The Radial Point Interpolation Mixed Collocation (RPIMC) Method for the Solution of the Reaction-Diffusion Equation in Cardiac Electrophysiology

Abstract

The Radial Point Interpolation Mixed Collocation (RPIMC) method is developed for the solution of the reaction-diffusion equation in cardiac electrophysiology simulations. RPIMC is an efficient and purely meshfree technique which is expected to be a valuable alternative to the Finite Element Method (FEM) for cardiac electrophysiology applications where models with large number of degrees of freedom and high geometric complexity are common. We propose applying the operator splitting technique to solve the decoupled reaction-diffusion equation. In this way, the reaction (cardiac cell dynamics) and diffusion (action potential propagation) terms are solved independently. We evaluate the RPIMC in a simulation of the cardiac action potential (AP) propagation in a two-dimensional square tissue composed of human ventricular epicardium cells. The state-of-art O’Hara Rudy cell dynamics model is used to solve the reaction term while the diffusion term is solved using the standard forward Euler method. The simulation of the AP propagation using the RPIMC method is compared against a FEM simulation using isoparametric bilinear elements. Comparable results between RPIMC and FEM are obtained for both normal AP propagation and spiral wave generation conditions (expected in arrhythmic events). The convergence of the RPIMC solution to the FEM solution is evaluated for varying nodal spacing and varying dilatation coefficient during support domain nodes identification.