High Order Methods for Solving Cardiac Models

Abstract

Modeling of the heart became of great interest for both mathematicians and bioengineers over the past 25 years for the increasing impact of cardiovascular system problems in our every day lives. The bidomain model and its simplified version, the monodomain model, are the most used mathematical models for simulation of the electrical activity of the heart. These models consist of a system of non-linear partial differential equations coupled with a set of ordinary differential equations describing the electrochemical reaction in the cardiac cell. These models are computationally demanding and developing accurate numerical methods are needed. This thesis proposes methods to solve such systems with a higher order of accuracy, (order 3 and 4), for both space and time. For space discretization, the proposed method is based on an Alternating Direction Implicit (ADI) finite difference method, while for the time discretization, the Semi-Implicit Backward Difference Method (SBDF) is used to simplify the non-linearity in the cardiac model. The performance of our techniques is presented and tested using three cases: the planar wave, the regular wave, and the spiral wave.