An efficient finite element method is developed to model the spreading of excitation in ventricular myocardium by treating the thin region of rapidly depolarizing tissue as a propagating wavefront. The model is used to investigate the excitation sequence in the full canine ventricular myocardium. The solution to an eikonal–curvature equation for excitation time is shown to satisfy a reaction–diffusion equation for the bidomain myocardial model at the wavefront, while the solution to an eikonal–diffusion equation approximately satisfies the reaction–diffusion equation in the vicinity of the wavefront. The features of these two eikonal equations are discussed. A Petrov–Galerkin finite element method with cubic Hermite elements is developed to solve the eikonal–diffusion equation. The oscillatory errors seen when using the Galerkin weighted residual method with high mesh Péclet numbers are avoided by supplementing the Galerkin weights with C⁰ continuous functions based on derivatives of the interpolation functions. The ratio of the Galerkin and supplementary weights is a function of the Péclet number such that, for one-dimensional propagation, the error in the solution is within a small constant factor of the optimal error achievable in the trial space. An additional noinflow boundary term is developed to prevent spurious excitation initiating on the boundary. The need for discretization in time is avoided by using a continuation method to gradually introduce the non-linear term of the governing equation. A small amount of artificial diffusion is sometimes necessary. Simulations of excitation are performed using a model of the anisotropic canine ventricular myocardium with 23.55 degrees of freedom for the dependent variable, and results are compared with reported experimental observations. When it was assumed that Purkinje fibres influence propagation only on the endocardial surface, excitation of the entire myocardium was completed in 56 ms. Altering material parameters to represent penetration of the Purkinje fibres beneath the left endocardial surface reduced the completion time to 48 ms. Modelling the effects of the laminar structure of myocardium by reducing the propagation speed by 40% in the direction normal to the layers delayed completion of excitation by only 4%.

In this thesis, the forward problem of electrocardiography is investigated from a cellular level through to potentials on the surface of the torso. This integrated modelling framework is based on three spatial scales. At the smallest spatial resolution, several cardiac cellular models are implemented that are used to represent the underlying cellular electrophysiology. A bidomain framework is used to couple multiple individual cells together and this provides a mathematical model of the myocardial tissue. The cardiac geometry is described using finite elements with high order cubic Hermite basis function interpolation. An anatomically based description of the fibrous laminar cardiac microstructure is then defined relative to the geometric mesh. Within the local element space of the cardiac finite elements, a fine collocation mesh is created on which the bidomain equations are solved. Each collocation point represents a continuum cell and contains a cellular model to describe the local active processes. This bidomain implementation works in multiple coordinate systems and over deforming domains, in addition to having the ability to spatially vary any parameter throughout the myocardium. On the largest spatial scale the passive torso regions surrounding the myocardium are modelled using a generalised Laplace equation to describe the potential field and current flows. The torso regions are discretised using either finite elements or boundary elements depending on the electrical properties of each region. The cardiac region is coupled to the surrounding torso through several methods. A traditional dipole source approach is implemented that creates equivalent cardiac sources through the summation of cellular dipoles. These dipoles are then placed within a homogeneous cardiac region and the resulting potential field is calculated throughout the torso. Two new coupling techniques are developed that provide a more direct path from cellular activation to body surface potentials. One approach assembles all of the equations from the passive torso regions and the equations from the extracellular bidomain region into a single matrix system. Coupling conditions based on the continuity of potential and current flow across the myocardial surfaces are used to couple the regions and therefore solving the matrix system yields a solution that is continuous across all of the solution points within the torso. The second approach breaks the large system into smaller subproblems and the continuity conditions are iii iv imposed through an iterative approach. Across each of the myocardial surfaces, a fixed point iteration is set up with the goal of converging towards zero potential and current flow differences between adjacent regions. All of the numerical methods used within the integrated modelling framework are rigorously tested individually before extensive tests are performed on the coupling techniques. Large scale simulations are run to test the dipole source approach against the new coupling techniques. Several sets of simulations are run to investigate the effects of using different ionic current models, using different bidomain model simplifications, and the role that the torso inhomogeneities play in generating body surface potentials. The main question to be answered by this study is whether or not the traditional approach of combining a monodomain heart with an equivalent cardiac source in a two step approach is adequate when generating body surface potentials. Comparisons between the fully coupled framework developed here and several dipole based approaches demonstrate that the resulting sets of signals have different magnitudes and different waveform shapes on both the torso and epicardial surface, clearly illustrating the inadequacy of the equivalent cardiac source models. It has been found that altering the modelling assumptions on each spatial scale produces noticeable effects. At the smallest scale, the use of different cell models leads to significantly different body surface potential traces. At the next scale the monodomain approach is unable to accurately reproduce the results from a full bidomain framework, and at the largest level the inclusion of different torso inhomogeneities has a large effect on the magnitude of the torso and epicardial potentials. Adding a pair of lungs to the torso model changes the epicardial potentials by an average of 16% which is consistent with the experimental range of between 8 and 20%. This provides evidence that only a complex, coupled, biophysically based model will be able to properly reproduce clinical ECGs.

