Rita Levi Montalcini

Thierry Goudon

Nicolas Jesús

Guillaume

Lise

Pierre Lucas

Grazie

E Alle Amiche Pazze

Mari Luli

Silvs

Keywords: tumour-immune interactions, immunotherapy, discrete models, continuum models, numerical simulations interactions tumeur-système immunitaire, immunothérapie, modèles discrets, modèles continus, simulations numériques Mathematical glossary

e uniche coloro che usano entrambi.

Résumé

Au cours de la dernière décennie, les progrès technologiques ont permis la conception d'immunothérapies qui, contrairement aux thérapies anticancéreuses classiques, ciblent les interactions entre cellules tumorales et cellules immunitaires, dans le but de renforcer l'efficacité de la réponse immunitaire. Cependant, ces interactions reposent sur des mécanismes complexes, ce qui rend difficile la conception de traitements efficaces. Par conséquent, les modèles mathématiques sont des outils utiles pour reproduire la dynamique spatio-temporelle des interactions entre les cellules tumorales et les cellules immunitaires, afin de tester le potentiel de nouvelles techniques thérapeutiques de manière flexible et non coûteuse. Dans cette thèse, nous développons des modèles discrets et continus pour décrire la dynamique spatiotemporelle des interactions entre une tumeur solide et les cellules T cytotoxiques, dans le but d'étudier les paramètres biologiques qui permettent l'élimination, ou bien l'échappement, de la tumeur. Les modèles discrets développés dans ce travail décrivent la dynamique de chaque cellule, permettant ainsi la représentation de mécanismes à l'échelle cellulaire. De plus, ils sont suffisamment détaillés et spécifiques pour reproduire qualitativement les résultats d'études expérimentales. Quant aux modèles continus, ils ne sont pas formulés sur la base d'arguments phénoménologiques, qui peuvent limiter une description mathématique précise d'aspects biologiques et biophysiques cruciaux, mais ils sont dérivés formellement des modèles discrets par le biais de méthodes asymptotiques appropriées. Les résultats des simulations numériques des modèles discrets montrent qu'il existe un excellent accord quantitatif entre eux et les solutions des modèles continus correspondants, et clarifient les conditions de réussite, ou bien d'échec, de la surveillance immunitaire. Enfin, les modèles mathématiques présentés dans cette thèse peuvent fournir un cadre pour aider les biologistes et les cliniciens à mieux comprendre les mécanismes par lesquels les tumeurs échappent au système immunitaire, et peuvent être des outils prometteurs pour explorer des stratégies thérapeutiques conçues pour améliorer l'efficacité de la réponse immunitaire anti-tumorale.

Biological glossary

antigen Any substance that causes the body to make an immune response against that substance. Antigens include toxins, chemicals, bacteria, viruses, or other substances that come from outside the body. Body tissues and cells, including tumour cells, also have antigens on them that can trigger an immune response. 1, 2, 3, 5, 6, 7, 8, 9, 10, [START_REF] Arciero | łA mathematical model of tumorimmune evasion and siRNA treatmentž[END_REF][START_REF] Ardaševa | łComparative study between discrete and continuum models for the evolution of competing phenotype-structured cell populations in dynamical environmentsž[END_REF][START_REF] Atsou | łA size and space structured model describing interactions of tumor cells with immune cells reveals cancer persistent equilibrium states in tumorigenesisž[END_REF][START_REF] Beatty | łChimeric antigen receptor T cells are vulnerable to immunosuppressive mechanisms present within the tumor microenvironmentž[END_REF][START_REF] Besse | łStability analysis of a model of interaction between the immune system and cancer cells in chronic myelogenous leukemiaž[END_REF][START_REF] Boissonnas | łIn vivo imaging of cytotoxic T cell inőltration and elimination of a solid tumorž[END_REF][START_REF] Boon | łTumor antigens recognized by T lymphocytesž[END_REF][START_REF] Boon | łHuman T cell responses against melanomaž[END_REF][START_REF] Breart | łTwo-photon imaging of intratumoral CD8+ T cell cytotoxic activity during adoptive T cell therapy in micež[END_REF][START_REF] Bruni | łThe immune contexture and Immunoscore in cancer prognosis and therapeutic efficacyž[END_REF] antigen presenting cell (APC) Heterogeneous group of immune cells that mediates the cellular immune response by processing and presenting antigens for recognition by certain lymphocytes such as T cells. Classical APCs include dendritic cells and macrophages 4, 5, 7

chemoattractant A chemical (chemotactic) agent that induces an organism or a cell to migrate towards it. In the context of tumours, the chemoattractant can represent a mixture of small soluble proteins called chemokines (e.g. CCL3, CCL5, CCL20, and CXCL10). During the anti-tumour immune response, cytotoxic T lymphocytes migrate via chemotaxis towards the tumour, in the directions of stronger gradients of the chemokines produced by both tumour and immune cells. [START_REF] Almeida | łDiscrete and continuum models for the coevolutionary dynamics between CD8+ cytotoxic T lymphocytes and tumour cellsž[END_REF][START_REF] Beatty | łChimeric antigen receptor T cells are vulnerable to immunosuppressive mechanisms present within the tumor microenvironmentž[END_REF][START_REF] Blattman | łEstimating the precursor frequency of naive antigen-speciőc CD8 T cellsž[END_REF][START_REF] Bruni | łThe immune contexture and Immunoscore in cancer prognosis and therapeutic efficacyž[END_REF] cytokine A type of protein produced by certain immune and non-immune cells which has an effect on the immune system. Some cytokines, the pro-inŕammatory cytokines, stimulate the immune system, helping the body őght tumours, infections, and other diseases. Other cytokines, the anti-inŕammatory cytokines, slow down the immune system, inhibiting its function. Examples of cytokines are interleukins, such as IL-2 and IL-10, interferons, such as INF-γ, and growth factors. [START_REF] Alden | łSpartan: a comprehensive tool for understanding uncertainty in simulations of biological systemsž[END_REF][START_REF] Alhakeem | łChronic lymphocytic Leukemia-Derived IL-10 suppresses antitumor immunityž[END_REF][START_REF] Anagnostou | łEvolution of neoantigen landscape during immune checkpoint blockade in non-small cell lung cancerž[END_REF][START_REF] Anderson | Single-cell-based models in biology and medicine[END_REF][START_REF] Arciero | łA mathematical model of tumorimmune evasion and siRNA treatmentž[END_REF][START_REF] Ardaševa | łComparative study between discrete and continuum models for the evolution of competing phenotype-structured cell populations in dynamical environmentsž[END_REF][START_REF] Atsou | łA size and space structured model describing interactions of tumor cells with immune cells reveals cancer persistent equilibrium states in tumorigenesisž[END_REF][START_REF] Bertolaso | Philosophy of cancer[END_REF][START_REF] Besse | łStability analysis of a model of interaction between the immune system and cancer cells in chronic myelogenous leukemiaž[END_REF] cytotoxic A substance, a cell, a drug that kills cells, including tumour cells. Cytotoxic agents may stop tumour cells from dividing and growing and may cause tumours to shrink in size. For example: cytotoxic T lymphocyte, cytotoxic chemotherapy. 1, 5, 6, 22 cytotoxic T lymphocyte (CTL) A type of cells of the adaptive immune system that has the capacity to directly kill other cells. It plays a major role in host defense against tumour cells 3,[START_REF] Alhakeem | łChronic lymphocytic Leukemia-Derived IL-10 suppresses antitumor immunityž[END_REF][START_REF] Almeida | łDiscrete and continuum models for the coevolutionary dynamics between CD8+ cytotoxic T lymphocytes and tumour cellsž[END_REF][START_REF] Almeida | łTreatment-induced shrinking of tumour aggregates: A nonlinear volume-őlling chemotactic approachž[END_REF][START_REF] Anagnostou | łEvolution of neoantigen landscape during immune checkpoint blockade in non-small cell lung cancerž[END_REF][START_REF] Anderson | Single-cell-based models in biology and medicine[END_REF][START_REF] Angell | łFrom the immune contexture to the Immunoscore: the role of prognostic and predictive immune markers in cancerž[END_REF][START_REF] Aponte-Serrano | łMulticellular Multiscale Spatial Modeling of the Immune Response to Pathogens and Cancerž[END_REF][START_REF] Arciero | łA mathematical model of tumorimmune evasion and siRNA treatmentž[END_REF][START_REF] Ardaševa | łComparative study between discrete and continuum models for the evolution of competing phenotype-structured cell populations in dynamical environmentsž[END_REF][START_REF] Atsou | łA size and space structured model describing interactions of tumor cells with immune cells reveals cancer persistent equilibrium states in tumorigenesisž[END_REF][START_REF] Beatty | łChimeric antigen receptor T cells are vulnerable to immunosuppressive mechanisms present within the tumor microenvironmentž[END_REF][START_REF] Bertolaso | Philosophy of cancer[END_REF][START_REF] Besse | łStability analysis of a model of interaction between the immune system and cancer cells in chronic myelogenous leukemiaž[END_REF][START_REF] Blattman | łEstimating the precursor frequency of naive antigen-speciőc CD8 T cellsž[END_REF][START_REF] Boissonnas | łIn vivo imaging of cytotoxic T cell inőltration and elimination of a solid tumorž[END_REF][START_REF] Boon | łTumor antigens recognized by T lymphocytesž[END_REF][START_REF] Boon | łHuman T cell responses against melanomaž[END_REF][START_REF] Breart | łTwo-photon imaging of intratumoral CD8+ T cell cytotoxic activity during adoptive T cell therapy in micež[END_REF][START_REF] Bruni | łThe immune contexture and Immunoscore in cancer prognosis and therapeutic efficacyž[END_REF] dendritic cell A special type of immune cell that boosts immune responses by showing antigens on its surface to other cells of the adaptive immune system. A dendritic cell is a type of phagocyte and also an antigen-presenting cell (APC). [START_REF] Alden | łSpartan: a comprehensive tool for understanding uncertainty in simulations of biological systemsž[END_REF][START_REF] Alhakeem | łChronic lymphocytic Leukemia-Derived IL-10 suppresses antitumor immunityž[END_REF][START_REF] Almeida | łTreatment-induced shrinking of tumour aggregates: A nonlinear volume-őlling chemotactic approachž[END_REF][START_REF] Anagnostou | łEvolution of neoantigen landscape during immune checkpoint blockade in non-small cell lung cancerž[END_REF][START_REF] Boissonnas | łIn vivo imaging of cytotoxic T cell inőltration and elimination of a solid tumorž[END_REF] immune checkpoint A type of protein produced by some types of immune system cells, such as cytotoxic T lymphocytes, and some tumour cells, which helps prevent immune responses xix xx Biological glossary from being too strong. During tumour development, immune checkpoints can keep T cells from killing tumour cells. Examples of checkpoint proteins found on T cells or tumour cells include PD-1/PD-L1 and CTLA-4/B7-1/B7-2. 8, 11 immune checkpoint inhibitor A type of drug that blocks proteins called immune checkpoints. When these checkpoints are blocked, T cells can kill tumour cells more efficiently. Examples of immune checkpoint inhibitors include anti-PD-L1 or anti-PD-1 (which block the binding of PD-L1 to PD-1) and anti-CTLA-4 antibody (which block the binding of CTLA-4 to B7-1/B7-2) 2, 8, 11, 13

immunogenic Relating to or denoting substances able to produce an immune response. For example: immunogenic tumours, immunogenic antigens. 12, 18, 20, 25 intra-tumour heterogeneity (ITH) It refers to distinct tumour cell populations (with different molecular and phenotypical proőles) within the same tumour specimen. It differs from inter-tumour heterogeneity, which refers to the differences found between similar tumours in different patients. 8, 10, 25

macrophage A type of cell of the innate immune response that surrounds and kills microorganisms, removes dead cells, and stimulates the action of other immune system cells, acting as antigen-presenting cell (APC). Some macrophages have a tumour-promoting phenotype, and can drive pathological phenomena including tumour cell proliferation, tumour angiogenesis, invasion and metastasis, immunosuppression, and drug resistance. 4, 7

major histocompatibility complex (MHC) A group of genes that codes for proteins found on the surface of cells that helps the immune system recognize foreign substances. During an immune response, it allows T lymphocytes to detect antigen presenting cells that have ingested infectious microorganisms and display peptide fragments on their surface. 5, 7, 8, 9, 10 natural killer cell (NK cell) A type of cell of the innate immune system that has granules (small particles) with enzymes that can kill tumour cells or cells infected by a virus. 4, 21

solid tumour An abnormal mass of tissue that usually does not contain cysts or liquid areas. Solid tumours may be benign (not cancer), or malignant (cancer). Different types of solid tumours are named for the type of cells that form them. Examples of solid tumours are sarcomas, carcinomas, and lymphomas. Leukemias (cancers of the blood) generally do not form solid tumours. [START_REF] Alden | łSpartan: a comprehensive tool for understanding uncertainty in simulations of biological systemsž[END_REF][START_REF] Aponte-Serrano | łMulticellular Multiscale Spatial Modeling of the Immune Response to Pathogens and Cancerž[END_REF][START_REF] Arciero | łA mathematical model of tumorimmune evasion and siRNA treatmentž[END_REF][START_REF] Boissonnas | łIn vivo imaging of cytotoxic T cell inőltration and elimination of a solid tumorž[END_REF][START_REF] Bruni | łThe immune contexture and Immunoscore in cancer prognosis and therapeutic efficacyž[END_REF] T cell receptor (TCR) A group of proteins found on T cells. TCRs bind to certain antigens found on abnormal cells, tumour cells, cells from other organisms. This interaction causes the T cells to attack these cells and helps the body őght infection, tumour cells, or other diseases. 5, 6, 9, 16, 24

telomere The ends of a chromosome. Each time a cell divides, telomeres lose a small amount of DNA and become shorter. Over time, the chromosomes become damaged and the cells die. However, tumour cells express high levels of telomeraze, an enzyme preventing the shortening of the telomere, allowing for a longer or inőnite survival of cells. 3, 4

Chapter 1

Introduction

1

.1 Motivation

Cancer can be described as a disease in which some abnormal or damaged cells of the body grow uncontrollably and spread to other parts of the body1 . The őrst, possibly disputable, evidence of this disease was found in the fossil records of a human ancestor as old as 1.7 million years [START_REF] Odes | łEarliest hominin cancer: 1.7-million-year-old osteosarcoma from Swartkrans Cave, South Africaž[END_REF], and that of a dinosaur, dated back more than 75 million years ago [START_REF] Taya | łCT and histopathology used to diagnose osteosarcoma in a dinosaurž[END_REF]. Cancer was a quite rare disease in ancient human societies, but today in France it is the őrst cause of death in men and the second in women 2 . There are two probable reasons for that: (1) the harmful environmental and lifestyle factors that trigger its emergence, and (2) the increase in life expectancy, since the risk of developing a tumour increases with age. Today, cancer results in approximately ten million annual deaths, only taking into account official data [START_REF] Naghavi | regional, and national age-sex speciőc mortality for 264 causes of death, 1980-2016: a systematic analysis for the Global Burden of Disease Study 2016[END_REF]. Despite technological developments and major advances in cancer research, it is clear that this disease is still enduring. It seems that the best choice of treatment that humans used for centuries, was the surgical removal of malignant tumours, and even nowadays it frequently remains the best treatment option. However, it was already known to ancient Romans and Greeks that surgery could only be successful for superőcial tumours, and that in many cases it could even worsen the conditions of the patient [START_REF] David | łCancer: an old disease, a new disease or something in between?[END_REF]. The őrst major breakthrough happened at the turn of the 19th and 20th centuries with the discovery of X-rays and the introduction of radiotherapy treatments, which use high doses of radiation to kill tumour cells and shrink tumours. In fact, at high doses, radiation therapy is able to kill tumour cells or slow their growth by damaging their DNA. In the 1940s, the introduction of chemotherapy to treat tumours proved to be another considerable breakthrough in oncology. Chemotherapy uses one or more cytotoxic anti-cancer drugs (chemotherapeutic agents), which inhibit cell mitosis (i.e. division) or induce DNA damage. Surgery, radiotherapy and chemotherapy are still the most widespread modalities in tumour treatment. However, the side effects of these traditional treatment approaches can be severe. They include: sustained damage to healthy cells, nausea, fertility issues, hair loss and psychological issues.

More recently, the focus of cancer therapy research has turned to developing targeted and more personalised treatments. The development of some of these new treatment approaches relies on the fact that the immune system plays a primary and critical role in the prevention and eradication of tumours. In fact, tumour cells express antigens on their surface which allow them not only to be identiőable by cells of the immune system but also enable an effective immune response to be triggered. This őnding implies that clinically observable, and often 2 CHAPTER 1. Introduction aggressive, tumours are those that have escaped control of the immune system. In this context, designing immunomodulatory treatments to enhance or restore anti-tumour immune responses may be beneőcial in the treatment of multiple types of tumours. The development of these new therapeutic approaches is based on understanding the key mechanisms, or hallmarks, that enable the tumour to avoid immune destruction. There can be multiple mechanisms that lead to the suppression of the immune response; moreover, these immunosuppressive mechanisms can vary between patients. Consequently, immunotherapy techniques have been, and continue to be, developed by targeting the key mechanisms that are responsible for immune escape, to offer a more personalised form of cancer treatment. These techniques can be active, i.e. speciőcally targeting tumour cells, like cancer vaccines that contain speciőc antigens, or passive, i.e. enhancing the ability of the immune system to attack tumour cells, like immune checkpoint inhibitors, whose authors were awarded the Nobel Prize in Medicine in 2018. However, many of the biological phenomena that foster the escape of tumour cells from the action of immune cells and the failure of immunotherapy treatments are still not fully understood to this day. And the causes of this lack of understanding are diverse. One reason is the complex nature of the interactions between tumour cells and immune cells, which involve various molecules, proteins, receptors and cells that promote or inhibit the immune action. Moreover, the development and testing of new therapy techniques can be an expensive and time-consuming process, and often associated with ethical difficulties, as the result of changes in clinical protocols may reduce the overall treatment efficacy.

One option to tackle these problems is to develop mathematical models capable of describing and synthesising these complex phenomena in order to carry out numerical experiments (known as in silico). The added value of mathematical modelling and computer simulations are their reproducibility and ŕexibility. On the one hand, their reproducibility makes it possible to replicate various experiment in shorter periods of time than those required for clinical trials. On the other hand, their ŕexibility allows to add various biological phenomena as desired, in order to describe increasingly complex mechanisms or to vary unknown parameters. This last quality will allow us, in this thesis, to describe different mechanisms involved in tumour-immune interactions and to analyse the duality of the anti-tumour immune response, as well as the role of treatment strategies based on immunotherapy. Mathematical models represent a simpliőcation of reality and, therefore, do not allow to draw absolutely accurate conclusions on the underlying biological dynamics. However, their study allows to assess the value of certain biological parameters, as well as the reliability of certain hypotheses.

The inherent complexity, not only of the interactions between tumour cells and the immune system, but also in the mathematical modelling of this phenomenon, requires a clear deőnition of the underlying biological framework. Thus, in this chapter, a őrst part will be devoted to the description of the biological context. After a brief introduction on tumour development, we will describe the mechanisms governing the interactions between malignant tumour cells and cells of the immune system. We will then introduce some mechanisms leading the tumour cells to evade the immune response, and provide a short review of some current immunotherapies. In the second part, we will give an overview of the existing mathematical models in the literature used to describe tumour-immune interactions, and discuss how they can be adapted to include the effects of immunotherapies. Finally, a third part will be devoted to a summary of the contributions of the thesis. 1.2. Biological overview of the immune response to tumours 3

Biological overview of the immune response to tumours

In this section, we introduce the key biological mechanisms involved in the immune response to solid tumours. As previously said, the interactions between tumour cells and cells of the immune system constitute a very complex system, which involves cells and molecules of different types. Therefore, some of the concepts described in the following sections may represent a simpliőcation of the real mechanisms underlying the immune response to tumours. These simpliőcations are directly related to the assumptions that have led to the development of our mathematical models.

The őrst part brieŕy describes the general mechanisms that must be overcome in order for a normal cell to develop into a tumour cell. In the second part, we present an overview of the immune system, focussing on different types of cells which take part in the immune response in tumour development.

While being aware of the fact that a variety of different cells and molecules take part in the anti-tumour immune response, in this thesis we are mainly interested on a particular type of immune cells: cytotoxic T lymphocytes (CTLs). Such cells are the major antitumour effector cells and represent the immune cells that are most commonly stimulated by immunotherapies.

Additionally, we describe some of the key biological and immunological mechanisms involved in the recognition of tumour cells, such as the expression of tumour antigens, and how the immune system is activated to őght tumour cells. In the third part, we present some of the properties of tumour cells which allow them to evade the immune response. We will conclude the section by considering some immunotherapy techniques which have been developed to target these aspects.

Tumour development

Normal cells grow and divide, but have many mechanisms that control their growth. They only grow when stimulated by growth factors. If they are damaged, a molecular brake stops them from dividing until they are repaired. If they cannot be repaired, they commit programmed cell death (apoptosis) [START_REF] Evan | łProliferation, cell cycle and apoptosis in cancerž[END_REF]. Moreover, thanks to a speciőc enzyme, the telomerase, which shortens their telomeres at the end of the chromosomes at each cell division, normal cells can only divide a limited number of times [START_REF] Kim | łSpeciőc association of human telomerase activity with immortal cells and cancerž[END_REF]. Cells are part of a tissue structure, and remain where they belong. Each of these mechanisms is controlled by several proteins, which ensure the proper functioning of all our cells [START_REF] Evan | łProliferation, cell cycle and apoptosis in cancerž[END_REF].

All these mechanisms must be overcome in order for a normal cell to develop into a tumour cell. A critical protein must malfunction in each of those mechanisms. These proteins become non-functional when they acquire somatic mutations (mutations that are not inherited but occur after conception). This occurs in a series of steps, which in [START_REF] Hanahan | łHallmarks of cancer: the next generationž[END_REF] have been referred to as 'hallmarks of cancer'. These include the ability of tumour cells to: sustain or enhance replication and growth (cell proliferation), avoid growth suppressors, avoid or suppress the immune response, be able to divide/replicate indeőnitely, recruit immune cells to cause inŕammation, invade another area of the body (formation of metastasis), create new blood vessels (angiogenesis), mutate, resist apoptosis (cell death) and alter their metabolic processes to avoid the need for oxygen.

Through cell division, the altered cells cluster together to form a growing tumour. In the early stages of tumorigenesis, tumour cells stay close to the original site where they have formed [START_REF] Hanahan | łHallmarks of cancer: the next generationž[END_REF]. However, once the tumour has developed and is large enough, it can begin to create a blood supply through angiogenesis. This allows single cells to leave the initial site and travel to other sites of the body, through the blood stream, which can ultimately lead to the formation of metastasis [START_REF] Ellis | łAngiogenesis and metastasisž[END_REF]. Furthermore, most tumour cells express high levels of telomerase, an enzyme preventing the shortening of the telomeres and allowing for a longer or inőnite division, and thus survival, of cells [START_REF] Kim | łSpeciőc association of human telomerase activity with immortal cells and cancerž[END_REF].

In this thesis, we focus on solid tumours in the early stages of development (i.e. small pre-angiogenic tumours) and, in particular, on the ability of tumours to evade immune destruction.

In the following sections we will brieŕy describe the mechanisms governing immune surveillance and some mechanisms used by tumour cells to escape the control of the immune system.

Tumour immunology

The immune system is the body's natural defence mechanism against any foreign attack. Through the immune response, the body is able to attack these foreign and harmful materials and destroy them. It has also become evident that the immune system plays a role in the protection from harmful substances within the body, including tumour cells. This makes the immune system a very important mechanism for controlling and treating tumours. However, both the functioning of the immune system and the growth of tumours involve highly complex processes, making the interactions between tumour cells and immune cells an elaborate system that is not yet fully understood by either experimentalists or theoreticians [START_REF] Mallet | łA cellular automata model of tumor-immune system interactionsž[END_REF].

In the case of the human immune system, there are two different types of barriers, namely, the innate and adaptive immune systems. Both of them have their own individual methods and abilities to help our bodies stay healthy, and interact with one another to kill harmful invaders, including tumour cells.

Innate immune system and tumour antigen presentation Brieŕy, the innate immune system is the őrst line of defence against pathogens and malignant cells and its role is to recognize and respond to them in a generic way. It recruits speciőc immune cells at the site of infection, provides an immediate defense mechanism through the initiation of inŕammation, removes foreign substances, and activates the adaptive immune response [START_REF] Chaplin | łOverview of the immune responsež[END_REF]. Innate immune cells include different types of cells. In the context of tumours, the two most important one are the natural killer cells (NK cells) and the professional phagocytes.

NK cells are part of a class of white blood cells called lymphocytes. When NK cells receive signals from other immune cells, they secrete proteins called cytokines, which in turn signal other immune cells to come and help defend an infected area. Also, NK cells can destroy tumour cells without requiring activation by other cells via cell lysis (i.e. disintegration) which results in cell debris within the microenvironment [START_REF] Martıénez-Lostao | łHow do cytotoxic lymphocytes kill cancer cells?[END_REF].

The professional phagocytes consist of different types of cells, whose role is to engulf and remove invaders through a process called łphagocytosisž. Moreover, particular types of professional phagocytes, the dendritic cells and the macrophages, play an important role in the initial steps of the immune response to tumours. Whilst undergoing phagocytosis, dendritic cells and macrophages secrete pro-inŕammatory cytokines, such as the tumour necrosis factor (TNF), which can kill tumour cells, as well as activate other immune cells [START_REF] Sompayrac | How the immune system works[END_REF]. Secondly, dendritic cells and macrophages can act as antigen presenting cells (APCs) [START_REF] Chaplin | łOverview of the immune responsež[END_REF]. When the defences are put under pressure and APCs have engulfed an invader, they present speciőc proteins, called antigens, on their surface [START_REF] Messerschmidt | łHow cancers escape immune destruction and mechanisms of action for the new signiőcantly active immune therapies: Helping nonimmunologists decipher recent advancesž[END_REF]. In immunology, an antigen can represent a protein, a molecule or any substance that causes the body to make an immune response against that substance. In the context of tumours, such antigens are usually composed of fragments of proteins belonging to tumour cells, reŕecting changes or mutations that the latter have undergone during cancer formation. Once collected, the APCs can process the antigens and load them on their cell surface through major histocompatibility complex (MHC) molecules, allowing them to be recognised by the adaptive immune system [START_REF] Boon | łHuman T cell responses against melanomaž[END_REF][START_REF] Boon | łTumor antigens recognized by T lymphocytesž[END_REF].

Activation of the adaptive immune system While the innate immune response is a general defence reaction, the adaptive immune response is highly speciőc to the particular pathogen that induced it, and it provides long-lasting protection. The adaptive immune system requires more time to develop an effective response. However, it recognizes and remembers speciőc pathogens, and, therefore, its efficacy increases each time the same pathogen is encountered [START_REF] Sompayrac | How the immune system works[END_REF]. In the case of an anti-tumour immune response, the most important adaptive immune cell types are the T lymphocytes, the speciőc functions of which will be described later in this paragraph.

Adaptive immune cells must be activated to produce an effective anti-tumour immune response. The activation of the adaptive immune response requires the expression of antigens on the surface of tumour cells, and their presentation on the surface of APCs by MHC molecules. Once the APCs have processed the antigens, they move to the lymph nodes to present them to adaptive immune cells, speciőcally to naive T lymphocytes (or T cells). Naive T lymphocytes, which can be distinguished in naive CD4+ and naive CD8+ lymphocytes, are immature T cells that have differentiated and been released by the thymus, but have not yet encountered their corresponding antigens and are not activated. These cells are endowed with particular receptors, the T cell receptor (TCR), on the surface of their membrane, which allow them to recognise speciőc antigens, activate, and initiate an antigen-speciőc immune response [START_REF] Coulie | łTumour antigens recognized by T lymphocytes: at the core of cancer immunotherapyž[END_REF][START_REF] Messerschmidt | łHow cancers escape immune destruction and mechanisms of action for the new signiőcantly active immune therapies: Helping nonimmunologists decipher recent advancesž[END_REF]. The activation process relies on co-stimulatory (e.g CD28) and co-inhibitory (e.g CTLA-4, PD-1) proteins [START_REF] Martıénez-Lostao | łHow do cytotoxic lymphocytes kill cancer cells?[END_REF]. Depending on such complementary proteins and the antigen-MHC complex, subsequently, naive T lymphocytes can mature into distinct groups. The most important groups of T lymphocytes participating in the anti-tumour immune response include: the helper CD4+ T lymphocytes, the cytotoxic CD8+ T lymphocytes, and the regulatory T lymphocytes. The őrst group promotes the activation of a larger number of APCs and CD8+ T cells by secreting a variety of proinŕammatory cytokines, like interleukin 2 (IL-2) and interferon gamma (INF-γ), and displaying a variety of co-stimulatory proteins on their surface. The cytotoxic CD8+ T lymphocytes are the main players of the anti-tumour immune response, as they are able to directly eliminate tumour cells causing their apoptosis. The third type, the regulatory T lymphocytes, suppresses other cells of the immune system, preventing them from attacking normal tissue [START_REF] Sompayrac | How the immune system works[END_REF].

Particular proteins, the CD8 glycoproteins, on the surface of naive CD8+ T cells enhance the interaction with the B7 receptor expressed by dendritic cells and results in the maturation of CD8+ T lymphocytes, also known as CTLs [START_REF] Sompayrac | How the immune system works[END_REF].

In the hypothesis that lead to the development of our models, we also take into account the fact that the level of presentation of the antigens to the MHC may play a critical role in the effectiveness of the anti-tumour immune response [START_REF] Coulie | łTumour antigens recognized by T lymphocytes: at the core of cancer immunotherapyž[END_REF][START_REF] Messerschmidt | łHow cancers escape immune destruction and mechanisms of action for the new signiőcantly active immune therapies: Helping nonimmunologists decipher recent advancesž[END_REF]. By 'level of antigen presentation' we refer to different mechanisms that may affect the recognition of antigens by APCs, the activation of CTLs, and the efficacy of immune action. On the one hand, if the level of presentation of the antigens is too low, APCs will hardly recognize them, and CTLs will be poorly activated [START_REF] Gejman | łRejection of immunogenic tumor clones is limited by clonal fractionž[END_REF]. In addition, we assume that a low or incorrect presentation of antigens on the surface of tumour cells may result in potential difficulties for CTLs to bind to them, thus decreasing the efficacy of their immune action. On the other hand, if the level of antigen presentation is sufficiently high, CHAPTER 1. Introduction CTLs will be well activated and may eventually mount a more efficient immune response.

Immune action After their activation, CTLs proliferate and migrate to the tumour site through blood vessels. Once in the tumour microenvironment (TME), the presence of small soluble proteins called chemokines (e.g. CCL3, CCL5, CCL20, and CXCL10) produced by both tumour cells and immune cells, will help the movement and inőltration of CTLs into the tumour (see [START_REF] Boissonnas | łIn vivo imaging of cytotoxic T cell inőltration and elimination of a solid tumorž[END_REF][START_REF] Franciszkiewicz | łRole of Chemokines and Chemokine Receptors in Shaping the Effector Phase of the Antitumor Immune Responsež[END_REF][START_REF] Zlotnik | łChemokines and cancerž[END_REF]). This particular form of movement is governed by the ability of cells that are sensitive to an attractive chemical signal (e.g. the chemokines) to move in the directions of stronger gradients of it. Throughout this thesis we will refer to this form of movement as chemotaxis, while the chemical signal will be referred to as chemoattractant.

Following the inŕux of CTLs in the TME and their inőltration inside the tumour, the CTLs recognise tumour cells through the interaction of their TCR with an antigen-MHC complex on the surface of the tumour cell. Note that CTLs only produce one type of TCR, which recognises and binds speciőcally to a certain antigen (i.e. the cognate antigen) [START_REF] Coulie | łTumour antigens recognized by T lymphocytes: at the core of cancer immunotherapyž[END_REF], and possibly other antigens within a certain affinity range [START_REF] Mason | łA very high level of crossreactivity is an essential feature of the T-cell receptorž[END_REF][START_REF] Wooldridge | łA single autoimmune T cell receptor recognizes more than a million different peptidesž[END_REF]. This enables CTLs to exert an antigenspeciőc cytotoxic activity against tumour cells, whose efficacy may depend on the affinity range of TCRs and the level of antigen presentation. Upon recognition of the cognate antigen, CTLs can trigger tumour cell death by direct interaction with tumour cells, releasing cytotoxic factors (i.e. granzyme B, INF-γ) [START_REF] Kim | łCancer immunoediting: from immune surveillance to immune escapež[END_REF], causing their apoptosis. The apoptosis of tumour cells results in the release of additional antigens, which enhance the immune action.

In general, in the őnal step of immune response, CTLs must undergo apoptosis to prevent any damage to normal cells. The apoptosis of CTLs prevents the risk of autoimmune diseases occurring and the formation of a memory population which allow for a faster response if the same antigen is recognised in the future [START_REF] Klebanoff | łCD8+ T-cell memory in tumor immunology and immunotherapyž[END_REF].

Overall, many experimental studies have concluded that the immune system is able to őght tumour cells and remove early stage tumours by a loop called the cancer-immunity cycle [START_REF] Chen | łOncology meets immunology: the cancer-immunity cyclež[END_REF] (see Figure 1.1). However, through antigenic changes and other evolutionary mechanisms this cycle is subject to various obstacles, and tumour cells can adapt to evade the immune response through a process known as immunoediting.

In this thesis, we focus on the interaction dynamics occurring between tumour cells and CTLs. We use different properties of these cell types, e.g. the level of presentation of tumour antigens, the affinity between tumour antigens and TCRs and the movement of CTLs, to develop the mathematical models described in Chapters 2-5.

Immunoediting

Despite the important role played by the immune system in eliminating tumour cells, it is clear that, most of the time, the cancer-immunity cycle does not perform optimally in cancer patients. This is because, as tumour cells grow, they inhibit the protective functions of the immune system and use some mechanisms that promote tumour development.

In [START_REF] Dunn | łCancer immunoediting: from immunosurveillance to tumor escapež[END_REF], Dunn and collaborators proposed a new theory to describe the three different stages of the immune response to tumours: elimination, equilibrium and escape. These stages have been named the łthree Es of cancer immunoeditingž. Elimination refers to the inŕammatory response of the immune system to tumour cells and subsequent tumour eradication. If the tumour mass is able to evade immune eradication in the elimination stage, it enters the equilibrium stage. Here, the tumour is kept at roughly the same size and is prevented from inducing an angiogenic CHAPTER 1. Introduction response. The last łEž of immunoediting is escape. Whilst in the equilibrium stage, interactions between tumour cells and immune cells can produce two possible outcomes: tumour elimination or tumour escape. In the őrst, the tumour is eliminated by the immune system, making this outcome equivalent to the elimination stage. The second outcome is when the tumour cells have managed to, through different factors, escape the immune action. The interaction between immune cells and tumour cells over a long period of time neutralises immune attack, leading the tumour to grow uncontrollably.

In the following paragraphs, we highlight some of the key evasion mechanisms which may lead tumour cells to avoid immune destruction, which are also schematically illustrated in Figure 1.2. Then, a new classiőcation of tumours, based on the immune context, rather than on the tumour anatomical extent, is discussed.

Immune checkpoints Immune checkpoints are regulators of the immune system. In general, these pathways are crucial for self-tolerance, which prevents the immune system from attacking cells indiscriminately. However, in the context of cancers, some tumour cells can evade immune response by upregulating these immune checkpoint targets [START_REF] Darvin | łImmune checkpoint inhibitors: recent progress and potential biomarkersž[END_REF][START_REF] Vaddepally | łReview of indications of FDA-approved immune checkpoint inhibitors per NCCN guidelines with the level of evidencež[END_REF].

For example, tumours may evade immune responses using negative feedback mechanisms to prevent apoptosis and downregulate the immune response either before or after its activation [START_REF] Darvin | łImmune checkpoint inhibitors: recent progress and potential biomarkersž[END_REF][START_REF] Vaddepally | łReview of indications of FDA-approved immune checkpoint inhibitors per NCCN guidelines with the level of evidencež[END_REF]. These include immunosuppressive cytokines such as IL-10 and tumour growth factor (TGF)-β, inhibitory cell types such as regulatory T lymphocytes (Treg) or Myeloid-derived suppressor cells (MDCSs) and inhibitory receptors such as PD-1 and CTLA-4 (cf. Figure 1.2(B)). For example, the interaction between the CTLA-4 (cytotoxic T lymphocyte-associated antigen 4) protein with the activating receptor B7 expressed by dendritic cells may prevent CTL activation in secondary lymphoid organs, therefore limiting the number of activated CTLs and their efficiency at eliminating tumour cells. Similarly, after the arrival of CTLs in the TME, increased expression of PD-L1 (programmed death ligand 1) allows tumour cells to bind to the PD-1 (programmed death 1) receptor on CTLs, causing immune cell death or inhibition (cf. Figure 1.2(C)).

Alternatively, tumour cells can evade cell-cell recognition by immune cells through manipulating the binding process. The binding between the MHC molecule and the tumour antigen is the őrst step in antigen recognition. However, tumour cells can cause downregulation of MHC molecules, preventing presentation of the antigen (cf. left and central panel of Figure 1.2(D)). This may limit the activation of CTLs and their ability to recognise tumour cells, ultimately resulting in tumour escape [START_REF] Garrido | łłHardž and łsoftž lesions underlying the HLA class I alterations in cancer cells: implications for immunotherapyž[END_REF][START_REF] Garrido | łThe urgent need to recover MHC class I in cancers for effective immunotherapyž[END_REF][START_REF] Leone | łMHC class I antigen processing and presenting machinery: organization, function, and defects in tumor cellsž[END_REF].

With the aim of targeting the immune resistant mechanisms of tumour cells, immune checkpoint inhibitors have been developed as a method of treatment to allow for a stronger immune response.

Presence of sub-populations of tumour cells expressing sub-clonal antigens As tumours grow, they acquire mutations that can aid in immune resistance and immune evasion. In particular, these mutations may create neoantigens, potentially preventing CTLs from recognising them. There are many possible reasons for this, such as loss of antigen expression, creation of antigen variants not recognized by CTLs, etc. It has recently been reported that many of these neoantigens arise from sub-clonal branching mutations and could potentially increase intratumour heterogeneity (ITH) [START_REF] Mcdonald | łTumor heterogeneity correlates with less immune response and worse survival in breast cancer patientsž[END_REF][START_REF] Mcgranahan | łClonal neoantigens elicit T cell immunoreactivity and sensitivity to immune checkpoint blockadež[END_REF][START_REF] Reuben | łTCR repertoire intratumor heterogeneity in localized lung adenocarcinomas: an association with predicted neoantigen heterogeneity and postsurgical recurrencež[END_REF]. ITH describes the existence of distinct tumour cell populations with speciőc genetic, epigenetic and phenotypic features within the same tumour. It has recently been found to affect the effectiveness of the anti-tumour immune response in several tumour types, such as lung adenocarcinomas [START_REF] Reuben | łTCR repertoire intratumor heterogeneity in localized lung adenocarcinomas: an association with predicted neoantigen heterogeneity and postsurgical recurrencež[END_REF], breast cancer [START_REF] Mcdonald | łTumor heterogeneity correlates with less immune response and worse survival in breast cancer patientsž[END_REF] and melanoma [START_REF] Mcgranahan | łClonal neoantigens elicit T cell immunoreactivity and sensitivity to immune checkpoint blockadež[END_REF].

Tumours with high ITH are characterised by clonal antigens (presented by all tumour cells), and sub-clonal antigens (presented only by sub-populations of tumour cells) [START_REF] Mcgranahan | łClonal neoantigens elicit T cell immunoreactivity and sensitivity to immune checkpoint blockadež[END_REF]. Moreover, such sub-clonal antigens may be associated with a reduced level of antigen presentation by the MHC molecules, decreasing or preventing presentation of the antigen. This may limit the ability of CTLs to activate or recognise tumour cells, possibly leading to tumour survival [START_REF] Gejman | łRejection of immunogenic tumor clones is limited by clonal fractionž[END_REF] (cf. right panel of Figure 1.2(D)). In contrast, more homogeneous tumours (i.e. tumours with low ITH) express few clonal antigens in all tumour cells and appear to trigger a more efficient immune response across a wide range of tumour types [START_REF] Fennemann | łAttacking tumors from all sides: Personalized multiplex vaccines to tackle intratumor heterogeneityž[END_REF][START_REF] Mcgranahan | łClonal neoantigens elicit T cell immunoreactivity and sensitivity to immune checkpoint blockadež[END_REF]. Furthermore, CTLs activated against clonal antigens are more commonly found at the tumour site than CTLs reactive to sub-clonal antigens [START_REF] Mcgranahan | łClonal neoantigens elicit T cell immunoreactivity and sensitivity to immune checkpoint blockadež[END_REF]. These őndings suggest that ITH may strongly affect the effectiveness of the anti-tumour immune response.

In Chapter 3, we present a spatial stochastic individual-based model of tumour-immune interaction dynamics that can be used to explore the effect of ITH on immune surveillance. The model makes it possible to dissect the speciőc impact of two expressions of ITH on antitumour immune response. Through computational simulations of our model, the effect of ITH is investigated at different levels, in order to assess its impact on anti-tumour immunity in a controlled manner.

Immune contexture and Immunoscore The tumour site is a complex and continuously evolving entity, which include immune cells, signalling molecules and extracellular matrix. In particular, it has been observed that type, density and location of immune cells within the tumour site correlate with prognosis in different types of cancer [START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF]. As a result, the immune contexture, which is determined by the type, density, immune functional orientation and localization of immune cells within a tumour, can yield information that is relevant to prognosis and prediction of a treatment response [START_REF] Angell | łFrom the immune contexture to the Immunoscore: the role of prognostic and predictive immune markers in cancerž[END_REF][START_REF] Bruni | łThe immune contexture and Immunoscore in cancer prognosis and therapeutic efficacyž[END_REF]. 'Contexture' refers to the act of assembling parts into a whole, as an arrangement of interconnected parts.

Derived from the immune contexture, a simple and powerful immune-classiőcation has been termed as the 'Immunoscore' and is now used as a prognostic marker in cancer patients [START_REF] Angell | łFrom the immune contexture to the Immunoscore: the role of prognostic and predictive immune markers in cancerž[END_REF][START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF][START_REF] Galon | density, and location of immune cells within human colorectal tumors predict clinical outcomež[END_REF][START_REF] Galon | łImmunoscore and Immunoproőling in cancer: an update from the melanoma and immunotherapy bridge 2015[END_REF]. The Immunoscore provides a score that increases with the density of CD8+ and CD3+ T cells both in the centre and at the margin of the tumour. In this vein, a new immunebased, rather than a cancer-based, classiőcation of tumours based on the Immunoscore has been proposed in [START_REF] Galon | density, and location of immune cells within human colorectal tumors predict clinical outcomež[END_REF], where the authors have classiőed tumours in four categories. The "hot" category is characterized by tumours inőltrated with a large number of CTLs in the centre of it and, therefore, have a high Immunoscore. Such tumours present dysfunctions in the last steps of the cancer-immunity cycle (see Figure 1.1). In particular, inőltrated CTLs are exhausted or dysfunctional and express a number of inhibitory receptors, most notably CTLA-4 and PD-1 [START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF]. The category "altered-immunosuppressed" denotes tumours with a small amount of inőltrated CTLs. Here, the lack of adequate innate immune response constitutes a limiting factor restraining the development of an effective, adaptive anti-tumour response, therefore originating an immunosuppressed tumour. Tumours in the "altered-excluded" category are characterized by two different regions: their margin is inőltrated by activated CTLs while the centre is not. One reason for T cell exclusion is the lack of T cell-recruiting signals, such as chemokines directing T cell trafficking, or the presence of physical barriers. Tumours in this two categories have an intermediate Immunoscore. Moreover, such tumours present dysfunctions in the intermediate steps of the cancer-immunity cycle. Finally, "cold" tumours have a low Immunoscore and are often correlated to a poor response to immunotherapies since CTLs are absent both within the centre of the tumour and at its margin. Such tumours completely lack a pre-existing immune response.

