Chemotaxis, the phenomenon through which cells respond to external chemical signals, is one of the most important and universally observable in nature. It has been the object of considerable modelling effort in the last decades. The models for chemotaxis available in the literature cannot reconcile the dynamics of external chemical signals and the intracellular signalling pathways leading to the response of the cells. The reason is that models used for cells do not contain the distinction between the extracellular and intracellular domains. The work presented in this dissertation intends to resolve this issue. We set up a numerical hybrid simulation framework containing such description and enabling the coupling of models for phenomena occurring at extracellular and intracellular levels. Mathematically, this is achieved by the use of the fictitious domain method for finite elements, allowing the simulation of partial differential equations on evolving domains. In order to make the modelling of the membrane binding of chemical signals possible, we derive a suitable fictitious domain method for Robin boundary elliptic problems. We also display ways to minimise the computational cost of such simulation by deriving a suitable preconditioner for the linear systems resulting from the Robin fictitious domain method, as well as an efficient algorithm to compute fictitious domain specific linear operators. Lastly, we discuss the use of a simpler cell model from the literature and match it with our own model. Our numerical experiments show the relevance of the matching, as well as the stability and accuracy of the numerical scheme presented in the thesis.

Mathematical models have been used to capture the behaviour of biological systems, from low-level biochemical reactions to multi-scale whole-organ models. Models are typically based on experimentally-derived data, attempting to reproduce the observed behaviour through mathematical constructs, e.g. using Ordinary Differential Equations (ODEs) for spatially-homogeneous systems. These models are developed and published as mathematical equations, yet are of such complexity that they necessitate computational simulation. This computational model development is often performed in an ad hoc fashion by modellers who lack extensive software engineering experience, resulting in brittle, inefficient model code that is hard to extend and reuse. Several Domain Specific Languages (DSLs) exist to aid capturing such biological models, including CellML and SBML; however these DSLs are designed to facilitate model curation rather than simplify model development. We present research into the application of techniques from software engineering to this domain; starting with the design, development and implementation of a DSL, termed Ode, to aid the creation of ODE-based biological models. This introduces features beneficial to model development, such as model verification and reproducible results. We compare and contrast model development to large-scale software development, focussing on extensibility and reuse. This work results in a module system that enables the independent construction and combination of model components. We further investigate the use of software engineering processes and patterns to develop complex modular cardiac models. Model simulation is increasingly computationally demanding, thus models are often created in complex low-level languages such as C/C++. We introduce a highly-efficient, optimising native-code compiler for Ode that generates custom, model-specific simulation code and allows use of our structured modelling features without degrading performance. Finally, in certain contexts the stochastic nature of biological systems becomes relevant. We introduce stochastic constructs to the Ode DSL that enable models to use Stochastic Differential Equations (SDEs), the Stochastic Simulation Algorithm (SSA), and hybrid methods. These use our native-code implementation and demonstrate highly-efficient stochastic simulation, beneficial as stochastic simulation is highly computationally intensive. We introduce a further DSL to model ion channels declaratively, demonstrating the benefits of DSLs in the biological domain. This thesis demonstrates the application of software engineering methodologies, and in particular DSLs, to facilitate the development of both deterministic and stochastic biological models. We demonstrate their benefits with several features that enable the construction of large-scale, reusable and extensible models. This is accomplished whilst providing efficient simulation, creating new opportunities for biological model development, investigation and experimentation.

