Applied Mathematics, 2012, 3, 864-872 http://dx.doi.org/10.4236/am.2012.38128 Published Online August 2012 (http://www.SciRP.org/journal/am) Modified Fletcher-Reeves and Dai-Yuan Conjugate Gradient Methods for Solving Optimal Control Problem of Monodomain Model Kin Wei Ng, Ahmad Rohanin Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Johor Bahru, Malaysia Email: kwng3@live.utm.my, rohanin@utm.my Received May 17, 2012; revised June 17, 2012; accepted June 24, 2012 ABSTRACT In this paper, we present the numerical solution for the optimal control problem of monodomain model with Rogers- modified FitzHugh-Nagumo ion kinetic. The monodomain model, which is a well-known mathematical model for simulation of cardiac electrical activity, appears as the constraint in our problem. Our control objective is to dampen the excitation wavefront of the transmembrane potential in the observation domain using optimal applied current. Various conjugate gradient methods have been applied by researchers for solving this type of optimal control problem. For the present paper, we adopt the modified Fletcher-Reeves method and modified Dai-Yuan method for computing the opti- mal applied current. Numerical results show that the excitation wavefront is successfully dampened out by the optimal applied current when the modified Fletcher-Reeves method is used. However, this is not the case when the modified Dai-Yuan method is employed. Numerical results indicate that the modified Dai-Yuan method failed to converge to the optimal solution when the Armijo line search is used. Keywords: Monodomain Model; Conjugate Gradient Method; Galerkin Finite Element Method; Optimal Control 1. Introduction Sudden cardiac death is a leading cause of death among adults in most countries. In the United States, sudden cardiac death episodes affect 450,000 people each year [1]. In Singapore, about 23% of approximately 16,000 deaths that occur every year are reported as cardiac death [2]. Also, a recent study in China indicates that sudden cardiac death takes the lives of over 544,000 people an- nually [3]. Sudden cardiac death occurs when the elec- trical system of the heart malfunctions, causing an ir- regular heart rhythm. This irregular heart rhythm cause the heart muscle to quiver and the heart is no longer able to pump blood to the body and brain. Consequently, death can occurs within minutes unless the normal heart rhythm is restored through defibrillation. Nowadays, the implantable cardioverter defibrillator (ICD) is increas- ingly being used by patients who are at significant risk of sudden cardiac death. If any life-threatening arrhythmia is detected by ICD, an energy electrical shock will be delivered to the heart to restore normal heart rhythm. The optimal control approach to the defibrillation pro- cess was first proposed by Nagaiah et al. [4], with the objective to determine the minimal applied current which can help in the defibrillation process. More specifically, the control objective is to dampen the excitation wave- front of the transmembrane potential in the observation domain using optimal applied current. The monodomain model was employed by Nagaiah et al. [4] to represent the electrical behavior of the cardiac tissue. The mono- domain model consists of a parabolic partial differential equation (PDE) coupled to a system of nonlinear ordi- nary differential equations (ODEs), which is a well- known mathematical model for simulation of cardiac electrical activity [5-7]. Consequently, the optimization problem of defibrillation process can be generally known as optimal control problem of monodomain model. In the literature, the nonlinear conjugate gradient me- thods have been frequently applied by researchers for solving the optimal control problem of the monodomain model. Nagaiah et al. [4] employed the Polak-Ribière- Polyak (PRP) method [8,9], the Hager-Zhang (HZ) me- thod [10] and the Dai-Yuan (DY) method [11] to solve the optimal control problem of this monodomain model. Later, Ng and Rohanin [12] utilized the modified version of the DY method as proposed by Zhang [13] to solve the optimal control problem of monodomain model. On the other hand, the Hestenes-Stiefel (HS) method [14] has been applied by Ainseba et al. [15] for solving the optimal control problem of tridomain model. For the present pa- per, we present the numerical solution for the optimal Copyright © 2012 SciRes. AM