was recommended by the EFSUMB for quanti cation of tumor perfusion in clinical studies [ ]. e TIC inside the i th region of the tumor, C i T (t), is tted according to equation ( . ):

C i T (t) = A i T √ πσ i T t exp - ln(t)-µ i T σ i T if t ≥ , = otherwise.
( . )

Regional parameters A i T , µ i T , and σ i T were estimated for each sub-region. Semi-quantitative parameters are then derived from the model, including AUC i and WIR i . ese parameters depend, in a non-linear way, on the parameters of the LN model, A i , µ i , and σ i . Relations have been established, both analytically and experimentally, between parameters derived from the LN model and physiological parameters, showing AUC i is related to blood volume, and W IR i to blood ow [ ].

Normalized log-normal model (rLN)

Regional relative parameters: rAUC i and rWIR i were derived from the parameters estimated with the (LN) model in the tumor sub-regions (AUC i T , WIR i T ) and those estimated in the RT region (AUC R , WIR R ). ey were de ned as follows:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ rAUC i = AUC i T AUC R , rWIR i = WIR i T WIR R .
( . )

Simple one-compartment model (rLin)

e ese approaches have also been adapted to relax the need for blood sampling or AIF measurement and the kinetics inside a RT region was then used (see for instance [ ]). is resolution was adapted to CEUS data, considering the ultrasound contrast agent is strictly intra-vascular. Considering a onecompartment model to describe ow exchanges inside the tissue, the following equations can be written for each sub-region, i = , . . . , N T :

dC i T (t) dt = F i T .C A (t) - F i T V i T C i T (t) . ( . )
In ( . ) C A (t) represents the arterial input function feeding the tissue, while V i T stands for the fractional blood volume, and F i T for the blood ow in the sub-region i. Considering jointly the TIC inside the RT region and the TICs in the tumor, and assuming a common feeding input for the RT region and the tumor, we have the following set of equations:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ dC R (t) dt = F R .C A (t) -F R V R C R (t) , dC i T (t) dt = F i T .C A (t) - F i T V i T C i T (t) , ∀i. ( . )
Rearranging the rst equation, C A (t) can be isolated and expressed as a function of C R (t), and then replaced by its new expression in the N T following equations, yielding the next system:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ C A (t) = F R dC R (t) dt + V R C R (t) , dC i T (t) dt = F i T F R dC R (t) dt + F i T V R C R (t) - F i T V i T C i T (t) .
( . )

A er integration over time (from to t), and de nition of the parameters

rF i = F i T F R , rV i = V i T V R , and k i T = F i T V i
T , the last equations of the system become: ) . . METHODS For each sub-region i, a sub-system of N linear equations ( . ) is computed, obtained for N successive values of t. N is the total number of dynamic frames. Solving this sub-system of N linear equations, the parameters rF i , rV i , and k i T can thus be estimated in the leastsquares sense. Using this approach, the N T linear equations corresponding to the di erent sub-regions are thus solved independently.

C i T (t) = rF i C R (t) + rV i k i T ∫ t C R (τ) dτ -k i T ∫ t C i T (τ) dτ. ( .

Regularization of the one-compartment model (rLinReg)

Using the previously described rLin model, N T di erent values of k R = F R V R can be derived, using the estimation of rF i , rV i , and k i T and the relation between the four parameters:

k R = F R F i T F i T V i T V i T V R = rV i .k i T rF i for i = , . . . , N T . ( . )
As the k R values do not depend on sub-region i, the simple estimation proposed by rLin can introduce some inconsistencies k R and possible biases in some rF i , rV i , and k i T values. To solve this issue, and consider one single value for k R (whatever the number of sub-regions in the tumor), the regularized approach solves the system of equations (Eq. . ), under the following constraints:

rV i .k i T rF i = K, ∀i = , . . . , N T , ( .
)
where K is a constant (equal to k R ). e system is solved globally for the N T sub-regions.

Brie y, the value of k R is successively modi ed, thus providing a unique least-squares solution for the N T parameters (rF i , rV i , and k i T ), until the optimization of the t for the whole set of N T sub-regions.

. . Data analysis

For each model, a vector (Θ) of M perfusion parameters (θ m ) was estimated in each tumor sub-region (i = , . . . , ) of each mouse ( j = , . . . , ) for each repeated acquisition (k = , . . . , ), providing results of curve tting, Θ each mouse by computing coe cients of variation CV (θ m ) i j de ned for the four repeated studies, as the ratio between the standard deviation and the mean value µ i j m of the parameter

(θ m ) i j k : CV (θ m ) i j = ∑ k= ((θ m ) i j k -µ i j m ) µ i j m .
( . )

Parameters corresponding to poor quality ts (FMI > %) were replaced using multivariate imputation by chained equations with the R module {mice} [ ]. is strategy was de ned to compute the CV using four values systematically.

Statistical tests were nally applied to compare the CV of the parameters estimated using the four models. Signi cant di erences in the CV distributions were assessed using the Friedman test and the associated post-hoc analysis for multiple comparisons. Distribution means were considered as signi cantly di erent when p-values were less than . .

. Results

Model

LN rLN rLin rLinReg

FMI

. % . % . % . % N rem Table . : Median values of FMI obtained for the four models and number of sub-regions N rem , out of , for which FMI < %.

Table . shows the median values of the FMI obtained for the four models and the number of regions excluded from further statistical analysis because of bad t quality. In terms of blood volume parameters, the LN model is the most variable with a median value of the coe cient of variation (CV) equal to . %. Using the rLin model, the median CV tends to be lower ( . %), however the di erence is not statistically signi cant. Models with median CV values of . % and . %, respectively. For the blood ow parameters, models rLin and LN appear to the most variable parameters with medians of CV equal to . % and . %, respectively. e rLN model tends to yield lower CV, with a median value of . %. Finally the mean CV of blood ow using the rLinReg model is equal to . %. It is signi cantly lower than the CV of blood ow obtained with the three other models.

. Discussion e number of sub-regions was chosen to reveal some spatial heterogeneity in the vascular network of the tumor, while ensuring regions were large enough to guarantee reasonable signal to noise ratios in regional TICs. Increasing the number of regions would reveal spatial heterogeneity more nely, at the expense of the accuracy of the estimates.

Both physiological and experimental variations get in the way of accurate quanti cation and exam comparison, a ecting blood circulation, as well as measurements accuracy [ ].

e semi-quantitative parameters of the LN model, recommended for tumor quanti cation, were found highly sensitive to inter-exam changes in our study and resulted in the least reproducible parameters.

When compared to the LN model, the normalized version, the rLN model reduces the variability of parameters. If the reduction of variability for blood ow parameters was not statistically signi cant, it was signi cant for blood volume parameters. us normalization using a RT region has a real potential to improve exam comparison.

Similarly to the rLN approach, the rLin model uses the RT region, but in addition, it assumes a one-compartment model to describe contrast exchanges between large vessels and micro-vascular areas in tissue. e rst resolution method, which was tested in the present study and proposes to estimate three unknown parameters per sub-region, yields highly variable parameters, especially in terms of blood ow. However, the median CV of blood volume was reduced when compared to AUC CV, estimated with the LN model. is method was implemented in a naive way, resulting in inconsistent values of parameter k R in the di erent tumor sub-regions.

e rLinReg model was built to overcome these inconsistencies, ensuring a single value of k R . Enforcing a common value of k R in sub-regions comes down to impose a xed ratio between the rst and second terms of Eq. . . e number of degrees of freedom was thus reduced. e combined use of normalization through a RT region and regularization respects the compartmental modeling paradigm while yielding the most reproducible parameters in our study.

. Conclusion

Using the LN model, derived parameters have high coe cients of variation. e positive impact of normalization using a reference tissue region on parameter reproducibility was . . ONE VASCULAR COMPARTMENT MODEL (OVC)

established. e rLinReg approach takes into account the di erent sub-regions involved in the quanti cation, yielding a single value of parameter k R common to all tumor sub-regions.

In addition, this spatial regularization signi cantly reduces coe cients of variations of the blood ow parameter and should therefore be preferred to estimate spatially-distributed perfusion parameters.

Appendix

.

One vascular compartment model (OVC)

A vascularized tissue is considered an homogeneous compartment fed by an artery. e vascular compartment is parameterized by tissue blood volume V , and tissue blood ow F, since the distribution of microbubbles is restricted to the vascular space [ , ].

Ċ (t -D) = F ⋅ C A (t) -F V ⋅ C (t -D) , ∀t ≥ D, = otherwise. ( . )
where C A is the arterial input function, C is the modeled TIC inside the tissue of interest, and Ċ is the time derivative of C. is equation integrates as

C (t) = F ∫ t C A (τ) e -F V (t-D-τ) dτ, ∀t ≥ D, else. ( . )
Given an AIF C A (t), and the set of three perfusion parameters V , F, and D, the associated TIC C(t) can be simulated.

Relative OVC model (rOVC)

A relative OVC model can be derived from the previously presented OVC model, considering conjointly one tissue of interest with TIC C i T (t), and one reference tissue with TIC C R (t):

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ĊR (t -D R ) = F R ⋅ C A (t) -F R V R ⋅ C R (t -D R ) , ∀t ≥ D R , = otherwise ; Ċi T t -D i T = F i T ⋅ C A (t) - F i T V i T ⋅ C i T t -D i T , ∀t ≥ D i T , = otherwise. ( . )
e rst equation of the system of equations ( . ) can be rearranged as Replacing C A (t) in the second equation of system ( . ) by its expression in equation ( . ), Ċi T (t) can be expressed as

C A (t) = F R ⋅ ĊR (t -D R ) + V R ⋅ C R (t -D R ) ∀t ≥ D R , = otherwise. ( . )
Ċi T t -D i T = F i T F R ⋅ ĊR (t -D R ) + F i T V R ⋅ C R (t -D R ) - F i T V i T ⋅ C T i (t -D T i ) , ∀t ≥ D i T , = otherwise.
( . )

De ning the relative ow as rF i = V i T V R , the relative volume as rV i = V i T V R , and the rate

C A (t) C T (t) F T k T = F T / V T Figure -a: One-compartment model. . . ONE VASCULAR COMPARTMENT MODEL (OVC)
constant in the i th tissue of interest as

k i T = F i T V i T , the previous equation rewrites Ċi T t -D i T = rF i ⋅ ĊR (t -D R ) + rV i ⋅ k i T ⋅ C R (t -D R ) -k i T ⋅ C i T t -D i T , ∀t ≥ D i T , = otherwise.
( . )

Assuming initial concentrations are equal to zero in both tissues, Ċi T in Eq. . integrates in exponential form [ ], yielding

C i T t -D i T = rF i ⋅ k R -k i T ⋅ ∫ t C R (τ -D R ) ⋅ e -k i T ⋅(t-D R -τ) dτ +rF i ⋅ C R (t -D R ) ∀t ≥ D i T , = otherwise, ( . )
where k R = F R V R is the rate constant in the reference tissue.

Using such a formulation, Equation ( . ) is not linearly solvable. A non-linear resolution method must therefore be used in order to estimate vascular parameters rF i , rV i , k i T , and k R , in each of the N T tissues of interest, as well as the time-delay parameters D i T , and D R . is approach has been investigated in Chapters and , and was therefore not included in the study. Instead we used the linear formulation presented in the following Section.

C A (t)

F T k T = F T / V T Δ T C T (t)
F T F T / V T F R F R / V R C T (t) C R (t) C A (t) Δ T Δ R Figure -c: Block diagram of the relative one-compartment model in case of a single region of interest. C A (t) C T,1 (t) F T,1 F T,1 / V T,1 C R (t) F R F R / V R C T,2 (t) F T,2 / V T,2 F T,2 ... C T,N T (t) F T,N T / V T,N T F T,N T Δ T,1 Δ T,2 Δ T,N T Δ R Figure -d: Block diagram
of the relative one-compartment model for regional quanti cation.

Linear resolution of the rOVC model (rLin)

Alternatively, under similar assumptions, Ċi T in Eq. . can be integrated over time, yielding the following expression of

C i T [ ] C i T t -D i T = rF i ⋅ C R (t -D R ) + rV i ⋅ k i T ⋅ ∫ t C R (τ -D R ) dτ -k i T ⋅ ∫ t C i T τ -D i T dτ, ∀t ≥ D i T , = otherwise.
( . )

Assuming time delay parameters D i T and D R are estimated beforehand using the method presented in Section . , TICs can be time-shi ed, mimicking an ideal case with no delay in bolus arrival. Variables x i (t), y i (t) are time-shi ed versions of the TIC in the i th tissue of interest and its integral. ey are de ned ∀t ∈ , L -D i T , as

x i (t) = C i T t + D i T , y i (t) = -∫ t C i T τ + D i T dτ. ( . )
Similarly, variables u (t), v (t) are time-shi ed versions of the reference TIC and its integral.

ey are de ned ∀t ∈ [ , L -D R ], as

u (t) = C R (t + D R ) , v (t) = ∫ t C R (τ + D R ) dτ. ( . )
Eq. . can be interpreted as an overdetermined system of N S linear equations [ ],

i.e. one equation for each time sample t. It can therefore be written

x i (t) = a i ⋅ u (t) + b i ⋅ v (t) + c i ⋅ y i (t) ∀t ≥ D i T , ( . )
where coe cients a i , b i , and c i are de ned as

a i = rF i , b i = rV i ⋅ k i T , c i = k i T .
( . )

e system can be solved using a linear least-squares resolution method, yielding esti- mates of parameters a i , b i , and c i by minimization of the squared t error ε i arg min

{a i ,b i ,c i } ε i , where ε i = t x i (t) -a i ⋅ u (t) -b i ⋅ v (t) -c i ⋅ y i (t) . ( . )
Vascular parameters of the rLin model can then be derived easily using

rF i = a i , rV i = b i c i , and k i T = c i . ( . )
e linear resolution of the rOVC model will be referred to as rLin in the following. For the case of a single region, N T = , the rLin model is equivalent to the method proposed by

Cárdenas-Rodríguez et al. [ ].

Regularized linear resolution of the rOVC model (rReg)

Estimating rF i , rV i , and k i T in N T tissues using the rLin model, N T values of parameter k R can be derived as a linear combination of the rLin model parameters:

k R = F R F i T F i T V i T V i T V R = rV i ⋅ k i T rF i = b i a i ( . )
When there are more than one tissue of interest (N T > ), N T values of parameter k R can be derived from the parameters of the rLin model. However, these N T values of k R characterize the same reference tissue, associated to a single reference TIC C R (t). A unique value of k R should therefore be estimated per exams in order to avoid discrepancies between the N T tissues of interest.

e linear relation between parameters of the rLin model provided by Eq. . can be used as a constraint to ensure the N T derived values of k R are consistent across tissues. Substituting in Eq. . yields

x i (t) = a i ⋅ (u (t) + k R ⋅ v (t)) + c i ⋅ y i (t) , ( . )
which rewrites

x i (t) = a i ⋅ w (t) + c i ⋅ y i (t) ( . )
where w (t) is de ned by

w (t) = u (t) + k R ⋅ v (t) , = C R (t -D R ) + k R ⋅ ∫ t C R (τ -D R ) dτ. ( . )
When a value of parameter k R is provided, the N T linear system equations de ned in Eq. . are independently solvable using a linear least-squares resolution method, minimizing the squared error, e i : arg min

{a i ,c i } e i
, where

e i = t x i (t) -a i ⋅ w (t) + c i ⋅ y i (t) . ( . )
Since the value of k R is necessary to de ne w (t), its value must be determined. We proposed a non-linear iterative optimization scheme that estimates the value of k R by iteratively minimizing the normalized mean squared error, E:

arg min k R E, where E = i e i N i S x i (t) ∞ , ( . )
N i S being the number of samples in x i (t), i.e. the number of time samples verifying t ∈ , L -D i T , and x i (t) ∞ is the uniform norm of x i (t) de ned as the maximum absolute value in the regional time-shi ed curve. Vascular parameters were then derived from the model estimates as

rF i = a i , rV i = k R ⋅ a i c i , and k i T = c i . ( . )

Estimation of time-delay parameters

As stated in the presentation of the rLin and rReg models, time-delay parameters are known beforehand, and TICs shi ed in time in order to correct for time-delays prior to solving the linear system of equations. e determination method of the time-delay parameter, noted D, of a generic TIC, noted C (t), was de ned in order to determine the time of arrival of the rst microbubbles in the tissue of interest. is is a di cult task because of the noise present in data. We propose here an empirical estimation method adapted to contrast-enhanced ultrasound data.

D ≤ t % ∧ Ċ f (D) = . × Ċ f (t % ) . ( . )
Figure -illustrates the time-delay estimation process on an example contrast-enhanced ultrasound time-intensity curve.

Chapter

Impact of Recirculation in Dynamic

Contrast-Enhanced Ultrasound: a

Simulation Study . Abstract

Objectives e impact of recirculation on the quanti cation of perfusion is o en neglected.

It can however introduce a bias or some variability in the estimation of perfusion parameters and thus hamper comparison between exams. Methods Time-intensity curves (TICs)

were simulated using a one-compartment model fed by an arterial input function (AIF). A simple model was developed to simulate recirculation in the AIF. Using AIF with and without recirculation, and sets of regional perfusion parameters, TICs corresponding to di erent tissue regions were simulated by convolution of the AIFs with the transfer function associated to each region. simulations for each of the noise levels were then computed. For each simulated study, six quanti cation methods based on either Log-Normal modeling or relative compartmental modeling were tested. Variations of the conventional Log-Normal model were also investigated, including using parameters estimated in a reference tissue for normalization purposes, and tting only the rst phase of the TIC to avoid recirculation. Results e impact of recirculation varies according to the quanti cation method. Restricting parameter estimation to the rst samples of the TICs, before recirculation occurs, appears to be the worst strategy. Errors are largely minimized when using a reference tissue to establish relative parameters. e most robust approach is the compartmental modeling based on a reference tissue and applied to multiple regions with a regularization constraint. Conclusion is simulation study demonstrates the in uence of recirculation on the estimation of perfusion parameters. To reduce the impact of this unavoidable e ect, the quanti cation method based on compartmental modeling and using a reference tissue appear to be the most reliable strategy.

. Introduction

With the advent of contrast agents, perfusion imaging has been developed for di erent medical imaging modalities, including PET, CT, MRI, and more recently ultrasound. Perfusion parameters including regional tissue blood volume and tissue blood ow are functional indices which can help in the diagnosis of some vascular abnormalities, such as ischemia.

Vascular modi cation in tumors is also a key application of perfusion imaging and can be used in order to assess tumor diagnosis or tumor monitoring [ ].

A widely used approach to estimate perfusion parameters relies on bolus injections of contrast agent and dynamic recording of frames. However the quanti cation of signal and the estimation of perfusion parameters through mathematical modeling remains a hard task and has generated a lot of research work [ ]. An accurate and robust estimation of perfusion parameters is of course crucial to compare perfusion imaging exams meaningfully. is is primordial in order to allow inter-subject exams or to perform monitoring. Among the di erent mathematical models that have been proposed in contrast-enhanced ultrasound (CEUS) studies, little attention has been devoted to compartmental modeling, despite its wide use in PET or MRI studies. Indeed, explicit modeling using for instance a Log-Normal function is o en recommended to analyze dynamic data [ , ]. Of course di erent reasons can explain this restricted use of the compartmental approach; among them the di culty in estimating a correct arterial input function in dynamic ultrasound images can be cited.

To get rid of this di culty which occurs also while using other imaging techniques, some authors in PET imaging and more recently in MRI have proposed to use a reference tissue in order to de ne relative perfusion parameters [ , ], de ned as the ratio between the perfusion parameters in the tissue of interest and the perfusion parameters de ned in the reference tissue. Our group has recently shown the practical interest of this approach in a test-retest protocol applied to a murine tumor model [ , ].

As no absolute gold-standard exists for preclinical or clinical studies, simulations can be used to assess the performance of di erent models and compare them. Of course, as it is quite complex to reproduce in silico the complexity of in vivo, the extrapolation of simulations to real cases should be done very carefully. However they can be used to focus on one speci c trait and to quantify its impact. In the present study, the studied trait was recirculation, since this process is o en overlooked when quantifying CEUS exams. is is especially true in small animals, where recirculation occurs quickly and can overlap with the rst pass of the bolus of micro-bubbles in tissues, a ecting the time-intensity curves (TICs) used for quanti cation.

For the present study, a one-compartmental model was assumed to be representative of the underlying physiology that is observable at a regional scale. Di erent values of perfusion parameters (tissue blood ow, tissue blood volume and time-delays) were simulated in order to better apprehend the spatial heterogeneity that can be observed inside a tumor.

e values of these parameters were derived from results obtained in a preclinical study in order to be coherent with practical observations. In addition to recirculation, the impact of signal to noise ratio was studied. For the modelling approach, two versions of the Log-Normal model (absolute and relative), and two versions of the relative one-compartment model (one based on a single region, one taking advantage from the existence of multiple regions) were considered. In addition, in order to limit the impact of recirculation while estimating perfusion parameters with the Log-Normal model, a simple and popular strategy was tested which consists in using the rst samples of TICs, i.e. samples acquired before re- ecting the transit time of the contrast agent from the feeding artery to the tissue was also considered. e mathematical relationship between the tissue TIC and the TIC in its feeding artery is given by equation . :

C T i = C A * h F i ,V i ,D i ( . )
where

h F i ,V i ,D i (t) = F i ⋅ e - F i V i
(t-D i ) ∀t ≥ D i , else, represents the transfer function of the i th tissue region.

. . Simpli ed recirculation model

A er injection in a vein, the bolus of microbubbles travels through the lungs and heart chambers before being distributed in the whole body through the arterial system. A er this rst pass in the tissues, microbubbles return to the venous system for another circulation loop.

During each loop, the bolus is attenuated by the natural disruption of microbubbles, and their ltration through the lungs and the liver. Additionally, the bolus length spreads in time because of the inhomogeneous path length of the individual microbubbles [ ].

