Bio-fuel cells are driven by diverse and abundant bio-fuels and biological catalysts. The production/consumption cycle of bio-fuels is considered to be carbon neutral and, in principle, more sustainable than that of conventional fuel cells. The cost benefits over traditional precious-metal catalysts, and the mild operating conditions represent further advantages. It is important that mathematical models are developed to reduce the burden on laboratory based testing and accelerate the development of practical systems. In this study, recent key developments in bio-fuel cell technology are reviewed and two different approaches to modelling biofuel cells are presented, a detailed physics-based approach, and a data-driven regression model. The current scientific and engineering challenges involved in developing practical bio-fuel cell systems are described, particularly in relation to a fundamental understanding of the reaction environment, the performance and stability requirements, modularity and scalability. New materials and methods for the immobilization of enzymes and mediators on electrodes are examined, in relation to performance characteristics and stability. Fuels, mediators and enzymes used (anode and cathode), as well as the cell configurations employed are discussed. New developments in microbial fuel cell technologies are reviewed in the context of fuel sources, electron transfer mechanisms, anode materials and enhanced O2 reduction. Multi-dimensional steady-state and dynamic models of two enzymatic glucose/air fuel cells are presented. Detailed mass and charge balances are combined with a model for the reaction mechanism in the electrodes. The models are validated against experimental results. The dynamic performance under different cell voltages is simulated and the evolution of the system is described. Parametric studies are performed to investigate the effect of various operating conditions. A data-driven model, based on a reduced-basis form of Gaussian process regression, is also presented and tested. The improved computational efficiency of data-driven models makes them better candidates for modelling large complex systems.

Multiscale mathematical modelling is one of the major driving forces behind the systems biology revolution. The inherently interdisciplinary nature of its study and the multiple spatial and temporal scales which characterise its dynamics make cell migration an ideal candidate for a systems biology approach. Due to its ease of analysis and its compatibility with the type of data available, phenomenological continuum modelling has long been the default framework adopted by the cell migration modelling community. However, in recent years, with increased computational power, complex, discrete, cell-level models, able to capture the detailed dynamics of experimental systems, have become more prevalent. These two modelling paradigms have complementary advantages and disadvantages. The challenge now is to combine these two seemingly disparate modelling regimes in order to exploit the benefits offered by each in a comprehensive, multiscale equivalence framework for modelling cell migration. The main aim of this thesis is to begin with an on-lattice, individual-based model and derive a continuum, population-based model which is equivalent to it in certain limits. For simple models this is relatively easy to achieve: beginning with a one-dimensional, discrete model of cell migration on a regular lattice we derive a partial differential equation for the evolution of cell density on the same domain. We are also able to simply incorporate various signal sensing dynamics into our fledgling equivalence framework. However, as we begin to incorporate more complex model attributes such as cell proliferation/death, signalling dynamics and domain growth we find that deriving an equivalent continuum model requires some innovative mathematics. The same is true when considering a non-uniform domain discretisation in the one-dimensional model and when determining appropriate domain discretisations in higher dimensions. Higher-dimensional simulations of individual-based models bring with them their own computational challenges. Increased lattice sites in order to maintain spatial resolution and increased cell numbers in order to maintain consistent densities lead to dramatic reductions in simulation speeds. We consider a variety of methods to increase the efficiency of our simulations and derive novel acceleration techniques which can be applied to general reaction systems but are especially useful for our spatially extended cell migration algorithms. The incorporation of domain growth in higher dimensions is the final hurdle we clear on our way to constructing a complex discrete-continuum modelling framework capable of representing signal-mediated cell migration on growing (possibly non-standard) domains in multiple dimensions.

