Fractionated Electrograms From a Computer Model of Heterogeneously Uncoupled Anisotropic Ventricular Myocardium
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Abstract
Background The relation between heterogeneously coupled myocardium and fractionated electrograms is incompletely understood. The purpose of this study was to use a detailed computer model of nonuniformly anisotropic myocardium to test the hypothesis that spatial variation of morphology of electrograms recorded simultaneously from multiple sites increases with increasing heterogeneity of intercellular coupling.
Methods and Results A sheet of elements with BeelerReuter ionic kinetics was coupled with cytoplasmic resistivity to model cells. Gap junctional resistance values were assigned by recursive randomization to produce a fractal pattern of heterogeneous coupling, simulating damage resulting from infarction. The correlation dimension of the pattern, D, measured heterogeneity of intercellular coupling. The peaktopeak amplitude, duration, minimum derivative (steepest downslope), number of inflections, frequency of peak power, and bandwidth of unfiltered unipolar electrograms were calculated. Linear regressions indicate (P<.001) that the coefficient of variation of five electrogram metrics increases with increasing substrate heterogeneity and that the distance over which electrogram morphology decorrelates decreases with increasing heterogeneity of intercellular coupling.
Conclusions These findings confirm our hypothesis that the spatial variation of morphology of electrograms recorded simultaneously from multiple sites increases with increasing heterogeneity of intercellular coupling.
Sudden cardiac death resulting from ventricular arrhythmias is a significant cause of mortality in this country.^{1} The technology for percutaneous catheter ablation has improved dramatically over the last several years for patients with supraventricular arrhythmias and can now be used to treat selected patients who have survived an episode of ventricular tachycardia.^{2} ^{3} ^{4} ^{5} For ablation to be successful, it is necessary to determine the location of arrhythmogenic tissue. Previous research indicated that heterogeneous, nonuniformly anisotropic myocardium constitutes the substrate for ventricular tachycardia.^{6} ^{7} Specifically, arrhythmogenic tissue is frequently located in heterogeneously coupled tissue at the border zone of a healed infarction where relatively normal cells interdigitate with scar. Furthermore, in a general sense, fractionated electrograms can be recorded from such regions^{8} and can be observed in sinus rhythm or during tachycardia. Because fractionated electrograms are a nonspecific marker for ventricular tachycardia substrate, there is a need to better understand how heterogeneously coupled myocardium leads to the generation of fractionated electrograms.
Previous studies examined the characteristics of individually recorded electrograms.^{9} ^{10} ^{11} ^{12} Comparison of electrograms recorded simultaneously from multiple neighboring locations may provide information on the heterogeneity of intercellular coupling. In normal, homogeneously coupled tissue, simultaneously recorded electrograms from multiple sites during uniform propagation of an electrical impulse should be quite similar. In heterogeneously coupled tissue, however, the substrate myocardium varies with location, and this variation may be reflected in electrogram morphology. We hypothesize that the spatial variation of morphology of electrograms recorded simultaneously from multiple sites increases with increasing heterogeneity of intercellular coupling. Specifically, the coefficient of variation of various electrogram metrics, including amplitude, duration, minimum derivative, number of inflections, frequency of peak power, and bandwidth, is hypothesized to increase with increasing substrate heterogeneity, and the correlation distance is hypothesized to decrease with increasing heterogeneity. To test this hypothesis under conditions more controllable than a biological preparation so that we could explicitly state the underlying assumptions, we used a detailed computer model.
Methods
Computer Model
We implemented a computer model of heterogeneously coupled ventricular myocardium (Fig 1⇓). A rectangular sheet of cells was simulated with elements with BeelerReuter^{13} ionic kinetics. To include the spatial heterogeneity present even in normal myocardium, BeelerReuter elements were connected to model cells 75 to 125 μm long. Each modeled cell consisted of three to five elements, a number chosen by a random number generator. Elements were coupled by intracellular resistances, in agreement with published values of cytoplasmic resistivity.^{14} To give normal tissue a longitudinal propagation velocity of 0.5 m/s,^{15} the modeled cells were connected end to end by a resistance (R′_{j}) of 530 kΩ to simulate gap junctions along the fiber axis of the muscle. The cells were connected transversely by gap junctional resistances that were spaced randomly, averaging one every 100 μm, to model the anisotropy known to occur in ventricular muscle. Table 1⇓ gives details of simulated tissue parameters.
