The role of pulse timing in cardiac defibrillation

Figures

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  FIGURE 1. An example for an unsuccessful termination attempt for chaotic cardiac excitation using N pert = 500 randomly distributed perturbation sites (visible as yellow dots immediately after the shock at t = 0 ms). White circles indicate phase singularities. At t = 0 ms the phase singularities were computed immediately before the perturbation. After 500 ms, the state still exhibits phase singularities and since they are considered to be the organizing centers of the spiral wave dynamics, they clearly indicate that the termination attempt failed.

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  FIGURE 2. Successful termination of spiral wave dynamics of cardiac excitation by applying N pert = 1,000 local stimuli to the same state as shown in Figure 1. White circles indicate phase singularities and were computed for t = 0 ms before the perturbation was applied. Even though the number of phase singularities increases shortly after applying the perturbations (not shown here), they died out already 30 ms after applying the perturbations and the remaining excitation converges quickly to the steady state as shown exemplary at t = 100 ms.

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  FIGURE 3. Dose-response curve. The dependence of the probability S of a successful termination of the chaotic dynamics on the number of perturbation sites, N pert, corresponding to the strength of the applied shocks. Each point corresponds to 20 × 50 = 1,000 simulations. The simoid fit S = 100/(1 + c exp (−kN pert)) is obtained using nonlinear least squares (NLS) to determine the parameters c = 5,632 and k = 0.01106. The inflection point, as well as the average success rate of 4.3 % for the case of using N pert = 500 perturbation sites are indicated, the latter being referred to in the rest of this study.

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  FIGURE 4. Distributions of success rates (color coded) for different numbers of activated perturbation sites, N pert. Note that the colorbar, indicating the occurrence of a success rate range, is chosen logarithmically and normalized to the most frequently occurring success rate range per N pert.

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  FIGURE 5. Time series of the success rate, S(t appl), for perturbing a trajectory of chaotic cardiac excitation. With a temporal resolution of dt appl = 10 ms, the states along the trajectory are perturbed using 100 different spatial configurations of N pert = 500 perturbation sites that were randomly chosen before starting the analysis and kept fixed along the trajectory. For each of the 100 defibrillation attempts, the success rate was evaluated according to Section 2 and the spiral wave dynamics termination success was estimated in terms of whether phase singularities still exist 500 ms after the perturbation. The temporal average of the success rate over the whole trajectory is S¯≈6.5% . The peaks have a certain width and are not just particularly favorable points in time for shocking, as shown exemplary for a higher resolved peak in Figure 6.

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  FIGURE 6. Higher-resolution plot of the peak highlighted in Figure 5 at 3.99 s for N pert = 500, using smaller time steps for the application of local stimuli, dt appl = 0.1 ms. The full width at half maximum (FWHM) is roughly 25 ms. However, the whole peak and thus, the time window with a success rate being (in parts significantly) higher than the average success rate S¯ has a width of about 40 ms.

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  FIGURE 7. Time series of the termination success rate of a single trajectory of chaotic cardiac excitation for different amounts of local stimuli applied N pert to the system. S¯ represents the corresponding average success rate of the time series S(t). Perturbations are applied as described in the caption of Figure 5.

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  FIGURE 8. Time series of the termination success rate of a single chaotic trajectory for different magnitudes of the perturbations, ΔVm2×2 , applied to the system. S¯ represents the average success rate of the respective time series.

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  FIGURE 9. Probability density of waiting times for the next peak (with specific minimal heights, respectively) to occur inside one trajectory, averaged over all states of 100 trajectories that do not correspond to a peak of the success rate, see Eq. 10.

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  FIGURE 10. Mean amplitude spectra of the success rate and the membrane voltage averaged over 100 realizations (trajectories). Note that the hat denotes the normalization of both quantities before the carrying out the FFT in order to obtain comparable magnitudes and does not itself denote an FFT.