This thesis investigates the high-order hierarchical finite element method, also known as the finite element p-version, as a computationally-efficient technique for generating numerical solutions to the cardiac monodomain equation. We first present it as a uniform-order method, and through an a priori error bound we explain why the associated cardiac cell model must be thought of as a PDE and approximated to high-order in order to obtain the accuracy that the p-version is capable of. We perform simulations demonstrating that the achieved error agrees very well with the a priori error bound. Further, in terms of solution accuracy for time taken to solve the linear system that arises in the finite element discretisation, it is more efficient that the state-of-the-art piecewise linear finite element method. We show that piecewise linear FEM actually introduces quite significant amounts of error into the numerical approximations, particularly in the direction perpendicular to the cardiac fibres with physiological conductivity values, and that without resorting to extremely fine meshes with elements considerably smaller than 70 micrometres, we can not use it to obtain high-accuracy solutions. In contrast, the p-version can produce extremely high accuracy solutions on meshes with elements around 300 micrometres in diameter with these conductivities. Noting that most of the numerical error is due to under-resolving the wave-front in the transmembrane potential, we also construct an adaptive high-order scheme which controls the error locally in each element by adjusting the finite element polynomial basis degree using an analytically-derived a posteriori error estimation procedure. This naturally tracks the location of the wave-front, concentrating computational effort where it is needed most and increasing computational efficiency. The scheme can be controlled by a user-defined error tolerance parameter, which sets the target error within each element as a proportion of the local magnitude of the solution as measured in the H^1 norm. This numerical scheme is tested on a variety of problems in one, two and three dimensions, and is shown to provide excellent error control properties and to be likely capable of boosting efficiency in cardiac simulation by an order of magnitude. The thesis amounts to a proof-of-concept of the increased efficiency in solving the linear system using adaptive high-order finite elements when performing single-thread cardiac simulation, and indicates that the performance of the method should be investigated in parallel, where it can also be expected to provide considerable improvement. In general, the selection of a suitable preconditioner is key to ensuring efficiency; we make use of a variety of different possibilities, including one which can be expected to scale very well in parallel, meaning that this is an excellent candidate method for increasing the efficiency of cardiac simulation using high-performance computing facilities.

Studies of avian navigation are making increasing use of miniature Global Positioning Satellite devices, to regularly record the position of birds in flight with high spatial and temporal resolution. I suggest a novel approach to analysing the data sets pro- duced in these experiments, focussing on studies of the domesticated homing pigeon (Columba Livia) in the local, familiar area. Using Gaussian processes and Bayesian inference as a mathematical foundation I develop and apply a statistical model to make quantitative predictions of homing pigeon flight paths. Using this model I show that pigeons, when released repeatedly from the same site, learn and follow a habitual route back to their home loft. The model reveals the rate of route learning and provides a quantitative estimate of the habitual route complete with associated spatio-temporal covariance. Furthermore I show that this habitual route is best described by a sequence of isolated waypoints rather than as a continuous path, and that these waypoints are preferentially found in certain terrain types, being especially rare within urban and forested environments. As a corollary I demonstrate an extension of the flight path model to simulate ex- periments where pigeons are released in pairs, and show that this can account for observed large scale patterns in such experiments based only on the individual birds’ previous behaviour in solo flights, making a successful quantitative prediction of the critical value associated with a non-linear behavioural transition.

Tissue engineering has great potential as a method for replacing or repairing lost or damaged tissue. However, progress in the field to date has been limited, with only a few clinical successes despite active research covering a wide range of cell types and experimental approaches. Mathematical modelling can complement experiments and help improve understanding of the inherently complex tissue engineering systems, providing an alternative perspective in a more cost- and time-efficient manner. This thesis focusses on one particular experimental setup, a hollow fibre membrane bioreactor (HFMB). We develop a suite of mathematical models which consider the fluid flow, solute transport, and cell yield and distribution within a HFMB, each relevant to a different setup which could be implemented experimentally. In each case, the governing equations are obtained by taking the appropriate limit of a generalised multiphase model, based on porous flow mixture theory. These equations are then reduced as far as possible, through exploitation of the small aspect ratio of the bioreactor and by considering suitable parameter limits in the subsequent asymptotic analysis. The reduced systems are then either solved numerically or, if possible, analytically. In this way we not only aim to illustrate typical behaviours of each system in turn, but also highlight the dependence of results on key experimentally controllable parameter values in an analytically tractable and transparent manner. Due to the flexibility of the modelling approach, the models we present can readily be adapted to specific experimental conditions given appropriate data and, once validated, be used to inform and direct future experiments.

