Critical mass hypothesis revisited: role of dynamical wave stability in spontaneous termination of cardiac fibrillation

Abstract

The tendency of atrial or ventricular fibrillation to terminate spontaneously in finite-sized tissue is known as the critical mass hypothesis. Previous studies have shown that dynamical instabilities play an important role in creating new wave breaks that maintain cardiac fibrillation, but its role in self-termination, in relation to tissue size and geometry, is not well understood. This study used computer simulations of two- and three-dimensional tissue models to investigate qualitatively how, in relation to tissue size and geometry, dynamical instability affects the spontaneous termination of cardiac fibrillation. The major findings are as follows: 1) Dynamical instability promotes wave breaks, maintaining fibrillation, but it also causes the waves to extinguish, facilitating spontaneous termination of fibrillation. The latter effect predominates as dynamical instability increases, so that fibrillation is more likely to self-terminate in a finite-sized tissue. 2) In two-dimensional tissue, the average duration of fibrillation increases exponentially as tissue area increases. In three-dimensional tissue, the average duration of fibrillation decreases initially as tissue thickness increases as a result of thickness-induced instability but then increases after a critical thickness is reached. Therefore, in addition to tissue mass and geometry, dynamical instability is an important factor influencing the maintenance of cardiac fibrillation.

Fig. 1

A: u vs. time for a single cell (left) and a snapshot of u during spiral wave breakup in a 42 × 42 tissue (right) for the Bär model. B: comparison of simulated fibrillation patterns in a 10 × 10-cm² tissue with the Luo and Rudy (LR1) model for an unperturbed and a randomly perturbed initial condition; at 400 ms, the patterns are very different. C: transient time (T_s) distribution in 2-dimensional (2-D) tissue with the Bär model (total 656 transients, ∊ = 0.075, tissue size = 26.25 × 26.25). D: averaged T_s (T_s) vs. number of data points used to calculate T_s for the Bär model in a 21 × 21 (left) and a 24.5 × 24.5 (right) tissue. E: T_s vs. number of data points used to calculate T_s for the LR1 model in a 7.5 × 7.5-cm² (left) and an 8.75 × 8.75-cm² (right) tissue.

Fig. 2

Snapshots showing self-termination of “multiple-wavelet” fibrillation from a simulation in a 10 × 10-cm² homogeneous tissue using the LR1 model. Column 1 (170–190 ms): 2 spiral waves (arrows) at 170 ms collided at 180 ms and disappeared at 190 ms. Column 2 (240–260 ms): spiral waves (arrows) at 240 and 250 ms ran into refractory tails of their previous waves and disappeared at 260 ms. Column 3 (1,610–1,650 ms): spiral pairs (arrows) collided and disappeared. Column 4 (1,660–1,820 ms): the only surviving spiral wave (arrow) moved off the tissue border, and the tissue became quiescent.

Fig. 3

Effects of increasing dynamical instability on spiral wave breakup transient time in multiple-wavelet fibrillation in 10 × 10-cm² homogeneous tissue using the LR1 model. A: 2 action potential duration (APD) restitution curves and their slopes (inset). DI, diastolic interval. Black curve: Ḡ_si = 0.052 mS/cm²; gray curve: Ḡ_si = 0.06 mS/cm², τ_d → 0.75τ_d, τ_f → 0.75τ_f (where Ḡ_si is slow inward conductance and τ_d and τ_f represent activation and inactivation time constants, respectively). B: probability that T_s is longer than time T for APD restitution curves in A. C: T_s for the 2 cases in A. D: number of spiral wave tips vs. time for the 2 cases in A.

Fig. 4

Relation between dynamical instability and spiral wave breakup transient time in homogeneous tissue using the Bär model. A: Lyapunov exponent (λ) vs. ∊. B: T_s vs. ∊ for 24.5 × 24.5 (□) and 26.25 × 26.25 (■) tissues. C: T_s vs. 1/λ for 24.5 × 24.5 (□) and 26.25 × 26.25 (■) tissues.

Fig. 5

Effects of tissue size and geometry on T_s in homogeneous tissue. A and B: T_s vs. tissue area and area-to-perimeter ratio, respectively, for the Bär model. ■, Square tissue; □, rectangular tissue with one side fixed at 21 arbitrary units (AU). C and D: T_s vs. tissue area and area-to-perimeter ratio, respectively, for the LR1 model with the black APD restitution curve in Fig. 3A. Fixed side for rectangular tissue is 6.25 cm. [Theoretically, the function y = a + exp(αx²) should not also be the function y = b + exp(βx), but for a small range of x, a data set may well fit both functions, which may be the case here.]

Fig. 6

Effects of obstacles on T_s. A: T_s vs. tissue area. ●, Replot of T_s in homogeneous square tissue of different sizes; red squares 10 × 10-cm² tissue with a circular hole in the center (radius = 0.5, 1.0, 1.5, or 2.0 cm). B: T_s vs. area-to-outer perimeter ratio [i.e., T_s vs. (100 cm² − πr²)/40 cm, red filled squares] and T_s vs. area-to-total perimeter ratio [i.e., T_s vs. (100 cm² − πr²)/(40 cm + 2πr²), blue open squares] for T_s in A. ●, Replot of T_s in homogeneous square tissue vs. area-to-perimeter ratio. C: snapshots illustrating disappearance of spiral waves due to the obstacle.

Fig. 7

Effects of tissue thickness (L_z) on T_s according to the Bär model with ∊ = 0.075. x- and y-dimensions were fixed as follows: L_x = L_y = 21 AU.