Numerical solutions for optimal control of monodomain equations in cardiac electrophysiology

Abstract

In this article, we present computational techniques for optimal control of monodomain equations which are a well established model for describing wave propagation of the action potential in the heart. The model consists of a non-linear parabolic partial differential equation of reaction-diffusion type, where the reaction term is a set of ordinary differential equations which characterize the dynamics of cardiac cells. Specifically, an optimal control formulation is presented for the monodomain equations with an extracellular current as the control variable which must be determined in such a way that wavefronts of transmembrane voltage are smoothed in an optimal manner. Numerical results are presented based on the optimize before discretize and discretize before optimize techniques. Moreover, the derivation of the optimality system is given for both techniques and numerical results are discussed for higher order methods to solve the optimality system. Finally, numerical results are reported which show superlinear convergence when using Newton's method. © 2010 Springer -Verlag Berlin Heidelberg.