# Application of Models of Defibrillation to Human Defibrillation Data

## Implications for Optimizing Implantable Defibrillator Capacitance

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## Abstract

*Background* Theoretical models predict that optimal capacitance for implantable cardioverter-defibrillators (ICDs) is proportional to the time-dependent parameter of the strength-duration relationship. The hyperbolic model gives this relationship for average current in terms of the chronaxie (t_{c}). The exponential model gives the relationship for leading-edge current in terms of the membrane time constant (τ_{m}). We hypothesized that these models predict results of clinical studies of ICD capacitance if human time constants are used.

*Methods and Results *We studied 12 patients with epicardial ICDs and 15 patients with transvenous ICDs. Defibrillation threshold (DFT) was determined for 120-μF monophasic capacitive-discharge pulses at pulse widths of 1.5, 3.0, 7.5, and 15 ms. To compare the predictions of the average-current versus leading-edge-current methods, we derived a new exponential average-current model. We then calculated individual patient time parameters for each model. Model predictions were validated by retrospective comparison with clinical crossover studies of small-capacitor and standard-capacitor waveforms. All three models provided a good fit to the data (*r*^{2}=.88 to .97, *P*<.001). Time constants were lower for transvenous pathways (53±7 Ω) than epicardial pathways (36±6 Ω) (t_{c}, *P*<.001; average-current τ_{m}, *P*=.002; leading-edge-current τ_{m}, *P*<.06). For epicardial pathways, optimal capacitance was greater for either average-current model than for the leading-edge-current model (*P*<.001). For transvenous pathways, optimal capacitance differed for all three models (*P*<.001). All models provided a good correlation with the effect of capacitance on DFT in previous clinical studies: *r*^{2}=.75 to .84, *P*<.003. For 90-μF, 120-μF, and 150-μF capacitors, predicted stored-energy DFTs were 3% to 8%, 8% to 16%, and 14% to 26% above that for the optimal capacitance.

*Conclusions *Model predictions based on measured human cardiac-muscle time parameter have a good correlation with clinical studies of ICD capacitance. Most of the predicted reduction in DFT can be achieved with ≈90-μF capacitors.

Optimized capacitive-discharge waveforms are an important goal in the design of ICDs. Quantitative models have been proposed to facilitate optimizing these waveforms over a broad range of capacitances and pathway resistances.^{1} ^{2} ^{3} ^{4} ^{5} ^{6} Although the models differ, each predicts that the DFT is minimized when the time constant of the shock waveform is close to the value of a parameter that characterizes the time-dependent response of cardiac tissue to the applied shock. These parameters may be determined from the strength-duration curve,^{7} which is a plot of shock strength at the DFT versus shock duration. Based on parameters determined from animal data, these models predict that optimal ICD capacitance for a 50-Ω pathway is 30 to 43 μF^{1} ^{2} ^{3} ^{6} and that reduction in capacitance will result in substantial improvement in DFT.^{6} Small capacitors have performed better than conventional capacitors^{8} ^{9} ^{10} ^{11} ^{12} in animals, but they have shown modest^{13} or no^{14} ^{15} ^{16} ^{17} ^{18} ^{19} benefit for the majority of clinical defibrillation pathways. The objective of this study was to test the hypothesis that these models predict results of clinical studies of ICD capacitance if human cardiac-muscle time parameters are used. A secondary objective was compare optimal capacitance predicted by average-current and leading-edge-current methods directly by use of the same data and parallel models.

## Methods

### Models of Defibrillation

Capacitive-discharge waveforms are characterized by a τ_{s}, which represents the exponential-decay time constant of the defibrillation pulse. τ_{s} is the product of pathway resistance and ICD capacitance. Theoretical models^{1} ^{2} ^{4} define the optimal defibrillation waveform as that which minimizes stored-energy DFT, because stored energy is a critical determinant of ICD pulse generator size. Each model predicts the optimum value of τ_{s} in terms of a time parameter that characterizes the response of cardiac tissue to the shock. Strength-duration equations are shown in the “Appendix.”

#### Monophasic Models

** Hyperbolic average-current model.** The empirical hyperbolic strength-duration relationship

^{20}

^{21}may be applied to capacitive-discharge defibrillation pulses by expressing shock strength as average current during the pulse.

^{22}

^{23}This hyperbolic average-current model gives the DFT in terms of t

_{c}and the average-current rheobase, which is the DFT at an infinite pulse width. The waveform optimization strategy is summarized in the “Appendix.” The optimal value of τ

_{s}is 0.796 t

_{c}

^{1}

^{2}and the optimum pulse duration is 1.045 t

_{c}.