In Chapter 4, we develop an hybrid discrete-continuum modelling approach to explore the role of T cell inőltration in the immune response against solid tumours. Abstracting from the 'immunoscore' based classiőcation of tumours developed by Galon and collaborators [START_REF] Galon | density, and location of immune cells within human colorectal tumors predict clinical outcomež[END_REF], we classify different tumour scenarios depending on the spatial distribution of CTLs within the tumour. Moreover, we investigate the impact of CTL inőltration on the response of tumour cells to different types of anti-cancer immunotherapy. Also, in Chapter 5, we develop a stochastic individual-based model to investigate the effects of physiological stress on immune inőltration. Moreover, we develop an 'inőltration score' to quantitatively evaluate immune cell inőltration into the tumour. The results of numerical simulations of this model are able to qualitatively reproduce the experimental results presented in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF].

Given the extremely large amount of data showing the prognostic and predictive power of speciőc immune components (such as degree of immune inőltration, presence of immune suppressive cells or T cell checkpoint, low expression of antigens), it is important to assess individual immune determinants and/or a combination of immune determinants. This would allow for a more accurate patient assessment that would guide the design of effective tumour therapies [START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF].

Immunotherapy

Traditional treatments for tumours include chemotherapy and radiotherapy, which aim at attacking all rapidly-dividing cells within the body, effectively targeting fast-growing tumour cells. These types of treatments have proven to be beneőcial in many cases and are widely used to treat most types of tumours. However, these treatments can potentially have severe side effects. For example, the damage caused to healthy cells as well as tumour cells can result in further illness or even fatalities [START_REF] Carelle | łChanging patient perceptions of the side effects of cancer chemotherapyž[END_REF].

Targeted therapies, like immunotherapy, may be able to reduce the damage to normal cells. Generally, the goal of immunotherapies is to initiate or reinitiate a self-sustaining cancer-immunity cycle (see Figure 1.3), enabling it to restore or amplify immune response, but not so much as to generate unrestrained autoimmune inŕammatory responses. Tumour immunotherapies must therefore be carefully conőgured to overcome the negative feedback mechanisms that block the proper functioning of the cancer-immunity cycle.

Monoclonal antibodies Monoclonal antibodies are antibodies designed to bind to speciőc proteins. The aim of therapies based on these antibodies is to block immune checkpoints [START_REF] Darvin | łImmune checkpoint inhibitors: recent progress and potential biomarkersž[END_REF][START_REF] Huang | łT-cell invigoration to tumour burden ratio associated with anti-PD-1 responsež[END_REF][START_REF] Ribas | łCancer immunotherapy using checkpoint blockadež[END_REF][START_REF] Tumeh | łPD-1 blockade induces responses by inhibiting adaptive immune resistancež[END_REF]. For this reason, they are also called immune checkpoint inhibitors. As previously mentioned, overexpression of the checkpoint proteins CTLA-4 and PD-1 by CTLs, as well as overexpression of PD-L1 ligands by tumour cells, is a major factor in suppressing the antitumour immune response. In particular, CTLs that have been in contact with tumour cells possessing PD-L1 ligands are inhibited and become exhausted, i.e. tolerant to the presence of the tumour. Therefore, treatments (such as nivolumab, pembrolizumab, and lambrolizumab) that block PD-1 and PD-L1 interactions are able to restore the effector activity of exhausted CTLs. Similarly, monoclonal antibodies capable of blocking interactions between the checkpoint proteins CTLA-4 and B7 are able to restore the mechanism of T cell activation in secondary 1.2. Biological overview of the immune response to tumours employing these monoclonal antibodies have exhibited durable responses and improved survival of patients across a range of cancers including advanced melanoma [START_REF] Wolchok | łOverall survival with combined nivolumab and ipilimumab in advanced melanomaž[END_REF], Non-small cell lung cancer (NSCLC) [START_REF] Hellmann | łCombinatorial cancer immunotherapiesž[END_REF] and renal cancer [START_REF] Motzer | łNivolumab plus ipilimumab versus sunitinib in advanced renal-cell carcinomaž[END_REF], and have thus been granted regulatory approval from the Food and Drug Administration [START_REF] Vaddepally | łReview of indications of FDA-approved immune checkpoint inhibitors per NCCN guidelines with the level of evidencež[END_REF] and the European Medicines Agency [START_REF] Grilo | łThe increasingly human and proőtable monoclonal antibody marketž[END_REF] for those indications.

Adoptive T cell therapy The effectiveness of monoclonal antibody therapies relies on the fact that the immune system already produces sufficient tumour antigen-speciőc CTLs. However, in many tumours the amount of these CTLs is very limited. Two immunotherapy strategies can address this problem: adoptive cell transfer and modiőed cell transfer therapies [START_REF] Fesnak | łEngineered T cells: the promise and challenges of cancer immunotherapyž[END_REF]. Adoptive cell transfer consists in taking tumour-inőltrating CTLs from the patient that are speciőc to the tumour antigens and then culturing them in the presence of particular cytokines (such as IL-2) to induce their proliferation. Once the required number of CTLs has been reached, the cultured cells are then re-injected into the patient. The use of the patient's own CTLs allows this therapy to become more adaptable and personalised for each patient.

However, in some more severe cases, it is very difficult to isolate cells speciőc to the tumour antigens. Additionally, by targeting common antigens there is a risk of attacking healthy cells, leading to severe side effects [START_REF] Gross | łTherapeutic potential of T cell chimeric antigen receptors (CARs) in cancer treatment: counteracting off-tumor toxicities for safe CAR T cell therapyž[END_REF]. Therefore, modiőed cell transfer therapy involves genetically modifying the patient's CTLs so that they can better recognise the tumour antigens. One of the best known treatment based on modiőed cell transfer therapy today is the one based on CAR-T cells [START_REF] Wang | łClinical manufacturing of CAR T cells: foundation of a promising therapyž[END_REF]. Chimeric antigen receptors (CARs, also known as chimeric T cell receptors or artiőcial T cell receptors) are receptor proteins that have been engineered to give CTLs the new ability to target a speciőc antigen. The receptors are chimeric because they combine both antigen-binding and T cell activating functions into a single receptor. There have been extensive experiments and clinical trials based on CAR-T cell therapies with successful initial results; however, very few have been approved as viable options for tumour treatment. This is caused by the potential risk of CAR-T cells attacking non-tumour cells that express the target antigen [START_REF] Gross | łTherapeutic potential of T cell chimeric antigen receptors (CARs) in cancer treatment: counteracting off-tumor toxicities for safe CAR T cell therapyž[END_REF].

Combinatorial immune parameters Despite the clinical success of immune checkpoint blockade and adoptive CAR T-cell therapy, these approaches have shown substantial beneőt to only some of the patients, while the rest have not responded due to immune evasion [START_REF] Pauken | łAdverse events following cancer immunotherapy: obstacles and opportunitiesž[END_REF]. For example, in cold tumours, the lack of CTLs in the TME can lead to a failure of immune checkpoint inhibitor therapies, such as the anti-PD-1/PD-L1 therapy. Also, the mutations that create new sub-clonal antigens or an immunosuppressed TME can inhibit or limit the effects of adoptive T cell therapy [START_REF] Beatty | łChimeric antigen receptor T cells are vulnerable to immunosuppressive mechanisms present within the tumor microenvironmentž[END_REF].

As previously said, the analysis of the main immune determinants shaping tumour development, as well as the Immunoscore and immune contexture parameters, are useful prognostic (associated with survival) and predictive (associated with response to treatment) tools that should be used to guide the choice of the most effective therapeutic strategy. In this sense, it is reasonable to assume that the colder the tumour is, the more approaches are needed [START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF]. For example, hot tumours, which display a high degree of T cell inőltration [START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF], are associated with increased response to anti-PD-1 or anti-PD-L1 monotherapy (cf. Figure 1.3). In altered tumours, the combination of therapeutic epigenetic modulation directed to increase the level of chemokines for T cell trafficking and inőltration, with anti-PD-L1 treatment, has shown to slower tumour progression in preclinical models [START_REF] Nagarsheth | łPRC2 epigenetically silences Th1-type chemokines to suppress effector T-cell trafficking in colon cancerž[END_REF]. Finally, cold tumours, characterized by low Immunoscore, are the most challenging to eradicate and are invariably associated with poor prognosis. A proposed approach to overcome the lack of a pre-existing immune response is to combine a priming therapy that enhances T cell responses (such as chemotherapy or radiotherapy) with the removal of co-inhibitory signals (e.g. immune checkpoint inhibitors) and/or the supply of co-stimulatory signals [START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF]. For example, in [START_REF] Gandhi | łPembrolizumab plus chemotherapy in metastatic non-small-cell lung cancerž[END_REF] the authors found that the combination of chemotherapy, which increases the release of tumour-associated antigens and stimulates the activation of CTLs, with anti-PD1 treatment, had clinical beneőts in NSCLC.

Given the pivotal role of CTLs against cancer, it is clear that the identiőcation of key features, immune-related, at the moment of diagnosis is needed to build a solid classiőcation strategy supporting subsequent therapy.

The mathematical model developed in Chapter 4 is used to explore potential causes for the emergence of cold, altered-immunosuppressed, altered-excluded and hot tumour scenarios. Then, it is used to investigate how the effects of combination immunotherapies, namely anti-PD-1 therapy, anti-PD1-CTLA4 dual therapy and chemotherapy combined with anti-PD1 therapy, may vary the success of the immune response against tumour cells in the different categories of tumour scenarios.

Review of modelling strategies

In this section, we discuss the literature on mathematical models that have been developed to describe the interactions between a nascent tumour and the immune system.

In this thesis, we will develop discrete and continuum models to describe the spatio-temporal dynamics of the interactions between a solid tumour and cytotoxic T cells. Therefore, we begin this section with a discussion on the different methods of mathematical modelling used to describe tumour-immune interactions, which include discrete individual-based models, deterministic homogeneous and spatial models, and combinations of the two approaches, i.e. hybrid discrete-continuum models. Then we discuss some of the key models, from the literature, that have been developed previously to describe tumour development and its interaction with the immune system.

Methods of mathematical modelling

Mathematical models for tumour-immune interactions have become increasingly popular over the past few decades and are used as a tool to aid in the understanding of the mechanisms of tumour escape. Incorporating in these models the effects of therapeutic strategies that boost anti-tumour immune response can help biologists and clinicians to predict the success of cancer treatment protocols, including immunotherapy protocols.

Cellular processes in the tumour environment exhibits multiscale properties and the interactions between tumour cells and immune cells can occur across various spatial and temporal scales. Generally, there are three spatial scales potentially considered in modelling: the molecular scale (i.e. nmµm), the microscopic scale (i.e. µmmm), and the macroscopic scale (i.e. mm-cm) [START_REF] Deisboeck | łMultiscale cancer modelingž[END_REF]. The molecular scale is used to model intracellular processes, such as cell signaling mechanisms (in the context of tumours: cytokines, chemokines, etc.). The microscopic scale is used to model extracellular processes, representing for example the interactions between tumour and immune cells. Models at the macroscopic scale focus on the dynamics of the total tumour behaviour occurring at the tissue level including morphology, shape, extent of vascularization, and invasion. Note that these scales are sometimes referred to, respectively, as microscopic, mesoscopic and macroscopic [START_REF] Masoudi-Nejad | łCancer systems biology and modeling: microscopic scale and multiscale approachesž[END_REF]. Since many biological mechanisms in the immune response to tumours involve processes spanning different spatio-temporal scales, in the past twenty years new multiscale models have been proposed, where these multiscale properties of tumour-immune interactions can be effectively incorporated [START_REF] Masoudi-Nejad | łCancer systems biology and modeling: microscopic scale and multiscale approachesž[END_REF].

Furthermore, the mathematical modelling of tumour-immune interactions can involve techniques that are discrete, continuum, or hybrid, i.e., the integration of both. Discrete models usually rely on an explicit representation of individual cells in space and time, tracking and updating their internal states according to a predeőned set of biological and biophysical rules. Therefore, they are often referred to as individual-based models. They can be stochastic and are particularly suited to describe processes occurring at the microscopic scale and over short timescales (minutes-hours). They involve the representation of populations of low cell numbers (usually < O(10 5 )), which makes it possible to represent stochasticity and heterogeneity of the dynamics observed in small-scale phenomena. However, their computational cost increases rapidly with the number of cells modelled and they are usually less amenable to mathematical analysis. On the other hand, continuum models are, generally, deterministic and less computationally expensive than discrete models, allowing for the investigation of larger cell population sizes, where the small scale stochastic effects can be neglected. For this reason continuum models are particularly popular for tissue level dynamics over longer timescales (days-years). They give a macroscopic description of the system under study, as the terms in the model equations provide a mean-őeld representation of the underlying cellular dynamics. However, they cannot easily capture the emergence of population-level phenomena that are induced by stochastic ŕuctuations in single-cell biophysical properties, which are relevant in the regime of low cellular densities. By using hybrid discrete-continuum models , we can integrate the strengths of both continuum and discrete descriptions. Usually, in hybrid models a multiscale description of the system under study is given, where individual cells are treated discretely, but extracellular factors, such as chemical signals, are often modeled as continuous quantities. Hybrid models, similarly to individual-based models, may still be computationally expensive and less amenable to analytical investigations.

Various mathematical methods are used to model tumour-immune interactions including deterministic continuum model formulated in terms of differential equations, stochastic discrete individual-based models and combinations of these. The choice of the modelling approach should depends on the biological phenomena under investigation, the way one would represent the system (discrete or continuous), the spatial and time scale, as well as the kind of information (from hypothesis-driven to data-driven) one has.

In this thesis, we develop discrete individual-based and hybrid discrete-continuum modelling techniques to describe tumour-immune interactions in Chapters 2-5. Moreover, in Chapters 2 and 4 we will formally derive the deterministic continuum limit of these models.

Deterministic continuum models

The speciőc interactions between tumour cells and immune cells have mostly been modelled using three classical deterministic continuum techniques: Ordinary, Integro and Partial Differential Equation systems.

A system of ordinary differential equations (ODEs) considers the change of continuous variables over either time or space. In the the tumour-immune setting, systems of ODEs are often used to investigate the importance of certain underlying dynamics, for example cell-cell interactions, whereas spatial interactions are neglected. Integro-differential equations (IDEs) are similar to ODEs, but also involve integrals of the unknown variables.

On the other hand, a system of partial differential equations (PDEs) considers the change of continuous variables which depend on both space and time. In the the tumour-immune setting, systems of PDEs can model cellular and chemical concentration at given times, allowing for the inclusion of a wider range of biological aspects, such as cell movement or chemical diffusion.

In the same way that space provides a structural variable to account for cell movement, other structural variables can be included in PDE systems. Such structure variables can represent the size of the cells, and, therefore, allow to model the mitosis of the cells (i.e. when one mother cell divides into two daughter cells), or internal traits (i.e. phenotypes), characteristic of a relevant diversity in a cell population. In the context of tumours, such internal trait can represent the antigenic expression of tumour cells or their sensitivity to a drug under study. Therefore, it can be used, for example, to study the evolution of cell traits with time under the effect of environmental pressures, e.g. due to the interaction with the immune system or the introduction of anti-cancer drugs.

The continuum model presented in Chapter 2 describes the coevolutionary dynamics between tumour cells and CTLs in a well-mixed system (i.e. spatial interactions are not incorporated into the model). It is structured by a variable representing a parameterisation of the antigen expression proőles for tumour cells and a parameterisation of the target antigens of T-cell receptors (TCRs) for CTLs. The model comprises a non-local PDE for the phenotype distribution of tumour cells coupled with an IDE for the phenotype distribution of CTLs. The continuum model presented in Chapter 4 describes spatial interactions between tumour cells and CTLs and explicitly takes into account the chemotactic movement of CTLs towards the tumour. This model comprises an IDE for the density of tumour cells coupled with a PDE for the density of CTLs and a PDE for the concentration of a chemoattractant.

Stochastic discrete individual-based models Among other types of techniques that describe the interactions of tumour cells and its local environment there are stochastic and discrete individual-(or agent-) based models. These models can be subdivided into two different categories based on their geometry: on lattice and off lattice. In on lattice models, cells are distributed in a grid system modelling physical space, whereas in off lattice models cells are not conőned by grid restriction allowing the representation of complex cell shapes. Individual-based models attempt to describe how cells behave and interact over time by specifying rules for the behaviour of each individual cell (individual-based), helping to address the role of diversity in cell populations and also within each individual cell. These rules are formulated in terms of the probability of events occurring and allow for the observation of any patterns in the dynamics of the biological system. Speciőcally, cellular automata models are a type of on lattice individual-based models where, usually, only one cell can occupy each grid position through volume exclusion. The movement of cells and interactions with other cells within the system then depend on the rules of the model and the cells occupying the grids in the cell's neighbourhood.

An extension to the cellular automata model has been created in the form of a Cellular Potts model [START_REF] Graner | łSimulation of biological cell sorting using a two-dimensional extended Potts modelž[END_REF]. Here, the model is comprised of pixels with each cell, or cellular compartment, made up of several connecting pixels. Therefore, cells are no longer modelled as single points, but as spatial objects whose size and shape can be altered. The simulation progresses through a series of Monte Carlo Steps (MCS), which attempt to minimise the overall effective energy within the system. At each MCS, attempts are made to transform neighbouring pixels at the boundary of each cell to increase its volume. If the transformation, known as a łŕipž, reduces the overall effective energy then the pixel in question will be changed. The use of this type of model allows to include many biological aspects, such as cell adhesion, cell growth and mitosis, cell movement, etc, where each individual element of the system can be modelled actively using numerous different rules. This type of model has been successfully transformed into user friendly software. CompuCell3D is one of them [START_REF] Izaguirre | łCompuCell, a multi-model framework for simulation of morphogenesisž[END_REF], which is a very useful tool when modelling tumourimmune interaction dynamics.

For the computational implementation of the individual-based models developed in Chapter 3 and Chapter 5 we use a Cellular Potts approach, and simulations were developed and run using the software CompuCell3D. However, in order to have a deeper understanding and control over the dynamic produced, for the models presented in Chapter 2 and Chapter 4, we develop our own computational codes.

Apart from CompuCell3D, several major computational frameworks are available for studying 3D multicellular systems, such as Chaste [START_REF] Mirams | łChaste: an open source C++ library for computational physiology and biologyž[END_REF], CellSys [START_REF] Hoehme | łA cell-based simulation software for multi-cellular systemsž[END_REF] or PhisyCell [START_REF] Ghaffarizadeh | łPhysi-Cell: an open source physics-based cell simulator for 3-D multicellular systemsž[END_REF].

Hybrid discrete-continuum models Since many biological systems involve processes spanning different spatiotemporal scales, in recent years, it has become common to combine discrete and continuum approaches to develop hybrid discrete-continuum models. Such models combine individual-based models or cellular automata with ODEs or PDEs in order to provide a more in-depth description of the biological system under study. In hybrid discrete-continuum models for tumour-immune interactions, tumour cells and immune cells are usually treated as discrete entities which interact with other chemical continuum őelds, representing for example the concentration of chemokines, cytokines or oxygen in the TME. Hybrid discrete-continuous models provide an appropriate method to study tumour-immune interactions; however, they present some limitations, related for example to the possibly low number of cells and to the usually large number of parameters, the values of at least some of which are difficult or impossible to estimate from experimental data.

Mathematical modelling of tumour-immune dynamics

There have been many attempts over the years to model the interaction of the immune system and a growing tumour to try and help clinicians and experimentalists carry out their studies. These include deterministic continuum models, in the form of ODEs and PDEs, stochastic discrete models and hybrid discrete-continuum models. In this section, we will brieŕy discuss some of the key models, from the literature, that have been developed previously to describe the immune response to tumours. In particular, the models presented henceforth use discrete or continuum approaches to describe the interactions between tumour cells and immune cells, and in particular CTLs.

Deterministic nonspatial models

The őrst models developed to describe tumour-immune interactions were nonspatial models formulated in terms of ordinary differential equations (ODEs) or integro-differential equations (IDEs). Based on the prey-predator nature of the interactions between tumour cells and immune cells, such models describe the tumour as prey, whose predators are cells of the immune system (in particular, CTLs).

One of the őrst two-species model which focus on tumour-immune interactions is presented in the pivotal work of Kuznetsov and collaborators in [START_REF] Kuznetsov | łNonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysisž[END_REF], where the authors developed a system of CHAPTER 1. Introduction two ODEs to describe the interactions between CTLs and an immunogenic (i.e. able to produce an immune response) tumour. The Kuznetsov model is given as

       dc dt = S + α c cn ζ c + n -γ c nc -µ c c c ≡ c(t) : CTLs dn dt = α n n(1 -µ n n) -γ n cn n ≡ n(t) : tumour cells (1.1)
where functions c and n denote, respectively, the densities of effector cells (i.e. CTLs) and tumour cells. CTLs enter the system with constant inŕux S, are recruited at rate αcn ζc+n , are killed or inactivated at a rate proportional to the density of tumour cells, with constant of proportionality γ c , and die at rate µ c due to natural death. Tumour cells grow at a logistic rate α n (1µ n n) with carrying capacity 1/µ n , and are killed at a rate proportional to the density of CTLs, with constant of proportionality γ n . The model is used to describe the kinetics of growth and regression of the B-lymphoma BCL 1 in the spleen of mice. By comparing the model with experimental data, numerical estimates of parameters describing processes that cannot be measured in vivo are derived. Through numerical simulations of this model, the established behaviour of łsneaking throughž, in which larger tumours are eliminated but smaller ones remain and grow, is observed. Moreover, the analytical and numerical results of the model show the presence of stable limit cycles, meaning that the immune system and the tumour coexist and undergo oscillations. For tumours with low antigenicity (which is linked to the value of γ c ) these cycles may be relatively long, with the tumours kept under control in a dormant state for most of the time.

From this particular model, a general framework has been derived, deőned in [START_REF] Onofrio | łA general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferencesž[END_REF]. For particular choice of its terms, this general model has been used in the literature to describe different observed tumour-immune interaction dynamics [START_REF] Foryś | łAnti-tumor immunity and tumor anti-immunity in a mathematical model of tumor immunotherapyž[END_REF][START_REF] Kuznetsov | łNonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysisž[END_REF][START_REF] Sotolongo-Costa | łBehavior of tumors under nonstationary therapyž[END_REF][START_REF] Stepanova | łCourse of the immune reaction during the development of a malignant tumourž[END_REF]. In particular, this general framework is very helpful for elucidating basic and generic mechanisms that can induce observed behaviours, such as tumour regression and evasion, as well as tumour dormancy, a mechanism for which a small tumour continues to exist but is maintained at a restricted size by the immune system.

From these generic two equation models, many models have been developed by including one or more components representing speciőc aspects of the tumour-immune environment interactions such as: the interactions with normal tissue cells [START_REF] Deisboeck | łMultiscale cancer modelingž[END_REF], or with cytokines (e.g., IL-2, IFN-γ) and chemokines [START_REF] Kirschner | łModeling immunotherapy of the tumor-immune interactionž[END_REF].

In nonspatial models, the expression and recognition of tumour antigens is usually modelled by variation of the rates of activation and proliferation of CTLs and of killing of tumour cells [START_REF] Arciero | łA mathematical model of tumorimmune evasion and siRNA treatmentž[END_REF][START_REF] Besse | łStability analysis of a model of interaction between the immune system and cancer cells in chronic myelogenous leukemiaž[END_REF][START_REF] Frascoli | łA Dynamical Model of Tumour Immunotherapyž[END_REF]. More recently, these processes have been explicitly modelled by nonspatial models formulated in terms of either ODEs [START_REF] Aguadé-Gorgorió | łTumour neoantigen heterogeneity thresholds provide a time window for combination immunotherapyž[END_REF][START_REF] Balachandran | łIdentiőcation of unique neoantigen qualities in long-term survivors of pancreatic cancerž[END_REF][START_REF] Cattani | łMetamodeling the learning-hiding competition between tumours and the immune system: a kinematic approachž[END_REF][START_REF] Kolev | łNumerical simulations for tumor and cellular immune system interactions in lung cancer treatmentž[END_REF][START_REF] Łuksza | łA neoantigen őtness model predicts tumour response to checkpoint blockade immunotherapyž[END_REF] or IDEs [START_REF] Delitala | łRecognition and learning in a mathematical model for immune response against cancer[END_REF][START_REF] Delitala | łA mathematical model for immune and autoimmune response mediated by T-cellsž[END_REF][START_REF] Lorenzi | łMathematical model reveals how regulating the three phases of T-cell response could counteract immune evasionž[END_REF].

A good review of nonspatial models that study interactions between immune and tumour cells can be found in [START_REF] Eftimie | łInteractions between the immune system and cancer: a brief review of non-spatial mathematical modelsž[END_REF].

Deterministic spatial models While deterministic nonspatial models are a good basis for modelling research, key dynamics of tumour-immune interactions can be conőrmed or expanded in a wider range of biological aspects through the inclusion of spatial phenomena, e.g. random motility or chemotaxis, of the cells. This can be obtained by extending ODEs models into deterministic spatial models using partial differential equations [START_REF] Atsou | łA size and space structured model describing interactions of tumor cells with immune cells reveals cancer persistent equilibrium states in tumorigenesisž[END_REF][START_REF] Matzavinos | łMathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumourž[END_REF][START_REF] Matzavinos | łTravelling-wave analysis of a model of the immune response to cancerž[END_REF][START_REF] Al-Tameemi | łEvasion of tumours from the control of the immune system: consequences of brief encountersž[END_REF].

In particular, following the work of Kuznetsov and collaborators [START_REF] Kuznetsov | łNonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysisž[END_REF], in [START_REF] Matzavinos | łMathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumourž[END_REF], Matzavinos and collaborators used a Fisher-Kolmogorov type model to describe the logistic growth of a tumour coupled with a chemotaxis model to describe the movement of CTLs into the tumour microenvironment and an equation describing the dynamics of tumour cell-CTLs complexes. The Matzavinos model is given as

                       dc dt = β c ∆ x c -χ c ∇ x (c∇ x ϕ) + Sh(x) + α c r ζ c + n c ≡ c(x, t) : CTLs dϕ dt = β ϕ ∆ x ϕ + α ϕ r -κ ϕ ϕ ϕ ≡ ϕ(x, t) : chemoattractant dn dt = β n ∆ x n + ϕ n n(1 -µ n n) -γ n cn + α n r n ≡ n(x, t) : tumour cells dr dt = γ r cn -µ r r r ≡ r(x, t) : tumour cell-CTLs complexes (1.
2) Compared to model (1.1), the spatial structure included in this model allows one to describe the random motion of tumour cells and CTLs, as well as the chemotactic motion of CTLs towards the tumour. In particular, ϕ is the concentration of a speciőc substance, i.e. chemoattractant, along the gradient of which CTLs move with characteristic motility χ c . Tumour cells and CTLs move via undirected, random movement with diffusion coefficients β n and β c . Chemoattractant production is proportional to tumour cell-CTLs complex (denoted by the letter r). Once produced, the chemoattractant is assumed to diffuse throughout the domain with diffusion coefficient β ϕ and to decay with linear decay kinetics at rate κ ϕ . Moreover, the parameter S represents the inŕow of CTLs into the tissue (non-enhanced by the presence of tumour cells). The function h(x) models the blood vessels through which new CTLs can enter the domain. The other parameters represent the local kinetics between tumour cells, CTLs and tumour cell-CTLs complexes. Numerical simulations of the model show that eventually the tumour cells develop very small-amplitude oscillations about a 'dormant' state, indicating that the CTLs have successfully managed to keep the tumour under control. Moreover, they demonstrate the existence of cell distributions that are quasi-stationary in time and heterogeneous in space. A linear stability analysis of the underlying spatially homogeneous ODE kinetics coupled with a numerical investigation of the ODE system reveals the existence of a stable limit cycle. This is veriőed further by the authors undertaking a bifurcation analysis. These results allow to explain the complex heterogeneous spatio-temporal dynamics observed in the PDE system, and reveal the key parameter regimen in which the onset of tumour dormancy may occur.

In Chapter 4, we develop a spatial hybrid discrete-continuum modelling framework for the interaction dynamics between tumour cells and CTLs. We formally derive the deterministic continuum limit of this model. In the obtained deterministic continuum model, the motility of CTLs is described by modiőed forms of the terms of random motion and chemotactic motion of CTLs used in system (1.2). These modiőed forms of motility allow us to take into account volume-őlling effects [START_REF] Painter | łVolume-őlling and quorum-sensing in models for chemosensitive movementž[END_REF], which take into account possible reduction in cell motility at high cell densities.

Apart from purely spatial models, many others PDE models investigate the dynamics of the immune response to tumours, using other structured variables than space. For example, in [START_REF] Atsou | łA size and space structured model describing interactions of tumor cells with immune cells reveals cancer persistent equilibrium states in tumorigenesisž[END_REF], the authors have developed a system of PDEs structured in space and size to describe the early interactions between tumour cells and CTLs. In this model, the displacement of CTLs is governed by chemotaxis, according to signals emitted by the tumour. Numerical simulations of the model reveal the importance of space-structuration: space heterogeneities of the sources of naive immune cells, that provide, once activated, the CTLs that eliminate the tumour, dramatically inŕuence the immune response efficiency. In particular, it is shown that replacing the homogeneous distribution of immune cells by a few spots makes the immune response less efficient. Instead of the control of the tumour, that would be kept at a őxed mass, the authors observe that there is a periodic succession of rapid growth and remission phases. Discrete models By using discrete models, such as individual-based models and cellular automata models, a wider spectrum of biological phenomena can be translated into mathematical terms and described, such as cell-cell interactions, intracellular behaviour and individual cell properties. These models can be posed on a spatial domain (e.g. a grid), and a set of rules can be given to each cell with certain probabilities to achieve a detailed description of the interactions of tumour cells and the local tumour environment [START_REF] Christophe | łA biased competition theory of cytotoxic T lymphocyte interaction with tumor nodulesž[END_REF][START_REF] Kather | łIn silico modeling of immunotherapy and stroma-targeting therapies in human colorectal cancerž[END_REF][START_REF] Macfarlane | łModelling the immune response to cancer: an individual-based approach accounting for the difference in movement between inactive and activated T cellsž[END_REF][START_REF] Macfarlane | łA stochastic individual-based model to explore the role of spatial interactions and antigen recognition in the immune response against solid tumoursž[END_REF]. For example, in [START_REF] Macfarlane | łA stochastic individual-based model to explore the role of spatial interactions and antigen recognition in the immune response against solid tumoursž[END_REF] the authors used an individual-based approach to study the interactions between CTLs, dendritic cells and tumour cells. In their model, each tumour cell is characterised by an antigen proőle which can change over time due to either epimutations or mutations. In this model, the immune response against tumour cells is initiated by the dendritic cells that recognise the tumour antigens and present them to the CTLs. Consequently, CTLs become activated against the tumour cells expressing such antigens. Moreover, in the modelling strategies developed in [START_REF] Macfarlane | łA stochastic individual-based model to explore the role of spatial interactions and antigen recognition in the immune response against solid tumoursž[END_REF], the differences in movement between inactive and active immune cells are explicitly taken into account. The results obtained indicate that antigenic heterogeneity within a tumour determines the efficacy of immune action and highlight the complex interplay between spatial interactions and adaptive mechanisms that underpins the immune response against solid tumours.

CompuCell3D [START_REF] Izaguirre | łCompuCell, a multi-model framework for simulation of morphogenesisž[END_REF] is a very useful tool when modelling tumour-immune interactions. It allows the user to easily model an idealistic tumour environment where individual cells interact with one another. For example, in Tough PhD Thesis [START_REF] Tough | łMathematical modelling of the immune response to cancerž[END_REF], two Cellular Potts models have been developed and implemented within CompuCell3D to try and explore the dynamics between CTLs and tumour cells. The őrst model is based on the ODE and PDE models presented in [START_REF] Kuznetsov | łNonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysisž[END_REF][START_REF] Matzavinos | łMathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumourž[END_REF][START_REF] Matzavinos | łTravelling-wave analysis of a model of the immune response to cancerž[END_REF][START_REF] Al-Tameemi | łEvasion of tumours from the control of the immune system: consequences of brief encountersž[END_REF]; however, the extension of these works into a Cellular Potts model allows the inclusion of more precise biological aspects (such as cell adhesion, cell growth and mitosis, motility, competition for space, etc). Numerical simulations of the model are used to portray the three Es of immunoediting described in Section 1.2.3: elimination, equilibrium and escape. The model begins with a central tumour mass, which is attacked by CTLs. Tumour cells are allowed to divide when they reach a certain size, determined by how much oxygen is available to them. This creates different tumoural zones, whereby the tumour mass forms a proliferating rim, quiescent zone and necrotic core. CTLs are constantly supplied to the domain and can eliminate tumour cells at a certain success rate. As this success rate is varied, the three Es of immunoediting are realised. The second individual-based model try again to detect the three Es of immunoediting; however, this time considering two types of immune cells, c 1 and c 2 , and two types of tumour cells, n 1 and n 2 . Tumour cells n 1 are only attacked by immune cells c 1 , whereas tumour cells n 2 are killed by both c 1 and c 2 . These hypothesis try to investigate the idea that a tumour, when in interaction with a host immune system, can become less efficient. The idea stems from the hypothesis of immunoediting. As tumour cells are of mixed type, it is thought that the immune system actively kills cells that are more immunogenic, allowing cells that are less immunogenic to proliferate. Hence, the tumour evades łimmune destructionž. The model also investigate the impact of the łWarburg Effectž, i.e. the ability of tumour cells to use aerobic glycolysis to generate the energy needed for cellular processes, on tumour growth [START_REF] Vander Heiden | łUnderstanding the Warburg effect: the metabolic requirements of cell proliferationž[END_REF].

Hybrid discrete-continuum models Hybrid discrete-continuum models are also used to a great extent to study the immune response to tumours. In particular, the interaction between tumour cells and immune cells has been modelled using a combination between PDE and cellular automata models. In particular, in [START_REF] Mallet | łA cellular automata model of tumor-immune system interactionsž[END_REF] the authors use PDEs to describe the evolution of protumour nutrients within the tumour microenvironement and describe tumour cell, CTL and NK cell movement and interactions through the cellular automata approach. Using this framework, they explore the role of nutrients, cell-cell adhesion and immune cell inőltration capabilities on a growing tumour. The results of numerical simulations of the model demonstrate different types of tumour growth depending on key model parameters related to the tumour-immune system interactions. In the absence of the immune system, the model is able to reproduce tumours of both compact-circular and papillary (branchy) morphologies. Introducing the immune system to the model leads to various results, which display either stable or unstable oscillatory tumour growth, tumour elimination, or the inőltration of CTLs into the growing tumour.

Mathematical modelling of immunotherapies

Most of the works on modelling anti-tumour immunotherapy are based on systems of ODEs and are focused mainly on the last steps of cancer-immunity cycle, i.e., interaction of tumour cells with CTLs, death of tumour cells and inactivation of CTLs [START_REF] Besse | łStability analysis of a model of interaction between the immune system and cancer cells in chronic myelogenous leukemiaž[END_REF][START_REF] Kirschner | łModeling immunotherapy of the tumor-immune interactionž[END_REF][START_REF] Osojnik | łIdentifying and characterising the impact of excitability in a mathematical model of tumour-immune interactionsž[END_REF][START_REF] Shariatpanahi | łMathematical modeling of tumor-induced immunosuppression by myeloid-derived suppressor cells: Implications for therapeutic targeting strategiesž[END_REF]. The works in this direction are often based on the general prey-predator model presented in [START_REF] Onofrio | łA general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferencesž[END_REF], where the number or properties of competing immune cells change due to therapy [START_REF] Arciero | łA mathematical model of tumorimmune evasion and siRNA treatmentž[END_REF][START_REF] Konstorum | łAddressing current challenges in cancer immunotherapy with mathematical and computational modellingž[END_REF][START_REF] Frascoli | łA Dynamical Model of Tumour Immunotherapyž[END_REF][START_REF] Foryś | łAnti-tumor immunity and tumor anti-immunity in a mathematical model of tumor immunotherapyž[END_REF][START_REF] Sotolongo-Costa | łBehavior of tumors under nonstationary therapyž[END_REF]. The general outcomes of this type of models is that the immune system can effectively eradicate small tumours, whereas large enough tumours are able to escape immune surveillance. From a mathematical point of view, this corresponds to the presence of an unstable manifold, that separates the basins of attraction of two stable points, which correspond to the scenarios of immune clearance or immune escape. Under such mathematical formulation, an important question is how it can be possible, with the help of a therapeutic intervention, to move the initial state of the system, located in a immune escape area, to the immune clearance one. Such problems are in general formulated as optimal control problems and are solved analytically [START_REF] Castiglione | łOptimal control in a model of dendritic cell transfection cancer immunotherapyž[END_REF][START_REF] Hu | łModeling pancreatic cancer dynamics with immunotherapyž[END_REF].

Additionally, key markers for therapies to target, such as tumour antigens, can also be discovered through ODE models [START_REF] Balachandran | łIdentiőcation of unique neoantigen qualities in long-term survivors of pancreatic cancerž[END_REF][START_REF] Łuksza | łA neoantigen őtness model predicts tumour response to checkpoint blockade immunotherapyž[END_REF]. The account for the heterogeneity of antigens and the evolution of their expression proőle has also been realized through IDEs. Such approach are implemented for example in the work [START_REF] Lorenzi | łMathematical model reveals how regulating the three phases of T-cell response could counteract immune evasionž[END_REF], which develops a prey-predator model in which general T cell and target cell populations are structured by their respective target-antigenic and antigenic expression. The model is given as:

                 dc dt = [1 + T c1 (t)]α c c - µ c 1 + T c2 (t) ρ c (t)c + [1 + T c3 (t)] β c c 1 0 g(x, y; γ, θ)n dy c ≡ c(x, t) : CTLs dn dt = [α n -µ n ρ n (t)]n -β n n 1 0 g(y, x; γ, θ)c dx n ≡ n(y, t) : Target cells (1.3)
In system (1.3), variables x ∈ [0, 1] and y ∈ [0, 1] model the target-antigenic expressions of CTLs and the antigenic expressions of target cells. It is supposed that: (i) target cells proliferate at rate α n and die due to competition for limited space and resources with constant of proportionality µ n ; (ii) CTLs undergo antigen-independent proliferation at rate α c ; and (iii) CTL numbers are kept under control through homeostatic regulation mechanisms with constant of proportionality µ c . Functions ρ c (t) and ρ n (t) respectively represent the total number of CTLs and target cells. Moreover, interactions between antigen-speciőc CTLs and their targets result in antigen-driven CHAPTER 1. Introduction T-cell expansion at rate β c as well as selective action against target cells at rate β n . Furthermore, up to three hypothetical classes of immunotherapies are presented, that are respectively designed to enhance antigen-independent proliferation (T c1 -agents), interfere with homeostasis to promote self-renewal of antigen-speciőc CTLs (T c2 -agents), and stimulate antigen-driven expansion (T c3agents). First, the model is able to reproduce the three phases of the immune response orchestrated by CTLs: expansion, contraction and memory. Then, the model is devoted to the search for methods to counteract the dynamics of łchase and escapež, which can develop under heterogeneous time-varying expression of antigens. The results of numerical simulations show that the three hypothetical classes of immunotherapies under study can reduce the likelihood of immune evasion. Moreover, they suggest that therapeutic protocols relying on the simultaneous delivery of sufficiently high concentrations of T c1 -agents and T c2 -agents are the most effective of the therapeutic protocols considered here. This implies that the success of an immunotherapy protocol correlates strongly with its ability to shorten the duration of the contraction phase and stabilize as many CTLs as possible inside the long-lived memory reservoir.

In Chapter 2, we develop an individual-based model for the coevolutionary dynamics between tumour cells and CTLs. We show that a generalised version of the mathematical model (1.3) can be formally obtained as the deterministic continuum limit of such stochastic discrete model. In addition to the biological phenomena incorporated into the model (1.3), the continuum model obtained takes also into account the effect of changes in antigen expression proőles of tumour cells and more general forms of competitive feedback mechanisms regulating the growth of the numbers of tumour cells and CTLs.

Moreover, PDE models have been used, for example, to describe tumour growth in the presence of cytotoxic drugs, i.e. drugs that directly kill tumour cells, and cytostatic drugs, which instead slow down the velocity of the cell division cycle [START_REF] Cho | łModeling the dynamics of heterogeneity of solid tumors in response to chemotherapyž[END_REF][START_REF] Cho | łModeling the chemotherapy-induced selection of drugresistant traits during tumor growthž[END_REF][START_REF] Lorz | łModeling the effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumorsž[END_REF][START_REF] Stace | łDiscrete and continuum phenotype-structured models for the evolution of cancer cell populations under chemotherapyž[END_REF]. Many other discrete individual-based models and hybrid discrete-continuum models have also been utilized to predict the success of immunotherapy techniques [START_REF] Kather | łIn silico modeling of immunotherapy and stroma-targeting therapies in human colorectal cancerž[END_REF][START_REF] De Pillis | łSpatial tumor-immune modelingž[END_REF], such as PD-L1 inhibition therapies [START_REF] Gong | łA computational multiscale agent-based model for simulating spatio-temporal tumour immune response to PD1 and PDL1 inhibitionž[END_REF], antibody treatment [START_REF] Pappalardo | łSimB16: modeling induced immune system response against B16-melanomaž[END_REF] and cancer vaccines [START_REF] Kim | łModeling protective anti-tumor immunity via preventative cancer vaccines using a hybrid agent-based and delay differential equation approachž[END_REF].

Mathematical models in clinical oncology

Mathematical modelling of tumour-immune interactions has great potential. For example, it can help to reveal non-obvious or non-intuitive mechanisms which may lead to immune escape and allows new treatment strategies to be proposed and tested in an affordable way. Also, the study of such models can help to suggest optimization of anti-cancer therapies already introduced into clinical practice. However, compared with traditional, widely empirical methods of cancer research, methods based on mathematical modelling of tumour-immune interactions have not led to signiőcant success in clinical oncology. There are many reasons for this, such as őnding the right balance between the reductionalism of mathematical models and the level of complexity prevailing in the clinical environment. The interactions between tumour and immune cells can be mathematically reproduced with models of different types and complexity. Developing more and more detailed models can be a tempting activity, however, it is usually associated with certain difficulties. For example, models of increasing complexity are associated with the problem of incomplete calibration. On the other hand, due to the difficulty in estimating parameter values, many models consider simple systems of equations and instead identify potential parameter spaces of interest. However, mathematical modelling has already led to several predictions, validated using retrospective data [START_REF] Christophe | łA biased competition theory of cytotoxic T lymphocyte interaction with tumor nodulesž[END_REF][START_REF] Sherratt | łOncogenes, anti-oncogenes and the immune response to cancer: a mathematical modelž[END_REF][START_REF] Picco | łIntegrating models to quantify environment-mediated drug resistancež[END_REF][START_REF] Walker | łFrom Concept to Clinic: Mathematically Informed Immunotherapyž[END_REF], preclinical successes [START_REF] Almeida | łTreatment-induced shrinking of tumour aggregates: A nonlinear volume-őlling chemotactic approachž[END_REF][START_REF] Leonard | łNonlinear response to cancer nanotherapy due to macrophage interactions revealed by mathematical modeling and evaluated in a murine model via CRISPRmodulated macrophage polarizationž[END_REF] as well as initiated clinical trials [START_REF] Enriquez-Navas | łExploiting evolutionary principles to prolong tumor control in preclinical models of breast cancerž[END_REF][START_REF] Zhang | łIntegrating evolutionary dynamics into treatment of metastatic castrate-resistant prostate cancerž[END_REF].

