Percolation and tortuosity in heart-like cells

Abstract

In the last several years, quite a few papers on the joint question of transport, tortuosity and percolation have appeared in the literature, dealing with passage of miscellaneous liquids or electrical currents in different media. However, these methods have not been applied to the passage of action potential in heart fibrosis (HF), which is crucial for problems of heart arrhythmia, especially of atrial tachycardia and fibrillation. In this work we address the HF problem from these aspects. A cellular automaton model is used to analyze percolation and transport of a distributed-fibrosis inflicted heart-like tissue. Although based on a rather simple mathematical model, it leads to several important outcomes: (1) It is shown that, for a single wave front (as the one emanated by the heart's sinus node), the percolation of heart-like matrices is exactly similar to the forest fire case. (2) It is shown that, on the average, the shape of the transport (a question not dealt with in relation to forest fire, and deals with the delay of action potential when passing a fibrotic tissue) behaves like a Gaussian. (3) Moreover, it is shown that close to the percolation threshold the parameters of this Gaussian behave in a critical way. From the physical point of view, these three results are an important contribution to the general percolation investigation. The relevance of our results to cardiological issues, specifically to the question of reentry initiation, are discussed and it is shown that: (A) Without an ectopic source and under a mere sinus node operation, no arrhythmia is generated, and (B) A sufficiently high refractory period could prevent some reentry mechanisms, even in partially fibrotic heart tissue.

Figure 1

(A) Several generations of a typical case: Matrix 20 × 20, p = 60%. Generations presented: (a)-1, (b)-2, (c)-8 and (d)-9. Cells of type 1 are in red; those of type 2 in green; and those of type 3 in blue. (B) Exemplary active-cells through-path (in red) near percolation. Matrix 20 × 20, p = 60%. Other active cells during process are in yellow.

Figure 2

(Colors on line) Percolation curves for the four cases. Red (circles)—the 1 → 0 case. Green (triangles)—the 1 → 2 → 0 and the FFM cases. Blue (squares)—the "diagonal" unidirectional case.

Figure 3

(Colors on line) Output (arriving active cells at the last row) for p = 70%. (A) Three individual runs of 100 × 100 matrices with different random distributions of inactive cells. (B) Average output of forty 200 × 200 matrices and the Gaussian fit.

Figure 4

(Colors on line) Least square fits to Gaussian parameters of the output results near percolation threshold. (A) For $x_0$. (B) For AT. Critical behavior is observed.

Figure 5

(Colors on line) Active (AP) cells arrivals at the end row of the matrix for a specific trial of a 50 × 50 fibrotic tissue (p = 75%). Abscissa – No. of cell along the row, Ordinate—Time of arrival (time units from wave starting time).