^{2}

** Leading-edge-current exponential model.** A strength-duration relationship also can be derived from Blair’s resistor-capacitor model of the cardiac cell membrane.

^{24}

^{25}This exponential formulation of the strength-duration relationship was extrapolated to describe stimulation by capacitive-discharge pulses

^{26}and subsequently applied to defibrillation.

^{3}

^{5}This model assumes that defibrillation depends only on the peak cell response to a shock pulse rather than the average applied current during the pulse.

^{4}

^{5}

^{6}

^{26}The corresponding strength-duration relation gives leading-edge-current DFT for a capacitive-discharge pulse in terms of two time constants, τ

_{s}and Blair’s

^{24}

^{25}“cell membrane” τ

_{m}. The third model parameter is the leading-edge-current rheobase at an infinite value of τ

_{s}. The optimum value of τ

_{s}and the optimum pulse duration are both equal to τ

_{m}.

^{4}

^{5}

^{6}Fig 1⇓ shows the effect of varying the relationship between τ

_{s}and τ

_{m}on the predicted membrane-response and the corresponding strength-duration curves.

** Exponential average-current model.** The average-current model is based on the hyperbolic strength-duration relation, and the leading-edge-current model results in an exponential one. To permit direct comparison of predictions based on average current with those based on leading-edge current, we derived a new exponential average-current model. It gives the average-current DFT in terms of the average-current τ

_{m}and the average-current rheobase. The optimal value of τ

_{s}is τ

_{m}, and the optimum pulse duration is 1.337 τ

_{m}. Because average-current τ

_{m}and leading-edge-current τ

_{m}may differ, the predicted optimal values of τ

_{s}are not in general equal for the two exponential models. Fig 1⇑ shows that the exponential average-current model is the limit of the leading-edge-current model as τ

_{s}→∞.

#### Biphasic Models

The optimal first phase of the biphasic waveform is assumed to be the optimal monophasic waveform.^{3} ^{4} Thus, predicted optimal τ_{s} is equal for monophasic and biphasic models. The optimal second phase is assumed to prevent refibrillation by removing or “burping” the charge deposited on myocardial cells by phase 1.^{3} ^{4} ^{12} The initial conditions for phase 2 (cell-membrane voltage and ICD-capacitor voltage) are set equal to the corresponding values at the end of the optimal phase 1 as predicted by each monophasic model. The leading-edge-current model is then used to determine the optimal duration of phase 2. If phase 1 is optimized by an average-current method, the biphasic model becomes a hybrid model, using different methods for each phase.^{3}

### Experimental Methods

#### Patients

Patients undergoing ICD implantation participated in this study after giving written informed consent according to a protocol approved by the Human Subjects Committee. Because of the required number of fibrillation-defibrillation episodes, patients were excluded if their New York Heart Association class for heart failure was 3 or 4 after optimal therapy, their left ventricular ejection fraction was <.25, or a there was a proximal stenosis ≥70% in a major coronary artery supplying viable myocardium. Patients were excluded if they had ever received amiodarone, and other antiarrhythmic drugs were discontinued for 5 half-lives. However, therapy with digoxin (4 patients), β-blockers (7 patients), and calcium channel blockers (1 patient) was continued. Patients were studied at new implants of ICDs with a single transvenous electrode configuration (24 patients) or pulse generator change in patients with a single epicardial electrode configuration (12 patients). Six patients with transvenous ICDs were excluded from this study because the biphasic DFT was too high (see below), and the study was aborted in 1 patient with a transvenous ICD because of hypotension. Thus, complete data were available for 27 patients. They included 18 men and 9 women with a mean age of 59±15 years. The mean left ventricular ejection fraction was 0.39±0.11. Eighteen patients had coronary artery disease, 8 patients had myocardial or congenital disease, and 1 patient had idiopathic long-QT syndrome. The clinical arrhythmia was sustained monomorphic ventricular tachycardia in 16 patients and ventricular fibrillation in 11 patients.

#### Surgical Technique and Electrode Configurations

Patients were studied intraoperatively as described previously.^{18} The epicardial defibrillation pathway included a large patch electrode (Medtronic model 6921L) positioned posteriorly over the left ventricle and a large (5 patients) or medium (Medtronic model 6921M) (7 patients) patch electrode positioned anteriorly over the right ventricle. The left ventricular electrode was the cathode, and the right ventricular electrode was the anode. In patients with transvenous electrodes, a tripolar electrode with a 5-cm defibrillation coil (Medtronic model 6966 or 6936) was positioned in the right ventricular apex. The titanium shell of an ICD pulse generator (Active Can Emulator, 83 cm^{3}) was positioned in a retropectoral pocket. The right ventricular coil served as the cathode, and the titanium shell served as the anode.

