Cardiac motion is a highly complex and integrated process of vital importance as it sustains the primary function of the heart, that is pumping blood. Cardiac tissue microstructure, in particular the alignment of myocytes (also referred to as fibre direction) and their lateral organisation into laminae (or sheets), has been shown by both experimental and computational research to play an important role in the determination of cardiac motion patterns. However, current models of cardiac electromechanics, although already embedding structural information in the models equations, are not yet able to fully reproduce the connection between structural dynamics and cardiac deformation. The aim of this thesis was to develop an electromechanical modelling framework to investigate the impact of tissue structure on cardiac motion, focussing on left ventricular contraction in rat. The computational studies carried out were complemented with a preliminary validation study based on experimental data of tissue structure rearrangement during contraction from diffusion tensor MRI.

The present thesis explored ethological and geographical approaches for the investigation of vector-borne parasites. In the first part, the role of associative learning on vector preferences for hosts was investigated through a comprehensive series of behavioural experiments using the vector of dengue and yellow fever diseases, the mosquito Aedes aegypti. To this end, the possibility that the mosquitoes were able to associate unconditional stimuli with particular odours and visual patterns to which they were responsive was explored, but no evidence supporting the hypothesis that associative learning abilities are present in adults of this species was found. A critical review of the literature on learning in mosquitoes conducted afterward allowed the reinterpretation of findings in the field, narrowing the scope of evidence suggesting the existence of these cognitive abilities in some species. In the second part of the thesis, the distribution and evolution of mosquito-borne viruses was investigated with the use of geo-coded environmental data and spatial statistics. Initially, the eco-climates associated with the distribution of Japanese encephalitis virus were described and modelled, allowing the production of a worldwide predictive map defining the probability of each region to develop this disease in the future. Predominating amongst those areas shown to be under high risk were the equatorial regions of South America and Africa. The methodology used to infer such patterns – non-linear discriminant analysis – was subsequently explored with a number of simulations. Overall, differences in the choice of parameters required for the analysis were shown to lead to differences in the final outputs produced, basically in those cases where the environmental range for which predictions are generated is not rigorously limited. Finally, eco-climate surrogates for the evolution of the Japanese encephalitis serocomplex were investigated, but the current environmental distances between the viruses did not seem to be associated with the events leading to their speciation.

In this thesis we study three different modelling frameworks for biological systems of dispersal and combinations thereof. The three frameworks involved are individual-based models, group-level models in the form of partial differential equations (PDEs) and robot swarms. In the first two chapters of the thesis, we present ways of coupling individual based models with PDEs in so-called hybrid models, with the aim of achieving improved performance of simulations. Two classes of such hybrid models are discussed that allow an efficient simulation of multi-species systems of dispersal with reactions, but involve individual resolution for certain species and in certain parts of a computational domain if desired. We generally consider two types of example systems: bacterial chemotaxis and reaction-diffusion systems, and present results in the respective application area as well as general methods. The third chapter of this thesis introduces swarm robotic experiments as an additional tool to study systems of dispersal. In general, those experiments can be used to mimic animal behaviour and to study the impact of local interactions on the group-level dynamics. We concentrate on a target finding problem for groups of robots. We present how PDE descriptions can be adjusted to incorporate the finite turning times observed in the robotic system and that the adjusted models match well with experimental data. In the fourth and last chapter, we consider interactions between robots in the form of hard-sphere collisions and again derive adjusted PDE descriptions. We show that collisions have a significant impact on the speed with which the group spreads across a domain. Throughout these two chapters, we apply a combination of experiments, individual-based simulations and PDE descriptions to improve our understanding of interactions in systems of dispersal.

An increasingly important research area within the field of cardiac modelling is the development and study of methods of model-based parameter estimation from clinical measurements of cardiac function. This provides a powerful approach for the quantification of cardiac function, with the potential to ultimately lead to the improved stratification and treatment of individuals with pathological myocardial mechanics. In particular, the diastolic function (i.e., blood filling) of left ventricle (LV) is affected by its capacity for relaxation, or the decay in residual active tension (AT) whose inhibition limits the relaxation of the LV chamber, which in turn affects its compliance (or its reciprocal, stiffness). The clinical determination of these two factors, corresponding to the diastolic residual AT and passive constitutive parameters (stiffness) in the cardiac mechanical model, is thus essential for assessing LV diastolic function. However these parameters are difficult to be assessed in vivo, and the traditional criterion to diagnose diastolic dysfunction is subject to many limitations and controversies. In this context, the objective of this study is to develop model-based applicable methodologies to estimate in vivo, from 4D imaging measurements and LV cavity pressure recordings, these clinically relevant parameters (passive stiffness and active diastolic residual tension) in computational cardiac mechanical models, which enable the quantification of key clinical indices characterising cardiac diastolic dysfunction. Firstly, a sequential data assimilation framework has been developed, covering various types of existing Kalman filters, outlined in chapter 3. Based on these developments, chapter 4 demonstrates that the novel reduced-order unscented Kalman filter can accurately retrieve the homogeneous and regionally varying constitutive parameters from the synthetic noisy motion measurements. This work has been published in Xi et al. 2011a. Secondly, this thesis has investigated the development of methods that can be applied to clinical practise, which has, in turn, introduced additional difficulties and opportunities. This thesis has presented the first study, to our best knowledge, in literature estimating human constitutive parameters using clinical data, and demonstrated, for the first time, that while an end-diastolic MR measurement does not constrain the mechanical parameters uniquely, it does provide a potentially robust indicator of myocardial stiffness. This work has been published in Xi et al. 2011b. However, an unresolved issue in patients with diastolic dysfunction is that the estimation of myocardial stiffness cannot be decoupled from diastolic residual AT because of the impaired ventricular relaxation during diastole. To further address this problem, chapter 6 presents the first study to estimate diastolic parameters of the left ventricle (LV) from cine and tagged MRI measurements and LV cavity pressure recordings, separating the passive myocardial constitutive properties and diastolic residual AT. We apply this framework to three clinical cases, and the results show that the estimated constitutive parameters and residual active tension appear to be a promising candidate to delineate healthy and pathological cases. This work has been published in Xi et al. 2012a. Nevertheless, the need to invasively acquire LV pressure measurement limits the wide application of this approach. Chapter 7 addresses this issue by analysing the feasibility of using two kinds of non-invasively available pressure measurements for the purpose of inverse parameter estimation. The work has been submitted for publication in Xi et al. 2012b.

