The immune system’s vitality and function is of the upmost importance in the human body. The ingenuity and performance of this defence mechanism also plays a role in the prevention of mutated, transformed cells becoming malig- nant tumours (cancers). More recently, the subject of cancer immunology has been concerned with examining the local effects the immune system has on a pre-angiogenic tumour site. A recent immunological review article - “The Three Es of Cancer Immunoediting” - discusses the way the immune system interacts with cancer cells: elimination, equilibrium, and escape. This bio- logical explanation underpins the mathematical modelling in this PhD thesis where mathematical models of pre-angiogenic immune-tumour interactions are presented and analysed. Chapter two develops an individual-based model of immune-cancer cell interactions using the computational simulation plat- form, CompuCell3D, to extend an earlier spatio-temporal model of tumour dormancy (Matzavinos et al. [2004]). Chapter three investigates and analyses an ODE model of the interaction of two immune cells and two tumour cells. This model is extended to include spatial movement terms for the tumour and immune cells (in both one and two dimensions) and investigates the rich heterogeneous spatio-temporal dynamics of the system in the presence of a limit cycle in the reaction kinetics. Finally, chapter four extends the models in the previous two chapters by examining an individual based model of two immune cell populations interacting with two tumour cell populations.

Adhesion, which includes cell-to-cell and cell-to-extracellular-matrix adhesion, plays an important role in cancer invasion and metastasis. After undergoing morphological changes malignant and invasive tumour cells, i.e., cancer cells, break away from the primary tumour by loss of cell-cell adhesion, degrade their basement membrane and migrate through the extracellular matrix by enhancement of cell-matrix adhesion. These processes require interactions and signalling cross-talks between proteins and cellular components facilitating the cell adhesion. Although such processes are very complex, the necessity to fully understand the mechanism of cell adhesion is crucial for cancer studies, which may contribute to improving cancer treatment strategies. We consider mathematical models in an attempt to understand better the roles of cell adhesion involved in cancer invasion. Using mathematical models and computational simulations, the underlying complex biological processes can be better understood and their properties can be predicted that might not be evident in laboratory experiments. Cancer cell migration and invasion of the extracellular matrix involving adhesive interactions between cells mediated by cadherins and between cell and matrix mediated by integrins, are modelled by employing two types of mathematical models: a continuum approach and an individual-based approach. In the continuum approach, we use Partial Differential Equations in which cell adhesion is treated as non-local and formulated by integral terms. In the individual-based approach, we first develop pathways for cell-cell and cell-matrix adhesion using Ordinary Differential Equations and later incorporate the pathways in a simulation environment for multiscale computational modelling. The computational simulation results from the two different mathematical models show that we can predict invasive behaviour of cancer cells from cell adhesion properties. Invasion occurs if we reduce cell-cell adhesion and increase cell-matrix adhesion and vice versa. Changing the cell adhesion properties can affect the spatio-temporal behaviour of cancer cell invasion. These results may lead to broadening our understanding of cancer cell invasion and in the long term, contributing to methods of patient treatment.

Glioblastoma multiforme (GBM) is one of the most common and aggressive primary brain tumours, with a median patient survival time of 6-12 months in adults. It has been recently suggested that a typically small sub-population of brain tumour cells, in possession of certain defining properties of stem cells, is responsible for initiating and maintaining the tumour. More recent experiments have studied the interactions between this subpopulation of brain cancer cells and tumour microenvironmental factors such as hypoxia and high acidity. In this thesis a computational approach (based on Gillespie’s algorithm and cellular automata) is proposed to investigate the tumour heterogeneities that develop when exposed to various microenvironmental conditions of the cancerous tissue. The results suggest that microenvironmental conditions highly affect the characterization of cancer cells, including the self-renewal, differentiation and dedifferentiation properties of cancer cells.

Glioblastoma multiforme (GBM) is one of the most common and aggressive primary brain tumours, with a median patient survival time of 6-12 months in adults. It has been recently suggested that a typically small sub-population of brain tumour cells, in possession of certain defining properties of stem cells, is responsible for initiating and maintaining the tumour. More recent experiments have studied the interactions between this subpopulation of brain cancer cells and tumour microenvironmental factors such as hypoxia and high acidity. In this thesis a computational approach (based on Gillespie’s algorithm and cellular automata) is proposed to investigate the tumour heterogeneities that develop when exposed to various microenvironmental conditions of the cancerous tissue. The results suggest that microenvironmental conditions highly affect the characterization of cancer cells, including the self-renewal, differentiation and dedifferentiation properties of cancer cells.

