Multiphase modelling of tissue growth in dynamic culture conditions

In this thesis, a series of mathematical models suitable for describing biological tissue growth are developed. The motivation for this work is a bioreactor system which provides perfusion and compressive mechanical stimulation to a cell-seeded scaffold; however, the formulation is sufficiently general to be applied to a vast range of tissue engineering applications. Our models are used to investigate the influence of (i) cell-cell and cell-scaffold interactions, and (ii) the mechanical environment, on tissue growth. In the first part of the thesis, we extend a model due to Franks (2002) (in which the cell and culture medium phases are represented by viscous fluids) by including perfusion and coupling the cells' response to their environment. Specifically, we consider the effect of the cell density and pressure on tissue growth. We analyse the model using analytic and numerical techniques; numerical simulations suggest that comparison of construct morphology in the presence and absence of perfusion provides a means to identify the dominant regulatory growth stimulus. The solid characteristics of the construct and interactions between the cells and scaffold are necessarily neglected in the two phase model. Guided by this, we develop more complex three phase models. Using numerical simulations, the influence of cell-cell and cell-scaffold interactions is investigated and less porous scaffolds are shown to improve control over cell behaviour. We use the model to compare the cells' response to different regulatory stimuli, including flow-induced shear stress. Our results suggest that uniform initial cell seeding and stimulating cell movement are crucial in maintaining the mechanical integrity of tissue constructs. We also study the effect of scaffold compression on the mechanical environment of the cells contained within, developing both a classical Biot formulation and a multiphase model. We demonstrate that the bioreactor geometry introduces significant spatial variation in the mechanical stimuli relevant to tissue growth and that such considerations will play a key role in comprehensive models of mechanotransduction-affected growth

Multiscale analysis of pattern formation via intercellular signalling

Lateral inhibition, a juxtacrine signalling mechanism by which a cell adopting a particular fate inhibits neighbouring cells from doing likewise, has been shown to be a robust mechanism for the formation of fine-grained spatial patterns (in which adjacent cells in developing tissues diverge to achieve contrasting states of differentiation), provided that there is sufficiently strong feedback. The fine-grained nature of these patterns poses problems for analysis via traditional continuum methods since these require that significant variation takes place only over lengthscales much larger than an individual cell and such systems have therefore been investigated primarily using discrete methods. Here, however, we apply a multiscale method to derive systematically a continuum model from the discrete Delta-Notch signalling model of Collier \emph{et al.} (Pattern formation by lateral inhibition with feedback: a mathematical model of Delta-Notch intercellular signalling, \emph{J. Theor. Biol.}, 183, 1996, 429--446) under particular assumptions on the parameters, which we use to analyse the generation of fine-grained patterns. We show that, on the macroscale, the contact-dependent juxtacrine signalling interaction manifests itself as linear diffusion, motivating the use of reaction-diffusion-based models for such cell-signalling systems. We also analyse the travelling-wave behaviour of our system, obtaining good quantitative agreement with the discrete system

The isolation of spatial patterning modes in a mathematical model of juxtacrine cell signalling

Juxtacrine signalling mechanisms are known to be crucial in tissue and organ development, leading to spatial patterns in gene expression. We investigate the patterning behaviour of a discrete model of juxtacrine cell signalling due to Owen \& Sherratt (\emph{Math. Biosci.}, 1998, {\bf 153}(2):125--150) in which ligand molecules, unoccupied receptors and bound ligand-receptor complexes are modelled. Feedback between the ligand and receptor production and the level of bound receptors is incorporated. By isolating two parameters associated with the feedback strength and employing numerical simulation, linear stability and bifurcation analysis, the pattern-forming behaviour of the model is analysed under regimes corresponding to lateral inhibition and induction. Linear analysis of this model fails to capture the patterning behaviour exhibited in numerical simulations. Via bifurcation analysis we show that, since the majority of periodic patterns fold subcritically from the homogeneous steady state, a wide variety of stable patterns exists at a given parameter set, providing an explanation for this failure. The dominant pattern is isolated via numerical simulation. Additionally, by sampling patterns of non-integer wavelength on a discrete mesh, we highlight a disparity between the continuous and discrete representations of signalling mechanisms: in the continuous case, patterns of arbitrary wavelength are possible, while sampling such patterns on a discrete mesh leads to longer wavelength harmonics being selected where the wavelength is rational; in the irrational case, the resulting aperiodic patterns exhibit `local periodicity', being constructed from distorted stable shorter-wavelength patterns. This feature is consistent with experimentally observed patterns, which typically display approximate short-range periodicity with defects