An efficient finite element method is developed to model the spreading of excitation in ventricular myocardium by treating the thin region of rapidly depolarizing tissue as a propagating wavefront. The model is used to investigate the excitation sequence in the full canine ventricular myocardium. The solution to an eikonal–curvature equation for excitation time is shown to satisfy a reaction–diffusion equation for the bidomain myocardial model at the wavefront, while the solution to an eikonal–diffusion equation approximately satisfies the reaction–diffusion equation in the vicinity of the wavefront. The features of these two eikonal equations are discussed. A Petrov–Galerkin finite element method with cubic Hermite elements is developed to solve the eikonal–diffusion equation. The oscillatory errors seen when using the Galerkin weighted residual method with high mesh Péclet numbers are avoided by supplementing the Galerkin weights with C⁰ continuous functions based on derivatives of the interpolation functions. The ratio of the Galerkin and supplementary weights is a function of the Péclet number such that, for one-dimensional propagation, the error in the solution is within a small constant factor of the optimal error achievable in the trial space. An additional noinflow boundary term is developed to prevent spurious excitation initiating on the boundary. The need for discretization in time is avoided by using a continuation method to gradually introduce the non-linear term of the governing equation. A small amount of artificial diffusion is sometimes necessary. Simulations of excitation are performed using a model of the anisotropic canine ventricular myocardium with 23.55 degrees of freedom for the dependent variable, and results are compared with reported experimental observations. When it was assumed that Purkinje fibres influence propagation only on the endocardial surface, excitation of the entire myocardium was completed in 56 ms. Altering material parameters to represent penetration of the Purkinje fibres beneath the left endocardial surface reduced the completion time to 48 ms. Modelling the effects of the laminar structure of myocardium by reducing the propagation speed by 40% in the direction normal to the layers delayed completion of excitation by only 4%.