The simulated tissue was implemented numerically by a c program run on a Cray C90 supercomputer. The BeelerReuter equations were solved with the Euler method for integration with a time step of 0.0625 ms, which was sufficiently small that there was no distortion of the upstroke of the action potential, as verified by comparison with simulations using a smaller time step. Each simulation began from resting conditions. The tissue was excited by a 1ms current pulse in a 200μm square area at the middle of one edge to begin longitudinal propagation. Simulation continued for 100 to 400 ms, depending on propagation velocity.
Heterogeneity of Intercellular Coupling
In addition to the normal anisotropy just described, we developed a method for producing heterogeneous alterations in coupling between normal cells to simulate the observed structure of patchy infarction.^{8} An infarct is the result of coronary arterial occlusion. Thus, it is reasonable to propose that the pattern of cellular disruption after an infarction is in some way related to the branching pattern of the arterial system. Previous researchers observed that the branchings of coronary arteries and other cardiac structures are fractal.^{16} ^{17} ^{18} To simulate damage caused by infarction, we generated synthetic fractal patterns to give spatial distributions of gap junctional resistance values, which were superimposed on the normal anisotropic structure.
For each heterogeneously coupled simulated tissue, a synthetic fractal was constructed by random number generation and a multiplicative recursive process. Random numbers were generated from the Pareto distribution,^{19} which has the following distribution function:
With a=v−1 and v>1, this distribution generates numbers in the range from one to infinity with a “heavy tail.” Using a multiplicative process with numbers >1 ensured that gap junctional resistances were increased or unchanged by simulated damage, and the heavy tail resulted in a high frequency of large numbers that simulated completely disconnected or dead cells. The recursive process used the random numbers from the Pareto distribution as follows (Fig 2⇓). Each simulated tissue was divided into quadrants, and all gap junctional resistances in each quadrant were multiplied by a random number selected for that quadrant. Each quadrant was in turn subdivided into quadrants in which gap junctional resistance values were multiplied by another random number. This recursive subdivision simulates the random damage that might occur from ischemia along successively smaller branches in the arterial tree. The recursive process continued until no further divisions were possible. The effect of the subdivisions was to produce selfsimilarity across size scales, characteristic of a fractal structure. Thus, each gap junctional resistance (R_{j}) could be written as
where R′_{j} is normal gap junctional resistance and ρ_{1}, ρ_{2}. . .ρ_{N} are random multipliers increasing the resistance. Note that neighboring gap junctions were probably multiplied by a similar cascade, whereas distant gap junctions would have few multipliers in common. Thus, this approach produced heterogeneity of coupling marked by a decline in correlation of coupling resistance as a function of distance.
The heterogeneity of the coupling patterns was quantified by a single metric derived from the decline in correlation of coupling resistance as a function of distance. For uniform coupling, all resistances are identical, perfectly correlated, and fully described by a single value (or point with zero dimension). As heterogeneity increases, the description of the coupling pattern becomes more complex. An extremely heterogeneous, uncorrelated coupling pattern would require a twodimensional (2D) function defined at all points for a full description. The fractal coupling patterns used in this study were generated by a recursion that creates statistical similarity at different size scales. The correlation of these coupling patterns simplifies their descriptions, which require less information than a 2D function defined at all points. The heterogeneity of the coupling patterns was quantified by the minimum dimension (noninteger) describing the spatial distribution of coupling resistances. This dimension can be calculated from the spatial correlation of coupling resistances by evaluating the average normalized autocorrelation of the fractal pattern at any distance, d.
A linear regression was applied to log{C[R_{J}(d)]} versus log(d), and the slope of the regression line was defined as the correlation dimension, D.^{20} The correlation dimension is zero for uniform coupling resistance for which the correlation does not decline with distance, and it increases as heterogeneity reduces correlation.
Thirty synthetic, random fractals were created and used for coupling resistance patterns. Six fractals were generated for each of five different correlation dimensions determined by the parameter v in Equation 1. Using v=1000.0 resulted in virtually all random multipliers, ρ_{1}ρ_{2}. . .ρ_{N}, equal to one and uniform patterns of gap junctional resistances. Decreasing values of v yielded patterns with increasing heterogeneity, as quantified by the correlation dimension.
