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New experimental data have shown how the periodic exposure of cells to low oxygen levels (i.e., cyclic hypoxia) impacts their progress through the cell-cycle. Cyclic hypoxia has been detected in tumours and linked to poor prognosis and treatment failure. While fluctuating oxygen environments can be reproduced in vitro, the range of oxygen cycles that can be tested is limited. By contrast, mathematical models can be used to predict the response to a wide range of cyclic dynamics. Accordingly, in this paper we develop a mechanistic model of the cell-cycle that can be combined with in vitro experiments, to better understand the link between cyclic hypoxia and cell-cycle dysregulation. A distinguishing feature of our model is the inclusion of impaired DNA synthesis and cell-cycle arrest due to periodic exposure to severely low oxygen levels. Our model decomposes the cell population into five compartments and a time-dependent delay accounts for the variability in the duration of the S phase which increases in severe hypoxia due to reduced rates of DNA synthesis. We calibrate our model against experimental data and show that it recapitulates the observed cell-cycle dynamics. We use the calibrated model to investigate the response of cells to oxygen cycles not yet tested experimentally. When the re-oxygenation phase is sufficiently long, our model predicts that cyclic hypoxia simply slows cell proliferation since cells spend more time in the S phase. On the contrary, cycles with short periods of re-oxygenation are predicted to lead to inhibition of proliferation, with cells arresting from the cell-cycle in the G2 phase. While model predictions on short time scales (about a day) are fairly accurate (i.e, confidence intervals are small), the predictions become more uncertain over longer periods. Hence, we use our model to inform experimental design that can lead to improved model parameter estimates and validate model predictions.

Many microorganisms live within surface-associated consortia, termed biofilms, that can form intricate porous structures interspersed with a network of fluid channels. In such systems, transport phenomena, including flow and advection, regulate various aspects of cell behaviour by controllling nutrient supply, evacuation of waste products and permeation of antimicrobial agents. This study presents multiscale analysis of solute transport in these porous biofilms. We start our analysis with a channel-scale description of mass transport and use the method of volume averaging to derive a set of homogenized equations at the biofilmscale. We show that solute transport may be described via two coupled partial differential equations for the averaged concentrations, or telegrapher’s equations. These models are particularly relevant for chemical species, such as some antimicrobial agents, that penetrate cell clusters very slowly. In most cases, especially for nutrients, solute penetration is faster, and transport can be described via an advection-dispersion equation. In this simpler case, the effective diffusion is characterised by a second-order tensor whose components depend on: (1) the topology of the channels’ network; (2) the solute’s diffusion coefficients in the fluid and the cell clusters; (3) hydrodynamic dispersion effects; and (4) an additional dispersion term intrinsic to the two-phase configuration. Although solute transport in biofilms is commonly thought to be diffusion-dominated, this analysis shows that dispersion effects may significantly contribute to transport.

Information regarding the propagation dynamics of intramural electrical wavefronts in the 3D volume of the ventricles is critical in gaining a better understanding of normal and pathological cardiac function. However, investigation into such phenomena using cardiac modelling has so far been impaired due to limitations in the structural details included in current cardiac computational models. Here, we describe the technological pipeline required for the construction of realistic, highly-detailed and personalised whole ventricular models directly from high-resolution MR images and their use within a reliable, fully tested cardiac simulation software (Chaste). Simulations of cardiac propagation in the structurally-detailed model are presented that reveal the importance of complex structural geometry, fibre orientation, blood vessels and other heterogeneities in propagation of activation wavefronts through the ventricular volume. The tools and techniques presented in this study are expected to be key in the development and application of the next generation of cardiac models.

Chaste is a software library and set of test suites for modelling phenomena in the domain of computational biology. Specifically these the modelling problems arise in the fields of cancer modelling, cardiac simulation and soft-tissue engineering (Chaste stands for ‘Cancer, heart and soft-tissue environment’). It is released under the LGPL 2.1 licence. Chaste has been developed using agile programming methods. The project began 5 years when it was reasoned that the modelling of a variety of physio-logical phenomena requires both a generic mathematical modelling framework, and a generic computational/simulation framework. The Chaste project was formed as part of the larger Integrative Biology e-Science (IB) Project ([14]), an inter-institutional project aimed at developing a suitable IT infrastructure to support physiome-level computational modelling, with a focus on cardiac and cancer modelling.

