Analysis of damped oscillations during reentry: a new approach to evaluate cardiac restitution

Abstract

Reentry is a mechanism underlying numerous cardiac arrhythmias. During reentry, head-tail interactions of the action potential can cause cycle length (CL) oscillations and affect the stability of reentry. We developed a method based on a difference-delay equation to determine the slopes of the action potential duration and conduction velocity restitution functions, known to be major determinants of reentrant arrhythmogenesis, from the spatial period P and the decay length D of damped CL oscillations. Using this approach, we analyzed CL oscillations after the induction of reentry and the resetting of reentry with electrical stimuli in rings of cultured neonatal rat ventricular myocytes grown on microelectrode arrays and in corresponding simulations with the Luo-Rudy model. In the experiments, P was larger and D was smaller after resetting impulses compared to the induction of reentry, indicating that reentry became more stable. Both restitution slopes were smaller. Consistent with the experimental findings, resetting of simulated reentry caused oscillations with gradually increasing P, decreasing D, and decreasing restitution slopes. However, these parameters remained constant when ion concentrations were clamped, revealing that intracellular ion accumulation stabilizes reentry. Thus, the analysis of CL oscillations during reentry opens new perspectives to gain quantitative insight into action potential restitution.

Figure 1

Induction and resetting of reentry in a patterned cardiomyocyte ring. (A) Photographs of a microelectrode array (left) and of a patterned ring (right). (B) Schematic of the ring, covering 16 recording electrodes (dots) and four stimulation dipoles. S1 and S2 indicate the dipoles used to induce reentry. (C) Electrograms illustrating the induction of reentry. Stimulus S2 was applied during the vulnerable window of the S1-induced AP and resulted in unidirectional conduction, generating reentry. (D) Resetting of reentry by a stimulus (S) applied during the excitable gap with the stimulation dipole S2.

Figure 2

Damped oscillations of CL and CV of reentrant APs immediately after induction (A) and after resetting (B). Data point colors correspond to the schematic (insets). The oscillations were fitted with a damped sine function (shaded curve) to determine their spatial period P and decay length D (P = 1.58 and D = 6.55 in A; P = 1.67 and D = 0.98 in B, respectively). (C) In this example, upon the induction of reentry, CL exhibited a slow decreasing trend (left, dotted curve). The oscillatory dynamics of CL were therefore analyzed in the detrended CL series (right).

Figure 3

Comparison of P and D at induction of reentry and after resetting. (A) Values of P and D in the P-D parameter space, after induction (solid symbols) and after resetting (open symbols) of reentry (legend in the inset). Nonconnected solid squares represent preparations in which resetting was not followed by measurable CL oscillations. Solid circles correspond to preparations in which resetting attempts either failed or stopped reentry. Nonconnected open squares represent reentry resetting in preparations where spontaneous reentry was already present at the beginning of the experiment. (B) Compared to the induction of reentry, D was smaller after resetting (p < 0.05) and P was larger (p < 0.05).

Figure 4

(A) Induction and (B) resetting of reentry in a ring of 500 Luo-Rudy phase 1 model cells (length: 5 cm) and evaluation of restitution slopes from restitution curves and from P and D. The dotted curves represent the exponential envelopes of the CL oscillations. The slopes α_dyn, γ_dyn, and γ_dynCL_stable/L were computed from the plotted restitution curves and compared with those calculated from P and D.

Figure 5

Dependence of AP parameters, P, D, and restitution slopes on the density of I_K (A), I_Na (B), and the kinetics of the inactivation gate j of I_Na (C), after induction (solid) and resetting (shaded) of reentry in a ring of 200 Luo-Rudy phase-1 model cells. Initial model parameters (marked by vertical dotted lines) were: gK_max coefficient, 2; gNa_max, 8 mS/cm²; and τ_j divisor, 1. Solid lines indicate values obtained from the simulation. Dashed lines represent restitution slopes calculated from P and D. CL, APD, DI, and CV correspond to stable values after the damped oscillations had dissipated.

Figure 6

Effects of intracellular Na⁺ and Ca²⁺ accumulation on CL oscillations in a ring of 700 Luo-Rudy phase-2 model cells (length, 7 cm). (A) CL oscillations after induction and after resetting pulses applied at successive timings in the nominal model (top), during clamp of [Na⁺]_i and [K⁺]_i (middle), and during clamp of all ion concentrations (bottom). (B) Corresponding changes of [Na⁺]_i, diastolic [Ca²⁺]_i and stable values of CL, APD, and DI. (C) Corresponding changes in P and D. (D) Corresponding changes of restitution slopes.

Figure 7

Comparison of restitution slopes calculated from P and D of CL oscillations recorded in the cultured cardiomyocyte rings after induction and resetting of reentry. (A) Values of α and γCL_stable/L in the α−γCL_stable/L parameter space, after induction (solid symbols) and resetting of reentry (open symbols), for the data shown in Fig. 3 (same data and same symbols). (B) Compared to the induction of reentry, α, γCL_stable/L, and γ were all smaller after resetting (p < 0.05).

Figure 8

Spatiotemporal relationship of conduction parameters t, c, a, r, and p (defined in the Appendix). The thick curve illustrates the movement of the wavefront and the thin curve denotes the movement of the repolarization tail. The shaded area denotes the depolarized state. Cumulated distance x was wrapped for every circuit length L. Thus, x and x−L correspond to the same position in the circuit. The thick vectors represent conduction velocities c(x) and c(x−L), for two successive passages of the wavefront at distances x and x−L, respectively.

Figure 9

Behavior of the APD restitution slope α (blue) and the normalized CV restitution slope $\gamma p_{\text{st}}^2/L$ (red) as a function of P and D. Interestingly, certain combinations of P and D can exist only for negative values of α.