Termination of pinned vortices by high-frequency wave trains in heartlike excitable media with anisotropic fiber orientation

A variety of chemical and biological nonlinear excitable media, including heart tissue, exhibit vortices (spiral waves) that can anchor to nonexcitable obstacles. Such anchored vortices can be terminated by the application of high-frequency wave trains, as shown previously in isotropic excitable media. In this study, we examined the basic dependencies of the conduction velocities of planar waves and waves around curved obstacles as a function of anisotropy through numerical simulations of excitable media that mimic the fiber orientation in a real heart. We also investigated the unpinning of anchored spiral waves by high-frequency wave trains in an anisotropic excitable medium. Unlike the findings regarding the termination of spiral waves in isotropic excitable systems, we found a nonmonotonic relationship between the maximum unpinning period and the obstacle radius depending on the fiber orientation, where the formation of unwanted secondary pinned vortices or chaotic waves is seen over a wide range of parameters.

Figure 1

Examples of pattern formation in isotropic and anisotropic excitable media. (a) and (b) show free spiral waves, and (c) and (d) target waves that originate at the center of the media. (a) and (c) show wave propagation in isotropic media ($d_r=1$), and (b) and (d) show wave propagation in anisotropic media ($d_r=5$) with a fiber rotation angle of $\Phi ={45}^{\circ }$. The white arrows indicate the main fiber orientation. The wave-tip trajectory of spiral waves is illustrated with white dashed lines and shows a closed circle and an oval loop, respectively, which is due to the choice of parameters.

Figure 2

Plane wave propagation in anisotropic excitable media. (a) shows the restitution curves for waves in the direction $\Phi =0^{\circ }$, ${30}^{\circ }$, ${60}^{\circ }$, and ${90}^{\circ }$ with respect to the fiber orientation for an anisotropy ratio of $d_r=5$. $T_{\mathrm{crit}}$ is shown as red data points for $\Phi$ in steps of ${10}^{\circ }$. Corresponding $c_{\sin }$ and $c_{\mathrm{crit}}$ are shown in (b). The black solid line shows the analytic solution of $c_{\sin }\left(\Phi \right)$ according to Eq. (19), and the black dashed line shows the estimation of $c_{\mathrm{crit}}\left(\Phi \right)$ according to Eq. (20) with ${\lambda }_{\mathrm{pw}}=0.75$. (c) and (d) show $c_{\sin }$ and $c_{\mathrm{crit}}$, respectively, similar to (b) for different $\Phi$. The black dashed lines in (d) illustrate Eq. (20) with ${\lambda }_{\mathrm{pw}}=0.75$. Thus, ${\lambda }_{\mathrm{pw}}$ is shown to be an independent function of $d_r$ for plane propagating waves.

Figure 3

Example of wave propagation on finger-shaped obstacles. (a) and (b) show high-frequency paced wave propagation on a finger-shaped obstacle with a radius of 1 cm in an anisotropic medium ($d_r=5$ and $\Phi =0^{\circ }$). The white arrow in (b) indicates the main fiber orientation. The obstacle contour is emphasized by a white dotted line in (a). Waves originate at the top right of the obstacles using a dynamic stimulation protocol [see Eq. (17)]. The pacing period $T_p\left(n\right)$ is 123.8 and 92.2 ms in (a) and (b), respectively. Arrows in (a) illustrate the change in CV from $c_{\mathrm{l}}$ to $c_{\mathrm{c}}$ and back to $c_{\mathrm{l}}$ depending on the obstacle curvature. (b) shows a wave that detached from the obstacle. The wave tip is shown as a white dashed line from the time of detachment.

Figure 4

Conduction velocity on finger-shaped obstacles. (a) shows the CV of single propagating waves and (b) shows the critical CV of high-frequency paced waves as a function of the obstacle curvature for various $d_r$. The black solid and dashed lines in (a) and (b) show the respective linear approximations. (c) shows the slopes obtained from (a) and (b) for $c_{\sin }$ and $c_{\mathrm{crit}}$ as round and square data points, respectively. The black solid lines show the best fit of arbitrary power-law functions with the powers $-0.04$ and $-0.36$, respectively. (d) illustrates the effective diffusivities ($D_{\mathrm{eff}}$) following the relation in Eq. (26). The red and white squares show $D_{\mathrm{eff}}$ as a function of $d_r$ and $d_r^{-1}$, respectively. The black dashed line shows Eq. (26) considering the power-law functions obtained in (c).

Figure 5

Termination of a pinned spiral wave in an anisotropic excitable medium by a high-frequency wave train. The obstacle radius is 0.6 cm, the fiber orientation is $\Phi =0^{\circ }$, and the anisotropy ratio is $d_r=5$. The white arrow in (a) indicates the main fiber orientation. Planar waves originate on the top. Snapshots are illustrated at pacing periods of 180, 114, 101, 98, and 97 ms, respectively, following the dynamic pacing protocol with an iteration period of 1 ms [see Eq. (17)]. The maximum unpinning period is 101 ms, as illustrated in (c), and subsequent approaching waves push the free spiral wave to the boundary of the system (d), where it finally terminates (e).

Figure 6

Maximum unpinning period from a round obstacle in anisotropic excitable media ($d_r=5$). (a) shows the dependence of fiber orientation $\Phi$ and the maximum unpinning period $T_{\mathrm{unp}}$ for obstacle radii, $R$, of 0.3, 0.6, and 1.0 cm in red, blue, and green lines, respectively. The critical pacing period for planar waves ($R\to \infty$) is shown as a black line. (b) shows the the dependence of $R$ and $T_{\mathrm{unp}}$ for $\Phi =30$ and ${70}^{\circ }$ as red data points and blue squares, respectively. The dashed black line shows the approximation of Eq. (27) for $\Phi ={30}^{\circ }$ (see also Ref. [12]). The corresponding $\Phi$ are highlighted as black dashed lines in (a). (c) shows the dependence of $R$, $\Phi$, and the unpinning period $T_{\mathrm{unp}}$ as a heat map. The white solid line encircles the parameter space, where double-pinned spiral waves are formed temporarily before unpinning of the spiral wave or the development of chaotic wave formation.

Figure 7

Secondary pinned spiral waves due to an asymmetric conduction block. The obstacle radius is 1.3 cm, the fiber orientation is $\Phi ={70}^{\circ }$, and the anisotropy ratio is $d_r=5$. The white arrow in (a) indicates the main fiber orientation. Planar waves originate on the top. (a) shows the obstacle-detached wave arm of the approaching planar waves due to local conduction block. In (b), the detached wave is pushed toward the boundary and two pinned spiral waves remain on the obstacle, which is consistent with a topological charge of $+$2. The pacing period in (a) and (b) is 89 and 87 ms, respectively. The circle segment $\omega$ in (a) identifies the window for the possible induction of multiple pinned spiral waves, and the asterisk shows the position of highest diffusivity on the obstacle within the range of curling waves (lower half of the obstacle). This example ($\Phi ={70}^{\circ }$ and $R=1.3$ cm) corresponds to the example shown in Fig. 6b.