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.

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In order to understand biochemical networks we need to know not only how their parts work but also how they interact with each other. The goal of systems biology is to look at biological systems as a whole to understand how interactions of the parts can give rise to complex dynamics. In order to do this efficiently, new techniques have to be developed. This work shows how tools from mathematics are suitable to study problems in systems biology such as modeling, dynamics prediction, reverse engineering and many others. The advantage of using mathematical tools is that there is a large number of theory, algorithms and software available. This work focuses on how algebra can contribute to answer questions arising from systems biology. / Ph. D.

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.

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.

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.