An AIF with recirculation, C Aw , can therefore be approximated by a sum of consecutive passes of the bolus in the region of interest (equation . ):

C Aw (t) = C A (t) + N R r= R r (t) , ( .
)
where C A (t) is the TIC of the rst pass of the bolus, R r (t) is the TIC of the r th recirculation,

. . Noise model

A multiplicative noise model following a gamma distribution [ ] while constraining the mean intensity to be (unit mean). Indeed, a unit mean distribution for a multiplicative noise is the equivalent of a zero-centered distribution for additive noise. A gamma distribution is de ned by two parameters: its shape parameter κ, and its scale parameter θ. Enforcing a unit mean is equivalent to set θ = κ, the noise distribution p (v) is then de ned by equation . :

p (v) = Γ(κ) κ κ v κ-e -vκ , ∀ v ≥ . ( . )
e parameter κ controls the sharpness of the noise distribution, and is related to the standard deviation of the noise distribution by σ = √ κ.

. . Perfusion quanti cation methods

Six perfusion quanti cation methods (M -M ) were tested and compared. Among them, four relative approaches (M -M ) making use of an in-plane reference tissue (R) were proposed to make parameters more robust to inter-exam changes (due to unavoidable experimental or physiological varying conditions). Furthermore, the last method (M ) takes advantage of the multiple regions that can be de ned inside an image.

Methods

rAUC i = AUC i AUC R , rWIR i = WIR i WIR R , ∆ = τ i -τ R . ( . )
For the method M , all the time samples are analyzed while for the method M , the analysis is restricted to the rst pass of the bolus.

Methods

rF i = F i F R , rV i = V i V R , k i = F i V i , δ i = D i -D R . ( . )
When the time delay δ i is estimated (de ned as the in ection point a er temporal ltering), the convolution equation can be written as follows:

W i (t) = rF i ⋅ X(t) + rV i ⋅ k i ⋅ Y(t) -k i ⋅ Z i (t), ∀t ≥ δ i . ( . ) with W i (t) = C T i (t-δ i ), X(t) = C R (t-δ R ), Y(t) = ∫ t C R (τ -δ R ) dτ, and Z i (t) = ∫ t C T i (τ -δ i ) dτ.
e three parameters rF i , rV i , and k i can thus be estimated using a linear regression which minimizes the least-squares error. For that reason the method M is noted rLin. It was is approach was proposed in [ ] to overcome the limitations of the rLin model when it is applied to N (N being more than one) tissue regions. Indeed, the estimation of N values of rF i , rV i , and

k i provides N potentially di erent values of k R = F R V R , since k R = F R F i T F i T V i T V i T V R = rV i .k i rF i
. e discrepancy of the values of k R can be overcome by forcing this parameter to have the same value across the di erent regions, i.e. forcing a common ratio between rV i .k i T and rF i across all tissue regions. In summary, an iterative estimation method was proposed, each iteration being conducted in two steps : rst a value for k R = rV i .k i rF i is chosen, then the N values rF i , rV i , and k i are estimated by applying N linear optimization processes under constraints, this two-step procedure being repeated in order to minimize a global error term de ned as the sum of the N errors of the N ttings. As compartmental approaches take into account recirculation inherently, the truncation approach de ned for Log-Normal based models was not tested for models M and M . . were de ned and in each case, realizations were considered. A Log-Normal model was tted to each simulated noise-free C T i TIC (generated using C A (t)), yielding reference values for AUC i , WIR i and τ i .

. . Data analysis

For each simulated TIC C n j hi (t), associated with con guration h (for h = , the AIF is C A (t), for h = the AIF is C Aw (t)), region i (i = , ..., ), noise level n (n = , ..., ) and realization j ( j = , ..., ), the di erent perfusion parameters Θ n j hi (M m ) were estimated using the six methods (M m , m = , ..., ) presented in Section . . . As the methods M and M were de ned to be less sensitive to recirculation, the LN model was tted to the rst seconds following the time-delay estimated for each TIC, since γ = seconds was the recirculation period used for simulation.

For parameters related to the tissue blood ow or to the tissue blood volume, the relative estimation error, expressed in %, was de ned as follows:

E n j hi (M m ) = Θ n j hi (M m ) -Λ i (M m ) Λ i (M m ) ( . )
where Λ i (M m ) is the reference value of the perfusion parameter estimated in the i th tissue region using method M m . For time-delay parameters the absolute estimation error was de ned in seconds as:

E n j hi (M m ) = Θ n j hi (M m ) -Λ i (M m ), ( . )
.

Results

Fig

. . Model M

When focusing on data without recirculation, the LN model (model M ) is robust, it estimates accurate values of AUC i and WIR i , whatever the level of noise. In particular, the intermediate noise level yields relevant estimates in all tissue regions. e time delay seems to be the less robust parameter but does not impact the reliability of the other parameters.

When introducing the recirculation model, the estimation of AUC i (median error of about %), and in a less extent the estimation of WIR i (median error of about -%) are biased, but the bias does not vary with noise. For the estimation of time-delays, behaviors similar to the LN model without recirculation can be observed.

. . Model M

When using the LN model restricted to the earliest phase (model M ), both AUC i and WIR i parameters are respectively largely under and over estimated, whatever the con guration, i.e. without and with recirculation. Some disparities exist between regional parameters: the smallest values of tissue blood ow or tissue blood volume tend to provide larger relative estimation errors. First column: tissue blood volume related parameters, second column: tissue blood ow related parameters, third column: time-delay related parameters, fourth row: rate constants in the tissue region and reference tissue. Constant lines in black represent simulated values, blue lines the estimation corresponding to the LN model, red lines the estimation corresponding to the LN model restricted to earliest phase. Yellow color stands for rLin model, while purple color stands for rReg model. For all of the cases, lled symbols correspond to the con guration without recirculation, while empty symbols correspond to the con guration with recirculation. . . RESULTS

. . Model M

As expected, the rLN model (model M ) for data without recirculation is robust, and provides accurate values of rAUC i and rWIR i , whatever the level of noise and for the intermediate noise level, the estimation is relevant in all regions. As anticipated, when introducing recirculation, the estimation of rAUC i was rather accurate but the estimation of rWIR i was biased (bias invariant with noise), especially for smaller values of simulated tissue blood ow.

. . Model M

When using the rLN model restricted to the earliest phase (model M ), both rAUC i and rW IR i parameters were respectively largely under and over estimated, whatever the conguration: without and with recirculation. Results show similar trends to those observed when using the LN model restricted to the earliest phase (model M ).

. .

Discussion

is study aimed at comparing the behavior of di erent models suitable for quanti cation of perfusion in contrast-enhanced ultrasound studies. Our whole analysis was based on simulated studies in order to have an irrefutable gold standard and to compare the di erent methods in terms of precision and accuracy. e model describing contrast displacement relies on a one-compartment model, that has proved to be valid to describe contrast enhancement in a murine tumor model [ ]. In order to introduce some variations in the tissue blood ows, tissue blood volumes and time-delays, re ecting the regional heterogeneity among tissue regions, as observable with ultrasound. N regions inside the simulated tissue of interest were introduced, each one being characterized by its set of perfusion parameters. In addition an arterial input function was rst simulated using a single Log-Normal function, then approximated by a sum of modi ed Log-Normal functions to mimic recirculation. Although the deformation of the rst-pass is quite simplistic, this approach enables us to consider two con gurations: one idealized con guration without recirculation and one con guration closer to physiological conditions introducing recirculation. Finally di erent signal to noise conditions were simulated using a multiplicative noise model, as opposed to an additive Gaussian model, re ecting conditions encountered in contrast-enhanced ultrasound. Using noisy simulations for each condition guarantees a representative set of possibilities, allowing generalization of the results, as well as enabling the study of the precision of the estimations.

e use of median and quartile operators to assess the estimation errors was necessary to reduce the impact of outliers, which were mostly found in the estimates of the rLin model.

. . DISCUSSION

Using relative estimation errors (as opposed to absolute estimation errors) for tissue blood ow and tissue blood volume parameters allows easy comparison of the errors obtained for parameters with di erent simulated values.

As expected, results of this study exhibit lower precision of the estimated parameters with decreasing signal to noise ratio. Interestingly, most of the methods are not strongly a ected by the signal to noise ratio in terms of accuracy (re ected by the median error on parameters). In addition, perfusion parameters were found less accurate when estimated with recirculation, compared to the estimates without recirculation.

e Log-Normal model, when applied to the whole duration of the study is robust to noise. However this model is subject to recirculation error. To get rid of this dependency, a naive approach consists in limiting the estimation of Log-Normal model to the rst samples of the TIC (before recirculation occurs). However this solution appears to be unstable, providing estimates with huge discrepancies when compared to simulated parameters and also showing a large dependency to noise. is naive approach should therefore be absolutely avoided when dealing with dynamic perfusion data. Indeed results are not accurate for both con gurations, i. e present simulation study con rms the interest of normalization when looking at the accuracy and precision of the estimated parameters. Despite the use of the division operator that could be impacted by noise issues, normalized parameters are more accurate than "absolute" parameters when introducing recirculation in simulated TICs. As recirculation is physiological and cannot be suppressed experimentally, our results emphasis the need to address this question when dealing with real data. erefore, when quantifying CEUS data, we recommend using relative parameters, i.e. parameters normalized according to a reference tissue. e chosen reference tissue should be accessible and rather homogeneous. However the robustness of the rLin and rReg models to the choice of the reference tissue remains to be fully investigated.

.

Conclusion

is study was designed to investigate the impact of recirculation on the quanti cation of contrast-enhanced ultrasound exams, by means of simulations based on experimental data.

Fitting a Log-Normal model on the rst pass of the bolus implies a reduction of the number of points used to t the model and yields unstable estimates, especially on noisy data. is solution is thus inappropriate. Modeling methods such as compartmental modeling, that account for recirculation intrinsically, are indeed the most robust to recirculation. Making use of a reference tissue, the estimation of relative parameters appears to be robust. Taking advantage of the multiple regions, and enforcing estimation of a single rate constant characterizing the reference tissue, provides stable estimates, especially when comparing parameter estimates across regions. is approach is therefore recommended because of its reduced sensitivity to recirculation, and better homogeneity of the estimates inside the considered eld of view.

and considering the formal de nitions of the area under the curve and of the wash-in rate, it can be deduced:

AUC = A WIR = A σ √ π y σ -e y-µ-y σ with y = σ + σ √ σ +
ese equations were directly used to compute the AUC and WIR parameters from a Log-Normal approximation.

AUC and WIR can also be derived when considering a one-compartment vascular model, their expression is computed for two cases in the following table: one simpli ed case assuming that C A (t) follows a gate function and the general case. In addition relative parameters rAUC and rWIR are formally computed when considering a reference tissue. Using the gate function for C A (t) shows a strong equivalence between AUC and tissue blood volume

V , but also between WIR and tissue blood ow F. us, for that case, rAUC = rV and rW IR = rF. Using the general shape for C A (t) shows that AUC is strictly proportional to V , and that rAUC = rV . Furthermore, WIR is related to F, but rWIR is not strictly identical to rF, since a corrective factor ρ is introduced. is factor depends of the time of in ection (denoted t I ) of each TIC and explains why the rWIR is generally not strictly equivalent to rF. A gamma distribution is parameterized by two parameters: the shape parameter k, and the scale parameter θ. Enforcing a unit mean is equivalent to set θ = k, the noise distribution p (v) is therefore parameterized by a single shape parameter, k, as . Materials and Methods

AIF Krect a (t) C A (t) AUC KV T V T ∫ +∞ C A (τ) dτ WIR KF T a F T C A (t I ) -V T C T (t I -D T ) {t I dC T dt (t I -D T ) = V T dC A dt (t I ) , dC A dt (t I ) > } rAUC rV rV rWIR rF ρ ⋅ rF
p (v) = Γ(k) k k v k-e -vk , ∀ v ≥ . ( .

. . Quanti cation models

. . Simulations of CEUS data Simulation process

Regional perfusion parameters were derived from experimental data tted with the OVC model presented in Section . . e arterial input function was derived from a preclinical study using the segmentation method presented in Chapter . A log-normal model was tted to the resulting arterial curve for noise-ltering purposes. Regional enhancement curves corresponding to di erent tissues of interest were simulated using the OVC model, along with the model parameters estimated in di erent regions of the tumor, and the arterial input

V R 3V R /2 2V R 2V R /3 V R /2 0 0.1 0.2 F R 3F R /2 2F R 2F R /3 F R /2 0 0.005 0.01 k R 3k R /2 2k R 2k R /3 k R /

Varying factors

Noise level e in uence of noise was investigated by varying parameter σ, i.e. the standard deviation of the multiplicative noise model. σ was varied linearly with increments of . from , corresponding to a noiseless conditions, to . , corresponding to high noise conditions. For each noise level, random noise sets following the multiplicative gamma noise model were generated.

Exam duration Various exam durations were investigated by varying the number of samples in the simulated data. e exam duration was varied from seconds to seconds with seconds increments.

Sampling period Simulated noiseless enhancement curves were resampled using varying sampling periods to study the impact of this parameter on the accuracy and precision of the estimation. e sampling period was varied from . to . second with . increments, this range being representative of the acquisition settings that can be found in contrast-enhanced ultrasound studies.

Reference tissue e reference tissue is characterized by the parameters of the OVC model, i.e. V R , F R (and thus k R = F R V R

). e impact of these parameters on the accuracy and the precision of the quanti cation process using the rLin and the rReg models was investigated. In ters of the OVC model were used to derive the parameters of the rOVC model using the de nitions of Eq. . . e derived parameters were then used as ground truth to evaluate the accuracy of the perfusion parameters estimated using the rLin and the rReg models.

. . Data analysis

e accuracy and the precision of the parameters estimated using the rLin and the rReg models were respectively investigated through the median value, and either the standard deviation or the interquartile range, of the relative estimation error over random noise

samples. e relative estimation error of parameter θ, noted rE θ , is expressed in percent and de ned as . ) i.e. the di erence between the estimated parameter θ est and the simulated parameter θ sim , normalized by the simulated value. e estimation error E D = D est -D sim is used to assess

rE θ = × θ est -θ sim θ sim (

. Results

In this section we present the results of our experiments. Figure -provides the correspondance between regions, depending on the type of data to be presented. Indeed, color-coded bullseyes were used when a single value had to be represented per region, but grids of plots were used to show the regional results as a function of a simulation parameter, i.e. noise level, exam duration, and sampling period.

In Figures -a to -e, each plot represents a tumor region, i.e. the top row corresponds to the outer rim and the bottom row to the inner rim, the columns correspond to the clockwise ordering of the regions on a rim starting with the le region above the horizontal line. Regarding relative blood volume (see Figure -a), the rReg model was able to estimate the parameter accurately, and to reach a steady state with exams as short as seconds for the simulated noise level, i.e. σ = . . Oppositely, the rLin model only yielded accurate estimates using the whole exam duration, i.e. seconds, and at the exception of one region it overall underestimated rV . Comparably, the relative blood ow (see Figure -b) was steadily estimated using the rReg model with exams as short as seconds, despite an average overestimation of less than %. e estimates of rF given by the rLin were found more sensitive to the duration of the exam, and generally overestimated rF for very short exams, the estimates rst decreasing and then increasing linearly with the exam duration.

. . Noise level

Regarding the rate constant in the tumor (see Figure -c), the rLin model was unable to e regularization greatly improved the homogeneity of the estimation across regions.

In some cases the regularization actually induced stronger bias in the estimation, however the biases are more homogeneous across regions, allowing meaningful regional comparison of the perfusion parameters. Moreover, the rLin approach was largely outperformed by the rReg, especially regarding the accuracy of the rate constant estimates, k T and k R , but also regarding the relative tissue blood volume and ow parameters, rV and rF. e regularization also made the estimation more robust to the acquisition settings over the investigated range, studied via the exam duration and the sampling period.

eoretically, the number of regions included in the analysis should not have any impact on the parameters of the rLin model as every regional enhancement curve is modeled individually. In practice, increasing the number of regions and the underlying heterogeneity resulted in a large variability of the bias across regions. Varying the number of regions reveals the importance of regularizing parameter k R across tumor regions, as implemented by the rReg model, by comparison with the rLin model. e impact or regularization is measurable not only on k R , but also on the other parameters. In this study, simulations did not account for the variations of the signal to noise ratio depending on the size of the region.

CHAPTER . ERROR SOURCES AFFECTING RELATIVE QUANTIFICATION OF CEUS Instead it investigates solely the impact of the number of regions included in the analysis on the accuracy and precision of the estimates depending on whether the approach is regularized for parameter k R by keeping the noise level constant regardless of the number of regions.

e rate constant in the tumor, k T , appears to be related to the bias in the estimation of perfusion parameters using both models. In particular for the rReg model, the region with a simulated k T value larger than k R , and more generally regions with large k T values, yielded estimates of k T more biased than the other regions. Our study revealed that the choice of the reference tissue plays a crucial role in the accuracy and precision of the perfusion parameters estimated using the rReg model, it should therefore be further investigated. Indeed, it is necessary to identify the ideal reference tissue in order to improve the robustness of the estimation. A deeper and ner understanding of the relations between the characteristics of the reference tissue and the estimated perfusion parameters may allow correction of the estimation bias in case the ideal reference tissue does not exist in the image. Our study reveals that halving V R or doubling F R , which both result in doubling k R , has almost the same impact on the estimation bias. Actually, the least biased estimation overall was obtained using two thirds of the original k R value, i.e. k R = × . = .

. Additionally, increasing the values of both V R and F R , while enforcing xed values of k R , increased the accuracy of the estimates of the rReg model. is suggests that a well perfused tissue should be preferred, i.e. a tissue with a high tissue blood ow and high tissue blood volume.

Di erent simulation studies should be conducted to investigate the applicability of the rReg models to contrast-enhanced images acquired using other imaging modalities in case Indeed the choice of the reference tissue is crucial as it has a signi cant impact on the accuracy of the estimation, and should be further investigated. We were however able to draw some recommendations regarding its selection. e rate constant parameter is critical when selecting the reference tissue, a region exhibiting a rate constant larger than the rate constants in the regions of analysis yielded more accurate parameters in our experiments.

However an optimal value of k R seems to exist, and this issue needs to be further investigated on real examples. Furthermore, using a well perfused reference tissue, i.e. a tissue with high tissue blood volume and tissue blood ow, appears to provide more accurate perfusion parameters.

A erword

In the third part of this thesis, we proposed and assessed a new quanti cation model based on a one-compartment model using a reference tissue that allows simultaneous, robust, and homogeneous estimation of perfusion parameters in multiple regions. e model ensures that a single value of the rate constant characterizing the reference tissue is estimated across the various regions of analysis through regularization of this parameter.

In Chapter we rst demonstrated the superiority of the approach in terms of parameter In Chapter and Chapter , we assessed the accuracy and precision of the estimates of the rLin and rReg models through simulation experiments. In these two studies, the robustness of perfusion parameters was assessed in terms of accuracy, through the median estimation error, and precision, through interquartile range of the estimation error.

Chapter focuses on the impact of recirculating contrast agent on the estimated perfusion parameters. e accuracy and the precision of the LN and rLN models were assessed, and the simplistic approach consisting in tting only the part of the kinetics acquired before the recirculation occurs was also investigated. e absolute parameters of the LN model were expectedly a ected by recirculation, and the simplistic strategy to alleviate the impact of recirculation yielded strongly biased parameters. Regarding the rLN model parameters, rAUC was found robust to recirculation, however rWIR was not, and applying the simplistic strategy actually resulted in more biased perfusion parameters. is study revealed the increased robustness of the rReg model to recirculation compared to the rLin model.

Indeed, both models yielded biased perfusion parameters, however the median bias largely varies from one region to another using the rLin model. ese variations were considerably reduced using the rReg model, showing the ability of the model to quantify perfusion homogeneously across tumor regions, and therefore allowing meaningful intra-exam parameter comparison.

In Chapter we presented the rReg in more details, and studied the impact of varying data characteristics, including the noise, the exam duration, and the sampling period, as well as the impact of quanti cation choices, including the characteristics of the reference tissue, and the number of tissue regions. is study focused on the rLin and rReg models, as these two approaches were found the most accurate and robust in Chapter . Compared to the parameters of the rLin model, the parameters of the rReg model were found more precise in strong noise conditions, especially regarding the relative tissue blood ow parameter rF. Moreover, the rReg model was able to accurately quantify perfusion exams as short as seconds, while the rLin model required the entire seconds kinetics to accurately estimate parameters. Overall, the median bias in the estimation of the rReg model parameters was more homogeneous across tumor regions than for the rLin model. e characteristics of the reference tissue in uenced the estimation accuracy, but while our results suggest that CHAPTER . CONCLUSION a well perfused reference tissue would result in a more accurate quanti cation, further investigation is necessary to better understand this phenomenon and better characterize the ideal reference tissue.

Overall, the rReg model proved a promising perfusion quanti cation tool, and its applicability to contrast-enhanced ultrasound data was demonstrated in this thesis. e main outcome of this work is to take into account some redundancies when dealing with multiple regions which can be used to improve the estimation. e rReg model could be applied to reveal tumor functional heterogeneity at a ner scale, considering pixels or macro-pixels as regions for instance. In order to alleviate the impact of noise in local contrast enhancement curves on the robustness of the estimation, one should consider noise-ltering techniques, or alternatively estimating the local perfusion parameters with a xed value of parameter k R resulting from prior regional analysis. e lack of pixel-to-pixel correspondence in our testretest experiments motivated the regional cutout used in this work to ensure meaningful inter-exam parameter comparison.

