ABSTRACT In this paper, we present the numerical solution for the optimal control problem of monodomain modelwith 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 methodfor computing the optimal 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.
Cite this paper
K. Ng and A. Rohanin, "Modified Fletcher-Reeves and Dai-Yuan Conjugate Gradient Methods for Solving Optimal Control Problem of Monodomain Model," Applied Mathematics, Vol. 3 No. 8, 2012, pp. 864-872. doi: 10.4236/am.2012.38128.
 Z. J. Zheng, J. B. Croft, W. H. Giles and G. A. Mensah, “Sudden Cardiac Death in the United States, 1989 to 1998,” Circulation, Vol. 104, 2001, pp. 2158-2163.
 E. H. M. Ong, “Proposal for Establishment of a National Sudden Cardiac Arrest Registry,” Singapore Medical Journal, Vol. 52, No. 8, 2011, pp. 631-633.
 S. Zhang, “Sudden Cardiac Death in China,” Pacing and Clinical Electrophysiology, Vol. 32, No. 9, 2009, pp. 1159-1162. Hdoi:10.1111/j.1540-8159.2009.02458.x
 C. Nagaiah, K. Kunisch and G. Plank, “Numerical Solution for Optimal Control of the Reaction-Diffusion Equations in Cardiac Electrophysiology,” Computational Optimization and Applications, Vol. 49, No. 1, 2011, pp. 149-178. Hdoi:10.1007/s10589-009-9280-3
 S. M. Shuaiby, M. A. Hassan and M. El-Melegy, “Modeling and Simulation of the Action Potential in Human Cardiac Tissues Using Finite Element Method,” Journal of Communications and Computer Engineering, Vol. 2, No. 3, 2012, pp. 21-27.
 Y. Belhamadia, A. Fortin and Y. Bourgault, “Towards Accurate Numerical Method for Monodomain Models Using a Realistic Heart Geometry,” Mathematical Biosciences, Vol. 220, No. 2, 2009, pp. 89-101.
 C. Nagaiah and K. Kunisch, “Higher Order Optimization and Adaptive Numerical Solution for Optimal Control of Monodomain Equations in Cardiac Electrophysiology,” Applied Numerical Mathematics, Vol. 61, 2011, pp. 53-65. Hdoi:10.1016/j.apnum.2010.08.004
 E. Polak and G. Ribière, “Note Sur la Convergence de Méthodes de Directions Conjuguées,” Revue Francaise d’Informatique et de Recherche Opérationnelle, Vol. 16, 1969, pp. 35-43.
 B. T. Polyak, “The Conjugate Gradient Method in Extreme Problems;” USSR Computational Mathematics and Mathematical Physics, Vol. 9, 1969, pp. 94-112.
 W. W. Hager and H. Zhang, “A New Conjugate Gradient Method with Guaranteed Descent and an Efficient Line Search,” SIAM Journal on Optimization, Vol. 16, 2005, pp. 170-192. Hdoi:10.1137/030601880
 Y. H. Dai and Y. Yuan, “A Nonlinear Conjugate Gradient Method with a Strong Global Convergence Property,” SIAM Journal on Optimization, Vol. 10, No. 1, 1999, pp. 177-182. Hdoi:10.1137/S1052623497318992
 K. W. Ng and A. Rohanin, “Numerical Solution for PDE-Constrained Optimization Problem in Cardiac Electrophysiology,” International Journal of Computer Applications, Vol. 44, No. 12, 2012, pp. 11-15.
 L. Zhang, “Two Modified Dai-Yuan Nonlinear Conjugate Gradient Methods,” Numerical Algorithms, Vol. 50, 2009, pp. 1-16.Hdoi:10.1007/s11075-008-9213-8
 M. R. Hestenes and E. L. Stiefel, “Methods of Conjugate Gradients for Solving Linear Systems,” Journal of Research of the National Bureau of Standards, Vol. 49, No. 6, 1952, pp. 409-436.
 B. Ainseba, M. Bendahmane and R. Ruiz-Baier, “Analysis of an Optimal Control Problem for the Tridomain Model in Cardiac Electrophysiology,” Journal of Mathematical Analysis and Applications, Vol. 388, No. 1, 2012, pp. 231-247.Hdoi:10.1016/j.jmaa.2011.11.069
 L. Zhang, W. J. Zhou and D. H. Li, “Global Convergence of a Modified Fletcher-Reeves Conjugate Gradient Method with Armijo-Type Line Search,” Numerische Mathematik, Vol. 104, 2006, pp. 561-572.
 L. Zhang, “Nonlinear Conjugate Gradient Methods for Optimization Problems,” Ph.D. Thesis, Hunan University, Changsha, 2006.
 J. M. Rogers and A. D. McCulloch, “A Collocation-Galerkin Finite Element Model of Cardiac Action Potential Propagation,” IEEE Transactions on Biomedical Engineering, Vol. 41, No. 8, 1994, pp. 743-757.
 K. Kunisch and M. Wagner, “Optimal Control of the Bidomain System (I): The Monodomain Approximation with the Rogers-McCulloch Model,” Nonlinear Analysis: Real World Applications, Vol. 13, No. 4, 2012, pp. 1525-1550.
 Z. Qu and A. Garfinkel, “An Advanced Algorithm for Solving Partial Differential Equation in Cardiac Conduction,” IEEE Transactions on Biomedical Engineering, Vol. 46, No. 9, 1999, pp.1166-1168.
 P. C. Franzone, P. Deuflhard, B. Ermann, J. Lang and L. F. Pavarino, “Adaptivity in Space and Time for Reaction-Diffusion Systems in Electrocardiology,” SIAM Journal on Scientific Computing, Vol. 28, No. 3, 2006, pp. 942-962. Hdoi:10.1137/050634785