#### Defibrillation Waveforms

Models of defibrillation permit estimation of optimal biphasic waveforms from monophasic strength-duration data.^{3} ^{4} We used monophasic waveforms to minimize the number of variables that might influence calculation of the defibrillation time parameters. Study waveforms were fixed-duration, truncated exponential pulses delivered by an external defibrillator with a nominal 120-μF output capacitance (model 2394, Medtronic Inc). Clinically indicated testing was performed using biphasic pulses with 65% tilt in each phase. Polarity was reversed for phase 2 of biphasic pulses.

#### Clinical Testing

First, the DFT was determined with biphasic pulses by a step-down or step-up method. The first programmed leading-edge voltage was 400 V, and the step size was 100 V. Patients were excluded from the study if the biphasic DFT exceeded 500 V to avoid the possibility that the monophasic DFT at a pulse width of 1.5 ms would exceed the maximum output of the external defibrillator. This excluded 6 patients with transvenous electrodes.

#### DFT Testing

Four monophasic pulse durations were tested in random order in each patient: 1.5, 3, 7.5, and 15 ms. The DFT_{50} was estimated by a previously described delayed three-step up-down algorithm^{27} ^{28} with 50-V steps. The method for selecting the strength of the first defibrillation test shock is described below. If this shock succeeded, the strength of the second defibrillation test shock was decreased by 50 V. If it failed, the strength of the second shock was increased by 50 V. This process was repeated until there was a reversal of response from success to failure or from failure to success. Then the strength of the next defibrillation test shock was changed by 50 V in the opposite direction. The shock strength before the first reversal of response was the first data point, and the strengths of the subsequent two shocks were the second and third data points. The fourth data point was predicted from the outcome of the third defibrillation shock but not tested. The average of these four data points was taken as the DFT_{50}.

In a delayed up-down algorithm, the number of fibrillation-defibrillation episodes is minimized if the strength of the first test shock is near the DFT_{50}. To estimate the strength of this first test shock, we determined the upper limit of vulnerability^{29} for the first of the four test waveforms. This approximates the DFT_{90}.^{29} The method for determining the upper limit of vulnerability was modified to set the strength of the first monophasic upper-limit test shock at 200 V greater than the biphasic DFT and the step size to 100 V. We then set the strength of the first defibrillation test shock to 100 V below the upper limit of vulnerability. In the first 5 patients, the initial test shock for the second waveform tested was equal to the DFT_{50} for the first waveform. For the third and fourth waveforms, the initial test shock was equal to the average of the DFT_{50}s for the preceding waveforms. Beginning with the sixth patient, the initial test voltage for shocks with duration of 1.5 ms was set 20% higher than that of the other three pulse durations. Defibrillation test shocks were given after 10 seconds of induced ventricular fibrillation. Overall, patients had 14.2±1.4 monophasic defibrillation shocks in this study. They also had three or four biphasic defibrillation shocks for clinical reasons.

#### Data Acquisition

This method has been described previously.^{18} Voltage and current waveforms were digitized at 100 kHz with the Mac-Adios Board (GW Instruments) and recorded on a Macintosh computer. A custom-modified oscilloscope emulation program (SuperScope II, GW Instruments) was used to record voltage and current waveforms and to detect the leading- and trailing-edge voltages and currents.

#### Data Analysis

Mean resistance was determined by averaging the point-by-point quotient of the voltage waveform divided by the current waveform. Pulse duration was calculated as the difference in timing of the leading-edge and trailing-edge voltages. Previously described methods were used to calculate stored energy^{16} and average current^{2} ^{18} for shock pulses with a measured capacitance of 122.2±1.6 μF.^{18} Fig 2⇓ shows equivalent average-current waveforms for the truncated exponential waveforms tested. We constructed strength-duration curves for each patient by the method of least squares using Matlab 4.2 for the Macintosh (The MathWorks Inc). Average-current strength-duration data were fit to the hyperbolic average-current model of Equation 6 in the “Appendix” to determine t_{c} and to the exponential average-current model of Equation 9 to determine average-current τ_{m}. To characterize these two strength-duration relations quantitatively, we compared their rheobases and chronaxies. For a constant-current pulse, the relationship between exponential chronaxie and cell-membrane time constant is given by t_{c}=τ_{m} ln 2=0.693τ_{m}.^{30} Note that this relationship does not apply to the hyperbolic chronaxie and that the chronaxie is not an intrinsic parameter of the exponential model. Unless specified as the exponential chronaxie, we use the term chronaxie in its common usage, given by the hyperbolic model. Leading-edge-current data were fit to a composite curve to determine leading-edge-current τ_{m}. The left side of this curve corresponding to t<t_{opt} was Equation 4a. The right side had a constant current for t>t_{opt}. Because of the possibility that time-dependent processes might falsely elevate the leading-edge-current DFT for long pulses,^{12} ^{31} ^{32} these curves were fit with (four points) and without (three points) the 15-ms data point. The three-point fits are used unless specifically indicated. Calculated time constants were then applied to the corresponding models to estimate optimal waveform parameters for each patient.

