Utilizing the eikonal relationship in strategies for reentrant wave termination in excitable media

Obstacle-anchored vortices can be terminated by the application of high-frequency wave trains in excitable media. We theoretically derived the dependency between the obstacle radius and the maximum unpinning period through reinterpretation of the well-known eikonal equation. Our theoretical result was confirmed by experiments with cardiomyocyte monolayers. This result may be useful for improving the stimulation protocol of implantable cardiac pacemakers.

Figure 1

(Color online) Restitution curves for a round obstacle. (a) The solid blue line is a restitution curve of 2D-plane paced waves. The dotted blue lines are restitution curves of waves on an obstacle with radius $R$. Critical CVs, $v_p$, of the restitution curves of obstacles are marked by blue plus signs for radii between 2.1 and 0.4 cm. The dashed black line is the fit of $v_p\left(T_p,R\right)$. (b) Illustration of the linear dependency of $v_p\left(T_p,R\right)$. An example is given in (c), where a wave train with period $T_e$ is stimulated. Plane waves with CV $v_{1,e}$ propagate toward the round obstacle with radius $R_e$ on the upper plane side, start to curl, reduce the CV to $v_{2,e}$, and finally accelerate back to $v_{1,e}$ on the lower plane side. The change in CV is indicated by a black arrow in (b).

Figure 2

(Color online) Phase diagram of the termination of a pinned spiral wave. The maximum unpinning period, $T_p$, at which the unpinning of a pinned spiral wave on an obstacle with radius $R$, is possible for CS and DS is given in red crosses and blue plus signs, respectively. Above and beneath each curve the spiral wave is still pinned and unpinned, respectively, after the application of stimulation. CS and DS are illustrated below.

Figure 3

(Color online) Numerical results for the maximum unpinning period. (a) Restitution curves for $g_{fi}$ values of 2.35, 2.40, and $2.50\text{mmho}/{\text{cm}}^2$ shown in solid green, blue, and red lines, respectively. Critical pacing periods are shown as squares, plus signs, and crosses in the respective color for radii of 0.4–2.1 mm in steps of 0.1 mm. The slope of Eq. (12) is shown in dashed black lines for each $g_{fi}$ parameter. (b) The phase diagram of the maximum unpinning period, $T_p$, for the three $g_{fi}$ parameters in the same color scheme as in (a). Equation (15) is plotted in dashed black lines.

Figure 4

(Color online) Obstacle curvature of critical velocities. The dependencies of $g_{fi}$ values of 2.35, 2.40, and $2.50\text{mmho}/{\text{cm}}^2$ are shown as green squares, blue plus signs, and red crosses, respectively. The black solid line represents the slope of $-1$.

Figure 5

(Color online) Experimental results with cardiomyocyte monolayers. (a) $T_p$ and the respective $v_p$ of successful unpinning shown in blue squares. Red data points and lines show the restitution curves. Equation (12) is plotted by solid and dashed black lines, by approximating $T_{crit}=300\pm 40\text{ms}$, respectively, $v_{crit}=5.8\text{cm}/\text{s}$ and $\overline{v}=122\text{cm}/\text{s}$. (b) Phase diagram of the maximum unpinning period, $T_p$, for obstacles of different sizes. Equation (15) is plotted with the value obtained in (a). The diffusion constant was assumed to be $1.0{\text{cm}}^2/\text{s}$.