Multiphase modelling of tissue growth in dynamic culture conditions

In this thesis, a series of mathematical models suitable for describing biological tissue growth are developed. The motivation for this work is a bioreactor system which provides perfusion and compressive mechanical stimulation to a cell-seeded scaffold; however, the formulation is sufficiently general to be applied to a vast range of tissue engineering applications. Our models are used to investigate the influence of (i) cell-cell and cell-scaffold interactions, and (ii) the mechanical environment, on tissue growth. In the first part of the thesis, we extend a model due to Franks (2002) (in which the cell and culture medium phases are represented by viscous fluids) by including perfusion and coupling the cells' response to their environment. Specifically, we consider the effect of the cell density and pressure on tissue growth. We analyse the model using analytic and numerical techniques; numerical simulations suggest that comparison of construct morphology in the presence and absence of perfusion provides a means to identify the dominant regulatory growth stimulus. The solid characteristics of the construct and interactions between the cells and scaffold are necessarily neglected in the two phase model. Guided by this, we develop more complex three phase models. Using numerical simulations, the influence of cell-cell and cell-scaffold interactions is investigated and less porous scaffolds are shown to improve control over cell behaviour. We use the model to compare the cells' response to different regulatory stimuli, including flow-induced shear stress. Our results suggest that uniform initial cell seeding and stimulating cell movement are crucial in maintaining the mechanical integrity of tissue constructs. We also study the effect of scaffold compression on the mechanical environment of the cells contained within, developing both a classical Biot formulation and a multiphase model. We demonstrate that the bioreactor geometry introduces significant spatial variation in the mechanical stimuli relevant to tissue growth and that such considerations will play a key role in comprehensive models of mechanotransduction-affected growth

Spreading dynamics on spatially constrained complex brain networks

The study of dynamical systems defined on complex networks provides a natural framework with which to investigate myriad features of neural dynamics, and has been widely undertaken. Typically, however, networks employed in theoretical studies bear little relation to the spatial embedding or connectivity of the neural networks that they attempt to replicate. Here, we employ detailed neuroimaging data to define a network whose spatial embedding represents accurately the folded structure of the cortical surface of a rat and investigate the propagation of activity over this network under simple spreading and connectivity rules. By comparison with standard network models with the same coarse statistics, we show that the cortical geometry influences profoundly the speed propagation of activation through the network. Our conclusions are of high relevance to the theoretical modelling of epileptic seizure events, and indicate that such studies which omit physiological network structure risk simplifying the dynamics in a potentially significant way

Spreading dynamics on spatially constrained complex brain networks

The study of dynamical systems defined on complex networks provides a natural framework with which to investigate myriad features of neural dynamics and has been widely undertaken. Typically, however, networks employed in theoretical studies bear little relation to the spatial embedding or connectivity of the neural networks that they attempt to replicate. Here, we employ detailed neuroimaging data to define a network whose spatial embedding represents accurately the folded structure of the cortical surface of a rat brain and investigate the propagation of activity over this network under simple spreading and connectivity rules. By comparison with standard network models with the same coarse statistics, we show that the cortical geometry influences profoundly the speed of propagation of activation through the network. Our conclusions are of high relevance to the theoretical modelling of epileptic seizure events and indicate that such studies which omit physiological network structure risk simplifying the dynamics in a potentially significant way