Electrogram Calculation
Electrograms were calculated at nine points 100 μm above the 10×2.5mm simulated tissue. The points were 1 mm apart along a line beginning 1 mm from the pacing site, simulating unipolar recordings from a multielectrode catheter with the tip used as the stimulating electrode (Fig 3⇓). Assuming that the conducting medium is unbounded, homogeneous, and isotropic, the transmembrane current of each discrete element, I_{M}, contributes to an electrode at distance r, and electrograms, E, were calculated from the following equation^{21} :
Equation 4 was used with ρ_{e}=150 Ω · cm.^{22} Calculated electrograms were sampled at a rate of 1000 samples per second, and no filtering was applied.
Electrogram Metrics
The following six metrics were determined for all calculated electrograms: peaktopeak amplitude, duration, minimum derivative, number of inflections, frequency of peak power, and bandwidth. To eliminate pacing artifacts, the first 2 ms of all electrograms was truncated. The peaktopeak amplitude was the difference between the maximum and minimum values, and the duration was the time from the first excursion 0.1 mV from baseline to the last excursion 0.1 mV from baseline, excluding T waves.^{9} The first time derivative of each electrogram was calculated by the following equation:
The minimum derivative (or steepest downslope present) was recorded for each electrogram. The number of inflections was the number of fluctuations in the electrogram >0.5 mV, and electrograms longer than 60 ms with more than two inflections were considered to be fractionated.^{10} To determine frequency of peak power and bandwidth, a fast Fourier transform (FFT) of 256 ms of data was calculated from each electrogram, yielding a frequency resolution of 3.9 Hz. The frequency of peak power was the FFT coefficient with the largest squared magnitude. Bandwidth was determined to be the minimum frequency, so that the summed power of all lower frequency components was 90% of the total power.
Given the above metrics, the similarity of electrograms was evaluated by calculating the coefficient of variance for each electrogram metric. We calculated the mean, μ, and the SD, ς, of each metric for the nine electrograms recorded from each simulated tissue. The coefficient of variance for a tissue was calculated as the ratio ς/μ.
As an additional measure of electrogram similarity, the correlation coefficient for pairs of simultaneously recorded electrograms was determined. The voltage at each electrode (v or w) is considered to be a function of time (sample number, n), and the correlation coefficient for all time shifts, t, was calculated as follows:
We used correlation coefficients to compare the electrogram recorded farthest from the pacing site with electrograms from all other recording sites. The correlation coefficient had a magnitude of 1.0 for identical electrograms, and its magnitude approached zero for electrograms with few similarities. To avoid artifacts caused by differences in activation time, the peak value of the correlation coefficient as a function of time shift was used. Thus, electrograms were compared when time shifted to correlate best, thereby avoiding differences resulting from activation times.
Correlation coefficients decreased from 1.0 as the distance between recording sites increased. The decrease of the correlation coefficients with distance was observed to decay approximately exponentially with distance. Thus, exponential regression (linear regression of log[y]) was used to fit a function of the form r(t)=γ^{t} to the data for all nine electrode sites. From this fit, the correlation distance, defined as the distance between electrodes resulting in a correlation coefficient of .5, was calculated.
Statistical Analysis
There were six simulated tissues with different random number seeds at each of the five heterogeneity levels. Each of the simulations produced nine electrograms, creating a total of 270 electrograms. Determination of correlation distances and coefficients of variation used all nine electrograms from each simulated tissue, producing a total of 30 observations. Linear regression (statview, version 4.02) was done for each metric to detect changes as a function of heterogeneity, measured by the fractal dimension, D. A value of P=.05 was considered significant. Results are given as mean±SD.
Results
The five values of the distribution parameter, v, produced tissues with correlation dimensions of 0.000 (uniform), 0.006, 0.091, 0.223, and 0.591 (most heterogeneous). Correlation dimensions >0.6 were considered, but sustained propagation would not occur because of the high heterogeneity and a large population of uncoupled and functionally dead cells.