Despite a present great interest in modelling for immunotherapy, most of the relevant mathematical models are of purely theoretical interest. What greatly complicates their development is the fact that immunotherapy itself often lacks a precise understanding of when it works, when it does not, and when it worsens the clinical scene. However, there is a small number of works in which biological experiments have been feed into the model, which can help to predict the next step for new experiments, and can then be tested [START_REF] Radunskaya | łMathematical modeling of tumor immune interactions: a closer look at the role of a PD-L1 inhibitor in cancer immunotherapyž[END_REF][START_REF] Wang | łCD8+ T cell response to adenovirus vaccination and subsequent suppression of tumor growth: modeling, simulation and analysisž[END_REF].

Inspired by experimental observations in co-culture between breast cancer spheroids and activated immune cells, which reproduce in vivo behaviours as described in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF], in Chapter 5 we explore a simple setting to study the effect of psychological stress on immune inőltration. To study the effect of stress on immune inőltration, we use a simpliőed version of the mathematical model developed in Chapter 3, which has been calibrated to reproduce in silico the results presented in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF].

Formal derivation of continuum models

In their nature, deterministic continuum models are amenable not only to numerical simulations but also to analytical approaches, which enable a complete exploration of the model parameter space. This allows a precise identiőcation of the validity domain of the results obtained and ensures higher robustness and precision of the conclusions to be drawn. This in turn provides a more in-depth theoretical understanding of the underlying cellular dynamics. The analysis and numerical simulation of deterministic continuum models for tumour-immune interactions, which may help identifying the determinants of tumour development, progression and response to therapy, pose a series of mathematical challenges. Overcoming these challenges may require the use of a wide range of methods and techniques from different research areas of mathematics. In fact, the study of the qualitative and quantitative properties of the solutions of ODEs and PDEs, motivated by the need to address open questions in cancer research, has stimulated the extension of existing mathematical tools as well as the development of novel analytical techniques and numerical methods [START_REF] Murray | Mathematical biology II: spatial models and biomedical applications[END_REF][START_REF] Perthame | łParabolic equations in biologyž[END_REF][START_REF] Perthame | Transport equations in biology[END_REF].

Ideally, instead of deőning such continuum models for tumour-immune interactions on the basis of population-and tissue-scale phenomenological assumptions, one wants to derive them from őrst principles, that is, as the deterministic continuum limits of stochastic discrete models that track the dynamics of single cells [START_REF] Anderson | Single-cell-based models in biology and medicine[END_REF][START_REF] Van Liedekerke | łSimulating tissue mechanics with agent-based models: concepts, perspectives and some novel resultsž[END_REF]. This is to ensure that the terms comprised in the model equations provide a faithful representation of the underlying cellular dynamics. In fact, although being computationally intensive to simulate for large cell numbers and, for most of them, inaccessible to analytical techniques, single-cell-based models allow the representation of őne details of cell-scale mechanisms and capture stochastic intercellular variability in spatial and evolutionary trajectories [START_REF] Lorenzi | łCancer modelling as fertile ground for new mathematical challenges. Comment on" Improving cancer treatments via dynamical biophysical models[END_REF]. These aspects, which cannot be directly incorporated into deterministic continuum models, become especially important in scenarios where cell numbers and densities are low (e.g. in the early stages of tumour development or when tumour size is severely reduced after therapy), due to the stronger impact that single-cell processes and demographic stochasticity have on cellular dynamics.

For this reason, the derivation of continuum models formulated in terms of PDEs or IDEs CHAPTER 1. Introduction from underlying individual-based models has attracted the attention of a considerable number of mathematicians and physicists. Examples in this active őeld of research, whereby a range of asymptotic techniques, probabilistic methods and limiting procedures are used to systematically derive continuum models from their stochastic discrete counterparts, can be found, for instance, in [START_REF] Byrne | łIndividual-based and continuum models of growing cell populations: a comparisonž[END_REF][START_REF] Champagnat | łFrom individual stochastic processes to macroscopic models in adaptive evolutionž[END_REF][START_REF] Chisholm | łEvolutionary dynamics of phenotype-structured populations: from individual-level mechanisms to population-level consequencesž[END_REF][START_REF] Johnston | łModelling the movement of interacting cell populations: a moment dynamics approachž[END_REF][START_REF] Lorenzi | łFrom individual-based mechanical models of multicellular systems to free-boundary problemsž[END_REF].

Contributions of the thesis

As mentioned in Section 1.2, a strong connection may exist between tumour progression and immune dysfunction that, however, remains to be fully explained. Mathematical modelling and numerical simulations may help us to describe different mechanisms involved in tumourimmune interactions, as well as to analyse possible ways in which tumour cells may escape immune surveillance. This in turn may provide possible different frameworks to help biologists and clinicians design immunotherapy strategies which signiőcantly improve the effectiveness of the overall anti-tumour immune response. In this thesis, we develop discrete and continuum models to describe the spatio-temporal dynamics of the interactions between a solid tumour and cytotoxic T cells, with the goal to investigate the biological settings which allow for the clearance or the escape of the tumour. This thesis is organized in four chapters. Each chapter focusses on a different mechanism involved in tumour-immune interactions and is composed of three main sections: the őrst one brieŕy resumes relevant biological background to inform the reader on the empirical evidence motivating the model assumptions; the second one is dedicated to the presentation of the mathematical framework together with analytical and numerical results; the őnal section is dedicated to the discussion of the model and research perspectives, both in terms of applicability to empirical systems and mathematical tractability. As shown in Section 1.3, in the literature many models of tumour-immune interactions exist. However, the mathematical models developed in this thesis, include a new or more precise representation of some mechanisms involved in tumour-immune interactions.

Throughout the different chapters of the thesis, the contributions at the mathematical and biological level are diverse.

Chapter 2 -Modelling the coevolutionary dynamics between cytotoxic T lymphocytes and tumour cells In Chapter 2, we develop an individual-based model for the coevolutionary dynamics between CTLs and tumour cells in a well-mixed system (i.e. spatial interactions are not incorporated into the model). In this model, the phenotypic state of each cell is modelled by a discrete variable, which represents a parameterisation of the antigen expression proőles for tumour cells and a parameterisation of the target antigens of T-cell receptors (TCRs) for CTLs. We formally derive the deterministic continuum limit of this stochastic discrete model, which comprises a non-local partial differential equation for the phenotype distribution of tumour cells coupled with an integro-differential equation for the phenotype distribution of CTLs. We őnd the biologically relevant homogeneous steady-state solutions of the continuum model equations, and study their linear stability, in order to identify possible conditions on the model parameters leading to different outcomes of immune competition and to the emergence of patterns of phenotypic coevolution between tumour cells and CTLs. The results presented in this chapter support the idea that TCR-tumour antigen binding affinity may be a good intervention target for immunotherapy and offer a theoretical basis for the development of anti-cancer therapy, such as adoptive T cell therapy, aiming at engineering TCRs in order to shape their affinity for cancer targets.

Contributions of the thesis

For the sake of simplicity, in this őrst model we ignore spatial interactions between tumour cells and CTLs. Moreover, we consider a unique tumour cell population, characterized by the same antigen expression proőle. However, solid tumours can be heterogeneous, characterized by different sub-populations of tumour cells expressing different antigens. Moreover, such antigens can be presented at different levels. These aspects, as well as spatial interactions between tumour cells and CTLs, may affect the efficacy of immune action and the outcomes of immune response, making it an interesting problem to investigate on its own. Therefore, these aspects are further included in the mathematical model presented in Chapter 3.

Chapter 3 -Modelling the impact of intra-tumour heterogeneity on immune response

In Chapter 3, we extend the individual-based model developed in Chapter 2 further to explicitly include a spatial structure and tumour antigen expression and presentation. This allows us to investigate the impact of intra-tumour heterogeneity (ITH) on the immune response to tumours. Therefore, in Chapter 3, we develop a new spatially individual-based model of the interaction dynamics between tumour cells and CTLs which makes it possible to dissect the speciőc impact of two expressions of ITH on anti-tumour immune response. Such expressions of ITH are (i ) the number of sub-populations of cancer cells expressing different antigens and (ii ) the percentage of immunogenic cells (i.e. tumour cells that are effectively targeted by immune cells). The originality of this model lies in the characterisation of antigen presentation levels by tumour cells, which drive, by the means of a chemoattractant, the inŕux of CTLs in the tumour microenvironment and their movement towards tumour cells. In our model, the effectiveness of the anti-tumour immune response is directly linked to the level of presentation of tumour antigens. The set-up of numerical simulations of the model is deőned so as to mimic scenarios considered in previous experimental studies reported in the literature [START_REF] Gejman | łRejection of immunogenic tumor clones is limited by clonal fractionž[END_REF][START_REF] Wolf | łUvb-induced tumor heterogeneity diminishes immune response in melanomaž[END_REF]. First, the results of numerical simulations of this model indicate that the presence of a larger number of sub-populations of tumour cells that express different antigens is associated with a reduced ability of CTLs to mount an effective anti-tumour immune response. Secondly, the presence of a larger percentage of tumour cells that are not effectively targeted by CTLs may reduce the effectiveness of antitumour immunity.

One potential limitation of individual-based models is their lack of amenability to mathematical analysis and their huge computational cost. To overcome this, we aimed to derive the continuum counterparts of the individual-based model introduced in this chapter. However, due to the complexity of the biological mechanisms included, in Chapter 4 we őrst consider a simpler biological situation.

Chapter 4 -Modelling the role of T cell infiltration in the immune response In Chapter 4, we present a spatial hybrid discrete-continuum modelling framework for the interaction dynamics between tumour cells and CTLs. Building on the modelling strategies developed in the previous chapter, in this framework, a stochastic individual-based model for cell dynamics is coupled with a reaction-diffusion equation for the evolution of a chemoattractant, which dictates the movement of CTLs towards the tumour. In order to take into account possible alterations in the inőltration capabilities of CTLs within the tumour, we let the probability of CTL movement be modulated by a decaying function of the densities of tumour cells and CTLs. We formally derive the deterministic continuum limit of this model, which comprises an integro-differential equation for the density of tumour cells coupled with a partial differential equation for the density of CTLs. We report on computational results of the hybrid model, and show that there is an excellent agreement between them and numerical results of the corresponding continuum model. The results presented in this chapter shed light on the mechanisms that underlie the emergence of different levels of inőltration of CTLs into the tumour and elucidate how CTL inőltration shapes anti-tumour immune response. Moreover, exploiting the computational efficiency of the continuum model, we carry out extensive numerical simulations to investigate the impact of CTL inőltration on the response of tumour cells to different types of anti-cancer immunotherapy.

The different models of tumour-immune interactions developed in Chapters 2, 3 and 4 attempt to describe or explore different mechanisms of the anti-tumour immune response. The different models have been parametrised using parameter values drawn from the extant literature wherever possible and aim to reproduce or investigate observed behaviours. However, at this stage, they have not been calibrated to any particular type of data. Hence, they cannot be employed to generate predictions that can directly be used in the clinic. In Chapter 5, we address this, by proposing a simpliőed version of the mathematical model presented in Chapter 3, and calibrating it to qualitatively reproduce experimental observations.

Chapter 5 -Modelling the effect of psychological stress on immune infiltration In Chapter 5, we develop, in collaboration with biologists, a simple individual-based model to study the effect of physiological stress on immune inőltration. The development of this model is motivated by in vitro experimental observations in co-culture between breast cancer spheroids and activated immune cells, which study the effect of the glucocorticoid stress hormone, cortisol, on immune inőltration [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF]. In this in vitro study, it is found that cortisol decreases the levels of the pro-inŕammatory cytokine IFN-γ and increases the levels of the anti-inŕammatory cytokine IL-10. Therefore, in our model, we suppose that the effect of stress on the anti-tumour immune response is directly linked to the change of levels of these two cytokines in the tumour microenvironment. Using a method similar to the one employed in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF], we deőne a score to quantify the effects of stress on immune inőltration in a controlled manner. The results of numerical simulations of this model are able to qualitatively reproduce the results of in vitro experiments presented in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF], and demonstrate the importance of including the effect of different factors when exploring the impact of physiological stress on immune inőltration into tumours. -E. L., Tommaso Lorenzi, Shensi Shen, Luis Almeida, Chloe Audebert, A mathematical model to study the impact of intra-tumour heterogeneity on anti-tumour CD8+ T cell immune response, Journal of Theoretical Biology, 2022.

-Luis Almeida, Chloe Audebert, E. L. and Tommaso Lorenzi, A hybrid discrete-continuum modelling approach to explore the impact of T-cell infiltration on anti-tumour immune response, Bulletin of Mathematical Biology, 2022.

In preparation

-Luis Almeida, Chloe Audebert, Melanie S. Flint, E. L., Tommaso Lorenzi and Chandrasekhar Venkataraman, A mathematical model to investigate the effect of psychological stress on immune infiltration.

Chapter 2

Discrete and continuum models for the coevolutionary dynamics between cytotoxic T lymphocytes and tumour cells

Motivation

In this chapter, we present an individual-based model for the coevolutionary dynamics between cytotoxic T lymphocytes (CTLs) and tumour cells. In this model, every cell is viewed as an individual agent whose phenotypic state is modelled by a discrete variable. For tumour cells this variable represents a parameterisation of the antigen expression proőles, while for CTLs it represents a parameterisation of the target antigens of T-cell receptors (TCRs).

In general, individual-based models are useful for observing emergent dynamics from interactions between single cells. However, they can be computationally time consuming and do not allow for analysis to be completed on the model. Considering the continuum counterpart of stochastic discrete individual-based models may allow for mathematical analysis. Therefore, we formally derive the deterministic continuum limit of this individual-based model, which comprises a non-local partial differential equation for the phenotype distribution of tumour cells coupled with an integro-differential equation for the phenotype distribution of CTLs. The biologically relevant homogeneous steady-state solutions of the continuum model equations are found. The linear-stability analysis of these steady-state solutions is then carried out in order to identify possible conditions on the model parameters that may lead to different outcomes of immune competition and to the emergence of patterns of phenotypic coevolution between tumour cells and CTLs.

We report on computational results of the individual-based model, and show that there is a good agreement between them and analytical and numerical results of the continuum model. These results shed light on the way in which different parameters affect the coevolutionary dynamics between tumour cells and CTLs. Moreover, they support the idea that TCR-tumour antigen binding affinity may be a good intervention target for immunotherapy and offer a theoretical basis for the development of anti-cancer therapy, such as adoptive T cell therapy, aiming at engineering TCRs so as to shape their affinity for cancer targets.

CHAPTER 2. Discrete and continuum models for tumour-immune interactions

The models described in this chapter and the results shown have been published in Luis Almeida, Chloe Audebert, E. L. and Tommaso Lorenzi, Discrete and continuum models for the coevolutionary dynamics between CD8+ cytotoxic T lymphocytes and tumour cells, Mathematical biology and medicine, 2022.

Background

Biological background

As explained in Section 1.2, cytotoxic T lymphocytes (CTLs) play a key role in the immune response against cancer. CTLs carry a speciőc receptor on their surface, the T-cell receptor (TCR), which can recognise and bind to non-self antigens expressed by tumour cells [START_REF] Messerschmidt | łHow cancers escape immune destruction and mechanisms of action for the new signiőcantly active immune therapies: Helping nonimmunologists decipher recent advancesž[END_REF]. Each TCR recognises and binds speciőcally to a certain antigen (i.e. the cognate antigen) [START_REF] Coulie | łTumour antigens recognized by T lymphocytes: at the core of cancer immunotherapyž[END_REF], and possibly other antigens within a certain affinity range [START_REF] Mason | łA very high level of crossreactivity is an essential feature of the T-cell receptorž[END_REF][START_REF] Wooldridge | łA single autoimmune T cell receptor recognizes more than a million different peptidesž[END_REF]. This enables CTLs to exert an antigen-speciőc cytotoxic activity against tumour cells, whose efficacy may depend on the affinity range of TCRs and the strength of TCR-tumour antigen binding (i.e. TCR-tumour antigen binding affinity) [START_REF] Kastritis | łOn the binding affinity of macromolecular interactions: daring to ask why proteins interactž[END_REF][START_REF] Stone | łT-cell receptor binding affinities and kinetics: impact on T-cell activity and speciőcityž[END_REF].

The presence of tumour cells expressing non-self antigens triggers the clonal expansion of CTLs with matching TCRs. Thereupon, CTL numbers are kept under control by self-regulation mechanisms [START_REF] Garrod | łDissecting T cell contraction in vivo using a genetically encoded reporter of apoptosisž[END_REF][START_REF] Min | łSpontaneous T cell proliferation: a physiologic process to create and maintain homeostatic balance and diversity of the immune systemž[END_REF][START_REF] Stockinger | łThe concept of space and competition in immune regulationž[END_REF][START_REF] Yu | łRecent advances in CD8+ regulatory T cell researchž[END_REF][START_REF] Troy | łCutting edge: homeostatic proliferation of peripheral T lymphocytes is regulated by clonal competitionž[END_REF], which play a key role in the prevention of autoimmunity.

Furthermore, epigenetic and genetic processes inducing stochastic and heritable changes in the antigen expression proőles of tumour cells foster dynamical intercellular variability in the expression levels of tumour antigens [START_REF] Campoli | łHLA antigen changes in malignant cells: epigenetic mechanisms and biologic signiőcancež[END_REF][START_REF] Sigalotti | łIntratumor heterogeneity of cancer/testis antigens expression in human cutaneous melanoma is methylation-regulated and functionally reverted by 5-Aza-2'-deoxycytidinež[END_REF][START_REF] Urosevic | łExpression of melanomaassociated antigens in melanoma cell culturesž[END_REF]. Due to limitations posed by self-regulation mechanisms upon the numbers of CTLs targeted against different antigens at the same tumour site, such a form of intratumour heterogeneity creates the substrate for adaptation of tumour cells to the antigen-speciőc cytotoxic activity of CTLs and triggers adaptive changes in the repertoire of CTLs. This results in coevolutionary dynamics whereby CTLs dynamically sculpt the antigenic distribution of tumour cells, and tumour cells concurrently reshape the repertoire of CTLs [START_REF] Phillips | łImmunology taught by Darwinž[END_REF].

The observation that the numbers of CD8+ and CD3+ T lymphocytes at the tumour site correlate with prognosis in different types of cancer led to the development of the 'immunoscore' as a prognostic marker in cancer patients [START_REF] Angell | łFrom the immune contexture to the Immunoscore: the role of prognostic and predictive immune markers in cancerž[END_REF][START_REF] Galon | density, and location of immune cells within human colorectal tumors predict clinical outcomež[END_REF][START_REF] Galon | łImmunoscore and Immunoproőling in cancer: an update from the melanoma and immunotherapy bridge 2015[END_REF][START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF]. The immunoscore provides a score that increases with the density of CD8+ and CD3+ T lymphocytes both in the centre and at the periphery of the tumour. A possible tumour classiőcation based on the immunoscore has been proposed in [START_REF] Galon | density, and location of immune cells within human colorectal tumors predict clinical outcomež[END_REF], where tumours with a high immunoscore are classiőed as 'hot tumours', tumours with an intermediate immunoscore are classiőed as 'altered tumours', and tumours with a low immunoscore are classiőed as 'cold tumours'.

The mathematical model

In this vein, in this chapter, we develop an individual-based model for the coevolutionary dynamics between tumour cells and CTLs in a well-mixed system (i.e. spatial interactions are not incorporated into the model). Every cell is viewed as an individual agent whose phenotypic state is modelled by a discrete variable, which represents a parameterisation of the antigen expression proőles for tumour cells and a parameterisation of the target antigens of the TCRs for CTLs.

The model takes into account the effects of the following biological processes: proliferation and death of tumour cells and CTLs; heritable, spontaneous phenotypic changes of tumour cells resulting in variation of antigenic expression; antigen-driven expansion of CTLs (i.e. in situ clonal expansion following antigen recognition); death of tumour cells due to antigen-speciőc cytotoxic activity of CTLs. These processes are incorporated into the model through a set of rules that correspond to a discrete-time branching random walk on the space of phenotypic states [START_REF] Chisholm | łEvolutionary dynamics of phenotype-structured populations: from individual-level mechanisms to population-level consequencesž[END_REF][START_REF] Hughes | Random walks and random environments: random walks[END_REF].

We show that a generalised version of the mathematical model of immune competition presented in [START_REF] Lorenzi | łMathematical model reveals how regulating the three phases of T-cell response could counteract immune evasionž[END_REF] can be formally obtained as the deterministic continuum limit of the individualbased model presented here. This continuum model comprises a non-local partial differential equation (PDE) for the phenotype distribution of tumour cells coupled with an integro-differential equation (IDE) for the phenotype distribution of CTLs, and shares some similarities with nonlocal predator-prey models such as those considered in [START_REF] Delitala | łEvolutionary branching patterns in predator-prey structured populationsž[END_REF][START_REF] Génieys | łAdaptive dynamics: modelling Darwin's divergence principlež[END_REF][START_REF] Segal | łPattern formation in a model of competing populations with nonlocal interactionsž[END_REF][START_REF] Tian | łNonlocal interaction driven pattern formation in a prey-predator modelž[END_REF]. In addition to the biological phenomena incorporated into the model considered in [START_REF] Lorenzi | łMathematical model reveals how regulating the three phases of T-cell response could counteract immune evasionž[END_REF], the deterministic continuum counterpart of the individual-based model developed here takes also into account the effect of changes in antigen expression proőles of tumour cells and more general forms of competitive feedback mechanisms regulating the growth of the numbers of tumour cells and CTLs.

The biologically relevant homogeneous steady-state solutions of the continuum model equations are found. Then, their linear stability analysis is studied in order to identify possible conditions on the model parameters leading to different outcomes of immune competition and to the formation of patterns of phenotypic coevolution between tumour cells and CTLs. We report on computational results of the individual-based model, and show that there is an excellent agreement between them and analytical and numerical results of the continuum model. Moreover, we explore possible scenarios in which differences between the outcomes of the discrete and continuum models may emerge. The results obtained disentangle the role of different cell parameters in the coevolutionary dynamics between tumour cells and CTLs.

Structure of the chapter

In Section 2.3, the individual-based model is introduced. In Section 2.4, the deterministic continuum counterpart of this model is presented (a formal derivation is provided in Appendix B.1). In Section 2.5, the homogeneous steady-state solutions of the continuum model equations are identiőed and their linear stability is investigated. In Section 2.6, computational results of the individual-based model are discussed and integrated with numerical solutions of the continuum model. In Section 2.7, key biological implications of the main őndings of this study are summarised and directions for future research are outlined.

The individual-based model

We model the coevolutionary dynamics between a population of tumour cells and a population of CTLs in a well-mixed system. The population of tumour cells is labelled by the letter n, while the population of CTLs is labelled by the letter c. Building on the modelling approach developed in [START_REF] Delitala | łA mathematical model for immune and autoimmune response mediated by T-cellsž[END_REF][START_REF] Delitala | łRecognition and learning in a mathematical model for immune response against cancer[END_REF][START_REF] Lorenzi | łMathematical model reveals how regulating the three phases of T-cell response could counteract immune evasionž[END_REF], at any time t ∈ [0, t f ] ⊂ R + , the phenotypic state of every tumour cell is modelled by a variable x ∈ I, where I := [-L, L] ⊂ R is the closure of the set I := (-L, L) ⊂ R with L > 0, and the phenotypic state of every CTL is modelled by a variable y ∈ I. We make the following modelling assumptions.

Assumption 1. The variable x represents a parameterisation of the antigen expression proőles of tumour cells, while the variable y represents a parameterisation of the target antigens of the TCRs (see Figure 2.1). As a result, CTLs in the phenotypic state y will be primarily capable of recognising tumour cells in the phenotypic state x = y. Assumption 2. Tumour cells will have higher antigenic similarity if their phenotypic states are modelled by closer value of x. Hence, depending on the range of TCR affinity, CTLs in the of the two cell populations) are deőned, respectively, as

ρ k n ≡ ρ n (t k ) := i N k i and ρ k c ≡ ρ c (t k ) := j C k j . (2.2)
In the mathematical framework of our model, the function

I h ≡ I(t k ) := ρ c (t k ) ρ n (t k ) (2.
3) provides a possible simpliőed measure of the immune score at the k th time-step in the well-mixed system considered here. In particular, abstracting from the immune-score based classiőcation of tumours recalled in Section 2.2, throughout the chapter we will refer to situations in which the average value of I, i.e. the quantity

I = τ t f h I h , (2.4)
is smaller than 1 or at least about one order of magnitude larger than 1 as 'cold tumour-like scenarios' and 'hot tumour-like scenarios', respectively, whereas the remaining situations will be classiőed as 'altered tumour-like scenarios'.

Remark 1. A more precise deőnition of the immunoscore, based on the spatial distribution of CTLs within the tumour and not just by their number, will be given in Chapter 4.

As mentioned earlier, we focus on a biological scenario whereby: (i) cells in the two populations divide and die due to intra-population competition (i.e. clonal competition amongst tumour cells and self-regulation of CTLs); (ii) tumour cells undergo heritable, spontaneous phenotypic changes which result in variation of antigen expression proőles; (iii) CTLs undergo antigen-driven expansion (i.e. in situ clonal expansion following antigen recognition); (iv) tumour cells die due to the antigen-speciőc cytotoxic activity of CTLs. These biological phenomena are incorporated into the model via the modelling strategies described in the following subsections, which are also schematised in Figure 2.2.

Mathematical modelling of cell division and death due to intrapopulation competition

We assume that a dividing cell is replaced by two identical cells that inherit the phenotypic state of the parent cell (i.e. the progenies are placed on the same lattice site as their parent), while a dying cell is removed from the population.

At every time-step k, we allow tumour cells and CTLs to undergo cell division at rates α n > 0 and α c > 0, respectively. Moreover, on the basis of Assumption 3, at every time-step k, we allow tumour cells in the phenotypic state x i and CTLs in the phenotypic state y j to die due to intra-population competition at rates µ n K k ni and µ c K k cj , respectively, where µ n , µ c > 0 and

K k ni ≡ K n (x i , t k ) := h g(x i , x h ; θ n ) N k h , K k cj ≡ K c (y j , t k ) := h g(y j , y h ; θ c ) C k h . (2.5) τ α n 1-τ(α n + µ n K ni & + γ𝜁 n J ni & ) Cell division Cell death Quiescence τ (µ n K ni & + γ𝜁 n J ni & )

Tumour cell division and death

τ (α c + γ𝜁 c J cj & ) 1-τ (α c +μ c K cj & + γ𝜁 c J cj & ) Cell division Cell death Quiescence τ μ c K cj &

CTL division and death

Phenotypic changes in tumour cells

1 -𝜆 n

-τ (α c + ζ c γJ k cj + µ c K k cj )
. Panel c.: a tumour cell in the phenotypic state x i that undergoes a phenotypic change can either enter into the phenotypic state x i-1 with probability λn 2 , enter into the phenotypic state x i+1 with probability λn 2 or remain in its current phenotypic state with probability 1λ n .

The function g is deőned as follows (2.7)

g(x, y; ξ) :=    1 |L ξ (x)| if |y -x| ⩽ ξ 0 if |y -x| > ξ, for (x, y; ξ) ∈ I × I × (0, |I|], (2.6
The quantity K k ni deőned via (2.5)-(2.7) represents the number of tumour cells whose phenotypic states are modelled by values of the variable x h at a distance smaller than or equal to θ n from x i , rescaled to |L θn (x i )|. Similarly, the quantity K k cj deőned via (2.5)-(2.7) represents the number of CTLs whose phenotypic states are modelled by values of the variable y h at a distance smaller than or equal to θ c from y j , rescaled to |L θc (y j )|. Hence, the inverse of the parameter 0 < θ n ⩽ |I| (i.e. 1/θ n ) provides a measure of the level of selectivity of clonal competition amongst tumour cells and the inverse of the parameter 0 < θ c ⩽ |I| (i.e. 1/θ c ) provides a measure of the level of selectivity of self-regulation mechanisms acting on CTLs (i.e. the smaller θ n and θ c , then the higher the corresponding levels of selectivity). Furthermore, the parameters µ n and µ c represent the rates of death of tumour cells and CTLs due to these forms of intra-population competition.

Mathematical modelling of phenotypic changes in tumour cells

We account for spontaneous, heritable phenotypic changes which result in variation of antigen expression proőles by allowing tumour cells to update their phenotypic states according to a random walk. More precisely, between the time-steps k and k + 1, every tumour cell enters a new phenotypic state, with probability 0 < λ n < 1, or remains in its current phenotypic state, with probability 1λ n . When a tumour cell in the phenotypic state x i undergoes a phenotypic change, it enters into either the phenotypic state x i-1 or the phenotypic state x i+1 with probability λ n /2. This models the fact that phenotypic changes occur randomly due to non-genetic instability, rather than being induced by selective pressures [START_REF] Huang | łGenetic and non-genetic instability in tumor progression: link between the őtness landscape and the epigenetic landscape of cancer cellsž[END_REF]. No-ŕux boundary conditions are implemented by aborting any attempted phenotypic variation of a tumour cell if it requires moving into a phenotypic state outside the interval I.

Mathematical modelling of tumour-immune competition

Similarly to cell division, we assume that a CTL undergoing antigen-driven expansion is replaced by two identical cells that inherit the phenotypic state of the parent cell. Moreover, similarly to cell death due to intra-population competition, we assume that a tumour cell dying due to the antigen-speciőc cytotoxic activity of CTLs is removed from the population.

On the basis of Assumptions 1 and 2, at every time-step k we allow CTLs in the phenotypic state y j to undergo antigen-driven expansion at rate ζ c γ J k cj , while tumour cells in the phenotypic state x i will die due to antigen-speciőc cytotoxic activity of CTLs at rate ζ n γ J k ni . Here, ζ n , ζ c , γ > 0 and

J k cj ≡ J c (y j , t k ) := i g(y j , x i ; η) N k i , J k ni ≡ J n (x i , t k ) := j g(x i , y j ; η) C k j , (2.8)
where the function g is deőned via (2.6) and (2.7). The quantity J k cj deőned via (2.6)-(2.8) represents the number of tumour cells whose phenotypic states are modelled by values of the variable x i at a distance smaller than or equal to η from y j , rescaled to |L η (y j )|. Similarly, the quantity J k Ci deőned via (2.6)-(2.8) represents the number of CTLs in phenotypic states which are modelled by values of the variable y j at a distance smaller than or equal to η from x i , rescaled to |L η (x i )|. Hence, the parameter 0 < η ⩽ |I| provides a measure of the affinity range of TCRs. Furthermore, the parameter γ provides a measure of the TCR-tumour antigen binding affinity (i.e. the strength of TCR-tumour antigen binding), while the parameter ζ c represents the rate at which a CTL undergoing antigen-driven expansion divides (i.e. the rate of cell division corresponding to in situ clonal expansion), and the parameter ζ n represents the rate at which a CHAPTER 2. Discrete and continuum models for tumour-immune interactions tumour cell can die due to the antigen-speciőc cytotoxic activity of a CTL.

Computational implementation of the individual-based model

Numerical simulations of the individual-based model are performed in Matlab. At each timestep, we follow the procedures described hereafter to simulate phenotypic variation, cell division and death of tumour cells, and cell division and death of CTLs. All random numbers mentioned below are real numbers drawn from the standard uniform distribution on the interval (0, 1) using the built-in function rand.

Computational implementation of spontaneous, heritable phenotypic changes of tumour cells For each tumour cell, a random number, r 1 , is generated and used to determine whether the cell undergoes a phenotypic change (i.e. 0 < r 1 < λ n ) or not (i.e. λ n ⩽ r 1 < 1). If the cell undergoes a phenotypic change, then a second random number, r 2 , is generated. If 0 < r 2 < 1/2, then the cell moves into the phenotypic state to the left of its current state (i.e. a cell in the phenotypic state x i will move into the phenotypic state x i-1 = x iχ), whereas if 1/2 ⩽ r 2 < 1 then the cell moves into the phenotypic state to the right of its current state (i.e. a cell in the phenotypic state x i will move into the phenotypic state x i+1 = x i + χ). No-ŕux boundary conditions are implemented by aborting attempted phenotypic changes that would move a cell into a phenotypic state outside the interval I.

Computational implementation of cell division and death of tumour cells and CTLs

For each population, the number of cells in each phenotypic state is counted. The quantities K n and K c are computed via (2.5) and the quantities J n and J c are computed via (2.8), and the following deőnitions are used to calculate the probabilities of cell division, death and quiescence (i.e. no division nor death) for every phenotypic state of cells in the two populations, respectively,

P b n := τ α n , P d n := τ µ n K k ni + ζ n γ J k ni , P q n := 1 -P b n + P d n (2.9)
and

P b c := τ α c + ζ c γ J k cj , P d c := τ µ c K k cj , P q c := 1 -P b c + P d c .
(2.10)

Notice that we are implicitly assuming that the time-step τ is sufficiently small that 0 < P k n < 1 and 0 < P k c < 1 for all h ∈ {b, d, q}. For each cell, a random number, r 3 , is generated and the cells' fate is determined by comparing this number with the probabilities of division, death and quiescence corresponding to the phenotypic state of the cell. In more detail, for a cell in population C: if 0 < r 3 < P d n then the cell is considered dead and is removed from the population; if P d n ⩽ r 3 < P d n + P b n then the cell undergoes division and an identical daughter cell is created; whereas if P d n + P b n ⩽ r 3 < 1 then the cell remains quiescent (i.e. does not divide nor die). Similarly, for a cell in population T : if 0 < r 3 < P d c then the cell is considered dead and is removed from the population; if P d c ⩽ r 3 < P d c + P b c then the cell undergoes division and an identical daughter cell is created; whereas if P d c + P b c ⩽ r 3 < 1 then the cell remains quiescent.

Corresponding deterministic continuum model

In the case where cell dynamics are governed by the rules described in Sections 2.3.1-2.3.3, between time-steps k and k + 1 a tumour cell in the phenotypic state x i may divide, die or remain quiescent (i.e. not divide nor die) with probabilities deőned via (2.9), while a CTL in the phenotypic state y j may divide, die or remain quiescent with probabilities deőned via (2.10).

Hence, recalling that between time-steps k and k + 1 a tumour cell in the phenotypic state x i may also enter into either of the phenotypic states x i-1 and x i+1 with probabilities λ n /2, the principle of mass balance gives the following system of coupled difference equations for the population densities n k i and c k j :

       n k+1 i = 2 P b n + P q n λ n 2 n k i+1 + n k i-1 + (1 -λ n ) n k i , (x i , t k ) ∈ I × (0, t f ], c k+1 j = 2 P b c + P q c c k j , (y j , t k ) ∈ I × (0, t f ].
(2.11)

The difference equation (2.11) 1 for n k i is subject to no-ŕux boundary conditions due to the fact that, as mentioned in Section 2.3.2, any attempted phenotypic variation of a tumour cell is aborted if it requires moving into a phenotypic state outside the interval I.

Starting from the system of coupled difference equations (2.11), letting the time-step τ → 0 and the phenotype-step χ → 0 in such a way that

λ n χ 2 2τ → β n with 0 < β n < ∞, (2.12)
where the parameter β n is the rate of spontaneous, heritable phenotypic changes of tumour cells, using the method employed in [START_REF] Ardaševa | łComparative study between discrete and continuum models for the evolution of competing phenotype-structured cell populations in dynamical environmentsž[END_REF][START_REF] Chisholm | łEvolutionary dynamics of phenotype-structured populations: from individual-level mechanisms to population-level consequencesž[END_REF][START_REF] Stace | łDiscrete and continuum phenotype-structured models for the evolution of cancer cell populations under chemotherapyž[END_REF], it is possible to formally show (see Appendix B.1) that the deterministic continuum counterpart of the stochastic discrete model comprises the following PDE-IDE system for the cell population density functions n(x, t) and c(y, t)

                     ∂ t n -β n ∂ 2 xx n = α n -µ n K n (x, t) -ζ n γ J n (x, t) n, (x, t) ∈ I × (0, t f ], ∂ t c = α c -µ c K c (y, t) + ζ c γ J c (y, t) c, (y, t) ∈ I × (0, t f ],
J n (x, t) := with I = (-L, L). Here, the function g is deőned via (2.6) and (2.7), and the non-local PDE (2.13) 1 for n is subject to the following no-ŕux boundary conditions

∂ x n(-L, t) = 0 and ∂ x n(L, t) = 0 for all t ∈ (0, t f ].
(2.14)

Steady-state and linear-stability analyses of the continuum model equations

In this section, we őrst identify the biologically relevant homogeneous steady-state solutions of the continuum model equations. Then, we carry out linear-stability analysis to: (i) determine conditions that may lead to the eradication of tumour cells by CTLs or to the coexistence between the two cell populations, and (ii) identify sufficient conditions for the emergence of patterns of phenotypic coevolution between tumour cells and CTLs.

CHAPTER 2. Discrete and continuum models for tumour-immune interactions

Biologically relevant steady-state solutions

A biologically relevant steady-state solution of the PDE-IDE system (2.13) subject to the boundary conditions (2.14) is given by a pair of real, non-negative functions n * (x) and c * (y) that satisfy the following system of differential equations

     -β n ∂ 2 xx n * = α n -µ n K * n (x) -ζ n γ J * n (x) n * , x ∈ I, α c -µ c K * c (y) + ζ c γ J * c (y) c * = 0, y ∈ I, (2.15)
with I = (-L, L), subject to the boundary conditions

∂ x n * (-L) = 0 and ∂ x n * (L) = 0.
(2.16)

In the system (2.15),

J * n (x) := I g(x, y; η) c * (y) dy, K * n (x) := I g(x, z; θ n ) n * (z) dz (2.17) and J * c (y) := I g(y, x; η) n * (x) dx, K * c (y) := I g(y, z; θ c ) c * (z) dz. (2.18)
The components of homogeneous steady-state solutions satisfy the following system of equations

     α n -µ n K * n (x) -ζ n γ J * n (x) n * = 0, x ∈ I, α c -µ c K * c (y) + ζ c γ J * c (y) c * = 0, y ∈ I (2.19)
and are of the form

n * (x) = ρ * n |I| ∀x ∈ I and c * (y) = ρ * c |I| ∀y ∈ I, (2.20)
where ρ * n ⩾ 0 and ρ * n ⩾ 0 satisfy the following system of algebraic equations

     α n |I| -µ n ρ * n -γ n ρ * c ρ * n = 0, α c |I| -µ c ρ * c + γ c ρ * n ρ * c = 0, with γ n := ζ n γ and γ c := ζ c γ.
(2.21)

The system of algebraic equations (2.21) is obtained by őrst integrating the PDE (2.13) 1 over I and imposing the boundary conditions (2.14), then integrating the IDE (2.13) 2 over I, next substituting ansatz (2.20) into the resulting equations and equating to zero their right-hand sides, and őnally using the fact that, when the function g is deőned via (2.6) and (2.7),

I g(x, y; ξ) dy = 1, ∀(x; ξ) ∈ I × (0, |I|]. (2.22)
In particular, since we are studying tumour-immune competition, we are interested in solutions of the system of equations (2.21) whose ρ * c component is strictly positive. There exist two solutions of this type, that is, the semitrivial solution

(ρ * n1 , ρ * c1 ) = 0, |I|α c µ c , (2.23)
and, provided that the following condition on the model parameters is met

γ < µ c α c α n ζ n , (2.24)
the nontrivial solution

(ρ * n2 , ρ * c2 ) = |I| (α n µ c -α c γ n ) γ c γ n + µ n µ c , |I| (α c µ n + α n γ c ) γ c γ n + µ n µ c . (2.25)
The semitrivial steady-state solution given by (2.20) and (2.23) corresponds to biological scenarios whereby tumour cells are eradicated by CTLs, while the nontrivial steady-state solution given by (2.20) and (2.25) corresponds to situations where coexistence between tumour cells and CTLs occurs. Notice that condition (2.24) indicates that lower TCR-tumour antigen binding affinity (i.e. smaller values of γ) make it more likely that tumour cells survive the cytotoxic action of CTLs, thus promoting coexistence between the two cell populations.

Linear-stability analysis

Linearising the PDE-IDE system (2.13), subject to the boundary conditions (2.14), about a steady-state of components n * (x) and c * (y), and using the conditions given by equations (2.15), we obtain the following PDE-IDE system for the perturbations ñn (x, t) and ñT (y, t)

     ∂ t ñ -β n ∂ 2 xx ñ = [α n -µ n K * n (x) -γ n J * n (x)] ñ -µ n Kn (x, t) + γ n Jn (x, t) n * , (x, t) ∈ I × (0, t f ], ∂ t c = [α c -µ c K * c (y) + γ c J * c (y)] c -µ c Kc (y, t) -γ c Jc (y, t) c * , (y, t) ∈ I × (0, t f ],
(2.26) subject to the boundary conditions

∂ x ñ(-L, t) = 0 and ∂ x ñ(L, t) = 0 for all t ∈ (0, t f ].
(2.27)

In
     ∂ t ñ -β n ∂ 2 xx ñ = α n -γ n J * n (x) ñ, (x, t) ∈ I × (0, t f ], ∂ t c = -µ c Kc (y, t) -γ c Jc (y, t) ρ * c1 |I| , (y, t) ∈ I × (0, t f ], (2.30
       ∂ t ñ -β n ∂ 2 xx ñ = -µ n Kn (x, t) + γ n Jn (x, t) ρ * n2 |I| , (x, t) ∈ I × (0, t f ], ∂ t c = -µ c Kc (y, t) -γ c Jc (y, t) ρ * c2 |I| , (y, t) ∈ I × (0, t f ].
(2.31)

Conditions for eradication of tumour cells by CTLs or coexistence between the two cell populations

In order to determine conditions on the model parameters that may lead to the eradication of tumour cells by CTLs or to the coexistence between the two cell populations, we study the stability of the steady-state solutions given by (2.20) and (2.23) or (2.25) to perturbations of the form

ñ(x, t) = ϵ n e λt ∀x ∈ I and c(y, t) = ϵ c e λt ∀y ∈ I with ϵ n , ϵ c ∈ R * , λ ∈ C. (2.32)
Substituting the ansatz (2.32) into the PDE-IDE system (2.30) and using property (2.22) along with the expression (2.23) of ρ * c1 gives the following system of algebraic equations

     λϵ n = α n -γ n α c µ c ϵ n , λϵ c = -µ c ϵ c -γ c ϵ n α c µ c , (2.33)
which can be written in matrix form as

  α n -γ n α c µ c -λ 0 γ c α c µ c -α c -λ   ϵ n ϵ c = 0 0 .
For a non-trivial solution of system (2.33) to exist, the determinant of the above matrix must be zero. This leads to the following quadratic equation for λ

λ 2 -Bλ + C = 0 with B := α n -γ n α c µ c -α c and C := α c γ n α c µ c -α n .
Hence, the semitrivial steady-state solution given by (2.20) and (2.23) is locally asymptotically stable if the reverse of condition (2.24) holds, that is if

γ > µ c α c α n ζ n , (2.34)
since in this case B < 0 and C > 0 (i.e. Re(λ) < 0). On the other hand, performing similar calculations on the PDE-IDE system (2.31) gives the the following system of algebraic equations

       λϵ n = -µ n ϵ n + γ n ϵ c (α n µ c -α c γ n ) γ c γ n + µ n µ c , λϵ c = -µ c ϵ c -γ c ϵ n (α c µ n + α n γ c ) γ c γ n + µ n µ c , (2.35)
which can be written in matrix form as

   -µ n (α n µ c -α c γ n ) γ c γ n + µ n µ c -λ -γ n (α n µ c -α c γ n ) γ c γ n + µ n µ c γ c (α c µ n + α n γ c ) γ c γ n + µ n µ c -µ c (α c µ n + α n γ c ) γ c γ n + µ n µ c -λ    ϵ n ϵ c = 0 0 .
For a non-trivial solution of system (2.35) to exist, the determinant of the above matrix must be zero. This leads to the following quadratic equation for λ

λ 2 -Bλ + C = 0 with B := -µ n (α n µ c -α c γ n ) γ c γ n + µ n µ c + µ c (α c µ n + α n γ c ) γ c γ n + µ n µ c
and

C := (α n µ c -α c γ n )(α c µ n + α n γ c ) (γ c γ n + µ n µ c ) 2 [µ n µ c + γ n γ c ] .
Hence, if condition (2.24) is met, then the nontrivial steady-state solution given by (2.20) and (2.25) is locally asymptotically stable, since in this case B < 0 and C > 0 (i.e. Re(λ) < 0).