Centrosomin (Cnn) is an integral centrosomal protein in Drosophila with orthologues in several species, including humans. The human orthologue of Cnn is required for brain development with Cnn hypothesised to play a similar role in Drosophila. Control of Cnn incorporation into centrosomes is crucial for controlling asymmetric division in certain types of Drosophila stem cells. FRAP experiments on Cnn show that Cnn recovers in a pe- culiar fashion, which suggest that Cnn may be incorporated closest to the centrioles and then spread radially outward, either diffusively or ad- vectively. The aim of this thesis is to understand the mechanism of Cnn incorporation into the Drosophila centrosomes, to determine the mode of transport of the incorporated Cnn, and to obtain parameter estimates for transport and biochemical reactions. A crucial unknown in the modelling process is the distribution of Cnn receptors. We begin by constructing coupled partial differential equation models with either diffusion or advection as the mechanism for incorpo- rated Cnn transport. The simplest receptor distribution we begin with involves a spherical, infinitesimally thick, impermeable shell. We refine the diffusion models using the insights gained from comparing the model out- put with data (gathered during mitosis) and through careful assessment of the behaviour of the data. We show that a Gaussian receptor distribution is necessary to explain the Cnn FRAP data and that the data cannot be explained by other simpler receptor distributions. We predict the exact form of the receptor distribution through data fitting and present pre- liminary experimental results from our collaborators that suggest that a protein called DSpd2 may show a matching distribution. Not only does this provide strong experimental support for a key prediction from our model, but it also suggests that DSpd2 acts as a Cnn receptor. We also show using the mitosis FRAP data that Cnn does not exhibit appreciable radial movement during mitosis, which precludes the use of these data to distinguish between diffusive and advective transport of Cnn. We use long time Cnn FRAP data gathered during S-phase for this purpose. We fit the S-phase FRAP data using the DSpd2 profiles gath- ered for time points corresponding to the Cnn FRAP experiments. We also use data from FRAP experiments where colchicine is injected into the embryos to destroy microtubules (since microtubules are suspected to play a role in advective transport of Cnn). From the analysis of all these data we show that Cnn is transported in part by advection and in part by diffusion. Thus, we are able to provide the first mechanistic description of the Cnn incorporation process. Further, we estimate parameters from the model fitting and predict how some of the parameters may be altered as nuclei progress from S-phase to mitosis. We also generate testable predic- tions regarding the control of the Cnn incorporation process. We believe that this work will be useful to understand the role of Cnn incorporation in centrosome function, particularly in asymmetrically dividing stem cells.

Understanding the perfusion of blood and drugs in tumours is fundamental to foreseeing the efficacy of treatment regimes and predicting tumour growth. In particular, the dependence of these processes on the tumour vascular structure is poorly established. The objective of this thesis is to derive effective equations describing blood, and drug perfusion in vascular tumours, and specifically to determine the dependence of these on the tumour vascular structure. This dependence occurs through the interaction between two different length scales - that which characterizes the structure of the vascular network, and that which characterizes the tumour as a whole. Our method throughout is to use homogenization as a tool to evaluate this interaction. In Chapter 1 we introduce the problem. In Chapter 2 we develop a theoretical model to describe fluid flow in solid tumours through both the vasculature and the interstitium, at a number of length scales. Ultimately we homogenize over a network of capillaries to form a coupled porous medium model in terms of a vascular density. Whereas in Chapter 2 it is necessary to specify the vascular structure to derive the effective equations, in Chapter 3 we employ asymptotic homogenization through multiple scales to derive the coupled equations for an arbitrary periodic vascular network. In Chapter 4, we extend this analysis to account for advective and diffusive transport of anticancer drugs delivered intravenously; we consider a range of reaction properties in the interstitium and boundary conditions on the vascular wall. The models derived in Chapters 2–4 could be applied to any drug type and treatment regime; to demonstrate their potential, we simulate the delivery of vinblastine in dorsal skinfold chambers in Chapter 5 and make quantitative predictions regarding the optimal treatment regime. In the final Chapter we summarize the main results and indicate directions for further work.

Biological tissue differs from other materials in many ways. Perhaps the most crucial difference is its ability to grow. Growth processes may give rise to stresses that exist in a body in the absence of applied loads and these are known as residual stresses. Residual stress is present in many biological systems and can have important consequences on the mechanical response of a body. Mathematical models of biological structures must therefore be able to capture accurately the effects of differential growth and residual stress, since greater understanding of the roles of these phenomena may have applications in many fields. In addition to residual stresses, biological structures often have a complex morphology. The theory of 3-D elasticity is analytically tractable in modelling mechanical properties in simple geometries such as a cylinder. On the other hand, rod theory is well-suited for geometrically-complex deformations, but is unable to account for residual stress. In this thesis, we aim to develop a map between the two frameworks. Firstly, we use 3-D elasticity to determine effective mechanical properties of a growing cylinder and map them into an effective rod. Secondly, we consider a growing filament embedded in an elastic foundation. Here, we estimate the degree of transverse reinforcement the foundation confers on the filament in terms of its material properties. Finally, to gain a greater understanding of the role of residual stress in biological structures, we consider a case study: the chameleon's tongue. In particular we consider the role of residual stress and anisotropy in aiding the rapid projection of the tongue during prey capture. We construct a mechanical model of the tongue and use it to investigate a proposed mechanism of projection by means of an energy balance argument.

