Uniqueness and stability of action potential models during rest, pacing, and conduction using problem-solving environment

Abstract

Development and application of physiologically detailed dynamic models of the action potential (AP) and Ca2+ cycling in cardiac cells is a rapidly growing aspect of computational cardiac electrophysiology. Given the large scale of the nonlinear system involved, questions were recently raised regarding reproducibility, numerical stability, and uniqueness of model solutions, as well as ability of the model to simulate AP propagation in multicellular configurations. To address these issues, we reexamined ventricular models of myocyte AP developed in our laboratory with the following results. 1), Recognizing that the model involves a system of differential-algebraic equations, a procedure is developed for estimating consistent initial conditions that insure uniqueness and stability of the solution. 2), Model parameters that can be used to modify these initial conditions according to experimental values are identified. 3), A convergence criterion for steady-state solution is defined based on tracking the incremental contribution of each ion species to the membrane voltage. 4), Singularities in state variable formulations are removed analytically. 5), A biphasic current stimulus is implemented to completely eliminate stimulus artifact during long-term pacing over a broad range of frequencies. 6), Using the AP computed based on 1-5 above, an efficient scheme is developed for computing propagation in multicellular models.

Figure 1

Ventricular myocyte model. Symbols are defined in the Supporting Material and in the research section of http://rudylab.wustl.edu.

Figure 2

Time course to steady state of ion concentrations and membrane potential in a quiescent canine cell model for different initial conditions. (A) (Clockwise) membrane potential (V_m), myoplasmic concentration of Cl⁻ ([Cl⁻]_i), Na⁺ ([Na⁺]_i), and K⁺ ([K⁺]_i). (B) (Clockwise) myoplasmic concentration of Ca²⁺ ([Ca⁺]_i), total Ca²⁺ concentrations in junctional SR ([Ca⁺]_j,t), total Ca²⁺ concentration in the subspace ([Ca⁺]_s), and Ca²⁺ concentration in network SR ([Ca⁺]_n). (1, solid line) [K⁺]_i = 135 mM, [Na⁺]_i = 8.65 mM, [Cl⁻]_i = 19.23 mM; (2, dashed line) [K⁺]_i = 137 mM [Na⁺]_i = 12.65 mM, [Cl⁻]_i = 19.23; (3, dotted line) [K⁺]_i = 138 mM, [Na⁺]_i = 5.65 mM, [Cl⁻]_i = 19.23; and (4, shaded line) [K⁺]_i =138 mM, [Na⁺]_i = 10.65 mM, [Cl⁻]_i = 18.23 mM; all other initial conditions are the same for all traces.

Figure 3

(A) Ion concentration and membrane potential in a quiescent guinea-pig cell model. Time course starting from assignment of three sets of initial ion concentrations: (1, solid line) [K⁺]_i = 142.2 mM and [Na⁺]_i = 5.6 mM; (2, dashed line) [K⁺]_i = 147.2 mM and [Na⁺]_i = 15.6 mM; and (3; dotted line) [K⁺]_i = 137.2 mM and [Na⁺]_i = 10.6 mM; all other state variables have the same initial values for all traces. (B) Effects of sodium background conductance (G_Na,b) on the steady-state values of ion concentrations and V_m. G_Na,b is normalized to its normal value of 0.004 mS/μF.

Figure 4

Dynamic transient response of guinea pig (A) and canine (B) cell models to perturbations of ion concentrations from a dynamic steady state during pacing at 1 Hz. [K⁺]_i was increased while [Na⁺]_i was decreased by 2 mM at the 10th beat (arrow); for this perturbation, Eq. 6 is conserved. (Top to bottom) APD₉₀, total end-diastolic [Ca²⁺]_j,t, minimum [Ca²⁺]_i in myoplasm [Ca²⁺]_i,min, and maximum [Ca²⁺]_i in myoplasm [Ca²⁺]_i,max. All concentrations recovered the same values they attained before the perturbation (see text for details).

Figure 5

Ionic balance during a cardiac cycle in the cell model at periodic steady state during pacing at CL = 400 ms. (A) Canine and (B) guinea pig. ΔV_Na (dashed gray), ΔV_Ca (dotted black), ΔV_K (solid gray), and ΔV_Cl (dotted gray) are contributions of Na⁺, Ca²⁺, K⁺, and Cl⁻ ions, respectively, to the membrane voltage change (ΔV_m; solid black) as defined in Eq. 8.

Figure 6

Rate dependence during pacing by monophasic (shaded) and biphasic (solid) stimulus current. (A) Canine and (B) guinea pig. Steady-state end-diastolic values of [Na⁺]_i, [K⁺]_i, V_m,rest, and APD₉₀ are shown. Note bifurcation of APD₉₀ into alternans mode at fast rate (>3 Hz).

Figure 7

AP propagation in one-dimensional fiber. (A) Guinea pig. (B) Canine. Both models were simulated with the same geometrical and electrical parameters. Diffusion coefficient D = 0.000856 cm²/ms results in conduction velocity of 46 cm/s in both models. At the simulated pacing frequency of 5 Hz (CL = 200 ms), AP exhibits alternans in both models. APD alternans in the canine (APD_long = 160 ms; APD_short = 130 ms) is of greater amplitude than in the guinea pig (APD_long = 90 ms; APD_short = 80 ms).