Two-dimensional data still represents the majority of contrast-enhanced ultrasound exams nowadays, making inter-exam comparison on the same individual di cult. Indeed it is particularly di cult to ensure that the exact same imaging plane is selected in two exams performed days apart, and this is especially true when monitoring evolving tissues where structures change in shape and size, e.g. tumor growth, treatment response. erefore we would highly recommend the use of three-dimensional perfusion imaging whenever possible, in particular when it comes to exam comparison. e recent performance improvements of three-dimensional ultrasound scanners is therefore promising for tumor monitoring. Indeed ultrasound imaging will be able to catch up on tomographic imaging modalities, giving access to a more relevant information regarding the shape, size, structure, and function of the lesions, while retaining its real-time, non-ionizing, and cost-e ective characteristics.

ree-dimensional perfusion imaging is vastly available in PET, and X-ray CT, and possibly in MRI. e application of the rReg model to other perfusion imaging modalities could be further investigated. Of course it would require some adaptations to account for the underlying compartmental model (with multiple compartments instead of a single compartment). Further adaptation would also be necessary in order to study perfusion in the liver, an appropriate model would account for both the arterial and portal blood supplies of the tissue. Similarly, the vascular network of the kidney is complex and results in multiple perfusion phases. Using the kidney as a reference tissue without taking this spatial heterogeneity into account may provide erroneous conclusions. e incorporation of the multiple phases in the kidney remains to be fully developed. Choosing the appropriate quanti cation method to assess tissue perfusion through contrastenhanced exams remains a complex problem that heavily depends on the data and on the goal of the study. Exam comparison is particularly di cult because of variations occurring between exams at both the experimental and physiological levels. is thesis demonstrated the ability of one-compartment models to quantify contrast-enhanced ultrasound exams reproducibly and robustly at a regional scale when using a reference tissue to overcome the di culties in the estimation of the arterial input function. Indeed, reference tissue models proved more reproducible and robust than arterial input function models overall, however this may not hold if accurate image-based estimation of the arterial input function is achievable. We emphasis the necessity to consider the relations between the perfusion parameters of various tissues, or tissue regions, as this proved to considerably improve the accuracy, robustness, and reproducibility of perfusion parameters in the rReg approach. It additionally shows the functional heterogeneity of tumors while enabling meaningful intra-exam and inter-exam comparison. However, using reference tissues to monitor the perfusion of tumors undergoing therapy raises a question regarding the e ect of the treatment on the reference tissue, especially in the case of anti-angiogenic treatments.

Future work includes the confrontation of the perfusion parameters estimated using the rReg model to histological ndings for further validation of the method. e perfusion

L

  'impact de la fonction d' entrée sur les valeurs des paramètres absolus et relatifs du modèle AIF a été étudié en comparant les valeurs obtenues pour un examen donné, avec deux fonctions d' entrée estimées dans les images suite à deux types de seuils. Les paramètres absolus se sont montré extrêmement sensibles à la fonction d' entrée artérielle, V et F pouvant varier du simple au double dans certaines régions. Les paramètres relatifs du modèle rAIF se sont montré plus robustes aux variations de la fonction d' entrée artérielle, en particulier rV AIF est pratiquement insensible à la fonction d' entrée utilisée dans notre étude. Le Chapitre présente les relations existant entre les paramètres des di érents modèles étudiés dans le Chapitre , i.e. LN, rLN, AIF et RT, sous l'hypothèse d'un modèle compartimental alimenté par une fonction d' entrée artérielle. Les relations sont d'abord établies théoriquement en établissant les dé nitions analytiques des paramètres d'un modèle en fonction des paramètres d'un autre modèle, puis empiriquement à travers une étude de corrélation des paramètres estimés par les di érents modèles. En particulier, les paramètres des modèles LN et rLN ont été exprimés en fonction des paramètres du modèle AIF pour trois fonctions d' entrée di érentes, i.e. dans les deux cas idéalisés que sont l' entrée de type Dirac et de type porte, mais également dans le cas général. Ce chapitre démontre notamment les relations entre le paramètre AUC et le paramètre de volume sanguin tissulaire V d'une part, et entre ix le paramètre WIR et le paramètre de ux sanguin tissulaire F d'autre part. Ce constat a motivé notre sélection de paramètres issus du modèle LN dans le Chapitre puisque d'autres paramètres auraient pu être étudiés (WOR, MTT), mais les relations avec les paramètres de ux sanguin tissulaire et de volume sanguin tissulaire étaient moins directes. Il montre également la sensibilité des paramètres semi-quantitatifs à la fonction d' entrée artérielle, et con rme l'intérêt de la normalisation par une région de référence pour s'absoudre de cette dépendance. L'intérêt de la normalisation est également con rmé empiriquement, en e et la corrélation entre les paramètres relatifs des di érents modèles est beaucoup plus forte que celle entre les paramètres absolus. Partie III. Proposition et évaluation d'une nouvelle méthode de quantication Dans cette troisième partie, nous présentons d'abord une approche d' estimation reposant sur une formulation linéaire du modèle RT qui a été proposée initialement pour d'autres modalités d'imagerie de contraste (TEP et IRM), notée rLin. Les méthodes de résolution linéaires ont l'avantage de procéder à l' estimation directe des paramètres minimisant la fonction d' erreur, permettant ainsi d' éviter les problèmes liés à l'initialisation des paramètres tout en accélérant l' estimation. Par ailleurs, ce modèle permet l' estimation d'un paramètre supplémentaire, i.e. la constante de transfert du tissu de référence k R . Cependant, cette formulation conduit potentiellement à di érentes valeurs de ce paramètre qui bien évidemment devrait être unique. Pour pallier à cet inconvénient, nous avons proposé une nouvelle méthode d' estimation régularisée, notée rReg, exploitant le fait que plusieurs régions soient analysées conjointement en s'assurant qu'une valeur unique du paramètre k R soit estimée pour toutes les régions d'un même examen. Nous avons par la suite étudié la robustesse de ces deux modèles rLin et rReg aux variations inter-examens, à des phénomènes physiologiques comme la recirculation, à des paramètres d'acquisition, et à di érentes stratégies dans l'analyse des données. À notre connaissance ce type d'approche n'a été proposé dans aucune des modalités d'imagerie de perfusion. Les équations des modèles rLin et rReg sont d'abord développées brièvement dans le x Chapitre . En e et, le Chapitre qui étudie la reproductibilité des paramètres de perfusion estimés avec les deux approches a fait l' objet d'une publication courte suite à la présentation de ces travaux lors du congrès IEEE International Ultrasonics Symposium (IUS) qui s' est tenu à Tours en septembre . Le Chapitre étudie spéci quement l'impact de la recirculation des microbulles sur la justesse et la précision des estimations réalisées avec di érents modèles, en utilisant des cinétiques de perfusion simulées de façon réaliste à l'aide du modèle AIF. Cette étude a fait l' objet d'une publication intitulée 'Impact of Recirculation in Dynamic Contrast-Enhanced Ultrasound : A Simulation Study ' et publiée dans la revue Innovation and Research in BioMedical Engineering (IRBM). Dans le Chapitre sont détaillés les développements théoriques qui ont amené le développement du modèle rReg, puis nous avons étudié l'impact de divers paramètres sur la justesse et la précision des estimations. Nous nous somme notamment intéressés aux paramètres d'acquisition comme la durée des examens ou la fréquence d' échantillonage, mais aussi à des stratégies d'analyse comme le nombre de régions d'intérêt ou le choix du tissu de référence. Nous prévoyons de soumettre une version revue de ce Chapitre pour publication dans la revue Medical Image Analysis. Dans le Chapitre , les paramètres de perfusion estimés à l'aide des méthodes rLin et rReg sont comparés, en terme de reproductibilité, aux paramètres des modèles LN et rLN, en se basant la même étude test-retest que celle détaillée dans le Chapitre . Le modèle rLin s' est révélé être l'approche relative la moins robuste aux variations inter-examens, notamment en terme de paramètre de ux puisque nous avons obtenu un coe cient de variation médian supérieur à % pour rF rLin . Cette forte variabilité montre la faible identi abilité du paramètre k R et son impact sur les autres paramètres du modèle, notamment sur rF rLin . Elle explique également la pratique courante qui consiste à ne pas estimer k R mais à utiliser une valeur xe provenant de la littérature ou d' expériences préalables pour stabiliser l' estimation des autres paramètres, comme nous l'avons fait avec le modèle RT présenté dans le Chapitre . L'approche régularisée rReg s' est montrée la plus reproductible dans cette étude, puisque nous avons obtenu des coe cients de variation médians inférieurs à % pour rV rReg , et inférieurs à % pour rF rReg , dans notre étude test-retest. Dans le Chapitre , nous avons étudié la sensibilité à la recirculation des paramètres estimés à l'aide des modèles LN, rLN, rLin et rReg. En e et la recirculation de l'agent de xi contraste dans le tissu étudié est un problème connu mais rarement pris en compte dans les approches de quanti cation de la perfusion in vivo en échographie de contraste. Pour s'absoudre de ce phénomène, une approche simpliste consiste à 'couper ' les données avant que la recirculation n'intervienne. Nous avons donc également étudié la capacité de cette technique à estimer des paramètres de perfusion en l'appliquant aux modèles LN et rLN, dont les variantes sont notées LN W I et rLN W I . Les paramètres absolus du modèle LN se sont révélés particulièrement sensibles à la recirculation en raison de leur forte dépendance à la fonction d' entrée artérielle. En ce qui concerne le modèle relatif rLN, l' estimation du paramètre rAUC s' est montrée robuste à la recirculation, cependant le paramètre rWIR reste sensible à la recirculation. Les approches LN W I et rLN W I se sont révélées peu robustes, en e et malgré l'accord entre les paramètres estimés avec et sans recirculation, le nombre réduit d' échantillons utilisés durant l'ajustement du modèle ne permet pas d'ajuster le modèle correctement à la phase descendante de la cinétique, et rend l' estimation particulièrement sensible au bruit. Les paramètres de ux et de volume sanguin tissulaires relatifs des modèles rLin et rReg se sont révélés peu sensibles à la recirculation, avec un léger avantage pour le modèle rLin, ce qui con rme la robustesse structurelle des modèles compartimentaux à cet égard. En revanche, l' estimation des paramètres k T et k R par le modèle rLin est sujette à de fortes erreurs variant d'une région à l'autre, et ce même en l'absence de recirculation. Le modèle rReg n' est pas a ecté par ce phénomène et montre une erreur d' estimation très faible pour ces deux paramètres. En outre, si les paramètres rV rReg et rF rReg sont globalement plus a ectés par la recirculation que les paramètres rV rLin et rF rLin , l' erreur est plus homogène entre les di érentes régions d'intérêt en utilisant le modèle rReg, permettant ainsi une comparaison des paramètres régionaux plus robuste.

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  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx . . Quanti cation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulations of CEUS data . . . . . . . . . . . . . . . . . . . . . . . . . . . Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noise level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exam duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sampling period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference tissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number of regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of semi-quantitative parameters are illustrated on the mean kinetics (black dots) inside the perfused area of a murine tumor observed in contrast-enhanced ultrasound and tted with a log-normal model (red dashed line): the wash-in rate (WIR, in green), the time-to-peak (T TP, in cyan), the peak enhancement (PE, in orange), and the area under the curve (AUC, in dark blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  Right: Mean kinetics associated with the non-necrotic part of the tumor, the reference tissue, and the renal cortex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -Automated detection of the AIF: parametric maps T TP and PE inside the artery region; segmentation results and associated AIF with: (a) rPE * = % and ∆T TP * = s (in green color); (b) rPE * = % and ∆T TP * = . s (in blue color). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -Comparison of the volume-based and ow-based parameters obtained for the four test-retest exams (R , R , R , and R ) of the mouse m : linear regressions between (a) rV RT and AUC, (b) rV RT and V , (c) rV RT and rAUC, (d) rV RT and rV AIF , (e) rF RT and WIR, (f) rF RT and F, (g) rF RT and rWIR, (h) rF RT and rF AIF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -Boxplot showing the coe cients of variation of blood volume parameters (le ) and blood ow parameters (right) estimated with the aLN, rLN, aAIFd, rAIFd, and rRTd models. For each box, the bold line represents the median value, the bottom and top lines the rst and third quartiles. Dotted lines extend to the most extreme data points which are less than . times the interquartile range. Outlier points are displayed with empty circles. Two groups of parameters were built (horizontal lines below the parameter names) such that there were no signi cant intra-group di erences while there were statistically signi cant inter-group di erences (marked by * ). . . . . . . . . . xxiv -(a-b) Median (of values) coe cient of determination (R ) of the leastsquares linear regression between pairs of parameters θ i , θ j computed for the sub-regions of one exam: (a) parameters derived from the aLN approach; (b) volume (AUC, V and rV RT ), ow (WIR, F and rF RT ) and time delay (∆ T , d t and D RT ) parameters respectively computed with aLN, aAIFd and RTd models. (c-d) Coe cients of determination (R ) of the least-squares linear regression computed when pooling the sub-regions together: (c) R between pairs of volume parameters computed with aLN, aAIFd, rLN, rAIFd and RTd models, (d) R between pairs of ow parameters computed with aLN, aAIFd, rLN, rAIFd and RTd models. . . . . . . . . . . . . . . . . . -Illustration of the data pre-processing steps. Le : e contours of the perfused tumor area have been overlaid on a contrast-enhanced image (in blue color). is area was automatically divided into radial layers and angular sectors as shown by the spiderweb patterns. A RT region (in orange color) was also delineated. Right: Mean TICs associated with the perfused area of the tumor, and the RT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -Boxplot showing the CV of blood volume (le ) and blood ow (right) estimated with the LN, rLN, rLin, and rLinReg models. . . . . . . . . . . . . . . -a One-compartment model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -b One-compartment model with additional time-delay parameter. . . . . . . . -c Block diagram of the relative one-compartment model in case of a single region of interest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -d Block diagram of the relative one-compartment model for regional quanti cation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv -Top: original C(t) (blue dots) and noise ltered C f (t) (orange line) timeintensity curves, the horizontal line shows the % of maximum intensity threshold, and the vertical line shows t % . Middle: ltered time-intensity curve C f (t) (orange line) cropped to t ≤ %, and the tangent to C f (t) with maximum upslope gradient for t ≤ t % , and the vertical line shows the estimated time delay D. Bottom: gradient of the cropped noise ltered timeintensity curve Ċ f (t) (pink line), the horizontal black line shows the % of maximum gradient threshold, and the vertical line shows the estimated time-delay D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -Simulated TICs with (orange) and without recirculation (blue) corresponding to noise-free AIF (top), examples of noise-free and noisy TICs in the fourth tissue region (middle) and in the reference tissue (bottom). e rst hundred seconds are displayed here. . . . . . . . . . . . . . . . . . . . . . . . . -Bull's-eyes representation of the perfusion parameters used to simulate the regional TICs, C T i (large circle), and the reference TIC, C R (bottom rightdisk). From le to right: tissue blood volume (V ), tissue blood ow (F), time-delay (D), and rate constant (k). e scale displayed in red color shows relative parameters: rV , rF, δ as de ned by equation ( . ). . . . . . . . . . . . -Median values (large symbols), rst and third quartiles (small symbols) of parameters estimated for the fourth tissue region C T (outer ring, upper halve, right octant). First column: tissue blood volume related parameters, second column: tissue blood ow related parameters, third column: timedelay related parameters, fourth row: rate constants in the tissue region and reference tissue. Constant lines in black represent simulated values, blue lines the estimation corresponding to the LN model, red lines the estimation corresponding to the LN model restricted to earliest phase. Yellow color stands for rLin model, while purple color stands for rReg model. For all of the cases, lled symbols correspond to the con guration without recirculation, while empty symbols correspond to the con guration with recirculation. xxvi -Bull's-eyes of the median estimation errors obtained by the LN model at the intermediate noise level. From le to right: estimation errors corresponding to tissue blood volume, tissue blood ow, time delay. From top to bottom: M without recirculation, M with recirculation, M without recirculation, M with recirculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -Bull's-eyes of the median estimation errors obtained by the rLN model at the intermediate noise level. From le to right: errors corresponding to tissue blood volume, tissue blood ow, time delay. From top to bottom: M without recirculation, M with recirculation, M without recirculation, M with recirculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -Bull's-eyes of the median estimation errors obtained by the rLin and rReg models at the intermediate noise level. From le to right: errors corresponding to tissue blood volume, tissue blood ow, time delay, rate constant in tissue regions, rate constant in the reference tissue deduced from the di erent estimations inside tissue regions. From top to bottom: M without recirculation, M with recirculation, M without recirculation, M with recirculation. -Absolute perfusion parameters used for simulation with the OVC model, i.e. tissue blood volume, V T and V R (dimensionless), tissue blood ow, F T and F R (in s -), tissue rate constant, k T and k R (expressed in s -), and time delay, D T and D R (in s). Bullseye view of the parameters in the tumor regions. e bottom disks represent the parameters used to simulated the reference tissue region, the middle disk being the original value used for all experiments. e other disks are, from le to right, the half, two thirds, three halves, and double of the original value, used to study the in uence of the reference tissue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -Time-intensity curves inside of each of the tumor regions (top grid), and in the reference tissue (bottom), simulated using the OVC. Each plot shows the simulated noiseless curve (orange line) in the region, i.e. σ = , as well as the curve with simulated multiplicative noise (blue dots), i.e. σ = . . . . xxvii -Relative perfusion parameters used for simulation varying the number of tumor regions N T from (le ) to (right), i.e. relative tissue blood volume, rV , relative tissue blood ow, rF, tissue rate constant, k T . . . . . . . . . . . . -Correspondence of the regions between bullseye (le ) and grid (right) representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -a Median (large symbols) and rst and third quartiles (small symbols) of the relative estimation error for the relative blood volume (rV ) estimated in the tumor regions using the rLin (yellow diamonds) and rReg (purple squares) models, as a function of the noise level. . . . . . . . . . . . . . . . . . -b Median (large symbols) and rst and third quartiles (small symbols) of the relative estimation error for the relative blood ow (rF) estimated in the tumor regions using the rLin (yellow diamonds) and rReg (purple squares) models, as a function of the noise level. . . . . . . . . . . . . . . . . . . . . . . -c Median (large symbols) and rst and third quartiles (small symbols) of the relative estimation error for the rate constant in the tumor (k T ) estimated in the tumor regions using the rLin (yellow diamonds) and rReg (purple squares) models, as a function of the noise level. . . . . . . . . . . . . . . . . . -d Median (large symbols) and rst and third quartiles (small symbols) of the relative estimation error for the rate constant in the reference tissue (k R ) estimated in the tumor regions using the rLin (yellow diamonds) and rReg (purple squares) models, as a function of the noise level. . . . . . . . . . -e Median (large symbols) and rst and third quartiles (small symbols) of the estimation error (in seconds) for the relative time-delay (D) estimated in the tumor regions using the rLin (yellow diamonds) and rReg (purple squares) models, as a function of the noise level. . . . . . . . . . . . . . . . . . -a Median (large symbols) and rst and third quartiles (small symbols) of the relative estimation error for the relative blood volume (rV ) estimated using the rLin (yellow diamonds) and rReg (purple squares) models, as a function of the exam duration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviii -b Median (large symbols) and rst and third quartiles (small symbols) of the relative estimation error for the relative blood ow (rF) estimated using the rLin (yellow diamonds) and rReg (purple squares) models, as a function of the exam duration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -c Median (large symbols) and rst and third quartiles (small symbols) of the relative estimation error for the rate constant in the tumor (k T ) estimated using the rLin (yellow diamonds) and rReg (purple squares) models, as a function of the exam duration. . . . . . . . . . . . . . . . . . . . . . . . . . . . -d Median (large symbols) and rst and third quartiles (small symbols) of the relative estimation error for the rate constant in the reference tissue (k R ) estimated using the rLin (yellow diamonds) and rReg (purple squares) models, as a function of the exam duration. . . . . . . . . . . . . . . . . . . . . . . . . . -e Median (large symbols) and rst and third quartiles (small symbols) of the estimation error (in seconds) for the relative time-delay (D) estimated using the rLin (yellow diamonds) and rReg (purple squares) models, as a function of the exam duration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -a Median (large symbols) and rst and third quartiles (small symbols) of the relative estimation error for the relative tissue blood volume (rV ) estimated using the rLin (yellow diamonds) and rReg (purple squares) models, depending on the sampling period used for simulation. . . . . . . . . . . . . . . -b Median (large symbols) and rst and third quartiles (small symbols) of the relative estimation error for the relative tissue blood ow (rF) estimated using the rLin (yellow diamonds) and rReg (purple squares) models, depending on the sampling period used for simulation. . . . . . . . . . . . . . . . . . -c Median (large symbols) and rst and third quartiles (small symbols) of the relative estimation error for the rate constant in the tumor (k T ) estimated using the rLin (yellow diamonds) and rReg (purple squares) models, depending on the sampling period used for simulation. . . . . . . . . . . . . . . xxix -d Median (large symbols) and rst and third quartiles (small symbols) of the relative estimation error for the rate constant in the reference tissue (k R ) estimated using the rLin (yellow diamonds) and rReg (purple squares) models, depending on the sampling period used for simulation. . . . . . . . . . . . . . -e Median (large symbols) and rst and third quartiles (small symbols) of the estimation error (in seconds) for the relative time-delay (D) estimated using the rLin (yellow diamonds) and rReg (purple squares) models, depending on the sampling period used for simulation. . . . . . . . . . . . . . . . . . . . -a Bullseyes of the median relative estimation error for the relative tissue blood volume (rV ) estimated using the rLin (top) and rReg (bottom) models depending on the characteristics of the reference tissue used for simulation. . . -b Bullseyes of the median relative estimation error for the relative tissue blood ow (rF) estimated using the rLin (top) and rReg (bottom) models depending on the characteristics of the reference tissue used for simulation. . . . . . -c Bullseyes of the median relative estimation error for the rate constant in the tumor (k T ) estimated using the rLin (top) and rReg (bottom) models depending on the characteristics of the reference tissue used for simulation. -d Bullseyes of the median relative estimation error for the rate constant in the reference tissue (k R ) estimated using the rLin (top) and rReg (bottom) models depending on the characteristics of the reference tissue used for simulation. -a Bullseyes of the median value and the standard deviation of the relative estimation error for the relative tissue blood volume (rV ) estimated using the rLin (top) and rReg (bottom) models depending on the number of regions N T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -b Bullseyes of the median value and the standard deviation of the relative estimation error for the relative tissue blood volume (rF) estimated using the rLin (top) and rReg (bottom) models depending on the number of regions N T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxx -c Bullseyes of the median value and the standard deviation of the relative estimation error for the rate constant in the tumor (k T ) estimated using the rLin (top) and rReg (bottom) models depending on the number of regions N T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -d Bullseyes of the median value and the standard deviation of the relative estimation error for the rate constant in the reference tissue (k R ) estimated using the rLin (top) and rReg (bottom) models depending on the number of regions N T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . di erent models tested. e rst three models propose absolute quanti cation. e last ve models propose relative quanti cation. . . . Median [ rst-third quartiles] values of N MRSE and FMI (in %) obtained for the di erent models. N is the number of sub-regions where FMI < %. Signi cant di erences between aLN and any other model are indicated by ⋆ . In addition, signi cant di erences between aAIF (resp. rRT) and aAIFd (resp. rRTd) are indicated by † (resp. ‡ ). e symbol ○ indicates that comparisons were not reported due to the high number of missing data. . . . . . . Mean ± standard deviation of the parameters estimated with the aAIFd and rAIFd models, using two di erent sets of cut-o s to generate the AIF functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean ± standard deviation of the volume, ow and delay parameters estimated in the di erent sub-regions of the tumor, for the four test-retest exams, a er multiple imputation of missing values due to poor t quality. Values of WIR and F are multiplied by . . . . . . . . . . . . . . . . . . . . . Mean ± standard deviation of the coe cients of variation (CV), expressed in percentage, of volume and ow parameters estimated for each sub-region a er multiple imputation of missing values due to poor t quality. CV were not computed for time delays, since their values can be either positive or negative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closed-form expressions of perfusion parameters using the aLN model, WOR being the absolute value of the maximum negative slope. . . . . . . . . . . . . xxxiii . Closed-form expressions of perfusion parameters using a one-compartment model (aAIFd) and assuming three di erent shapes of AIF: impulse function (δ), rectangle function of width a and height a, rect a (t), and general case C A (t). In the rst two cases, K stands for the injected concentration. In the general case, MT T C A stands for the mean transit time of C A (t). . . . . . . . Closed-form expressions of the relative perfusion parameters using a relative one-compartment model (rAIFd) and assuming three di erent shapes of AIF: impulse function (δ), rectangle function of width a and height a, rect a (t), and general case C A (t). In the rst two cases, K stands for the injected concentration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Median values of FMI obtained for the four models and number of subregions N rem , out of , for which FMI < %. . . . . . . . . . . . . . . . . . Introduction Chapter Introduction is thesis addresses the quanti cation of tumor perfusion using contrast-enhanced ultrasound imaging. In this chapter, we present the biological and technical context that motivated this thesis. e research was performmed at the Laboratoire d'Imagerie Biomédicale (LIB), and was nanced by the Fondation pour la Recherche Médicale (FRM) through grant DBS . e global project aims at developing a multiparametric tumor tissue classi cation tool, based on multiple ultrasound imaging modalities, including quantitative ultrasound, elastosonography, and contrast-enhanced ultrasound. e data should be used to develop a realistic tumor growth model, as well as an appropriate treatment response prediction model [ ]. e rst step of this project was therefore the accurate estimation of parameters from ultrasound data in order to obtain reproducible results and use them in longitudinal studies.

