^{1,a)}, Ana Simic

^{1}, Jorge Elorza

^{1}, Roman O. Grigoriev

^{2}, Elizabeth M. Cherry

^{3}, Robert F. Gilmour Jr.

^{4}, Niels F. Otani

^{3,5}and Flavio H. Fenton

^{2}

### Abstract

In this article, we compare quantitatively the efficiency of three different protocols commonly used in commercial defibrillators. These are based on monophasic and both symmetric and asymmetric biphasic shocks. A numerical one–dimensional model of cardiac tissue using the bidomain formulation is used in order to test the different protocols. In particular, we performed a total of 4.8 × 10^{6} simulations by varying shock waveform, shock energy, initial conditions, and heterogeneity in internal electrical conductivity. Whenever the shock successfully removed the reentrant dynamics in the tissue, we classified the mechanism. The analysis of the numerical data shows that biphasic shocks are significantly more efficient (by about 25%) than the corresponding monophasic ones. We determine that the increase in efficiency of the biphasic shocks can be explained by the higher proportion of newly excited tissue through the mechanism of direct activation.

In the present paper, we show how numerical simulations can be used to understand the efficiency of different defibrillation protocols. Fibrillation is a rapid, irregular electrical activity of the heart. This fatal medical condition is usually treated by the application of an external electric shock to the patient chest through external paddle electrodes. The shape of the electric waveforms that are usually applied are either monophasic or biphasic. This means that in the latter the polarity is switched at some point in the course of the application of the shock. Empirical observations suggest that biphasic shocks are more efficient than monophasic shocks in terminating fibrillation. In this paper, by using a simplified mathematical model of cardiac tissue, which, however, includes a realistic response of the cells to large electric fields, we confirm and explain this experimental observation. The model developed here could be used in subsequent studies in order to design and test more complex waveforms, which could be done systematically because the model is simple and not very computationally costly. The next goal is to find the optimal waveform that reduces the energy needed for defibrillatory shocks. This would be of great benefit for patients undergoing defibrillation by limiting the damage to the heart tissue caused by such a strong electric shock.

The financial support from the “Salvador Madariaga” program PR2011-0168 (J.B.) and the research grant FIS2011-28820-C02-02 from the Ministry of Education and Sciences of Spain are acknowledged. A portion of this work (N.F.O., F.H.F., and R.F.G.) was supported by the National Heart, Lung and Blood Institute of the National Institutes of Health, Award No. R01HL089271. This material is also partially based upon work supported by the National Science Foundation under Grant Nos. 1028133 (R.O.G.) and CMMI–1028261 (E.M.C. and F.H.F.). The content is solely the responsibility of the authors, and does not necessarily represent the official views of the NIH.

I. INTRODUCTION

II. MODEL AND NUMERICAL SIMULATIONS

A. The model

B. Comparing three standard defibrillation protocols

C. Parameters influencing the defibrillation outcome

D. Mechanisms for elimination of reentrant dynamics

III. MONTE–CARLO SIMULATIONS

A. Automatic classification of the simulation data by artificial neural networks

B. Dose-response curves

C. Relation between the dynamics, timing and shock outcome

IV. CONCLUSIONS AND FURTHER RESEARCH

### Key Topics

- Action potential propagation
- 18.0
- Electric fields
- 15.0
- Electrodes
- 12.0
- Heart
- 12.0
- Shock wave effects
- 8.0

## Figures

Schematic of the annular ring of cardiac tissue. Two diametrically opposed actuating electrodes are connected to the extra-cellular domain (shown by the two arrows) and can inject or subtract electrical charges during defibrillatory shocks. A traveling action potential is also shown.

Schematic of the annular ring of cardiac tissue. Two diametrically opposed actuating electrodes are connected to the extra-cellular domain (shown by the two arrows) and can inject or subtract electrical charges during defibrillatory shocks. A traveling action potential is also shown.

Space–time plot showing the wave dynamics on the ring. The color scale represents the membrane potential Vm ranging from −90 to +40 mV. Note that the hyperpolarized—(around −136 mV) and depolarized—(around +155 mV) regions are out of scale at time t = 0. The undisturbed dynamics (t < 0) represents discordant alternans. At t = 0 a monophasic shock of 8 ms duration with a corresponding electric field intensity of E = 2 V/cm is applied. In this particular case, the shock leads to a suppression of the wave propagation. The two electrodes are located at π/2 and 3π/2 along the ring (shown by thick white segments in the vertical axis).