Simulation of cardiac electrical activity using the bidomain equations can be a massively computationally demanding problem. This study provides a comprehensive guide to numerical bidomain modelling. Each component of bidomain simulations-discretisation, ODE-solution, linear system solution, and parallelisation-is discussed, and previously-used methods are reviewed, new methods are proposed, and issues which cause particular difficulty are highlighted. Particular attention is paid t

Competition in nature is traditionally conceptualised as a battle between individual organisms in the constant struggle for survival. The triumph of one organism results in the demise of the other. Cooperation on the other hand involves individuals acting in a mutually beneficial manner and is therefore often regarded as the opposite of competition. From the perspective of individual cells, multicellular life is perhaps the most extreme form of cooperation. However, even within multicellular organisms, cells compete for space and survival. Genetically damaged “loser” cells that grow at a slower rate than their neighbouring “winner” cells are eliminated from the body by apoptosis. Importantly, cell competition is context-dependent; loser cells are perfectly viable if the whole organism is composed of them. The demise of loser cells is triggered specifically by the presence of cells that are perceived to be more fit. Multiple triggers and pathways involved with cell competition have been discovered, but the mechanism by which cells measure and communicate their relative fitness remains elusive. Current models of cell competition assert a priori winner or loser status to competing cell types. By their nature, such models cannot explain how the winner or loser identity is attained. In this thesis, we develop a modelling framework to study the emergence of winners and losers in cell competition. Specifically, we construct models of heterotypic populations where cell types can only vary in their model parameters. Using this approach, we show that i) variations in mechanical parameters are not sufficient for cell competition in a vertex-based model, and ii) winner or loser status in a competition system based on the exchange of death signals is determined by the emission rate of death signals and the tolerance to death signals. Finally, we make concrete suggestions for the experimental validation of our predictions.

In vivo observations show that oxygen levels in tumours can fluctuate on fast timescales. As a result, cancer cells can be periodically exposed to pathologically low oxygen levels; a phenomenon known as cyclic hypoxia. Even though in vitro models of cyclic hypoxia exist, they fail to capture the complex and heterogeneous oxygenation dynamics of real tumours. Mathematical models can help to overcome current experimental limitations and, by doing so, offer new insights into the biology of cyclic hypoxia by predicting cell responses to a wide range of cyclic dynamics. Recent experimental evidence suggests that cyclic hypoxia alters the progression of cancer cells along the cell-cycle. In this thesis, we use a range of mathematical approaches to study cell-cycle dysregulation in cyclic hypoxia and its impact on the growth and survival of cell cultures. Accordingly, we develop and analyse a series of mathematical models that can be used, alongside in vitro experiments, to increase understanding of cell responses to cyclic hypoxia. First, we present a deterministic cell-cycle model to describe the dynamics of a population of cells growing in different oxygen environments, in the absence of competition for space. Our model comprises a system of 6 linear differential equations with two time delays and describes the evolution of the number of cells in different cell-cycle states. The impact of hypoxia on cell-cycle progression is captured by introducing arrest (or checkpoint compartments) and allowing the rates at which cells transition between different cell-cycle states to depend on the oxygen environment they experience. We then develop a corresponding individual-based model to study the impact of cyclic hypoxia on cell survival in the presence and absence of treatment. We apply the models we develop in two different ways. On the one hand, we use a combination of analytical and numerical approaches to identify cell-cycle control strategies that favour proliferation and/or cell survival in different oxygen environments. Specifically, we identify environmental regimes in which cells may, or may not, benefit from having defective cell-cycle checkpoints. Furthermore, we illustrate how our models can be used as a tool to complement and design in vitro experiment to uncover the biological mechanisms that drive cell responses to constant and cyclic hypoxia.