In this thesis, the forward problem of electrocardiography is investigated from a cellular level through to potentials on the surface of the torso. This integrated modelling framework is based on three spatial scales. At the smallest spatial resolution, several cardiac cellular models are implemented that are used to represent the underlying cellular electrophysiology. A bidomain framework is used to couple multiple individual cells together and this provides a mathematical model of the myocardial tissue. The cardiac geometry is described using finite elements with high order cubic Hermite basis function interpolation. An anatomically based description of the fibrous laminar cardiac microstructure is then defined relative to the geometric mesh. Within the local element space of the cardiac finite elements, a fine collocation mesh is created on which the bidomain equations are solved. Each collocation point represents a continuum cell and contains a cellular model to describe the local active processes. This bidomain implementation works in multiple coordinate systems and over deforming domains, in addition to having the ability to spatially vary any parameter throughout the myocardium. On the largest spatial scale the passive torso regions surrounding the myocardium are modelled using a generalised Laplace equation to describe the potential field and current flows. The torso regions are discretised using either finite elements or boundary elements depending on the electrical properties of each region. The cardiac region is coupled to the surrounding torso through several methods. A traditional dipole source approach is implemented that creates equivalent cardiac sources through the summation of cellular dipoles. These dipoles are then placed within a homogeneous cardiac region and the resulting potential field is calculated throughout the torso. Two new coupling techniques are developed that provide a more direct path from cellular activation to body surface potentials. One approach assembles all of the equations from the passive torso regions and the equations from the extracellular bidomain region into a single matrix system. Coupling conditions based on the continuity of potential and current flow across the myocardial surfaces are used to couple the regions and therefore solving the matrix system yields a solution that is continuous across all of the solution points within the torso. The second approach breaks the large system into smaller subproblems and the continuity conditions are iii iv imposed through an iterative approach. Across each of the myocardial surfaces, a fixed point iteration is set up with the goal of converging towards zero potential and current flow differences between adjacent regions. All of the numerical methods used within the integrated modelling framework are rigorously tested individually before extensive tests are performed on the coupling techniques. Large scale simulations are run to test the dipole source approach against the new coupling techniques. Several sets of simulations are run to investigate the effects of using different ionic current models, using different bidomain model simplifications, and the role that the torso inhomogeneities play in generating body surface potentials. The main question to be answered by this study is whether or not the traditional approach of combining a monodomain heart with an equivalent cardiac source in a two step approach is adequate when generating body surface potentials. Comparisons between the fully coupled framework developed here and several dipole based approaches demonstrate that the resulting sets of signals have different magnitudes and different waveform shapes on both the torso and epicardial surface, clearly illustrating the inadequacy of the equivalent cardiac source models. It has been found that altering the modelling assumptions on each spatial scale produces noticeable effects. At the smallest scale, the use of different cell models leads to significantly different body surface potential traces. At the next scale the monodomain approach is unable to accurately reproduce the results from a full bidomain framework, and at the largest level the inclusion of different torso inhomogeneities has a large effect on the magnitude of the torso and epicardial potentials. Adding a pair of lungs to the torso model changes the epicardial potentials by an average of 16% which is consistent with the experimental range of between 8 and 20%. This provides evidence that only a complex, coupled, biophysically based model will be able to properly reproduce clinical ECGs.

Medical imaging is an essential tool for improving the diagnoses, understanding and treatment of a large variety of diseases. Over the last century technology has advanced from the discovery of x-rays to a variety of 3D imaging tools such as magnetic resonance imaging, computed tomography, positron emission tomography and ultrasonography. As a consequence the size and complexity of medical data sets has increased tremendously making it ever more difficult to understand, analyze, compare and communicate this data. Visualization is an attempt to simplify these tasks according to the motto "An image says more than a thousand words". This thesis introduces a toolkit for visualizing biomedical data sets with a particular emphasis on second-order tensors, which are mathematically described by matrices and can be used to express complex tissue properties such as material de-formation and water diffusion. The toolkit has a modular design which facilitates the comparison and exploration of multiple data sets. A novel field data structure allows the interactive creation of new measures and boolean filters are introduced as a universal visualization tool. Various new visualization methods are presented including new colour mapping techniques, ellipsoid-based textures and a line integral convolution texture for visualizing tensor fields. To motivate the design and to assist in the use of the toolkit, guidelines for creating effective visualizations are derived by using perceptual concepts from cognitive science. A new classification for visual attributes according to representational accuracy, perceptual dimension and spatial requirements is presented and the results are used to derive values for the information content and information density of each attribute. A review and a classification of visualization icons completes the theoretical background. The thesis concludes with two case studies. In the first case study the toolkit is used to visualize the strain tensor field in a healthy and a diseased human left ventricle. New insight into the cardiac mechanics is obtained by applying and modifying techniques traditionally used in solid mechanics and computational fluid dynamics. The second case study explores ways to obtain in vivo information of the brain anatomy by visualizing and systematically exploring Diffusion Tensor Imaging (DTI) data. Three new techniques for the visualization of DTI data are presented: Barycentric colour maps allow an integrated view of different types of diffusion anisotropy. Ellipsoid-based textures and Anisotropy Modulated Line Integral Convolution create images segmented by tissue type and incorporating a texture representing the 3D orientation of nerve fibers. The effectiveness of the exploration approach and the new visualization techniques are demonstrated by identifying various anatomical structures and features from a diffusion tensor data set of a healthy brain.