Cardiovascular disease is the primary cause of death globally. Although this group encompasses a heterogeneous range of conditions, many of these diseases are associated with abnormalities in the cardiac electrical propagation. In these conditions, structural abnormalities in the form of scars and fibrotic tissue are known to play an important role, leading to a high individual variability in the exact disease mechanisms. Because of this, clinical interventions such as ablation therapy and CRT that work by modifying the electrical propagation should ideally be optimized on a patient specific basis. As a tool for optimizing these interventions, computational modelling and simulation of the heart have become increasingly important. However, in order to construct these models, a crucial step is the estimation of tissue conduction properties, which have a profound impact on the cardiac activation sequence predicted by simulations. Information about the conduction properties of the cardiac tissue can be gained from electrophysiological data, obtained using electroanatomical mapping systems. However, as in other clinical modalities, electrophysiological data are often sparse and noisy, and this results in high levels of uncertainty in the estimated quantities. In this dissertation, we develop a methodology based on Bayesian inference, together with a computationally efficient model of electrical propagation to achieve two main aims: 1) to quantify values and associated uncertainty for different tissue conduction properties inferred from electroanatomical data, and 2) to design strategies to optimise the location and number of measurements required to maximise information and reduce uncertainty. The methodology is validated in several studies performed using simulated data obtained from image-based ventricular models, including realistic fibre orientation and conduction heterogeneities. Subsequently, by using the developed methodology to investigate how the uncertainty decreases in response to added measurements, we derive an a priori index for placing electrophysiological measurements in order to optimise the information content of the collected data. Results show that the derived index has a clear benefit in minimising the uncertainty of inferred conduction properties compared to a random distribution of measurements, suggesting that the methodology presented in this dissertation provides an important step towards improving the quality of the spatiotemporal information obtained using electroanatomical mapping.

I develop social evolution theory to study the evolution of cooperation as follows: (1) Many organisms undergo a dispersal phase prior to breeding; I demonstrate that knowing ones dispersal status aids the evolution of helping (by non-dispersers) and harming (by dispersers). (2) Policing driven by group-benefits may be selected to enforce cooperation in human and animal societies. I extend existing theory to show that policing may be harder to evolve that previously thought, but that it is maintained more readily than it evolves. (3) Archeological and anthropological evidence suggests that warfare was prevalent during our evolution. I show that, contrary to previous suggestions, between-group competition can favour any social behaviour (pro-social or anti-social) so long as it helps the group compete, and that such traits can be altruistic or mutually beneficial. (4) Reproductive leveling is analogous to policing; in the human literature there is doubt as to whether it can evolve. I extend my previous work to consider the coevolution of culturally and genetically inherited traits for reproductive leveling and selfishness. I find that cooperation can evolve between non-kin if they share the same culture. (5) Monogamy is thought to favour the evolution of cooperative breeding. I show that in the simplest case, because of the cost of competition between non-dispersing siblings, the level of promiscuity has little or no effect on the evolution of cooperation. (6) Spatial structure (limited dispersal) is thought to favour the evolution of inter-specific mutualisms as it aligns the partners’ interests. I consider the case of plant-fungi mutualisms and show that spatial structure can disfavour cooperation if it limits the potential fungal partners available to the plant.