Conditions for the emergence of patterns of phenotypic coevolution between tumour cells and CTLs

In order to identify sufficient conditions for the emergence of patterns of phenotypic coevolution between tumour cells and CTLs, we study the stability of the nontrivial steady-state solution given by (2.20) and (2.25) to perturbations of the form

ñ(x, t) = ϵ n e λt φ h (x) and c(y, t) = ϵ c e λt φ h (y) with ϵ n , ϵ c ∈ R * , λ ∈ C. (2.36)
Here, {φ h } h⩾1 are the eigenfunctions of the Laplace operator on I with homogeneous Neumann boundary conditions indexed by the wavenumber h, that is,

φ h (x) = cos (h x) with h = mπ |I| , m ∈ N, x ∈ I. (2.37)
Substituting the ansatz given by (2.36) and (2.37) into the PDE-IDE system (2.31), using the fact that

I g(x, y; ξ)φ h (y) dy = sin(hξ) h Ψ(x; ξ) φ h (x) with Ψ(x; ξ) := 2 |L ξ (x)| CHAPTER 2.
Discrete and continuum models for tumour-immune interactions for all x ∈ I and ξ ∈ (0, |I|], we obtain the following inőnite system of algebraic equations

       λ ϵ n = -h 2 β n ϵ n -µ n sin(hθ n ) h Ψ(x; θ n ) ϵ n + γ n sin(hη) h Ψ(x; η) ϵ c ρ * n2 |I| , λ ϵ c = -µ c sin(hθ c ) h Ψ(x; θ c ) ϵ c -γ c sin(hη) h Ψ(x; η) ϵ n ρ * c2 |I| , (2.38)
which can be written in matrix form as

    -h 2 β n -µ n sin(hθ n ) h Ψ(x; θ n ) ρ * n2 |I| -λ -γ n sin(hη) h Ψ(x; η) ρ * n2 |I| γ c sin(hη) h Ψ(x; η) ρ * c2 |I| -µ c sin(hθ c ) h Ψ(x; θ c ) ρ * c2 |I| -λ     ϵ n ϵ c = 0 0 .
For each x ∈ I, for a non-trivial solution of the system of algebraic equations (2.38) to exist the determinant of the above matrix must be zero. For each x ∈ I, this leads to the following quadratic equation for λ λ 2 -Bλ + C = 0 where

B ≡ B(h, x) := -h 2 β n -µ n sin(hθ n ) h ρ * n2 |I| Ψ(x; θ n ) -µ c sin(hθ c ) h ρ * c2 |I| Ψ(x; θ c ) (2.39)
and

C ≡ C(h, x) : = h 2 β n µ c sin(hθ c ) h ρ * c2 |I| Ψ(x; θ c )+ + ρ * n2 ρ * c2 |I| 2 γ n γ c sin(hη) h Ψ(x; η) 2 + µ n µ c sin(hθ n ) h sin(hθ c ) h Ψ(x; θ n )Ψ(x; θ c ) .
A sufficient condition for the nontrivial steady-state solution given by (2.20) and (2.25) to be driven unstable by perturbations of the form (2.36) (i.e. for patterns of phenotypic coevolution between tumour cells and CTLs to be formed) is that B > 0 and/or C < 0 so that Re(λ) > 0 for all x ∈ I. In particular, in the case where

θ n = θ c = θ, θ ∈ (0, |I|], (2.40)
since k is deőned via (2.37), for the condition B(h, x) > 0 to hold for all x ∈ I it suffices that

β n < 1 |I| min hθ - sin(hθ) h ρ * n2 µ n + ρ * c2 µ c h 2 min x∈I Ψ(x; θ),
where

H := h = mπ |I| , m ∈ N : sin(hθ) < 0 . Since, under deőnition (2.7), min x∈I Ψ(x; θ) = 2 max x∈I |L θ (x)| = 1 θ ,
the above condition on β n reduces to

β n < 1 |I| min hθ - sin(hθ) hθ ρ * n2 µ n + ρ * c2 µ c h 2 .
(2.41)

Numerical simulations

In this section, we report on computational results of the individual-based model along with numerical solutions of the corresponding continuum model given by the PDE-IDE system (2.13) and subject to the boundary conditions (2.14). Simulations are integrated with the results of steady-state and linear-stability analyses of the continuum model equations presented in Section 2.5. In particular, we investigate the way in which the outcomes of the models are affected by key parameters whose impact on the coevolutionary dynamics between tumour cells and CTLs is of particular biological interest. Such key parameters are: the TCR-tumour antigen binding affinity, γ, the level of selectivity of self-regulation mechanisms acting on CTLs, 1/θ c , the level of selectivity of clonal competition amongst tumour cells, 1/θ n , and the affinity range of TCRs, η. Moreover, we explore the existence of scenarios in which differences between the outcomes produced by the two models can emerge due to effects which reduce the quality of the approximation of the individual-based model provided by the continuum model.

Set-up of numerical simulations

Without loss of generality we choose L = 1, so that I = [-1, 1] and |I| = 2, and consider a discretisation of the interval [-1, 1] consisting of 1500 points (i.e. the phenotype-step is χ ≈ 0.0013). Furthermore, we use the time-step τ = 0.05 and, unless otherwise speciőed, we choose the őnal time t f = 30 days. Building on the results of steady-state and linear-stability analyses of the continuum model equations presented in Section 2.5, we carry out simulations using the following initial condition for the individual-based model

n 0 (x i ) := 10 4 (1 + a cos(A x i )), c 0 (y j ) := 10 4 (2 + a cos(A y j )), a ⩾ 0, A > 0. (2.42)
In Appendix B.2, we provide a detailed description of the methods employed to numerically solve the PDE-IDE system (2.13) complemented with the boundary conditions (2.14) and the continuum analogue of the initial condition (2.42), i.e. the initial condition

n 0 (x) := 10 4 (1 + a cos(A x)), c 0 (y) := 10 4 (2 + a cos(A y)), a ⩾ 0, A > 0. (2.43)
Unless otherwise speciőed, we use the parameter values listed in Table 2.1. Here, the values of the parameters α n , α c , ζ n and ζ c are consistent with previous measurement and estimation studies on the dynamics of tumour cells and CTLs [START_REF] Christophe | łA biased competition theory of cytotoxic T lymphocyte interaction with tumor nodulesž[END_REF][START_REF] De Pillis | łMathematical model creation for cancer chemo-immunotherapyž[END_REF][START_REF] Kuznetsov | łNonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysisž[END_REF][START_REF] Schlesinger | łCoevolutionary immune system dynamics driving pathogen speciationž[END_REF]. The values of the parameters µ n and µ c and the range of values of the parameters θ n and θ c are chosen so as to ensure that the equilibrium sizes and phenotype distributions of the two cell populations in isolation are biologically relevant. The range of values of the parameter η is consistent with experimental estimations of the precursor frequency of CTLs [START_REF] Blattman | łEstimating the precursor frequency of naive antigen-speciőc CD8 T cellsž[END_REF], while the values of the parameter γ are consistent with those used in [START_REF] Stromberg | łRobustness and fragility in immunosenescencež[END_REF][START_REF] Stromberg | łDiversity of T-cell responsesž[END_REF]. The value of the parameter λ n is taken from [START_REF] Stace | łDiscrete and continuum phenotype-structured models for the evolution of cancer cell populations under chemotherapyž[END_REF] and corresponds to values of β n that are consistent with experimental data reported in [START_REF] Doerŕer | DNA methylation: development, genetic disease and cancer[END_REF][START_REF] Duesberg | łExplaining the high mutation rates of cancer cells to drug and multidrug resistance by chromosome reassortments that are catalyzed by aneuploidyž[END_REF].

Main results

Eradication of tumour cells When γ is high enough so that condition (2.34) is satisőed (i.e. condition (2.24) does not hold), after initial growth, the total number of tumour cells decreases steadily over time until the tumour cell population is completely eradicated (cf. Figure 2.3a). This is due to the fact that, in response to a rapid growth in the size of the tumour cell population, the high TCR-tumour antigen binding affinity allows the population of CTLs to embark on rapid expansion in size that continues until CTLs have reached the critical mass required to push the population of tumour cells towards extinction. The expansion of the CTL population is followed by the transition to a contraction phase, which is characterised by a decline of the total number of CTLs to a level corresponding to the maintenance of a form of immunological memory. Moreover, the plots in Figure 2.3a demonstrate that there is a good agreement between numerical simulations of the individual-based and continuum models and, coherently with the analytical results presented in Section 2.5.2, the total numbers of tumour cells and CTLs converge to the steady-state values given by (2.23).

Hot tumour-like scenarios When γ satisőes condition (2.24) but is still sufficiently high, the total number of CTLs attains a value large enough to keep the total number of tumour cells steadily low. After initial growth, the total number of tumour cells decreases over time until it stabilises itself around a relatively small value (cf. Figure 2 ) given by (2.25) occurs, in the long run, the value of the average immune score I deőned via (2.4) reŕects the value of the ratio ρ * c2 /ρ * n2 . Therefore, in the framework of our tumour classiőcation based on the average immune score I (see page 5), if condition (2.24) is met: cold tumour-like scenarios and hot tumour-like scenarios will emerge when the values of the model parameters are such that the ratio ρ * T 2 /ρ * C2 is smaller than 1 or at least about one order of magnitude larger than 1, respectively, whereas altered tumour-like scenarios will emerge in the remaining cases. This has been conőrmed by the results of additional numerical simulations (results not shown). Hence, independently of the speciőc values of the model parameters, provided that assumption (2.24) is satisőed, cell dynamics qualitatively similar to those of Figure 2.3, and corresponding to hot, altered or cold tumour scenarios, will be observed depending on the value of the ratio ρ * c2 /ρ * n2 . This testiőes to the robustness of the numerical results presented here. Sample temporal dynamics of such patterns are summarised by the plots in Figure 2.6, which show that clonal expansion leads to a rapid proliferation of CTLs that are targeted to the antigens mostly expressed by tumour cells, whereas self-regulation mechanisms induce formerly stimulated CTLs to decay. In turn, the antigen-speciőc cytotoxic action of CTLs causes the selection of those tumour cells that are able to escape immune recognition. As a result, immune competition induces the formation of multiple peaks in the phenotype distribution of tumour cells. This concurrently shapes the phenotype distribution of CTLs with a time shift corresponding to the time required for the CTLs to adapt to the antigenic distribution of tumour cells. The plots in Figure 2.6 demonstrate that there is again a good agreement between numerical simulations of the individual-based and continuum models. ues of η correlate with the emergence of oscillations in the total numbers of tumour cells and CTLs, that is, CTLs undergo a succession of expansion and contraction phases that result in an alternate decay and growth of tumour cells (cf. bottom panel in Figure 2.7c). The plots in Figure 2.7 demonstrate that there is a good agreement between numerical simulations of the individual-based and continuum models. The results presented in this study shed light on the way in which different parameters shape the coevolutionary dynamics between tumour cells and CD8+ cytotoxic T lymphocytes. In particular, we demonstrated that, ceteris paribus, higher values of the TCR-tumour antigen binding affinity (i.e. the parameter γ in the model) promote the eradication of tumour cells by CTLs, while lower values facilitate the coexistence between tumour cells and CTLs. Speciőcally, progressively reducing the TCR-tumour antigen binding affinity brings about the emergence of: hot tumour-like scenarios, which are characterised by a large number of in situ CTLs and a low number of tumour cells, and thus represent a more fertile ground for anticancer therapeutic intervention; altered tumour-like scenarios, which reŕect the intrinsic ability of the immune system to effectively mount a CTL-mediated immune response and the ability of tumour cells to partially escape such a response; cold tumour-like scenarios, which are characterised by an insufficient number of in situ CTLs and are invariably associated with poor prognosis [START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF]. This classiőcation of tumours is also supported by experimental works showing that in situ immune reaction might be the strongest parameter inŕuencing clinical outcome, regardless of the local tumour extension and its spread to lymph nodes [START_REF] Pagès | łThe essential role of the in situ immune reaction in human colorectal cancerž[END_REF][START_REF] Galon | density, and location of immune cells within human colorectal tumors predict clinical outcomež[END_REF][START_REF] Galon | łImmunoscore and Immunoproőling in cancer: an update from the melanoma and immunotherapy bridge 2015[END_REF]. Moreover, our őndings support the idea that TCR-tumour antigen binding affinity may be a good intervention target for immunotherapy that aims to turn cold or altered tumours into hot ones by enhancing CTL response. In this regard, our őndings are in agreement with the conclusions of previous experimental articles indicating that a strong binding affinity of T cells to tumour antigens may play a key role in the overall immune response to the disease [START_REF] Gerdemann | łCytotoxic T lymphocytes simultaneously targeting multiple tumor-associated antigens to treat EBV negative lymphomaž[END_REF].In particular, in altered tumours, increasing antigenicity, via the removal of co-inhibitory signals and/or the supply of co-stimulatory signals [START_REF] Galon | łImmunoscore and Immunoproőling in cancer: an update from the melanoma and immunotherapy bridge 2015[END_REF][START_REF] Whiteside | łEmerging opportunities and challenges in cancer immunotherapyž[END_REF], may enhance in situ CTLs activity, and has proven to be effective in the treatment of advanced-stage melanoma [START_REF] Wolchok | łOverall survival with combined nivolumab and ipilimumab in advanced melanomaž[END_REF], renal cell carcinoma [START_REF] Motzer | łNivolumab plus ipilimumab versus sunitinib in advanced renal-cell carcinomaž[END_REF] and non-small cell lung cancer [START_REF] Hellmann | łNivolumab plus ipilimumab in lung cancer with a high tumor mutational burdenž[END_REF]. In cold tumours, a proposed approach to overcome the lack of a pre-existing immune response consists in combining a priming therapy that boosts CTL responses with the removal of co-inhibitory signals through approaches such as immune checkpoint [START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF]. The therapeutic success achieved by combining immune checkpoint therapy with chemotherapy in metastatic NSCLC has demonstrated the potential strength of this dual approach [START_REF] Gandhi | łPembrolizumab plus chemotherapy in metastatic non-small-cell lung cancerž[END_REF].

Patterns of phenotypic coevolution between tumour cells and CTLs

Patterns of phenotypic coevolution between tumour cells and CTLs

Moreover, the results presented here indicate that the affinity range of TCRs (i.e. the parameter η in the model), the selectivity of clonal competition amongst tumour cells (i.e. the inverse of the parameter θ C in the model) and the selectivity of self-regulation mechanisms acting on CD8+ cytotoxic T lymphocytes (i.e. the inverse of the parameter θ T in the model) play a pivotal role in the formation of patterns of phenotypic coevolution, which create the substrate for the emergence of less regular cell phenotype distributions with multiple peaks. Such patterns are underpinned by some form of immunoediting whereby the population of CTLs evolves and continuously adapts its receptor repertoire in order to recognise and effectively eliminate tumour cells and, in turn, the antigen-speciőc selective pressure exerted by CTLs leads to the selection of those tumour clones that are able to evade immune recognition [START_REF] Dunn | łCancer immunoediting: from immunosurveillance to tumor escapež[END_REF]. The adaptability of tumour cells and CTLs and the selective pressure they mutually exert on each other during cancer development are emerging as crucial factors in determining cancer evolutionary trajectories. This has been shown in the context of chronic lymphocytic leukemia [START_REF] Purroy | łCoevolution of leukemia and host immune cells in chronic lymphocytic leukemiaž[END_REF] and other cancer types, as reviewed in [START_REF] Levine | łImplications of tumor-immune coevolution on cancer evasion and optimized immunotherapyž[END_REF]. Our results offer also a theoretical basis for the development of anti-cancer therapy aiming at engineering TCRs so as to shape their affinity for cancer targets [START_REF] Border | łAffinity-enhanced T-cell receptors for adoptive T-cell therapy targeting MAGE-A10: strategy for selection of an optimal candidatež[END_REF][START_REF] Crean | łMolecular Rules Underpinning Enhanced Affinity Binding of Human T Cell Receptors Engineered for Immunotherapyž[END_REF][START_REF] Li | łGenetically engineered T cells for cancer immunotherapyž[END_REF][START_REF] Zhao | TCR-T Cell Immunotherapy in Anticancer Precision Medicine: Pros and Consž[END_REF] and adaptive therapy aiming at altering intratumour clonal competition [START_REF] Gatenby | łAdaptive therapyž[END_REF][START_REF] West | łTowards multidrug adaptive therapyž[END_REF], in order to control the coevolutionary dynamics between tumour cells and CD8+ cytotoxic T lymphocytes. In this respect, one of the best known treatment based on engineering speciőc TCRs is based on CAR-T cells [START_REF] Wang | łClinical manufacturing of CAR T cells: foundation of a promising therapyž[END_REF], which confer CTLs the ability to target speciőc antigens. It has been demonstrated that this therapeutic strategy has several potential advantages over conventional therapies, including speciőcity, rapidity, high success rate and long-lasting effects [START_REF] Carl H June | łCAR T cell immunotherapy for human cancerž[END_REF][START_REF] Gomes | łCancer immunotherapy using CAR-T cells: from the research bench to the assembly linež[END_REF].

The good agreement between the results of numerical simulations of the individual-based and continuum models, along with the quantitative information given by (2.23) and (2.25) and the precise conditions given by (2.34) and (2.41), testiőes to the robustness of the biological insight gained in this work. We also showed that possible differences between cell dynamics produced by the individual-based and continuum models can emerge under parameter settings that correspond to less regular cell phenotype distributions and more pronounced demographic stochasticity. In fact, these cause a reduction in the quality of the approximations employed in the formal derivation of the deterministic continuum model from the individual-based model (cf. Appendix B.1). This demonstrates the importance of integrating individual-based and continuum approaches when considering mathematical models for tumour-immune competition.

Research perspectives From a mathematical point of view, we plan to carry out a systematic investigation of the conditions on the affinity range of TCRs that may lead to the emergence of oscillations in cell numbers observed in the numerical simulations presented in this work. Moreover, from a modelling point of view, our individual-based modelling framework for the coevolutionary dynamics between tumour cells and CD8+ cytotoxic T lymphocytes, along with the formal derivation of the corresponding continuum model, can be developed further in several ways. For instance, a myriad of immunosuppressive strategies, the so-called immune checkpoints, help tumour cells acquiring features that enable them to evade immune detection, which may ultimately induce the exhaustion of CTLs in the tumour micro-environment, which impairs the immune response. The modelling approach presented here does not capture this aspect. However, exhaustion mechanisms could be incorporated into the individual-based model by, for example, allowing CTLs to enter a suppressed state (i.e. CTLs would become exhausted and thus would no longer able to eliminate tumour cells). In the continuum model, this would result in the presence of an additional loss term in the IDE (2.13) 2 along with a third equation for the dynamics of exhausted CTLs. Another track to follow to further enrich our model would be to include a spatial structure, for instance by embedding the tumour cells in the geometry of a solid tumour, and to take explicitly into account the effect of both spatial and antigen-speciőc interactions between tumour cells and CTLs, as similarly done in [START_REF] Kather | łIn silico modeling of immunotherapy and stroma-targeting therapies in human colorectal cancerž[END_REF][START_REF] Macfarlane | łModelling the immune response to cancer: an individual-based approach accounting for the difference in movement between inactive and activated T cellsž[END_REF][START_REF] Macfarlane | łA stochastic individual-based model to explore the role of spatial interactions and antigen recognition in the immune response against solid tumoursž[END_REF]. Including a spatial structure would make it possible, inter alia, to introduce a more precise deőnition of the immune score that incorporates the level of CTL inőltration. Furthermore, at this stage, the mathematical representation of the phenotypic state of tumour cells and CTLs employed in our modelling framework is rather abstract. This might make it difficult to carry out precise quantitative comparisons between the results of numerical simulations and experimental data. This limitation could be overcome by employing a mathematical representation of tumour antigens and TCRs similar to the one that we proposed in [START_REF] Leschiera | łA mathematical model to study the impact of intra-tumour heterogeneity on anti-tumour CD8+ T cell immune responsež[END_REF], whereby a discrete set of tumour antigens that can be recognised by a unique repertoire of TCRs is considered. Finally, it would be interesting to incorporate explicitly into the model the effects of immunotherapeutic agents or other therapeutic agents. These are all lines of research that we will be pursuing in the future.

Chapter 3

A mathematical model to study the impact of intra-tumour heterogeneity on anti-tumour T cell immune response

Motivation

In the previous chapter we have developed a discrete model, and formally derived its continuum counterparts, of tumour-immune competition to describe the interactions between cytotoxic T lymphocytes (CTLs) and tumour cells. In the model, we considered that tumour cells and CTLs were structured by a variable representing a parameterisation of the antigen expression proőles and a parameterisation of the target antigens of T-cell receptors.

For the sake of simplicity, in this őrst work we did not consider spatial aspects of tumourimmune interaction dynamics. Moreover, we considered that the tumour cell population was characterized by the same antigen expression proőle, which could eventually be recognized by every CTL. However, solid tumours can be characterized by different antigen proőles and different levels of antigen presentation by the major histocompatibility complex 1 (MHC-I), possibly preventing their recognition by CTLs. These aspects, as well as spatial dynamics between tumour cells and CTLs (such as cell spatial growth and movement, spatial heterogeneity within the tumour), can affect the efficacy of immune action and the outcomes of immune response. In particular, intra-tumour heterogeneity (ITH) has a strong impact on the efficacy of the immune response against solid tumours. The number of sub-populations of cancer cells expressing different antigens and the percentage of immunogenic cells (i.e. tumour cells that are effectively targeted by immune cells) in a tumour are both expressions of ITH.

In this chapter, we aim to capture these two expressions of ITH using mathematical modelling techniques. As mentioned in Section 1.3.1, stochastic discrete models, such as individual-based models, are appropriate modelling approaches for addressing the role of diversity in cell populations and also within each individual cell. Therefore, we develop a spatially explicit stochastic individual-based model of the interaction dynamics between tumour cells and CTLs, which makes it possible to dissect out the speciőc impact of these two expressions of ITH on anti-tumour immune response.

CHAPTER 3. A model to study the impact of ITH on CTL immune response

The originality of this model lies in the characterisation of antigen presentation levels by tumour cells, which drive the inŕux of CTLs in the tumour micro-environment and their movement towards tumour cells. As explained in 1.2.2, by 'level of antigen presentation' we refer to different mechanisms that may affect the recognition of antigens by antigen presenting cells, the activation of CTLs, and the efficacy of immune action. The characterization of antigen presentation levels allows us to include different level of immunogenicity in the tumour and to consider two different subtypes of tumour cells, that is: immunogenic cells (i.e. tumour cells that are effectively targeted by CTLs) and non-immunogenic cells (i.e. tumour cells that are poorly targeted by CTLs). In our model, the effectiveness of the anti-tumour immune response is directly linked to the level of presentation of tumour antigens.

The set-up of numerical simulations of the model is deőned so as to mimic scenarios considered in previous experimental studies. Moreover, the ability of the model to qualitatively reproduce experimental observations of successful and unsuccessful immune surveillance is demonstrated. First, the results of numerical simulations of this model indicate that the presence of a larger number of sub-populations of tumour cells that express different antigens is associated with a reduced ability of CTLs to mount an effective anti-tumour immune response. Secondly, the presence of a larger percentage of tumour cells that are not effectively targeted by CTLs may reduce the effectiveness of anti-tumour immunity. Ultimately, the mathematical model presented in this chapter may provide a framework to help biologists and clinicians to better understand the mechanisms that are responsible for the emergence of different outcomes of immunotherapy.

This study is a joint work in collaboration with:

• Shensi Shen, Institute of Thoracic Oncology, West China Hospital, Sichuan University, Chengdu, China.

The model described in this chapter and the results shown have been published in E. L., Tommaso Lorenzi, Shensi Shen, Luis Almeida, Chloe Audebert, A mathematical model to study the impact of intra-tumour heterogeneity on anti-tumour CD8+ T cell immune response, Journal of Theoretical Biology, 2022.

Background

Biological background

Cytotoxic T lymphocytes (CTLs) are capable of detecting and eliminating tumour cells by recognising cancer-associated antigens expressed by tumour cells. The effectiveness of the immune response depends on the level of presentation of such antigens by the major histocompatibility complex 1 (MHC-I) [START_REF] Coulie | łTumour antigens recognized by T lymphocytes: at the core of cancer immunotherapyž[END_REF][START_REF] Messerschmidt | łHow cancers escape immune destruction and mechanisms of action for the new signiőcantly active immune therapies: Helping nonimmunologists decipher recent advancesž[END_REF]. In particular, CTLs express T cell receptors (TCRs) and, once activated, they migrate via chemotaxis in response to concentration gradients of chemical signals towards the tumour cells expressing the matching antigens [START_REF] Miller | łAutonomous T cell trafficking examined in vivo with intravital two-photon microscopyž[END_REF]. The inŕux and movement of CTLs are dictated by the spatial distribution of tumour antigens and by the level of chemokines in the tumour micro-environment [START_REF] Boissonnas | łIn vivo imaging of cytotoxic T cell inőltration and elimination of a solid tumorž[END_REF]. Upon intratumoural inőltration, CTLs can trigger tumour cell death by direct interaction with tumour cells, releasing cytotoxic factors (i.e. granzime B, interferon gamma) [START_REF] Kim | łCancer immunoediting: from immune surveillance to immune escapež[END_REF]. Oncogenic mutation-driven cancers harbor neoantigens that can be recognized by CTL receptors [START_REF] Gubin | łCheckpoint blockade cancer immunotherapy targets tumour-speciőc mutant antigensž[END_REF]. A high mutational burden and neoantigen load in tumours have been associated with an enhanced response to immunotherapy [START_REF] Chan | łDevelopment of tumor mutation burden as an immunotherapy biomarker: utility for the oncology clinicž[END_REF][START_REF] Germano | łInactivation of DNA repair triggers neoantigen generation and impairs tumour growthž[END_REF][START_REF] Hellmann | łGenomic features of response to combination immunotherapy in patients with advanced non-small-cell lung cancerž[END_REF][START_REF] Samstein | łTumor mutational load predicts survival after immunotherapy across multiple cancer typesž[END_REF][START_REF] Snyder | łGenetic basis for clinical response to CTLA-4 blockade in melanomaž[END_REF][START_REF] Van Allen | łGenomic correlates of response to CTLA-4 blockade in metastatic melanomaž[END_REF]. However, it has recently been reported that many of these neoantigens arise from sub-clonal branching mutations and could potentially increase intratumour heterogeneity (ITH) [START_REF] Mcdonald | łTumor heterogeneity correlates with less immune response and worse survival in breast cancer patientsž[END_REF][START_REF] Mcgranahan | łClonal neoantigens elicit T cell immunoreactivity and sensitivity to immune checkpoint blockadež[END_REF][START_REF] Reuben | łTCR repertoire intratumor heterogeneity in localized lung adenocarcinomas: an association with predicted neoantigen heterogeneity and postsurgical recurrencež[END_REF]. These tumours are characterised by clonal antigens (presented by all tumour cells), and sub-clonal antigens (presented only by sub-populations of tumour cells). Moreover, such sub-clonal antigens may be associated with decreased level of antigen presentation by the MHC-I, leading to a weaker antigen-speciőc CTL response [START_REF] Gejman | łRejection of immunogenic tumor clones is limited by clonal fractionž[END_REF]. In contrast, more homogeneous tumours express few clonal antigens in all tumour cells and appear to have a better response to immunotherapy across a wide range of tumour types [START_REF] Fennemann | łAttacking tumors from all sides: Personalized multiplex vaccines to tackle intratumor heterogeneityž[END_REF][START_REF] Mcgranahan | łClonal neoantigens elicit T cell immunoreactivity and sensitivity to immune checkpoint blockadež[END_REF]. Furthermore, CTLs activated against clonal antigens are more commonly found at the tumour site than CTLs reactive to sub-clonal antigens [START_REF] Mcgranahan | łClonal neoantigens elicit T cell immunoreactivity and sensitivity to immune checkpoint blockadež[END_REF]. These őndings suggest that ITH may strongly affect the effectiveness of the anti-tumour immune response.

The mathematical model

In light of these considerations, we present a spatial stochastic individual-based model of tumourimmune interaction dynamics that can be used to explore the effect of ITH on immune surveillance. There is a variety of individual-based model approaches (e.g., cellular automata, Cellular Potts models, hybrid discrete/continuous models). In our study, we used a Cellular Potts model and the CompuCell3D open-source simulation environment [START_REF] Izaguirre | łCompuCell, a multi-model framework for simulation of morphogenesisž[END_REF]. The originality of this model lies in the characterisation of antigen presentation levels by tumour cells, which drive the inŕux of CTLs in the tumour micro-environment and their movement towards tumour cells. In our model, the effectiveness of the anti-tumour immune response is directly linked to the level of presentation of tumour antigens. In addition, the model takes into account biological phenomena that are driven by stochastic aspects of the interaction dynamics between tumour cells and CTLs.

The effect of ITH on immune surveillance is investigated at two different levels through computational simulations of this model. First, we explore the outcomes of the immune response considering different number of sub-populations of cancer cells constituting the tumour. Then, we asses the efficiency of the immune response by varying the immunogenicity of tumour cells. We study the impact of these two characteristics on tumour progression independently and together, assessing their inŕuence on anti-tumour immunity in a controlled manner.

Structure of the chapter

The chapter is organised as follows. In Section 3.3, we present the individual-based model and the mathematical description of each biological process included in the model. Section 3.4 summarises the set-up of computational simulations and presents some preliminary results of computational simulations. Full details of model implementation and model parametrisation are provided in Appendix A and Appendix C.1, respectively. In Section 2.6.2 we present the main computational results and we discuss them in view of previous biological works. Finally, Section 3.6 concludes the chapter and provides a brief overview of possible research perspectives.

Model and methods

We consider two cell types in our model: tumour cells, characterised by an antigen proőle and a level of antigen presentation, and antigen-speciőc CTLs. To describe the interactions occurring between the two cell types we use an on-lattice individual-based model posed on a 2D spatial domain partitioned into square elements of side ∆x. In our model, this domain biologically represents the tumour micro-environment. At each time step of length ∆t, the states of the cells are updated according to the probabilistic and deterministic rules described below.

In the remainder of this section, we őrst present the modelling framework in a general setting, along with the underlying biological hypotheses and assumptions. Then, we detail how each biological mechanism is mathematically described. A detailed description of the computational implementation of the model, which relies on a Cellular Potts approach, can be found in Appendix A.

Modelling framework

To include different level of immunogenicity in the tumour, two different subtypes of tumour cells are considered: immunogenic cells (i.e. tumour cells that are effectively targeted by CTLs) and non-immunogenic cells (i.e. tumour cells that are poorly targeted by CTLs). On the one hand, we deőne immunogenic cells as cells expressing one or more clonal antigens, considered as immunodominant, and presented at a normal level by the MHC-I. On the other hand, we assume that non-immunogenic cells have experienced, through mutations, a deterioration of their level of antigen presentation, and have acquired new antigens. These new antigens are presented only by a subset of tumour cells, and will be denoted as sub-clonal antigens [START_REF] Mcgranahan | łClonal neoantigens elicit T cell immunoreactivity and sensitivity to immune checkpoint blockadež[END_REF]. Therefore, we deőne non-immunogenic cells as cells expressing clonal and sub-clonal antigens, both presented at a low level by the MHC-I. The system is initially composed of tumour cells only, which grow and proliferate through mitosis. Tumour cells secrete different chemoattractants that trigger the inŕux of speciőc CTLs into the domain. When they arrive in the domain, CTLs move via chemotaxis towards tumour cells expressing the matching antigens and, upon contact, try to eliminate them.

The modelling strategies used to reproduce these dynamics are described in detail in the following subsections, and are also schematically illustrated with an example in Figure 3.1 and Figure 3.2.

Dynamics of tumour cells

Antigen expression

We let N T (t) denote the number of tumour cells in the system at time t = h∆t, with h ∈ N 0 , and we label each cell by an index n = 1, . . . , N T (t). We let each tumour cell express one or more antigens, and we characterise the antigen proőle of the tumour by means of a vector A = (a 1 , . . . , a f ), a 1 , . . . , a f ∈ N,

where a i denotes an antigen and f is the total number of antigens expressed by the tumour [see Figure 3.1(a)]. Using phylogenetic tree representations [see Figure 3.1(b)-(c)], we deőne each antigen a i ∈ A, i = 1, . . . , f , of the tumour as clonal if it belongs to the trunk of the phylogenetic tree, or sub-clonal if it belongs to one of the branches of the phylogenetic tree. We let A C and A SC denote the sets of clonal and the sub-clonal antigens, whereby:

A C , A SC ⊂ A, A C ∪ A SC = A and A C ∩ A SC = ∅. (3.2)
Then, based on the phylogenetic tree representation, we divide the tumour in f different subpopulations of tumour cells labelled by the last antigen a i ∈ A acquired [see Figure 3.

1(d)].

Following this notation, in this model, cells in the same sub-population express the same set of antigens [see Figure 3.1(a, b, d)]. Moreover, if a i ∈ A C , cells in the sub-population labelled by the antigen a i express only clonal antigens, whereas if a i ∈ A SC , cells in the sub-population labelled by the antigen a i express both clonal and sub-clonal antigens. Therefore, we deőne cells in sub-populations labelled by a clonal antigen a i ∈ A C as immunogenic cells, whereas cells in sub-populations labelled by a sub-clonal antigen a i ∈ A SC are deőned as non-immunogenic cells.

Antigen presentation by MHC-I We incorporate antigen presentation into our model by letting each tumour cell present its antigens at a certain level. There can be high variability in each antigen's presentation between patients with the same type of tumour and even within tumour cell samples from the same patient [START_REF] Anagnostou | łEvolution of neoantigen landscape during immune checkpoint blockade in non-small cell lung cancerž[END_REF][START_REF] Mcgranahan | łClonal neoantigens elicit T cell immunoreactivity and sensitivity to immune checkpoint blockadež[END_REF]. Therefore, for the n th tumour cell, we characterise the level of presentation of each one of its antigens a i ∈ A by the normalized variable

l n ai ∈ [0, 1] (3.3)
whereby the value l n ai = 0 corresponds to a tumour cell that lost the expression of the antigen a i , while l n ai = 1 corresponds to a tumour cell presenting the antigen a i at the highest level. To capture the idea that immunogenic cells present their antigens at a higher level than non-immunogenic cells, we introduce the discrete sets In particular, if the n th tumour cell is an immunogenic cell, it presents each antigen a i at a normal level l n ai ∈ L I , whereas if the n th tumour cell is a non-immunogenic cell, all of its antigens a i are presented at a low level l n ai ∈ L N I .

L I = {m I , . . . , M I } ⊂ [0,
Tumour cell growth and division At each time-step, we let tumour cells grow at a random rate drawn from a uniform distribution; the parameters of the bounds of the uniform distribution are chosen to match the mean duration of a tumour cell cycle length. Mitosis occurs when a tumour cell grows to a critical size and then divides along a randomly orientated axis. Upon division at the time t, the n th tumour cell is replaced by two cells [see Figure 3.1(e)], one labelled by the parent index n and the other one labelled by the index N T (t) + 1. The daughter cell will inherit most of the properties of the parent cell, including the antigens expressed by the parent cell, so the fact that the cell is immunogenic or not [see Figure 3.1(e)]. For each antigen a i expressed by the daughter cell, a random level of antigen presentation l N T (t)+1 ai will be chosen. This level of antigen presentation can then be different from the one of the parent cell. Another property not inherited by the daughter cell is the intrinsic lifespan of the cell, which is randomly drawn from a uniform distribution. In this model, we do not take into account the appearance of new antigens due to the occurrence of mutations.

Tumour cell death If a tumour cell exhausts its lifespan (which is drawn when the cell is created), it dies (i.e. it undergoes apoptosis) at the end of the time-step and it is removed from the domain. A tumour cell can also die due to intra-tumour competition, with a rate proportional to the total number of tumour cells, or because of the cytotoxic action of CTLs. More details about tumour cell death due to the cytotoxic action of CTLs will be given in Section 3.3.1.

Secretion of chemoattractants

We let tumour cells at the border of the tumour (the region where cytokines and immune cells are more abundant [START_REF] Boissonnas | łIn vivo imaging of cytotoxic T cell inőltration and elimination of a solid tumorž[END_REF]) secrete different chemoattractants for each expressed antigen a i ∈ A. The secretion of a chemoattractant by a tumour cell expressing antigen a i is proportional to the level of presentation of such antigen a i . Therefore, we model the chemoattractant secretion rate s n ai by the n th tumour cell expressing antigen a i using the following deőnition:

s n ai := C 1 l n ai , (3.5)
where C 1 ∈ R + is a scaling factor of units

[mol] [time] [space]
, where [mol], [time] and [space] denote respectively the number of chemoattractant molecules and the units of time and of the size of a grid site, and l n ai is the level of presentation of antigen a i by the n th tumour cell. The total amount of chemoattractant secreted by tumour cells expressing antigen a i induces the arrival of CTLs speciőc to antigen a i into the domain. More details about the mathematical modelling of the different chemoattractant dynamics will be discussed in Section 3.3.1.

Dynamics of CTLs

Influx of CTLs Following [START_REF] Gong | łA computational multiscale agent-based model for simulating spatio-temporal tumour immune response to PD1 and PDL1 inhibitionž[END_REF], to model the tumour vessels that allow the arrival of CTLs in the tumour micro-environment, we generate a set of points in the domain. In order not to rely on a detailed angioarchitecture, we generate 5 entry points, equidistant from each other and from the centre of the domain. At each time step, a CTL speciőc to antigen a i ∈ A can be supplied to the domain from one of the 5 entry points, provided that the entry point is not occupied by other cells. The probability 0 < p(t) ⩽ 1 of inŕux of a CTL speciőc to antigen a i into the domain is proportional to the total amount S tot ai (t) of chemoattractant associated to antigen a i secreted at time t. Therefore, we deőne p(t) as

p(t) := C 2 S tot ai (t),
with C 2 ∈ R + a scaling factor of units [time] [mol] . Since the secretion of chemoattractants by tumour cells is proportional to the level of antigen presentation (see Eq. (3.5)), the total amount of chemoattractants secreted by non-immunogenic cells is lower than the total amount of chemoattractants secreted by immunogenic cells. Therefore, the inŕux of CTLs targeted to sub-clonal antigens, which are expressed only by nonimmunogenic cells, is lower than the inŕux of CTLs targeted to clonal antigens.

TCR expression and CTL death

We denote by N C (t) the number of CTLs in the system at time t, and we label each of them by an index m = 1, ..., N C (t). Every CTL has a unique TCR [see Figure 3.2(a)], and we suppose that each TCR is speciőc to a unique tumour antigen [see Figure 3.2(b)]. When the m th CTL with a TCR targeted against antigen a i ∈ A arrives into the domain it undergoes chemotactic movement towards tumour cells expressing the matching antigen a i .

CTL division occurs mostly in the lymph nodes [START_REF] Delves | łThe immune systemž[END_REF] and cells then move to the tumour site. CTLs can also proliferate at the tumour site but this is not the main site of proliferation. We thus neglect the effects of CTL proliferation at the tumour site and consider only the effects of proliferation outside the spatial domain of the model, leading to a varying inŕux of CTLs. A CTL undergoes apoptosis when it reaches the end of its intrinsic lifespan, which is drawn from a uniform distribution upon its arrival in the domain.

Elimination of tumour cells by CTLs

Upon contact, CTLs can interact only with tumour cells expressing the matching antigen [see Figure 3.2(c)], and can induce their death, on the condition that the matching antigen is presented at a sufficiently high level. If a CTL is in contact with more than one tumour cell expressing the matching antigen, it will try to eliminate the one presenting the antigen at the highest level. In particular, when the m th CTL interacts with the n th tumour cell expressing the matching antigen a i , we let the tumour cell be removed from the system, provided that µ l n ai > (1r).

(3.6)

Here µ is a random variable drawn from the standard uniform distribution, l n ai is the level of presentation of antigen a i by the n th tumour cell and 0 < r ⩽ 1 is the intrinsic TCR-recognition probability, which we suppose to be equal for every CTL. If the tumour cell satisőes the conditions Each TCR is able to recognize a particular tumour antigen. In the model, the number of TCRs is equal to the number of expressed tumour antigens. (c) Purple circles represent tumour cells. CTLs can eliminate tumour cells, upon contact, under certain conditions. A tumour cell is eliminated if it presents the antigen matching the CTL receptor at a sufficiently high level. In this example, a tumour cell expressing antigen a 1 and a 2 cannot be eliminated by a CTL with TCR matching antigen a 4 . On the other hand, the same tumour cell may be eliminated by a CTL expressing the TCR matching antigen a 2 , under a condition on the level l a2 of presentation of such antigen a 2 . The parameter r is the intrinsic TCR-recognition probability and µ is a random variable drawn from a standard uniform distribution.

to be eliminated, it undergoes apoptosis. The parameter r determines the range of tumour cells the CTL population can interact with: large values of r represent a CTL population able to eliminate tumour cells presenting their antigens at a low level, whereas low values of r model the scenario where the CTLs can only eliminate tumour cells presenting their antigens at a high level.

Tumour cell elimination by CTLs takes approximately 6 hours to be completed in vitro [START_REF] Caramalho | CTL/melanoma cell interactions: multiple hits must be delivered for tumour cell annihilationž[END_REF] and in vivo [START_REF] Breart | łTwo-photon imaging of intratumoral CD8+ T cell cytotoxic activity during adoptive T cell therapy in micež[END_REF]. Accordingly, we require that an elimination event keeps a CTL engaged for 6 hours and only after this time the CTL can eliminate again [START_REF] Kather | łIn silico modeling of immunotherapy and stroma-targeting therapies in human colorectal cancerž[END_REF]. If the condition (3.6) is not satisőed, the CTL is not engaged and can try, in the next time step, to eliminate again a tumour cell.

Chemoattractant field

As mentioned earlier, we let the n th tumour cell at the border of the tumour secrete a different chemoattractant for each antigen a i that it expresses. Denoting by ϕ ai the concentration of the chemoattractant secreted by tumour cells expressing antigen a i , we let the dynamic of ϕ ai be described by the following reaction-diffusion equation:

∂ϕ ai ∂t = D∆ϕ ai -γϕ ai + n∈N BT (t) s n ai , a i ∈ A. (3.7)
In Eq. (3.7), D is the diffusion constant and γ is the rate of natural decay; these two parameters are assumed to have the same value for each chemoattractant. On the other hand, we recall that the secretion rate s n ai is speciőc to the n th tumour cell, because it is proportional to the level of presentation of antigen a i by the tumour cell (see Eq. (3.5)). N BT (t) denotes the set of tumour cells in contact with the surrounding medium at time t.

Numerical simulations and preliminary results

61

We add to Eq. (3.7) zero-ŕux boundary conditions and an initial concentration ϕ init a which is set to be zero everywhere in the domain but at the border of the tumour.

Numerical simulations and preliminary results

Set-up of simulations

For numerical simulations of our individual-based model, we use a Cellular Potts approach on a 2D spatial grid with a total of 400×400 lattice sites. Simulations were developed and run using the software CompuCell3D [START_REF] Izaguirre | łCompuCell, a multi-model framework for simulation of morphogenesisž[END_REF] on a standard workstation (Intel i7 Processor, 4 cores, 16 GB RAM, macOS 11.2.2), with one time-step chosen to be ∆t = 1 min. The computational implementation of Cellular Potts models is described in Appendix A, while full details of the model parametrisation are provided in Appendix C.1. Files to run a simulation example with Compucell3D software [START_REF] Izaguirre | łCompuCell, a multi-model framework for simulation of morphogenesisž[END_REF] are available at: https://plmlab.math.cnrs.fr/audebert/cc3dmodeltumourcd8.