Coronary blood flow through the ventricular contraction cycle has been investigated in this thesis using an anatomically accurate computational model. Using Strahler ordered morphological data and an avoidance algorithm a three dimensional finite element model has been constructed of the largest six generations of the coronary arterial network within an anatomically accurate finite element model of the left and right ventricles. Segment radii, lengths and connectivity are consistent with the literature, local network branch angles are consistent with the principle of minimum shear stress at bifurcations, and an even spatial distribution of vessel segments throughout the myocardium has been achieved. A finite difference collocation grid has been generated on the coronary finite element mesh. The Navier Stokes equations governing blood flow through elastic vessels have been reduced to one dimension and are solved on this finite difference grid using the two step Lax Wendroff method. Blood flows at bifurcations are calculated using an iterative method ensuring conservation of mass and momentum. The microcirculation networks are modelled using a lumped parameter model incorporating the nonlinear variation of resistance and compliance with pressure by fitting results from published anatomical data. The venous network is assumed to parallel the generated arterial model. The calculated blood flow through the network model demonstrates physiological pressure drops, flow rates and a regional distribution within the ventricular geometry consistent with experimental data. The intramyocardial pressure (IMP) exerted on the coronary vasculature during contraction is calculated from quasi-static solutions of the equations of finite deformation applied to the ventricular model with a nonlinear anisotropic constitutive law. IMP is found to vary approximately linearly between ventricular pressure at the endocardium and atmospheric pressure at the epicardium. IMP and vessel stretch are included in the transmural pressure radius relationship to model the effect of myocardial deformation on coronary flow. The calculated coronary blood flow through the contraction cycle shows that arterial flow is predominantly diastolic while venous flow is significantly increased during systole. Calculated time varying velocity profiles in the large epicardial vessels compare well with published experimental results. Regionally averaged velocities in small vessels show that arterial inflow is most significantly impeded at the left ventricular endocardium. Furthermore, the large time constant associated with the capillary and venule networks limits the filling of these vessels during diastole particularly at the endocardium. An increase in heart rate, modelled by reducing the time for diastole causes an increase in small vessel epicardial blood flow and a decrease in blood flow through small vessels within the myocardium. The decrease in flow is most pronounced at the left ventricular endocardium.

In this thesis, the forward problem of electrocardiography is investigated from a cellular level through to potentials on the surface of the torso. This integrated modelling framework is based on three spatial scales. At the smallest spatial resolution, several cardiac cellular models are implemented that are used to represent the underlying cellular electrophysiology. A bidomain framework is used to couple multiple individual cells together and this provides a mathematical model of the myocardial tissue. The cardiac geometry is described using finite elements with high order cubic Hermite basis function interpolation. An anatomically based description of the fibrous laminar cardiac microstructure is then defined relative to the geometric mesh. Within the local element space of the cardiac finite elements, a fine collocation mesh is created on which the bidomain equations are solved. Each collocation point represents a continuum cell and contains a cellular model to describe the local active processes. This bidomain implementation works in multiple coordinate systems and over deforming domains, in addition to having the ability to spatially vary any parameter throughout the myocardium. On the largest spatial scale the passive torso regions surrounding the myocardium are modelled using a generalised Laplace equation to describe the potential field and current flows. The torso regions are discretised using either finite elements or boundary elements depending on the electrical properties of each region. The cardiac region is coupled to the surrounding torso through several methods. A traditional dipole source approach is implemented that creates equivalent cardiac sources through the summation of cellular dipoles. These dipoles are then placed within a homogeneous cardiac region and the resulting potential field is calculated throughout the torso. Two new coupling techniques are developed that provide a more direct path from