Space–time plot showing the wave dynamics on the ring. The color scale represents the membrane potential Vm ranging from −90 to +40 mV. Note that the hyperpolarized—(around −136 mV) and depolarized—(around +155 mV) regions are out of scale at time t = 0. The undisturbed dynamics (t < 0) represents discordant alternans. At t = 0 a monophasic shock of 8 ms duration with a corresponding electric field intensity of E = 2 V/cm is applied. In this particular case, the shock leads to a suppression of the wave propagation. The two electrodes are located at π/2 and 3π/2 along the ring (shown by thick white segments in the vertical axis).

The three shock waveforms analyzed in this paper: monophasic, biphasic I (symmetric), and biphasic II (asymmetric). Iext is the current injection term that appears in Eq. (2) . The currents shown here are the ones that are applied by the electrode that is located at position ; the currents applied by the other electrode (located at ) have the same magnitude but opposite polarity.

The three shock waveforms analyzed in this paper: monophasic, biphasic I (symmetric), and biphasic II (asymmetric). Iext is the current injection term that appears in Eq. (2) . The currents shown here are the ones that are applied by the electrode that is located at position ; the currents applied by the other electrode (located at ) have the same magnitude but opposite polarity.

Color space-time plots of Vm showing the four different mechanisms for reentrant dynamics removal: (a) mechanism of Direct Block (DB) (E = 1 V/cm, monophasic); (b) mechanism of Annihilation (An) of two counter-propagating fronts (E = 3 V/cm, biphasic I); (c) mechanism of Delayed Block (De) showing that a single wave encounters a refractory region and is finally blocked (E = 4 V/cm, monophasic); (d) mechanism of Direct Activation (DA) showing that a large proportion of tissue is excited and then relaxed to the rest state (E = 6 V/cm, biphasic II). Note that for all the four plots (a)–(d), the horizontal time scale is not constant. The time resolution is enlarged by one order of magnitude up to 18 ms in order to highlight the effect of the shock. The shock is always initiated at t = 0.

Color space-time plots of Vm showing the four different mechanisms for reentrant dynamics removal: (a) mechanism of Direct Block (DB) (E = 1 V/cm, monophasic); (b) mechanism of Annihilation (An) of two counter-propagating fronts (E = 3 V/cm, biphasic I); (c) mechanism of Delayed Block (De) showing that a single wave encounters a refractory region and is finally blocked (E = 4 V/cm, monophasic); (d) mechanism of Direct Activation (DA) showing that a large proportion of tissue is excited and then relaxed to the rest state (E = 6 V/cm, biphasic II). Note that for all the four plots (a)–(d), the horizontal time scale is not constant. The time resolution is enlarged by one order of magnitude up to 18 ms in order to highlight the effect of the shock. The shock is always initiated at t = 0.

Fitted logistic curves (see Eq. (9) ) for the three different shock protocols: Monophasic (black); Biphasic I (red) and Biphasic II (green). Also depicted are the box plots showing the dispersion in the results due to the heterogeneities in the internal conductivity. In order to avoid overlap of the box plots, we have shifted to the left (by 1/3 V/cm) all the box plots associated with the monophasic protocol (in black) and we have shifted to the right (also by 1/3 V/cm) all the box plots associated with the biphasic II protocol (in green). The box plots associated with the biphasic I protocols (in red) as well as all the logistic curves have not been shifted. The horizontal dashed lines at 50% and 90% are plotted to ease the comparison between the three protocols. The information about the defibrillation mechanisms at work at selected values (E = 1; 3; 5; 7 V/cm) of the energy is also displayed. The color coding is the following: DB (purple); An (yellow); De (blue); DA (orange).

Fitted logistic curves (see Eq. (9) ) for the three different shock protocols: Monophasic (black); Biphasic I (red) and Biphasic II (green). Also depicted are the box plots showing the dispersion in the results due to the heterogeneities in the internal conductivity. In order to avoid overlap of the box plots, we have shifted to the left (by 1/3 V/cm) all the box plots associated with the monophasic protocol (in black) and we have shifted to the right (also by 1/3 V/cm) all the box plots associated with the biphasic II protocol (in green). The box plots associated with the biphasic I protocols (in red) as well as all the logistic curves have not been shifted. The horizontal dashed lines at 50% and 90% are plotted to ease the comparison between the three protocols. The information about the defibrillation mechanisms at work at selected values (E = 1; 3; 5; 7 V/cm) of the energy is also displayed. The color coding is the following: DB (purple); An (yellow); De (blue); DA (orange).