Motility of flagellated bacteria has been a topic of increasing scientific interest over the past decades, attracting the attention of mathematicians, physicists, biologists and engineers alike. Bacteria and other micro-organisms cause substantial damage through biofilm growth on submerged interfaces in water cooling systems, ship hulls and medical implants. This gives social and economic motivations for learning about how micro-organisms swim and behave in different environments. Fluid flows on such small scales are dominated by viscosity and therefore behave differently from the inertia-dominated flows that we are more familiar with, making bacterial motility a physically intriguing phenomenon to study as well. We use the boundary element method (BEM) to simulate the motion of singly flagellated bacteria in a viscous, Newtonian fluid. One of our main objectives is to investigate the influence of external surfaces on swimming behaviour. We show that the precise shape of the cell body and flagellum can be important for determining boundary behaviour, in particular, whether bacteria are attracted or repelled from surfaces. Furthermore, we investigate the types of motion that may arise between two parallel plates and in rectangular channels of fluid and show how these relate to the plane boundary interactions. As an extension to original models of flagellar propulsion in bacteria that assume a rotation of the rigid helical flagellum about an axis fixed relative to the cell body, we consider flexibility of the bacterial hook connecting the aforementioned parts of the swimmer. This is motivated by evidence that the hook is much more flexible than the rest of the flagellum, which we therefore treat as a rigid structure. Elastic dynamics of the hook are modelled using the equations for a Kirchhoff rod. In some regimes, the dynamics are well described by a rigid hook model but we find the possibility of additional modes of behaviour.

Angiogenesis, the process by which new vessels form from existing ones, is a key event in the development of a large and malignant vascularised tumour. A rapid expansion in in vivo and in vitro angiogenesis research in recent years has led to increased knowledge about the processes underlying angiogenesis and to promising steps forward in the development of anti-angiogenic therapies for the treatment of various cancers. However, substantial gaps in knowledge persist and the development of effective treatments remains a major challenge. In this thesis we study tumour-induced angiogenesis within the context of a highly controllable experimental environment: the cornea micropocket assay. Using a multidisciplinary approach that combines experiments, image processing and analysis, and mathematical and computational modelling, we aim to provide mechanistic insight into the action of two angiogenic factors which are known to play central roles during tumour-induced angiogenesis: vascular endothelial growth factor A (VEGF-A) and basic fibroblast growth factor (bFGF). Image analysis techniques are used to extract quantitative data, which are both spatially and temporally resolved, from experimental images. These data are then used to develop and parametrise mathematical models describing the evolution of the corneal vasculature in response to both VEGF-A and bFGF. The first models developed in this thesis are one-dimensional continuum models of angiogenesis in which VEGF-A and/or bFGF are released from a pellet implanted into a mouse cornea. We also use an object-oriented framework, designed to facilitate the re-use and extensibility of hybrid multiscale models of angiogenesis and vascular tumour growth, to develop a complementary three-dimensional hybrid model of the same system. The hybrid model incorporates a new non-local cell sensing model which facilitates the formation of well-perfused and functional vascular networks in three dimensions. The continuum models are used to assess the utility of the cornea micropocket assay as a quantitative assay for angiogenesis, to characterise proposed synergies between VEGF-A and bFGF, and to generate experimentally testable predictions regarding the effect of anti-VEGF-A therapies on bFGF-induced angiogenesis. Meanwhile, the hybrid model is used to provide context for the comparison that is drawn between the continuum models and the data, to study the relative distributions of perfused and unperfused vessels in the evolving neovasculature, and to investigate the impact of tip cell sensing dysregulation on the angiogenic response in the cornea. We have found that by exploiting a close link with quantitative data we have been able to extend the predictive and hypothesis-testing capabilities of our models. As such, this thesis demonstrates the potential for integrating mathematical modelling with image analysis techniques to increase insight into the mechanisms underlying angiogenesis.