  Surgery, radiotherapy, and chemotherapy directly target cancerous cells, and an e cient treatment should have a direct impact on the size of the tumor. erefore, morphological criteria were proposed to assess tumor response to therapy. e classical RECIST and WHO criteria are based on the changes in tumor diameter, and give an indication on the evolution of the disease, i.e. stable, regressive, or progressive. Oppositely, anti-angiogenic treatments do not target cancerous cells, but rather the neovascularization of the tumor. ey do not have a direct impact on the size of the tumor, especially at the early stages of the treatment. erefore, classical morphological criteria fail CHAPTER . INTRODUCTION to reveal the e ciency of such treatments. Quanti cation of tumor angiogenesis and of the response to anti-angiogenic treatments requires the development of functional criteria assessing the microvascularization of tumor tissues. Microvascularization can be observed in vivo using functional imaging, and in particular through perfusion imaging.

  Nuclear medicine regroups imaging modalities using radioactive tracers. Planar scintigraphy uses gamma-emitting tracers and a single gamma camera, yielding projections images of the tracer concentration. Single-photon emission computed tomography (SPECT) uses the same tracers as scintigraphy, but uses a rotating gamma camera to create three-dimensional volumes of the tracer concentration. Positron emission tomography (PET) uses a positron-emitting tracer, and a coincident detection of the two gamma rays resulting from the annihilation of the positron emitted during the beta decay of the radioactive isotope.Radioactive isotopes are usually attached to a biological molecule, forming a radiotracer that follows the distribution of the biological molecule and enables its tracking in biological systems.X-ray computed tomography (CT) X-ray computed tomography uses a combination of multiple planar X-ray projections with varying angles to produce three-dimensional images, the resulting volumes show the absorption of X-rays.e contrast agents used for X-ray computed tomography are usually iodinated compounds that were chosen for their high . . PERFUSION IMAGING radiodensity, resulting in a strong absorption of the X-ray beams, and for their fast renal elimination. Additionally, they exhibit a linear relation between the X-ray attenuation and the concentration of contrast-agent. Magnetic resonance imaging (MRI) MRI is based on nuclear magnetic resonance that exploits the ability of some atoms to absorb and emit radio frequencies when placed in an external magnetic eld. Typically, the imaged nuclides are protons present in tissues composed of water molecules. In this case, magnetic resonance images are formed by measuring the spin-lattice (T ) and spin-spin (T ) relaxation times of the protons inside the studied tissue. Magnetic resonance contrast agents are usually paramagnetic substances that shorten the T relaxation time of the protons inside the observed tissues, for instance Gadolinium chelates are widely used. Most Gadolinium-based contrast agents di use from the blood pool to the interstitial space through the capillary surface. However intravascular contrast agents were also developed for magnetic resonance imaging, but their usage is mainly restricted to research. Endogenous tracers are also being developed for MRI, for instance arterial spin labelling (ASL) allows tracking of spin-labelled arterial blood and alleviates the need for injection of a exogenous contrast agent. Contrast-enhanced ultrasound (CEUS) Ultrasound images are formed by sending sequences of ultrasound pulses inside the tissue using a transducer, the ultrasound pulses are re ected by scatterers in the tissue, and the re ected pulses are then recorded by the transducer. e recorded signals are processed to retrieve the time of ight of the echo, determining the location of the scatterer in the image, as well as the strength of the echo, determining the image intensity associated to the scatterer. Coated gas-lled microbubbles are used as ultrasound contrast agents for their ability to oscillate asymmetrically when exposed to acoustic waves, and for their size that makes them resonate in the ultrasound frequency domain [ ]. eir size being nearly the same as the size of a red blood cell, they can go everywhere in the vascular system, from the largest arteries to the smallest capillaries, and they do not leak in the interstitial space. Microbubbles CHAPTER . INTRODUCTION exhibit a strong and speci c non-linear response to ultrasound pulses. is non-linearity is exploited by ultrasound scanners using speci c background-cancelling sequences to produce contrast-speci c images, e.g. harmonic imaging [ ], subharmonic imaging [ ], contrast pulse sequencing (CPS) [ ].

  I -Quanti cation of perfusion: state of the art Chapter presents a state of the art of the methods that have been proposed for the quanti cation of perfusion, and in particular using medical imaging. Quanti cation approaches are classi ed into three categories: semi-quantitative, deconvolution-based, or compartmental models. Indeed, the two last categories consider an arterial input function, and compartmental modeling is a speci c case of deconvolution. Some main studies from the literature were reviewed, focusing on earlier CHAPTER . INTRODUCTION work at the origin of methodological developments. Part II -Reproducibility of the existing methods and the relations between them Chapter studies the impact of mathematical modeling on the reproducibility of perfusion parameters through preclinical test-retest experiments. is study revealed the sensitivity of absolute semi-quantitative parameters to inter-exam variations in experimental or physiological conditions. It also showed the superiority of of normalized parameters, and more precisely of the reference tissue model, in terms of reproducibility. Chapter extends the work from Chapter by rst establishing theoretically and then verifying experimentally the relations between the parameters of the various models. e existence of these relations shows the ability of semi-quantitative parameters to reveal relative variations of the vascular function. Part III -Proposition and assessment of a new quanti cation method Chapter rst presents a new regularized linear estimation method for the reference tissue model, and compares its parameters to those obtained using the classical linear estimation method in terms of reproducibility. is study proves the superior robustness of the regularized estimates to inter-exam variations. en Chapter compares the robustness of the two models to contrast-agent recirculation through simulation experiments. And Chapter studies the accuracy and the precision of the models when varying intrinsic characteristics of the data,

  ey are however not directly related to any physiological function and are prone to changes in experimental or physiological conditions. ey are therefore o en used as relative indicators of perfusion and contrast agent transit time [ ], allowing intra-exam comparisons. Examples of semi-quantitative parameters include the peak enhancement [ , , , ], the time to peak enhancement [ , , ], maximum upslope gradient, also known as wash-in rate [ , ], the area under the enhancement curve [ ]. We also considered semiquantitative the perfusion parameters based on indicator dilution theory that were derived from the previously cited parameters [ , , , , , , , ]. Figure -shows graphical representations of the above mentioned semi-quantitative parameters.

Figure - :

 - Figure -:Examples of semi-quantitative parameters are illustrated on the mean kinetics (black dots) inside the perfused area of a murine tumor observed in contrast-enhanced ultrasound and tted with a log-normal model (red dashed line): the wash-in rate (WIR, in green), the time-to-peak (T TP, in cyan), the peak enhancement (PE, in orange), and the area under the curve (AUC, in dark blue).

  For instance, Hilson et al. [ ] and later Peters et al. [ ] both proposed an enhancementbased perfusion index, which is commonly de ned as the ratio of the tissue blood ow to the cardiac output. On the one hand, Hilson et al. [ ] de ned the perfusion index as the ratio of the area under the arterial curve to the area under the renal curve, both curves are integrated up to the arterial peak. On the other hand, Peter et al. de ned the perfusion index as the ratio of the maximum upslope of the renal curve normalized for the injected quantity to the maximum upslope of the integrated arterial curve normalized for the area under the arterial enhancement curve, and multiplied by a constant converting the number of received gamma photons into a unit of activity [ , ]. Nally et al. [ ] reported signi cant di erences in the wash-in rate, curve width at % of the peak enhancement, and maximum enhancement in normal and stenosed kidney in a canine model. . . X-ray imaging Derived from the work of Peters et al. [ ], semi-quantitative parameters were also proposed by Miles et al. in to quantify renal cortical and medullary perfusion in X-ray computed tomography (CT) with bolus injection of Iodine [ , ]. e proposed perfusion index was de ned as the ratio of the maximum slope in the tissue curve and the peak enhancement of the arterial curve. e method was successively used to quantify perfusion in the pancreas [ ], solitary pulmonary nodules [ ], lymphoma masses [ ], lung adenocarcinoma [ ], and more generally to study tumor angiogenesis [ , ].

,

  Wei et al. [ ] proposed an explicit model to quantify myocardial perfusion. For this method, microbubbles were injected as a continuous infusion. When the micro-bubble concentration reached a steady state, high mechanical index ultrasound pulses were used to disrupt microbubbles in the myocardium with increasing pulsing interval. is technique is known as intermittent imaging. e video intensity, which was assumed proportional to the concentration in microbubbles, was plotted as a function of the pulsing interval. en an exponential function with plateau value A and rate constant β was tted. Assuming a constant beam elevation, Wei et al. [ ] demonstrated that blood ow F was proportional to the slope at the origin, i.e. Aβ. e authors validated their approach in vitro, but also ex vivo and in vivo in a canine study. e method was later validated for perfusion quanti cation of the kidney, estimating both cortical and medullary blood ow in another canine study [ ]. e development of power pulse inversion imaging by Simpson and Burns [ ] in , allowing real-time imaging of low microbubble concentration at low mechanical index [ ]. Tiemann et al. [ ] in and Masugata et al. [ ] in demonstrated the use of power pulse inversion to image the replenishment of tissue following a single series of microbubbledisrupting pulses in real-time. Both studies also reveal that the model proposed by Wei et al. [ ] could be used to accurately t real-time replenishment curves. In , Schlosser et al. [ ] applied the same model to quantify renal perfusion, and compared the estimated parameters according to the acquisition sequence, i.e. power pulse inversion [ ] vs. pulse inversion [ ]. e authors disclosed highly di erent perfusion parameter values according to the acquisition scheme, which yielded signi cantly di erent values of both parameters. However, they were able to distinguish large arteries in the renal hilum from smaller arteries in the renal cortex, using either of the schemes, and either of the parameters. In , Krix et al. [ ] presented a hyperbolic model to quantify perfusion using either intermittent or real-time imaging, relying on physiological considerations, and accounting

  well [ ].Potdevin et al. [ ] investigated the mis t of the exponential model to re ll curves and proposed the use of the error function instead. ey introduced a time-delay parameter in their model in order to better re ect experimental data. ey also investigated the impact of observing multiple blood velocities in the region of interest, as well as the e ect of the CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW point spread function of the imaging system through a simulation study and concluded that replenishment curves contain more information that just mean transit time. For instance they were able to reveal the presence of abnormal vascular structures, such as shunting, i.e. direct ow from the arterial system to the venous system without passing through the capillary bed. In a latter study, they proposed an adaptation of the model to quantify local replenishment curves in tissues with complex vascular structures. Replenishment curves are rst normalized according to the estimated depth-dependent pixel intensity in a pixel containing only blood. e model accounts for the speci c angles, lengths, and velocities of the various vascular structures in the studied tissue [ ]. ey also used their model as a tissue classi er tool, determining tissue type as the prede ned vascular model that best tted the enhancement curve in the least squares sense. However, applying the simpler exponential model with the average tissue mean transit time on normalized data yielded accurate tissue classi cation maps, suggesting that the major factor di erentiating the replenishment curves of the various tissue types is actually mean transit time. e additional information could be used to characterize vascular properties more nely. In spite, while their is no physical evidence why the exponential model should t replenishment curves, it remains a good approximation and allows accurate di erentiation among tissue types. is is especially true when applied to noisy replenishment curves.Arditi et al. [ ] introduced a new formalism for the quanti cation of real-time destruction-replenishment acquisitions using a low mechanical index. e authors emphasize the importance of linearizing ultrasound data according to the type and settings of the imaging equipment, as opposed to grey levels intensities directly extracted from the log-compressed images visible on the monitor. More importantly, they present a perfusion model accounting for the variety in blood ow velocity and direction, assuming a model accurately describing the distribution of transit times is known (e.g. log-normal distribution).e method achieves a better description of the S-shaped replenishment curves that can be observed.In, Quaia et al. [ ] proposed a model re ecting the drag (related to ow) and diffusion (not related to ow) of microbubbles in blood, and accounting for the variety of blood velocities and directions. is variety is modeled as the sum of a variable number of piecewise linear replenishment curves. e model was validated through in vitro experiments

  be seen as the probability distribution of the contrast particles transit times [ ]. Parameters describing the systemic response of the vascular system can therefore be derived from the impulse-response function estimated by deconvolution. In , Zierler [ ] established the relevance of the impulse response to assess indicatordilution curves. He deeply investigated the theoretical aspects of indicator-dilution theory, CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW the conditions for its validity, and the consequences when they are not respected, including when a sudden injection is not truly instantaneous. is section established the theoretical basis for deconvolution of indicator-dilution curves. In another section addressing the impact of recirculation, he established the role of the Laplace transform when solving convolution integrals, as found in his theoretical framework. In , Coulam et al. [ ] proposed deconvolution to estimate the impulse response of coronary and renal circulatory systems, as well as the global circulatory impulse response, in a canine study with one dog. Because a perfect impulse injection is not achievable in practice, the impulse response cannot be estimated directly from the downstream time-concentration curve. erefore bolus injections of indocyanine green dye were performed in a pulmonary artery and blood was sampled at various downstream sites. Deconvolution of the downstream curves by the upstream curve was performed in the frequency domain by simple division of the Fourier coe cients, and the inverse transform was applied to obtain the impulse response in the time domain. e accuracy of the estimation was assessed both visually and numerically, through time-domain convolution of the estimated impulse response with the upstream curve, yielding an estimate of the downstream curve. Using simulations, the role of the number of harmonics present in the upstream curves was investigated. Indeed, for low numbers of harmonics, oscillations could occur in the impulse response, yielding inaccurate estimation. e authors suggest intra-ventricular injection would alleviate this issue, as the upstream curve would contain more harmonics. In , Maseri et al. [ ] used deconvolution to estimate the impulse response of the pulmonary circulatory system. Two tracers were injected simultaneously, the rst one in the pulmonary artery, and the second one in the le ventricle. Both tracers were sampled at the same site, at the aortic root, yielding two time-concentration curves. e authors used Laplace transforms to demonstrate that deconvolving the rst time-concentration curve by

  pulse response shape was proposed byKnopp et al. [ ] to assess the impulse response of the coronary bed. e method was tested in a canine study with intra-ventricular injection dye injection, and simultaneous blood sampling at the aortic root and in the coronary sinus. e technique rst models the impulse response as a weighted sum of delayed statistical distributions with unit area, which allowed predetermination of the convolved curves and aleviated the signal periodicity condition, and then iteratively minimized the t error using a gradient descent algorithm. Impulse functions were rst investigated, however other distribution functions were chosen to reduce the computational load while making the estimation more robust and ensuring similar quality of t. Experiments reveal right-skewed impulse responses with prolonged tails, suggesting non-monoexponential downslopes expected using classical unimodal functions, i.e. lagged-normal, gamma-variate, lognormal. e authors compared their estimates of mean and standard deviation of transit times to the estimates from previous studies, performed by the same group, but focused on other vascular structures, i.e. lungs [ ], descending aorta [ ], and femoral artery [ ]. Despite the reported parametric overlap, authors observed a rise of the relative variation of transit times correlated with the complexity of the vascular structures, which they justi ed by reminding that di erent pathways have di erent mean transit times and ows. As soon as , Gamel et al. [ ] published a review of the various pitfalls a ecting the deconvolution process when estimating tissue impulse responses from indicator-dilution CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW

  et al. [ ], Kenny et al. [ ] used a deconvolution technique rst presented in [ ], based on the Laplace transform, in an attempt to make the estimation of the kidney function independent on the spread of the bolus when it enters the kidney, and assessed it in a clinical

  Appledorn [ ] responded to the publication and expressed his disagreement regarding the formulation of the clearance function, arguing that the input to the renal system depends on the negative time derivative of the cardiac curve. He explained his reasoning mathematically, assuming a two-compartment model previously presented in the litterature [ ], and proposed a correction for the renal input function. Di ey et al. [ ] proposed the use of the matrix-based deconvolution method, rst generically introduced by Valentinuzzi and Montaldo Volachec [ ], to quantify the impulse responses of the renal parenchyma and renal pelvis to a bolus injection of m Tc-DTPA. ey used the iterative data-bounding technique prior to deconvolution for noise removal, the technique is detailed in [ ].Non-renal activity is assumed qualitatively similar to arterial activity, thus the correction is simply applied by setting the rst term of the impulse response equal to the second one. e impulse response of the renal parenchyma was obtained by deconvoluting the parenchymal curve by the arterial curve. e impulse reponse of the renal pelvis was obtained by deconvoluting the pelvic curve by the parenchymal curve. Accounting for the regional variations of tracer retention enabled the authors to di erentiate between patients with and without pelvic obstruction, and proved to be an e cient tool to assess the renal function. Authors regret that the spatial resolution of the gamma-camera did not enable to distinguish between cortical and medulary structures. In , a Fourier-based deconvolution technique, relying on the same computational technique as [ ], was proposed by Alderson et al. [ ] to quantify le -to-right cardiac shunts from time-activity curves from scintigraphy following bolus injection of [ m Tc]pertechnetate in a canine study. e shunt quanti cation technique used the gamma-tting technique rst presented by Maltz and Treves [ ], and was preceded by either low-pass ltering in the frequency domain, or smoothing of the deconvolved curves in the time domain, in order to reduce the high frequency components, and thus the oscillations observed in the obtained impulse responses. e ability of the technique to estimate the impulse response in case of either prolonged or fragmented bolus was investigated. e technique was able to signicantly improve the accuracy of the shunt estimation in case of prolonged bolus, and while time-domain smoothing improved the accuracy of the estimates, frequency domain ltering proved more e cient. However none of the techniques yielded satisfactory results in case of fragmented bolus. . . X-ray imaging In , Axel [ ] proposed a deconvolution technique to estimate the mean transit time of contrast-enhanced X-ray CT acquired in the brain, where iodinated tracers do not di use in the extravascular space. Because the only considered parameter under investigation is the mean transit time, the authors consider the shape of the impulse response unimportant, with only a measure of its width being required. Authors demonstrates this by estimating the impulse response as an exponential function, a Fermi function or a step function. e Fermi and the step functions yielded similar estimations of the mean transit time, however the exponential function yielded lower values of the parameter. Additionally, the Fermi and step function achieved slightly better ts than the exponential function, but the di erence was not found signi cant. e method previously in presented in [ ], was used as a reference. e mean transit time was de ned as the di erence of the rst moments of the tissular and vascular curves following gamma-variate tting. e mean transit time estimated using this method seem to best agree with the mean transit time values obtained using the exponential impulse response. e author assumes that the impulse response function in most tissues are likely to have a squared shape with rounded corner, which the Fermi function can easily approximate. ey comment on the insensitivity of the method to noise compared to transform-based methods, as well as recirculation without needing curve tting and extrapolation. In , Yeung et al. [ ] presented a single photon absorptiometry method to quantify arterial concentration of iodinated contrast agents using blood sampling, and investigated it through both preclinical and clinical experiments. A deconvolution method was used for . . DECONVOLUTION METHODS the quanti cation, it was proposed by the research group the same year, however while the paper was submitted it seems it has never been published. e only given details are that it uses a non-negative least square algorithm [ ] to ensure that only physiologically relevant positive values can occur in the impulse response function, and that estimated function must be smooth. In , Cenic et al. [ ] reused the deconvolution method to quantify dynamic perfusion computed-tomography following bolus injection of iopamidol in white rabbits. ree regions of interest were de ned in the brain, i.e. two in the parietal regions, and one in the basal ganglia. Multiple arteries were present in the image, i.e. ear arteries, postcommunicating arteries, and internal carotid arteries. In order to limit the impact of partial volume e ects, the chosen artery was the one with the largest cross section seen in the image which resulted in the highest peak average intensity, typically an ear artery. Additionally, partial volume e ects were corrected by rst tting a gaussian curve to the non-contrast pro le of the artery to obtain its standard deviation. is parameter was used to estimate a partial volume scaling factor, using a formula resulting from phantom experiments of tubes with known inner diameters and tracer concentrations. Finally the arterial intensities were divided by this factor to correct for partial volume e ect. Non-contrast scan intensities were subtracted from the average curves in the three regions of interest, as well as from the average arterial curve. Deconvolution allowed the estimation of the impulse response, which was then used to derive the mean transit time as the ratio of the area under the curve to its plateau value [ ]. e cerebral blood volume was de ned as the ratio of the area under the enhancement curve inside the tissue of interest to the area under the arterial input function [ ]. Finally, the cerebral blood ow was de ned using the central volume principle as the ratio of the cerebral blood volume to the tracer mean transit time [ ]. For comparison

