Mechanisms of vortices termination in the cardiac muscle

Figure 1.. Parallel termination of two pinned vortices with geometrical locations and time positions of the critical (vulnerable) windows (VW) both unknown. The slow vortex 1 (period Tv1=87 ms, pinned to the 1.2 mm defect 1, is entrained by the fast vortex 2 (period Tv2=83 ms, 1.0 mm defect 2. They are paced with electric field directed from top to bottom, |E|=1.3 V cm−1, pulses 2 ms duration, period 100 ms. This induces the phase scanning with the time step s=17 ms. Colour code: red is a wave, green is the wavefront. Time is measured from the start of pacing at t=0 ms. 196 ms: a wave W emitted by vortex 2 enslaves vortex 1. 204 ms: an E-pulse delivered at t=200 ms induces a wave S. 216 ms: the right wavebreak of wave S annihilates with the tip of vortex 1 (they have opposite topological charges). 232 ms: vortex 1 is unpinned and terminated. The left wavebreak of S created a free vortex F. 336…376 ms: F disappears on the boundary. 408…440 ms: Next E-pulse similarly terminates vortex 2. Barkley model, parameters a=0.8, b=0.09, ϵ=0.02.

Figure 2.. Phase scanning. (a) u1(t), v1(t) are recordings from the point just above defect 1, figure 1, and u2(t), v2(t) are the same for defect 2. The bold black lines indicate timing of the delivered E-pulses. Shaded areas are VWs, defined as time intervals where v∈(0.0871,0.18), u<b/a. Seen that in spite of small phase disturbances produced by E-pulses, the topological features of the scanning are not disturbed, scanning successfully terminates the vortices. E-pulse 3 (t=200 ms) reaches VW of vortex 1 and terminates it, compare with figure 1, t=204 ms, where is seen a wave S induced by an E-pulse delivered at t=200 ms. E-pulse 5 (t=400 ms) reaches VW of vortex 2 and terminates it. Compare with figure 1, t=408…440 ms: E-pulse delivered at (t=400 ms) terminates vortex 2. (b,c) Schematic. Superimposed action potentials (AP) are shown. Red arrows indicate timing of the delivered E-pulses; ‘e’ is an excitable gap, s is the scanning step, s=T−Tv. (b) s>0 for T>Tv, scanning reaches the VW. (c) s<0 for faster pacing T<Tv, the scanning moves in the opposite direction. E-pulse reaches the excitable gap ‘e’, excites an AP thus resetting the rotation phase, and all subsequent pulses get into the…

Figure 3.. Termination of a free vortex by an E-pulse. 147 ms: a free vortex and a defect (white). 307 ms: a semicircular wave (with wavebreaks A and B) emitted from the defect by an E-pulse, electric field directed from right to left. 323 ms: wavebreak A fused with the vortex tip. 368 ms: after annihilation of wavebreak B with the border, only a wave without wavebreaks is left in the medium. Barkley model, parameters a=0.6, b=0.075, ϵ=0.02.

Figure 4.. Change of topological charge, creation of phase singularities and vulnerable window (VW). (a–d) One-dimensional mechanism. (a) Nullclines of FHN equations. M is the Maxwell point. The topological charge of a wave pattern is changed by an E-pulse only when an image of a nucleated wave contains the Maxwell point. (b) Wave front velocity θ versus the slow variable v. The value of v corresponding to velocity θ=0 is the ordinate vM of the Maxwell point on panel (a). (c) The topological charge conservation in one dimension. Generic case: an electric pulse 3 ms in duration is delivered far from the tail of an action potential (AP). t=10 ms after pacing: a nucleated wave, very narrow, and the electrode (black square) below it. t=75 ms: the nucleated wave developed into two counter propagating APs. Their total topological charge is zero. (d) Violation of the topological charge conservation. t=10 ms: an electric pulse is delivered closer to the tail of the AP, inside VW. t=75 ms: only one AP is induced. It propagates to the left only. The topological charge is changed. Cardiac ionic model by Majahan et al. [29]. (e,f) two-dimensional mechanism. (e) No phase singularities are created. An…

Figure 5.. Success rate of two vortices termination. Success rate versus normalized frequency f^=f/fd where fd is the dominant frequency. On the image, ‘different’ and ‘close’ frequencies mean significant and insignificant difference in frequencies as per equation (2.2). Numerical calculations with the normalized VW¯ =0.2.

Figure 6.. Fibrillation termination in the isolated pig hearts. The success rate of defibrillation in 486 experiments by five biphasic E-pulses. (a) Success rate versus normalized frequency f^=f/fd where fd is the dominant frequency of fibrillation. Error bars: the standard deviation. The blue curve is obtained by mixture (50 : 50) of two theoretical curves shown in figure 5. (b,c) Success rate for defibrillation energies not exceeding e^, for frequencies f^ shown near each curve. Normalized energy e^=e/e1, where e1 is the threshold E50 energy of defibrillation by 1 shock. (b) shows graphs for f^<1, (c) shows graphs for f^≥1. Graphs (b,c) and the experimental curve in (a) are calculated from data in [38]. Image (a) indicates that in about a half of fibrillation experiments, the frequencies of the vortices were not significantly different. The optimal pacing frequency f^=0.77 is below the arrhythmia frequency (f^<1) as it should be for terminating pinned vortices. These experiments provide evidence that pinned vortices, hidden from direct observation, are significant in fibrillation.