At the initial time point of the simulation, a certain number of tumour cells are already present in the domain, while CTLs arrive only when the simulation starts. At the beginning of simulations there is a total of 400 tumour cells, tightly packed in a circular conőguration positioned at the centre of the domain, reproducing the geometry of a solid tumour.

All quantities we present in this section and in Section 3.5 are obtained by averaging over the results of 10 simulations, with parameter values kept constant and equal to those listed in Table C.1 and Table C.2. Unless otherwise explicitly stated, we carry out numerical simulations for 28800 time-steps, corresponding to 20 days.

The next two subsections describe two preliminary computational results of our model which will be used to guide the simulations leading to the main results presented in Section 3.5.

Baseline scenario: tumour development in the absence of CTLs

We őrst establish a baseline scenario where tumour cells grow, divide and die via the modelling rules described in Section 3.3.1, in the absence of CTLs. For this case, we carry out numerical simulations for 36000 time-steps, corresponding to 25 days. Figure 3.3 shows the growth over time of the number of tumour cells. The growth of the tumour cell number is of logistic type, as expected by the rules that govern tumour cell death. Logistic growth has been used by a number of authors to model the temporal evolution of the size of solid tumours [START_REF] Drasdo | łIndividual-based approaches to birth and death in avascu1ar tumorsž[END_REF][START_REF] Kirschner | łModeling immunotherapy of the tumor-immune interactionž[END_REF][START_REF] Kuznetsov | łNonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysisž[END_REF]. The carrying capacity, i.e. the saturation value reached by the number of tumour cells due to intra-population competition, is numerically estimated to be of about 1100 cells.

In the following subsections, we explore the immune response to tumours characterised by different degrees of ITH. Each simulation is carried out by keeping all parameter values őxed (and equal to this baseline scenario) and changing only the initial compositions of the different tumours.

Tumours with larger number of sub-populations of cancer cells lead to lower immune response efficacy

In a recent in vivo study in mice, the volume of UVB irradiated tumours after 20 days is found to be linked to the number of sub-populations of cancer cells constituting each inoculated tumour [START_REF] Wolf | łUvb-induced tumor heterogeneity diminishes immune response in melanomaž[END_REF]. In our next simulations, we attempt to verify that our model reproduces such phenomenon, exploring the outcomes of immune response to 7 different tumours characterised by an increasing number of sub-populations of tumour cells. At this preliminary stage, we simplify our model. We consider that each tumour consists of the same type of cells, and we do not of Section 3.4.2, we see that the immune response is almost inefficient, as it is not able to really limit the growth of the two tumours. Our computational results are in agreement with experimental results presented in [START_REF] Wolf | łUvb-induced tumor heterogeneity diminishes immune response in melanomaž[END_REF]. In this study, the induction of UVB-derived tumour, which lead to an increase in the number of sub-populations of cancer cells, results in aggressive tumours with reduced anti-tumour immune activity. However, when different single-cell-clone derived tumours (characterised by a unique sub-population of tumour cells) are considered, the immune system is able to effectively eradicate them.

Figure 3.4(h) also shows that, when the tumour is constituted of more than 3 sub-populations of tumour cells, the total number of CTLs at the end of simulations remain almost constant in the different tumours (around 200 cells). A consequence of this is that the average size of each speciőc CTL sub-population decreases as we consider tumours with increased number of subpopulations of tumour cells. This leads to a less efficient anti-tumour immune response. As highlighted in [START_REF] Wolf | łUvb-induced tumor heterogeneity diminishes immune response in melanomaž[END_REF], these computational results also suggest that increasing the number of subpopulations of tumour cells reduce the exposition of each antigen to the łfront-linež, thus making more difficult for immune cells to detect them. The resulting outcome is a reduced inŕux of speciőc CTLs in the tumour micro-environment and a weaker anti-tumour immune response.

Initial composition of two tumours inspired by biological studies

In the previous subsections we investigated simple cases of tumour growth with and without the action of the immune system. We will now explore further the effect of ITH on immune surveillance considering two tumours inspired by biological studies, in order to effectively capture more layers of biological complexity. For the two tumours we consider different initial antigenic compositions, corresponding to different degrees of ITH. In particular, following the experiments presented in [START_REF] Wolf | łUvb-induced tumor heterogeneity diminishes immune response in melanomaž[END_REF], we dissect out two characteristics of ITH: the number of sub-populations of cancer cells constituting a tumour and the percentage of immunogenic and non-immunogenic cells within it. With our model, we wish to investigate the effect of these two expressions of ITH on tumour aggressiveness independently and together, evaluating their inŕuence on anti-tumour immunity in a controlled manner. To this end, őrst we generate two tumours with different number of sub-populations of cancer cells. Following the experiments presented in [START_REF] Wolf | łUvb-induced tumor heterogeneity diminishes immune response in melanomaž[END_REF] and the results of Section 3.4.3, we consider, respectively, tumours with 3 and 7 sub-populations of cancer cells. For simplicity, we denote the őrst tumour as tumour-3a and the second one as tumour-7a. The antigenic composition of the two tumours and their corresponding phylogenetic tree representation are inspired by experiments presented in [START_REF] Wolf | łUvb-induced tumor heterogeneity diminishes immune response in melanomaž[END_REF]. More details about the two tumours are given in the next paragraphs. Next, for each tumour we consider different initial percentages of immunogenic and non-immunogenic cells. When different sub-populations of immunogenic (or non-immunogenic) cells are considered, the total percentage of immunogenic (or non-immunogenic) cells is equally distributed in each sub-population. This enables us to decouple antigen heterogeneity and antigen immunogenicity, and study their inŕuence on tumour aggressiveness in a causal, systematic manner.

Tumour-3a

The őrst tumour we consider expresses three different antigens, one of which is clonal and the other two are sub-clonal (see Figure 3.5(a)). With the notation introduced in Section 3.3.1, we denote respectively by

A = {4, 5, 7}, A C = {5} and A SC = {4, 7} (3.8)
the antigen proőle of the tumour, the clonal antigens and the sub-clonal antigens.

Main results

Large number of sub-populations of cancer cells constituting a tumour reduces the effectiveness of the immune response

To investigate how the immune response is affected by different degrees of heterogeneity, we start by comparing two situations in which the initial tumours are characterised by different number of sub-populations of tumour cells. We consider as initial conditions tumour-3a, with 3 different sub-populations of tumour cells, and tumour-7a, with 7 different sub-populations of tumour cells, deőned as in Section 3.4.4. For each tumour, we consider the same initial percentage of immunogenic and non-immunogenic cells, corresponding to 75% of immunogenic cells and 25% of non-immunogenic cells. Note that these two tumours are different from those considered in the results presented in Section 3.4.3. In fact, in tumour-3a and tumour-7a cells can express clonal and sub-clonal antigens and, therefore, be either of immunogenic or non-immunogenic type. On the other hand, in the results presented in Section 3.4.3, each tumour cell presents a unique antigen, which is shared by all the cells in the same sub-population of tumour cells. The situation considered here provides a more faithful representation of biological complexity, as a tumour cell can express more than one antigen presented at different levels. Figure 3.6(a)-(c) show the time evolution of the total number of tumour cells, along with the corresponding time evolution of immunogenic and non-immunogenic cell number. Figures 3.6(d)-(f ) also display the spatial cell distributions observed at different times of two simulations. As shown by Figure 3.6(a), the two tumours have similar dynamics from the beginning of simulations until day 10, with an initial increase of the cell number followed by a steep decrease. After day 10, in tumour-3a, the number of tumour cells continues to decrease until it reaches a low, almost constant level. Figure 3.6(b)-(c), along with the corresponding panel of Figure 3.6(f ), show that, at the end of simulations, all the immunogenic cells are eliminated by the CTLs, and only few non-immunogenic cells remain in the system. On the other hand, for tumour-7a, after day 10 the tumour cell number increases steadily over time. This dynamic leads to a őnal tumour size similar to the initial one. Moreover, as shown by Figure 3.6(b), the number of immunogenic cells tends to decrease over time, whereas the number of non-immunogenic cells, after being initially kept under control by immune cells, increases steeply (cf. C.1, we observe a qualitative similar behaviour (i.e. similar mean and standard deviation), conőrming the reliability of the tumour-7a results presented in Figure 3.6. As shown by Figure 3.6, we observe a larger variability in the simulation of tumour-7a than in the one of tumour-3a. This variability is probably due to the larger number of antigens that are present in tumour-7a. This result indicates that the number of antigens can be an important source of variability in the model.

Comparing these results with the baseline scenario of Section 3.4.2, for both tumours we clearly see the effects of the action of immune cells on tumour growth, which is no longer simply logistic and saturating to carrying capacity. However, the effectiveness of the immune response depends on the tumour considered. For tumour-3a, the immune response is efficient and almost eliminates all tumour cells. On the other hand, for tumour-7a, the higher heterogeneity leads to a less effective immune response and the tumour eventually grows again. These results suggest that, even if characterised by equal percentages of immunogenic and non-immunogenic cells, tumours with a larger number of sub-populations of tumour cells, which express a wider spectrum of antigens, are more aggressive. This was already suggested by the results presented in Section of tumour cells are present, non-immunogenic cells have a better chance of escaping immune surveillance. The outcome is a weaker anti-tumour immune response. Overall, our results are in agreement with the recent hypothesis presented in [START_REF] Wolf | łUvb-induced tumor heterogeneity diminishes immune response in melanomaž[END_REF] that, because of increased antigenic variability, the relative expression of each antigen is weaker in tumours composed of a larger number of sub-populations of tumour cells. In particular, clonal antigens undergo łdilutionž within the tumour, and, therefore, the chance for CTLs to identify immunogenic cells is reduced. This leads to a diminished ability of CTLs to mount a sufficient cytotoxic response.

Different initial percentages of immunogenic and non-immunogenic cells can cause variations in anti-tumour immune response

The results discussed in the previous subsection illustrate how the effectiveness of the immune response can decreases in tumours with larger number of sub-populations of tumour cells. We investigate the effects of ITH further, focusing on the role of the percentage of immunogenic and non-immunogenic cells. We őx the number of sub-populations of tumour cells considering only tumour-3a, and vary the initial percentage of immunogenic and non-immunogenic cells. The plot in Figure 3.7 displays the number of tumour cells remaining at the end of simulations (after 20 days), for different initial percentages of immunogenic and non-immunogenic cells. For low percentages of non-immunogenic cells (⩽ 25%), none or very few tumour cells survive after 20 days. On the contrary, for tumours initially composed of more than 50% of non-immunogenic cells, the number of tumour cells after 20 days is larger than the initial one. In addition, the őnal number of cells increases as we increase the initial percentage of non-immunogenic cells. These results suggest that the anti-tumour immune action is efficient only when the percentage of non-immunogenic cells is low compared to the percentage of immunogenic cells. Moreover, the larger the percentage of non-immunogenic cells, the weaker the immune response is.

Compared to the baseline scenario of Section 3.4.2, we see the effects of the immune system on tumour growth. In fact, for each scenario the number of cells at the end of simulations is lower than the tumour carrying capacity shown in Figure 3.3. However, for larger percentages of non-immunogenic cells, the immune response is not efficient enough to reduce the initial tumour size.

Taken together, our results qualitatively reproduce key őndings of experiments performed in in vivo syngeneic mice tumour models [START_REF] Gejman | łRejection of immunogenic tumor clones is limited by clonal fractionž[END_REF]. The results presented in [START_REF] Gejman | łRejection of immunogenic tumor clones is limited by clonal fractionž[END_REF] indicate that a noneffective immune response may occur when the percentage of immunogenic cells in the tumour is low. Our computational results provide an explanation for such emergent behaviour. Since subclonal antigens are presented at a low level by the MHC-I, non-immunogenic cells trigger a poor CTL response. Thus, tumours characterised by a major percentage of non-immunogenic cells result in a weaker overall immune response. Furthermore, the experimental results presented in [START_REF] Gejman | łRejection of immunogenic tumor clones is limited by clonal fractionž[END_REF] put forward the idea that the threshold percentage of immunogenic cells that is required to trigger an antigen-speciőc CTL response may vary depending on the antigens. In order to address this point, such a feature could be implemented in the model, for example by considering antigen presentation levels or chemotactic responses speciőc to each antigen.

The role of the immunogenic cell percentage within the tumour is further analysed as we observe a gap between the results obtained considering 25% and 50% of non-immunogenic cells (see Figure 3.7). This is investigated by performing simulations considering percentages of nonimmunogenic cells between these two values. Figure 3.8 displays the time evolution of the number of tumour cells for 10 different realisations of the same simulation, considering the same initial condition with 33% of non-immunogenic cells and 67% of immunogenic cells. In this case, we carried out numerical simulations for 38800 time-steps (corresponding to 27 days).

Under this choice of the initial condition, we observe a large variability in the tumour-immune cell number eventually saturates at a certain value (see Figure 3.9(c1)-(e1)). In these cases, the saturation value of the number of tumour cells is larger than the initial tumour cell number. Moreover, the saturation value attained increases as we increase the level of heterogeneity of the tumour (respectively, the number of sub-populations of tumour cells and the percentage of non-immunogenic cells). Such results indicate that in these cases CTLs are present in the tumour micro-environment but do not produce an effective immune response. Persistent antigen presentation has been proven to cause continuous TCR stimulation that could directly induce CTL dysfunction and exhaustion [START_REF] Yi | łT-cell exhaustion: characteristics, causes and conversionž[END_REF][START_REF] Zuniga | łT-cell exhaustion due to persistent antigen: Quantity not quality?[END_REF]. The model presented in this work does not include this aspect, but it could be easily extended to do so.

The outcomes of our model recapitulate the main results of in in vivo clonal mixing experiments in mice models presented in [START_REF] Wolf | łUvb-induced tumor heterogeneity diminishes immune response in melanomaž[END_REF], who studied the combined effect of these two characteristics of ITH in mediating tumour growth and eradication. Wolf and collaborators have demonstrated that tumours with increased number of clones and large genetic diversity are more aggressive. In our model, the number of clones can be linked to the number of sub-populations of tumour cells, while genetic diversity may be linked to the immunogenicity of the tumour. Moreover, our őndings are in agreement with an experimental work indicating that patients whose tumours are highly heterogeneous have increased levels of relapse after an initial response to immunotherapy and worse survival expectations than patients with more homogeneous tumours [START_REF] Mcgranahan | łClonal neoantigens elicit T cell immunoreactivity and sensitivity to immune checkpoint blockadež[END_REF].

We next analyse the evolution over time of immunogenic and non-immunogenic cells. When tumours are initially composed of 25% of non-immunogenic cells, Figure 3.9(b2) shows that the two tumours evolve in different ways. While the number of non-immunogenic cells is considerably reduced in tumour-3a, the őnal number of non-immunogenic cells increases up to four times its initial number in tumour-7a. On the other hand, when tumours are initially composed of more than 50% of non-immunogenic cells, independently of the tumour considered, we observe a similar trend in the evolution of immunogenic and non-immunogenic cells (see Figure 3.9(c2)-(e2)). In particular, for both tumour types, the number of immunogenic cells tends to remain stable or decreases slightly. On the other hand, the number of non-immunogenic cells increases and grows to up to twice its initial value.

These results suggest that, beyond a certain non-immunogenic cell percentage threshold, the immune system becomes inefficient in both tumour types independently of the number of sub-populations of tumour cells. Moreover, they suggest that the selective pressure of the immune response can lead to more aggressive tumours, characterised by larger percentages of nonimmunogenic cells. In this regard, our results follow the same behaviour of previous experimental works demonstrating that, under cancer therapeutics (e.g. chemotherapy or radiotherapy), the population of tumour cells is exposed to the selective stress induced by the treatment [START_REF] Greaves | łClonal evolution in cancerž[END_REF][START_REF] Iwasa | łEvolution of resistance during clonal expansionž[END_REF][START_REF] Trédan | łDrug resistance and the solid tumor microenvironmentž[END_REF]. Therefore, more resistanCTLs acquire a competitive advantage over more sensitive cells and induce a weaker response to treatment in the long run. The resulting outcome is a more aggressive tumour, which may ultimately grow again [START_REF] Nowell | łThe clonal evolution of tumor cell populationsž[END_REF].

Discussion, conclusions and research perspectives

Discussion and conclusions

The number of sub-populations of cancer cells constituting a tumour and the percentage of immunogenic and non-immunogenic cells within it are two major components of ITH, and play a key role in the immune response against solid tumours. Mathematical models make possible to asses the inŕuence of these two components of ITH on anti-tumour immunity in a controlled manner.

have a better chance of escaping immune surveillance. This outcome may be explained by the fact that clonal antigens undergo łdilutionž within the tumour relative to other antigens, diminishing the ability of CTLs to mount a sufficient cytotoxic response (cf. Figure 3.6).

The outcomes of our model support the idea that varying the initial percentage of immunogenic and non-immunogenic cells leads to variations on the effectiveness of the immune response and results in distinct scenarios, from immune clearance of the tumour to immune escape (cf. Figure 3.7). We have also observed that for certain intermediate percentages of immunogenic and non-immunogenic cells, stochasticity in cell dynamics plays an important role, and can lead both scenarios close to tumour eradication and to scenarios where a large number of tumour cells persists over time (cf. Figure 3.8).

We have also studied the effects of ITH on anti-tumour immune response by varying both the number of sub-populations of cancer cells and the initial percentage of immunogenic and nonimmunogenic cells (cf. Figure 3.9). For equal percentage of immunogenic and non-immunogenic cells, tumours with increased number of sub-populations of cancer cells are more aggressive than tumours with lower number of sub-populations of cancer cells. However, beyond a certain threshold value of the percentage of non-immunogenic cells, the immune system becomes inefficient against both types of tumours, independently of the number of sub-populations of cancer cells. In addition, we found that increasing initial percentages of non-immunogenic cells always led to a less effective CTL response. When the tumours are not eradicated, the őnal percentage of non-immunogenic cells is larger than the initial one. This suggests that the immune system may act as a bottleneck which selects and eliminates immunogenic cells, thus allowing the tumours to escape immune regulation.

In summary, our őndings demonstrate the importance of ITH as a possible predictor of the outcome of immune action. Our results support the idea that patients with tumours bearing few clonal antigens are expected to be more likely to exhibit a durable beneőt from immune response than patients with heterogeneous tumours characterised by many different sub-clonal antigens [START_REF] Mcgranahan | łClonal neoantigens elicit T cell immunoreactivity and sensitivity to immune checkpoint blockadež[END_REF]. On the other hand, our results disbelieve the fact that highly heterogeneous tumours, characterised by the expression of many different antigens, can enhance the efficacy of immune response. In fact, our results indicate that excessive antigen heterogeneity may, conversely, actively impair anti-tumour CTL immune response. This is also supported by a recent clinical work which found that excessive mutagenesis, directed to enhance the tumour mutational burden, may decrease the efficacy of immunotherapy [START_REF] Wolf | łUvb-induced tumor heterogeneity diminishes immune response in melanomaž[END_REF].

Research perspectives

The current version of our model can be developed further in several ways. We could incorporate extended aspects of the tumour micro-environment, such as the expression of immunosuppressive factors (e.g. PD1 or CTLA4), which affect the effectiveness of anti-tumour immune response. In fact, these inhibitory factors induce the exhaustion of CTLs in the tumour micro-environment impairing the immune response [START_REF] Jiang | łT-cell exhaustion in the tumor microenvironmentž[END_REF][START_REF] Wherry | łT cell exhaustionž[END_REF]. The inclusion of CTL exhaustion caused by inhibitory factors could give further explanations for other mechanisms of immune escape. The exhaustion mechanism could be included in the model by, for example, altering the value of the parameter governing the efficiency of the CTL population in eliminating tumour cells.

The spatial dimension and the ŕexibility of our model would also allow for the study of the spatial distribution of CTLs within the tumour and the role of immune inőltration on the tumour dynamics [START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF]. Moreover, by posing the model on a 3D domain, a deeper understanding of the spatial dynamics of tumour-immune interactions could be achieved.

All results we have reported on were obtained by averaging over the results of 10 realisations, in order to present in a synthetic way the outcomes of our model which includes inherent stochastic variations. When the standard deviation between these 10 realisations was relatively small, we presented the results of these realisations along with the mean and the standard deviation. This is the case of the results displayed in Figures 3.3, 3.4, 3.7, 3.9 and part of Figure 3.6. When the standard deviation was relatively large, we presented the results of each realisation individually (cf. Figure 3.8). This allowed us to investigate the different emerging behaviours of our stochastic model. Moreover, in order to check the robustness of the results, for one simulation that had a relatively large standard deviation (tumour-7a in Figure 3.6), we reported on the results of 100 realisations of this simulation (cf. Figure C.1). We observed that the overall dynamics of the 100-realisation case were qualitatively similar to those of the 10-realisation case. Consistency analysis would be another option to establish the number of replicate runs needed to obtain a desired level of numerical robustness. This could be performed using, for example, the tool presented in [START_REF] Alden | łSpartan: a comprehensive tool for understanding uncertainty in simulations of biological systemsž[END_REF].

We managed to estimate some parameters of the model (see )) from the literature and deőne them on the basis on precise biological assumptions. However, there are some parameters (e.g r, C 1 , C 2 and the parameters related to the chemoattractants) whose values were simply chosen with an exploratory aim and to qualitatively reproduce essential aspects of the experimental results obtained in [START_REF] Wolf | łUvb-induced tumor heterogeneity diminishes immune response in melanomaž[END_REF]. In order to minimise the impact of this limitation on the conclusions of our study, we carried out simulations by keeping all parameter values őxed and changing only the initial composition of the tumours, and then comparing the simulation results so obtained.

Finally, from a modelling point of view, although more tailored to capture őne details of the dynamics of single cells, individual-based models are not amenable to analytical studies, which may support a more in-depth theoretical understanding of the application problems under study. For this reason, in future work we plan to derive a continuum model from a simpliőed version of our individual-based model by using mean őled methods similar to those employed in [START_REF] Ardaševa | łComparative study between discrete and continuum models for the evolution of competing phenotype-structured cell populations in dynamical environmentsž[END_REF][START_REF] Chisholm | łEvolutionary dynamics of phenotype-structured populations: from individual-level mechanisms to population-level consequencesž[END_REF][START_REF] Painter | łNavigating the ŕow: individual and continuum models for homing in ŕowing environmentsž[END_REF].

In its present form, our modelling framework qualitatively reproduces scenarios of successful and unsuccessful immune surveillance reported in experimental studies [START_REF] Gejman | łRejection of immunogenic tumor clones is limited by clonal fractionž[END_REF][START_REF] Mcgranahan | łClonal neoantigens elicit T cell immunoreactivity and sensitivity to immune checkpoint blockadež[END_REF] and [START_REF] Wolf | łUvb-induced tumor heterogeneity diminishes immune response in melanomaž[END_REF]. However, at this stage, the model has not been calibrated using any particular type of data. Hence, it cannot be employed to generate predictions that can directly be used in the clinic. By őtting its parameters to a speciőc type of clinical data, our model could, in principle, be used to assess different levels of ITH as potential biomarkers for comparing and predicting outcomes in tumour immunotherapy treatments. Integrating the model with tumour biopsies from patients could offer insight into potential outcomes of treatments. Finally, our model may be a promising tool to explore therapeutic strategies designed to decrease tumour heterogeneity and improve the overall anti-tumour immune response.

Chapter 4

A hybrid discrete-continuum modelling approach to explore the impact of T cell infiltration on anti-tumour immune response

Motivation

Individual-based models track the dynamics of single cells, thus permitting the representation of single cell-scale mechanisms, and account for possible stochastic ŕuctuations in single-cell biophysical properties. However, they can be computationally time consuming and do not allow for analysis to be completed on the model. Considering the continuum counterpart of our stochastic discrete individual-based model developed in Chapter 3 would allow us to carry out numerical simulations that require computational times smaller than those required by the numerical exploration of the individual-based model.

In the mathematical model developed in Chapter 3, cells grow, die and interact, and we characterized antigen presentation levels by tumour cells, which drive the inŕux of CTLs in the tumour micro-environment and their movement toward tumour cells. In our study, we used a Cellular Potts model and the CompuCell3D open-source simulation environment. Due to the complexity of the modelling approach used in the previous chapter, the derivation of a continuum version of this model would be potentially challenging. The derivation of a continuum description from Cellular Potts model has been studied in [2,3,[START_REF] Lushnikov | łMacroscopic dynamics of biological cells interacting via chemotaxis and direct contactž[END_REF]. These works of increasing complexity studied, both in 1D and 2D, the continuous limit of Cellular Potts models describing the motion of an homogeneous cell population in a medium and in the presence of an external őeld with contact cell-cell interactions. Moreover, in [START_REF] Lushnikov | łMacroscopic dynamics of biological cells interacting via chemotaxis and direct contactž[END_REF] the őnite size of cells in the Cellular Potts model was taken into account, using exclusion volume principle (meaning that two cells can not occupy the same volume). However, these models did not consider many of the biological aspects included in the modelling framework developed in Chapter 3, such as cell growth and mitosis, cell death, interactions between cells of different types, etc. Therefore, these methods may not be applicable for deriving the continuum counterpart of our stochastic discrete individual-based model developed in Chapter 3.

Therefore, in this chapter we start by considering a simpler model of tumour-immune inter-actions. In this work, we present a spatial hybrid discrete-continuum modelling framework for the interaction dynamics between tumour cells and CTLs. Building on the modelling strategies developed in the previous chapter, in this framework, a stochastic individual-based model for the dynamics of tumour cells and CTLs is coupled with a balance equation for the concentration of a chemoattractant, which dictates the movement of CTLs towards the tumour. In order to take into account possible alterations in the inőltration capabilities of CTLs within the tumour, we let the probability of CTL movement be modulated by a decaying function of the densities of tumour cells and CTLs We formally derive the deterministic continuum counterpart of this model, which is given by a coupled system that comprises an integro-differential equation for the density of tumour cells, a partial differential equation for the density of CTLs, and a partial differential equation for the concentration of chemokines.

We report on computational results of the hybrid model and show that there is an excellent quantitative agreement between them and numerical solutions of the corresponding continuum model. These results shed light on the mechanisms that underlie the emergence of different levels of inőltration of CTLs into the tumour and elucidate how CTL inőltration shapes anti-tumour immune response. Moreover, exploiting the computational efficiency of the continuum model, we carry out extensive numerical simulations to investigate the impact of CTL inőltration on the response of tumour cells to different types of anti-cancer immunotherapy.

The model described in this chapter and the results shown have been published in Luis Almeida, Chloe Audebert, E. L. and Tommaso Lorenzi, A hybrid discrete-continuum modelling approach to explore the impact of T-cell infiltration on anti-tumour immune response, Bulletin of mathematical biology, 2022.

Background

Biological background

Experimental and clinical evidence indicates that the immune system plays a critical role in the prevention and eradication of tumours, by detecting immunogenic tumour cells through mutational or abnormally expressed genes and mounting an adaptive immune response [START_REF] Coulie | łTumour antigens recognized by T lymphocytes: at the core of cancer immunotherapyž[END_REF]. In an efficient immune response, cytotoxic T lymphocytes (CTLs), are activated in secondary lymphoid organs draining the tumour site. Then, these CTLs migrate to the tumour microenvironment (TME) in an attempt to eliminate the tumour [START_REF] Boissonnas | łIn vivo imaging of cytotoxic T cell inőltration and elimination of a solid tumorž[END_REF][START_REF] Miller | łAutonomous T cell trafficking examined in vivo with intravital two-photon microscopyž[END_REF].

However, a myriad of immunosuppressive strategies, the so-called immune checkpoints, help tumour cells acquiring features that allow them to evade immune detection, which may ultimately result in tumour escape. One important route towards this escape is created as tumour cells hijack the regulatory pathways of the immune system to suppress its functionality [START_REF] Rabinovich | łImmunosuppressive strategies that are mediated by tumor cellsž[END_REF]. The programmed cell death protein-1 (PD1) and its ligand PD-L1 are amongst these inhibitory pathways [START_REF] Hegde | łThe where, the when, and the how of immune monitoring for cancer immunotherapies in the era of checkpoint inhibitionž[END_REF][START_REF] Iwai | łInvolvement of PD-L1 on tumor cells in the escape from host immune system and tumor immunotherapy by PD-L1 blockadež[END_REF]. Under protracted immune stress, PD-L1 expression can be induced on tumour cells, leading to CTL exhaustion and resistance to anti-tumour immune action in many cancers, such as melanoma [START_REF] Spranger | łMelanoma-intrinsic β-catenin signalling prevents anti-tumour immunityž[END_REF][START_REF] Wieland | łT cell receptor sequencing of activated CD8 T cells in the blood identiőes tumor-inőltrating clones that expand after PD-1 therapy and radiation in a melanoma patientž[END_REF] and non-small cell lung cancer (NSCLC) [START_REF] Mcgranahan | łClonal neoantigens elicit T cell immunoreactivity and sensitivity to immune checkpoint blockadež[END_REF]. Moreover, the engagement of various oncogenic pathways results in the expression of cytokines and chemokines that mediate the exclusion of CTLs from the TME [START_REF] Kato | łProspects for personalized combination immunotherapy for solid tumors based on adoptive cell therapies and immune checkpoint blockade therapiesž[END_REF] or, alternatively, the repression of factors that facilitate CTL trafficking and inőltration into the tumour [START_REF] Spranger | łMelanoma-intrinsic β-catenin signalling prevents anti-tumour immunityž[END_REF]. In this context, the design of immune checkpoint therapies which target regulatory pathways in CTLs to enhance anti-tumour immune responses may be beneőcial to the treatment of multiple types of cancer [START_REF] Hegde | łThe where, the when, and the how of immune monitoring for cancer immunotherapies in the era of checkpoint inhibitionž[END_REF][START_REF] Kato | łProspects for personalized combination immunotherapy for solid tumors based on adoptive cell therapies and immune checkpoint blockade therapiesž[END_REF][START_REF] Ribas | łCancer immunotherapy using checkpoint blockadež[END_REF].

The observation that type, density and location of immune cells within the tumour may be associated with prognosis in different types of cancer led to the development of the 'immunoscore' as a prognostic marker in cancer patients [START_REF] Angell | łFrom the immune contexture to the Immunoscore: the role of prognostic and predictive immune markers in cancerž[END_REF][START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF][START_REF] Galon | density, and location of immune cells within human colorectal tumors predict clinical outcomež[END_REF][START_REF] Galon | łImmunoscore and Immunoproőling in cancer: an update from the melanoma and immunotherapy bridge 2015[END_REF]. The immunoscore provides a score that increases with the density of CD8+ and CD3+ CTLs both in the centre and at the margin of the tumour. In this vein, a new immune-based, rather than a cancer-based, classiőcation of tumours that relies on the immunoscore has been proposed in [START_REF] Galon | density, and location of immune cells within human colorectal tumors predict clinical outcomež[END_REF], where the authors have classiőed tumours in four categories. The łhot" category comprises tumours which are highly inőltrated by CTLs and thus have a high immunoscore. The category łaltered-immunosuppressed" is for tumours with a small amount of inőltrated CTLs. Tumours in the łaltered-excluded" category are characterised by two different regions: their margin is T cell-inőltrated while the centre is not. Tumours in these two categories have an intermediate immunoscore. Finally, łcold" tumours have a low immunoscore and are often associated with a poor response to immunotherapies, since CTLs are absent both in the centre of the tumour and at its margin.

The mathematical model

In this chapter, we develop a spatial hybrid discrete-continuum model for the interaction dynamics between tumour cells and CTLs. In this framework, a stochastic individual-based model tracking the dynamics of single tumour cells and CTLs is coupled with a balance equation for the concentration of chemokines (e.g. CXCL9/10) which are secreted by tumour cells and drive the chemotactic movement of CTLs towards the tumour [START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF][START_REF] Gorbachev | łCXC chemokine ligand 9/monokine induced by IFN-γ production by tumor cells is critical for T cell-mediated suppression of cutaneous tumorsž[END_REF].

In this model, cell dynamics are governed by a set of rules that result in a discrete-time branching random walk on a regular lattice [START_REF] Hughes | Random walks and random environments: random walks[END_REF]. Using methods similar to those we have previously employed in [START_REF] Almeida | łDiscrete and continuum models for the coevolutionary dynamics between CD8+ cytotoxic T lymphocytes and tumour cellsž[END_REF][START_REF] Bubba | łFrom a discrete model of chemotaxis with volume-őlling to a generalized Patlak-Keller-Segel modelž[END_REF][START_REF] Macfarlane | łA hybrid discrete-continuum approach to model Turing pattern formationž[END_REF], we formally derive the deterministic continuum counterpart of the hybrid model, which is given by a coupled system that comprises an IDE for the density of tumour cells, a PDE for the density of CTLs, and a PDE for the concentration of chemokines. We report on computational results of the hybrid model and show that there is an excellent quantitative agreement between them and numerical solutions of the corresponding continuum model. These results shed light on the mechanisms that underlie the emergence of different levels of inőltration of CTLs into the tumour and elucidate how CTL inőltration shapes anti-tumour immune response. Moreover, exploiting the computational efficiency of the continuum model, we carry out extensive numerical simulations to investigate the impact of CTL inőltration on the response of tumour cells to different types of anti-cancer immunotherapy.

Structure of the chapter

The chapter is organised as follows. In Section 4.3, the hybrid discrete-continuum model is introduced. In Section 4.4, the deterministic continuum counterpart of this model is presented (a formal derivation is provided in Appendix D.1). In Section 4.5, computational results of the hybrid model are discussed and integrated with numerical solutions of the continuum model. In Section 4.6, biological implications of the main őndings of this study are summarised and directions for future research are outlined.

Hybrid discrete-continuum model

In our model, each cell is seen as an agent that occupies a position on a lattice, while the concentration of chemokines, to which we will refer as łchemoattractantž in the remainder of the chapter, is described by a discrete, non-negative function. Each tumour cell can proliferate or die at certain rates. In the vein of [START_REF] Atsou | łA size and space structured model describing interactions of tumor cells with immune cells reveals cancer persistent equilibrium states in tumorigenesisž[END_REF][START_REF] Leschiera | łA mathematical model to study the impact of intra-tumour heterogeneity on anti-tumour CD8+ T cell immune responsež[END_REF][START_REF] Macfarlane | łA stochastic individual-based model to explore the role of spatial interactions and antigen recognition in the immune response against solid tumoursž[END_REF], here we focus on tumours in the early stages of development (i.e. small pre-angiogenic tumours) and, therefore, we neglect the effects of the movement of tumour-cell. As similarly done in [START_REF] Atsou | łA size and space structured model describing interactions of tumor cells with immune cells reveals cancer persistent equilibrium states in tumorigenesisž[END_REF], we let the chemoattractant be secreted by tumour cells and undergo both natural decay and linear diffusion. CTLs enter the spatial domain where the tumour is located through blood vessels at a rate that is proportional to the total amount of chemoattractant. Upon entering the domain, CTLs undergo undirected, random movement and chemotactic movement towards regions of higher concentration of the chemoattractant (i.e. cells migrate towards the tumour), and exert a cytotoxic action against tumour cells.

For ease of presentation, we let the cells and the chemoattractant be distributed across a ddimensional domain Ω, with d = 1 or d = 2. In particular, we consider the case where the spatial domain is represented by the set Ω := [0, ℓ] d , with ℓ ∈ R * + , where R * + is the set of positive real numbers not including zero. The position of the cells and the molecules of chemoattractant at time t ∈ R + is modelled by the variable x ∈ [0, ℓ] when d = 1 and by the vector

x = (x, y) ∈ [0, ℓ] 2 when d = 2.
We discretise the time variable t as t k = kτ with k ∈ N 0 and the space variables x and y as x i = iχ and y j = jχ, with (i, j) ∈ [0, N ] 2 ⊂ N 2 0 , where τ ∈ R * + and χ ∈ R * + are the time-and space-step, respectively, and N := 1 + ⌈ ℓ χ ⌉ , where ⌈•⌉ denotes the ceiling function. Here, N 0 is the set of natural numbers including zero. Throughout this section we use the notation i ≡ i and x i ≡ x i when d = 1, and i ≡ (i, j) and x i ≡ (x i , y j ) when d = 2.

We denote by n k

i the density of tumour cells, which is deőned as the number of tumour cells at position x i and at time t k , N k i ∈ N 0 , divided by the size of a lattice site, that is

n k i ≡ n(x i , t k ) := N k i χ d . (4.1)
Furthermore, we denote by c k i the density of CTLs, which is deőned as the number of CTLs at position x i and at time t k , C k i ∈ N 0 , divided by the size of a lattice site, that is

c k i ≡ c(x i , t k ) := C k i χ d . (4.2)
Finally, the concentration of chemoattractant on the lattice site i and at time-step k is modelled by the discrete, non-negative function ϕ k i ≡ ϕ(x i , t k ). In the mathematical framework of our model, the quantity

I k ≡ I(t k ) := |Ω c | |Ω tum | I Ωc (t k ) + |Ω m | |Ω tum | I Ωm (t k ) (4.3)
provides a possible simpliőed measure of the immunoscore I at time t k . In (4.3), I Ωc ∈ N 0 is the number of CTLs within the set Ω c ⊂ Ω deőned as the 'centre of the tumour', I Ωm ∈ N 0 is the number of CTLs within the set Ω m ⊂ Ω deőned as the 'margin of the tumour', and

Ω tum = Ω c ∪ Ω m is
Ω c := {x i ∈ Ω : ∥x i -x cm ∥ d < R} and I Ωc (t k ) := i c k i χ d 1 Ωc (x i ) (4.4)
while ). The immunoscore is here deőned as a weighted sum of the number of CTLs in the centre of the tumour and the number of CTLs in the margin of the tumour. In more detail, following [START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF], scenarios for which the value of the immunoscore is low will be classiőed as 'cold tumour scenarios'; scenarios with an intermediate value of the immunoscore will be classiőed as 'altered tumour scenarios', which will then be further classiőed as 'altered-immunosuppressed tumour scenarios' or 'altered-excluded tumour scenarios' based on the distribution of CTLs in the centre and at the margin of the tumour; őnally, scenarios characterised by a high value of the immunoscore will be classiőed as 'hot tumour scenarios'.

Ω m := Ω tum \ Ω c and I Ωm (t k ) := i c k i χ d 1 Ωm (x i ). ( 4
Here, x cm ∈ Ω is the initial centre of mass of the tumour, which is computed as

x cm = 1 ρ 0 n i n 0 i χ d x i , (4.6)
where ρ 0 n is the initial number of tumour cells. Moreover, 1 Ωm and 1 Ωc are the indicator functions of the sets Ω m and Ω c , respectively. Note that, in the deőnition of sets Ω c and Ω m , we are supposing that the radius R is őxed and, therefore, the two sets do not change over time. This is coherent with the fact that, as mentioned earlier, tumour-cell movement is neglected.

Abstracting from the 'immunoscore'-based classiőcation of tumours recalled in Section 4.2.1, throughout the chapter we will classify different tumour scenarios depending on the value of I at the end of numerical simulations, i.e. the quantity In particular, scenarios for which the value of I f is low will be classiőed as 'cold tumour scenarios'; scenarios with an intermediate value of I f will be classiőed as 'altered tumour scenarios', which will then be further classiőed as 'altered-immunosuppressed tumour scenarios' or 'alteredexcluded tumour scenarios' based on the distribution of CTLs at the centre and margin of the tumour; őnally, scenarios characterised by a high value of I f will be classiőed as 'hot tumour scenarios'. This tumour classiőcation is illustrated by the schematics presented in Fig. 4.1.

I f ≡ I(t f ) = |Ω c | |Ω tum | I Ωc (t f ) + |Ω m | |Ω tum | I Ωm (t f ).
The strategies used to model the dynamics of the cells and the chemoattractant when d = 1 are described in detail in the following subsections, and are also schematically illustrated in Fig. 4.2. Analogous strategies are used in the case where d = 2.

Dynamics of tumour cells

We consider a scenario where tumour cells proliferate (i.e. undergo cell division) and die due to intra-tumour competition as well as to the cytotoxic action of CTLs. We assume that a dividing tumour cell is replaced by two identical cells that are placed on the same lattice site as their parent, while a dying cell is removed from the system.

Tumour cell proliferation and death induced by intra-tumour competition

At every time-step k, we allow tumour cells to undergo cell division with probability

τ α n > 0, (4.8)
where α n ∈ R * + represents the rate of tumour cell proliferation. In order to capture the effect of cell death induced by intra-tumour competition, we let tumour cells die at a rate proportional to their number, which is deőned as

ρ k n ≡ ρ n (t k ) := i n k i χ, (4.9)
with constant of proportionality µ n ∈ R * + . Hence, between the time-step k and the time-step k + 1, we let a tumour cell die due to intra-tumour competition with probability

τ µ n ρ k n ⩾ 0. (4.10)
Notice that we are implicitly assuming that the time-step τ is sufficiently small that the quantities deőned via (4.8) and (4.10) are between 0 and 1.

Cytotoxic action of CTLs against tumour cells

When CTLs are sufficiently close to a tumour cell, they release cytotoxic substances which eventually lead to the death of the tumour cell [START_REF] Boissonnas | łIn vivo imaging of cytotoxic T cell inőltration and elimination of a solid tumorž[END_REF]. Therefore, building on the modelling strategies proposed in [START_REF] Atsou | łA size and space structured model describing interactions of tumor cells with immune cells reveals cancer persistent equilibrium states in tumorigenesisž[END_REF], we let a tumour cell die due to the cytotoxic action of CTLs at a rate proportional to the number of CTLs in a sufficiently close neighbourhood of the tumour cell. This mimics the fact that CTLs can interact with a tumour cell up to a certain distance, and that beyond such a distance the tumour cell can no longer be induced to death. Hence, we introduce K k i ∈ N 0 which models, at each time-step t k , the number of CTLs that can exert a cytotoxic action against a tumour cell at position x i . In particular, we deőne K k i as

K k i ≡ K(x i , t k ) := p η(x i , x p ; θ)c k p χ. (4.11)
The function η is deőned as follow

η(x, z; θ) := 1 if |x -z| ⩽ θ 0 if |x -z| > θ for (x, z; θ) ∈ Ω × Ω × (0, |Ω|], (4.12)
where |Ω| denotes the size of the set Ω (i.e. |Ω|= ℓ if Ω := [0, ℓ]). The quantity K k i deőned via (4.11) and (4.12) represents the number of CTLs at positions x p at a distance smaller than or equal to θ from x i . The parameter 0 < θ ⩽ |Ω| regulates the maximum radius of interaction between a tumour cell at position x i and the CTLs in its neighbourhood. Therefore, we deőne the probability of death of tumour cells at position x i and time t k due to the cytotoxic action of CTLs as

τ ζ n K k i ⩾ 0. (4.13)
The parameter ζ n ∈ R * + is linked to the level of efficiency of CTLs at eliminating tumour cells: lower values of ζ n correspond to scenarios in which this cytotoxic action of CTLs is less effective, due for example to a higher expression of PD1/PD-L1 inhibitory receptors on the surface of CTLs/tumour cells [START_REF] Hegde | łThe where, the when, and the how of immune monitoring for cancer immunotherapies in the era of checkpoint inhibitionž[END_REF][START_REF] Iwai | łInvolvement of PD-L1 on tumor cells in the escape from host immune system and tumor immunotherapy by PD-L1 blockadež[END_REF].

Remark 2. Note that (4.8), (4.10) and (4.13) implicitly require the time-step τ to be sufficiently small that τ α

n + τ µ n ρ k n + ζ n K k i
is less than or equal to 1.

