High-frequency pacing of scroll waves in a three-dimensional slab model of cardiac tissue

Vortices in excitable media underlie dangerous cardiac arrhythmias. One way to eliminate them is by stimulating the excitable medium with a period smaller than the period of the vortex. So far, this phenomenon has been studied mostly for two-dimensional vortices known as spiral waves. Here we present a first study of this phenomenon for three-dimensional vortices, or scroll waves, in a slab. We consider two main types of scroll waves dynamics: with positive filament tension and with negative filament tension and show that such elimination is possible for some values of the period in all cases. However, in the case of negative filament tension for relatively long stimulation periods, three-dimensional instabilities occur and make elimination impossible. We derive equations of motion for the drift of paced filaments and identify a bifurcation parameter that determines whether the filaments orient themselves perpendicular to the impeding wave train or not.

Figure 1

Low-voltage cardioversion of a scroll wave with positive filament tension $(a=0.03)$. Pacing period $T_{\mathrm{stim}}=12$. (a) Situation before pacing (red filament, $u$, on bottom surface) and filaments at later moments in time. (b) Average filament length during pacing ($Y$ axis) against medium thickness ($X$ axis). (c) Filaments shown at $t=745$ for different simulations when the pacing was switched off at $t=647$, $t=659$, $t=671$, $t=683$, and $t>745$. The longer the pacing was kept on, the smaller the filament at the fixed observation time. After switching off the pacing, positive tension caused the filament to further shrink and disappear. The bottom face shows $u$ at $t=745$ when pacing was switched off at $t=647$.

Figure 2

Low-voltage cardioversion of a scroll wave with negative filament tension $(a=0.08)$. (a) Situation before pacing (red filament, $u$ on bottom surface), and filaments at later moments (orange), for $T_{\mathrm{stim}}=12$. (b) Average filament length during pacing ($Y$ axis) against medium thickness ($X$ axis). Pacing period $T_{\mathrm{stim}}=21$. (c) Unsuccessful cardioversion for $T_{\mathrm{stim}}=23$. Filaments are shown at $t=100$ (purple), $t=500$ (red), $t=800$ (green), and $t=900$ (blue, also shown in bottom surface).

Figure 3

Evolution of a scroll wave with one end of its filament on the pacing electrode for negative filament tension ($a=0.08$), $L_z=48$, and $T_{\mathrm{stim}}=18$. The electrode is formed by the entire bottom face. Side faces show snapshots of $u$. (a) Before the pacing began ($t=69$); (b) temporary breakup induced by the pacing ($t=148$); (c) a wave with a long curved solitary filament that connects the left and front faces ($t=300$). The superseding ended successfully at $t=490$.

Figure 4

Angle between the filament tangent and the traveling direction of the wave train over time, shown along a filament extending from $z=0$ to $z=L_z$, as shown in Fig. 1. The point electrode is located at $z=0$ such that $\beta ={90}^{\circ }$ there, while it can never reach that value on the opposite medium boundary ($z=L_z$).

Figure 5

Successive snapshots of the interaction between a 2D spiral wave and a wave train from the lower left corner ($a=0.03$, $T_{\mathrm{stim}}=12$, frame spacing is 4 time units). The spiral tip is shown as a black dot.

Figure 6

Orthogonal triad $(\vec{T},{\vec{e}}_{\left|\right|},{\vec{e}}_{\perp })$ and angle $\beta$ in 3D interaction between a wave train with propagation direction $\vec{n}$ and a filament. The vector ${\vec{e}}_{\left|\right|}$ lies in the plane spanned by $\vec{T}$ and $\vec{n}$.