,

  Wintermark et al. [ ] compared the results of two deconvolution methods to quantify perfusion computed-tomography exams, using stable xenon computed-tomography as a reference. e rst deconvolution is performed using a commercial so ware, known as CT Perfusion (GE Medical Systems). e so ware basically uses an improvement of the method proposed by Cenic et al. [ ], allowing processing of pixel-by-pixel parametric maps with improved stability, by means of a singular value decomposition [ ] enforcing noise ltering through thresholding of the singular values. Additionally, the computation time is considerably reduced using this method, which makes it usable in clinical routine. e second method uses a least squares minimization technique with improved stability, taking advantage of prior rank determination through QR decomposition with pivoting, i.e. decomposition of the matrix of shi ed observations as the product of an orthogonal matrix, an upper triangular matrix, and a permutation matrix [ ]. e authors report a good agree-

  CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW cal aspects of the deconvolution method were presented in more details in [ ]. is study actually investigates the use of two contrast agents, a classical iodinated contrast agent that di uses in the extravascular space, i.e. iobitridol, and a macromolecular iodinated contrast agent that mostly remains in the intravascular space with little capillary leakage, i.e. carboxyl dextran. e authors report larger estimates of the mean transit time using the macromolecular contrast agent. Moreover, when comparing hepatic perfusion indices in normal liver tissue with metastatic tissue, they observed much larger values in the metastasis, as well as a large reduction in hepatic blood ow, and lower tissue blood volume. e method proved robust to recirculation, which alleviates the necessity for gamma-variate tting to get rid of recirculation. is is particulary important in the liver, since arterial and portal components partially overlap. In , Kudo et al. [ ] investigated variants of the singular value decomposition deconvolution method, rst developed for delay-insensitive deconvolution of magnetic resonance data, and their sensitivity to the tracer time delay for quanti cation of contrast-enhanced computed tomography data. e authors compared regional blood ow, regional blood volume, and mean transit time estimates of the standard singular value decomposition algorithm, to the estimates of a delay-insensitive block-circulant deconvolution method introduced by Wu et al. [ ], as well as a delay-corrected deconvolution method proposed by Ibaraki et al. [ ].

  al. [ ] proposed a review of the contrast agents usable for perfusion imaging using magnetic resonance, depending on the property involved, i.e. relaxivity or susceptibility, and adressed the di erent perfusion quanti cation techniques for contrastenhanced magnetic resonance data, o en inspired by the advances in the other perfusion imaging modalities. e main goal the authors wanted to reach was the estimation of tissue blood ow, and they rely on the central blood volume principle established by Stewart [ ], which de nes tissue blood ow as the ratio of the tissue blood volume to the mean transit time. ey recommended estimating the mean transit time through deconvolution, suggesting the use of Fourier-based methods [ ], and proposed the tissue blood volume as the ratio of the areas under the tissular and arterial curves, as proposed by Lassen et al. [ ] for indicator-dilution theory, and Axel [ ] for perfusion computed-tomography imaging. In , Rempp et al. [ ] investigated susceptibility Gd-DTPA -enhanced MRI to quantify cerebral perfusion in a clinical study with twelve healthy patients with intact blood-brain barrier. A dual FLASH sequence was used, allowing interleaved acquisition in two parallel

  In , Østergaard et al. [ ] reviewed the various deconvolution approaches which had already been applied for the quanti cation of perfusion using contrast-enhanced imaging using simulated magnetic resonance data, especially in the case of cerebral imaging where the tracers remain intravascular thanks to the blood-brain barrier. e methods were divided between model-based and model-free approaches, the former referring to a onecompartment model with exponential impulse response [ ], the latter referring to either transform approaches through the Fourier method [ ], algebraic approaches through a matrix-based method [ ] with an additional regularization term enforcing decreasing impulse response, or SVD approaches through a classical method with ltering of components which singular values are close to zero [ ].

  shorter mean transit time values, which the authors justi ed as a consequence of the arterial input function varying slowly compared to the mean transit time, which determines the time-scale of the impulse response to be estimated.e second part of the study [ ] confronts the models assessed through simulations to experimental contrast-enhanced magnetic resonance data acquired in six healthy subjects, as well as in four clinically relevant cases, following automatic bolus injection of paramagnetic contrast-agents, i.e. Dy-DTPA-BMA, and Gd-DTPA respectively. An additional model-based approach was introduced in the experimental part of the study, i.e. the decreasing part of a Gaussian was proposed as an ad-hoc intermediate between the exponential (one-compartment model) and the square (plug-ow model) impulse responses. All image sequences were ltered using a × averaging kernel prior to deconvolution. e arterial input function was estimated in pixels located around the middle cerebral artery. e timeconcentrations curves used for quanti cation were not converted in actual units, instead a constant hematocrit throughout the brain was assumed, and white matter was used as a CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW reference. Additionally, because time-delays tended to bias the estimation of blood ow in the simulation study discussed above, the in uence of this parameter was investigated in multiple slices by comparing the estimation of the arterial input function along the middle central artery. is study revealed very little dispersion and delay along major arteries, and

  Axel [ ] for quanti cation of perfusion computed-tomography, in Jerosch-Herold et al. [ ] proposed a Fermi function to model the myocardial impulse response in an attempt to quantify the myocardial perfusion reserve from the rst pass of a bolus of Gd-DTPA imaged using a multislice MRI technique. Interestingly, the convolution operation was performed as a product in the frequency domain, following Fourier transform, but the non-linear least-square tting was performed in the time-domain, following inverse Fourier transform of the convolved functions. Since two measurements are necessary to estimate myocardial perfusion reserve, two magnetic resonance sequences were acquired to assess this parameter in nine patients with suspected microvascular disease. e

  , Mischi et al. [ ] proposed a semi-quantitative method to assess pulmonary blood volume in cardiac contrast-enhanced exams. e semi quantitative aspects of the methods are discussed in details in Section . . . However, prior to being tted by either the local density random walk or rst passage time models, the time-intensity curves were deconvolved CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW using a Wiener lter to correct for the non-instantaneous bolus injection. Because the bolus was injected using an automatic system, the arterial input function was not estimated from image measurements in this study, instead it was modeled as a . second rectangle function corresponding to the settings of the electronic injector. e ability of the Wiener lter to accurately recover the original or simulated time-intensity curve from noise-corrupted convolved curves was assessed. e two investigated methods yielded stable and reproducible volume estimates over the investigated range of ows, however the local density random walk was generally more accurate, and the rst passage time model tended to overestimate the distribution volume in their experiments. In , Mischi et al. [ ] used the same deconvolution method, but this time applied it to the estimation of the impulse response of a vascular system located between two measurement sites. In particular, the goal of the study was to propose an accurate measurement of the forward ejection fraction, even in case of leakage through the mitral valve, but avoiding the invasive intra-ventricular injection and replacing it by intravenous injection. To meet this goal, the authors propose estimating the impulse response from the attenuationcorrected time-intensity curve measured in the le ventricle, and using the signal measured in the le atrium as an input function to their deconvolution algorithm. e method yielded a noisy estimation of the le -ventricular impulse response. Assuming a one-compartment model for the le ventricle, an exponential curve was tted to its downslope, from which the forward ejection fraction can be derived. e accuracy and precision of the deconvolution method was assessed through simulations with varying signal to noise ratio and forward ejection fraction, and good agreement between the simulated and estimated values of both blood volume (R = . ) obtained using the methodology previously described in [ ] (see Section . . ). Experimental contrast-enhanced ultrasound data were acquired in twenty patients with leaking mitral valve following the injection of a small quantity of microbubbles, therefore limiting commonly observed saturation artifacts. e forward ejection fraction was estimated with and without deconvolution, i.e. tting the exponential function to either the deconvolved or original le -ventricular time-intensity curves, and the results were compared to the ejection fraction estimated using an established bi-plane echocardiographic method [ ]. e results without deconvolution clearly underestimated the ejection fraction . . DECONVOLUTION METHODS

  , Gauthier et al. [ ] used a deconvolution method based on singular value decomposition, but enforcing Tikhonov regularization to reduce the rapid oscillations in the impulse response function, and nally the impulse response was tted by the explicit model proposed by Elie et al. [ ] and presented in Section . . . e blood ow and the blood volume were then derived from the model tted to the estimated impulse response, and were respectively de ned as the peak value, and the area under the curve. e additional regularization term was weighted, and the determination of this weighting parameter was determined using the L-curve method described in [ ]. e reproducibility of the method was assessed through both in vitro and in vivo experiments, and was compared to the semiquantitative parameters extracted from the time-intensity curve tted with the same explicit model, without deconvolution. e in vitro experiments consisted in ve acquisition following bolus injection of microbubbles in phantom composed of a feeding pipe and three interwined pipes, injecting two di erent volumes of contrast agent.

  , Jirik et al. [ ] proposed a contrast-enhanced ultrasound acquisition scheme referred to as bolus-and-burst for it combines bolus injection and destruction-replenishment, as well as a blind deconvolution method, speci c to their custom acquisition scheme, for si-CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW multaneous estimation of the arterial input function and the tissue impulse response. As the name suggests, the acquisition is divided in two main parts. e rst part consists of a classical low mechanical index image acquisition to image the passage of a bolus of microbubbles injected intravenously. en, at the end of the bolus-phase, when the concentration of contrast agent in the tissue decays rather slowly, a sequence of high mechanical index pulses are sent to disrupt the circulating microbubbles. e second part is another classical low mechanical index acquisition to image the replenishment of contrast agent in the tissue. A blind deconvolution algorithm is then applied to the average time-intensity curve in the tissue of interest. e quanti cation is divided in two parts as well. First a rough estimation of the tissue impulse response is performed from the destruction-replenishment curve using the quantitative formulation of the popular monoexponential model proposed by [ ]. is initial impulse response estimate is then used to roughly estimate the arterial input function by deconvolution of the bolus-phase. In the second part, the arterial input function and the tissue impulse response are re ned by multichannel deconvolution, considering the two parts of the acquisition are independent measurements with a common impulse response. To ensure physiologically relevant and smooth estimation of the arterial input function, a positivity constraint as well as a Tikhonov regularization are enforced. e authors investigated several techniques to address the problem of absolute quanti cation, and retained the use of a population-based scaling constant resulting from previous acquisitions in feeding arteries with lower contrast agent doses, as proposed by Taxt et al. [ ] for MRI. e estimates of the mean time obtained using proposed method proved less sensitive to the number of pixels in the regions of interest than the estimates of the destruction-replenishment method in terms of accuracy and precision. . Compartmental models . . Generalities e major assumption underlying compartmental modeling used to model contrast agents time-concentration curves is that the tissue under investigation consists of three well-mixed . . COMPARTMENTAL MODELS

Figure - :

 - Figure -:Block diagrams and rst-order di erential equations of compartmental models with (a) one tissue compartment, (b) two tissue compartments, (c) three tissue compartments. C A (t) is the contrast-agent concentration in arterial blood, C V (t) is the tissue vascular concentration, C I (t) is the tissue interstitial concentration, C C (t) is the tissue cellular concentration. K is the unidirectional transfer rate of contrast-agent from blood to tissue vascular space and is related to blood ow and capillary permeability. k is the unidirectional transfer rate of contrast-agent from the tissue vascular space to blood and is de ned as K V D , where V D is the contrast-agent fractional distribution volume. Similarly k and k are the unidirectional transfer rates of contrast-agent between tissue vascular space and interstitial space, and k and k are the unidirectional transfer rates of contrast-agent between interstitial space and intracellular space. K is classically expressed in milliliter of blood per minute per milliliter of tissue (mL.min -.mL -), and the other rate constants k , k , . . . , k in fraction of contrast-agent per minute (min -).

  Figure -shows examples of classical compartmental tissue models with increasing number of compartments, along with the associated rst-order di erential equations de ning the transport of contrast-agent between compartments in terms of transfer constants. Compartmental models estimate the tissue impulse response of a tissue making assumptions on its shape, depending on the characteristics of the tissue, as well as the characteritics of the contrast agent, i.e. ow limitation, membrane limitation, transport mechanism, binding, excretion, metabolism [ ]. Resolving the di erential equations yields explicit parametric formulations for the tissue impulse response, and allows direct estimation of the phys-CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW iological parameters. e explicit formulation of the impulse response function improves the stability of the estimation by reducing the number of degrees of freedom compared to deconvolution. Early use of compartmental models to t indicator-dilution curves was based on the Fickś principle that relates the blood ow to the arterial and venous measurements [ , ]. In , Kety and Schmidt [ ] derived a method to estimate cerebral blood ow from arterial and ven'ous blood sampling during inhalation of nitrous oxyde, and assessed it in a clinical experiment with eleven subjects. e method corrected for the rise of the timeconcentration curve using a xed ratio of the amplitude of the monoexponential model tted to the decay phase. Following these developments, the authors proposed a one-compartment model based on the same principle to assess fractional blood ow and fractional blood volume assuming a perfectly di usible tracer [ ].In , Renkin [ ] studied capillary permeability to K in skeletal muscles using another di usion model based on Fick's law through in vivo radioactivity measurements of the arterial and venous blood. e authors were among the rst to use the permeability surface product as a combined parameter of the model, and established its correspondance to the maximum capillary clearance possible for a given substance assuming an in nite ow. However they proposed a method to quantify the clearance at any ow rate assuming the permeability surface product is known. ey suggested that variations of the estimated permeability surface product were due to hemodynamic factors following either of the three following pattern: arteriovenous shunting, inhomogeneous ow distribution, or ow-dependant perfusion network. ey were however unable to determine which pattern was actually explaining the correlation between blood ow and permeability surface product in their data. In , Crone [ ] developed the indicator di usion method that relates the capillary permeability to blood ow, capillary surface area, and initial extraction which corresponds to the fractional reduction of the arterial concentration, however litterature values for the two rst parameters were used and only the initial extraction was estimated. e authors performed blood sampling experiments in dogs with injection of Evan's blue dye or a combination of inulin and sucrose to study the capillary permeability of these substances in multiple tissues, i.e. brain, kidney, liver, lung, hind limb, by means of invasive catheterization.. . COMPARTMENTAL MODELS eir study contradicts the results of Renkin [ ], and suggest that the discrepencies are due to a conceptual di erence between the two studies regarding the de nition of 'extraction '.

  In, Gjedde [ ] used a formula from Crone [ ], and the previous developments of the group presented in [ ], to derive two integral methods to estimate rate constants of glucose from blood to brain tissues in rats using scintigraphy measurements of blood sampled following the injection of a mixture of a H-labelled substance (D-glucose, L-glucose, D-mannitol, or sucrose), with C-butanol and InCl. e methods consider the exchanges between the capillary and extracellular spaces, as well as the metabolic pools re ecting consumption of glucose by brain cells. e rst method estimated the initial rate constant as the CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW di erence of the total amount of tracer in the brain and the amount of plasma in brain twenty seconds a er injection, normalized by the area under the arterial curve up to twenty seconds.e second method exploited dynamic measurements to derive a graphical method, allowing the conjoint estimation of initial rate constant, and plasma volume, respectively as the slope, and intercept in the linear regression analysis of the ratio of the total amount of tracer in the brain to the arterial concentration, and the ratio of the total amount of tracer that passed in the artery to the arterial concentration. eir experiments led them to the conclusion that the two methods estimated two di erent transfer rates, re ecting at least two transport mechanisms, i.e. a high a nity system with low capacity, and a low a nity system with high capacity.Similarly, Patlak et al. [ ] proposed a graphical analysis method in to estimate transfer constants across the blood-brain barrier for irreversible tracers. e method is able to estimate the in ux rate in any membrane system composed of at least one region with reversible exchanges without metabolisation where the tracer rapidly equilibrates with the plasma and one region with irreversible exchanges. e methods consists in ploting the ratio of the curve in the tissue of interest to the plasmatic curve vs. the ratio of the integral of the plasmatic curve to the plasmatic curve, and tting a a ne model to the linear portion of the resulting curve that corresponds to the steady-state of the system. e slope of the tted linear model yields the in ux rate from plasma to tissue, while the intercept yields an upper bound to the reversible plus plasmatic fractional volumes. Patlak and Blasberg [ ] then generalized the method to analyze time-intensity curves using a reversible reference tissue instead of the arterial measurement, yielding an estimate of the ratio of the in ux rate of the tissue to the sum of the reversible and plasmatic fractional volumes. ey also addressed cases when tracer binding is incomplete in the irreversible compartment, for both arterial and reference measurements, and presented speci c applications of the method in the case of one reversible region (with or without reÂŋversible binding) and one irreversible region. In , Logan et al. [ ] proposed a graphical analysis method to analyze the kinetics of reversible tracers, binding reversibly to receptors or enzyme, and applied it to estimate the parameters of the three-compartment model to characterize the binding of a radiolabeled . . COMPARTMENTAL MODELS cocaine complex in a PET study.

  case of noisy data, this linear formulation of the compartmental modeling can yield underestimated distribution volume estimates because the errors are cumulative in the integral terms [ ]. Logan et al. [ ] proposed a strategy to remove the bias in the graphical analysis method, using the generalized linear least-squares algorithm proposed by Feng et al. [ ] to determine the distribution volume directly for one-compartment models or as a smoothing technique prior to graphical analysis for more complex model structures. e authors demonstrated the improved accuracy of the new estimation method, applied to evaluate either the distribution volume using an arterial input function, or the distribution volume ratio using a refrence tissue, in PET experiments following injection of C-labelled raclopride or d-threo-methylphenidate. To estimate the distribution volume, an arterial in-CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW put function was measured (for both tracers) from sampled blood. In , Hawkins et al. [ ] used a two-compartment model based on the equation of the graphical analysis with a vascular component to evaluate the blood-brain barrier permeability in human brains using PET exams and Ga-EDTA as a tracer. However they directly estimated the parameters of the model using a non-linear least-squares algorithm to t the model, a er resolution of the di erential equations that relates the curve in the tissue of interest to the arterial blood sampling measurements. e non-linear resolution of the equations yields a convolution formulation assuming a monoexponential decay in the tissue impulse reponse. e method was used to estimate in ux and out ux rate constants, as well as the blood volume fraction from which estimates of the regional cerebral blood volume corrected for the di erences in hematocrit were derived, and compared it to direct estimation obtained using a three-compartment model previously described by Phelps et al. [ ]. In , Ziegler et al. [ ] compared a dual-input and a single-input one-compartment model regarding their ability to quantify blood ow in dog liver using dynamic PET data following injection of a bolus of O-labelled water. e dual-input model accounts for the arterial and portal supplies of the liver, using direct measurements resulting from blood sampling in the aorta and in the portal vein. In the single-input model the determination of the portal supply is part of the tting procedure, and is derived from the arterial measurement resulting from blood sampling by considering a gut compartment, based on a onecompartment model, which distribution volume was xed and derived from post-mortem experiments in dogs. Additionally the models correct for time-delays and dispersion using the method presented in [ ]. Both models make the assumption that the blood supplies di use instantaneously from vasculature to the liver tissue, and that the volume of distribution of the tracer is the same for arterial and portal blood. For both models, the volume of distribution in the liver was not estimated but xed for all experiments using the average value of the ratio of the arterial to liver concentrations in the late phase, assuming an equlibrium was reached. e authors investigated the impact of this parameter on the estimation process, and reported a high sensitivity of liver blood ow, dispersion and t accuracy to the xed distribution volume value. e single-input model additionally enforces a xed ratio of gut tissue to liver tissue volume. Poor agreement of the blood ow estimates of the dual-input and single-input models was found with microsphere measurements, however the single-input model performed slighlty better. In , Blomqvist [ ] proposed a linear formulation of a three-compartment model, accounting for both metabolized and unmetabolized tracer, presented by Raichle et al. [ ] obtained by rearranging and integrating twice the second-order di erential equations relating the arterial concentration to the tissue concentration. e method was used to derive functional maps from dynamic PET data of various tracers labelled with C, i.e. Cmethionine, C-deoxyglucose, C-glucose. e authors did not report the method used to acces the arterial tracer concentration. e three-compartment model was composed of a vascular compartment, and two extravascular compartments, one representing the extravascular space exchanging with the plasma, where the tracer is unmetabolized, and the other representing the metabolized compartment. e authors estimated the tracer transfer rate

  e authors di erentiated micro parameters, i.e. the actual rate constant parameterizing the model, from macro parameters, i.e. global parameters de ned as a function of the model rate constants, and showed that all the micro parameters of a perfusion model cannot be estimated individually, instead one should interpret the macro parameters in terms of the micro parameters, as the estimation of the latter was shown to be more stable. Additionally, CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW investigating models using a reference tissue region they showed that the macro parameters are expressed relatively to the reference region, and proposed a correction for the bias in graphical estimation methods when the contribution of blood in the reference and studied tissues cannot be neglected. In , Watabe et al. [ ] also proposed a review of compartmental models for quanti cation of dynamic PET data, especially addressing the models for the quanti cation of binding tracers. e classical model counts four compartments, i.e. an arterial compartment exchanging with a free compartment, that exchanges with a speci c binding compartment, and a non-speci c binding compartment. e speci c binding compartment is generally the compartment of interest when studying binding tracers. If the non-speci c binding com-

  In, Yeung et al. [ ] used a two-compartement model to quantify the blood-to-brain impulse response of iopadimol, as well as plasmatic blood volume, using dynamic X-ray CT in a clinical study of brain tumors. e two compartments correspond to the cerebral plasma space and the extravascular extracellular space, exchanging tracer with each other through the blood brain barrier assuming permeability limiting conditions. e arterial measurements resulted from blood sampling measurements in a radial artery. e clinical experiments revealed the ability of the estimated parameters to di erentiate between healthy white and gray matter, tumor tissue, and metastatic tissue. Indeed, higher values of both parameters were found in tumors and metastases. Additionally, parametric maps of blood to brain impulse response and plasmatic blood volume were derived from the dynamic perfusion images, revealing the tumor functional heterogeneity.In, Tsushima [ ] used a graphical analysis technique based on the work of Patlak et al. [ ] to estimate the fractional blood volume and the contrast clearance rate per unit volume of tissue (which is closely related to glomerular ltration), using perfusion X-ray CT of healthy (n = ) and diabetic (n = ) human kidneys following the intravenous injection of a bolus of iodinated contrast agent. e model uses an image-based arterial input function de ned as the average enhancement curve in the manually segmented abdominal aorta region. e authors reported a negative correlation of the patient age with the fractional clearance rate, but not with the fractional blood volume despite a tendency to decrease with age. A linear relation was found between X-ray CT estimates and the creatinine clearance rate obtained from urine sampling, which was the clinical reference method to estimate the glomerular ltration rate. In , Harvey et al. [ ] used the same method to estimate the contrast clearance rate and the fractional blood volume of various tumor types in an attempt to assess tumor response to radiotherapy. Acquisitions were performed before treatement for baseline, then early and late response exams were acquired one or two, and six to twelve weeks a er treatement, respectively. Authors showed an increase in the clearance rate as well as in the fractional blood volume in the early response exams, indicating an hyperemic response to radiotherapy. Later exams revealed a decrease in parameter values compared to the early response, but still exhibited larger values when compared to the baseline experiments. In , Materne et al. [ ] used a dual-input compartmental model, similar to the model proposed by Ziegler et al. [ ] for PET, to characterize liver perfusion from dynamic contrast-enhanced X-ray CT experiments accounting for both arterial and portal blood supplies. e proposed method accounted for time-delays, but not for dispersion as originally proposed by Ziegler et al. [ ]. e arterial and portal curves were extracted directly from the dynamic images, which was not possible with PET data due to the poor spatial resolution of the imaging modality. Two models were actually considered, in the rst one only the liver tissue component was considered, and in the second one the measured kinetics was considered a linear combination of the tissular, arterial, and portal components. Hepatic, arterial, and portal perfusion indices were derived from the estimated model parameters.