#### Retrospective Validation of Models

Model predictions were validated by retrospective comparison with results from clinical crossover studies of small-capacitor and standard-capacitor waveforms. We first performed a literature search for clinical crossover studies that compared DFTs for a standard 120- to 125-μF capacitor and a smaller capacitor. Studies were excluded if they did not include information required for model predictions, such as pathway resistance. When clinical data were available for both monophasic and biphasic waveforms, we used monophasic data. When only biphasic waveforms were used, we compared their predicted performance on the basis of the first phase. Waveforms with equal tilt were compared for each capacitance. The ratio of stored-energy DFTs of the experimental waveform to the stored-energy DFT of the standard waveform±SEM was determined for each study. To determine the corresponding predicted ratio, we first calculated the ratio of the lowest DFT for an arbitrary capacitance to the lowest DFT for the optimal capacitance for each model. This permitted a prediction of the expected ratio of stored-energy DFTs for the two capacitance values in each clinical study.

#### Statistical Analysis

We assessed the effect of pulse duration on DFT_{50} using one-factor repeated-measures ANOVA with pulse duration as the factor. Post hoc analysis was performed by Scheffé’s test. The relationship between electrode configuration (epicardial or transvenous) and each time constant was assessed by the unpaired *t* test. The relationship between pathway resistance and each time constant was assessed by linear regression. In this analysis, resistance was the average value for all four pulse widths. We used SuperANOVA 1.11 for the Macintosh (Abacus Concepts) for ANOVA calculations. Goodness of fit was compared for different models by the paired *t* test. The relationship between observed DFT ratios in previous clinical studies and predicted ratios was assessed by linear regression. Data are presented as mean±SD. When multiple comparisons were performed, we required a value of *P*<.05 divided by the number of comparisons. Basic statistics were calculated with the paired two-tailed *t* test.

## Results

Table 1⇓ shows group mean values for the DFT_{50} at each of the four pulse widths. DFT_{50} is higher at all pulse widths for transvenous pathways than epicardial pathways (*P*<.01). Table 2⇓ shows group mean time constants for each model. Fig 3⇓ shows best-fit strength-duration curves for each model.

### Average-Current Models

Average-current DFT_{50} decreased monotonically as pulse width increased. Fig 3⇑ shows that both average-current models provide a good fit to the data for transvenous and epicardial pathways. For the group as a whole, *r*^{2}=.97±.02 for the hyperbolic model and *r*^{2}=.94±.04 for the exponential model. The curves diverge at very long and very short pulse widths. The exponential curves lie above the corresponding hyperbolic curves for very short pulses. As pulse width increases into the clinical range, the negative slope of the exponential curve is steeper, and it falls below the hyperbolic curve. It then makes a sharper bend, crossing the hyperbolic curve to reach a higher rheobase (epicardial: 3.2±0.4 versus 2.4±0.3 A, *P*<.001; transvenous: 4.8±0.9 versus 3.8±0.7 A, *P*<.001). This difference in curve shapes is reflected in the lower exponential chronaxie for both epicardial and transvenous pathways (epicardial: 2.7±0.3 versus 4.7±0.8 ms, *P*<.001; transvenous: 2.2±0.4 versus 3.5±0.5 ms; *P*<.001).

### Leading-Edge-Current Model

The leading-edge-current DFT_{50} is highest for 1.5-ms pulses for both transvenous and epicardial pathways. The transvenous curve reaches a minimum at 7.5 ms and is unchanged at 15 ms. The epicardial curve has a minimum at 3.0 ms. The epicardial DFT_{50} is ≈15% higher at 15 ms than at 3 ms or 7.5 ms. These differences are significant by the paired *t* test (*P*<.01) but not by Scheffé’s test (3.0 versus 15 ms: *P*=.24; 7.5 versus 15 ms: *P*=.30). The fit to the data (*r*^{2}=.88±.14 for three points, *r*^{2}=.79±.24 for four points) is not as close as that for the hyperbolic average-current model (*P*<.005) or exponential average-current model (*P*<.03). When all four data points were used for curve fitting, the value of τ_{m} was lower for epicardial pathways (2.4±0.8 ms, *P*=.003) but unchanged for transvenous pathways (2.3±0.4 ms, *P*=.86). Table 2⇑ shows that τ_{m} is lower for the leading-edge-current model than the average-current model for both epicardial pathways (*P*<.001) and transvenous pathways (*P*<.001).