cellular activation to body surface potentials. One approach assembles all of the equations from the passive torso regions and the equations from the extracellular bidomain region into a single matrix system. Coupling conditions based on the continuity of potential and current flow across the myocardial surfaces are used to couple the regions and therefore solving the matrix system yields a solution that is continuous across all of the solution points within the torso. The second approach breaks the large system into smaller subproblems and the continuity conditions are iii iv imposed through an iterative approach. Across each of the myocardial surfaces, a fixed point iteration is set up with the goal of converging towards zero potential and current flow differences between adjacent regions. All of the numerical methods used within the integrated modelling framework are rigorously tested individually before extensive tests are performed on the coupling techniques. Large scale simulations are run to test the dipole source approach against the new coupling techniques. Several sets of simulations are run to investigate the effects of using different ionic current models, using different bidomain model simplifications, and the role that the torso inhomogeneities play in generating body surface potentials. The main question to be answered by this study is whether or not the traditional approach of combining a monodomain heart with an equivalent cardiac source in a two step approach is adequate when generating body surface potentials. Comparisons between the fully coupled framework developed here and several dipole based approaches demonstrate that the resulting sets of signals have different magnitudes and different waveform shapes on both the torso and epicardial surface, clearly illustrating the inadequacy of the equivalent cardiac source models. It has been found that altering the modelling assumptions on each spatial scale produces noticeable effects. At the smallest scale, the use of different cell models leads to significantly different body surface potential traces. At the next scale the monodomain approach is unable to accurately reproduce the results from a full bidomain framework, and at the largest level the inclusion of different torso inhomogeneities has a large effect on the magnitude of the torso and epicardial potentials. Adding a pair of lungs to the torso model changes the epicardial potentials by an average of 16% which is consistent with the experimental range of between 8 and 20%. This provides evidence that only a complex, coupled, biophysically based model will be able to properly reproduce clinical ECGs.

An efficient finite element method is developed to model the spreading of excitation in ventricular myocardium by treating the thin region of rapidly depolarizing tissue as a propagating wavefront. The model is used to investigate the excitation sequence in the full canine ventricular myocardium. The solution to an eikonal–curvature equation for excitation time is shown to satisfy a reaction–diffusion equation for the bidomain myocardial model at the wavefront, while the solution to an eikonal–diffusion equation approximately satisfies the reaction–diffusion equation in the vicinity of the wavefront. The features of these two eikonal equations are discussed. A Petrov–Galerkin finite element method with cubic Hermite elements is developed to solve the eikonal–diffusion equation. The oscillatory errors seen when using the Galerkin weighted residual method with high mesh Péclet numbers are avoided by supplementing the Galerkin weights with C⁰ continuous functions based on derivatives of the interpolation functions. The ratio of the Galerkin and supplementary weights is a function of the Péclet number such that, for one-dimensional propagation, the error in the solution is within a small constant factor of the optimal error achievable in the trial space. An additional noinflow boundary term is developed to prevent spurious excitation initiating on the boundary. The need for discretization in time is avoided by using a continuation method to gradually introduce the non-linear term of the governing equation. A small amount of artificial diffusion is sometimes necessary. Simulations of excitation are performed using a model of the anisotropic canine ventricular myocardium with 23.55 degrees of freedom for the dependent variable, and results are compared with reported experimental observations. When it was assumed that Purkinje fibres influence propagation only on the endocardial surface, excitation of the entire myocardium was completed in 56 ms. Altering material parameters to represent penetration of the Purkinje fibres beneath the left endocardial surface reduced the completion time to 48 ms. Modelling the effects of the laminar structure of myocardium by reducing the propagation speed by 40% in the direction normal to the layers delayed completion of excitation by only 4%.