Treatment in advanced cancers often provides only temporary improvements due to the emergence of drug-resistant tumour subpopulations. Standard-of-care regimens seek to maximise cell kill in order to achieve a cure, but thereby inadvertently also free resistant cells from the intra-tumoral competition which usually constrains their growth. An emerging alternative strategy, called adaptive therapy (AT), aims to tackle this problem by prioritising tumour control over cure. The idea is to leverage intra-tumoral competition to slow, or even avoid, the expansion of resistance. In this thesis we present an integration of mathematical modelling, clinical data analysis, and biological experiments in order to investigate: i) the factors which determine whether a patient will benefit from AT, and ii) how to best adjust treatment to control resistance. We begin by studying a simple two-population model, which we formulate first as an ordinary differential equation (ODE), and subsequently as an agent-based, model. We dissect the impact of various model parameters on the benefit of AT, and identify cell turnover as an important, but previously overlooked, factor. Furthermore, we find that the tumour’s spatial organisation reflects, as well as modulates, intra-tumoral competition, and we discuss implications for intermittent androgen deprivation treatment in prostate cancer. In the second part of this thesis, we investigate whether adaptive scheduling of poly-adenosine di-phosphate ribose polymerase inhibitors (PARPis) can improve the treatment of ovarian cancer. We develop a framework which uses an ensemble of six ODE models to dynamically guide AT decision-making. In collaboration with an experimentalist, we test our framework in an in vitro spheroid model, and find that it compares favourably to other adaptive dosing algorithms. However, this work also shows that high-dose, continuous treatment achieves the best overall outcome. To conclude we therefore develop an ODE model of PARPi treatment, and discuss implications for PARPi scheduling. Overall, this thesis provides novel insights into the eco-evolutionary underpinnings of AT, and highlights some of the challenges involved in translating AT to a new cancer type.

Signalling pathways are subcellular networks that enable the cell to respond to its microenvironment. Abnormal function of a signalling pathway is a hallmark of carcinogenesis and developmental diseases. The signalling cascades in these pathways involve biochemical interactions that can lead to highly nonlinear cellular responses. Studies of signalling pathways often involve the proposal of models based on experimental data that are difficult to quantify and reproduce. Due to the nature of the experimental procedures, the experimental results can often be qualitative or at best semi-quantitative. As a result, the data and models proposed in different studies can often appear inconsistent. The canonical Wnt pathway, overactive in 90% of colorectal cancer cases, is not immune to these problems. A review of the literature reveals many seemingly inconsistent experimental results and models regarding the pathway dynamics. This makes the Wnt pathway a natural candidate for investigation using a new model selection framework using qualitative experimental data. The steps involved in our model selection framework are as follows. We propose a set of candidate mathematical models that may be consistent with experimental observations. Using a qualitative cost function, we explore the output of these models to investigate whether they can reproduce qualitatively similar outputs to the experimental observations. Any consistent candidate models are analysed further using sen- sitivity analysis and in silico experiments, before new in vitro experiments are proposed to discriminate between remaining models. Our literature review of the Wnt pathway revealed that a major source of discrepancy is the way in which Wnt stimulation leads to the initial accumulation of the Wnt-target β-catenin. Before studying this problem, we introduce a set of candidate models to study the first steps in the Wnt signalling cascade involving the Axin scaffolding protein that degrades β-catenin and the LRP6 co-receptor, which facilitates deactivation of this process. Using our framework, we reduce the number of suitable candidate models from twelve to four and propose new in vitro experiments to discriminate between these models further. In a multi-disciplinary study, we carry out these experiments in the laboratory to discriminate between the remaining candidate models. Our results suggest that Axin can interact with LRP6 independently of Wnt, and that Wnt stimulation leads to Axin-driven LRP6 phosphorylation. We build on our initial candidate models and introduce β-catenin and further details of the pathway in a new set of candidate mathematical models. We compare these models to experimental results from three different studies of the pathway, which have proposed different models for the pathway. We establish that one of our candidate models can qualitatively reproduce the results from all three studies. According to this model, the first steps that lead to accumulation of β-catenin upon Wnt stimulation is sequestration of Axin at the membrane. Beyond the first half hour, dephosphorylation of Axin allows levels of β-catenin to increase further. The existence of seemingly inconsistent experimental results and models is not unique to the Wnt pathway. Thus we believe that the framework developed in this thesis could be adapted for use in other studies of signalling pathways to understand the underlying networks of individual pathways and their crosstalk.