### Correlation of Electrode Configuration, Pathway Resistance, and DFT With Model Parameters

Table 2⇑ shows that time parameters for all models were greater for epicardial pathways than for transvenous pathways: t_{c} (*P*<.001), average-current τ_{m} (*P*<.001), and leading-edge-current τ_{m} (*P*=.06). For the group as a whole, there was a significant inverse correlation between resistance and the average-current time parameters (t_{c}: *r*^{2}=.48, *P*<.001; average-current τ_{m}: *r*^{2}=.56, *P*<.001) but not leading-edge-current τ_{m}: *r*^{2}=.04, *P*=.34. This correlation between resistance and the average-current time parameters was significant for transvenous pathways (t_{c}=6.531−0.058×R: *r*^{2}=.57, *P*=.001; average-current τ_{m}=6.457−0.061×R: *r*^{2=}.72, *P*<.001) but not for epicardial pathways (t_{c}: *r*^{2}=.001, *P*=.94; average-current τ_{m}: *r*^{2}=.01, *P*=.73).

There was no correlation between current, voltage, or stored-energy DFT and time parameter for any model. For example, *r*^{2} for leading-edge-current DFT varied from.004 to.12 for transvenous pathways (*P*=.21 to.83) and from.02 to.03 for epicardial pathways (*P*=.64 to.69).

### Predicted Optimal Waveforms

Table 2⇑ shows predicted optimal ICD waveforms. For epicardial pathways, the two average-current models predict similar optimal τ_{s}, whereas the leading-edge-current model predicts a lower value (*P*<.001). For transvenous pathways, optimal τ_{s} was greatest for the exponential average-current model, intermediate for the hyperbolic average-current model, and lowest for the leading-edge-current model. All pairwise differences were significant at the level of *P*<.001. The leading-edge-current model predicts shorter optimal durations for phase 1 than the average-current models (*P*<.001) and for phase 2 than the hybrid biphasic models (*P*<.001).

### Penalty for Suboptimal Capacitance

Fig 4⇓ shows the predicted effect on stored-energy DFT of varying capacitance (or τ_{s}) for transvenous pathways, provided that the best waveform is used for each capacitance. The curves show each model’s predicted “stored-energy penalty” for suboptimal capacitance. They have a steep descending limb for lower-than-optimal capacitance, a relatively flat valley with a nadir at the optimal capacitance, and a gradually sloping ascending limb for higher values. The range of penalties is 9% to 21%, 2% to 6%, 8% to 18%, and 20% to 45% for capacitance values that are 0.5, 1.5, 2, and 3 times optimal, respectively. The penalty for underestimating optimal capacitance by 50% approximates that for overestimating it by 100%. Expressed as multiples of optimal capacitance, the stored-energy penalty over this range is approximately twice as high for the exponential average-current model as either the hyperbolic average-current or leading-edge-current model. From a different perspective, the predicted reduction in DFT achieved by optimizing capacitance for current ICDs ranges from 14% to 21% for a 150-μF capacitor and 5% to 10% for a 100-μF capacitor. The predicted reduction is greatest for the leading-edge-current model, which has the lowest optimal capacitance.

### Comparison of Model Predictions With Results of Experimental Studies

We identified five clinical studies that met the criteria for retrospective analysis.^{13} ^{14} ^{15} ^{18} ^{19} Table 3⇓ shows observed and predicted DFT ratios. Most studies used biphasic waveforms. All three models provide reasonable agreement with observed results for transvenous pathways with experimental capacitance values of 60 to 90 μF. The leading-edge-current model overestimates the performance of 60-μF waveforms for epicardial pathways. The correlation coefficients are comparable for each model: hyperbolic average-current model *r*^{2}=.80, *P*=.0011; exponential average-current model *r*^{2}=.75, *P*=.0027; leading-edge-current model *r*^{2}=.84, *P*=.0005. If the intercept is required to be zero, the correlation coefficients are higher: hyperbolic average-current model *r*^{2}=.98; exponential average-current model *r*^{2}=.99; leading-edge-current model *r*^{2}=.98 (*P*<.001 for each model). The slope of the zero-intercept regression line (observed DFT ratio/predicted DFT ratio) was 1.04 for the hyperbolic average-current model, 1.04 for the exponential average-current model, and 1.09 for the leading-edge-current model. Thus, all three models underestimated the effect of capacitance on DFT, and the degree of underestimation was greatest for the leading-edge-current model.

