Recent Numerical Methods in Electrocardiology

Abstract

Heart diseases are the leading cause of death in the world. Many questions have not yet been answered regarding the electrical waves propagation in cardiac tissue, and the mechanism of ventricular fibrillation that is produced by one or many spiral propagation waves of the excitation cardiac wall. Numerical modeling can play a crucial role and provides the necessary tools to answer some of these questions. However, the mathematical models, which give the best reflection of electrophysiological waves in cardiac tissue, are extremely complicated and present a significant computational challenges. The bidomain model is considered as the mathematical equations that have been used for simulating cardiac electrophysiological waves for many years (see Sundnes (2002), and Pierre (2006) and the reference therein). This model represents the cardiac tissue at a macroscopic scale by relating the transmembrane potential, the extracellular potential, and the ionic currents. The biodomain model consists of a system of two nonlinear partial differential equations coupled to a system of ordinary differential equations. From the numerical point of view, the model is computationally very expensive. The major difficulties are due to the computational grids size that must be very fine to get a realistic simulation of cardiac tissue. Indeed, the action potential is a wave with sharp depolarization and repolarization fronts and this wave travels across the whole computational domain calling for a very fine uniform mesh. One popular way of reducing the computational challenges of the bidomain model is the use of the monodomain model. This model considers a single nonlinear partial differential equation coupled with the same system of ordinary differential equations for the ionic currents. Although, it has been reported that the CPU requirements are reduced when simplifying the bidomain model to a monodomain model (see Sundnes et al. (2006)), both models still encounter computational difficulties because of the need for fine meshes and small time-steps. Many methods have been introduced in the literature to overcome these difficulties. The operator splitting is usually performed to separate the large non-linear system of ODEs and thus introduces subproblems easier to solve. A first-order (Godunov method) and a second-order (Strang method) accurate splitting technique can be employed. For more details the reader is referred to Sundnes et al. (2005), Lines, Buist, Grottum, Pullan, Sundnes & Tveito (2003); Lines, Grottum & Tveito (2003), and Weber Dos Santos et al. (2003)). To reduce the computational time at each time step, parallel computing techniques are used (see Colli Franzone & Pavarino (2004), Karpoukhin et al. (1995) and Weber dos Santos et al. (2004)). Several timestepping strategies have also been used, fully implicit ( Bourgault et al. (2003), and Murillo & Cai (2004)), and semi-implicit ( Franzone & Pavarino (2004), Ethier & Bourgault (2008))