  In, To s et al. [ ] introduced compartmental models for the quanti cation of perfusion parameters using di usible tracers in dynamic contrast-enhanced MRI, as well as a set of standardized notations for these parameters. e authors adapted the methods developed for quanti cation of PET based on an arterial input function[ , , ] to MRI assuming the contribution of intravascular contrast-agent to signal intensity can be ignored. ey then derived a generalized model parameterized by the transfer constant from plasma to tissue noted K trans , the extravascular extracellular space volume fraction noted V e , and de ned the rate constant as the ratio of these two parameters, noted k e p . e transfer constant is equivalent to the permeability surface area product under permeability-limited conditions, to blood ow under ow-limited conditions, and to extraction ratio under mixed conditions. ey also showed that the rate constant parameter is the decay rate of the exponential impulse response by solving the rst-order di erential equation of the generalized model.is paper is much theoretical, however it assumes the arterial input function is estimated from the dynamic MR images, which measures whole blood contrast-agent concentration (as opposed to blood plasma) and shoud therefore be corrected for the hematocrit. is paper became a standard in the eld of perfusion imaging for MRI, for it introduced a set of clear and general notations that can be used to model the kinetics of any di usible contrast agent.In, Murase [ ] proposed a linear formulation of the generalized kinetics model presented by To s et al. [ ], additionally accounting for the vascular component of the signal. e main di erence with the non-linear formulation originates in the resolution of the rst-order di erential equation, for which the author integrates both sides of the di erential equation. e tracer concentration in the tissue of interest is de ned as the linear combination of the integral of the arterial measurement, the integral of the tissue measurement, and the arterial measurement itself, respectively weighted by the sum of the blood to tissue transfer constant with the product of the tissue to blood transfer constant with the volume . . COMPARTMENTAL MODELS fraction of blood in the tissue, the tissue to blood transfer constant, and the volume fraction of blood in the tissue. e non-linear and linear resolution methods were compared through simulations in terms of accuracy and precision of the estimates, but also in terms of sensitivity to the sampling frequency. e linear method was extensively faster, and more accurate in low to moderate noise conditions, and less sensitive to the sampling interval. Additionally, the linear resolution method does not require initial values for the parameters, making the estimation method more robust. In , Balvay et al. [ ] investigated the sensitivity of two compartmental models for quanti cation of contrast-enhanced MRI to the duration of the acquisition using a new criteria to assess the quality of t of the models in a preclinical study in mice with human prostatic tumors implantes subcutaneously. A one-compartment model accounting for tissue perfusion only, and a two-compartment model accounting for both perfusion and permeability were tted non-linearly to the mean enhancement curve in the tumor region using an image-based plasma measurement from the heart le ventricle. e one-compartment model was not able to accurately t the full-length mean enhancement curve in the tumor region and yielded biased tissue blood ow and fractional blood volume, but it was able to accurately t shorter curves where permeability can be neglected and yielded parameter estimates comparable to those of the two-compartment model for full-length data. In , Kovar et al. [ ] faced limitations regarding the duration of their contrastenhanced MRI acquisitions, and proposed a compartmental model using the contrast-agent kinetics in a reference tissue, such as a muscle, as well as litterature values of perfusion rate, extraction fraction, and extracellular volume in the chosen reference tissue to estimate the vascular input function. is scheme was developed to reduce the high sampling frequency necessary to obtain a well resolved arterial input function, while avoiding invasive procedures, e.g. blood sampling. ey then used the estimated input function to estimate the product of the perfusion rate and the extraction fraction in the tissue of interest. e equations of the model were derived from the di erential form of the model proposed by Kety [ ]. e method was investigated in a preclinical magnetic resonance study of rats with implanted mammary and prostate tumors to assess the distribution of a di usible paramagnetic tracer. CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW Later, in Yankeelov et al. [ ] proposed a method similar to that of Kovar et al. [ ], they however derived their equations from the integral form of the model proposed by Kety [ ] and used the standardized notations of To s et al. [ ]. is formulation avoids the estimation of the arterial input function, and litterature values of the transfer constant from plasma to tissue and the distribution volume of the tracer in the reference tissue were used to directly derive estimates of these parameters in the tissue of interest. e model was investigated through simulations to reveal the sensitivity of the model parameters to noise and to reference values. e distribution volume parameter proved robust to noise using the proposed method, however the transfer constant from blood to tissue decreased with rising noise amplitude. e authors also discussed the possibility to avoid the use of litterature values characterizing the reference tissue by de ning relative parameters as the ratio of a parameter in the tissue of interest to its value in the reference tissue. In , Faranesh and Yankeelov [ ] proposed a modi cation of the model presented by Yankeelov et al. [ ] to account for the intravascular tracer contribution to signal intensity in either or both of the studied and reference tissues. Given litterature values of the vascular parameters characterizing the reference tissue, the arterial input function was estimated using two di erent formula depending on whether the vascular component was considered. e estimated arterial input function was then used to estimate the perfusion parameters of the studied tissue assuming a two-compartment model, with or without a vascular compartment, and the linear formulation of the model introduced by Murase [ ]. Simulations were conducted incorporating the vascular term, and the accuracy and precision of the parameters estimated using the modi ed reference region model were assessed using the model accounting for intravascular tracer in both the studied and reference tissues as a gold standard. e simulations revealed that accounting for the vascular term in both the studied and reference tissues made the estimation exponentially dependent on the noise amplitude, although accounting for the fractional plasma volume in the studied tissue only yielded inaccurate estimation of all parameters. e e ect of the sampling frequency on the accuracy of the vascular parameter estimation revealed complex behavior. e authors gave recommendations on whether the vascular term should be incorporated in the model depending on the information that can be found in the litterature about the expected range of . . COMPARTMENTAL MODELS vascular volumes in the studied and reference tissues, as well as on the ability of the sampling frequency to capture the vascular peak in the reference tissue kinetics. In , Cárdenas-Rodríguez et al. [ ] derived a linear formulation of the reference region model when the contribution of vascular tracer to signal intensity can be neglected in both the studied and reference tissues, as proposed by Yankeelov et al. [ ]. e linear formulation is obtained by integrating both sides of the equation derived by Yankeelov et al. [ ] instead of solving it by part. e concentration of tracer in the tissue of interest is expressed as the linear combination of the tracer kinetics in the reference tissue, of its integral, and of the integral of the kinetics in the tissue of interest. e weights of this linear model are respectively the relative blood to tissue transfer constant, the ratio of the transfer constant in the tissue of interest to the fractional distribution volume, and the rate constant in the tissue of interest. ese weights can therefore be estimated using a linear least-square tting algorithm. e authors compared the linear and non-linear estimation methods through simulation experiments, and studied the accuracy and the precision of the estimates, as well as the sensitivity of the parameters to the sampling frequency, the noise amplitude, the values of the parameters, and the shape of the arterial input function used to simulate the studied and reference kinetics. Expectedly, in addition to alleviate the need for initialization values, the linear resolution method proved more accurate, especially for low sampling frequencies and high noise amplitudes.

  and irregular in size, yielding poor resistance to injection and ultrasound pulses, as well as inconsistencies of the ultrasound signal. All these characteristics made the quanti cation process of early studies extremely di cult.Additionally, early contrast-enhanced imaging techniques were not contrast speci c and exhibited low sensitivity, especially in the capillaries. Moreover, contrast-enhanced ultrasound data was o en acquired using a high mechanical index, causing the disruption of a large number of microbubbles. Intermittent, or transient, imaging accounted for the disruptive nature of the high mechanical index but intrinsically forbit real-time perfusion imaging.While microbubbles increased the sensitivity of Doppler techniques, allowing visualization of small and deep vessels, it fails to image the capillaries, and was generally of poor help to characterize moving structures. Real-time non-destructive contrast-speci c ultrasound imaging only arose with the development of pulse inversion sequences using low mechanical index in the late s.Most quanti cation approaches developed for contrast-enhanced ultrasound rely on indicatordilution theory. One of the main assumption underlying indicator-dilution theory is that the mass of the indicator is conserved throughout the experiment. In vitro studies usually ap-. . DISCUSSION

  quantitative parameters using a compartmental model using multiples enhancement curves, one should consider time-delay parameters. Indeed, time-delay parameters are o en overlooked in the litterature, however they play a crucial role in the estimation of perfusion parameters as tting a model without a time-delay parameter or prior delay correction can induce a bias in the estimation. A few methods were presented as delay-insensitive, some other methods actually include time-delay parameters in their equations, and commercial CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW so wares certainly correct for delay, but to our knowledge no study reported on the absolute or relative values of such parameters, or exploited parametric maps revealing the timeline of the bolus arrival in the studied tissue.A erword e development of quanti cation methods o en came along with the development of the imaging modality itself. Semi-quantitative approaches are the most intuitive approaches, they are generally used to characterize perfusion data in the early stage of the imaging technique, they are however subject to inter-exam physiological and experimental changes. en physiological parameters are o en derived from semi-quantitative parameters characterizing the tracer kinetic in the tissue of interest and in an artery, they however su er from the same limitations as semi-quantitative parameters, but are also a ected by the di culty of estimating the arterial curve.Ultrasound contrast agents are not anymore in their early development stages, on the contrary they reached a point where their behavior is well understood and where they can be used routinely for some clinical applications. While the ultrasound imaging technique is not new at all, it was using analog signal processing for a long period of time. It however reached a turning point with the development of numerical ultrasound scanners, exploiting the development of high-end graphical hardware which allows parallel computing to allow complex real-time signal processing. is turning point suggest future development of the imaging technique, in particular with the joint development of ultrafast plane wave imaging and three-dimensional ultrasound probes.

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  photometric analysis of two-dimension echo-contrast studies. American Journal of Cardiology, ( ): , February .

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  time taken by the tracer to arrive in the tissue of interest on the reproducibility of the estimated perfusion parameter was studied. e experimental setup and the results are reported in Chapter which is the content of the article that we published in Physics in Medicine and Biology [ ]. In Chapter , we propose an extention of this work establishing the relations between the parameters of the di erent models mentioned above, rst theoretically, and then experimentally through a correlation study. N.B. All the references cited in this part of the thesis are pooled together and presented at the end of Part II. Ultrasound has been proposed to monitor tumor therapy, in complement to volume measurements. To assess the variability of perfusion parameters in ideal conditions, four consecutive test-retest studies were acquired in a tumor model of mouse, using controlled injections. e impact of mathematical modeling on parameter variability was then investigated. Coe cients of variation (CV) of tissue blood volume (BV) and tissue blood ow (BF) based-parameters were estimated inside sub-regions of the tumors, comparing the log-normal (LN) model with a one-compartment model fed by an arterial input function (AIF) and improved by the introduction of a time delay parameter. Relative perfusion parameters were also estimated by normalization of the LN parameters and normalization of the one-compartment parameters estimated with the AIF, using a reference tissue (RT) region. A direct estimation (rRTd) of relative parameters based on the one-compartment model without using the AIF was also obtained by using the kinetics inside the RT region. Results on test-retest studies show that absolute regional parameters CHAPTER . QUANTIFICATION OF TUMOR PERFUSION USING DCE-US: IMPACT OF MATHEMATICAL MODELING

  cancer diagnosis and to monitor therapy. Tumor perfusion can be assessed through various imaging modalities, including PET, Dynamic Contrast Enhanced (DCE) MR, CT, and ultrasound (DCE-US). Compared to DCE-MRI, DCE-CT and PET, the main advantages of DCE-US are its real-time, non-ionizing, and cost-e ective characteristics. Moreover, as micro-bubbles do not di use in the extra-vascular space, DCE-US studies re ect only the tissue vasculature. Di erent acquisition protocols are available including bolus and destruction-replenishment during infusion [ ]. e present work focuses on bolus injections, since this acquisition mode is the most widely used [ ]. It is currently recommended for bolus DCE-US studies to estimate semi-quantitative parameters using explicit models, such as the Log-Normal model [ ]. However, it was shown that these semi-quantitative parameters were sensitive to various factors [ ]: scanner-related (e.g. frequency, mechanical index, dynamic range, focal length), patient-related (e.g. blood pressure, tissue motion, physiological interaction), and bubble-related (e.g. bubble type, concentration, preparation, injection) factors. e quanti cation of bolus DCE-US studies thus remains a major challenge. In an attempt to make quanti cation of tissue perfusion less sensitive to external factors, quantitative approaches based on indicator dilution theory and commonly used for PET, DCE-MRI or DCE-CT exams [ , , ] could be applied to DCE-US data. ese methods estimate tissue blood volume and tissue blood ow parameters by performing a speci c de-. . MATERIALS convolution of the tissue time-intensity curve inside the tumor by an Arterial Input Function (AIF) measured in the imaging plane. Similarly to what was done in above cited imaging modalities, a compartmental approach was recently proposed for DCE-US [ ]. Precisely, a one-compartment model was proposed since the contrast agent remains in the blood. is approach de nes a set of admissible curves for the transfer function (mono-exponential functions), and the estimation of parameters is then regularized intrinsically. It can thus be distinguished from blind deconvolution, as recently proposed in DCE-US [ , ]. Indeed, these approaches require establishing strong constraints on the transfer functions, due to their large number of unknown parameters. e present study aimed at comparing di erent modeling approaches and at studying the reproducibility of perfusion parameters in test-retest measurements on a mouse tumor model, acquired using controlled injections [ ]. ree absolute modeling approaches including a Log-Normal model and a one-compartment model without and with a time delay were rst compared. In a second time, relative perfusion parameters were de ned by normalizing the values obtained in the tumor with values obtained in a reference tissue region, and ve relative derived models were studied. (CT ) were used. Tumor fragments ( -mm ) were implanted subcutaneously days prior to the experiment in the right ank of four Balb/C mice. All experiments were conducted in accordance with the institutional guidelines and the recommendations for the care and use of laboratory animals.

  previously described [ ]. In order to further reveal spatial heterogeneity inside the tumor, a regional analysis of the tumor area was performed. Dividing the non-necrotic tumor region into sub-regions according to radial layers and angular sectors (Figure -) provided a good compromise between showing heterogeneity while preserving the signal to noise ratio of the time-intensity curves and the spatial matching of the sub-regions between the four test-retest studies.

Figure - :

 - Figure -:Illustration of the data pre-processing steps. Le : e contours of the tumor and its necrotic core have been overlaid on a contrast enhanced image (in ochre color). e perfused tumor area was divided into radial layers and angular sectors. A reference tissue region (in green color) and a renal cortex region (in blue color) were also delineated. Right: Mean kinetics associated with the non-necrotic part of the tumor, the reference tissue, and the renal cortex.

  cording to anatomical considerations and high values of enhancement (see Figure -). Peak Enhancement (PE) and Time To Peak (T TP) parametric maps were then computed for each pixel of the bounding box. e maximal value of PE (PE max ) and the minimal value of T TP (T TP min ) were extracted. Pixels verifying PE PE max ≥ rPE * and T TP -T TP min ≤ ∆T TP * were considered as part of the artery region (Figure -), where rPE * and ∆T TP * are empirically chosen cut-o values, equal to % and s, unless speci ed di erently. e AIF, C A (t), was computed as the geometric mean of the kinetics inside the artery region and modeled using the LN model ( . ).

Figure - :

 - Figure -: Automated detection of the AIF: parametric maps T TP and PE inside the artery region; segmentation results and associated AIF with: (a) rPE * = % and ∆T TP * = s (in green color); (b) rPE * = % and ∆T TP * = . s (in blue color).

  Figure -illustrates for one speci c mouse (m ) the comparison between the parameters estimated by the di erent models. It shows a high correlation between the volume-based parameters: AUC, rAUC, V , rV AIF , and rV RT as well as a

FinallyFigure - :

 - Figure -: Comparison of the volume-based and ow-based parameters obtained for the four test-retest exams (R , R , R , and R ) of the mouse m : linear regressions between (a) rV RT and AUC, (b) rV RT and V , (c) rV RT and rAUC, (d) rV RT and rV AIF , (e) rF RT and WIR, (f) rF RT and F, (g) rF RT and rWIR, (h) rF RT and rF AIF .

Figure - :

 - Figure -:Boxplot showing the coe cients of variation of blood volume parameters (le ) and blood ow parameters (right) estimated with the aLN, rLN, aAIFd, rAIFd, and rRTd models. For each box, the bold line represents the median value, the bottom and top lines the rst and third quartiles. Dotted lines extend to the most extreme data points which are less than . times the interquartile range. Outlier points are displayed with empty circles. Two groups of parameters were built (horizontal lines below the parameter names) such that there were no signi cant intra-group di erences while there were statistically signi cant inter-group di erences (marked by * ).

  e most commonly used methods require the estimation of an arterial input function. Deconvolution approaches were recently proposed to quantify tissue perfusion in DCE-US [ ].ese approaches estimate the transfer function of the system (depending on a large number of parameters), and to avoid aberrant solutions, this estimation needs to be regularized. Following this idea, the one-compartment model with time delay is a deconvolution depending on three parameters only. When the time delay is set to CHAPTER . QUANTIFICATION OF TUMOR PERFUSION USING DCE-US: IMPACT OF MATHEMATICAL MODELING

  can be di cult to estimate the AIF robustly (seeFigure -). Indeed, AIF measurements in small vessels can be a ected by partial volume e ects, yielding underestimation of the signal intensity. us coe cients of variation deduced from the aAIFd model can be high. Note that some problems could also occur in larger vessels, including non-linearities between concentration and measured signal, and attenuation artifacts [ ]. e use of relative parameters was suggested to overcome the di culties of estimating the AIF for a compartmental model in DCE-MRI [ ]. Clinical studies have reported the interest of estimating normalized perfusion parameters in DCE-US [ , , ]. Using systematically three models (rLN, rAIFd, rRTd), our study reinforces the interest of estimatingrelative parameters. e choice of a reference tissue region is less critical than the segmentation of an artery. Indeed, a larger structure can be used, reducing segmentation errors and the impact of partial volume e ect. Furthermore, as the contrast concentration is lower, the quanti cation errors due to non-linearity are reduced. Kidney regions were initially tested but nally excluded because of the overlap of cortical, proximal tubular and distal tubular compartments. Muscular regions that could be delineated on the four exams were nally chosen.

Figure

  Figure -reveals strong correlations between the di erent parameters computed inside a same sub-region, and a large variation of these parameters according to the tumor subregions.us, whatever the model used, all the ow-based or volume-based parameters can reveal spatial tumor heterogeneity. However, when it comes to the comparison of longitudinal exams, it is crucial to have comparable parameter values. us, the estimation of relative parameters seem to be the most powerful solution, provided that the reference tissue characteristics are not modi ed between exams. Some interesting results have been recentlyshown [ ] for a longitudinal study using a D DCE-US approach. Compared with the D approach, the D approach enables assessment of the whole tumor and should be preferred for tumor monitoring.

  a complement to the work presented in Chapter , and relies on the same experimental data, mathematical models, and notations. It aims at establishing the relations between the semi-quantitative perfusion parameters commonly derived from the Log-Normal model, and the relations between these parameters and the quantitative parameters of the one-compartment model.e relations between parameters were rst established theoretically, and then experimentally through correlation studies.