Effective equations governing an active poroelastic medium

In this work we consider the spatial homogenization of a coupled transport and fluid-structure interaction model, to the end of deriving a system of effective equations describing the flow, elastic deformation, and transport in an active poroelastic medium. The `active' nature of the material results from a morphoelastic response to a chemical stimulant, in which the growth timescale is strongly separated from other elastic timescales. The resulting effective model is broadly relevant to the study of biological tissue growth, geophysical flows (e.g. swelling in coals and clays) and a wide range of industrial applications (e.g. absorbant hygiene products). The key contribution of this work is the derivation of a system of homogenized partial differential equations describing macroscale growth, coupled to transport of solute, that explicitly incorporates details of the structure and dynamics of the microscopic system, and, moreover, admits finite growth and deformation at the pore-scale. The resulting macroscale model comprises a Biot-type system, augmented with additional terms pertaining to growth, coupled to an advection-reaction-diffusion equation. The resultant system of effective equations is then compared to other recent models under a selection of appropriate simplifying asymptotic limits

Cascading failures in networks of heterogeneous node behavior

Variability in the dynamical function of nodes comprising a complex network impacts upon cascading failures that can compromise the network's ability to operate. Node types correspond to sources, sinks or passive conduits of a current ow, applicable to renewable electrical power micro-grids containing a variable number of intermittently operating generators and consumers of power. The resilience to cascading failures of ensembles of synthetic networks with di_erent topology is examined as a function of the edge current carrying capacity and mix of node types, together with exemplar real-world networks. Whilst a network with homogeneous node type can be resilient to failure, one with identical topology but heterogeneous node function can be strongly susceptible to failure. For networks with similar numbers of sources, sinks and passive nodes the mean resilience decreases as networks become more disordered. Nevertheless all network topologies have enhanced regions of resilience, accessible by manipulation of node composition and functionality

Microstructural influences on growth and transport in biological tissue—a multiscale description

The detailed understanding of growth and transport dynamics within biological tissue is made particularly challenging by the complex and multiscale nature of this medium. For this reason so-called effective descriptions are frequently sought. These offer coarse-scale models that still accommodate aspects of microscale dynamics. When considering tissue growth, such formulations must accommodate the continuous growth and remodeling processes that occur in response to environmental cues. As a model system for investigating relevant phenomena, in this chapter we consider nutrient-limited growth of a porous medium (with broad application to vascularized tumor growth). Using asymptotic homogenization we derive the macroscale equations that describe a ‘double porous medium’ whose flow is influenced by both the tissue microstructure and growth that occurs in response to nutrient transport governed by an advection–reaction equation. The coupled flow and transport dynamics are demonstrated by numerical experiments indicating the influence of microscale structure and transport phenomena on the macroscale dynamics. The importance of slip, tortuosity, and of nutrient-limited growth are considered

Stability analysis of electrical microgrids and their control systems

The drive towards renewable energy generation is causing fundamental changes in both the structure and dynamics of power grids. Their topology is becoming increasingly decentralised due to distributed, embedded generation, and the emergence of microgrids. Grid dynamics are being impacted by decreasing inertia, as conventional generators with massive spinning cores are replaced by DC renewable sources. This leads to a risk of destabilisation and places an upper limit on the volume of renewable power sources that can be installed. A wide variety of different control schemes have been proposed to overcome this problem. Such schemes fall into two broad categories: so-called 'grid-following' controllers that seek to match output AC power with grid frequency, and 'grid-forming' systems that seek to boost grid stability. The latter frequently work by providing synthetic inertia, enabling DC renewable sources to emulate conventional generators. This paper uses the master stability function methodology to analyse the stability of synchrony in microgrids of arbitrary size and containing arbitrary control systems. This approach provides a powerful and computationally efficient framework in which to benchmark the impact of any number of renewable sources on grid stability and thereby to support microgrid design strategies. The method is demonstrated by computing stability bounds for two different grid-forming systems, providing bounds on the feasible number of generators that can be accommodated. In addition, we contrast our results with predictions from a simplistic but widely-used phase oscillator model, finding that such descriptions significantly overestimate the grid stability properties