Dynamics of the chemoattractant

We denote by ϕ k i the concentration of chemoattractant at position x i and at time t k . The dynamic of ϕ k i is governed by the following discrete balance equation

ϕ k+1 i = ϕ k i + τ β ϕ (Lϕ k ) i + τ α ϕ n k i -τ κ ϕ ϕ k i , (i, k) ∈ [1, N -1] × N, (4.14)
subject to discrete zero-ŕux boundary conditions, i.e.

ϕ k 0 = ϕ k 1 and ϕ k N = ϕ k N -1 , k ∈ N. (4.15)
In the balance equation (4.14), L is the second-order central difference operator on the lattice

{x i } i , i.e. ( Lϕ
k ) i = 1 χ 2 ϕ k i+1 + ϕ k i-1 -2ϕ k i (4.16)
Moreover, the parameter

β ϕ ∈ R * + is the diffusion coefficient of the chemoattractant and κ ϕ ∈ R * +
is the rate at which the chemoattractant undergoes natural decay. Finally, the parameter α ϕ ∈ R * + represents the per capita production rate of the chemoattractant by tumour cells. The balance equation (4.14) is simply a standard discretisation of a reaction-diffusion equation of the type that is commonly used to describe the dynamics of molecular species, see for example [START_REF] Maini | łSpatial pattern formation in chemical and biological systemsž[END_REF]. Moreover, we will assume that

0 < ϕ k i ⩽ ϕ max ∀ (i, k) ∈ [0, N ] × N 0
for some maximal concentrations ϕ max ∈ R * + (see comments below deőnition (4.25)).

Dynamics of CTLs

Following [START_REF] Gong | łA computational multiscale agent-based model for simulating spatio-temporal tumour immune response to PD1 and PDL1 inhibitionž[END_REF], we consider a scenario where CTLs are recruited from different sources corresponding to blood vessels that are located in the tissue surrounding the tumour. CTLs can change their position according to a combination of undirected, random movement and chemotactic movement, which are seen as independent processes. Finally, CTLs can die at a certain rate due to homeostatic regulation mechanisms, and dying cells are removed from the system. This results in the following rules for the dynamics of CTLs.

Inflow and death of T cells

We let ω ⊂ Ω be the set of points in the tissue surrounding the tumour that are occupied by blood vessels, through which new T cells can enter the domain. Since we do not consider the formation of new blood vessels, we assume that ω is given and does not change in time. We denote by r k i the term controlling the inŕow of T cells from blood vessels, which is deőned as

r k i ≡ r(x i , t k ) := ϕ k tot 1 ω (x i ), (4.17)
where 1 ω is the indicator function of the set ω and ϕ k tot is the total amount of chemoattractant at time t k , that is,

ϕ k tot ≡ ϕ tot (t k ) := i ϕ k i χ. (4.18)
We then let the inŕux rate of T cells from blood vessels at position x i and time t k be proportional to r k i with constant of proportionality α c ∈ R * + . Hence, between the time-step k and the time-step k + 1, we let a density of T cells equal to

τ α c r k i ⩾ 0 (4.19)
enter a blood vessel at position x i . Finally, we let T cells die due to homeostatic regulation mechanisms. In analogy with the case of tumour cells, we suppose the rate of death of T cells to be proportional to the number of T cells

ρ k c ≡ ρ c (t k ) := i c k i χ, (4.20)
with constant of proportionality µ c ∈ R * + . Therefore, between the time-step k and the time-step k + 1, we let a T cell die with probability

τ µ c ρ k c ⩾ 0. (4.21)
Remark 3. Note that (4.21) implicitly requires the time-step τ to be sufficiently small that the corresponding quantity is less than or equal to 1.

Chemotactic movement of CTLs

We now turn to modelling the chemotactic movement of CTLs (i.e. the movement of CTLs up the gradient of the chemoattractant ϕ k i ). Building on [START_REF] Bubba | łFrom a discrete model of chemotaxis with volume-őlling to a generalized Patlak-Keller-Segel modelž[END_REF], chemotactic movement is here modelled as a biased random walk whereby the movement probabilities depend on the difference between the concentration of chemoattractant at the site occupied by a CTL and the concentration of chemoattractant at the neighbouring sites. To take into account possible reduction in cell motility at high cell densities [START_REF] Painter | łMathematical models for chemotaxis and their applications in self-organisation phenomenaž[END_REF][START_REF] Slaney | łTrafficking of T cells into tumorsž[END_REF][START_REF] Van Der Woude | łMigrating into the tumor: a roadmap for T cellsž[END_REF], we incorporate into the model volume-őlling effects [START_REF] Painter | łVolume-őlling and quorum-sensing in models for chemosensitive movementž[END_REF], whereby CTL movement is allowed only to site locations x i where the total cell density w k i := n k i + c k i is smaller than a threshold value w max ∈ R * + , which corresponds to a cell tight-packing state. Therefore, we modulate the movement probabilities of CTLs by a monotonically decreasing function of the total cell density at the neighbouring sites. Speciőcally, as similarly done in [START_REF] Wang | łClassical solutions and pattern formation for a volume őlling chemotaxis modelž[END_REF], we deőne this function as

ψ(w k i ) :=    1 - w k i w max , 0 ⩽ w k i ⩽ w max , 0, otherwise. (4.22)
Hence, for a CTL on the lattice site x i and at time step t k , we deőne:

1. the probability of moving to the lattice site x i-1 (i.e. the probability of moving left) via 84 CHAPTER 4. Exploring the impact of CTL inőltration on anti-tumour immune response chemotaxis as

q k Li := ν ψ(w k i-1 ) ϕ k i-1 -ϕ k i 2ϕ max + for i ∈ [1, N ] and q k L0 = 0 (4.23)
where (•) + denotes the positive part of (•);

2. the probability of moving to the lattice site x i+1 (i.e. the probability of moving right) via chemotaxis as

q k Ri := ν ψ(w k i+1 ) ϕ k i+1 -ϕ k i 2ϕ max + for i ∈ [0, N -1] and q k RN = 0; (4.24)
3. and the probability of not undergoing chemotactic movement as

1 -q k Li -q k Ri for i ∈ [0, N ]. (4.25)
Here, the parameter ν ∈ R + , with 0 ⩽ ν ⩽ 1, is directly proportional to the chemotactic sensitivity of CTLs. Moreover, the parameter ϕ max > 0 is directly proportional to the maximal value that can be attained by the concentration of chemoattractant. In particular, on the basis of the considerations drawn in [START_REF] Bubba | łFrom a discrete model of chemotaxis with volume-őlling to a generalized Patlak-Keller-Segel modelž[END_REF], we deőne ϕ max as

ϕ max := max max i ϕ 0 i , A w max ,
where w max is given by (4.22) and A > 0 is a scaling factor that ensures unit consistency. Dividing by ϕ max ensures that the values of the quotients in (4.23)- (4.25) are all between 0 and 1.

Undirected, random movement of CTLs

To model the effect of undirected, random movement, we allow CTLs to update their position according to a random walk with movement probability λ ∈ R * + , where 0 < λ ⩽ 1. In particular, we assume that a CTL on the lattice site x i can move via undirected, random movement into either the lattice site x i-1 or the lattice site x i+1 with probability λ 2 . As similarly done in the case of chemotactic movement, in order to capture a possible reduction in CTL motility at higher cell densities [START_REF] Painter | łMathematical models for chemotaxis and their applications in self-organisation phenomenaž[END_REF][START_REF] Slaney | łTrafficking of T cells into tumorsž[END_REF][START_REF] Van Der Woude | łMigrating into the tumor: a roadmap for T cellsž[END_REF], we modulate the movement probability by a decreasing function of the density of tumour cells and CTLs at the neighbouring sites. In particular, for a CTL on the lattice site i and at the time-step k, recalling the notation w k i := n k i + c k i for the total cell density and deőnition (4.22) of the modulating function ψ(w k i ), we deőne: 1. the probability of moving to the lattice site i -1 via undirected, random movement as

T k Li := λ 2 ψ(w k i-1 ) for i ∈ [1, N ] and T k L0 = 0; (4.26)
2. the probability of moving to the lattice site i + 1 via undirected, random movement as

T k Ri := λ 2 ψ(w k i+1 ) for i ∈ [0, N -1] and T k RN = 0; (4.27)
3. and the probability of not undergoing undirected, random movement as

1 -T k Li -T k Ri for i ∈ [0, N ]. ( 4

Corresponding continuum model

Letting the time-step τ → 0 and the space-step χ → 0 in such a way that

λχ 2 2d τ → β c ∈ R + * and νχ 2 2d ϕ max τ → γ c ∈ R + * , (4.29)
using the formal method employed in [START_REF] Almeida | łDiscrete and continuum models for the coevolutionary dynamics between CD8+ cytotoxic T lymphocytes and tumour cellsž[END_REF][START_REF] Bubba | łFrom a discrete model of chemotaxis with volume-őlling to a generalized Patlak-Keller-Segel modelž[END_REF][START_REF] Macfarlane | łA hybrid discrete-continuum approach to model Turing pattern formationž[END_REF], it is possible to formally show (see Appendix D.1) that the deterministic continuum counterpart of the hybrid model described in Section 4.3 comprises the following coupled IDE-PDE-PDE system for the density of tumour cells n(x, t), the density of CTLs c(x, t) and the chemoattractant concentration ϕ(x, t)

                         ∂ t n = α n n -µ n ρ n (t) n -ζ n K(x, t)n ∂ t c -∇ • β c ψ(w)∇c -γ c ψ(w)c∇ϕ -β c cψ ′ (w)∇w = -µ c ρ c (t)c + α c r(x, t) ∂ t ϕ -β ϕ ∆ϕ = α ϕ n -κ ϕ ϕ w(x, t) := n(x, t) + c(x, t), (4.30)
where the IDE (4.30) 1 is posed on Ω × R * + , while the PDEs (4.30) 2 and (4.30) 3 are posed on Ω\∂Ω×R * + and are subject to zero-ŕux boundary conditions on ∂Ω. The IDE-PDE-PDE system (4.30) is complemented with the following deőnitions

K(x, t) := Ω η(x, x ′ ; θ)c(x ′ , t) dx ′ , r(x, t) := ϕ tot (t)1 ω (x) ρ n (t) := Ω n(x, t) dx, ρ c (t) := Ω c(x, t) dx, ϕ tot (t) := Ω ϕ(x, t) dx.
In system (4.30), β c ∈ R * + deőned via (4.29) is the diffusion coefficient (i.e. the motility) of CTLs, while γ c ∈ R + deőned via (4.29) is the chemotactic sensitivity of CTLs to the chemoattractant.

Numerical simulations

In this section, we report on computational results of the hybrid discrete-continuum model along with numerical solutions of the corresponding continuum model given by the IDE-PDE-PDE system (4.30). First, we establish a baseline scenario in which the level of efficiency of CTLs at eliminating tumour cells (i.e. the parameter ζ n ) is sufficiently high so as to lead to tumour eradication. Then, we reduce the level of CTL efficiency in order to avoid tumour eradication, and we explore the mechanisms that underlie the emergence of different levels of inőltration of CTLs into the tumour, which correspond to cold, altered-immunosuppressed, altered-excluded and hot tumour scenarios. In particular, we carry out sensitivity analysis to two parameters that we expect to play a key role in determining the spatial distribution of CTLs: the secretion rate of chemoattractant by tumour cells (i.e. the parameter α ϕ ) and the threshold value of the total cell density above which CTL movement is impaired (i.e. the parameter w max in deőnition (4.22)). Finally, we investigate the impact of CTL inőltration on the response of tumour cells to different types of anti-cancer immunotherapy.

Set-up of numerical simulations

The hybrid and continuum models are parameterised using parameter values retrieved from the literature, wherever possible. The full list of parameter values and related references are provided in Table 4.1. For the numerical simulations we report on, we use the 2D spatial domain Ω := [0, 1] 2 . Under the parameter choice of Table 4.1, this is equivalent to considering a square region of a 2D cross-section of a tumour tissue of area 1 cm 2 . Furthermore, to carry out numerical simulations of the hybrid model, we use the space-step χ = 0.016 cm and the time-step τ = 1 × 10 -4 days. Finally, unless otherwise speciőed, we choose the őnal time t f = 15 days. All simulations are performed in Matlab [146].

Initial conditions and blood vessel distribution Fig. 4.3 displays the initial conditions chosen to carry out numerical simulations. In particular, for the hybrid model we have

n 0 ij = 800 exp[-200(iχ -x * 11 ) 2 -200(jχ -y * 11 ) 2 ], (4.31)
c 0 ij = 60 4 p=1 exp[-300(iχ -x * 2p ) 2 -300(jχ -y * 2p ) 2 ], (4.32)
and

ϕ 0 ij = 90 exp[-200(iχ -x * 11 ) 2 -200(jχ -y * 11 ) 2 ], (4.33) with (x
(x, y, 0) = 800 exp[-200(x -x * 11 ) 2 -200(y -y * 11 ) 2 ], (4.34)
c(x, y, 0) = 60

4 p=1 exp[-300(x -x * 2p ) 2 -300(y -y * 2p ) 2 ], (4.35)
A description of the algorithmic rules that underlie computational simulations of the hybrid model is provided in Appendix D.2.1, while the methods employed to numerically solve the IDE-PDE-PDE system (4.30), subject to suitable initial conditions and no-ŕux boundary conditions, are detailed in Appendix D.2.2.

Given the initial conditions of the two models, we compute the coordinates of the centre of mass of the tumour (x * cm , y * cm ) = (0.5, 0.5), and we deőne the set Ω c (i.e. the 'centre of the tumour') via deőnition (4.4) as

Ω c := x i ≡ (x i , y j ) ∈ [0, 1] 2 : (x i -x * cm ) 2 + (y j -y * cm ) 2 < 0.144 .
The set Ω c corresponds to approximately 65% of the region initially occupied by the tumour and, therefore, the set Ω m = Ω \ Ω c (i.e. the 'margin of the tumour') comprises the remaining 35% of the tumour region.

Parameter values Unless otherwise speciőed, we use the parameter values listed in Table 4.1.

Here, the value of the parameter α n is consistent with previous measurement and estimation studies on the dynamics of tumour cells [START_REF] Christophe | łA biased competition theory of cytotoxic T lymphocyte interaction with tumor nodulesž[END_REF], where the authors calculated the estimated proliferation rate of a tumour cell using the average duplication time of melanoma cells. The values of the diffusion coefficient β ϕ and decay rate κ ϕ of the chemoattractant correspond to those used in [START_REF] Cooper | łA Cellular Automata and a Partial Differential Equation Model of Tumor-Immune Dynamics and Chemotaxisž[END_REF][START_REF] Matzavinos | łMathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumourž[END_REF]. Moreover, the range of values of the secretion rate α ϕ is consistent with those used in [START_REF] Atsou | łA size and space structured model describing interactions of tumor cells with immune cells reveals cancer persistent equilibrium states in tumorigenesisž[END_REF]. To explore a wide range of biological situations corresponding to different degrees of immune inőltration, we use an arbitrary range of values for the parameter w max . Finally, the values of the parameters λ and ν correspond to values of β c and γ c that are consistent with those reported in [START_REF] Matzavinos | łMathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumourž[END_REF] and [START_REF] Atsou | łA size and space structured model describing interactions of tumor cells with immune cells reveals cancer persistent equilibrium states in tumorigenesisž[END_REF], respectively. Given the values of the parameters β c and γ c chosen to carry out numerical simulations of the continuum model, the following deőnitions are used for the hybrid model

λ := β c 4τ χ 2 and ν := γ c 4ϕ max τ χ 2
so that conditions (4.29) are met.

Baseline scenario corresponding to tumour eradication

As mentioned earlier, we őrst establish a baseline scenario where the parameter ζ n is high enough so that CTLs are able to eradicate the tumour. The plots in Fig. 4.4 and Fig. 4.5 summarise the results of simulations of the hybrid and continuum models obtained under this scenario. After initial growth, the number of tumour cells decreases steadily over time until tumour cells are completely eliminated (cf. Fig. 4.4). The chemoattractant produced by tumour cells triggers the inŕow of CTLs through blood vessels and the movement of CTLs towards the tumour (cf. Fig. 4.5). Since the value of ζ n is sufficiently large, once CTLs are close enough to tumour cells they start eliminating them until eradication (cf. Fig. 4.5). When the number of tumour cells decreases, the total amount of chemoattractant decays as well, thus triggering a reduction in the inŕow of CTLs, which initiates a decrease in the number of CTLs (cf. Chemoattr.

Concentration at position x φ(x, t) ⩾ 0 (mol/cm 2 ) Total amount φtot(t) ⩾ 0 (mol) Diffusion coefficient β φ = 10 -1 (cm 2 /day) [148]
Secretion rate α φ ∈ [0.001, 1.5](mol/cell/day) [START_REF] Atsou | łA size and space structured model describing interactions of tumor cells with immune cells reveals cancer persistent equilibrium states in tumorigenesisž[END_REF] Decay rate κ φ = 2 (1/day) [START_REF] Cooper | łA Cellular Automata and a Partial Differential Equation Model of Tumor-Immune Dynamics and Chemotaxisž[END_REF] Maximal concentration φmax = Awmax, with A = 1 (c) show that increasing the value of α ϕ leads to a progressive increase in the total amount of chemoattractant. This in turn increases the inŕow of CTLs and facilitates their movement towards the tumour. The spatial distribution of CTLs within the tumour varies depending on the value of w max . Fig. 4.9 shows that smaller values of w max lead to an accumulation of CTLs at the margin of the tumour, which corresponds to an altered-excluded tumour scenario. On the other hand, larger values of w max promote the inőltration of CTLs into the tumour and lead to an altered-immunosuppressed tumour scenario (see Fig. 4.1. The results from the hybrid model correspond to the average over three simulations and the related variance is displayed by the coloured areas surrounding the curves. and continuum models presented in the previous subsections, here we carry out the numerical simulations of the continuum model only, since they require computational times much smaller than those required by the numerical exploration of the corresponding hybrid model. Moreover, to obtain these results, we carried out numerical simulations only for 10 days, i.e. t f = 10.

Anti-PD1 monotherapy Fig. 4.12(b) displays the number of tumour cells at the end of numerical simulations of the continuum model for parameter settings corresponding to anti-PD1 monotherapy (i.e. when only the value of ζ n in increased). Comparing these results with those displayed in Fig. 4.12(a), we see that, in general, for the same values of parameters α ϕ and w max , increasing the value of ζ n leads to a decrease in the numbers of tumour cells at the end of simulations. However, when the value of α ϕ is too small (i.e. in cold tumour scenarios) or when the value of w max is too small (i.e. in altered-excluded tumour scenarios), increasing ζ n has no beneőt on the action of CTLs against tumour cells. Finally, when the values of α ϕ and w max are sufficiently large (i.e. in hot tumour scenarios) anti-PD1 monotherapy is more effective.

Anti-PD1-CTLA4 dual therapy Fig. 4.12(c) displays the number of tumour cells at the end of numerical simulations of the continuum model for parameter settings corresponding to anti-PD1-CTLA4 dual therapy (i.e. when both the value of ζ n and the value of α c are increased). Comparing these results with those displayed in Fig. 4.12(b), we see that increasing the value of α c along with the value of ζ n improves immune efficacy only when the values of α ϕ and w max are large enough (i.e. in hot tumour scenarios). Moreover, for intermediate values of α ϕ (i.e. Chemotherapy combined with anti-PD1 therapy Fig. 4.12(d) displays the number of tumour cells at the end of numerical simulations of the continuum model for parameter settings corresponding to chemotherapy in combination with anti-PD1 therapy (i.e. when the value of α n is decreased and the values of ζ n and α c are increased). Compared to the other heat maps of Fig. 4.12, these results show that the number of tumour cells decreases even for small values of α ϕ or w max . Moreover, when the values of α ϕ or w max are small (i.e. in cold and altered-excluded tumour scenarios), the numbers of tumour cells at the end of simulations are similar. On the other hand, from intermediate to large values of α ϕ and w max , the numbers of tumour cells at the end of simulation decrease as the values of these two parameters increase. As expected, the larger the values of α ϕ and w max (i.e. the łhotter" the tumour scenario considered), the more effective the combined action of chemotherapy and anti-PD1 therapy. count the spatial dynamics leading to inőltration of CTLs into the tumour. We used a discrete mass-balance equation for the evolution of a chemoattractant, which drives the movement of CTLs towards the tumour, while cell dynamics were described by stochastic individual-based models. From our hybrid model, a continuum model comprising a PDE for the CTL density, an IDE for the tumour cell density and a PDE for the concentration of chemoattractant has been formally derived. Through comparison of both models we found that the results of computational simulations of the hybrid model faithfully mirror the qualitative properties of the solutions to the corresponding continuum model. The results of computational simulations of the hybrid model, which are in excellent agreement with numerical solutions of the continuum model, shed light on the way in which different parameters affect the spatial distribution of CTLs within the tumour and allow us to assess the impact of CTL inőltration on the immune response against tumours.

CHAPTER 4. Exploring the impact of CTL inőltration on anti-tumour immune response

The results that we have presented demonstrate that the level of efficiency of CTLs at eliminating tumour cells (i.e. the parameter ζ n ) plays a key role in tumour-immune competition. In fact, when the value of ζ n is large enough, our results indicate that tumour eradication can occur, while lower values of ζ n may result in tumour cell survival. This is consistent with experimental and clinical data which point to a key role of immune check-points in immunosuppressing CTL responses. In fact, the presence of immunosuppressive components in the TME, such as PD-1 and/or PD-L1 inhibitory receptors, decreases the efficiency of CTLs at eliminating tumour cells, and can ultimately result in tumour escape [START_REF] Huang | łT-cell invigoration to tumour burden ratio associated with anti-PD-1 responsež[END_REF][START_REF] Tumeh | łPD-1 blockade induces responses by inhibiting adaptive immune resistancež[END_REF].

Moreover, our numerical results indicate that when tumour eradication does not occur (i.e. when the value of ζ n is sufficiently small), the secretion rate of the chemoattractant by tumour cells (i.e. the parameter α ϕ ) and the threshold value of the total cell density above which CTL movement is impaired (i.e. the parameter w max ) have a strong impact on the level of inőltration of CTLs into the tumour, and different combinations of the values of these parameters bring about the emergence of four immune-based tumour scenarios. Hot tumour scenarios emerge for high values of α ϕ and w max , and are characterised by a large number of CTLs in the centre of the tumour. By displaying a high degree of CTL inőltration, these tumours provide a fertile ground for immune checkpoint therapies. Altered tumour scenarios emerge for intermediate values of α ϕ , and reŕect the intrinsic ability of the immune system to effectively mount a CTL -mediated immune response and the ability of tumour cells to partially escape such a response. This can either be due to an insufficient number of inőltrated CTLs (the immunosuppressed tumour scenarios, which emerge for intermediate to large values of w max ) or to the presence of physical barriers that hinder CTL inőltration (the excluded tumour scenarios, which emerge for small values of w max ). Finally, cold tumour scenarios emerge for sufficiently small values of α ϕ and w max . These tumours are characterised by an insufficient number of CTLs both in the centre of the tumour and at its margin, and are invariably associated with poor prognosis.

We also explored how the outcomes of different immunotherapy protocols can vary in these four immune-based tumour scenarios. In particular, our results suggest that increasing the level of efficiency of CTLs (i.e. the value of the parameter ζ n ), which is associated to the effects of anti-PD1 monotherapy, is not sufficient for treating all types of tumour scenarios, and it is particularly ineffective in altered-excluded and cold tumour scenarios. This őnding is coherent with experimental observations indicating that anti-PD1 monotherapy is effective only in the context of hot or altered-immunosuppressed tumours, as a certain number of CTLs is already inőltrated into the tumour [START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF].

Moreover, the results of our model indicate that, in these two categories of tumours, increasing both the level of efficiency of CTLs and their rate of inŕow (i.e. the value of the parameters ζ n and α c ), which are associated with the combined effects of anti-PD1 and anti-CTLA4 therapy, may lead to a better therapeutic outcome. This conclusion is also supported by experimental work showing that anti-PD1-CTLA4 dual therapy may be successful in treating advanced-stage melanoma [START_REF] Wolchok | łOverall survival with combined nivolumab and ipilimumab in advanced melanomaž[END_REF], renal-cell carcinoma [START_REF] Motzer | łNivolumab plus ipilimumab versus sunitinib in advanced renal-cell carcinomaž[END_REF] and non-small-cell lung cancer (NSCLC) [START_REF] Hellmann | łNivolumab plus ipilimumab in lung cancer with a high tumor mutational burdenž[END_REF], resulting in regulatory approval. However, our results suggest that prognosis in altered-excluded and cold tumour scenarios may not beneőt from the combined effects of these two immune checkpoint inhibitors. Nevertheless, our results indicate that therapeutic strategies promoting the inőltration of CTLs could turn altered-excluded tumours into altered-immunosuppressed or hot tumours, helping to decrease the resistance of tumours to the combination of anti-PD1 and anti-CTLA4 therapy. This őnding is coherent with experimental observations suggesting that a synergistic effect can be achieved by combining anti-angiogenic therapies, which act on vascular abnormalities facilitating CTL inőltration, with immune checkpoint therapies [START_REF] Tian | łMutual regulation of tumour vessel normalization and immunostimulatory reprogrammingž[END_REF].

Finally, the outputs of our model suggest that increasing both the level of efficiency of CTLs and their rate of inŕow (i.e. the values of the parameters ζ n and α c ) and decreasing the pro-liferation rate of tumour cells (i.e. the parameter α n ), which may represent the combination of anti-PD1 therapy with chemotherapy, a stronger immune response may be induced. In fact, a proposed approach to overcome the lack of a pre-existing immune response consists in combining a priming therapy that enhances CTL responses (such as chemotherapy) with the removal of coinhibitory signals (through approaches such as immune checkpoint therapies) [START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF]. For example, the success of the combination of anti-PD-1 therapy with chemotherapy in metastatic NSCLC has demonstrated the strength of this dual approach [START_REF] Gandhi | łPembrolizumab plus chemotherapy in metastatic non-small-cell lung cancerž[END_REF].

Research perspectives

We conclude with an outlook on possible extensions of the present work. While here we focused on the role of the secretion rate of the chemoattractant by tumour cells and the threshold value of the total cell density above which T cell movement is impaired, it would be interesting to investigate how other model parameters (e.g. the chemotactic sensitivity of T cells) may affect the level of inőltration of T cells into the tumour. Carrying out a more extensive exploration of the model parameter space would ultimately allow more robust biological conclusions to be drawn.

Moreover, our hybrid modelling framework for the spatial dynamics of tumour cells and cytotoxic T cells, along with the formal derivation of the corresponding continuum model, can be developed further in several ways. For instance, a key factor of the immune response is that T cells express a unique repertoire of T cell receptors (TCRs) [START_REF] Coulie | łTumour antigens recognized by T lymphocytes: at the core of cancer immunotherapyž[END_REF], and are capable of detecting and eliminating tumour cells by recognising speciőc cancer-associated antigens. The model presented here does not include this aspect, but it could easily be extended to do so by introducing, for instance, a variable representing the antigens expressed by tumour cells and the TCR expressed by T cells. This would make it possible to take explicitly into account the effects of both spatial and antigen-speciőc interactions between tumour cells and T cells, as similarly done in [START_REF] Kather | łIn silico modeling of immunotherapy and stroma-targeting therapies in human colorectal cancerž[END_REF][START_REF] Leschiera | łA mathematical model to study the impact of intra-tumour heterogeneity on anti-tumour CD8+ T cell immune responsež[END_REF][START_REF] Macfarlane | łModelling the immune response to cancer: an individual-based approach accounting for the difference in movement between inactive and activated T cellsž[END_REF][START_REF] Macfarlane | łA stochastic individual-based model to explore the role of spatial interactions and antigen recognition in the immune response against solid tumoursž[END_REF], and then study the effects of antigen presentation or intra-tumour heterogeneity on immune surveillance.

Only a simpliőed representation of the action of different types of immunotherapy was considered in this work, but it would be important to carry out a more detailed study of the impact of T-cell inőltration on the dynamics of tumour cells under different immunotherapeutic protocols. In particular, by using optimal control methods for the continuum model, we could investigate the best delivery schedule of therapeutic agents (i.e. the best delivery times and dosages) that make it possible to minimise the number of tumour cells at the end of the treatment and achieve the best therapeutic outcomes [START_REF] Jarrett | łOptimal control theory for personalized therapeutic regimens in oncology: Background, history, challenges, and opportunitiesž[END_REF]. These are all lines of research that we will be pursuing in the near future.

Chapter 5

Integrating the model with data: a discrete model to investigate the effect of psychological stress on immune infiltration

Motivation

In the previous chapters, we have developed discrete and continuum models of tumour-immune interactions that attempt to describe different aspects of the anti-tumour immune response. The different models have been parametrised using parameter values obtained from published works wherever possible and aim to investigate, or qualitatively reproduce, observed cell behaviours. However, at this stage, these models have not been calibrated using any particular type of data. Hence, they cannot be employed to generate predictions that can directly be used in the clinic.

Motivated by experimental observations in co-culture between breast cancer spheroids and activated immune cells, which reproduce in vivo behaviours as described in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF], in this chapter we explore a simple setting to study the effect of psychological stress on immune inőltration. In comparison to the models developed in the previous chapters, the mathematical framework presented here is developed in collaboration with the biologists who designed and conducted the in vitro experiment presented in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF]. Compared to in vivo studies, which are performed on a whole living organism, in vitro experiments are performed in a controlled environment, allowing for a more detailed analysis and examination of biological effects. For this reason, mathematical models can more easily be calibrated on in vitro studies, in order to reproduce the speciőc in vitro experiment.

To study the effects of stress on immune inőltration, we use a mathematical model based on a simpliőed version of the one developed in Chapter 3, and we calibrate it to qualitatively reproduce, in silico, the experimental results obtained in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF]. In this experimental study, the authors found that the introduction of cortisol in the co-culture resulted in a decrease in immune cell inőltration into cancer spheroids, a decrease in the levels of the pro-inŕammatory cytokine IFN-γ and an increase in the levels of the anti-inŕammatory cytokine IL-10. In our model, we assume that stress replaces the role of cortisol in the experiment, and investigate which mechanisms are most likely to be affected by stress and impact immune inőltration. Using a of inhibitory cytokines, such as the pro-tumourigenic chemokine CXCL10 and granzyme B, contributing to suppression of anti-tumour immune response. In [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF], the results from the analysis of IFN-γ and IL-10 demonstrated that stress decreased the levels of IFN-γ and increased the levels of IL-10. IFN-γ is a pro-inŕammatory cytokine that stimulates immune response, such as T cell trafficking in the tumour-microenvironment and inőltration [START_REF] Castro | łInterferongamma at the crossroads of tumor immune surveillance or evasionž[END_REF][START_REF] Schiltz | łEffects of IFN-γ and Interleukin-1 β on Major Histocompatibility Complex Antigen and Intercellular Adhesion Molecule-1 Expression by 9L Gliosarcoma: Relevance to Its Cytolysis by Alloreactive Cytotoxic T Lymphocytesž[END_REF], whereas IL-10 is an anti-inŕammatory cytokine that inhibits immune response, such as T cell proliferation [START_REF] Alhakeem | łChronic lymphocytic Leukemia-Derived IL-10 suppresses antitumor immunityž[END_REF][START_REF] Couper | łIL-10: the master regulator of immunity to infectionž[END_REF].

From a biological and medical perspective, it is difficult to investigate the connection between stress, immune inőltration and the underlying molecular and cellular processes. The challenge is to integrate theoretical and empirical knowledge to better understand the mechanisms and factors that contribute to failure of the anti-tumour immune response. In this context, mathematical models provide useful tools towards identifying dependencies between different phenomena and how these may affect the efficacy of the immune response.

The mathematical model

In light of these considerations, and motivated by the results presented in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF], in this chapter we develop a spatial stochastic individual-based model of the interaction dynamics between tumour cells and cytotoxic T lymphocytes (CTLs), to explore the role played by psychological stress in immune inőltration. In particular, our goal is to investigate which mechanisms are most likely to be inŕuenced by psychological stress and impact immune inőltration. The individual-based model considered here is based on a modiőed and simpliőed version of the Cellular Potts model developed in Chapter 3. In the in vitro experiment only one tumour cell line is used. Therefore, compared to the model presented in Chapter 3, here we are not interested in intra-tumour heterogeneity, and we do not incorporate antigen expression in the model.

In [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF], the authors found that cortisol decreased the levels of the pro-inŕammatory cytokine IFN-γ and increased the levels of the anti-inŕammatory cytokine IL-10. Therefore, to explore which mechanisms are inŕuenced by psychological stress, in our model we take into account the effects that these two cytokines may have on immune response. On the one hand, the effect of INF-γ is taken into account by letting tumour cells at the border of the tumour (the region where cytokines and immune cells are more abundant in the tumour-microenvironment) secrete a chemoattractant, which drives the movement of CTLs towards the tumour. Moreover, we let INF-γ also affect the capability of CTLs to inőltrate through tumour cells. On the other hand, the effect of IL-10 is taken into account by modulating the growth rate of CTLs, which directly affects their proliferation rate.

In this work, we explore different scenarios. We suppose that in normal conditions the levels of IFN-γ are high, the levels of IL-10 are low, and CTLs are able to inőltrate into the tumour. In stressed conditions however, we suppose that the levels of IFN-γ decrease, the levels of IL-10 increase, and CTL inőltration is reduced. By varying the secretion rate of the chemoattractant, the capability of CTL to inőltrate through tumour cells and the growth rate of CTLs, we study the impact of stress on immune inőltration. Finally, using a method similar to the one employed in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF], we deőne a score to quantify the effects of stress on immune inőltration in a controlled manner.

Structure of the chapter

The chapter is organised as follows. In Section 5.3, we describe the in vitro experiments and the main experimental observations that motivated our modelling assumption. The model developed in collaboration with the biologists is described in Section 5.4. The set-up of computational simulations is summarised in Section 5.5, where some preliminary results of computational sim-5.3. Description of the experiments 105 there were 5 times as many T cells as tumour cells at time -7. Spheroids and splenocytes were later co-cultured, imaged daily using confocal microscopy, and split into 7 groups. Group-1 contained spheroids only and group-2 comprised spheroids and splenocytes. The other 5 groups were treated daily with corticosterone (cortisol). Group-3 contained spheroids, splenocytes and cortisol. Groups-4-5-6 contained spheroids, splenocytes and 3 different glucocorticoid receptor antagonists. Group-7 contained spheroids and cortisol. Spheroids from each group were treated daily for 4 days (i.e. from day 0 to day 4), and the media was saved for further analysis.

Implementation of two independent trafficking indexes to measure infiltration levels

To test whether cortisol decreased immune inőltration in the co-culture model, the co-culture of each group was imaged daily for 4 days (i.e. from day 0 to day 4) using confocal microscopy. Two trafficking indexes were developed by the authors and applied every day to quantify immune inőltration levels. Both approaches sought to measure immune cell inőltration into the spheroid, which was interpreted as the number of immune cells into the tumour spheroid boundary.

The őrst trafficking index, which the authors refer to as segmentation-based trafficking index (TIS), relies on a two-step approach. In the őrst step, the tumour boundary is identiőed using image segmentation techniques. Alongside this, in the second step, connected regions of high red channel intensity (which correspond to the presence of immune cells) are segmented. The TIS is a dimensionless number computed as the sum over each pixel that is in both the segmented regions (i.e. in regions with both tumour and immune cells) of the red channel intensity divided by the green channel intensity (which characterize the presence of only tumour cells). The TIS is the largest in pixels within the tumour (and segmented immune cell regions) at which the red channel intensity is high, and the green channel intensity is low, respectively. In fact, these pixels unambiguously correspond to immune cells within the tumour, indicating successful trafficking.

The second alternative independent algorithm to measure trafficking, which the authors refer to as K-means classification-based trafficking index (TIC), does not involve segmentation, and is based on a k-means classiőer. The assumption underlying the classiőcation is that the red and green levels of a pixel in a given image characterise the cell type that occupies the region, respectively immune and tumour cells. The TIC is also a two-step approach in which őrst a machine learning algorithm is used to classify each pixel in the image into different groups obtaining four classes, background, tumour cell, immune cell or colour saturated. The identiőcation of the colour saturated class allows for the exclusion of pixels that cannot be quantiőed appropriately, since the colour saturation hinders any comparison between the red and green levels. The TIC is then based on the number of pixels classiőed as immune cells that are completely surrounded by pixels classiőed as tumour cells. The resulting statistics yields a number in the interval [0, 1].

From day 0 to day 4: investigation of the effects of cortisol on immune infiltration For 4 days (i.e. from day 0 to day 4), the co-culture in each group was imaged, and the TIS and TIC trafficking indexes described above were applied on the images.

The authors found that, compared to the group with untreated spheroids and splenocytes only (i.e. group-2), the introduction of cortisol signiőcantly decreased immune cell inőltration in the spheroids in group-3 [see Figure 5. 2(a)-(d)]. Moreover, the 3 glucocorticoid receptor antagonists, respectively introduced in groups-4-5-6, reversed the effects of cortisol and signiőcantly increased immune inőltration in spheroids compared to the group of cells treated with cortisol (i.e. group-3 -results not shown). The results also showed that cortisol did not affect spheroid area and did not alter their roundness [see between neighboring cells in Cellular Potts models have an effective energy, which characterises the strength of cell-cell adhesion [START_REF] Izaguirre | łCompuCell, a multi-model framework for simulation of morphogenesisž[END_REF] (see Appendix A for a detailed description of the implementation of Cellular Potts models). Therefore, in our model, the effective energy between cells can directly affect the capability of CTLs to inőltrate through tumour cells. In particular, small values of the energy at the interface between tumour cells and CTLs correspond to a high adhesivity between the two cells types and, therefore, lead to scenarios in which CTLs display a high capability to inőltrate through tumour cells. On the other hand, large values of the energy at the interface between tumour cells and CTLs correspond to a low adhesivity between the two cells types and, therefore, lead to scenarios in which CTLs will tend to accumulate at the margin of the tumour, without inőltrating it. From here on, we use the term łtumour cell-CTL adhesion strengthž (TC-CTL adhesion strength) to refer to the parameter associated with the energy at the interface between tumour cells and CTLs and regulating CTL inőltration. With this notation, we will assume that high values of the TC-CTL adhesion strength facilitate the inőltration of CTLs through tumour cells, while low values of this parameter lead the CTLs to accumulate at the margin of the tumour, without inőltrating it.

Our model is posed on a 2D spatial domain partitioned into square elements of side ∆x. In our modelling framework, this domain biologically represents the media in which the co-culture is analysed. At each time step of length ∆t, the states of the cells are updated according to the probabilistic and deterministic rules described below.

From day -7 to day 0: growth of tumour cells We let N T (t) denote the number of tumour cells in the system at time t = h∆t, with h ∈ N 0 , and we label each cell by an index n = 1, . . . , N T (t). Motivated by the experiments described in Section 5.3, at the initial time point of the simulation (corresponding to day -7 of the experiments described in Section 5.3), a certain number of tumour cells is already present in the domain. Tumour cells are tightly packed in a circular conőguration positioned at the centre of the domain, reproducing a 2D cross-section of the geometry of the spheroids.

At each time-step, we let tumour cells grow at a certain rate drawn from a uniform distribution; the parameters of the bounds of the uniform distribution are chosen to match the mean duration of the spheroids cycle length. Mitosis occurs when a tumour cell grows to a critical size and then divides. We refer the reader to Chapter 3 for a detailed description of cell division modelling strategy.

A tumour cell can die due to intra-tumour competition, at a rate proportional to the total number of tumour cells. This modelling rule allows us to obtain a logistic growth in the number of tumour cells.

Day 0: introduction of CTLs We denote by N C (t) the number of CTLs in the system at time t, and we label each of them by an index m = 1, ..., N C (t). At day 0 of our simulations, a certain number of CTLs is randomly distributed across the spatial domain. To match the experiments, the number of CTLs introduced in the domain at day 0 corresponds to 5 times the number of tumour cells introduced at day -7. Once in the domain, we let CTLs grow, divide by mitosis and die due to intra-population competition with rules similar to those used for tumour cells. If a CTL exhausts its lifespan (which is drawn when the cell is created), it dies (i.e. it undergoes apoptosis) at the end of the time-step and it is removed from the domain.

Tumour cells at the border of the tumour secrete a chemoattractant (which models the effects of INF-γ) that triggers the movement of CTLs towards tumour cells. Denoting by ϕ the concentration of the chemoattractant secreted by tumour cells, we let the dynamic of ϕ be described by the following reaction-diffusion equation:

∂ϕ ∂t = D∆ϕ -γϕ + α n∈N T B (t)
δ n∈N T B (t) .

(5.1)

In Eq. (5.1), D is the diffusion rate of the chemoattractant, γ is the rate of natural decay and α is the secretion rate. Moreover, δ n∈N T B (t) = 1 if n ∈ N T B (t), and δ n∈N T B (t) = 0 otherwise, where N T B (t) denotes the set of tumour cells contact with the surrounding medium at time t. This function allows to take into account the fact that only the tumour cells at the border of the tumour secrete the chemoattractant. Once in the domain, CTLs move up the gradient of the chemoattractant in direction of tumour cells. The chemotaxis of CTLs towards the tumour is modulated by a parameter λ chem , which determines the strength and direction of chemotaxis (see Appendix A for a detailed description of the implementation of Cellular Potts models).

Implementation of our algorithm to measure infiltration levels In a similar way as the authors of [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF] developed two trafficking indexes to measure the level of inőltration of splenocytes into the spheroids, in our work we deőne an 'inőltration score'. This score allows us to quantify the level of CTL inőltration into the tumour and is deőned in a numerically equivalent way of the TIC algorithm developed in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF]. In fact, we deőne the inőltration score as the number of CTLs surrounded by tumour cells, divided by the number of tumour cells and CTLs surrounded by tumour cells, and can be written as

I(t) := N T (t) n=1 N C (t) m=1 δ m∈N CS (t) δ m∈N CS (t) + δ n∈N T S (t) . (5.2)
In Eq. (5.2), functions δ m∈N CS (t) and δ n∈N T S (t) are deőned in a similar way of function δ n∈N T B (t) in Eq. (5.1). Here, these functions allow to respectively count the number of CTLs in set N CS (t), and the number of tumour cells in set N T S (t), where N CS (t) and N T S (t) respectively denote the sets of CTLs and tumour cells surrounded by tumour cells at time t. This computation gives a number in the interval [0, 1].

From day 0 to day 4: investigation of the effects of stress on immune infiltration Between day 0 and day 4, CTLs grow and divide, move via chemotaxis towards the tumour and inőltrate it. Accordingly to the experiments, CTLs are activated against the tumour cells. Therefore, we suppose that upon contact, CTLs can induce tumour cell death with a certain probability. We refer to this probability as the łimmune success ratež. If the tumour cell satisőes the conditions to be eliminated, it undergoes apoptosis. Building on the same modelling framework developed in Chapter 3, we require that an elimination event keeps a CTL engaged for 6 hours and only after this time the CTL can eliminate again.

Figure 5.3 describes the strategies adopted in our model to study the effects of stress on immune inőltration and qualitatively reproduce the experimental results obtained in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF]. In this study, to test whether stress decreased inőltration levels in the co-culture model, cortisol was added to the co-culture and the outcomes were compared to those obtained for the group of spheroids and activated splenocytes only. Cortisol signiőcantly decreased the levels of INF-γ in the co-culture media and signiőcantly increased the levels of IL-10 (cf. step (1) of Figure 5.3). In our model, we explore the way in which immune inőltration is affected by three parameters which can be associated to the effects of cortisol in the co-culture. Such parameters are: the secretion rate of the chemoattractant, the TC-CTL adhesion strength and the growth rate of CTLs. The őrst two parameters are associated to the role played by INF-γ in the co-culture model, while the last one is associated to the role played by IL-10 (cf. step (2) of Figure 5.3). Varying these parameters affects both the number of CTLs at the end of the simulation, as well as their movement towards and within the tumour. In this work, we explore different scenarios. We suppose that in normal conditions (i.e. in our control scenario) the levels of IFN-γ are high, the levels of IL-10 are low, and CTLs are able to inőltrate into the tumour. In stressed conditions however, we suppose that the levels of IFN-γ decrease, the levels of IL-10 increase, and CTL inőltration is reduced. By considering a range of values of the three aforementioned parameters, we study their impact on tumour-immune dynamics independently and together, assessing their inŕuence on immune inőltration in a controlled manner (cf. step (3) of Figure 5.3).