Activation Through Simulated Myocardium
Activation patterns were observed to be increasingly irregular as heterogeneity increased (Fig 4⇓). For uniform coupling resistances (D=0.000), there were almost smooth, elliptical isochrones, indicating normal activation. The slight deviation from completely smooth isochrones was due to the normal tissue heterogeneity and anisotropy built into the simulations. For mild coupling heterogeneity (D=0.006 and D=0.091), isochrones were closer together, indicating reduced propagation velocity, and were no longer smooth, indicating that propagation was disturbed. Moderate coupling heterogeneity (D=0.223) resulted in isochrones that were quite irregular, indicating that propagation was saltatory with velocity, varying greatly with location. For high coupling heterogeneity (D=0.591), activation was incomplete and even more serpiginous. Cells in large areas of the tissue were completely uncoupled and, as a result, not excited during propagation, simulating areas of fixed scar. Isochrones indicate that propagation varied greatly in velocity and direction with zones of slow conduction, block, and circuitous pathways around areas of block.
Electrogram Morphology and Metrics
The electrograms recorded from the nine sites above the simulated tissues reflected the coupling heterogeneity and resulting patterns of propagation. Fig 5⇓ shows examples. For uniformly coupled tissues, the electrograms were biphasic and normal in appearance, with very little variation between recording sites.
As coupling heterogeneity increased, electrograms were increasingly fractionated, as shown by decreases in mean peaktopeak amplitude, minimum derivative, frequency of peak power, and bandwidth accompanied by increases in mean duration and number of inflections (Table 2⇓). As heterogeneity increased, peaktopeak amplitude decreased 93% from a mean of 66±11 mV in the homogeneous case to a mean of 4.5±2.2 mV in the most heterogeneous cases. Similarly, minimum derivative decreased 93% from 32±4.9 to 2.3±1.6 V/s, frequency of peak power decreased 89% from 71±42 to 8.0±8.4 Hz, and bandwidth decreased 60% from 309±44 to 124±65 Hz. Duration and number of inflections increased with increasing heterogeneity, with duration increasing more than sixfold from 50±11 to 324±34 ms and number of inflections increasing from 2.8±0.6 to 5.0±2.2.
Spatial Variation in Electrogram Morphology
In addition, the coefficients of variation of these metrics showed that variation of electrograms from different sites increased with increasing coupling heterogeneity (Fig 6⇓). The coefficient of variation of electrogram peaktopeak amplitude, bandwidth, and frequency of peak power increased with increasing correlation dimension (P<.0001). The coefficient of variation of number of inflections increased significantly (P=.0090). The coefficient of variation of duration did not change significantly with increasing coupling heterogeneity (P=.4).
Electrogram variation between recording sites was also reflected by the correlation coefficients. As the distance between sites increased, the correlation coefficients between electrograms decreased approximately exponentially with increasing distance between electrodes (Fig 7⇓). The correlation distance was a measure of how much distance between electrodes was required to observe electrograms that were distinctly different. As a result of increasing variation between electrograms recorded at different sites, the correlation distance decreased with increasing heterogeneity as measured by the correlation dimension of the coupling resistances (P<.0001) (Fig 8⇓).
Discussion
Summary of Results
The results of our computer simulations show that, with increasing heterogeneity of cellular uncoupling, electrogram peaktopeak amplitude, minimum derivative, frequency of peak power, and bandwidth decrease, whereas duration and number of inflections increase. Furthermore, our results confirmed our hypothesis that spatial variation of morphology of electrograms recorded simultaneously from multiple sites increases with increasing heterogeneity of intercellular coupling. Specifically, the coefficient of variation of electrogram peaktopeak amplitude, minimum derivative, number of inflections, frequency of peak power, and bandwidth increases with increasing substrate heterogeneity. In addition, the correlation distance, defined as the distance between electrodes producing electrograms with a correlation coefficient of .5, decreases with increasing heterogeneity.