  Figure -correspond to R θ i θ j coe cients from the linear regressions between pairs of parameters θ i , θ j . Figure -(a) shows the R coe cients between all pairs of parameters derived from the aLN model. It reveals strong linear relationships between some of the parameters, especially median R AUC,PE , median R PE,W IR , median R W IR,WOR .e formal non-linear relationships between these parameters (expressed in Table

  CHAPTER . RELATIONS BETWEEN PERFUSION PARAMETERS: THEORETICAL AND EXPERIMENTAL CONSIDERATIONSshows the formal relations between the relative value of AUC (rAUC) and WIR (rWIR)and the parameters of the aAIFd model. It reveals that rAUC and rV are equivalent, and that rWIR is linearly related to rF in the general case of input function, the proportionality coe cient depending on C A (t). ese theoretical identi cations were con rmed experimentally by Figure-(b) which shows strong linear relationships between volume ( rst rows/columns) and ow parameters (middle rows/columns). For instance, the linear regressions between AUC, V and rV RT all yielded median R values greater than . . e same trend is observed for WIR, F and rF RT . e correlations between volume parameters and ow parameters are medium (median R V ,F < . ). Finally for the time parameters (last rows/columns), there is a high correlation between d and D but there is no correlation between ∆ and the other time delays (median (R ) < . ).

FigureFigure - :

 - Figure -(c) andFigure -(d) show the trends observed on ow and volume parameters when pooling parameters issued from all the exams. ere is a very high correlation (R > . ) between the relative volume parameters rAUC, rV AIF , and rV RT and a high correlation (R > . ) between the relative ow parameters rWIR, rF AIF , and rF RT . Correlations are poor (R < . ) between absolute and relative volume parameters, and medium between absolute and relative ow parameters (R < . ). Since the intra-exam correlations between volume (resp. ow) parameters are high, the correlation over pooled data re ects the inter-exam consistency of linear regression slopes: it is much higher for relative volumes (or ows) than for absolute volumes (or ows). Figure -from Chapter illustrates this trend for one speci c mouse (m ).

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RELATIONS BETWEEN PERFUSION PARAMETERS: THEORETICAL AND EXPERIMENTAL CONSIDERATIONS from dynamic contrast-enhanced T( )-weighted MRI of a di usable tracer: standardized quantities and symbols. J Magn Reson Imaging, ( ): -, . [ ] A Tudorica, H Fang Li, F Hospod, E Delucia-Deranja, W Huang, CS Patlak, and GC Newman. Cerebral blood volume measurements by rapid contrast infusion and T *-weighted echo planar MRI. Magn Reson Med, ( ): -, June . [ ] Stef van Buuren and Karin Groothuis-Oudshoorn. mice: Multivariate Imputation by Chained Equations in R. J Stat So w, ( ): -, December . [ ] Huaijun Wang, Dimitre Hristov, Jiale Qin, Lu Tian, and Jürgen K Willmann. reedimensional dynamic Contrast-enhanced US imaging for early antiangiogenic treatment assessment in a mouse colon cancer model. Radiology, Bonthala Wedam, Jennifer A Low, Sherry X Yang, Catherine K Chow, Peter Choyke, David Danforth, Stephen M Hewitt, Arlene Berman, Seth M Steinberg, David J Liewehr, Jonathan Plehn, Arpi Doshi, Dave omasson, Nicole McCarthy, Hartmut Koeppen, Mark Sherman, JoAnne Zujewski, Kevin Camphausen, Helen Chen, and Sandra M Swain. Antiangiogenic and antitumor e ects of bevacizumab in patients with in ammatory and locally advanced breast cancer. J Clin Oncol, Wei, A R Jayaweera, S Firoozan, A Linka, D M Skyba, and S Kaul. Quanti cation of myocardial blood ow with ultrasound-induced destruction of microbubbles administered as a constant venous infusion. Circulation, ( ): -, February . [ ] omas E Yankeelov, Je rey J Luci, Martin Lepage, Rui Li, Laura Debusk, P Charles Lin, Ronald R Price, and John C Gore. Quantitative pharmacokinetic analysis of DCE-MRI data without an arterial input function: a reference region model. Magn Reson Imaging, , Je rey J Luci, Martin Lepage, Rui Li, Laura Debusk, P Charles Lin, Ronald R Price, and John C Gore. Quantitative pharmacokinetic analysis of DCE-MRI data without an arterial input function: a reference region model. Magnetic resosuperior reproducibility of the relative perfusion parameters estimated by the reference tissue model. However, as discussed in the a erword of the second part of this thesis, the formulation of the reference tissue model used in this study was achieved using a xed value of parameter k R ,i.e. the rate constant characterizing the timeintensity curve in the reference tissue. In the third and last part of this thesis, we propose a linear formulation of the reference tissue model alleviating the need for a xed value of k R [ ], and assess it using both experimental and simulated data. Using the linear formulation for regional analysis, i.e. including multiple regions of interest, a di erent value of parameter k R should be estimated for each region. Such discrepancies in the values of k R can also a ect the value of the other perfusion parameters. is issue is further discussed in Chapter , where a new regularized estimation method ensuring a single value of k R is estimated for all the regions of interest of an exam. e reproducibility of the linear resolution method and of the proposed regularized method are assessed using the same test-retest experiments as presented in Chapter , and compared to the absolute and normalized parameters of the Log-Normal model, i.e. AUC, rAUC, WIR, rWIR. is work was published in the proceedings of the IEEE International Ultrasonics Symposium (IUS) [ ]. e reviewers of [ ] raised an interesting question regarding the impact of recirculation on the perfusion parameters estimated in our contrast-enhanced ultrasound test-retest study. Chapter addresses this issue in depth and studies the impact of recirculating microbubbles on the accuracy and precision of the perfusion parameters through a simulation study. A simple recirculation model was used, it was however able to re ect the various passes of the bolus, as well as its dispersion and attenuation. An intuitive approach to limit the impact of recirculation consist in tting the model to the part of data acquired before recirculation occurs, thus a classical and a cropped version of the Log-Normal model were assessed. Oppositely, compartmental models are build to account for recirculation. e robustness of the linear reference tissue model and of its regularized version to recirculation was assessed. A revised version of this work was submitted to Innovation and Research in BioMedical Engineering (IRBM) [ ].

Figure - :

 - Figure -: Illustration of the data pre-processing steps. Le : e contours of the perfused tumor area have been overlaid on a contrast-enhanced image (in blue color). is area was automatically divided into radial layers and angular sectors as shown by the spiderweb patterns. A RT region (in orange color) was also delineated. Right: Mean TICs associated with the perfused area of the tumor, and the RT.

  resolution of the one-compartment model follows the graphical analysis technique introduced by Patlak et al. for the quanti cation of irreversible tracers in PET. e method, based on compartmental modeling, estimates blood-related physiological parameters by means of linear regression, assuming the arterial input function (AIF) is known [ ]. is linear approach is generalized to reversible tracers [ ].

  model. e t quality was assessed quantitatively, using the fraction of modeled information, FMI, according to [ ]. e reproducibility of the perfusion parameters was then deduced for each sub-region of CHAPTER . REGULARIZED LINEAR RESOLUTION OF A ONE-COMPARTMENT MODEL TO IMPROVE THE REPRODUCIBILITY OF PERFUSION PARAMETERS IN CEUS

Figure

  Figure -displays a boxplot of the coe cients of variation of blood volume parameters and blood ow parameters obtained for the four models LN, rLN, rLin, and rLinReg. e p-values obtained a er the post-hoc analysis of the Friedman test are shown in Table . , signi cant di erences in parameter distributions are emphasized in bold.

Figure - :

 - Figure -: Boxplot showing the CV of blood volume (le ) and blood ow (right) estimated with the LN, rLN, rLin, and rLinReg models.

  Figure -a shows a diagram representing the structure of the model. An additional time-delay parameter D, re ecting the transit time of the contrast agent from the feeding artery to the tissue of interest, was introduced to describe experimental data more accurately, thus avoiding a bias in the estimation of the vascular parameter. Figure -b shows a diagram of the timedelayed one-compartment model. e mathematical expression of this model is given by Equation ( . ):

Figures -c

  Figures -c and -d respectively show a diagram of the rOVC model in case of a single tissue of interest, and in case of N T tissues of interest.

Figure - b :

 b Figure -b: One-compartment model with additional time-delay parameter.

  circulation occurs [ ]. ese six perfusion quanti cation methods were tted to simulated TICs to study the precision and the accuracy of the estimated perfusion parameters. CHAPTER . IMPACT OF RECIRCULATION IN DYNAMIC CONTRAST-ENHANCED ULTRASOUND: Consider N vascularized tissue regions T i , i = , .., N in a spatial domain, each region being an homogeneous compartment fed by the same arterial input function (AIF), C A . is mono-compartmental hypothesis is realistic since the distribution of microbubbles is restricted to the vascular space [ ]. Each tissue TIC, C T i , is characterized by a tissue blood volume V i , and a tissue blood ow F i . Since introducing a time-delay parameter in this model was shown to improve the quality of t in tumor tissues [ ], a parameter D i re-

Figure - :

 - Figure -: Simulated TICs with (orange) and without recirculation (blue) corresponding to noise-free AIF (top), examples of noise-free and noisy TICs in the fourth tissue region (middle) and in the reference tissue (bottom). e rst hundred seconds are displayed here.

  M and M -Log-Normal model (LN)e Log-normal function is an explicit model that depends on four parameters, it is frequently used to t TICs, in particular from dynamic contrast-enhanced ultrasound studies [ ]. From this model, the area under the curve AUC i , which is proportional to the tissue blood volume (see Appendix for proof) and τ i a time parameter re ecting the delay between the beginning of the acquisition and the arrival of the rst microbubbles in the tissue of interest are directly estimated. In addition, the wash-in rate (WIR i ), that is the maximal slope of the uptake part of the TIC, a parameter related to the tissue blood ow (see Appendix for proof), is commonly derived. Appendix shows the analytical expression of the AUC and W IR parameters, using the conventional expression of the Log-Normal model. For the rst method (M ), all the time samples are analyzed while for the second model (M ), the analy-. . THEORY sis is restricted to the rst pass of the bolus, which roughly corresponds to the wash-in and the start of the wash-out phases.MethodsM and M -relative Log-Normal model (rLN) e relative Log-Normal models propose the comparison of the LN model parameters estimated in the tissue region i (AUC i , WIR i , and τ i ) with the corresponding values estimated in the reference tissue R (AUC R , WIR R , and τ R ), following equation . :

  M -relative one-compartment model (rLin) e model M is derived from the one-compartment model presented in Section . . . It was proposed to take into account the multiple cases for which the estimation of the AIF is tricky, see for instance [ ]. It assumes that the tissue region and the reference tissue are parallel single compartments, fed by a common AIF. Writing equation ( . ) respectively for C T i and C R , and rearranging them, a convolution equation that is independent of the AIF can be deduced. Four related perfusion parameters [ ] can then be estimated as de ned by equation . :

Figure - :

 - Figure -: Bull's-eyes representation of the perfusion parameters used to simulate the regional TICs, C T i (large circle), and the reference TIC, C R (bottom right disk). From le to right: tissue blood volume (V ), tissue blood ow (F), time-delay (D), and rate constant (k).e scale displayed in red color shows relative parameters: rV , rF, δ as de ned by equation ( . ).

  Fig. -shows statistical results related to the perfusion parameters estimated inside one speci c region (i = ) using the six models (M m ) described in section . . , Θ n h (M m ). Indeed, the simulated values Λ (M m ) and the median, rst, and third quartile values over the simulations of parameters are represented as a function of the noise level (index n, proxied by σ). ese results are displayed for simulations without (h = ) and with recirculation (h = ). In complement to Fig. -, Fig. -, -, and -display bull's-eye representations of the median estimation errors in the regions for an intermediate noise level (σ = . , n = ), E hi (M m ), for the six quanti cation methods and the two conditions of recirculation.

  Figure -:Median values (large symbols), rst and third quartiles (small symbols) of parameters estimated for the fourth tissue region C T (outer ring, upper halve, right octant). First column: tissue blood volume related parameters, second column: tissue blood ow related parameters, third column: time-delay related parameters, fourth row: rate constants in the tissue region and reference tissue. Constant lines in black represent simulated values, blue lines the estimation corresponding to the LN model, red lines the estimation corresponding to the LN model restricted to earliest phase. Yellow color stands for rLin model, while purple color stands for rReg model. For all of the cases, lled symbols correspond to the con guration without recirculation, while empty symbols correspond to the con guration with recirculation.

Figure - :

 - Figure -: Bull's-eyes of the median estimation errors obtained by the LN model at the intermediate noise level. From le to right: estimation errors corresponding to tissue blood volume, tissue blood ow, time delay. From top to bottom: M without recirculation, M with recirculation, M without recirculation, M with recirculation.

  e. with or without recirculation. us the present simulation study also emphasizes the need for acquisitions with su ciently long durations in order a reliably estimate perfusion parameters, and in particular when relying on the Log-Normal model for quanti cation. e use of a reference tissue to normalize perfusion parameters was already recommended following a test-retest study that was conducted on dynamic contrast-enhanced ultrasound acquisitions performed on small animals [ ]. Normalization was also proposed by a clinical study in order to enable the comparison of perfusion parameters estimated using contrast-enhanced ultrasound data and contrast-enhanced computerized tomography [ ].

  Figure -. Method M enforces a single value of k R across tissue regions. As a consequence, the approach yields spatially regularized estimates of k T , slightly biased. As expected, the impact of recirculation on estimates of models M and M is rather low, however not negligible. is e ect could be partly explained by the approximation of the TIC inside the reference tissue by the Log-Normal model which does not account for recirculation. is prior modeling was performed in order to reduce the impact of noise on the relative perfusion parameters. e median relative errors on these parameters are all less than %, even in presence of recirculation, which acceptable. is is illustrated in Figure -for an intermediate noise level.

  ) e shape parameter k is related to the standard deviation by the relation σ = √ k, allowing modulation of the noise level in simulated TICs. Fig. -shows an example of multiplicative random noise on simulated TICs for k = , corresponding to σ = . . Unless speci ed di erently, this value of σ was used as the default standard deviation of the noise distribution and random noise sequences were generated from this distribution.

  Figure -.

Figure - :Figure - :

 -- Figure -: Absolute perfusion parameters used for simulation with the OVC model, i.e. tissue blood volume, V T and V R (dimensionless), tissue blood ow, F T and F R (in s -), tissue rate constant, k T and k R (expressed in s -), and time delay, D T and D R (in s). Bullseye view of the parameters in the tumor regions. e bottom disks represent the parameters used to simulated the reference tissue region, the middle disk being the original value used for all experiments. e other disks are, from le to right, the half, two thirds, three halves, and double of the original value, used to study the in uence of the reference tissue.

Figure - :

 - Figure -: Relative perfusion parameters used for simulation varying the number of tumor regions N T from (le ) to (right), i.e. relative tissue blood volume, rV , relative tissue blood ow, rF, tissue rate constant, k T .

Figure - :

 - Figure -: Correspondence of the regions between bullseye (le ) and grid (right) representation.

Figures -aFigureFigure

  Figures -a to-e show the relative estimation error of the relative tissue blood volume (Figure -a), relative tissue blood ow (Figure -b), the rate constants in the tumor (Figure -c) and reference (Figure -d) tissues, and the relative time-delay (Figure -e) obtained using the rLin and rReg models, as a function of the noise level simulated in the data, i.e. the standard deviation of the multiplicative noise model.Expectedly, for the four parameters of both models, the interquartile range of the relative estimation bias increased with the noise level, and no relation was found between the

Figure

  Figure -a:Median (large symbols) and rst and third quartiles (small symbols) of the relative estimation error for the relative blood volume (rV ) estimated using the rLin (yellow diamonds) and rReg (purple squares) models, as a function of the exam duration.

Figures -aFigureFigure

  Figures -a to -e show the impact of the sampling period on the relative estimation error of the perfusion parameters of the rLin and rReg models, i.e. relative tissue blood volume (Figure -a), relative tissue blood ow (Figure -b), the rate constants in the tumor (Figure -c) and in the reference tissue (Figure -d), and the relative time-delay (Figure -e).Varying the sampling period over the investigated range, i.e. from . to second, did not have a signi cant impact on the accuracy of the estimation of the relative tissue blood volume parameter using both the rLin and rReg models (see Figure-a), and only a slight decrease in precision was observed for larger sampling period. Similarly to the varying noise level experiments, the bias in the estimation bias of rV was more consistent across regions using the rReg model, with the median relative estimation errors generally inferior to %. e estimates of rF, the relative tissue blood ow, exhibited a tendency to decrease with increasing sampling period, especially in regions with large simulated k T values. While in most regions the estimates of rF from the rReg model get closer to zero for large sampling periods, one region underestimated the parameter by more than %. e estimates of rF from the rLin model exhibited a similar trend, but in addition they were overall more sensitive to the sampling period in terms of precision, and less consistent across regions. In terms of rate constant in the tumor (see Figure-c), the rReg model yielded consistent negative biases across regions, and generally yielded more precise estimates than the rLin model. For both models, a slight decrease of the estimates of k T with the sampling period was observed for both models in most regions, but stronger negative slopes were observed in regions with larger simulated k T . e rate constant in the reference tissue was estimated consistently across tumor regions using the rReg model (see Figure-d), and the estimation was overall more accurate and precise with this model. Indeed, the estimation of k R using the rLin model was extremely imprecise in some regions.

  is phenomenon was reported by Cárdenas-Rodríguez et al. [ ] in a simulation study assessing the ability of the rLin model to accurately quantify perfusion in contrast-enhanced magnetic resonance imaging data.e authors of this study applied the rLin model pixel by pixel, overlooking the relations between the local perfusion parameters. However, regions with large k T values yielded accurate estimates of rV , while regions with small k T values yielded more biased estimates of the parameter. Our experiments also reveal that estimation biases can either be positive or negative in a given region, depending on the value of k R .

  , or in permeability-limited and ow-limited conditions with di using tracers, i.e. when permeability or blood ow is much larger than the other one [ ], or when the data only re ects one of the processes [ ]. Indeed, the range of investigated parameters should be adapted to the imaging modality of interest, for instance the sampling period and the exam duration are usually longer in other modalities than in ultrasound. Additionally, the multiplicative noise model used for simulation is speci c of ultrasound data, and the data was simulated using the OVC model to re ect the intravascular characteristics of ultrasound contrast agents.. ConclusionA simulation study based on preclinical contrast-enhanced ultrasound experiments was conducted to assess and compare the accuracy and precision of two estimation methods for the rOVC model, a one-compartment model using a reference tissue. e rLin model relies on the linear formulation of the rOVC model, and estimates perfusion parameters in each region of analysis individually. A limitation of this approach lies in the existence of a perfusion parameter that characterizes the unique reference tissue di erently for each tumor region, i.e. the rate constant k R . However, since the same reference tissue is used to analyse all the regions, this parameter should be homogeneous across the di erent regions. e rReg model is based on the rLin model, but it takes advantage of the functional diversity of the regions under analysis, and ensures that the rate constant of the reference tissue is the same across all the regions. ese simulation studies demonstrated the superiority of the rReg model in terms of accuracy and precision. e regularization of k R also made the biases more homogeneous across regions, making comparison of regional parameters more meaningful. Additionally, the rReg model relaxes the requirements for temporal resolution and exam duration, allowing accurate estimation of perfusion parameters in shorter acquisitions with low temporal resolution. Our experiments suggest that the regularization improve the accuracy and precision of the estimation if at least four regions are included in the analysis. Regarding the number and the size of the regions under analysis, a compromise should be made to re-CHAPTER . ERROR SOURCES AFFECTING RELATIVE QUANTIFICATION OF CEUS veal the spatial heterogeneity of the tissue, while limiting the noise level to ensure accurate estimation of the perfusion parameters.

  reproducibility through preclinical test-retest experiments. Indeed, the regularized model yielded the most reproducible perfusion parameters among the investigated methods. en, the robustness of the model to recirculation of the contrast agent throughout the course of the acquisition was assessed on simulated data in Chapter .is study con rmed the robustness of compartmental models to recirculation, and revealed the sensitivity of semiquantitative parameters to this physiological phenomenon. Finally, in Chapter we investigated the impact of acquisition characteristics, i.e. noise level, exam duration, and sampling period, as well as analysis strategies, i.e. number of regions, choice of reference tissue. e regularized model proved robust to noise, and relaxed the requirements for temporal resolution and exam duration. Additionally, regularizing the estimation of k R made the estimation more homogeneous across regions, allowing meaningful comparison of perfusion parameters between regions. e choice of the reference tissue was proved to a ect parameter estimation, and the role of parameter k R was revealed. is rst analysis pushes us to recommend the use of a large and well perfused tissue as a reference. CHAPTER . ERROR SOURCES AFFECTING RELATIVE QUANTIFICATION OF CEUS Bibliography [ ] Daniel Balvay, Frédérique Frouin, Guillaume Calmon, Bertrand Bessoud, Edmond Kahn, Nathalie Siauve, Olivier Clément, and Charles A Cuenod. New criteria for assessing t quality in dynamic contrast-enhancedT -weighted MRI for perfusion and permeability imaging. Magnetic Resonance in Medicine, ( ): -, . , [ ] Guillaume Barrois, Alain Coron, omas Payen, Alexandre Dizeux, and S Lori Bridal. A Multiplicative Model for Improving Microvascular Flow Estimation in Dynamic eory and Experimental Validation. IEEE Trans. Ultrason., Ferroelect., Freq. Contr., Barrois, Alain Coron, omas Payen, Alexandre Dizeux, and S Lori Bridal. A multiplicative model for improving microvascular ow estimation in dynamic contrast-enhanced ultrasound (DCE-US): theory and experimental validation. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, Björck. Numerical methods for least squares problems, . [ ] M J Blomley and P Dawson. Bolus dynamics: theoretical and experimental aspects. Br J Radiol, ( ): -, April . [ ] Julio Cárdenas-Rodríguez, Christine M Howison, and Mark D Pagel. A linear algorithm of the reference region model for DCE-MRI is robust and relaxes requirements for temporal resolution. Magnetic resonance imaging, Dietrich, Michalakis Averkiou, J-M Correas, N Lassau, E Leen, and F Piscaglia. An EFSUMB introduction into Dynamic Contrast-Enhanced Ultrasound (DCE-US) for quanti cation of tumour perfusion. In Ultraschall in der Medizin (Stuttgart, Germany : ), pages -. Innere Medizin , Caritas-Krankenhaus, Bad Mergentheim. Christoph.Dietrich@ckbm.de, © Georg ieme Verlag KG, Doury, Alain De Cesare, S Lori Bridal, Claire Pellot-Barakat, and Frédérique Frouin. Impact of Recirculation in Dynamic Contrast-Enhanced Ultrasound: A Simulation Study. IRBM. [ ] Maxime Doury, Alexandre Dizeux, Alain De Cesare, Olivier Lucidarme, S Lori Bridal, and Frédérique Frouin. Regularized linear resolution of a one-compartment model to improve the reproducibility of perfusion parameters in CEUS. In IEEE International Ultrasonics Symposium (IUS), pages -Doury, Alexandre Dizeux, Alain De Cesare, Olivier Lucidarme, Claire Pellot-Barakat, S Lori Bridal, and Frédérique Frouin. Quanti cation of tumor perfusion using dynamic contrast-enhanced ultrasound: impact of mathematical modeling. Physics in medicine and biology, ( ): -,

