| This dissertation is divided into several chapters; in many cases, a chapter represents an individual paper published in a journal.;The conditionally stable numerical method (forward Euler) is used to solve the Fenton-Karma model of spiral wave breakup. I derive the stability relation for the bidomain model, which is made simpler by the positive definite nature of the conductivity tensors; furthermore, the uniqueness of the stability relationship arises precisely from the positive definite nature of the stability relation. I also separately derive the stability relation for the forward Euler numerical approximation for an anisotropic and heterogeneous monodomain.;One of the more useful mathematical tools is a numerically based technique to automate the tracking of phase singularities in the thousands of simulations (required here for statistical analysis) without having to visually inspect each individual simulation.;The general ability of the Fenton-Karma model of spiral wave activity to function as a model for electrical defibrillation is examined in the context of electrical defibrillation. The biological significance of the model is highlighted by relating the calculations to drugs that block particular ion channels. Of more importance to the dissertation, I examine how long the model can sustain fibrillation (in a real heart, fibrillation continues indefinitely so a useful mathematical model requires that fibrillation be sustained). The chapter examines a change to the model that is required for simulation of stimuli, a crucial modification for studying defibrillation. A novel, multi-electrode feedback pacing algorithm is developed that evinces a much higher success than in my previously published simulation studies. This strategy is separately optimized to the physical parameters of the pacing algorithm---namely, electrode spacing and the minimum interactivation interval.;A more realistic topology for defibrillation modeling that presents the heart as a sphere is developed. We make use of an alternative numerical technique for producing a spherical surface on which we simulate fibrillation and defibrillation, producing a more accurate representation of true cardiac topology than the bi-periodic boundary conditions. |