Medical imaging is an essential tool for improving the diagnoses, understanding and treatment of a large variety of diseases. Over the last century technology has advanced from the discovery of x-rays to a variety of 3D imaging tools such as magnetic resonance imaging, computed tomography, positron emission tomography and ultrasonography. As a consequence the size and complexity of medical data sets has increased tremendously making it ever more difficult to understand, analyze, compare and communicate this data. Visualization is an attempt to simplify these tasks according to the motto "An image says more than a thousand words". This thesis introduces a toolkit for visualizing biomedical data sets with a particular emphasis on second-order tensors, which are mathematically described by matrices and can be used to express complex tissue properties such as material de-formation and water diffusion. The toolkit has a modular design which facilitates the comparison and exploration of multiple data sets. A novel field data structure allows the interactive creation of new measures and boolean filters are introduced as a universal visualization tool. Various new visualization methods are presented including new colour mapping techniques, ellipsoid-based textures and a line integral convolution texture for visualizing tensor fields. To motivate the design and to assist in the use of the toolkit, guidelines for creating effective visualizations are derived by using perceptual concepts from cognitive science. A new classification for visual attributes according to representational accuracy, perceptual dimension and spatial requirements is presented and the results are used to derive values for the information content and information density of each attribute. A review and a classification of visualization icons completes the theoretical background. The thesis concludes with two case studies. In the first case study the toolkit is used to visualize the strain tensor field in a healthy and a diseased human left ventricle. New insight into the cardiac mechanics is obtained by applying and modifying techniques traditionally used in solid mechanics and computational fluid dynamics. The second case study explores ways to obtain in vivo information of the brain anatomy by visualizing and systematically exploring Diffusion Tensor Imaging (DTI) data. Three new techniques for the visualization of DTI data are presented: Barycentric colour maps allow an integrated view of different types of diffusion anisotropy. Ellipsoid-based textures and Anisotropy Modulated Line Integral Convolution create images segmented by tissue type and incorporating a texture representing the 3D orientation of nerve fibers. The effectiveness of the exploration approach and the new visualization techniques are demonstrated by identifying various anatomical structures and features from a diffusion tensor data set of a healthy brain.

Coronary blood flow through the ventricular contraction cycle has been investigated in this thesis using an anatomically accurate computational model. Using Strahler ordered morphological data and an avoidance algorithm a three dimensional finite element model has been constructed of the largest six generations of the coronary arterial network within an anatomically accurate finite element model of the left and right ventricles. Segment radii, lengths and connectivity are consistent with the literature, local network branch angles are consistent with the principle of minimum shear stress at bifurcations, and an even spatial distribution of vessel segments throughout the myocardium has been achieved. A finite difference collocation grid has been generated on the coronary finite element mesh. The Navier Stokes equations governing blood flow through elastic vessels have been reduced to one dimension and are solved on this finite difference grid using the two step Lax Wendroff method. Blood flows at bifurcations are calculated using an iterative method ensuring conservation of mass and momentum. The microcirculation networks are modelled using a lumped parameter model incorporating the nonlinear variation of resistance and compliance with pressure by fitting results from published anatomical data. The venous network is assumed to parallel the generated arterial model. The calculated blood flow through the network model demonstrates physiological pressure drops, flow rates and a regional distribution within the ventricular geometry consistent with experimental data. The intramyocardial pressure (IMP) exerted on the coronary vasculature during contraction is calculated from quasi-static solutions of the equations of finite deformation applied to the ventricular model with a nonlinear anisotropic constitutive law. IMP is found to vary approximately linearly between ventricular pressure at the endocardium and atmospheric pressure at the epicardium. IMP and vessel stretch are included in the transmural pressure radius relationship to model the effect of myocardial deformation on coronary flow. The calculated coronary blood flow through the contraction cycle shows that arterial flow is predominantly diastolic while venous flow is significantly increased during systole. Calculated time varying velocity profiles in the large epicardial vessels compare well with published experimental results. Regionally averaged velocities in small vessels show that arterial inflow is most significantly impeded at the left ventricular endocardium. Furthermore, the large time constant associated with the capillary and venule networks limits the filling of these vessels during diastole particularly at the endocardium. An increase in heart rate, modelled by reducing the time for diastole causes an increase in small vessel epicardial blood flow and a decrease in blood flow through small vessels within the myocardium. The decrease in flow is most pronounced at the left ventricular endocardium.