## Discussion

The results of this study support the hypothesis that application of human time parameters to models of defibrillation provides a reasonable estimate of the results of clinical studies of ICD capacitance. These models predict that use of 90- to 100-μF capacitors in the present generation of ICDs realizes most of the reduction in DFT that can be achieved by optimizing capacitance.

### New Findings Regarding Models of Defibrillation

There are three secondary findings.

1. The hyperbolic and exponential average-current models predict similar optimal waveforms. The exponential strength-duration curve has a higher rheobase and lower chronaxie.

2. Average-current models predict higher values of optimal τ_{s} and capacitance than the leading-edge-current model. Average-current and leading-edge-current models are based on different fundamental assumptions. The leading-edge-current model explicitly assumes that the response to a defibrillating pulse depends only on the instantaneous peak value of membrane voltage. In contrast, the average-current models implicitly assume that the cell membrane responds to the average value of the applied field over the pulse. To compare these two methods by use of parallel exponential models, we used a new average-current model with an exponential formulation rather than a hyperbolic one.

3. Time parameters for lower-resistance epicardial pathways are greater than those for higher-resistance transvenous pathways. We found an inverse correlation between pathway resistance (or τ_{s}) and both average-current time parameters (t_{c} and τ_{m}) for the group as a whole and for transvenous pathways. The minimal overlap in resistances for epicardial and transvenous pathways precluded an analysis to determine whether electrode configuration had an effect on time parameters independent of the effect of resistance. Further, because we studied acute transvenous electrodes and chronic epicardial electrodes, we do not know whether this difference contributed to the observed correlation between electrode configuration and defibrillation time parameters. This correlation was not present for leading-edge-current τ_{m}.

### Previous Estimates of Defibrillation Time Parameters

Estimates of t_{c} for the hyperbolic average-current model have used average values from several studies in animals: 2.0 ms,^{1} 2.0 ms,^{4} 2.4 ms,^{3} and 2.7 ms.^{2} For transvenous defibrillation in dogs, Geddes and Bourland^{7} used data from Wessale et al^{33} to calculate a leading-edge-current τ_{m} of 1.2, 1.5, and 1.8 ms for trapezoidal pulses with tilt of 50% to 80%. Cleland^{6} determined values for epicardial leading-edge-current τ_{m} of 1.5 ms based on data from Tang et al^{34} and 1.3, 1.4, and 1.7 ms using three different sets of data from Walker et al.^{35} Walcott et al^{4} used the equation t_{c}=0.693τ_{m}^{30} to calculate a value of 2.8 ms for leading-edge-current τ_{m} from reported values of hyperbolic average-current t_{c}. However, this equation relates exponential average-current t_{c} to exponential average-current τ_{m}, not hyperbolic average-current t_{c} to exponential leading-edge-current τ_{m}, as they apply it. Use of time parameters derived by one method in a model based on another method may result in significant error. Our results confirm the prediction of Block et al^{36} that average-current defibrillation time parameters are higher in humans than in animals. There is only one previous estimate for t_{c} in humans. In a preliminary report of the effect of waveform tilt on DFT, Shorofsky et al^{37} gave a value of 4.7 ms for pooled data from epicardial, transvenous, and hybrid electrode systems. This value is similar to t_{c} for epicardial pathways in the present study.

### Fit of Human Defibrillation Data to Models

Both hyperbolic and exponential average-current models provided a good fit over the range of pulse widths we tested. The better fit of our data to the average-current models than the leading-edge-current model may be due in part to fewer data points on the descending limb of the leading-edge-current curve or use of a composite curve-fitting method for this model. More data points near the anticipated value of τ_{m} might have permitted a better fit. We cannot distinguish limitations of curve fitting from limitations of the model.

### Comparison of Model Predictions to Experimental Studies of ICD Capacitance

All three models predict that optimal capacitance is proportional to the model’s time parameter and inversely proportional to pathway resistance. Experimental studies generally have found a greater beneficial effect of small capacitance on DFT in animals than in humans. In dogs and pigs, smaller 40- to 90-μF capacitors have performed better than conventional 120- to 140-μF capacitors.^{8} ^{9} ^{10} ^{11} ^{12} In humans, however, 60- to 90-μF capacitors have shown substantial benefit only for high-resistance transvenous pathways.^{14} ^{18} They have shown modest^{13} or no^{14} ^{15} ^{16} ^{17} ^{18} ^{19} benefit for the majority of current clinical transvenous pathways, and 60-μF capacitors have underperformed 120-μF capacitors for epicardial pathways.^{18}

