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Chaste is a software library and set of test suites for modelling phenomena in the domain of computational biology. Specifically these the modelling problems arise in the fields of cancer modelling, cardiac simulation and soft-tissue engineering (Chaste stands for ‘Cancer, heart and soft-tissue environment’). It is released under the LGPL 2.1 licence. Chaste has been developed using agile programming methods. The project began 5 years when it was reasoned that the modelling of a variety of physio-logical phenomena requires both a generic mathematical modelling framework, and a generic computational/simulation framework. The Chaste project was formed as part of the larger Integrative Biology e-Science (IB) Project ([14]), an inter-institutional project aimed at developing a suitable IT infrastructure to support physiome-level computational modelling, with a focus on cardiac and cancer modelling.

Simulation of cardiac electrical activity using the bidomain equations can be a massively computationally demanding problem. This study provides a comprehensive guide to numerical bidomain modelling. Each component of bidomain simulations-discretisation, ODE-solution, linear system solution, and parallelisation-is discussed, and previously-used methods are reviewed, new methods are proposed, and issues which cause particular difficulty are highlighted. Particular attention is paid t

The generalized minimal residual (GMRES) method is a common choice for solving the large nonsymmetric linear systems that arise when numerically computing solutions of incompressible nonlinear elasticity problems using the finite element method. Analytic results on the performance of GMRES are available on linear problems such as linear elasticity or Stokes' flow (where the matrices in the corresponding linear systems are symmetric), or on the nonlinear problem of the Navier-Stokes flow (where the matrix is block-symmetric/block-skew-symmetric); however, there has been very little investigation into the GMRES performance in incompressi

Soft tissue deformation is often modelled using incompressible non-linear elasticity, with solutions computed using the finite element method. There are a range of options available when using the finite element method, in particular the polynomial degree of the basis functions used for interpolating position and pressure, and the type of element making LIP the mesh. The effect of these choices oil the accuracy of the computed Solution is investigated, using a selection of model probl

When modelling tissue-level cardiac electrophysiology, a continuum approximation to the discrete cell-level equations, known as the bidomain equations, is often used to maintain computational tractability. The bidomain equations are derived from the discrete equations using a mathematical technique known as homogenisation. As part of this derivation conductivity tensors are specified for use in the continuum model. Analysing the derivation of the bidomain equations allows us to investigate how microstructure, in particular gap junctions that electrically connect cells, affect tissue-level conductivity properties and model solutions. We perform two distinct but related strands of investigation in this thesis. In the first, we consider the effect of including gap junctions on the results of both discrete and continuum simulations, and identify when the continuum model fails to be a good approximation to the discrete model. Secondly, we perform a comprehensive study into how cell-level microstructure properties, such as cell shape, impact the homogenised conductivities to be used in a tissue-level continuum model. This will allow us to predict how the onset of a disease or a change in cellular microstructure will affect the propagation of action potentials. To do this, we first derive a modified version of the bidomain equations that explicitly takes gap junctions into account. We then derive analytic solutions for the homogenised conductivity tensors on a simplified two-dimensional geometry and find that diseased gap junctions have a large impact on the results of homogenisation. On this same geometry we compare the results of discrete and continuum simulations and find a significant discrepancy between model conduction velocities when we introduce gap junctions with lower coupling strength, or when we consider elongated cells. From this, we conclude that the bidomain equations are less likely to give an accurate representation of the underlying discrete system when modelling diseased states whose symptoms include reduced gap junction coupling or an increase in myocyte length. We then use a more realistic two-dimensional geometry and numerically approximate the homogenised conductivity tensors on this geometry. We discover that the packing of cells has a substantial effect on conduction, with a brick-wall geometry particularly beneficial for fast propagation, and that gap junctions also have a large effect on conduction. Finally, we consider a three-dimensional cellular geometry and show that the effect of changing gap junction properties is different when compared to two dimensions, and discover that the anisotropy ratios of the tissue are highly sensitive to changes in gap junction parameters. Overall, we conclude that gap junctions and cell structure have a large effect on discrete and continuum model results, and on homogenised conductivity calculations in tissue-level cardiac electrophysiology.

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. For this reason cardiac motion abnormalities are often associated with severe pathologies. Clinical non-invasive techniques can assess this fundamental connection between motion aberrant behaviour and pathology only to a certain extent. Computational models of cardiac motion would thus be of great help in linking local to global motion abnormalities and to pathology. No current model of the heart though is able yet to realistically simulate motion patterns in the healthy or diseased heart and therefore prediction of clinically relevant parameters such as ejection fraction, stroke volume, wall thickening and wall motion are not yet totally reliable. It is well known that cardiac tissue microstructure, that is cardiac cells organisation into fibres and sheets, has an important role in cardiac motion, as suggested by experimental and computational research, but current models of cardiac function, although already embedding structural information in the models equations, are not yet able to correctly link structure to function and therefore predict realistic cardiac motion. The hypothesis underlying this project is that a more realistic representation of tissue structure within an electromechanical model of the heart, with fibre and sheet orientation extracted from data rather than mathematically defined, together with a more careful definition of tissue material properties, would better take into account the high heterogeneity of tissue structure, thus improving the predictive power of the model. The aim of the project is to investigate how different settings of tissue structural arrangement affect the motion prediction of an electromechanical model applied to murine left ventricular geometries obtained from magnetic resonance imaging. The project relies on the integration of cardiac imaging data and mathematical modelling in the belief that realistic models of cardiac function need to reach the best compromise between the level of modelling detail and the amount and quality of information that can be actually obtained from data and used either to instruct or validate such models.