Un problème majeur du cancer est l'apparition de métastases, difficiles à détecter par l'imagerie médicale et qui peuvent progresser rapidement. Par le biais de la modélisation mathématique, nous espérons développer de nouveaux outils capables d'anticiper l'état métastatique d'un patient.Les deux premières parties de cette thèse sont dédiées au développement d'un tel outil, l'objectif étant sonutilisation chez l'animal voire en clinique. Dû aux variabilités intra- et inter-individuelles, nous sommes amenés à utiliser des modèles statistiques coûteux en temps de calcul.Dans la partie 1, nous étendons une approche introduite par Iwata et al. et développée dans l'équipe. Nousproposons une résolution numérique plus efficace basée sur la reformulation du modèle sous formed'équation intégrale de Volterra de type convolution, qui s'avère également utile pour montrer despropriétés théoriques du modèle. En outre, nous étudions une extension stochastique de ce modèle déterministe.Dans la partie 2, nous montrons que notre approche est adaptée à la description de données souris. Utilisant le cadre statistique des modèles nonlinéaires à effets mixtes, nous construisons un modèle métastatique identifiable à partir des données et nous interprétons les résultats biologiquement.La partie 3 regroupe des résultats issus de collaborations avec des biologistes. Nous avons commencé àmodéliser la croissance tumorale à partir d'observations par imagerie SPECT en utilisant un modèle deGyllenberg et Webb. D'autre part, afin d'améliorer la précision des observations SPECT, nous testons des techniques dedétection de contours via des méthodes volumes finis basées sur des schémas DDFV. / Metastasis is one of the major problems of cancer because metastases areoften difficult to detect by clinical imaging and may develop rapidly. With the help of mathematical modelling, we hope to developnew tools capable of anticipating the metastatic state of a patient.The first two parts of this thesis are dedicated to developing such a tool, destined for a preclinical oreven clinical use. As tumour growth dynamics vary strongly between individuals and since observations are often sparse andnoisy, we need to consider computationally expensive statistical tools.In the first part, we extend an approach introduced by Iwata et al. and developed by Barbolosi et al. In particular, wepropose a more efficient numerical resolution based on a model reformulation into a Volterra integral equation of convolutiontype. This reformulation also permits to prove theoretical model properties (regularity and identifiability). Moreover, we study a stochastic generalisation of this deterministic model.In the second part, we will show that our approach is suitable for the description of experimental data on tumour-bearing mice.Using the statistical framework of nonlinear mixed-effects modelling, we build a metastatic model that is identifiable fromour data. We then interpret the results biologically.The last part of this thesis contains several results obtained in collaboration with biologists. We have started to model tumourgrowth with data obtained from SPECT imaging, using a model by Gyllenberg and Webb. Also, in order to improve the precision ofSPECT data, we have tested contour detection methods via finite volume methods based on DDFV schemes.

Colorectal cancer (CRC) arises in part from the dysregulation of cellular proliferation, associated with the canonical Wnt pathway, and differentiation, effected by the Notch signalling network. In this thesis, we develop a mathematical model of ordinary differential equations (ODEs) for the coupled interaction of the Notch and Wnt pathways in cells of the human intestinal epithelium. Our central aim is to understand the role of such crosstalk in the genesis and treatment of CRC. An embedding of this model in cells of a simulated colonic tissue enables computational exploration of the cell fate response to spatially inhomogeneous growth cues in the healthy intestinal epithelium. We also examine an alternative, rule-based model from the literature, which employs a simple binary approach to pathway activity, in which the Notch and Wnt pathways are constitutively on or off. Comparison of the two models demonstrates the substantial advantages of the equation-based paradigm, through its delivery of stable and robust cell fate patterning, and its versatility for exploring the multiscale consequences of a variety of subcellular phenomena. Extension of the ODE-based model to include mutant cells facilitates the study of Notch-mediated therapeutic approaches to CRC. We find a marked synergy between the application of γ-secretase inhibitors and Hath1 stabilisers in the treatment of early-stage intestinal polyps. This combined treatment is an efficient means of inducing mitotic arrest in the cell population of the intestinal epithelium through enforced conversion to a secretory phenotype and is highlighted as a viable route for further theoretical, experimental and clinical study.