The shape of the predicted “stored-energy DFT penalty” curves provides a possible explanation for the differential effect of capacitance on DFT in animals and humans. For example, the predicted optimal capacitance in both animal and human transvenous pathways is substantially less than values in ICDs, 32 μF^{1} ^{2} versus 57 μF for the hyperbolic average-current model and 30 μF^{6} ^{7} versus 45 μF for the leading-edge-current model. However, the corresponding predicted DFT penalties for a 120-μF capacitor are 29% and 37% in animals but only 8% and 16% in humans. Cleland^{6} shows a similar DFT penalty curve based on animal data and the leading-edge-current model in Fig 7b of that article. Interpolation of this graph gives the penalty for a 120-μF capacitor as ≈36%. Our results thus provide a conceptual basis for the observations that defibrillation-waveform studies in animals cannot be applied directly to humans. However, if animal time parameters are known, the human time parameters determined in this study may be applied to defibrillation models to estimate corresponding results in humans.

Retrospective comparison of model predictions with results of clinical crossover trials shows that all three models provide reasonable agreement with observed results for experimental capacitance values of 60 to 90 μF. All models underestimated the observed effect of capacitance on DFT. However, model predictions were for the best waveforms with each capacitance, whereas the tested waveforms were not in general optimal for each capacitance. For biphasic waveforms, the degree to which the second phase improved defibrillation efficacy might vary for different durations of each phase.^{3} ^{12}

### Implications for Design of ICDs

The stored-energy penalty curves have important implications for design of ICDs because stored energy is a critical determinant of the pulse generator size^{38} and current capacitor technology limits maximum voltage to ≈750 V. Suppose, for example, that the population DFT for an ICD with a 100-μF biphasic waveform is 10±4 J for a 50-Ω transvenous pathway.^{13} ^{39} ^{40} Because current technology limits the maximum voltage to ≈750 V, the maximum output of 28.1 J exceeds the mean DFT+2 SD by 10 J. Using a near-optimal 60-μF waveform will decrease the mean DFT to 9.0 to 9.5±4 J. However, the maximum 16.9-J output of the 60-μF ICD is 0.1 to 0.6 J less than the mean DFT+2 SD. Newer bidirectional transvenous pathways that include both a superior vena cava and active-can electrode have resistances of 30 to 40 Ω.^{17} ^{40} ^{41} For a 35-Ω pathway, the predicted optimal capacitance is 65 to 105 μF, and the stored-energy penalty for a 100-μF waveform is 0% to 3%. These considerations suggest that for biphasic waveform defibrillation with current electrode configurations, the penalty incurred by use of ≈100-μF capacitors in the present generation of ICDs is a small and appropriate price to ensure reliable defibrillation of the vast majority of patients. However, the penalty for 130- to 150-μF capacitors used in earlier ICDs is unnecessary.

### Limitations

A major limitation is that current models of defibrillation are considered to be first-order approximations. However, the measured time parameters for human defibrillation can be applied to future models. A second major limitation related to curve fitting for the leading-edge-current model has been discussed. A third major limitation is the accuracy with which the DFT can be determined at four different pulse widths in humans. The number of fibrillation-defibrillation episodes in this study approaches a prudent maximum for clinical protocols. We wish to emphasize three other limitations: (1) τ_{m} derived from the exponential strength-duration curves corresponds to a true cell-membrane time constant only in a simple resistor-capacitor model of the cell membrane.^{24} ^{25} (2) The patient population may not be representative of ICD recipients in general. Because of the number of fibrillation-defibrillation episodes required, we excluded the sickest patients. We also excluded transvenous patients whose DFTs at a pulse duration of 1.5 ms had a high probability of exceeding the maximum output of the defibrillator. Despite this, determination of the transvenous DFT_{50} at 1.5 ms required shock strengths >800 V (38 J) in 6 patients (40%) and 900 V (49 J) in 2 patients (13%). (3) This study used a single value of capacitance. We do not know how variations in τ_{s} caused by changes in capacitance might have affected our results.

### Conclusions

When human time parameters are applied to models of defibrillation, they provide a good estimate of the results of clinical studies of ICD capacitance in the range 60 to 125 μF and predict the better performance of small-capacitor waveforms in animals than in humans. Use of 90- to100-μF capacitors realizes most of the predicted reduction in DFT that can be achieved by optimizing capacitance while maintaining a sufficient safety margin for current transvenous electrode configurations.

