Proceedings of the I998 Winter Simulation Conference D.J. Medeiros, E.F. Watson, J.S. Carson and M.S. Manivannan, eds. MODELING CARDIAC ION CHANNEL CONDUCTIVITY: MODEL FITTING VIA SIMULATION John L. Maryak Richard H. Smith The Johns Hopkins University Applied Physics Laboratory 11100 Johns Hopkins Road Laurel, MD 20723-6099, U.S.A. ABSTRACT We describe a Markov state model for a cloned potassium channel of the human heart ( ZKvLeTl ). The parameters of the model are determined by a least-squares fit of predicted vs. measured data. The fitting process is achieved by using the “SPSA” optimizer to sequentially choose trial values of the parameters. At each choice of parameter value, a loss function is computed by simulating the action of the channel at that trial parameter value. When the optimizer has converged, the parameter value represents the best fit. 1 INTRODUCTION Recently, modeling approaches to the understanding of heart action have received increasing attention by researchers, e.g., Romey et al. (1997), Luo and Rudy (1994), Vandenberg and Bezanilla (1991), and Baker, Roden, and Bennett (1990). An important aspect of modeling the action of the human heart is modeling the electrical conductivity of ion channels within the heart. An understanding of cardiac electrical activity can lead to new diagnostic and treatment protocols, as well as facilitating the development of drug treatments for heart disease. In this paper, we describe a model for conductivity of a cardiac potassium channel. The model is based on eight key parameters that determine the transition rates among two closed (non-conducting) states and one open (conducting) state of the channel. We describe a simulation-based method to compute the least squares fit of these parameters to experimental data obtained when various voltages are applied across the channel. In this computation, the parameters are varied in a controlled fashion, and, at each setting of the parameters, a simulation of the conductivity of the channel is performed to obtain predicted output current across the membrane. By iteratively comparing the predicted currents with actual Raimond L. Winslow Department of Biomedical Engineering Center for Computational MedicineBiology The Johns Hopkins U. School of Medicine Rm. 41 1 Traylor Res. Bldg. 720 Rutland Ave. Baltimore, MD 21205, U.S.A. measured current values, the algorithm attempts to converge to the parameter vector that best fits with the measured data. The next sections describe the model, the simulation- based model fitting methodology, and the results of fitting the model to experimental data. 2 THE CONDUCTIVITY MODEL The model postulates three modes, designated CO, C1, and 02 of the KvLQTl potassium channel, CO and C1 being closed states and 02 being the open state of the channel. It is assumed that the state of the channel can transition between C1 and either CO or 02, but not between CO and 02, and that there are voltage-dependent quantities Kii = K,,-(V), where V is membrane potential, representing the transition rates between states, as shown schematically in Figure 1. A+ CO c1 02. < KlO K21 Figure 1. Schematic of Transitions Between States By the law of mass action, the probabilities CO, C, , or 0, that the channel is in state CO, C1, or 02, respectively, are governed by the three differential equations: 60 = K,&’,- Ko&’o , 02 = KiZC1- K2102, 6, = K0,CO + K,,O, - (K10 + K&, Y (1) where the derivatives are with respect to time, t , and the probabilities are, of course, functions of time 1587