| Numerical methods have become the viable tools for solving scientific and engineering problems which do not have any known analytical solution. The finite difference method is one of the most commonly used numerical methods which requires less computational resources than some other methods, e.g., finite element method, and is thus attractive for solving large problems. In general the computational requirements can increase significantly with the problem's dimension and the complexity of the material medium. This becomes especially critical for problems that are time dependent or need to be solved many times.; This work is motivated from the need of a computationally efficient method which can handle time-dependent problems involving large domains with complex mediums, e.g., inhomogeneous and anisotropic mediums. In this respect, a new finite difference formulation is used to solve the elliptic Laplace's equation in two and three dimensions for both continuously and discretely inhomogeneous anisotropic mediums. It is then extended to solve the coupled, time-dependent nonlinear bidomain equations, which are widely used to model bioelectric phenomenons in cardiac tissue.; Modeling biological systems is one of the most computationally challenging work due to the inherently complicated structure and complex nonlinear dynamics involved. The new formulations are used to model bioelectric phenomenons in two different inhomogeneous anisotropic systems. First, it is used to solve the governing Laplace's equation and obtain the potential distribution in a canine torso during electrical defibrillation. Then, it is used to solve the bidomain equations in a study of action potential propagation in cardiac tissue during electrical stimulation.; Previous finite difference studies on bioelectric phenomenons in the torso and cardiac tissue have disregarded the anisotropic inhomogeneity of these structures. The incorporation of the anisotropic inhomogeneity in the new numerical models allows a more realistic representation of these structures, which in turn will provide more accurate solutions of the bioelectric problems.; Since a large number of finite difference nodes are required to represent realistic sizes of the thoracic volume conductor or cardiac tissue, it is still beyond the capacity of any conventional computer to provide the computational resources required for the realistic models considered in this study. To resolve this issue, a massively parallel computer, the Connection Machine CM-5 is used to implement the models.; In the chapters of this dissertation, the new finite difference formulations are described and verified, and their applications in modeling the thoracic volume conductor and cardiac tissue are presented together with computational performance and physiological results. |
