Negative Curvature Boundaries as Wave Emitting Sites for the Control of Biological Excitable Media

Understanding the interaction of electric fields with the complex anatomy of biological excitable media is key to optimizing control strategies for spatiotemporal dynamics in those systems. On the basis of a bidomain description, we provide a unified theory for the electric-field-induced depolarization of the substrate near curved boundaries of generalized shapes, resulting in the localized recruitment of control sites. Our findings are confirmed in experiments on cardiomyocyte cell cultures and supported by two-dimensional numerical simulations on a cross section of a rabbit ventricle.

Figure 1

Boundary effects in two-dimensional cardiac cell culture experiments (homogeneous cell distribution). Cultured neonatal rat cardiac cells were stimulated with vertical electric fields (white arrows) of (a) 0.91 and (b) $1.37\mathrm{V}/\mathrm{cm}$, and activity was imaged by using the fluorescent ${\mathrm{Ca}}^{2+}$ indicator Fluo-8 (for details, see Supplemental Material [27]). Colors indicate time from pacing to activation; gray color marks regions without tissue. At higher field strengths, the flat parts of the boundary are excited as well as the tip of the $V$-shaped boundary. For the lower electric field strength, excitation can be induced only at the latter location. A series of $N=9$ experiments yielded a field-strength ratio of $1.49\pm 0.26$ between the two types of boundaries.

Figure 2

Stationary membrane potential distributions for infinitesimal electric fields. (a)–(d) Two-dimensional geometries. White color indicates regions that are not part of the active tissue domain $\mathcal{D}$. The arrows indicate the location of maximum depolarization. (e),(f) Three-dimensional geometries. The surfaces indicate the boundary of the excitable media. The mean curvature at the location of maximum depolarization is identical for all six cases and $\approx \pm 0.34$ in units of $1/\lambda$ (cf. Fig. 3). All length units on the axes are given in units of $\lambda$; the depolarization $e^{*}$ is in units of $\lambda |\mathbf{E}|$.

Figure 3

Maximum depolarizations occurring for the geometries shown in Fig. 2 (see the legend). Solid lines refer to analytical or semianalytical solutions of Eqs. (7, 8), dashed lines denote purely numerical solutions, and the dots are numerical values to confirm the analytical calculations.

Figure 4

Numerical simulation of the Fenton-Karma model on the complex anatomical geometry of a two-dimensional cross section of the left ventricle (rabbit, micro-CT). An electric field (field strength $E$, duration 5 ms) results in curvature-dependent activation. (a) $E=0.2\mathrm{V}/\mathrm{cm}$: Wave emission is observed at a protuberance with large negative curvature, while depolarization at the boundary remains below threshold. (b) $E=0.4\mathrm{V}/\mathrm{cm}$: Excitation of almost the entire tissue boundary results in a quasiplanar wave. (c) $E=1.0\mathrm{V}/\mathrm{cm}$: Internal boundaries with positive curvature (blood vessels indicated by arrows) emit waves.