Abstract:
The Iast forty years have seen an enormous growth in the physiological understanding of the structure and function of cardiac muscle, and in the creation of mathematical models which attempt to quantify the observed mechanical, electrical and biochemical behaviour of the heart. These models have, in general, been developed independently of one another, and there has been little attempt to integrate the various mechanisms, even though the processes are clearly interdependent. The work presented in this thesis illustrates the first stages of development of a unified model of cardiac structure and function, with particular reference to modelling cardiac activation. We are building on a base which has already been constructed describing the geometry and microstructure of ventricular muscle. Models of mechanical deformation are also under construction, and this work proposes a solution technique which allows the electrical processes to be integrated with these, and other, models of cardiac structure and function. A new collocation method is developed which constructs a grid of collocation points defined at specific material locations throughout the ventricular finite element mesh. A finite difference based solution technique uses local metric information to solve the activation equations on this grid. In general, the collocation grid is non-uniformly spaced, and will deform with the movement of the finite element mesh. This collocation method is used to simulate activation using a multiple-purpose solution program on a range of one, two and three-dimensional geometries, with a number of variations. Initial test simulations in two dimensions examine the performance of the method on a square, isotropic domain using a simple ionic current model to demonstrate convergence with both spatial and temporal resolution. The various parameters are then changed incrementally to incorporate such variations as an irregular domain, anisotropic conductivities using a range of fibre fields, and various ionic current models. Simulations using multiple stimuli show the effect of anisotropy on reentrant behaviour. The dimensionality of the solution domain is also altered, with a one-dimensional solution showing the interaction of the Purkinje network with a myocardial sheet, and a range of examples are defined on several three-dimensional geometries including a cube and an anatomically accurate ventricular mesh. We also investigate coupling of the myocardial activation model to two other models of cardiac function. Firstly, the bidomain model is used to couple the myocardial potential solution with a calculation of torso potential. The activation model is also coupled to a model of mechanical deformation. This electromechanical coupling is demonstrated in two dimensions using two ionic current models on both a square domain, and a mesh generated from a ventricular cross-section. A final three-dimensional solution presents preliminary results showing the activation model coupled to a model of deformation in an anatomically accurate ventricular mesh.