2D histograms of the probability of reentrant dynamics removal as a function of the two parameters and (see text for explanation) for E = 1 V/cm. The top, middle, and bottom rows are for monophasic, biphasic I, and biphasic II, respectively. Subscripts denote different reentrant dynamics removal mechanisms: DB = 1, An = 2, De = 3, and the total for all mechanisms = a. Note that the fourth mechanisms (DA) is not present at low energies and is therefore not shown in this figure.

2D histograms of the probability of reentrant dynamics removal as a function of the two parameters and (see text for explanation) for E = 1 V/cm. The top, middle, and bottom rows are for monophasic, biphasic I, and biphasic II, respectively. Subscripts denote different reentrant dynamics removal mechanisms: DB = 1, An = 2, De = 3, and the total for all mechanisms = a. Note that the fourth mechanisms (DA) is not present at low energies and is therefore not shown in this figure.

Same as Fig. 6 for E = 3 V/cm.

Same as Fig. 6 for E = 5 V/cm. Note that DB mechanism is not shown here because it is vanishingly small. The histogram for the DA mechanism (sub-index = 4) is shown instead.

Same as Fig. 6 for E = 5 V/cm. Note that DB mechanism is not shown here because it is vanishingly small. The histogram for the DA mechanism (sub-index = 4) is shown instead.

Same as Fig. 8 for E = 7 V/cm.

Histograms showing the time distribution of the disappearance of the last surviving wavefront in the simulations for four shock energy levels (a)–(d), corresponding to E = 1, 3, 5, and 7 V/cm, respectively. For each group (a)–(d), the upper, medium and lower sub-graphs indicate monophasic, biphasic I and biphasic II, respectively. The vertical scale of the histogram is in thousands of shock events. The bars all have 20 ms horizontal width. Note that the total number of events for all the subgraphs is the same (160 000). The bar colors indicate the mechanism by which reentrant dynamics removal occurred: DB (purple); An (yellow); De (blue); DA (orange).

Histograms showing the time distribution of the disappearance of the last surviving wavefront in the simulations for four shock energy levels (a)–(d), corresponding to E = 1, 3, 5, and 7 V/cm, respectively. For each group (a)–(d), the upper, medium and lower sub-graphs indicate monophasic, biphasic I and biphasic II, respectively. The vertical scale of the histogram is in thousands of shock events. The bars all have 20 ms horizontal width. Note that the total number of events for all the subgraphs is the same (160 000). The bar colors indicate the mechanism by which reentrant dynamics removal occurred: DB (purple); An (yellow); De (blue); DA (orange).

## Tables

Classification of the outcomes of reentrant dynamics removal obtained by the ANN analysis for shocks of four different levels of energy. The probability (in percents) and its standard deviation (in parentheses) is given for each outcome.

Classification of the outcomes of reentrant dynamics removal obtained by the ANN analysis for shocks of four different levels of energy. The probability (in percents) and its standard deviation (in parentheses) is given for each outcome.

This table gives the confidence intervals (with α = 0.01) for the electric fields needed to get 50% (E 50) and 90% (E 90) of successful reentrant dynamics removal, respectively. The second column gives the fitting parameters of all the simulation data with a logistic curve (see Eq. (9) ). The standard error for each of the fitting parameter is also given (small sub-indices in parentheses next to each parameter).

This table gives the confidence intervals (with α = 0.01) for the electric fields needed to get 50% (E 50) and 90% (E 90) of successful reentrant dynamics removal, respectively. The second column gives the fitting parameters of all the simulation data with a logistic curve (see Eq. (9) ). The standard error for each of the fitting parameter is also given (small sub-indices in parentheses next to each parameter).

A comparison of the medians of the distribution (corresponding to all the box plots shown in Fig. 5 ) for the three protocols at different energies. The statistical comparison is realized through a pairwise Wilcoxon rank sum test for equal medians. The comparison is then translated into a Z-score in order to see the significant differences more clearly.

A comparison of the medians of the distribution (corresponding to all the box plots shown in Fig. 5 ) for the three protocols at different energies. The statistical comparison is realized through a pairwise Wilcoxon rank sum test for equal medians. The comparison is then translated into a Z-score in order to see the significant differences more clearly.

The χ^{2} goodness-of-fit test of the default null hypothesis that the data are a random sample from a normal distribution with mean and variance estimated from the data. The notation “1” means that the null hypothesis can be rejected at the α = 0.05 significance level, while “0” means that the null hypothesis cannot be rejected. The p-values are also given in parentheses.

The χ^{2} goodness-of-fit test of the default null hypothesis that the data are a random sample from a normal distribution with mean and variance estimated from the data. The notation “1” means that the null hypothesis can be rejected at the α = 0.05 significance level, while “0” means that the null hypothesis cannot be rejected. The p-values are also given in parentheses.

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