By growing together in dense communities, microorganisms (microbes) have a huge impact on human life. Microbes also come in a wide variety of shapes, but we have yet to understand the importance of these shapes for community biology. How are multi- species communities, such as biofilms and colonies, affected by the morphologies of constituent cells? Which morphologies might these environments select for in turn? To address these questions, we use individual-based modelling to investigate the effects of cell shape on patterning and evolution within microbial communities. We develop a flexible simulation framework, coupling a continuum model of the biofilm chemical environment to a cellular-level description of biofilm growth mechanics. This modelling system allows competitions between different microbial cell shapes to be simulated and studied, in different community contexts. Our models predict that cell shape can strongly affect spatial structure and patterning within competitive communities. Rod cells perform better at colonising surfaces and the expanding edges of colonies, while round cells are better at dominating the upper surface of a community. Our predictions are supported by experiments using Escherichia coli and Pseudomonas aeruginosa bacteria, and demonstrate that particular shapes can confer a selective advantage in communities. In summary, the work presented in this thesis predicts and examines new mechanisms of self-organisation driven by cell shape, demonstrating a new significance for microbial morphology as a means for cells to succeed in a dense and competitive environment.

Sudden cardiac death following cardiac arrest is a major killer in the industrialised world. The leading cause of sudden cardiac death are disturbances in the normal electrical activation of cardiac tissue, known as cardiac arrhythmia, which severely compromise the ability of the heart to fulfill the body's demand of oxygen. Ventricular fibrillation (VF) is the most deadly form of cardiac arrhythmia. Furthermore, electrical defibrillation through the application of strong electric shocks to the heart is the only effective therapy against VF. Over the past decades, a large body of research has dealt with the study of the mechanisms underpinning the success or failure of defibrillation shocks. The main mechanism of shock failure involves shocks terminating VF but leaving the appropriate electrical substrate for new VF episodes to rapidly follow (i.e. shock-induced arrhythmogenesis).A large number of models have been developed for the in silico study of shock-induced arrhythmogenesis, ranging from single cell models to three-dimensional ventricular models of small mammalian species. However, no extrapolation of the results obtained in the aforementioned studies has been done in human models of ventricular electrophysiology. The main reason is the large computational requirements associated with the solution of the bidomain equations of cardiac electrophysiology over large anatomically-accurate geometrical models including representation of fibre orientation and transmembrane kinetics.In this Thesis we develop simulation technology for the study of cardiac defibrillation in the human heart in the framework of the open source simulation environment Chaste. The advances include the development of novel computational and numerical techniques for the solution of the bidomain equations in large-scale high performance computing resources. More specifically, we have considered the implementation of effective domain decomposition, the development of new numerical techniques for the reduction of communication in Chaste's finite element method (FEM) solver, and the development of mesh-independent preconditioners for the solution of the linear system arising from the FEM discretisation of the bidomain equations.The developments presented in this Thesis have brought Chaste to the level of performance and functionality required to perform bidomain simulations with large three-dimensional cardiac geometries made of tens of millions of nodes and including accurate representation of fibre orientation and membrane kinetics. This advances have enabled the in silico study of shock-induced arrhythmogenesis for the first time in the human heart, therefore bridging an important gap in the field of cardiac defibrillation research.

We introduce a Chaste plugin for the generation and the simulation of Gene Regulatory Networks (GRNs) in multiscale models of multicellular systems. Chaste is a widely used and versatile computational framework for the multiscale modeling and simulation of multicellular biological systems. The plugin, named CoGNaC (Chaste and Gene Networks for Cancer), allows the linking of the regulatory dynamics to key properties of the cell cycle and of the differentiation process in populations of cells, which can subsequently be modeled using different spatial modeling scenarios. The approach of CoGNaC focuses on the emergent dynamical behavior of gene networks, in terms of gene activation patterns characterizing the different cellular phenotypes of real cells and, especially, on the overall robustness to perturbations and biological noise. The integration of this approach within Chaste's modular simulation framework provides a powerful tool to model multicellular systems, possibly allowing for the formulation of novel hypotheses on gene regulation, cell differentiation, and, in particular, cancer emergence and development. In order to demonstrate the usefulness of CoGNaC over a range of modeling paradigms, two example applications are presented. The first of these concerns the characterization of the gene activation patterns of human T-helper cells. The second example is a multiscale simulation of a simplified intestinal crypt, in which, given certain conditions, tumor cells can emerge and colonize the tissue.