Biological invasions are devastating ecosystems and economies world-wide, while many native species' survival depends on their ability to track climate change. Characterising the spread of biological populations is therefore of utmost importance, and can be studied with spatially explicit, discrete-time integro-difference equations (IDEs), which reflect numerous species' processes of demography and dispersal. While spatial variation has often been ignored when implementing IDE models, real landscapes are rarely spatially uniform and environmental variation is crucial in determining biological spread. To address this, we use novel methods to characterise population spread in heterogeneous landscapes. Asymptotic analysis is used for highly fragmented landscapes, where habitat patches are isolated and smaller than the dispersal scale, and in landscapes with low environmental variation, where the ecological parameters vary by no more than a small factor from their mean values. We find that the choice of dispersal kernel determines the effect of landscape structure on spreading speed, indicating that accurately fitting a kernel to data is important in accurately predicting speed. For the low-variation case, the spreading speeds in the heterogeneous and homogeneous landscapes differ by ϵ², where ϵ governs the degree of variation, suggesting that in many cases, a simpler homogeneous model gives similar spread rates. For irregular landscapes, analytical methods become intractable and numerical simulation is needed to predict spread. Accurate simulation requires high spatial resolution, which, using existing techniques, requires prohibitive amounts of computational resources (RAM, CPU etc). We overcome this by developing and implementing a novel algorithm that uses adaptive mesh refinement. The approximations and simulation algorithm produce accurate results, with the adaptive algorithm providing large improvements in efficiency without significant losses of accuracy compared to non-adaptive simulations. Hence, the adaptive algorithm enables faster simulation at previously unfeasible scales and resolutions, permitting novel areas of scientific research in species spread modelling.

Filamentous fungi are important in an enormous variety of ways to our life, with examples ranging from bioremediation, through the food and drinks industry to human health. These organisms can form huge networks stretching metres and even kilometres. However, their mode of growth is by the extension of individual hyphal tips only a few microns in diameter. Tip growth is mediated by the incorporation of new wall building materials at the soft apex. Just how this process is controlled (in fungi and in cell elongation in other organisms) has been the subject of intense study over many years and has attracted considerable attention from mathematical modellers. In this thesis, we consider mathematical models of fungal tip growth that can be classified as either geometrical or biomechanical. In every model we examine, a 2-D axisymmetric semihemisphere-like curve represents half the medial section of fungal tip geometry. A geometrical model for the role of the Spitzenkorper in the tip growth was proposed by Bartnicki-Garcia et al (1989), where a number of problems with the mathematical derivation were pointed out by Koch (2001). A suggestion is given as an attempt to revise the derivation by introducing a relationship between arc length of a growing tip, deposition of wall-building materials and tip curvature. We also consider two types of geometrical models as proposed by Goriely et al (2005). The first type considers a relationship between the longitudinal curvature and the function used to model deposition of wall-building materials. For these types of models, a generalized formulae for the tip shape is introduced, which allows localization of deposition of wall-building materials to be examined. The second type considers a relationship between longitudinal and latitudinal curvatures and the function used to model deposition of wall-building materials. For these types of models, a new formulation of the function used to model deposition of wall-building materials is introduced. Finally, a biomechanical model as proposed by Goriely et al (2010). Varying arc length of the stretchable region on the tip suggests differences in geometry of tip shape and the effective pressure profile. The hypothesis of orthogonal growth is done by focusing only on the apex of a "germ tube". Following that, it suggests that material points on the tip appear to move in a direction perpendicular to the tip either when surface friction is increased or decreased.

Motivated by applications in tissue engineering, this thesis is concerned with the flow through and around a free-moving porous tissue construct (TC) within a high-aspect-ratio vessel (HARV) bioreactor. We formalise and extend various results for flow within a Hele-Shaw cell containing a porous obstacle. We also consider the impact of the flow on related nutrient transport problems. The HARV bioreactor is a cylinder with circular cross-section which rotates about its axis at a constant rate, and is filled with a nutrient-rich culture medium. The porous TC is modelled as a rigid porous cylinder with circular cross-section and is fully saturated with the fluid. We formulate the flow problem for a porous TC (governed by Darcy's equations) within a HARV bioreactor (governed by the Navier-Stokes equations). We couple the two regions via appropriate interfacial conditions which are derived by consideration of the intricate boundary-layer structure close to the TC surface. By exploiting various small parameters, we simplify the system of equations by performing an asymptotic analysis, and investigate the resulting system for the flow due to a prescribed TC motion. The motion of the TC is determined by analysis of the force and torque acting upon it, and the resulting equations of motion (which are coupled to the flow) are investigated. The short-time TC behaviour is periodic, but we are able to study the long-time drift from this periodic solution by considering the effect of inertia using a multiple-scale analysis. We find that, contrary to received wisdom, inertia affects TC drift on a similar timescale to tissue growth. Finally, we consider the advection of nutrient through the bioreactor and TC, and investigate the problem of nutrient advection-diffusion for a simplified model involving nutrient uptake.