The importance of coronary microcirculatory perfusion is highlighted by the severe impact of microvascular diseases such as diabetes and hypertension on heart function. Recently, highly-detailed three-dimensional (3D) data on ex vivo coronary microvascular structure have become available. However, hemodynamic information in individual myocardial capillaries cannot yet be obtained using current in vivo imaging techniques. In this thesis, a novel data-driven modelling framework is developed to predict tissue-scale flow properties from discrete anatomical data, which can in future be used to aid interpretation of coarse-scale perfusion imaging data in healthy and diseased states. Mathematical models are parametrised by the 3D anatomical data set of Lee (2009) from the rat myocardium, and tested using flow measurements in two-dimensional rat mesentery networks. Firstly, algorithmic and statistical tools are developed to separate branching arterioles and venules from mesh-like capillaries, and then to extract geometrical properties of the 3D capillary network. The multi-scale asymptotic homogenisation approach of Shipley and Chapman (2010) is adapted to derive a continuum model of coronary capillary fluid transport incorporating a non-Newtonian viscosity term. Tissue-scale flow is captured by Darcy's Law whose coefficient, the permeability tensor, transmits the volume-averaged capillary-scale flow variations to the tissue-scale equation. This anisotropic permeability tensor is explicitly calculated by solving the capillary-scale fluid mechanics problem on synthetic, stochastically-generated periodic networks parametrised by the geometrical data statistics, and a thorough sensitivity analysis is conducted. Permeability variations across the myocardium are computed by parametrising synthetic networks with transmurally-dependent data statistics, enabling the hypothesis that subendocardial permeability is much higher in diastole to compensate for severely-reduced systolic blood flow to be tested. The continuum Darcy flow model is parametrised by purely structural information to provide tissue-scale perfusion metrics, with the hypothesis that this model is less sensitive and more reliably parametrised than an alternative, estimated discrete network flow solution.

A major problem in the quantitative analysis of AC voltammetric data has been the variance in results between laboratories, often resulting from a reliance on "heuristic" methods of parameter estimation that are strongly dependent on the choices of the operator. In this thesis, an automatic method for parameter estimation will be tested in the context of experiments involving electron-transfer processes in solution-phase. It will be shown that this automatic method produces parameter estimates consistent with those from other methods and the literature in the case of the ferri-/ferrocyanide couple, and is able to explain inconsistency in published values of the rate parameter for the ferrocene/ferrocenium couple. When a coupled homogeneous reaction is considered in a theoretical study, parameter recovery is achieved with a higher degree of accuracy when simulated data resulting from a high frequency AC voltammetry waveform are used. When surface-confined reactions are considered, heterogeneity in the rate constant and formal potential make parameter estimation more challenging. In the final study, a method for incorporating these "dispersion" effects into voltammetric simulations is presented, and for the first time, a quantitive theoretical study of the impact of dispersion on measured current is undertaken.

Marine populations worldwide are in decline due to anthropogenic effects. Spatial management via marine reserves may be an effective conservation method for many species, but the requisite theory is still underdeveloped. Integrodifference equation (IDE) models can be used to determine the critical domain size required for persistence and provide a modelling framework suitable for many marine populations. Here, we develop a novel spatially implicit approximation for the proportion of individuals lost outside the reserve areas which consistently outperforms the most common approximation. We examine how results using this approximation compare to the existing IDE results on the critical domain size for populations in a single reserve, in a network of reserves, in the presence of alongshore currents, and in structured populations. We find that the approximation consistently provides results which are in close agreement with those of an IDE model with the advantage of being simpler to convey to a biological audience while providing insights into the significance of certain model components. We also design a stochastic individual based model (IBM) to explore the probability of extinction for a population within a reserve area. We use our spatially implicit approximation to estimate the proportion of individuals which disperse outside the reserve area. We then use this approximation to obtain results on extinction using two different approaches, which we can compare to the baseline IBM; the first approach is based on the Central Limit Theorem and provides efficient simulation results, and the second modifies a simple Galton-Watson branching process to include loss outside the reserve area. We find that this spatially implicit approximation is also effective in obtaining results similar to those produced by the IBM in the presence of both demographic and environmental variability. Overall, this provides a set of complimentary methods for predicting the reserve area required to sustain a population in the presence of strong fishing pressure in the surrounding waters.

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.

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.