The only metric for which the coefficient of variation did not increase with increasing coupling heterogeneity was electrogram temporal duration. This result may be explained by the reduction in propagation velocity owing to a family of electrograms with a large mean duration produced by uncoupling. A large mean by itself reduces the coefficient of variation, which is defined as ς/μ, and this effect may dominate. Although the coefficient of variation of duration appears to be a less useful metric of heterogeneity, the other five electrogram metrics used did show increased coefficients of variation, supporting our hypothesis that variation increases with increasing heterogeneity. Note that, although the coefficient of variation of duration for multiple sites did not increase, the mean duration did increase, which is consistent with biological data.^{9}
Previous Studies
In a previous study, MullerBorer et al^{23} used 10μm elements in an anatomically detailed model. Although their study did not examine electrograms, they showed that, in an anatomic model, propagation was irregular on a microscopic scale while appearing continuous on a macroscopic scale. They also showed that irregularities on a microscopic scale increase with uniform cellular uncoupling. The present study uses a similar anatomically detailed model. For uniform coupling, no fractionated electrograms were observed for electrode points as close as 1 μm from the tissue surface. Our preliminary report used 100μm elements to model entire cells and showed fractionated electrograms resulting from cellular uncoupling.^{24} These results suggest that, because electrograms are spatial averages of transmembrane currents, heterogeneity on a microscopic scale does not cause fractionation. Heterogeneity of uncoupling on a larger size scale is needed to produce fractionated electrograms.
Previous studies examined individually recorded electrograms. Richards et al^{9} demonstrated that electrograms from infarcted regions had longer duration and lower amplitude compared with those from normal regions. Fractionated electrograms with long duration and a large number of inflections recorded during sinus rhythm are associated with slow conduction, which may be the substrate for ventricular tachycardia.^{10} Spach and Dolber^{11} found that cellular uncoupling caused slower conduction and zigzag propagation reflected in electrograms with smaller amplitudes and smaller maximum slopes. Peak amplitude and maximum slope were successfully used to discriminate between normal and infarcted tissue.^{12} The results of the present study are in agreement, showing reductions in mean amplitude and minimum derivative (or maximum slope) and increases in mean number of inflections and duration for electrograms generated by heterogeneously coupled tissue simulating infarction.
Blanchard et al^{25} used correlation coefficients of electrograms recorded simultaneously from multiple sites. The correlation coefficients were evaluated in a computer model and in canine tissue paced in three directions, and the correlation coefficients were found to decrease below .90 for interelectrode distances between 1.4 and 15.6 mm, depending on wavefront orientation. This information was used to determine a maximum distance over which linear interpolation of electrograms was accurate. Although the present study uses a different threshold and examines the effects of coupling heterogeneity, the results are consistent in showing a decrease in correlation of electrograms over distance. In addition, the results of Blanchard et al suggest that the decrease in correlation depends on propagation direction. Although we did not examine transverse propagation in the present study, the sparser distribution of gap junctions would be expected to cause a greater spatial variation of electrograms than that observed for longitudinal propagation.
Previous research showed that “mottled” infarcts with close interspersion of normal and abnormal myocardium are susceptible to the initiation of sustained ventricular tachycardia.^{26} ^{27} Electrophysiological characteristics of these infarcts indicated that areas of slow conduction were heterogeneously distributed and that disruptions in celltocell coupling and decreased excitability may contribute to the slow conduction.^{28} Although changes in cellular excitability were observed soon after infarction, they tended to normalize in the chronic setting. Membrane dynamics were uniform throughout our simulated tissue, modeling a chronic, healed infarct in which any acute changes in cellular excitability are normalized. Reduced space constants also suggested that disruptions in celltocell coupling contribute to slow conduction in the infarcted area.^{29} Furthermore, disruption of celltocell coupling with heptanol led Spear et al^{30} to conclude that slow, dissociated conduction in the infarcted region was due to abnormal gap junctions or gap junctional distributions. Previous research also indicated that the degree of heterogeneity of an infarcted region correlates with the degree to which it is susceptible to ventricular tachycardia.^{7} The present model has heterogeneously distributed areas of slow conduction resulting from heterogeneous cellular uncoupling modeled by increases in gap junctional resistance, yielding results consistent with previous studies.
Limitations
Previous studies showed that electrogram morphology reflects substrate activation, and electrograms can be explained on the basis of the distribution of intercellular currents.^{31} The results of the present study indicate that variation in morphology of simultaneously recorded electrograms reflects spatial variation of substrate activation in heterogeneously coupled tissue. A fractal pattern of heterogeneous coupling was used to allow quantization of heterogeneity with a single metric, the correlation dimension. Coefficients of variation and correlation distances were found to be sensitive to this type of intercellular coupling heterogeneity. However, these metrics indicate statistical variation and do not uniquely identify the distribution of intercellular currents. As a result, these metrics should be sensitive to any form of intercellular coupling heterogeneity and may be sensitive to heterogeneity of tissue characteristics other than intercellular coupling.
