(Circulation. 1995;92:1619-1626.)
© 1995 American Heart Association, Inc.
Articles |
Fractionated Electrograms From a Computer Model of Heterogeneously Uncoupled Anisotropic Ventricular Myocardium
From the Bioengineering Graduate Group, University of California, Berkeley and San Francisco (W.S.E., D.M.A., M.D.L.); the Mechanical Engineering Department (D.M.A.), University of California, Berkeley; and the Cardiovascular Research Institute (M.D.L.), University of California, San Francisco.
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Abstract |
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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 Beeler-Reuter 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 peak-to-peak 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.
Key Words: electrophysiology • myocardium • potentials
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Introduction |
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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 regions8 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.
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Methods |
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Computer Model
We implemented a computer model of heterogeneously coupled ventricular myocardium (Fig 1
). A rectangular sheet of
cells was simulated with
elements with Beeler-Reuter13 ionic kinetics. To include
the spatial heterogeneity present even in normal
myocardium, Beeler-Reuter 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 k
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.
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The simulated tissue was implemented numerically by a C program run on a Cray C90 supercomputer. The Beeler-Reuter 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 1-ms current pulse in a 200-µm 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:
 | (1) |
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 self-similarity
across size scales, characteristic of a fractal structure. Thus, each
gap junctional resistance (Rj) could be written as
|
 | (2) |
where
R'j is normal gap junctional resistance and
1, 2. . .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 two-dimensional (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.
 | (3) |
A linear regression was applied to log{C[RJ(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,
12. . .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
10x2.5-mm 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,
IM, contributes to an electrode at distance r, and
electrograms, E, were calculated from the following
equation21 :
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 | (4) |
Equation
4 was used with e=150
· 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: peak-to-peak 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 peak-to-peak 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:
 |
 | (5) |
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, 
, of each metric for the
nine electrograms recorded from each simulated tissue. The
coefficient of variance for a tissue was calculated as the ratio
/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:
 | (6) |
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)=
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.
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Results |
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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.
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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.
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As coupling heterogeneity increased, electrograms were
increasingly fractionated, as shown by decreases in mean peak-to-peak
amplitude, minimum derivative, frequency of peak power, and bandwidth
accompanied by increases in mean duration and number of inflections
(Table 2). As heterogeneity increased,
peak-to-peak 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.
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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
peak-to-peak 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).
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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).
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Discussion |
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Summary of Results
The results of our computer simulations show that, with increasing heterogeneity of cellular uncoupling, electrogram peak-to-peak 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 peak-to-peak 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 /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, Muller-Borer et
al23 used
10-µm 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 100-µm 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 al9 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 Dolber11 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 al25 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 cell-to-cell 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 cell-to-cell coupling contribute to slow conduction in the infarcted area.29 Furthermore, disruption of cell-to-cell coupling with heptanol led Spear et al30 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 25-µm elements and full Beeler-Reuter 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 10x2.5-mm sheets of simulated myocardium.
A further consideration is accuracy of
activation and extracellular
potential calculation in the simulated tissues. Models more recent than
the Beeler-Reuter model are available, such as the
Luo-Rudy33 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
model34 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
three-dimensional areas of myocardium is necessary. The
results of this study indicate that biological experiments using
unipolar recordings from a decapolar catheter with 
1-mm
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.
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Acknowledgments |
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This work was supported by NIH grant HL-45664, by American Heart Association Grant-in-Aid 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.
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Footnotes |
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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 94143-1354.
Received February 27, 1995; accepted April 1, 1995.
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