Numerical simulations and preliminary results

Set-up of simulations

For numerical simulations of our individual-based model, we use a Cellular Potts approach on a 2D spatial grid with a total of 400 × 400 lattice sites. Simulations were developed and run using the software CompuCell3D [START_REF] Izaguirre | łCompuCell, a multi-model framework for simulation of morphogenesisž[END_REF] on a standard workstation (Intel i7 Processor, 4 cores, 16 GB RAM, macOS 11.2.2). The computational implementation of Cellular Potts models is described in Appendix A, while full details of the model parametrisation are provided in Appendix E.

At the initial time point of the simulation (i.e. at day -7 of the experiments), a small number of tumour cells is placed in the domain. Tumour cells proliferate for 7 days (i.e. from day -7 to day 0) and CTLs are introduced in the domain at day 0.

All quantities we present in this section and in Section 5.6 are obtained by averaging over the results of 5 simulations, with parameter values equal to those listed in Table E.1 and Table E.2. First we let tumour cells grow in absence of CTLs for 11 days, carrying out numerical simulation for 33000 time-steps. Then we introduce CTLs, and we let them interact with tumour cells for 4 days, carrying out numerical simulation for 1200 time-steps.

In the next two subsections, two preliminary computational results of our model are presented, which will be used to guide the simulations leading to the main results presented in Section 5.6.

Tumour development in the absence of CTLs

We őrst establish a preliminary scenario where tumour cells grow, divide and die via the modelling rules described in Section 5.4, in the absence of CTLs. For this case, we carry out numerical simulations for 33000 time-steps, corresponding to 11 days (i.e. from day -7 to day 4 of the experiments). Figure 5.4(a) shows the growth over time of the number of tumour cells. Figure 5.4(b) also displays an example of the spatial cell distribution observed at different times of one simulation. As shown by Figure 5.4(a), the number of tumour cells increases from day -7 to day 0. At day 0, the number of tumour cells reaches its carrying capacity, i.e. the saturation value attained due to intra-population competition. Therefore, from day 0 to day 4, the number of tumour cells remains stable around the value of carrying capacity. These results qualitatively reproduce the growth of the spheroids obtained in the experiments described in Section 5.3.

In the following subsection, we establish a control scenario in which CTLs are introduced into the domain and interact with tumour cells in non-stressed conditions. These simulations are performed by keeping the values of the parameters related to tumour cells őxed (and equal to this preliminary scenario) and by setting the initial number of tumour cells at its carrying we also explore different values of the TC-CTL adhesion strength. Next, for each scenario considered, we decrease the growth rate of CTLs. This parameter slows down the proliferation rate of CTLs and, therefore, may be associated with increased levels of IL-10 in the experiments.

In the next section, we investigate the effects of stress on immune inőltration. The obtained dynamics are compared to the control scenario. In the next simulations, we vary the values of the secretion rate of the chemoattractant, the TC-CTL adhesion strength, and the growth rate of CTLs, while the other parameters are kept constant to the values listed in Table E.1 and Table E.2. The values of the parameters are chosen so as to qualitatively reproduce essential aspects of the experimental results obtained by [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF]. For each scenario, we compute the inőltration score via (5.2), and compare it with the inőltration score obtained in the control scenario.

Main results

Decreasing the secretion rate of the chemoattractant and the TC-CTL adhesion strength reduces the infiltration of CTLs into the tumour

To investigate how immune inőltration is affected by the levels of INF-γ in the domain, we start by comparing our control scenario with different scenarios in which the secretion rate of the chemoattractant is progressively decreased. For each scenario considered, we also vary the value of the TC-CTL adhesion strength. Figure 5.6(a) displays the average value of the inőltration score at different times of the simulation, for different combined values of the secretion rate of the chemoattractant and the TC-CTL adhesion strength. In particular, Figure 5.6(a) shows that both parameters affect the inőltration of CTLs into the tumour, as the inőltration score decreases as soon as the TC-CTL adhesion strength or the secretion rate of the chemoattractant decrease. In addition, when the TC-CTL adhesion strength is sufficiently high, decreasing the secretion rate of the chemoattractant considerably reduces the inőltration score. On the other hand, when low values of the TC-CTL adhesion strength are considered, decreasing the secretion rate of the chemoattractant does not have an impact on the inőltration score, as its value is already small. These results suggest that the secretion rate of the chemoattractant has an impact on T cell inőltration only when CTLs display a sufficiently high capability to inőltrate through tumour cells. On the other hand, if CTLs have a low capability to inőltrate through tumour cells, independently on the dynamics of the chemoattractant, they will not be able inőltrate into the tumour.

We next analyse the spatial cell distributions observed at the end of simulations (i.e. at day 4 of the experiments). Taken together, our results qualitatively reproduce key őndings of the experimental results presented in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF]. The results presented in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF] indicate that cortisol decreased the levels of

Decreasing the growth rate of CTLs decreases the number of infiltrated CTLs

The results discussed in the previous subsection illustrate how the inőltration of CTLs into the tumour is affected by the levels of INF-γ in the domain. We further investigate the effects of stress, exploring the role played by IL-10 on immune inőltration. Therefore, for these simulations, we consider the same parameter values used in the previous subsection, and decrease the value of the growth rate of CTLs. The outcomes of our model indicate that increasing the levels of IL-10 may inhibit the proliferation rate of CTLs, which in turn diminish the number of CTLs in the domain. When the levels of INF-γ are high and CTLs display a high capability to inőltrate trough tumour cells, we observe a reduction in the number of inőltrated CTLs (i.e. value of the inőltration score decreases). On the other hand, if the levels of INF-γ are low and CTLs have a low capability to inőltrate through tumour cells, increasing the levels of IL-10 does not affect the inőltration score, as the number of inőltrated CTLs is already low. Therefore, our computational results suggest that IL-10 decreases immune inőltration only when the levels of INF-γ are high and CTLs display a high capability to inőltrate trough tumour cells.

Increasing the immune success rate has an impact on the infiltration score only when the TC-CTL adhesion strength is high

So far, we have investigated with our model the effects of stress on immune inőltration in a framework in which CTLs are not able to eliminate tumour cells (i.e. the value of the immune success rate is equal to 0). Now, we investigate tumour-immune dynamics and the effects of stress on immune inőltration in the scenario in which CTLs can eliminate tumour cells with a small probability (i.e. the value of the immune success rate is slightly greater then 0). 5.8(b), we see that, when the TC-CTL adhesion strength is high, increasing the value of the immune success rate decreases the inőltration score. This is probably due to the fact that, when CTLs can inőltrate through tumour cells, they are more likely to come into con-5.7 Discussion, conclusions and research perspectives

Discussion and conclusions

In this chapter, we have presented a spatially explicit stochastic individual-based model of the interaction dynamics between CTLs and tumour cells to study the effects of physiological stress on immune inőltration. Building on the in-vitro experiments presented in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF], we have investigated, in silico, the way in which two cytokines, namely INF-γ and IL-10 can impact on the inőltration of CTLs into the tumour. We explored different scenarios: őrst we supposed that, in normal conditions, the levels of IFN-γ were high, the levels of IL-10 were low and CTLs had a high capability to inőltrate through tumour cells. Then we supposed that, in stressed conditions, the levels of IFN-γ decreased, the levels of IL-10 increased and CTLs had a lower capability to inőltrate through tumour cells. By varying the secretion rate of the chemoattractant, the growth rate of CTLs and the TC-CTL adhesion strength, we studied the impact of stress on immune inőltration. Building an inőltration score on the basis of the TIC algorithm developed in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF], we quantiőed the effects of stress on immune inőltration in a controlled manner.

First, our numerical results are able to qualitatively reproduce the growth of the spheroids prior the introduction of activated immune cells (cf. Figure 5.4). The growth of the number of tumour cells is of logistic type, so that when CTLs are introduced, changes in the tumour surface can be attributed to the inőltration of CTLs. Then, we have established our control scenario. In the experimental results obtained by [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF], in absence of cortisol, splenocytes were able to inőltrate the spheroids. Our numerical results indicate that when the values of the secretion rate of the chemoattractant, the TC-CTL adhesion strength and the growth rate of CTLs are high, CTLs are able to inőltrate into the tumour (cf. Figure 5.5).

We then have investigated the effects of stress on immune inőltration. First, the outcomes of our model support the idea that decreasing the secretion rate of the chemoattractant and the TC-CTL adhesion strength, which are associated to a decrease in the levels of INF-γ, reduces the inőltration of CTLs into the tumour (cf. Figure 5.6). Our results also suggest that the secretion rate of the chemoattractant is more likely to have an impact on T cell inőltration when CTLs display a sufficiently high capability to inőltrate through tumour cells. We have also studied the effects of stress on immune inőltration by decreasing the growth rate of CTLs, which is associated to an increase in the levels of IL-10 (cf. Figure 5.7). Decreasing the growth rate of CTLs decreases the number of CTLs in the domain. This signiőcantly decreases the inőltration score only when the TC-CTL adhesion strength is high. Conversely, when the TC-CTL adhesion strength is low, decreasing the growth rate of CTLs does not have an impact on the inőltration score, as its value is already small. Finally, we have investigated the outcomes of our model by letting CTLs eliminate tumour cells with a small probability (i.e. we increase the value of the immune success rate) (cf. Figure 5.8). In the scenario in which CTLs are able to inőltrate into the tumour, increasing the value of the immune success rate decreases the inőltration score, as tumour cells in contact with CTLs are eliminated. This in turn leads to a slightly decreased number of tumour cells at the end of simulations. However, when the CTL inőltration level is low, increasing the value of the immune success rate does not have an impact on the inőltration score, nor does it decrease the őnal number of tumour cells.

In summary, our őndings demonstrate the importance of including the effect of different factors when exploring the role of stress on immune inőltration. Our results support the idea that a high inőltration score can be obtained only when the secretion rate of the chemoattractant and the TC-CTL adhesion strength are high, provided that the growth of CTLs is not inhibited. On the other hand, our results suggest that decreasing the value of these parameters can decrease immune inőltration in different ways. For example, our results indicate that the parameter having the strongest impact on immune inőltration is the TC-CTL adhesion strength, which is associated with the capability of CTLs to inőltrate through tumour cells. When the TC-CTL adhesion strength is low, independently of the value of the secretion rate of the chemoattractant and the growth rate of CTLs, we obtain a low inőltration score. On the other hand, when the TC-CTL adhesion strength is high, varying the value of the other two parameters has a greater impact on immune inőltration, and leads to a different scenarios with different inőltration scores.

Research perspectives

In the future, better quantitative comparison with experiments will allow for systematic choice of parameters and validation of the mechanisms we propose here. From a biological point of view, a natural sequel of this work consists in studying the effects of therapeutic strategies which counteract the effects of stress. In fact, in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF], the authors found that the administration of glucocorticoid receptor antagonists reversed the effects of cortisol and signiőcantly increased the immune inőltration in 66CL4 spheroids compared to those cells treated with cortisol. Finally, the effects of changing the spatial domain from 2D to 3D would need to be considered. It is probable that the change of dimension would alter the time it takes to run simulations of our mathematical model. However, in this way, a more realistic scenario, closer to the 3D co-culture model, could be explored.

Chapter 6

Conclusion and perspectives

The past decade's technological advances have led to the development of immunotherapies, which differ from conventional anti-cancer therapies by targeting tumour-immune cell interactions in order to enhance the effectiveness of the anti-tumour immune response. However, these interactions are based on complex mechanisms that make it difficult to design treatments to effectively boost the immune response. For this reason, mathematical models are useful tools for reproducing and predicting the spatio-temporal dynamics of interactions between tumour cells and immune cells, in order to test the potential of new therapeutic techniques in a ŕexible and affordable way.

Throughout this thesis, a series of mathematical models have been proposed to study different mechanisms involved in tumour-immune interactions, with the goal of investigating the biological settings which allow for the clearance or the escape of the tumour.

We began this thesis by describing the key biological processes and mechanisms governing the interactions between malignant tumour cells and immune cells, focussing on the role of one particular type of immune cells: the cytotoxic T lymphocytes (CTLs). CTLs are the main actors of the anti-tumour immune response, as they can actively target and destroy tumour cells. Along with a description of the immune response to tumours, we described processes that allow tumour cells to evade the immune response and some immunotherapy techniques developed and used to restore an effective immune action.

Mathematical models for tumour-immune interactions have become increasingly popular over the past few decades and are used as a tool to aid in the understanding of the mechanisms of tumour escape. We provided a short review of the existing mathematical models of tumourimmune interactions and their applications, with a focus on discrete stochastic individual-based models, deterministic continuum models and hybrid discrete-continuum models.

Following this background chapter, we provided the main body of this work. This thesis was organized in four chapters. Each chapter focussed on a mathematical model considering a speciőc aspect involved in the interactions between tumour cells and CTLs. In particular, these models have been developed to investigate:

(1) the coevolutionary dynamics between CTLs and tumour cells in a well mixed system;

(2) the impact of intra-tumour heterogeneity (ITH) on anti-tumour CTL immune response;

(3) the role of CTL inőltration on anti-tumour immune response; (4) the effect of psychological stress on immune inőltration.

To investigate these problems, different methods of mathematical modelling have been developed, which allowed to include different layers of biological complexity and different aspects of tumour-immune interactions. In particular, the mathematical models proposed in this thesis were formulated as: 122 CHAPTER 6. Conclusion and perspectives (A) discrete stochastic individual-based models, which relied on an explicit representation of individual cells in space and time, tracking and updating their internal states according to a predeőned set of biological and biophysical rules. The development of these models allowed to represent stochasticity and heterogeneity in the interaction dynamics between tumour cells and CTLs; (B) deterministic continuum models, which were not formulated on the basis of phenomenological arguments, which can hinder a precise mathematical description of crucial biological and biophysical aspects, but have been formally derived from the discrete models through suitable asymptotic methods. The continuum models were formulated as systems of nonlinear and nonlocal IDEs and PDEs, and provided a mean-őeld representation of the underlying cellular dynamics. Through linear stability analysis, we identiőed possible conditions on the model parameters leading to different dynamics of one continuum model, gaining a more in-depth theoretical understanding of the system under consideration.

Numerical simulations, relying on parameter values drawn from the extant literature, have been employed for in silico investigations of the underlying biological aspect, to qualitatively test the model assumptions against empirical observations. When the continuum model was derived from the discrete model, an excellent quantitative agreement between numerical simulations of the two models was obtained. Moreover: (A) we developed our own computational code to carry out computational simulations of two discrete individual-based models, in order to have a deeper understanding and control over the dynamic produced. For the computational implementation of the two other individual-based models, instead, we took advantage of the Cellular Potts modelling approach, and simulations were run using the CompuCell3D open-source simulation environment. This software demonstrated to be a very useful tool when modelling tumour-immune interaction dynamics, due to the easy inclusion of precise biological aspects (such as cell adhesion, cell growth and mitosis, motility, competition for space, etc); (B) appropriate numerical schemes were used for the numerical simulations of the continuum models, which prevented the emergence of spurious oscillations. Moreover, the computational efficiency of the continuum model with respect to its discrete counterpart has been exploited, to carry out extensive numerical simulations to investigate particular hypothesis.

In agreement with biological studies, the results obtained have shed light on: (1) the way in which different parameters shape the coevolutionary dynamics between tumour cells and CTLs and induce the formation of patterns of phenotypic coevolution;

(2) the speciőc impact of different expressions of ITH on immune surveillance, highlighting the importance of ITH as a possible predictor of the outcome of immune action;

(3) the mechanisms that underlie the emergence of different levels of inőltration of CTLs into the tumour, and their impact on the response to different types of anti-cancer immunotherapy; (4) the way in which a simple mathematical model can qualitatively reproduce an in vitro experiment, and demonstrate the importance of including the effect of different factors when exploring the impact of physiological stress on immune inőltration into tumours.

Each chapter of this thesis was ended with a speciőc discussion of the model and research perspectives. Below we discuss global potential future directions and perspectives.

Optimisation of anti-tumour treatments An important objective in oncology is the improvement of anti-tumour therapies and their combinations, with the goal of increasing the overall treatment efficacy. One option to achieve this goal is optimization of the schedule of drug administration or performing other medical actions. A key advantage of mathematical models is that they can be amenable to an analysis of their control (optimization and optimal control). Let us note that optimization and optimal control methods can be applied mainly to deterministic, CHAPTER 6. Conclusion and perspectives 123 both ODE and PDE models, having in mind that therapeutic optimization may resort to these methods. Optimization of an anti-tumour treatment usually implies determining what should be the best schedule, i.e. dose distribution and timing, for a treatment that affects both tumour cells and healthy (usually fast renewing) cells. Of course, from the point of view of clinical use, the goal is to achieve the best therapeutic results with the fewest possible side effects to healthy cells.

Optimal control theory is applied in a huge number of studies of different natures and has also become popular in mathematical oncology (see [START_REF] Kuznetsov | łImproving cancer treatments via dynamical biophysical modelsž[END_REF] for a review on examples of therapeutic problems in oncology and how they have been addressed theoretically).

In Chapter 4 we investigated the effects of immunotherapeutic protocols only when they were delivered at the initial time of simulations and treatment effect was constant in time. Therefore, it would be interesting to include in our models the effects of different types of therapies and to apply optimal control methods. This would allow to determine the best delivery schedule of different therapeutic agents that would achieve the best therapeutic outcome, minimising the number of tumour cells at the end of the treatment with the least possible side effects to healthy cells.

Data availability and model vadility One of the goals of mathematical modelling in oncology is to provide personalized predictions by taking into account patient-speciőc data. In clinical practice, cancer patients undergo several examinations such as blood sample analysis, clinical imaging (e.g. CT, MRI), biopsy sampling, etc. However, such clinical data are provided for very limited time-points. There are different reasons for that, e.g. the fact that data collection is restricted to the clinical presentation of the patient, the cost, the availability of the equipments, the side effects of the exams, etc. Moreover, the mechanistic connection between these data is elusive, and the way to connect all these variables in a single model can be complex. Therefore, many of these data sets are left out from the model development process. Moreover, even having all the required mechanistic knowledge to develop an appropriate model, this latter would involve a huge number of parameters, making the model untractable mathematically. In combination with the lack of clinical data with sufficient temporal resolution and speciőcity, the goal of personalized model calibration would have been a 'mission impossible'. Under this constraint, delivering personalized predictions becomes a difficult task.

However, the obsession for data őtting should not be the main driver for model development. One should keep in mind that the purpose of a model is to answer a speciőc question and that its domain of applicability and validity is always restricted. It is important to point out that the complexity of cancer makes it impossible to accurately model all aspects which allow it to escape immune surveillance. In this thesis, we aimed at showing that mathematical models are an excellent tool to describe underlying dynamics of tumour-immune interaction, and may elucidate some important biological actions that govern it. The models developed in the different chapters proved very useful in giving some insights into different mechanisms that allow the tumour to escape immune surveillance, as well as into some therapeutic aspects, to understand why and how therapeutic combinations can lead to the clearance or the escape of the tumour.

In 1976, George Box, a British statistician, wrote the famous line, łAll models are wrong, some are useful.ž His point was that we should focus more on whether something can be applied to everyday life in a useful manner rather than debating endlessly if an answer is correct in all cases. Therefore, regardless of being right or wrong, mathematical models for tumour-immune interactions can be very useful, and can open up new avenues and stimulate new experimental or theoretical researches.

The importance of interdisciplinary research In this thesis, we have developed different models of tumour-immune interactions which try to better understand some mechanisms that are responsible for immune escape. During the development of the different models, fruitful discussions with biologists and immunologists helped us to construct reasonable modelling assumptions, and to test them on interesting biological problems. The different models may provide possible insight to design immunotherapy strategies which effectively improve the effectiveness of the overall anti-tumour immune response. However, it is of paramount importance that more experimental clinical data is used when developing and implementing these models to try and enrich and validate the results.

More effort should be made to try to őnd a common language between mathematicians and medical workers, as well as to őnd the right balance between the rigor of mathematical models and the level of uncertainty prevailing in the clinical environment. Collaborations between biologists, clinicians and mathematicians can ensure that models are more in-depth and can increase the success of research on the development of new cancer therapies. Nevertheless, confronted with the undoubtable successes met in the last 50 years in clinical oncology, encountering more and more limitations as new treatments emerge, oncologists together with evolutionary biologists, immunologists, physicists and mathematicians, have begun to lower barriers between their disciplines. This trend of interdisciplinary research has recently reached even up to philosophers [START_REF] Bertolaso | Philosophy of cancer[END_REF] who have thus emerged as a community of łphilosophers of cancerž.

Interdisciplinary work has become more common in cancer research, and mathematical modelling has already led to several predictions, as well as to the discovery of some important biological actions that govern tumour dynamics, with a hope to develop better treatments to extend the host's life, or to eradicate the cancer completely.

"ŕip", in order to minimise the effective energy. If the index-copy attempt is successful the cell will increase in volume by a lattice site and the neighbouring cell will decrease in volume. The success of the index copy attempt is dependent upon a Boltzmann acceptance function which takes into account the change in energy. Once all of the boundary pixels in the simulation have performed an index-copy attempt, a Monte Carlo Step (MCS) has been completed.

In addition to generalized cells, CompuCell3D can also adequately model nutrients, chemical gradients or the extra cellular matrix by using a őeld component. Fields evolve due to secretion, absorption, diffusion, reaction and decay according to appropriate PDEs. Generalised-cell behaviors are affected by extracellular chemical őelds and subcellular networks (e.g. changes in cell target volume due to chemical absorption, chemotaxis in response to a őeld gradient or cell differentiation based on the internal state of a genetic network). Auxiliary ODE equations are used to describe cell interactions with őelds, such as absorption, and secretion, and state changes within the cells, such as mitosis and cell death.

A.2 Effective energy

The effective energy is the basis for operation of all Cellular Potts Models, including Compu-Cell3D, because it determines the behavior of the interactions between generalized cells. The energy is described in two ways: boundary energy or constraints. The most important boundary energy components of the effective energy equations governs the adhesion of cells. Adhesion is deőned by the boundary energy J(τ (σ i ), τ (σ j )), which describe the contact energy per unit area between the two cells (σ i , σ j ) of types (τ (σ i ), τ (σ j )). This is calculated by the sum over all neighbouring pixels i and j that form the boundary between two cells: H boundary = i,j neighbours J(τ (σ i ), τ (σ j ))(1δ(σ i , σ j ))

(A.1)

The delta function restricts the boundary energy contribution to cell-cell interfaces. J(τ (σ i ), τ (σ j )) is deőned as a matrix indexed by the cell types. Higher boundary energies between cells result in greater repulsion between cells and lower boundary energies imply a greater adhesion between cells.

As said before, the second way of using the effective energy in a model is by means of constraints on cell behaviour. Constraints are written in a general elastic form and are represented by: H constrain = λ(valuetarget value) 2 (A.2)

As we can see, if value = target value, the constraint is satisőed and, therefore, the energy is zero. This condition is known as the equilibrium condition. Since CompuCell3D tries to minimise the effective energy in a simulation the constraint will be driven to the equilibrium condition.

However, as the nature of the simulations is stochastic, the condition does not always have to be met and it is possible that two or more constraints conŕict with each other so that they are only able to be partially met, resulting in random ŕuctuations. Parameter λ, a positive real number known as the spring constraint, determines how far the value can deviate from the target. Large values of λ will result in a larger effective energy change, essentially preventing a index-copy attempt, whereas small values allow for greater deviation from the equilibrium condition.

The most common employment of constraint energies for biological cells in CompuCell3D is to restrict the size of a cell to a given target value. This is done through two constraints: the we can use their Taylor expansions ψ(w(x ± χ, t)) = ψ ± χ∂ x ψ + In the case where the chemoattractant dynamics are governed by the rules described in Section 4.3.2, using the same methods employed in [START_REF] Bubba | łFrom a discrete model of chemotaxis with volume-őlling to a generalized Patlak-Keller-Segel modelž[END_REF], considering (i, k) ∈ [1, N -1] × N 0 and assuming that relations (D.

D.2 Details of numerical simulations

The numerical simulations of our hybrid and continuum models are carried out on a twodimensional domain and are performed in Matlab. As summarised by Fig. D.2, at each time-step τ , each CTLundergoes a three-phase process: Phase A) undirected, random movement according to the probabilities deőned via (4.26) and (4.27); Phase B) chemotaxis according to the probabilities deőned via (4.23) and (4.24); Phase C) inŕow and death according to the probabilities deőned via equations (4.17) and (4.21). We let then each tumour cell proliferate with the probability deőned via (4.8), die due to intratumour competition with the probability deőned via (4.10), or die due to immune action with the probability deőned via (4.13). Finally, the tumour cell density at every lattice site is computed via (4.1) and inserted into (4.14) in order to update the concentration of the chemoattractant.

D.2.1 Details of numerical simulations of the hybrid model

In a two-dimensional setting, the positions of the single CTLs are updated following a procedure analogous to that explained in

D.2.2 Details of numerical simulations of the continuum model

To construct numerical solutions of the IDE-PDE-PDE system (4.30), we use a uniform discretisation consisting of N 2 = 3721 points of the square Ω := [0, 1] 2 as the computational domain of the independent variable x ≡ (x, y) (i.e. (x i , y j ) = (i∆x, j∆x) with ∆x = 0.016 and i, j = 0, . . . , N ) Moreover, we choose the time step ∆t = 1 × 10 -4 , and we perform numerical simulations for 15 × 10 4 time-steps (i.e. the őnal time of simulations is t f = 15). The method for constructing numerical solutions of the IDE-PDE-PDE system (4.30) is based on a őnite difference scheme whereby the discretised dependent variables are n k i,j := n(x i , y j , t k ), c k i,j := c(x i , y j , t k ) and ϕ k i,j := ϕ(x i , y j , t k ).

We solve numerically the IDE (4.30) 1 for n and the PDE (4.30) 3 for ϕ using the following schemes

n k+1 i,j -n k i,j ∆t = α n -µ n ρ k n -ζ n K k i,j n k i,j
i, j = 0, . . . , N, and

ϕ k+1 i,j -ϕ k i,j ∆t =β ϕ ϕ k i-1,j + ϕ k i+1,j -2ϕ k i,j (∆x) 2 + β ϕ ϕ k i,j-1 + ϕ k i,j+1 -2ϕ k i,j (∆x) 2
+ α ϕ n k i,jκ ϕ ϕ k i,j , i, j = 1, . . . , N -1, and impose zero-ŕux boundary conditions for ϕ by letting ϕ k+1 0,j = ϕ k+1 1,j and ϕ k+1 N,j = ϕ k+1 N -1,j , j = 0, . . . , N ϕ k+1 i,0 = ϕ k+1 i,1

and ϕ k+1 i,N = ϕ k+1 i,N -1 , i = 0, . . . , N Moreover, we solve numerically the PDE (4.30) 2 for c using the following explicit scheme, which is the same as the one used in [START_REF] Bubba | łFrom a discrete model of chemotaxis with volume-őlling to a generalized Patlak-Keller-Segel modelž[END_REF],

c

,

Mathematical models of tumour-immune interactions: discrete and continuum approaches

Abstract

The past decade's technological advances have led to the development of immunotherapies, which differ from conventional anti-cancer therapies by targeting tumour-immune interactions to enhance the effectiveness of the anti-tumour immune response. However, these interactions are based on complex mechanisms that make it difficult to design treatments to effectively boost the immune response. For this reason, mathematical models are useful tools for reproducing and predicting the spatio-temporal dynamics of interactions between tumour cells and immune cells, in order to test the potential of new therapeutic techniques in a flexible and affordable way. In this thesis, we develop discrete and continuum models to describe the spatio-temporal dynamics of the interactions between a solid tumour and cytotoxic T cells, with the goal to investigate the biological settings which allow for the clearance or the escape of the tumour. The discrete models developed in this work track the dynamics of single cells, thus permitting the representation of single cell-scale mechanisms, and are sufficiently detailed and specific to qualitatively investigate and reproduce empirical observations. The continuum models considered are not formulated on the basis of phenomenological arguments, which can hinder a precise mathematical description of crucial biological and biophysical aspects, but they are formally derived from the discrete models through suitable asymptotic methods. The results of computational simulations of the discrete models show that there is an excellent quantitative agreement between them and numerical solutions of the corresponding continuum models, and further clarify the conditions for successful and unsuccessful immune surveillance. Ultimately, the mathematical models presented in this thesis may provide a framework to help biologists and clinicians gain a better understanding of the mechanisms that are responsible for immune escape, and they may be promising tools in the exploration of therapeutic strategies to improve the effectiveness of the overall anti-tumour immune response.

Keywords: tumour-immune interactions, immunotherapy, discrete models, continuum models, numerical simulations

Résumé

Au cours de la dernière décennie, les progrès technologiques ont permis la conception d'immunothérapies qui, contrairement aux thérapies anticancéreuses classiques, ciblent les interactions entre cellules tumorales et cellules immunitaires, dans le but de renforcer l'efficacité de la réponse immunitaire. Cependant, ces interactions reposent sur des mécanismes complexes, ce qui rend difficile la conception de traitements efficaces. Par conséquent, les modèles mathématiques sont des outils utiles pour reproduire la dynamique spatio-temporelle des interactions entre les cellules tumorales et les cellules immunitaires, afin de tester le potentiel de nouvelles techniques thérapeutiques de manière flexible et non coûteuse. Dans cette thèse, nous développons des modèles discrets et continus pour décrire la dynamique spatiotemporelle des interactions entre une tumeur solide et les cellules T cytotoxiques, dans le but d'étudier les paramètres biologiques qui permettent l'élimination, ou bien l'échappement, de la tumeur. Les modèles discrets développés dans ce travail décrivent la dynamique de chaque cellule, permettant ainsi la représentation de mécanismes à l'échelle cellulaire. De plus, ils sont suffisamment détaillés et spécifiques pour reproduire qualitativement les résultats d'études expérimentales. Quant aux modèles continus, ils ne sont pas formulés sur la base d'arguments phénoménologiques, qui peuvent limiter une description mathématique précise d'aspects biologiques et biophysiques cruciaux, mais ils sont dérivés formellement des modèles discrets par le biais de méthodes asymptotiques appropriées. Les résultats des simulations numériques des modèles discrets montrent qu'il existe un excellent accord quantitatif entre eux et les solutions des modèles continus correspondants, et clarifient les conditions de réussite, ou bien d'échec, de la surveillance immunitaire. Enfin, les modèles mathématiques présentés dans cette thèse peuvent fournir un cadre pour aider les biologistes et les cliniciens à mieux comprendre les mécanismes par lesquels les tumeurs échappent au système immunitaire, et peuvent être des outils prometteurs pour explorer des stratégies thérapeutiques conçues pour améliorer l'efficacité de la réponse immunitaire anti-tumorale.

Mots clés : interactions tumeur-système immunitaire, immunothérapie, modèles discrets, modèles continus, simulations numériques Laboratoire Jacques-Louis Lions Sorbonne Université -Campus Pierre et Marie Curie -4 place Jussieu -75005 Paris -France

1 Introduction 1 . 1

 111 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Biological overview of the immune response to tumours . . . . . . . . . . . . . 1.2.1 Tumour development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Tumour immunology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Immunoediting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Immunotherapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Review of modelling strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Methods of mathematical modelling . . . . . . . . . . . . . . . . . . . . . 1.3.2 Mathematical modelling of tumour-immune dynamics . . . . . . . . . . . 1.3.3 Mathematical modelling of immunotherapies . . . . . . . . . . . . . . . . 1.3.4 Mathematical models in clinical oncology . . . . . . . . . . . . . . . . . . 1.3.5 Formal derivation of continuum models . . . . . . . . . . . . . . . . . . . 1.4 Contributions of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Accepted publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 In preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 6 -

 6 Potential future directions In Chapter 6, we present a global conclusion of the thesis and we comment possible research perspectives. 1.5. List of publications 1.5 List of publications 1.5.1 Accepted publications -Luis Almeida, Chloe Audebert, E. L. and Tommaso Lorenzi, Discrete and continuum models for the coevolutionary dynamics between CD8+ cytotoxic T lymphocytes and tumour cells, Mathematical biology and medicine, 2022.

Figure 2 . 2 :

 22 Figure 2.2: Schematic representation of the algorithmic rules governing the coevolutionary dynamics between tumour cells and CTLs in the stochastic individual-based model. Panel a.: for a tumour cell in the phenotypic state x i we let α n model the rate of cell division, µ n K k ni the rate of cell death due to intra-population competition, with the quantity K k ni deőned via (2.5)-(2.7). Moreover, tumour cells can die due to antigen-speciőc cytotoxic activity of CTLs at rate ζ n γJ k ni , with J k ni deőned via (2.6)-(2.8). Therefore, during the time interval of length τ ≪ 1, between the time step k and the time step k + 1 we let a tumour cell in a phenotypic state x i divide with probability τ α n or die with probability τ (µ n K k ni + ζ n γJ k ni ) or remain quiescent with probability 1τ (α n + µ n K k ni + ζ n γJ k ni ). Panel b.: for a CTL in the phenotypic state y j we let α c model the rate of cell division and µ c K k cj the rate of cell death due to intra-population competition, with the quantity K k cj deőned via (2.5)-(2.7). Moreover, CTLs can undergo antigen-driven expansion at rate ζ c γJ k cj , with J k cj deőned via (2.6)-(2.8). Therefore, between the time step k and the time step k +1 we let a CTL in a phenotypic state y j divide with probability τ (α c + ζ c γJ k cj ) or die with probability τ µ c K k cj or remain quiescent with probability 1τ (α c + ζ c γJ k cj + µ c K k cj). Panel c.: a tumour cell in the phenotypic state x i that undergoes a phenotypic change can either enter into the phenotypic state x i-1 with probability λn 2 , enter into the phenotypic state x i+1 with probability λn 2 or remain in its current phenotypic state with probability 1λ n .

  ) where |I| denotes the size of the interval I (i.e. |I| = 2L) and |L ξ (x)| denotes the size of the interval L ξ (x) := {y ∈ I : |y -x| ⩽ ξ}, (x; ξ) ∈ I × (0, |I|].

I

  g(x, y; η) c(y, t) dy, K n (x, t) := I g(x, z; θ n ) n(z, t) dz, J c (y, t) := I g(y, x; η) n(x, t)dx, K c (y, t) := I g(y, z; θ c ) c(z, t) dz, (2.13)

  : impact of the parameters θ n and θ c Figure2.4 displays the plots of the phenotype distributions of tumour cells (top panel) and CTLs (central panel) at the end of numerical simulations (i.e. close to numerical equilibrium) alongside the plots of the corresponding time evolution of the total cell numbers (bottom panel). In agreement with the analytical results presented in Section 2.5.2, when condition (2.24) is satisőed and conditions (2.40) and (2.41) are met as well, patterns of phenotypic coevolution between tumour cells and CTLs may emerge. Moreover, the top and central panels of Figure2.4 show that, coherently with the shape of the function B(h) deőned via (2.39) (cf. the plots in Figure2.5), smaller values of θ n and θ c correlate with the formation of more peaks in the phenotype distributions of the two cell populations. The plots in Figure2.4 also demonstrate that there is a good agreement between numerical simulations of the individualbased and continuum models.

Figure 2 . 4 :Figure 2 . 5 :

 2425 Figure 2.4: Patterns of phenotypic coevolution between tumour cells and CTLs: impact of the parameters θ n and θ c . Top panels display the plots of the population density function of tumour cells (n) and central panels display the plots of the population density function of CTLs (c) of the individual-based model (solid, coloured lines) and continuum model (dashed, black lines) at the end of numerical simulations (i.e. at t = 30) when conditions (2.24) and (2.41) are satisőed and progressively smaller values of θ n and θ c are considered -i.e. θ n = θ c = 0.5 (panels a.), θ n = θ c = 0.3 (panels b.) and θ n = θ c = 0.2 (panels c.). All the other parameters are as in Table 2.1 with γ = 1.5 and η = 0.7. Bottom panels display the corresponding plots of the time evolution of the total number of tumour cells (ρ n ) and CTLs (ρ c ). Initial conditions (2.42) and (2.43) with a = 1 and A = 5 were used to carry out numerical simulations. Analogous results were obtained when using different values of the parameter A (results not shown). The results from the individual-based model correspond to the average over two realisations of the underlying random walk and the related variance is displayed by the coloured areas surrounding the curves.

Figure 2 . 6 :

 26 Figure 2.6: Sample temporal dynamics of patterns of phenotypic coevolution between tumour cells and CTLs. Top panels display the plots of the population density function of tumour cells (n) and bottom panels display the plots of the population density function of CTLs (c) of the individual-based model (solid, coloured lines) and continuum model (dashed, black lines) at őve successive time instants -i.e. t = 0.4 (panels a.), t = 4 (panels b.), t = 10 (panels c.), t = 16 (panels d.), t = 30 (panels e.) -in the case where condition (2.24) is satisőed. Here, θ n = θ c = 0.3, γ = 1.5 and η = 0.7, and all the other parameters are as in Table 2.1. Initial conditions (2.42) and (2.43) with a = 1 and A = 5 were used to carry out numerical simulations. Analogous results were obtained when using different values of the parameter A (results not shown). The results from the individual-based model correspond to the average over two realisations of the underlying random walk and the related variance is displayed by the coloured areas surrounding the curves.

Figure 2 . 7 : 51 2. 7

 27517 Figure 2.7: Patterns of phenotypic coevolution between tumour cells and CTLs: impact of the parameter η. Top panels display the plots of the population density function of tumour cells (n) and bottom panels display the plots of the population density function of CTLs (c) of the individual-based model (solid, coloured lines) and continuum model (dashed, black lines) at the end of numerical simulations (i.e. at t = 30) when condition (2.24) is satisőed and progressively smaller values of η are considered -i.e. η = 1 (panels a.), η = 0.6 (panels b.) and η = 0.2 (panels c.). All the other parameters are as in Table 2.1 with γ = 1 and θ n = θ c = 0.7. Bottom panels display the corresponding plots of the time evolution of the total number of tumour cells (ρ n ) and CTLs (ρ c ). Initial conditions (2.42) and (2.43) with a = 1 and A = 5 were used to carry out numerical simulations. Analogous results were obtained when using different values of the parameter A (results not shown). The results from the individual-based model correspond to the average over őve realisations of the underlying random walk and the related variance is displayed by the coloured areas surrounding the curves.

1 ]

 1 and L N I = {m N I , . . . , M N I } ⊂ [0, 1], with M N I < M I . (3.4) They characterise the range of different values that can be taken by the variable l n ai [see Figure 3.1(d)].

Figure 3 . 2 :

 32 Figure 3.2: Schematic representation of the modelling assumptions for CTLs and their interaction with tumour cells. (a) Red circles represent CTLs, which express a unique TCR. (b) TCR are represented with different shapes and colors.Each TCR is able to recognize a particular tumour antigen. In the model, the number of TCRs is equal to the number of expressed tumour antigens. (c) Purple circles represent tumour cells. CTLs can eliminate tumour cells, upon contact, under certain conditions. A tumour cell is eliminated if it presents the antigen matching the CTL receptor at a sufficiently high level. In this example, a tumour cell expressing antigen a 1 and a 2 cannot be eliminated by a CTL with TCR matching antigen a 4 . On the other hand, the same tumour cell may be eliminated by a CTL expressing the TCR matching antigen a 2 , under a condition on the level l a2 of presentation of such antigen a 2 . The parameter r is the intrinsic TCR-recognition probability and µ is a random variable drawn from a standard uniform distribution.

Figure 3 .

 3 6(c)). The related panels of Figure 3.6(d)-(f ) show the progressive colonisation of the tumour by non-immunogenic cells. Due to the relatively large standard deviation observed in the tumour-7a results, we run 100 realisations of this simulation (see Figure C.1 in C.2), in order to check the robustness of our results. Comparing Figures 3.6

  and

  the whole region occupied by the tumour. Here, |(•)| is the measure of the set (•). Given the initial distribution of tumour cells, we deőne (cf. Fig 4.1)

. 5 )Figure 4 . 1 :

 541 Figure 4.1:Schematics describing the classification of cold, altered-excluded, alteredimmunosuppressed and hot tumour scenarios based on the immunoscore. In the mathematical framework of our model we classify different tumour scenarios depending on the value of the immunoscore at the end of numerical simulations, i.e. the quantity deőned via (4.7). The immunoscore is here deőned as a weighted sum of the number of CTLs in the centre of the tumour and the number of CTLs in the margin of the tumour. In more detail, following[START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF], scenarios for which the value of the immunoscore is low will be classiőed as 'cold tumour scenarios'; scenarios with an intermediate value of the immunoscore will be classiőed as 'altered tumour scenarios', which will then be further classiőed as 'altered-immunosuppressed tumour scenarios' or 'altered-excluded tumour scenarios' based on the distribution of CTLs in the centre and at the margin of the tumour; őnally, scenarios characterised by a high value of the immunoscore will be classiőed as 'hot tumour scenarios'.

Figure 4 . 2 :

 42 Figure 4.2: Schematics summarising the mechanisms and processes included in the hybrid discrete-continuum model. The governing rules for the dynamics of tumour cells are here represented by green solid arrows, the rules for the dynamics of CTLs are represented by blue solid arrows, and the processed underlying the dynamics of the chemoattractant are represented by grey solid arrows. At each time-step t k , tumour cells divide at rate α n , die due to intra-tumour competition at rate µ n ρ k n or die due to the cytotoxic action of CTLs at rate ζ n K k i . The quantities ρ k n and K k i are, respectively, deőned via (4.9) and (4.11). Tumour cells also secrete the chemoattractant at rate α ϕ . The chemoattractant diffuses through the domain at rate β ϕ and decays at rate κ ϕ . CTLs enter the domain at position x i at rate α c r k i , with r k i deőned via (4.17). CTLs change their position through a combination of undirected, random movement and chemotactic movement in response to the chemoattractant secreted by tumour cells, while tumour cell movement is neglected. Finally, CTLs die due to homeostatic regulation mechanisms at rate µ c ρ k c , with ρ k c deőned via (4.20).

Figure 4 . 3 :

 43 Figure 4.3: Initial conditions. Plots of the density of tumour cells n(x, y, t) (panel (a)), the density of CTLs c(x, y, t) (panel (b)), and the chemoattractant concentration ϕ(x, y, t) (panel (c)) at the initial time of the simulations (i.e. at t = 0) for the continuum model. Analogous initial conditions are used for the hybrid model.

  and ϕ(x, y, 0) = 90 exp[-200(xx * 11 ) 2 -200(yy * 11 ) 2 ].