  N Gunn, S R Gunn, and V J Cunningham. Positron emission tomography compartmental models. Journal of Cerebral Blood Flow & Metabolism, Pilleul, Sébastien Mulé, S Lori Bridal, Frédérique Frouin, Catherine Lombard-Bohas, omas Walter, Olivier Lucidarme, and Aymeric Guibal. Correlation and agreement between contrast-enhanced ultrasonography and perfusion computed tomography for assessment of liver metastases from endocrine tumors: normalization enhances correlation. Ultrasound Med Biol, ( ): -, . [ ] J Logan, J S Fowler, N D Volkow, A P Wolf, S L Dewey, D J Schlyer, R R MacGregor, R Hitzemann, B Bendriem, and S J Gatley. Graphical analysis of reversible radioligand binding from time-activity measurements applied to [N-C-methyl]-(-)-cocaine PET studies in human subjects. Journal of Cerebral Blood Flow & Metabolism, ( ): -, September . [ ] J Logan, J S Fowler, N D Volkow, Y S Ding, G J Wang, and D L Alexo . A strategy for removing the bias in the graphical analysis method. Journal of Cerebral Blood Flow & Metabolism, ( ): -, March . [ ] MR Lowerison, JJ Tse, MN Hague, AF Chambers, DW Holdsworth, and JC Lace eld. Compound speckle model detects anti-angiogenic tumor response in preclinical nonlinear contrast-enhanced ultrasonography. Med Phys, ( ): -, . [ ] Cli ord S Patlak, Ronald G Blasberg, and Joseph D Fenstermacher. Graphical Evaluation of Blood-to-Brain Transfer Constants from Multiple-Time Uptake Data. Journal of Cerebral Blood Flow & Metabolism, ( ): -, March . , [ ] Costas Strouthos, Marios Lampaskis, Vassilis Sboros, Alan McNeilly, and Michalakis Averkiou. Indicator dilution models for the quanti cation of microvascular blood ow with bolus administration of ultrasound contrast agents. IEEE Trans Ultrason Ferroelectr Freq Control, Tang, H Mulvana, T Gauthier, A K P Lim, D O Cosgrove, R J Eckersley, and E Stride. Quantitative contrast-enhanced ultrasound imaging: a review of sources of variability. Interface Focus, ( ): -, August . , [ ] Paul S To s, Gunnar Brix, David L Buckley, Je rey L Evelhoch, Elizabeth Henderson, Michael V Knopp, Henrik B Larsson, Ting-Yim Lee, Nina A Mayr, Geo rey J Parker, Ruediger E Port, June Taylor, and Robert M Weissko . Estimating kinetic parameters from dynamic contrast-enhanced T( )-weighted MRI of a di usable tracer: standardized quantities and symbols. Journal of Magnetic Resonance Imaging, ( ): -, . [ ] Simona Turco, Hessel Wijkstra, and Massimo Mischi. Mathematical Models of Contrast Transport Kinetics for Cancer Diagnostic Imaging: A Review. IEEE Reviews in Biomedical Engineering, : -, . [ ] Stef van Buuren and Karin Groothuis-Oudshoorn. mice: Multivariate Imputation by Chained Equations in R. Journal of Statistical So ware, ( ): -, December . [ ] E T Whittaker and G Robinson. e calculus of observations a treatise on numerical mathematics. . Chapter Conclusion e review of the semi-quantitative, deconvolution, and compartmental methods used to quantify tissue perfusion enables us to identify some common methods which are used in various imaging modalities. Semi-quantitative approaches are o en used as perfusion indicators, especially in combination with indicator dilution theory, indeed they are easily derived from contrast enhancement curve in the tissue of interest only. Deconvolution methods and classical compartmental models require the knowledge of the input function, which is an arterial input function and can either be obtained through blood sampling or from the image. However blood sampling is invasive, especially when arterial blood samples are drawn. Additionally, because of their small cross-section and their high contrast agent concentrations, image-based estimation su ers from various artifacts, e.g. partial volume, saturation. e di culties in the detection of the arterial input function yielded to the development of reference tissue models for relative quanti cation of perfusion. Indeed, a reference tissue can be selected in a large, well perfused area of the image, alleviating the risks of partial volume e ect, and saturation artifacts. While compartmental models and reference tissue models were used to quantify perfusion in PET, X-ray CT, and MRI exams, they were never applied to contrast-enhanced ultrasound data to the best of our knowledge. In Chapter we compared a semi-quantitative approach based on the log-normal model LN to a one-compartment model using an arterial input function (AIF) in terms of reproducibility through preclinical test-retest contrast-enhanced ultrasound experiments. is study revealed the higher reproducibility of AIF model compared to the semi-quantitative CHAPTER . CONCLUSION parameters of the LN model. But the study also revealed the di culties encountered in the estimation of the arterial input function and the impact of these variations on the estimated perfusion parameters. Normalizing perfusion parameters of the LN and AIF models according to a reference tissue improved inter-exam reproducibility. e direct estimation of relative perfusion parameters using a one-compartment reference tissue (RT) yielded the most reproducible parameters in our test-retest study. Moreover, the quality of t of the model was assessed for the three models, and regions with bad t quality were removed from further statistical analysis. e AIF and RT models were tted without accounting for the regional time-delays, resulting in a large number of regions with bad t quality. When accounting for time-delays, the LN model best tted the curves overall, but it was also found to be the model that yielded the most regions with bad t quality. And the AIF and RT yielded comparable t quality, and the same number of regions with bad t quality. In Chapter we established the relations between the parameters of the LN, AIF, and RT models. ese relations reveal the strong link between the semi-quantitative parameters of the LN model and the parameters of the compartmental approaches, but also explain the inter-exam variations observed in semi-quantitative parameters which are due to variations in the arterial input function between successive exams. Normalizing perfusion parameters according to a reference tissue which has similar perfusion characteristics between the different exams results in an improved robustness to inter-exam variations. ese analytical considerations were also veri ed experimentally in preclinical test-retest data. In Chapter we presented a linear formulation of the RT model derived from Patlak and revealed its limitations when considering multiple tissues of interest or multiple regions in a single tissue. We proposed a new regularized linear estimation method (rReg) for the relative perfusion parameters of RT model, and compared it to the standard non-regularized linear estimation method (rLin). e rReg model takes advantage of the functional heterogeneity of the tissue of interest to regularize the estimation according to the reference tissue. e reproducibility of the two models, rLin and rReg, was assessed on the same preclinical test-retest data as in our previous studies (Chapter and Chapter ). Regularization signi cantly improved the reproducibility of perfusion parameters, in particular the reproducibility of relative blood ow estimated by rReg.

e

  impact of the reference tissue on the perfusion parameters of the rReg model was shown in Chapter , however a ner study is necessary to better de ne the characteristics of the ideal reference tissue and characterize the impact of an non-ideal reference tissue. Accounting for multiple reference tissues may alleviate the sensitivity of the model estimates to the characteristics of the reference tissues. For instance, Yang et al. [ ] used multiple reference tissues to estimate the arterial input function in contrast-enhanced magnetic resonance exams.

  CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW voxel level, and does not give direct access to microscopic measurements, i.e. at the cellular or molecular level. erefore, an image-based measurement is really a mixture of multiple signals corresponding to the various vascular structures present inside the unitary volume of interest.Realistically, this review of perfusion quanti cation techniques cannot be close to comprehensive, instead it tackles the subject from the three angles mentioned therea er, and emphasis the major landmarks of the perfusion quanti cation landscape. Section . presents semi-quantitative methods, extracting parameters directly either from raw or noise-ltered

enhancement curves. en Section . presents deconvolution-based quanti cation methods, estimating the impulse response in the tissue of interest by means of blind or regularized deconvolution. Finally, compartmental models accounting for the various interactions of the contrast agent with the tissue are presented in Section . .

  . . SEMI-QUANTITATIVE METHODS image, it was not used due to peak attenuation by partial volume e ects. Instead, the peak venous enhancement in the superior sagittal sinus was used. e authors reported relatively

	small approximation error, with slightly attened enhancement curves, explained by the
	short transit time of the contrast agent in brain tissues. e cerebral blood ow, commonly
	noted CBF, is the equivalent of the perfusion index in brain tissues. e authors de ned it as
	the ratio of the maximum slope of the tissue enhancement curve to the peak enhancement in
	the superior sagittal sinus. e fractional cerebral blood volume, commonly noted CBV, was
	de ned as the ratio of the peak tissue enhancement to the peak enhancement in the superior
	sagittal sinus. In a following study [ ], the authors assess the linearity, the spatial resolution,
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				CHAPTER . QUANTIFICATION OF PERFUSION EXAMS: A REVIEW
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[

  Compartment with delay Yes/Yes ( . , . ) rV AIF , rF AIF , D AIF rRT Relative Reference Tissue No/Yes ( . , . ) rV RT , rF RT rRTd Relative Reference Tissue with delay No/Yes ( . , . ) rV RT , rF RT , D RT

	Acronym Model Name	Input data Eq. Parameters
			AIF/RT
	aLN	Log-Normal	No/No	( . ) AUC, WIR, ∆
	aAIF	One-Compartment	Yes/No	( . ) V , F
	aAIFd	One-Compartment with delay	Yes/No	( . ) V , F, d
	rLN	Relative Log-Normal	No/Yes ( . , . ) rAUC, rWIR, D
	rAIF	Relative One-Compartment	Yes/Yes ( . , . ) rV AIF , rF AIF
	rAIFd	Relative One-

based, ow-based, and time delays parameters.

Table . :

 . Synthesis of the di erent models tested. e rst three models propose absolute quanti cation. e last ve models propose relative quanti cation.

Absolute Log-normal model: aLN [ ].

Table . :

 . . ) θ hl (k) being the parameter θ hl estimated for the kth test-retest study (with k from to Median [ rst-third quartiles] values of N MRSE and FMI (in %) obtained for the di erent models. N is the number of sub-regions where FMI < %. Signi cant di erences between aLN and any other model are indicated by ⋆ . In addition, signi cant di erences between aAIF (resp. rRT) and aAIFd (resp. rRTd) are indicated by † (resp. ‡ ). e symbol ○ indicates that comparisons were not reported due to the high number of missing data.

	CHAPTER . QUANTIFICATION OF TUMOR PERFUSION USING DCE-US: IMPACT OF MATHEMATICAL
						MODELING
	Model	All data N RMSE	FMI	FMI > % N RMSE FMI	N
	aLN	. [ . -. ]		. [ . -. ]	. [ . -. ]	. [ . -. ]
	aAIF	. ⋆ † [ . -. ]	. ⋆ † [ . -. ] . ○ [ . -. ]	. ○ [ . -. ]
	aAIFd	. [ . -. ]		. ⋆ [ . -. ] . [ . -. ]	. ⋆ [ . -. ]
	rRT	. ⋆ ‡ [ . -. ]	. ⋆ ‡ [ . -. ] . ⋆ ‡ [ . -. ] . ⋆ ‡ [ . -. ]
	rRTd	. [ . -. ]		. ⋆ [ . -. ]	. [ . -. ]	. ⋆ [ . -. ]
	). As parameters θ hl (k) corresponding to poor quality ts were removed, missing values
	were replaced using multivariate imputation according to the R package {mice}, 'Multivari-
	ate Imputation by Chained Equations' [ ], in order to compute CV hl using four values
	systematically.

Statistical tests were performed to compare goodness of t criteria and coe cients of variation between the di erent models, using the R package {coin}, 'Conditional Inference Procedures in a Permutation Test Framework' [ ]. ey were considered as signi cant when p values were less than . . As all the tests were conducted on paired data, when goodness of t criteria were removed (due to poor quality ts), they were replaced with imputed data. e non-parametric Friedman test with post-hoc analysis (Tukey's HSD test) was chosen for dealing with multiple comparisons.

  these results are not reported in Table.. Additionally, quartile values are provided for the N sub-regions verifying FMI > %. Due to the large portion of missing data for the

	aAIF model when considering only good ts (the number of excluded regions, N, being
	equal to

Table

.

shows the quartile values of the quality of t criteria (N RMSE and FMI), which are computed for the ( × × ) tumor sub-regions for the ve models: aLN, aAIF, aAIFd, rRT, and rRTd. Since the N MRSE and FMI criteria obtained by the three relative methods rLN, rAIF and rAIFd are identical to those obtained by aLN, aAIF and aAIFd, ), results of hypothesis testing were not presented for that speci c case. e aLN model shows slightly better quality criteria than the other models (these di erences are signi cant for FMI in all cases, and signi cant for N RMSE in case of aAIF and rRT models). e introduction of the time delay parameter (aAIFd, rAIFd and rRTd models)

. . RESULTS

Table . :

 . Mean

		Absolute parameters	Relative parameters
		CV AUC	CV V	CV rAUC	CV rV AIF	CV rV RT
	m	. ± .	. ± .	. ± .	. ± .	. ± .
	m	. ± .	. ± .	. ± .	. ± .	. ± .
	m	. ± .	. ± .	. ± .	. ± .	. ± .
	m	. ± .	. ± .	. ± .	. ± .	. ± .
		CV W IR	CV F	CV rW IR	CV rF AIF	CV rF RT
	m	. ± .	. ± .	. ± .	. ± .	. ± .
	m	. ± .	. ± .	. ± .	. ± .	. ± .
	m	. ± .	. ± .	. ± .	. ± .	. ± .
	m	. ± .	. ± .	. ± .	. ± .	. ± .

± standard deviation of the volume, ow and delay parameters estimated in the di erent sub-regions of the tumor, for the four test-retest exams, a er multiple imputation of missing values due to poor t quality. Values of WIR and F are multiplied by .

Table . :

 . Mean ± standard deviation of the coe cients of variation (CV), expressed in percentage, of volume and ow parameters estimated for each sub-region a er multiple imputation of missing values due to poor t quality. CV were not computed for time delays, since their values can be either positive or negative.

Table . :

 . p-values obtained in the post-hoc analysis of the Friedman test. Signi cant results (p < . ) in bold.

rLN and rLinReg yield signi cantly more reproducible blood volume parameters than LN,

  CHAPTER . REGULARIZED LINEAR RESOLUTION OF A ONE-COMPARTMENT MODEL TO IMPROVE THE REPRODUCIBILITY OF PERFUSION PARAMETERS IN CEUS

  CHAPTER . REGULARIZED LINEAR RESOLUTION OF A ONE-COMPARTMENT MODEL TO IMPROVE THE REPRODUCIBILITY OF PERFUSION PARAMETERS IN CEUS

  Top: original C(t) (blue dots) and noise ltered C f (t) (orange line) timeintensity curves, the horizontal line shows the % of maximum intensity threshold, and the vertical line shows t % . Middle: ltered time-intensity curve C f (t) (orange line) cropped to t ≤ %, and the tangent to C f (t) with maximum upslope gradient for t ≤ t % , and the vertical line shows the estimated time delay D. Bottom: gradient of the cropped noise ltered time-intensity curve Ċ f (t) (pink line), the horizontal black line shows the % of maximum gradient threshold, and the vertical line shows the estimated time-delay D. C (t) was noise-ltered twice by a moving-average lter of width seconds, yielding C f (t). e time t % at which C f (t) reaches % of its maximum value sets an upper bound of D. C f (t) was then truncated, keeping only the part where t ≤ t % . e derivative of C f (t), noted Ċ f (t), was approximated by convolution of the TIC with the central di erence operator [ ]. Finally, assuming no oscillation occurred in the TIC prior to bolus arrival, D was de ned as the time at which the derivative Ċ f (t) reaches % of its value at t = t % :

	CHAPTER . REGULARIZED LINEAR RESOLUTION OF A ONE-COMPARTMENT MODEL TO IMPROVE THE
					REPRODUCIBILITY OF PERFUSION PARAMETERS IN CEUS
	0.025											C(t) C f (t)
	0.02
	0.015
	0.01
	0.005
		t 20%	20	40	60	80		100	120	140		t (sec)
	× 10 -3
	5
	4
	3
	2
	1
	0	1	2	3	4	5	D	6	7	8	9	C f (t), t ≤ t 20% t (sec)
	× 10	-4
	10
	8
	6
	4
	2
	0	1	2	3	4	5	D	6	7	8	9	Ċ f (t), t ≤ t 20% t (sec)
	Figure -:

  . Model M e rLin model (model M ) accurately estimates relative tissue blood volume and relative tissue blood ow parameters, exhibiting small biases, but the precision depends on the noise level. e estimation of time delays parameters appears to be robust. Biases are larger when taking into account studies with recirculation, but remain in most cases moderate (less than %). Considering rate constants, some heterogeneity in estimated k R values was found. is con rms our assumption, no constraints are applied to this speci c values. Moreover, related errors appear in the estimates of k T i .Compared with the model M , results using model M are improved. By construction, a constant value of k R is estimated for all the subregions (slightly underestimated in the case shown in Figure-), and consequently the relative error in the estimation of k T i is highly homogeneous across tissue regions and almost constant. Indeed the largest median relative CHAPTER . IMPACT OF RECIRCULATION IN DYNAMIC CONTRAST-ENHANCED ULTRASOUND: A SIMULATION STUDY error on k T i is observed for the highest value of simulated k T i . e rReg model estimates relative blood ow parameters and relative blood tissue parameters, with median relative errors generally less than % without recirculation for the intermediate noise level (see Fig. -). Of course, recirculation yields higher estimation errors, nevertheless they remain below % in most cases and appear to be more homogeneous across the subregions than the estimates of model M .

	. .	Model M

Table . :

 . Analytic expressions of perfusion parameters using a one-compartment model and assuming two di erent shapes of AIF: rectangle function of width a and height a, rect a (t), and general case C A (t). In the rst case, K stands for the injected concentration.

		CHAPTER . ERROR SOURCES AFFECTING RELATIVE QUANTIFICATION OF CEUS
	.	eory
	. . Simulation models
	In this section, the two models employed to simulate synthetic noisy CEUS data are pre-
	sented. First, the one vascular compartment model was used to generate noiseless time-Chapter intensity curves (TICs), with known physiology-related perfusion parameters. en, be-
	cause of the multiplicative nature of the noise in ultrasound data, a parametric multiplicative
	noise model was used to corrupt the simulated noiseless TICs.
	Error Sources A ecting Relative
	Quanti cation of CEUS
	. Introduction
	In Chapter we studied the impact of inter-exam changes on perfusion parameters esti-
	mated from CEUS data, whether occurring at the experimental or physiological level. Addi-
	tionally, a simulation study addressing the issue of recirculation in CEUS quanti cation was
	presented in Chapter . In these studies we showed the superiority of the one-compartment
	reference tissue model in terms of reproducibility. In particular, the linear formulation of
	this model and its regularized version yielded the most reproducible and robust perfusion
	parameters among the investigated methods. In this Chapter we investigate other potential
	sources of error a ecting quanti cation using these two models through a series of simu-

lation experiment with varying factors, and by assessing the accuracy and precision of the estimated perfusion parameters. ese include data intrinsic characteristics, i.e. noise level, exam duration, sampling time; as well as quanti cation strategy, i.e. analysis scale, estimation method, reference tissue selection.

Multiplicative noise model

A multiplicative noise model following a gamma distribution and enforcing unit mean was used, i.e. mean v p (v) = , inspired by

Barrois et al. [ ]

. A unit mean distribution for a multiplicative noise is the equivalent of a centered distribution for additive noise.

  MATERIALS AND METHODStissue of interest is noted C i T (t), where i ∈ [[ , N T ]], and the TIC in the chosen reference tissue is noted C R (t). All TICs are de ned for t ∈ [ , L] and count N S samples.

Two di erent relative quanti cation methods, making use of a reference tissue, derived from the previously described OVC model are described. e following methods are intended to estimate perfusion parameters from N T tissues in a single CEUS exam. e TIC in the i th . .

  CHAPTER . CONCLUSION parameters of the rReg model could then be used to classify tumor tissues into necrotic, hypoxic, and non-hypoxic classes. We then intend to use the tissue classi cation results to drive a realistic tumor growth model that accounts for treatment response inspired by Ribba et al. [ ]. Benjamin Ribba, Emmanuel Watkin, Michel Tod, Pascal Girard, Emmanuel Grenier, Benoît You, Enrico Giraudo, and Gilles Freyer. A model of vascular tumour growth in mice combining longitudinal tumour size data with histological biomarkers. European journal of cancer (Oxford, England : ), ( ): -, February . [ ] Cheng Yang, Gregory S Karczmar, Milica Medved, and Walter M Stadler. Multiple reference tissue method for contrast agent arterial input function estimation. Magnetic

	Resonance in Medicine, ( ):	-	, December	.

BIBLIOGRAPHY

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Appendix

Taking into account the generic expression of the Log-Normal model:

Figure -d: Bullseyes of the median relative estimation error for the rate constant in the reference tissue (k R ) estimated using the rLin (top) and rReg (bottom) models depending on the characteristics of the reference tissue used for simulation. In press In writing

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