## Selected Abbreviations and Acronyms

DFT | = | defibrillation threshold |

DFT_{50} | = | shock strength with 50% probability of defibrillation |

ICD | = | implantable cardioverter-defibrillator |

t_{c} | = | empirical time parameter chronaxie |

τ_{m} | = | time constant for leading-edge current |

τ_{s} | = | waveform or system time constant |

## Appendix A1

### Blair’s Model

Blair^{24} ^{25} developed a theoretical relation which describes the passive effects of a shock pulse applied to a cell. The relation is where V_{m}(t) is the cell’s membrane potential over time (t), and V_{s}(t) is the stimulus potential. For defibrillation, V_{s}(t) represents the field of the applied shock. Blair^{24} solved Equation 1 for a stimulus with a constant amplitude. The resultant strength-duration relationship is where *I*(t) is the applied current at the DFT (in this case a constant value) and *I*_{r} is the rheobase.

### Exponential Leading-Edge-Current Model

This model is derived by solving Blair’s relation of Equation 1 for a capacitively discharged shock pulse described by The optimal duration (t_{opt}) is the shortest duration that results in the maximum voltage on the cell membrane at the end of the pulse.^{4} ^{5} ^{6} For pulses shorter than this optimal duration, the strength-duration curve is given by^{4} ^{5} ^{26} where *I*(t) is the applied leading-edge current at the DFT and *In*_{r} is the leading-edge-current rheobase at an infinite value of τ_{s}. For longer pulses, *I*(t) remains constant at the value given by Equation 5. Note that Equation 4a reduces to Equation 2 as τ_{s}→∞. Intuitively, this can be appreciated by noting that a truncated-exponential waveform approaches a constant-current pulse as τ_{s}→∞.

### Average-Current Hyperbolic Model

The empirical strength-duration relation is where *I*_{avg} is the average-current DFT of a pulse with duration (t), t_{c} is the chronaxie, and *I*_{r} is the average-current rheobase. Note that τ_{s} does not appear explicitly as a term in the average-current hyperbolic model. Instead, it is implicit in the expression for average current of a truncated exponential pulse in Equation 7 where C is the capacitance of the ICD and V is the leading-edge voltage.

### Effective Current

Kroll developed the concept of effective current (*I*_{e}) to derive the optimal pulse width and τ_{s} (or capacitance) for the hyperbolic average-current model. For a shock pulse, *I*_{e} is defined as “the rheobase requirement that it can satisfy.”^{2} Kroll^{2} used this concept of *I*_{e} to optimize pulse width and capacitance or τ_{s} in two steps. First, he found the pulse width that maximized *I*_{e} for a given τ_{s}. Then, using this optimal pulse width, he determined the value of τ_{s} that maximized *I*_{e} for a fixed stored energy (E_{s}).

### Exponential Average-Current Model

This model is based on the strength-duration relation in Equation 2 if the constant current (*I*) is replaced by the average current over the pulse (*I*_{avg}). Note that as in the hyperbolic average-current model, τ_{s} is implicit in the expression for average current. The optimal value of pulse width and τ_{s} are derived by use of the effective-current method in analogy to Kroll’s approach for the hyperbolic average-current model. We insert the expression for average current of a truncated exponential pulse from Equation 7 into Equation 9, substitute *I*_{e} for *I*_{r}, and solve for *I*_{e}: We then calculate the optimal pulse width and τ_{s} by Kroll’s method for the hyperbolic average-current model. First, the pulse width that maximizes *I*_{e} for a given τ_{s} is determined by differentiating Equation 9 with respect to t and solving for its zero. This gives t_{opt} =1.337×τ_{m}. Then this optimal pulse width is substituted into Equation 9. Remembering that V=(2E_{s}/C), where E_{s} is stored energy, we differentiate *I*_{e} with respect to τ_{s} for a fixed stored energy (E_{s}) and solve for its zero. We find that τ_{s}=τ_{m} maximizes *I*_{e} with respect to τ_{s} for a fixed value of E_{s}.

## Acknowledgments

This study was supported in part by an American Heart Association Greater Los Angeles Affiliate Grant-in-Aid (1085-GI1) and by a grant from the PM foundation to Dr Swerdlow.

- Received April 3, 1997.
- Revision received June 16, 1997.
- Accepted June 19, 1997.

- Copyright © 1997 by American Heart Association

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- Application of Models of Defibrillation to Human Defibrillation DataCharles D. Swerdlow, James E. Brewer, Robert M. Kass and Mark W. KrollCirculation. 1997;96:2813-2822, published online before print November 4, 1997http://dx.doi.org/10.1161/01.CIR.96.9.2813
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- Application of Models of Defibrillation to Human Defibrillation DataCharles D. Swerdlow, James E. Brewer, Robert M. Kass and Mark W. KrollCirculation. 1997;96:2813-2822, published online before print November 4, 1997http://dx.doi.org/10.1161/01.CIR.96.9.2813

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