Fig. 4 . 88 CHAPTER 4 .ν = γc 4φmaxτ χ 2 λ = βc 4τ χ 2

 488422 4). Both Fig. 4.4 and Fig. 4.5 indicate that there is an excellent quantitative agreement between numerical solutions of the continuum model (4.30) and the results of numerical simulations of the hybrid model. Exploring the impact of CTL inőltration on anti-tumour immune response Table 4.1: Model parameters and related values used in numerical simulations. in the x and y direction χ = 0.016 cm Time-step τ = 10 -4 (days) Final time t f = 15 (days) Tumour cells Cell density at position x n(x, t) ⩾ 0 (cells/cm 2 ) Initial number ρn(0) = 45228 (cells) Proliferation rate αn = 1.5 (1/day) [43] Rate of death due to competition between tumour cells µn = 1.25 × 10 -5 (1/day/cell ) Level of efficiency of CTLs ζn ∈ [0.001, 1] (1/day/cell ) Radius of interaction between tumour cells and CTLs θ = 3 × 0.016 (cm) CTLs Cell density at position x c(x, t) ⩾ 0 (cells/cm 2 ) Initial number ρc(0) = 8960 (cells) inflow rate αc = 6 (cell/cm 2 /day/mol ) Chemotactic sensitivity (hybrid model) Chemotactic coefficient (continuum model) γc = 10 (cm 2 /day/mol ) [14] Random movement prob. (hybrid model) Diffusion coefficient (continuum model) βc = 1 × 10 -3 (cm 2 /day) [148] Total cell density above which CTL movement is impaired wmax = [0.74 × 10 5 , 8.88 × 10 5 ] (cells/cm 2 ) Rate of death due to competition between CTLs µc = 6 × 10 -6 (1/day/cell )

Figure 4 . 5 :

 45 Figure 4.5: Baseline scenario: evolution of the spatial distributions of cells over time. Panels (a)-(d)-(g) display the plots of the density of tumour cells n(x, y, t) and panels (b)-(e)-(h) display the plots of the density of CTLs c(x, y, t) of the continuum model at three successive time instants -i.e. t = 1, t = 4, and t = 15. The pink dashed lines highlight the 1D crosssection corresponding to y = 0.5. Panels (c)-(f )-(i) display the corresponding side on view plot of the densities of tumour cells n(x, y, t) and CTLs c(x, y, t) of the hybrid model (solid, coloured lines) and continuum model (dotted, black lines) (i.e. at y = 0.5 and t = 1, t = 4, and t = 15). Here, ζ n = 0.004 and all the other parameters are as in Table4.1 with α ϕ = 1.5 and w max = 2.96 × 10 5 . The results from the hybrid model correspond to the average over three simulations and the related variance is displayed by the coloured areas surrounding the curves.

Figure 4 . 6 :

 46 Figure 4.6: Immunoscore. The heat map in panel (a) displays the value of the immunoscore computed via (4.7) at the end of numerical simulations of the hybrid model for different combinations of α ϕ and w max . For each given value of α ϕ and w max , the values of the other parameters are as in Table 4.1, with ζ n = 0.00012. This heat map matches with the corresponding heat map obtained for the continuum model, which is displayed in panel (b). Sample dynamics of the numbers and densities of tumour cells and CTLs for the values of parameters α ϕ and w max corresponding to the dotted pink squares 1-4 are displayed in the plots of Fig. 4.7 and Figs. 4.8-4.11, respectively.

4 . 10 ). 92 CHAPTER 4 .

 410924 Exploring the impact of CTL inőltration on anti-tumour immune response

Figure 4 . 7 :

 47 Figure 4.7: Sample dynamics of the numbers of tumour cells and CTLs in hot, alteredimmunosuppressed, altered-excluded and cold tumour scenarios. Time evolution of the number of tumour cells ρ n (t), the number of CTLs ρ c (t), and the total amount of chemoattractant ϕ tot (t) (in the insets) of the hybrid model (solid, coloured lines) and the continuum model (dotted, black lines) for values of α ϕ and w max corresponding to the dotted pink squares 1-4 of Fig. 4.6. Sufficiently low values of α ϕ lead to the emergence of cold tumour scenarios (panel (a)); intermediate values of α ϕ and sufficiently high values of w max lead to the emergence of alteredimmunosuppressed tumour scenarios (panel (b)); intermediate values of α ϕ and sufficiently small values of w max lead to the emergence of altered-excluded tumour scenarios (panel (c)); and sufficiently high values of α ϕ and w max lead to the emergence of hot tumour scenarios (panel (d)). To obtain these results, we used the values of the parameters α ϕ and w max corresponding to the dotted pink squares 1 (panel (a)), 3 (panel (b)), 2 (panel (c)), and 4 (panel (d)) of Fig. 4.6. All the other parameters are as in Table 4.1, with ζ n = 0.00012. The results from the hybrid model correspond to the average over three simulations and the related variance is displayed by the coloured areas surrounding the curves.

Figure 4 . 10 :

 410 Figure 4.10: Sample spatial distributions of cells in altered-immunosuppressed tumour scenarios. Panels (a) and (b) display, respectively, the densities of tumour cells n(x, y, t) and CTLs c(x, y, t) of the continuum model at the end of numerical simulations (i.e. at t = 15) for a choice of parameters that results in the emergence of a altered-immunosuppressed tumour scenario (cf. dotted pink square 3 in Fig. 4.6). The pink dashed line highlights the 1D crosssection corresponding to y = 0.5. Panel (c) displays the corresponding side on view plot of the densities of tumour cells n(x, y, t) and CTLs c(x, y, t) of the hybrid model (solid, coloured lines) and continuum model (dotted, black lines) (i.e. at y = 0.5 and t = 15). Here, ζ n = 0.00012, α ϕ = 0.15, w max = 8.88 × 10 5 , and all the other parameters are as in Table4.1. The results from the hybrid model correspond to the average over three simulations and the related variance is displayed by the coloured areas surrounding the curves.

Figure 4 . 11 :

 411 Figure 4.11: Sample spatial distributions of cells in hot tumour scenarios. Panels (a) and (b) display, respectively, the density of tumour cells n(x, y, t) and CTLs c(x, y, t) of the continuum model at the end of numerical simulations (i.e. at t = 15) for a choice of parameters that results in the emergence of a hot tumour scenario (cf. dotted pink square 4 in Fig. 4.6). The pink dashed line highlights the 1D cross-section corresponding to y = 0.5. Panel (c) displays the corresponding side on view plot of the densities of tumour cells n(x, y, t) and CTLs c(x, y, t) of the hybrid model (solid, coloured lines) and continuum model (dotted, black lines) (i.e. at y = 0.5 and t = 15). Here, ζ n = 0.00012, α ϕ = 1.5, w max = 8.88 × 10 5 , and all the other parameters are as in Table4.1. The results from the hybrid model correspond to the average over three simulations and the related variance is displayed by the coloured areas surrounding the curves.

4. 6

 6 Discussion, conclusions and research perspectives4.6.1 Discussion and conclusionsIn this chapter, we have developed a hybrid discrete-continuum modelling framework for the interactions between tumour cells and cytotoxic T lymphocytes, which explicitly takes into ac-

Figure 4 . 12 :

 412 Figure 4.12: Immunotherapy effects. Comparison between the numbers of tumour cells ρ n (t) at the end of simulations (i.e. at t = 10) of the continuum model for different values of α ϕ and w max without therapy (panel (a)) and when the effects of different immunotherapies are incorporated by considering different values of the parameters ζ n , α c and α n (panel (b)-(d). In panel (a), ζ n = 0.00012 and the values of the other parameters are as in Table 4.1. In panel (b), ζ n = 0.001 and the values of the other parameters are as in Table 4.1. In panel (c), ζ n = 0.001, α c = 12 and the values of the other parameters are as in Table 4.1. In panel (d), ζ n = 0.001, α c = 12, α n = 0.75 and the values of the other parameters are as in Table 4.1.

Figure 5 .

 5 2(e)-(f )]. Finally, the authors measured the concentration of two cytokines in the co-culture media, namely: the pro-inŕammatory cytokine INF-γ and the anti-inŕammatory cytokine IL-10. They found that cortisol signiőcantly decreased the levels of INF-γ and signiőcantly increased the levels of IL-10 [see Figure 5.2(g)-(h)]. 5.4. Modelling framework 107

  Figure 5.6(c)-(d) show an example of the őnal spatial distributions of tumour cells and CTLs for two decreasing values of the TC-CTL adhesion strength, whereas Figure 5.6(e)-(f ) show similar plots for two decreasing values of the secretion rate of the chemoattractant. These plots are compared to the one displayed in Figure 5.6(b), which shows the őnal spatial distributions of tumour cells and CTLs obtained in the control scenario. In particular, Figure 5.6(b)-(d) show that decreasing the TC-CTL adhesion strength leads to a scenario in which CTLs accumulate around the tumour, because the secretion rate of the chemoattractant is high, but they do not inőltrate into it. On the other hand, Figure 5.6(b)-(e)-(f ) show that decreasing the secretion rate of the chemoattractant leads to a scenario in which CTLs away from the tumour are not sensitive to the gradient of the chemoattractant and, therefore, do not move towards the tumour.

Figures 5. 7

 7

  (a)-(b) compare the inőltration score obtained when the effects of IL-10 are not considered (i.e. we consider a normal value for the growth rate of CTLs), with those obtained when the effects of IL-10 are considered (i.e. the value of the growth rate of CTLs is decreased). Figures 5.7(c)-(d) also compare the number of tumour cells and CTLs at the end of numerical simulations (i.e. at day 4 of the experiments) for the two scenarios considered. Comparing the results of Figure 5.7(a) with those displayed in Figure 5.7(b), we see that, similarly to the results of the previous paragraph, decreasing the growth rate of CTLs decreases the inőltration score only when the TC-CTL adhesion strength is high. Moreover, for intermediate values of the TC-CTL adhesion strength, decreasing the growth rate of CTLs slightly decreases the inőltration score. Finally, when the TC-CTL adhesion strength is low, decreasing the growth rate of CTLs does not have an impact on the inőltration score, as its value is already small. As shown by Figures 5.7

  (c)-(d), decreasing the growth rate of CTLs leads to a decreased number of CTLs at the end of simulations. However, as CTLs are not able to eliminate tumour cells, the őnal number of tumour cells remains similar in the two scenarios.

Figures 5. 8

 8

  (a)-(b) compare the inőltration score obtained with the parameter values considered in Section 5.6.1, with the inőltration score obtained when the same parameter values are considered and CTLs can eliminate tumour cells with a small probability (i.e. we only increase the value of the immune success rate).Figures 5.8

  (c)-(d) show a comparison between the number of tumour cells and CTLs at the end of simulations (corresponding to day 4 of the experiments) for the two scenarios considered. Comparing the results of Figure 5.8(a) with those displayed in Figure

  Figure A.1 shows how a lattice conőguration evolves to minimize the total energy of the system.

χ 2 2 ∂ 2 + 2 ∂ 2 ∂

 2222 xx ψ + O(χ 3 ), where ψ ≡ ψ(w), ∂ x ψ = ψ ′ (w)∂ x w, ∂ xx ψ = ψ ′′ (w) ∂ x w ψ ′ (w)∂ xx w and c(x ± χ, t) = c ± χ∂ x c + χ 2 xx c + O(χ 3 ), ϕ(x ± χ, t) = ϕ ± χ∂ x ϕ + χ 2 xx c + O(χ 3

∂ 2 χ 2 τ → β c ∈ R + * and ν 2ϕ max χ 2 τ

 222 xx cψ(w) -c∂ xx ψ(w) = ∂ x ψ(w)∂ x c -∂ x cψ ′ (w)∂ x w = ∂ x (ψ(w)cψ ′ (w))∂ x ccψ ′ (w))∂ x n , (D.7)and using the elementary property (a) + -(-a) + = a for a ∈ R we also obtain-ψc ∂ xx ϕ + --∂ xx ϕ + -ψ∂ x c ∂ x ϕ + --∂ x ϕ + + ∂ x ψc -∂ x ϕ + -∂ x ϕ + = -∂ x ψc∂ x ϕ . (D.8)Substituting the computation into Equation (D.6), letting τ → 0 and χ → 0 in such a way thatλ → γ c ∈ R + * ,after a little algebra, considering (x, t) ∈ Ω \ ∂Ω × R * + , we őnd∂ t c -∂ x β c ψ(w)∂ x cγ c ψ(w)c∂ x ϕβ c cψ ′ (w)∂ x w = -µ c ρ c (t)c + α c rwhere ψ is given by (4.22) and w := n + c.Moreover, zero-ŕux boundary conditions easily follow from the fact that T-cell moves that require moving out of the spatial domain are not allowed.D.1.3 Formal derivation of the balance equation for the concentration of chemoattractant φ(x, t)

  The ŕowchart in Fig. D.1 illustrates the general computational procedure to carry out simulations of the hybrid model in a one-dimensional setting, while the ŕowchart in Fig. D.2 provides further details of the computational procedure to simulate cell dynamics in a one-dimensional setting. Analogous strategies are used in a two-dimensional setting. All random numbers mentioned in Fig. D.2 are real numbers drawn from the standard uniform distribution on the interval (0, 1), which in our case are obtained using the built-in Matlab function rand.

Figure D. 1 :Figure D. 2 :

 12 Figure D.1: Flowchart illustrating the computational procedure to simulate the hybrid model in a one-dimensional setting. A detailed summary of steps 2) and 3) is provided by the ŕowchart in Fig. D.2. A similar procedure is used in a two-dimensional setting.

k i+ 1 2 1 2 1 2 1 2

 1111 )µ c ρ k c c k i,jfor i, j = 0, . . . , N , whereF ,j := β c ψ(w k i+ ,j c k i,j ψ(w k i+1,j ) + b k,- i+ ,j c k i+1,j ψ(w k i,j), i = 0, . . . N -1, j = 0, . . . N, ψ(w k i,j ), i = 0, . . . N, j = 0, . . . N -1,

2

  Discrete and continuum models for the coevolutionary dynamics between cytotoxic T lymphocytes and tumour cells 2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Biological background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Structure of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The individual-based model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Mathematical modelling of cell division and death due to intra-population competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical modelling of phenotypic changes in tumour cells . . . . . .

	xvi	Contents
	2.3.2
	xv

  ) 40 CHAPTER 2. Discrete and continuum models for tumour-immune interactions whereas if condition (2.24) is met and the steady-state solution (n * , c * ) is given by (2.20) and (2.25) then the PDE-IDE system (2.26) reduces to

Table 2 .

 2 1: Parameter values used in numerical simulations and their sources

		Biological meaning	Value	Source
	αn	Rate of tumour cell proliferation	1.5/day	[43]
	αc	Rate of antigen-independent CTL proliferation	5 ×10 -2 /day	[176]
	µn	Rate of death of tumour cells due to clonal competition	1.5 ×10 -6 µl/day ad hoc
	µc	Rate of death of CTLs due to self-regulation mechanisms	5 ×10 -6 µl/day	ad hoc
	ζn	Killing rate of tumour cells by CTLs	5 ×10 -6 µl/day	[123]
	ζc	Rate of replication of CTLs following recognition	3 ×10 -5 µl/day	[184]
	η	Affinity range of TCRs	[0.1, 2]	[19]
	θn	Level of selectivity of clonal competition amongst tumour cells [0.1, 2]	ad hoc
	θc	Level of selectivity of self-regulation mechanisms of CTLs	[0.1, 2]	ad hoc
	γ	TCR-tumour antigen binding affinity	[0.1, 3.5]	[198, 199]
	λn	Probability of phenotypic variation of tumour cells	0.01	[194]

  Cold tumour-like scenarios For sufficiently small values of γ that satisfy condition(2.24), in the early stage of cell dynamics the total number of tumour cells overtakes the total number of CTLs, and keeps expanding until saturation (cf. Figure2.3d). Accordingly, the average value of the immune score I deőned via (2.3) and (2.4) is smaller than 1 (i.e. for the parameter values considered here I ≈ 0.7), which corresponds to the emergence of cold tumour-like scenarios in the framework of our model. model and numerical solutions of the corresponding continuum model. Moreover, consistently with the results of linear stability analysis of the continuum model presented in Section 2.5.2, these numerical results show that the total numbers of tumour cells and CTLs converge either to the steady-state values given by (2.23) (cf. Figure2.3a), or the steady-state values given by (2.25) (cf. Figure2.3b-d), depending on the fact that the choices of the model parameters are such that condition (2.34) or condition (2.24) holds, respectively. When convergence to the steady state (ρ * n2 , ρ * c2

.3b). As a result, the average value of the immune score I deőned via (2.3) and (2.4) is one order of magnitude larger than 1 (i.e. for the parameter values considered here I ≈ 12.7). In the framework of our model, this corresponds to the emergence of hot tumour-like scenarios.

Altered tumour-like scenarios For intermediate values of γ that satisfy condition (2.24), after initial growth, a certain number of tumour cells and a slightly larger number of CTLs stably coexist (cf. Figure

2

.3c). In this case, the average value of the immune score I deőned via (2.3) and (2.4) is just slightly larger than 1 (i.e. for the parameter values considered here I ≈ 1.6). In the framework of our model, this corresponds to the emergence of altered tumour-like scenarios.

  Table (C.1) and Table (C.2

  The points where CTLs are initially concentrated are the points where blood vessels are assumed to be located, that is, the set ω in (4.17) is deőned

	* 11 , y * 11 ) = (0.5, 0.5), (x * 21 , y * 21 ) = (0.26, 0.74), (x * 22 , y * 22 ) = (0.26, 0.26), (x * 23 , y * 23 ) = (0.74, 0.74) and (x * 24 , y * 24 ) = (0.74, 0.26). as ω := {(x * 21 , y * 21 ), (x * 22 , y * 22 ), (x * 23 , y * 23 ), (x * 24 , y * 24 )} .
	Similarly, for the continuum model we have
	n

  ).Therefore, after a little algebra we obtainψ c(x + χ, t) + c(xχ, t)ψ(w(xχ, t)) + ψ(w(x + χ, t)) c = χ 2 (∂ xx cψ -c∂ xx ψ)

	χ 2 2	∂ xx ϕ(x, t)
	χ 2 2	∂ xx ϕ(x, t)

ϕ(x, t)ϕ(xχ, t)

+ c(xχ, t) = χ∂ x ϕ(x, t) -+ [c(x, t) -χ∂ x c(x, t)], ϕ(x, t)ϕ(x + χ, t) + c(x + χ, t) = -χ∂ x ϕ(x, t) -+ [c(x, t) + χ∂ x c(x, t)], ϕ(xχ, t)ϕ(x, t) + c(x, t) = -χ∂ x ϕ(x, t) + χ 2 2 ∂ xx ϕ(

x, t) + c(x, t) and ϕ

(x + χ, t)ϕ(x, t) + c(x, t) = χ∂ x ϕ(x, t) + χ 2 2 ∂ xx ϕ(x, t)

  2)-(D.4) and relations (D.1.2)-(D.1.2) hold for τ, χ sufficiently small, the difference Equation (4.14) can be formally rewritten in the approximate formϕ(x, t + τ )ϕ(x, t) τ = β ϕ ϕ(xχ, t) + ϕ(x + χ, t) -2ϕ(x, t) χ 2 + α ϕ n(x, t)κ ϕ ϕ(x, t) (D.9)If the function ϕ is continuously differentiable with respect to the variable t and twice differentiable with respect to the variable x, letting τ, χ → 0 in the above equation gives∂ t ϕ(x, t)β ϕ ∂ xx ϕ(x, t) = α ϕ n(x, t)κ ϕ ϕ(x, t) (D.10)which is the balance equation for the chemoattractant concentration ϕ posed on Ω × R * + . We complement Equation (D.10) with zero ŕux boundary conditions.

https://www.cancer.gov

https://www.santepubliquefrance.fr

Acknowledgements

CTL number

CTLs Figure 4.4: Baseline scenario: evolution of the numbers of tumour cells and CTLs, and the total amount of chemoattractant. Plots of the time evolution of the number of tumour cells ρ n (t), the number of CTLs ρ c (t), and the total amount of chemoattractant ϕ tot (t) (in the inset) of the hybrid model (solid, coloured lines) and the continuum model (dotted, black lines) for a choice of parameters that results in the eradication of the tumour. Here, ζ n = 0.004 and all the other parameters are as in Table 4.1 with α ϕ = 1.5 and w max = 2.96 × 10 5 . The results from the hybrid model correspond to the average over three simulations and the related variance is displayed by the coloured areas surrounding the curves.

Emergence of hot, altered and cold tumour scenarios

We now consider a lower value of the parameter ζ n in order to explore biological scenarios in which the cytotoxic action of CTLs is less effective, for example due to high expression of PD1/PD-L1 inhibitory receptors on the surface of CTLs/tumour cells. As mentioned earlier, we wish to investigate how the spatial distribution of CTLs within the tumour varies depending on the value of the parameters α ϕ and w max . Therefore, we perform numerical simulations holding all parameters constant but considering different combinations of α ϕ and w max . For each pair of values considered, we stored the resulting dynamics of the densities of tumour cells and CTLs along with the dynamics of the corresponding cell numbers, and the őnal value of the immunoscore computed via (4.7). The results obtained are summarised by the heat maps in and (b) display, respectively, the densities of tumour cells n(x, y, t) and CTLs c(x, y, t) of the continuum model at the end of numerical simulations (i.e. at t = 15) for a choice of parameters that results in the emergence of a cold tumour scenario (cf. dotted pink square 1 in Fig. 4.6). The pink dashed line highlights the 1D cross-section corresponding to y = 0.5. Panel (c) displays the corresponding side on view plot of the densities of tumour cells n(x, y, t) and CTLs c(x, y, t) of the hybrid model (solid, coloured lines) and continuum model (dotted, black lines) (i.e. at y = 0.5 and t = 15). Here, ζ n = 0.00012, α ϕ = 0.0015, w max = 2.96 × 10 5 , and all the other parameters are as in Table 4.1. The results from the hybrid model correspond to the average over three simulations and the related variance is displayed by the coloured areas surrounding the curves. demonstrate that there is an excellent agreement between numerical simulations of the hybrid and continuum models. This testiőes to the robustness of the computational results presented here and the biological insight that they provide.

Immunotheraphy effects

The results presented in the previous subsection summarise how scenarios corresponding to different levels of CTL inőltration into the tumour can emerge under different combinations of the values of the parameters α ϕ and w max . We now investigate possible outcomes of immunotheraphy in these different scenarios.

In order to do this, we consider the same parameter settings used for the numerical simulations of Fig. 4.6, but now we allow the level of efficiency of CTLs at eliminating tumour cells to be higher (i.e. we increase the value of ζ n ). This corresponds to a biological scenario in which the tumour is treated with anti-PD1 monotherapy, which restores immune efficacy [START_REF] Topalian | łSafety, activity, and immune correlates of anti-PD-1 antibody in cancerž[END_REF]. We also investigate the effects of coupling anti-PD1 therapy with two other therapies. First we explore the effects of anti-PD1 therapy in combination with another immune checkpoint therapy, i.e. the anti CTLA-4 therapy [START_REF] Van Allen | łGenomic correlates of response to CTLA-4 blockade in metastatic melanomaž[END_REF]. To do so, we perform numerical simulations deőning all parameters as in the case of the anti-PD1 therapy but increasing the rate of inŕow of CTLs through blood vessels (i.e. the value of the parameter α c ). Then, we explore the effects of combining anti-PD1 therapy with chemotherapy, which inhibits tumour cell division, inŕames the TME with tumour antigens, and boosts the activation of CTLs [START_REF] Galon | łApproaches to treat immune hot, altered and cold tumours with combination immunotherapiesž[END_REF]. To do so, we perform numerical simulations deőning all parameters as in the scenario of the anti-PD1 therapy but decreasing the proliferation rate of tumour cells (i.e. the value of the parameter α n ) and increasing the rate of inŕow of CTLs through blood vessels (i.e. the value of the parameter α c ). The results obtained are displayed in Fig. 4.12, which shows a comparison between the numbers of tumour cells at the end of numerical simulation in the scenario łwithout treatmentž (i.e. with the parameter values considered in Section 4.5.3) and the three aforementioned scenarios in which the effects of different therapeutic protocols are considered.

Exploiting the excellent agreement between the results of numerical simulations of the hybrid 102 CHAPTER 5. Integrating the model with data: the role of stress on immune inőltration method similar to the one employed in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF], we deőne a score to quantify the effects of stress on immune inőltration in a controlled manner. The results of numerical simulations of this model are able to qualitatively reproduce the results of in vitro experiments presented in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF], and demonstrate the importance of including the effect of different factors when exploring the role played by psychological stress on immune inőltration. This study is a joint work in collaboration with:

• Melanie S. Flint, School of Pharmacy and Biomolecular sciences, University of Brighton, Centre for Stress and Age-related Diseases, Moulsecoomb, Brighton, BN2, 4GJ, UK,

• Chandrasekhar Venkataraman, School of Mathematical and Physical Sciences, University of Sussex, Department of Mathematics, Falmer, Brighton, BN1 9QH, UK, both authors of [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF]. This chapter contains preliminary results from an ongoing work.

Background

Biological background

The ability of psychological stress to induce immune suppression is widely recognised, but the mechanisms underlying the effects of stress on the adaptive immune system during tumour progression are not completely understood.

There has been increasing interest in detailing the mechanistic role that psychological stress may play in the context of initiation, progression, metastasis and recurrence of cancer. In particular, it has been reported that stress positively inŕuences carcinogenesis through mechanisms that promote proliferation, angiogenesis and metastasis, as well as mechanisms that protect tumour cells from apoptosis [START_REF] Lee | łSurgical stress promotes tumor growth in ovarian carcinomaž[END_REF][START_REF] Nilsson | łStress hormones regulate interleukin-6 expression by human ovarian carcinoma cells through a Src-dependent mechanismž[END_REF]. In addition to the direct effect on tumour cell biology, the negative role played by stress on the immune system has also been documented. Using a pre-clinical mouse model, in [START_REF] Budiu | łRestraint and social isolation stressors differentially regulate adaptive immunity and tumor angiogenesis in a breast cancer mouse modelž[END_REF] the authors showed that stress has a negative impact on T cell numbers and activation, as evidenced by a decrease in CD8+ and CD3+CD69+ T cells. The administration of beta-blockers made it possible to reverse the negative effect of stress on the adaptive immune system and slowed down tumour progression.

In the study that motivated our work [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF], the authors developed a co-culture in vitro model to test the effects of the stress hormone cortisol, with a particular emphasis in exploring its role on immune cell inőltration. Prior to testing the effect of cortisol on the in vitro co-culture model, the authors veriőed that this model was able to reproduce the outcomes of a 66CL4 syngeneic in vivo mouse model. Using two independent image analysis algorithms, they successfully quantiőed the effects of cortisol on immune inőltration, which was interpreted as the number of immune cells within the tumour spheroid boundary. The results from the in vitro co-culture model, as well as the in vivo experiments, showed that cortisol decreases immune inőltration. In this respect, the outcomes obtained in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF] recapitulate the conclusions of [START_REF] Budiu | łRestraint and social isolation stressors differentially regulate adaptive immunity and tumor angiogenesis in a breast cancer mouse modelž[END_REF], which also report that stress negatively affects immune inőltration into tumours.

The mixture of cytokines that is produced in the tumour microenvironment has an important role in tumour progression [START_REF] Dranoff | łCytokines in cancer pathogenesis and cancer therapyž[END_REF]. Pro-inŕammatory cytokines that are released in response to infection can inhibit tumour development and progression. Alternatively, tumour cells can produce anti-inŕammatory cytokines that promote growth, attenuate apoptosis and facilitate invasion and metastasis. In stressed conditions, the production of pro-and anti-inŕammatory cytokines can be altered. For example, in [START_REF] Budiu | łRestraint and social isolation stressors differentially regulate adaptive immunity and tumor angiogenesis in a breast cancer mouse modelž[END_REF], the authors found that stress increased the levels Step (1): in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF], to test whether stress decreased inőltration levels in the co-culture model, they compared the outcomes obtained between the group of spheroids and splenocytes only and the one of spheroids, splenocytes and cortisol. Cortisol signiőcantly decreased the levels of INF-γ and signiőcantly increased the levels of IL-10. IFN-γ is a pro-inŕammatory cytokine that stimulates immune response, such as T cell trafficking in the tumour-microenvironment and inőltration, whereas IL-10 is an anti-inŕammatory cytokine that inhibits immune response, such as T cell proliferation.

Step (2): in our model, we explore the way in which immune inőltration is affected by three parameters which can be associated to the effects of cortisol in the co-culture. Such parameters are: the secretion rate of the chemoattractant, the TC-CTL adhesion strength and the growth rate of CTLs. The őrst two parameters are associated to the role played by INF-γ in the co-culture model, while the last one is associated to the role played by IL-10.

Step (3): in this work, we explore different scenarios. We suppose that in normal conditions (i.e. in our control scenario) the levels of IFN-γ are high, the levels of IL-10 are low, and CTLs are able to inőltrate into the tumour. In stressed conditions however, we suppose that the levels of IFN-γ decrease, the levels of IL-10 increase, and CTL inőltration is reduced. By considering a range of values of the three aforementioned parameters, we study their impact on tumour-immune dynamics independently and together, assessing their inŕuence on immune inőltration in a controlled manner.

capacity. The number of CTLs introduced in the domain at day 0 corresponds to 5 times the number of tumour cells introduced at day -7, and the values of the parameters related to CTLs are chosen so as to qualitatively reproduce the experimental results obtained by [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF].

Control scenario: immune infiltration in non-stressed conditions

In the experimental results obtained by [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF], in absence of cortisol, splenocytes are able to inőltrate the spheroids. Moreover, the levels of INF-γ measured in the co-culture are high, Appendix A

Cellular Potts models and Compucell3D

The individual-based models developed in Chapter 3 and Chapter 5 have been numerically solved using the multicellular modelling environment CompuCell3D [START_REF] Izaguirre | łCompuCell, a multi-model framework for simulation of morphogenesisž[END_REF]. This software is an open source solver, which uses a Cellular Potts model [START_REF] Graner | łSimulation of biological cell sorting using a two-dimensional extended Potts modelž[END_REF] (also known as CPM, or Glazier-Graner-Hogeweg model). The system is written in the programming language C++, but allows users to create new models in a user-friendly manner in the XML and Python languages. The software runs on Windows, Mac and Linux platforms without change of model speciőcation.

CompuCell3D models have a range of different uses within mathematical biology, allowing rapid and compact speciőcation of cells, diffusing őelds and biochemical networks, all of this in two or three dimensions.

A.1 Lattice configuration

Each Cellular Potts model is made up of generalized objects, a description of their behaviours and appropriate initial conditions. The objects in each model are comprised of regular lattice sites, called pixels in 2D and voxels in 3D. The most fundamental object is a generalized cell, which may represent a biological cell, a subcellular compartment, a cluster of cells, or a piece of non-cellular material or surrounding medium [START_REF] Izaguirre | łCompuCell, a multi-model framework for simulation of morphogenesisž[END_REF]. Each generalized cell has an associated list of attributes, e.g. cell type, surface area and volume. The index of the cell occupying cell-lattice site i is uniquely deőned as σ(i) and the type of the cell is referred to as τ (σ(i)). Here, i is a vector of integer occupying pixel i. Either increasing or decreasing the number of pixels a cell contains can change its volume and surface. For example, in the model we developed in Chapter 3, to characterise the different sub-populations of cancer cells, we deőne as many types of tumour cells as sub-populations of cancer cells. Moreover, CTLs have as many types as TCRs considered for the simulation, which corresponds also to the number of antigens considered.

Interaction descriptions and dynamics between cells are modelled by means of the system's effective energy. This determines many characteristics such as cell size, shape, motility, adhesion strength and the reaction to gradients of chemotactic őelds. The effective energy deőne a way to produce a desired set of biological and physiological cell behaviors and does not represent the physical energy of the cells. The user deőnes the parameters for the effective energy for each cell type and these parameters determine the characteristics of each cell. During a simulation, each cell will attempt to extend its boundaries, through a series of index-copy attempts, known as V t (σ)) as the pressure inside the cell. When v < V t the cell has positive internal pressure, which encourages it to expand, and when v > V t the cell has negative internal pressure, which encourages it to shrink. The amplitudes of ŕuctuations of the cell's volume about is target volume are controlled by the value of parameter λ vol (σ).

In similar fashion the surface area constraint is given by:

where s(σ) is the surface area of the cell σ, S t (σ) is the cell's target surface area and λ surf (σ) is its inverse membrane compressibility. By combining together the boundary energy in Equation (A.1) and volume constraints in Equation (A.3), the basic GGH effective energy is modelled by:

A.3 A Monte Carlo Step

Time is measured in CompuCell3D simulations by Monte Carlo Steps (MCS), which indicate when every pixel on the lattice has made an index-copy attempt to a neighbouring pixel. To begin an index-copy attempt, the algorithm randomly selects a lattice site to be a target pixel i, and a neighbouring lattice site to be a source pixel i ′ . An attempt will then be made to switch the target pixel to the same generalised cell as the source pixel, thereby increasing the volume of the source cell and decreasing the volume of the target cell. If the source and target pixels belong to the same cell (i.e. σ(i) = σ(i ′ )), they do not need to attempt a ŕip and thus the effective energy will not be calculated.

If different generalized cells occupy these pixels, the algorithm sets σ(i) = σ(i ′ )) with probability P (σ(i)) → σ(i ′ ), given by the Boltzmann acceptance function [START_REF] Metropolis | łEquation of state calculations by fast computing machinesž[END_REF]:

where ∆H is the change in effective energy if the copy occurs and T m determines the amplitude of cell membrane ŕuctuations and the effective cell random motility. For an index-copy attempt, this probability means that if the change in effective energy is less than or equal to zero then the attempt will be successful. On the other hand, if the resulting change in effective energy is greater than zero the attempt may still be successful with the probability exp ∆H Tm . For biological simulations, the parameters ∆H and T m allow for a conversion between the number of MCS and experimental time. The conversion depends on the average values of ∆H Tm . In practice, in simulations, high values of the ratio ∆H Tm results in rigid, barely-or non-motile cells and little cell rearrangement. For low ∆H Tm , large ŕuctuations allow a high degree of cell random motility and rearrangement.

A.4 Subcellular dynamics and chemotaxis

In addition to generalized cells, Compucell3D simulations may contain other objects such as chemical őelds and subcellular networks as well as auxiliary equations to describe events at the cellular level like cell growth, division and rule-based differentiation. These chemical őelds and A.4. Subcellular dynamics and chemotaxis 129 subcellular networks evolve due to secretion, absorption, diffusion, reaction and decay according to appropriate PDEs and affect generalized-cell behaviors by modifying the effective energy.

A cellular behaviour easy to simulate with Compucell3D is chemotaxis, deőned as the cell motion induced by a presence of a chemical. In a simulation, chemotaxis is obtained biasing a cell's motion up or down a őeld gradient by changing the calculated effective-energy change used in the acceptance function. For a őeld ϕ(i):

where ϕ(i) is the chemical őeld at the index-copy target pixel i, ϕ(i ′ ) the őeld at the index-copy source pixel i ′ , and λ chem the strength and direction of chemotaxis. If λ chem > 0 and ϕ(i) > ϕ(i ′ ), then ∆H chem is negative, increasing the probability of accepting the index copy. The net effect is that the cell moves up the őeld gradient with a velocity ∼ λ chem ∇ϕ. If λ chem < 0 is negative, the opposite occurs, and the cell will move down the őeld gradient.

The change in concentration of the chemical őeld ϕ is obtained by solving a reaction-diffusion equation of the following general form:

where D, γ and S denote the diffusion constant, decay constant and secretion rates of the őeld, respectively. These three parameters may vary with position and cell-lattice conőguration, and thus be a function of cell σ and pixel i.

Appendix B

Appendix for Chapter 2

B.1 Formal derivation of the continuum model

Using a method analogous to that employed in [START_REF] Ardaševa | łComparative study between discrete and continuum models for the evolution of competing phenotype-structured cell populations in dynamical environmentsž[END_REF][START_REF] Chisholm | łEvolutionary dynamics of phenotype-structured populations: from individual-level mechanisms to population-level consequencesž[END_REF][START_REF] Stace | łDiscrete and continuum phenotype-structured models for the evolution of cancer cell populations under chemotherapyž[END_REF], we show that the PDE-IDE system (2.13) can be formally derived as the appropriate continuum limit of the individual-based model presented in Chapter 2.

Substituting deőnitions (2.9) of P b n and P q n into the difference equation (2.11) 1 for n k i and deőnitions (2.10) of P b c and P q c into the difference equation (2.11) 2 for c k j yields

where n k i ≡ n(x i , t k ) with (x i , t k ) ∈ I × (0, t f ] and c k j ≡ c(y j , t k ) with (y j , t k ) ∈ I × (0, t f ]. Using the fact that the following relations hold for τ and χ sufficiently small

where the function g is deőned via (2.6), the system of equations (B.1) can be formally rewritten 132 APPENDIX B. Appendix for Chapter 2 in the approximate form

where x ∈ I, y ∈ I and t ∈ (0, t f ]. Here,

If the function n(x, t) is twice continuously differentiable with respect to the variable x, for χ sufficiently small we can use the Taylor expansions

If, in addition, the functions n(x, t) and c(y, t) are continuously differentiable with respect to the variable t, letting τ → 0 and χ → 0 in such a way that condition (2.12) is met, from the latter system of equations we formally obtain

Substituting deőnitions (B.3) of R n (K n , J n ) and R c (K c , J c ) into the above system of equations gives the PDE-IDE system (2.13). Finally, the no-ŕux boundary conditions (2.14) follow from the fact that the attempted phenotypic variation of a tumour cell is aborted if it requires moving into a phenotypic state that does not belong to the interval I.

B.2 Details of numerical simulations of the continuum model

To construct numerical solutions of the PDE-IDE system (2.13) subject both to the no-ŕux boundary conditions (2.14) and to the initial condition (2.43), we use a uniform discretisation of step ∆x = 0.0013 of the interval I = [-L, L] as the computational domain of the independent variables x and y, and a uniform discretisation of step ∆t = 0.05 of the time interval (0, t f ].

B.2. Details of numerical simulations of the continuum model 133

We construct numerical solutions of the non-local PDE (2.13) 1 for n using a time-splitting approach, which is based on the idea of decomposing the original problem into simpler subproblems that are then sequentially solved at each time-step using an explicit Euler method with step ∆t. This leads to the following time-dicretisation of the PDE-IDE system (2.13) subject to the Neumann boundary conditions (2.14):

The system of equations (B.5) is numerically solved using a three-point őnite difference explicit scheme for the diffusion term [START_REF] Leveque | Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems[END_REF] and an implicit-explicit őnite difference scheme for the remaining terms [START_REF] Lorenzi | łDissecting the dynamics of epigenetic changes in phenotype-structured populations exposed to ŕuctuating environmentsž[END_REF][START_REF] Lorz | łPopulational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapiesž[END_REF], which leads to the following system of equations

Here

Given the values of the parameter τ , χ and λ n of the individual-based model, the value of the parameter β n is deőned so that condition (2.12) is met. The other parameter values are chosen to be coherent with those used to carry out numerical simulations of the individual-based model, which are speciőed in the main body of Chapter 2.

Appendix for Chapter 3 C.1 Model parameters

The individual-based model developed in Chapter 3 is parametrised using parameter values obtained from published biological data wherever possible. We use a 2D squared spatial domain with 400 × 400 lattice sites (pixels). We assume that a pixel of the domain corresponds to 3 × 3 µm 2 . As the CTL diameter is estimated to be between 10 µm and 12 µm [START_REF] Gao | łIL-2 sensitivity and exogenous IL-2 concentration gradient tune the productive contact duration of CD8+ T cell-APC: a multiscale modeling studyž[END_REF][START_REF] Gong | łA computational multiscale agent-based model for simulating spatio-temporal tumour immune response to PD1 and PDL1 inhibitionž[END_REF], the initial size of a CTL is 4 × 4 pixels. A tumour cell diameter is estimated to be about 20 µm [START_REF] Christophe | łA biased competition theory of cytotoxic T lymphocyte interaction with tumor nodulesž[END_REF], therefore we assume that each newly divided tumour cell is made of 5 × 5 pixels. In addition, the maximum CTL migration speed measured in the simulation is around 10 pixels/100 MCS. Therefore, using the CTL migration measurements in vivo (2 -25 µm/min, see [START_REF] Miller | łAutonomous T cell trafficking examined in vivo with intravital two-photon microscopyž[END_REF]), we choose 1 MCS ∼ 1 minute as a time scale. The parameters for the Cellular Potts model are listed in Table C.1, while all the other parameters with their related references are listed in Table C.2.

We now provide a discussion on how some of the parameters of the Cellular Potts model were chosen. Interactions between neighboring pixels in the Cellular Potts model have an effective energy, J (as it appears in Equation (A.5)), which characterises the strength of cell-cell adhesion (see Table C.1). A larger J means that more energy is associated with the interface between two cells, which is less energetically favourable, corresponding to weaker adhesivity. It can be observed that J CT and J T T are lower than J CC . We make the assumption that tumour cells stay in contact to compactly create the tumour mass and that when a CTL enters in contact with a tumour cell then strongly binds to it. When they migrate in the domain to search for tumour cells, CTLs are not in contact with each other.

Files to run a simulation example with Compucell3D software are available at: https:// plmlab.math.cnrs.fr/audebert/cc3dmodeltumourcd8.

C.2 Supplementary figures

Using the fact that the following relations hold for τ and χ sufficiently small

where the function η is deőned via (4.12), equation (D.1) can be formally rewritten in the approximate form

If, in addiction, the function n(x, t) is continuously differentiable with respect to the variable t, starting from equation (D.5), and letting the time-step τ → 0, one formally obtains the following IDE for the tumour cell density n(x, t):

D.1.2 Formal derivation of the PDE for the density of CTLs c(x, t)

The formal derivation of the PDE for the density of CTLs c(x, t) is built on the methods employed in [START_REF] Bubba | łFrom a discrete model of chemotaxis with volume-őlling to a generalized Patlak-Keller-Segel modelž[END_REF]. In the case where T cell dynamics are governed by the rules described in Section 4.3.3, considering (i, k) ∈ [1, N -1] × N 0 , between time-steps k and k + 1 the principle of mass balance gives the following difference equation for the CTLdensity c k i :

Using the fact that relations (D.2)-(D.4) and the following relations

If the function ψ(w(x, t)) is twice continuously differentiable and the functions c(x, t) and ϕ(x, t) are twice continuously differentiable with respect to the variable x, for χ sufficiently small

The discrete ŕuxes F k i-1 2 ,j for i = 1, . . . , N, j = 0, . . . , N and F k i,j-1 2 for i = 0, . . . , N, j = 1, . . . , N are deőned in an analogous way, and we impose zero-ŕux boundary conditions by using the deőnitions F k 0-1 2 ,j := 0 and F k N + 1 2 ,j := 0, for j = 0, . . . , N, F k i,0-1 2 := 0 and F k i,N + 1 2 := 0, for i = 0, . . . , N.

Appendix for Chapter 5

The individual-based model developed in Chapter 5 is based on a simpliőed version of the mathematical model developed in Chapter 3, which has been calibrated to qualitatively reproduce the experimental results presented in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF]. We refer the reader to Appendix C.1 for a description of the parameterisation of the original model. We describe here the way in which additional components of the model were calibrated using the parameter values reported in tables E.1 and E.2, to reproduce the qualitative behaviour of the experimental results presented in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF].

The value of the rate of death due to competition between tumour cells is chosen so that the number of tumour cells reaches its carrying capacity after 7 days of proliferation. The number of CTLs introduced in the domain at day 0 corresponds to 5 times the number of tumour cells introduced at day -7. The value of the rate of death due to competition between CTLs is chosen so that the value of our inőltration score computed at the end of simulations in the control scenario is similar to the value of the TIC algorithm obtained at day 4 in [START_REF] Al-Hity | łAn integrated framework for quantifying immune-tumour interactions in a 3D co-culture modelž[END_REF] when cortisol was not introduced in the co-culture.

The ratio between the energy at the interface between tumour cells and CTLs and the energy at the interface between tumour cells (i.e. the values of parameters J CT and J T T in Equation (A.5)) allow us to consider a wide range of biological scenarios corresponding to different degrees of immune inőltration. In particular, if the value of J CT is lower than the one J T T then CTLs inőltrate through tumour cells, whereas if the value of J CT is larger than the one of J T T then CTLs accumulate at the margin of the tumour, without inőltrating it. Therefore, to obtain different degrees of immune inőltration, we őx the value of J T T and we vary the value of J CT . In the control scenario we suppose that CTLs have a high capability of inőltrate through tumour cells. Therefore we suppose that J CT is lower than J T T . In stressed conditions however, we suppose that CTLs have a lower capability to inőltrate through tumour cells. Therefore, we increase the value of J CT to a value equal to or greater than the one chosen for J T T .

E.1 Model parameters