Novel infectious diseases in humans are of great concern to public health authorities and researchers in epidemiology. Zoonotic pathogens in particular have the potential to cause epidemics without any or little warning. In this thesis, I investigate evolutionary and environmental conditions, and the interactions between both, which facilitate the zoonotic emergence of novel pathogens. I start with a list of the mechanisms and processes which might influence a zoonotic emergence, and identify some unsolved problems. I address these with multiple, theoretical models. First, I use a village-city model with different adaptation scenarios to examine the influence of spatial heterogeneity on the emergence process. I derive general analytical results for the statistical properties of emergence events, including the probability distribution of outbreak sizes. My results suggest that, for typical connection strengths between communities, spatial heterogeneity has only a weak effect on outbreak size distributions, and on the risk of emergence per introduction. Next, I extend the research on environmental conditions by looking at pathogen specialisation in multi-host systems. I derive threshold connectivities for which generalist pathogens, which infect multiple species and might therefore be more dangerous to cross into the human species, can sustain transmission and are not dominated by specialists, which can only cause sustained transmission chains in a single host species, but are able to cause emergences with little warning. My third research chapter is interested in the effect of the loss of biodiversity. I analytically derive expected prevalences for fast growing and slow growing species. If fast growing species tend to perform better in degraded environments, my analytical results suggest that the overall prevalence level of infectious diseases will rise as environments degrade, which facilitates the chance of zoonotic jumps. In my last research chapter, I examine the actual impact of a novel, emerging infectious disease. I use data from the recent `Swine flu' epidemic in England to estimate epidemiological parameters of the infectious agent. My results suggest that the majority of infected cases showed no or only mild symptoms. This reveals that more data than just the estimated number of cases are necessary to fully evaluate the danger of a possible zoonotic, emerging infectious disease. I conclude by discussing my results and the implications which these might have.

Cerebrospinal fluid (CSF) fills a system of cavities at the centre of the brain, known as ventricles, and the subarachnoid space surrounding the brain and the spinal cord. In addition, CSF is in free communication with the interstitial fluid of the brain tissue. Disturbances in CSF dynamics can lead to diseases that cause severe brain damage or even death. So-called infusion tests are frequently performed in the diagnosis of such diseases. In this type of test, changes in average CSF pressure are related to changes in CSF volume through infusion of known volumes of additional fluid. Traditionally, infusion tests are analysed with single compartment models, which treat all CSF as part of one compartment and balance fluid inflow, outflow and storage through a single ordinary differential equation. Poroelastic models of the brain, on the other hand, have been used to simulate spatial changes with disease, particularly of the ventricle size, on larger time scales of days, weeks or months. Wirth and Sobey (2008) developed a two-fluid poroelastic model of the brain in which CSF pressure pulsations are linked to arterial blood pressure pulsations. In this thesis, this model is developed further and simulation results are compared to clinical data. At first, the functional form of the compliance, which governs the storage of CSF in single compartment models, is examined by comparison of two different compliance models with clinical data. The derivations of a single-fluid and a two-fluid poroelastic model of the brain in spherical symmetry are laid out in detail and some of the parameters are related to the compliance functions considered earlier. The finite element implementation of the two-fluid model is described and finally simulation results of the average CSF pressure response and the pressure pulsations are compared to clinical data.

Linear digital signal processing based around linear, time-invariant systems theory finds substantial application in speech processing. The linear acoustic source-filter theory of speech production provides ready biomechanical justification for using linear techniques. Nonetheless, biomechanical studies surveyed in this thesis display significant nonlinearity and non-Gaussinity, casting doubt on the linear model of speech production. In order therefore to test the appropriateness of linear systems assumptions for speech production, surrogate data techniques can be used. This study uncovers systematic flaws in the design and use of exiting surrogate data techniques, and, by making novel improvements, develops a more reliable technique. Collating the largest set of speech signals to-date compatible with this new technique, this study next demonstrates that the linear assumptions are not appropriate for all speech signals. Detailed analysis shows that while vowel production from healthy subjects cannot be explained within the linear assumptions, consonants can. Linear assumptions also fail for most vowel production by pathological subjects with voice disorders. Combining this new empirical evidence with information from biomechanical studies concludes that the most parsimonious model for speech production, explaining all these findings in one unified set of mathematical assumptions, is a stochastic nonlinear, non-Gaussian model, which subsumes both Gaussian linear and deterministic nonlinear models. As a case study, to demonstrate the engineering value of nonlinear signal processing techniques based upon the proposed biomechanically-informed, unified model, the study investigates the biomedical engineering application of disordered voice measurement. A new state space recurrence measure is devised and combined with an existing measure of the fractal scaling properties of stochastic signals. Using a simple pattern classifier these two measures outperform all combinations of linear methods for the detection of voice disorders on a large database of pathological and healthy vowels, making explicit the effectiveness of such biomechanically-informed, nonlinear signal processing techniques.