Our study did not attempt to actually induce sustained reentry to verify that heterogeneously coupled tissue correlates with ventricular tachycardia substrate, although our simulations were consistent with known characteristics of arrhythmogenic tissue. In previous modeling studies, we showed that reentry can occur in a similar computer model.^{32} For sustained reentry, however, a sufficient area of tissue is required so that the activation wavefront does not meet refractory tissue. When 25μm elements and full BeelerReuter ionic kinetics are used, available computer time limits the size of the tissue sheet that can be simulated. Thus, we did not attempt to induce reentry in the 10×2.5mm sheets of simulated myocardium.
A further consideration is accuracy of activation and extracellular potential calculation in the simulated tissues. Models more recent than the BeelerReuter model are available, such as the LuoRudy^{33} model. These two models differ mostly in repolarization, and because the present study is interested in depolarization, the computational expense of more recent models was not necessary. However, if future studies use premature beats or rapid pacing in which repolarization effects are important, the updated cellular models will be useful. For extracellular potential calculations, we assumed point electrodes in an unbounded, homogeneous, isotropic extracellular space. The bidomain model^{34} uses a more detailed representation of extracellular resistance than we used in the present study. However, even if heterogeneity of extracellular space were included in the model, we would still expect the metrics examined in this study to be sensitive to substrate heterogeneity, although the concept of “heterogeneity” would need to include heterogeneity of both intercellular and extracellular resistivities. We believe that, regardless of the specific model used, the present study demonstrates a fundamental effect of substrate heterogeneity on electrograms. To confirm our hypothesis, biological experimentation with multipolar catheters and threedimensional areas of myocardium is necessary. The results of this study indicate that biological experiments using unipolar recordings from a decapolar catheter with ≈1mm interelectrode spacing should be sufficient to observe significant variation in simultaneously recorded electrograms as a function of substrate heterogeneity.
Conclusions
A computer model of heterogeneously coupled myocardium mimics electrogram generation seen in association with the border zone of a healed infarction. Furthermore, the results of this study indicate that information about coupling heterogeneity may be obtained from the variation of simultaneously recorded electrograms. Specifically, the coefficient of variation of the electrogram amplitude, number of inflections, minimum derivative, frequency of peak power, and bandwidth increase with increasing heterogeneity. Furthermore, the correlation distance of electrograms decreases with increasing heterogeneity, supporting the conclusion that spatial variation of electrogram morphology increases with increasing heterogeneity of substrate cellular uncoupling. Although these results await confirmation in biological experiments, they indicate geometric relations that should be useful in the development of clinical methods to detect substrate heterogeneity from multiple simultaneously recorded electrograms.
Acknowledgments
This work was supported by NIH grant HL45664, by American Heart Association GrantinAid 9000914, and by the National Science Foundation/San Diego Supercomputer Center. W.S. Ellis was initially supported by a National Science Foundation graduate fellowship. This work was done during the tenure of a research fellowship from the American Heart Association, California Affiliate. We are grateful to Ashutosh Goel for his computer programming assistance.
Footnotes

Reprint requests to Michael D. Lesh, Bioengineering Graduate Group, University of California, Berkeley and San Francisco, Cardiovascular Research Institute, University of California, San Francisco, 500 Parnassus Ave, PO Box 1354, San Francisco, CA 941431354.
 Received February 27, 1995.
 Accepted April 1, 1995.
 Copyright © 1995 by American Heart Association
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 Fractionated Electrograms From a Computer Model of Heterogeneously Uncoupled Anisotropic Ventricular MyocardiumWillard S. Ellis, David M. Auslander and Michael D. LeshCirculation. 1995;92:16191626, originally published September 15, 1995http://dx.doi.org/10.1161/01.CIR.92.6.1619
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 Fractionated Electrograms From a Computer Model of Heterogeneously Uncoupled Anisotropic Ventricular MyocardiumWillard S. Ellis, David M. Auslander and Michael D. LeshCirculation. 1995;92:16191626, originally published September 15, 1995http://dx.doi.org/10.1161/01.CIR.92.6.1619
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