Session S23.2

Ensuring Stability in Models of Atrial Kinetics

A van Oosterom*

University of Lausanne
Lausanne, Switzerland

The membrane kinetics formulation of human atrial myocytes of Courtemanche, Ramirez and Nattel (CRN) [1] is a commonly used model in the studies of atrial electric activity, in particular those of conditions leading to atrial fibrillation. A notorious problem in such models is the slow drift of all state variables. In an application of 1 hr of stimulation of a single unit, the observed transmembrane potential Vm never reached a steady state. More significantly, even when no stimulus was applied, the major state variables like Vm, and the intracellular ion concentrations [Na+], [K+], and [Ca2+], slowly, but continuously wandered, with magnitude changes of up to 40% with respect to their initial values. Recently [2], Jacquemet showed that the CRN model does have stable (constant) behavior for specific choices of the initial conditions of the above four state variables. In this contribution an algorithm is presented for finding such combinations of initial settings for these four state variables that ensure a steady state in the absence of external stimuli on the basis of an assumed value of the intercellular charge density. In turn, these then specify initial values for the remaining ones of the total of the 21 state variables involved. The initial values found in this manner indeed retained a completely stable, constant baseline for all state variables. However, the response to periodic stimuli still exhibited a pronounced drift. This could be effectively solved by introducing a mild feed-back for [Na+] and [K+], easing these back to the original constant steady state values in the absence of stimuli. The algorithm may be used for testing the effect of any conceivable variation of the numerous model parameters on the dynamics involved, while for each of these variations, ensuring a return to constant steady state during any prolonged pause in the stimulation protocol. Various applications have been tested, such as the effect of the calcium conductance, gCal values, on the restitution curve, of the influence of the external value of extracellular[K+]as significant for studying the effect of hypokalemia.
1. Courtemanche, M., R.J. Ramirez, and S. Nattel, Ionic mechanisms underlying human atrial action potential properties: insights from a mathematical model. Am J Physiol, 1998. 275: p. 301-321. 2. Jacquemet, V., Steady-state solutions in mathematical models of atrial cell electrophysiology and their stability. Mathematical Biosciences, 2007. 208: p. 241-269.

(Abstract Control Number: 88)