Simple Model for Identifying Critical Regions in Atrial Fibrillation

Atrial fibrillation (AF) is the most common abnormal heart rhythm and the single biggest cause of stroke. Ablation, destroying regions of the atria, is applied largely empirically and can be curative but with a disappointing clinical success rate. We design a simple model of activation wave front propagation on an anisotropic structure mimicking the branching network of heart muscle cells. This integration of phenomenological dynamics and pertinent structure shows how AF emerges spontaneously when the transverse cell-to-cell coupling decreases, as occurs with age, beyond a threshold value. We identify critical regions responsible for the initiation and maintenance of AF, the ablation of which terminates AF. The simplicity of the model allows us to calculate analytically the risk of arrhythmia and express the threshold value of transversal cell-to-cell coupling as a function of the model parameters. This threshold value decreases with increasing refractory period by reducing the number of critical regions which can initiate and sustain microreentrant circuits. These biologically testable predictions might inform ablation therapies and arrhythmic risk assessment.

Figure 1

(a)–(c) A coupling is represented by a link between two cells. Cells are always coupled longitudinally and with frequency $\nu$ transversely. (a),(b) A resting cell (black) will become excited (white) in the next time step if at least one of its neighboring coupled cells are excited. (b),(c) Once a cell is excited it will enter a refractory state (gray scale) for a duration of time $\tau$. (d) The time course of a cell once it has been excited by an excited neighboring coupled cell

Figure 2

Initiation and self-termination of fibrillation in a simulation with $L=200$, $\tau =50$, $\epsilon =0.05$, $\delta =0.05$, and $\nu =0.18$. See Fig. 1 of the Supplemental Material [32] for the full image and animation. The pacemaker cells self-activate with a period of $T=220$. (a) A planar wave front induced by pacemaker cells along the left boundary at time $t_0$. (b) An opening occurs in the refractory wake (i.e., wave break) and a wave of excitation leaks back on itself. (c) A reentrant circuit forms as a result of this breakthrough. (d) The reentrant circuit self-terminates and planar wave propagation resumes. (e) A critical region which can induce a rotor. The dysfunctional cell (dotted boundary) in the bottom cable may block the propagation wave front. When the wave front in the top cable reaches the rightmost vertical connection, it induces activity in the bottom cable that will also propagate retrograde. If the relevant path length (red line) of the circuit is greater than the refractory period, the cell to the left of the dysfunctional cell will be in a resting state when the retrograde wave front arrives and a reentrant circuit of activation forms. The reentrant circuit terminates when the dysfunctional cell blocks the retrograde wave front. This is the simplest example of a critical region which will induce fibrillationlike behavior in the model.

Figure 3

(a) A phase diagram of the model with $L=200$, $\tau =50$, $\epsilon =0.05$, and $\delta =0.05$. Pacemaker cells self-activate with a period of $T=220$. As $\nu$ decreases from 1, a transition from planar wave fronts ($C$) to a few self-terminating reentrant circuits ($B$) occurs. As $\nu$ is decreased below ${\nu }^{\star }\approx 1-(\delta L^2{)}^{-1/\tau }\approx 0.14$ the system develops multiple self-sustaining reentrant circuits ($A$). The red line is the analytically calculated probability $P_{\text{risk}}=1-{[1-(1-\nu {)}^{\tau }]}^{\delta L^2}$ of having at least one fibrillation-inducing structure, see Fig. 2. The solid blue circles represent the average duration that a system displays nonplanar wave fronts, averaged over 50 realizations of duration ${10}^6$ time steps and error bars represent one standard error of the mean. (b)–(d) Realizations of systems with $\nu =0.9$—planar behavior, $\nu =0.2$—single self-terminating rotor, and $\nu =0.15$—multiple self-terminating rotors. See Fig. 2 of the Supplemental Material [32] for the full image.

Figure 4

(a) A rotor due to a critical structure. (b) An ablation is made by rendering a $20\times 20$ section of tissue around the critical structure unexcitable (cross-hatched red). (c)–(f) The ablation terminates the reentrant circuit and the system returns to a planar wave front phase. See Fig. 3 in the Supplemental Material [32] for the full image and animation.