Download Nonlinear effects in subthreshold virtual electrode polarization

Am J Physiol Heart Circ Physiol 284: H2368–H2374, 2003;
10.1152/ajpheart.00988.2002.
Nonlinear effects in subthreshold virtual electrode polarization
Aleksandre T. Sambelashvili, Vladimir P. Nikolski, and Igor R. Efimov
Department of Biomedical Engineering, Case Western Reserve University, Cleveland, Ohio 44106-7207
Submitted 18 November 2002; accepted in final form 18 February 2003
electrophysiology; stimulation; excitation; ion channels
RECENT INVESTIGATION OF ELECTRICAL stimulation of the
heart revealed an important role of tissue heterogeneity in the generation and propagation of muscle excitation. One of the important intrinsic forms of the
heterogeneity consists of inequality of anisotropic
properties of the extracellular and intracellular spaces
of the cardiac syncytium. This inequality was first
theoretically described within the framework of the
bidomain model (9, 18, 29), which represents cardiac
tissue as two interpenetrating extra- and intracellular
domains with different conductivities along and across
the direction of the fibers, which are coupled via membrane resistance. Using the bidomain model,
Sepulveda et al. (26) predicted the existence of adjacent
areas of hyperpolarization and depolarization during
unipolar stimulation of the tissue. These areas were
termed virtual electrode polarizations (VEPs). For a
point-size cathodal stimulus, the transmembrane po-
Address for reprint requests and other correspondence: I. R. Efimov, Wickenden Bldg., Rm. 520, Case Western Reserve Univ., 10900
Euclid Ave., Cleveland, OH 44106-7207 (E-mail: [email protected]).
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tential (Vm) distribution pattern had a central depolarized virtual cathode (VC) region with a characteristic
dog-bone shape and two elongated hyperpolarized virtual anode (VA) regions on the sides that were parallel
to the direction of the fibers. In the linear bidomain
model, for the opposite polarity of the stimulus, the
VEP pattern was symmetrically reversed.
The dog-bone VEP pattern induced by point stimulation was experimentally confirmed using electrode
mapping (31) and optical imaging (12, 19, 30). This
virtual electrode phenomenon resulted in the formulation of a unified theory of stimulation and suggested
solutions to the old puzzles of break-excitation and
anodal stimulation (24, 30). The concept of activating
function generalized VEP theory to electric fields of
any geometrical configuration (27). In particular,
strong electric shocks applied during defibrillation also
induce areas of adjacent positive and negative polarization, which can be considered VEPs (8). This concept
allowed the explanation of shock-induced arrhythmias
and defibrillation failure via virtual electrode-induced
phase singularities (7, 15, 16).
Despite the impressive success of VEP theory in a
wide range of investigations, the very existence of
VEPs remains controversial as it pertains to subthreshold stimulation (STS) (1). Measurements of subthreshold VEPs represent a significant experimental
challenge (1). Nevertheless, STS plays a crucial role in
the generation and propagation of electric pulses. STS
of Purkinje fibers was shown to interrupt ventricular
tachycardia by synchronization of ventricular excitation (25). Preconditioning by STS inhibited the excitability of Purkinje fibers, which is in contrast to the
facilitation of muscle cell fiber excitability (6). Finally,
STS is used in the assessment of tissue coupling (1, 2,
14, 22).
Using optical imaging and the active bidomain
model, we sought to determine the existence and dynamics of VEPs during unipolar STS.
METHODS
Experimental preparation, optical mapping setup, and protocol. This study conformed to the guidelines of the American
Heart Association. The experiments were performed in vitro
on hearts obtained from New Zealand White rabbits (n ⫽ 4).
In two additional experiments, guinea pig hearts (n ⫽ 2) were
used to check for possible interspecies differences. The hearts
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Sambelashvili, Aleksandre T., Vladimir P. Nikolski,
and Igor R. Efimov. Nonlinear effects in subthreshold virtual electrode polarization. Am J Physiol Heart Circ Physiol
284: H2368–H2374, 2003; 10.1152/ajpheart.00988.2002.—
Introduction of the virtual electrode polarization (VEP) theory suggested solutions to several century-old puzzles of
heart electrophysiology including explanation of the mechanisms of stimulation and defibrillation. Bidomain theory predicts that VEPs should exist at any stimulus strength. Although the presence of VEPs for strong suprathreshold
pulses has been well documented, their existence at subthreshold strengths during diastole remains controversial.
We studied cardiac membrane polarization produced by subthreshold stimuli in 1) rabbit ventricular muscle using highresolution fluorescent imaging with the voltage-sensitive dye
pyridinium 4-{2-[6-(dibutylamino)-2-naphthalenyl]-ethenyl}1-(3-sulfopropyl)hydroxide (di-4-ANEPPS) and 2) an active
bidomain model with Luo-Rudy ion channel kinetics. Both in
vitro and in numero models show that the common dog-boneshaped VEP is present at any stimulus strength during both
systole and diastole. Diastolic subthreshold VEPs exhibited
nonlinear properties that were expressed in time-dependent
asymmetric reversal of membrane polarization with respect
to stimulus polarity. The bidomain model reveals that this
asymmetry is due to nonlinear properties of the inward
rectifier potassium current. Our results suggest that active
ion channel kinetics modulate the transmembrane polarization pattern that is predicted by the linear bidomain model of
cardiac syncytium.
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steps in the directions both across and along the fibers were
chosen to be 0.1 mm, which was less than half of the smallest
space constant. The time step was 10 ␮s. To solve for the
extracellular potential, the fast Fourier transform method
(28) was employed at each step as the most robust and the
fastest for our case. Repeating the simulations for the twicesmaller space and time steps confirmed the stability and
accuracy of our calculations to ⬍0.001 mV. All calculations
were performed on a Pentium 800 MHz personal computer.
The results were obtained in terms of Vm values and ionic
currents and were plotted and analyzed with MATLAB software (MathWorks; Natick, MA).
RESULTS
ST-Vm distribution exhibits classical dog-boneshaped pattern. Pacing thresholds determined in our
experiments ranged from ⫺0.15 to ⫺0.60 mA for
cathodal stimulation and from ⫹0.85 to ⫹1.76 mA for
anodal stimulation. A representative example of optical maps of the Vm recorded at different times (2, 4, and
10 ms) during 20-ms cathodal and anodal STS is shown
in Fig. 1. Even for subthreshold stimuli, the VEP
pattern is dynamic. Small VEPs are created around the
electrode tip almost instantaneously (faster than our
temporal resolution) after the application of the stimulus, and the pattern then rapidly spreads out with a
simultaneous increase in magnitude of both depolarization and hyperpolarization. The most pronounced
VEP pattern is observed at ⬃4 ms (13), which is determined by the time constant for the passive cardiac
tissue resistor-capacitor network. Later VEPs continue
to grow slowly; in the case of the cathodal stimulus,
this occurs by spreading of the two regions of hyperpolarization, and in the case of the anodal stimulus, it
occurs by spreading of the two regions of depolarization. In both cases, the maximum polarization at these
two regions diminishes, which makes it more challenging to detect the characteristic VEP pattern at times
⬎10 ms after the onset of the stimulus.
Resting tissue at diastole cannot be considered totally
passive. A remarkable asymmetry exists in the VEPs
for anodal and cathodal stimuli at 10 ms in Fig. 1.
First, the central hyperpolarized area for the anodal
pulse is substantially smaller than the corresponding
depolarized area for the cathodal pulse. The distance
from the electrode tip to the closest point with no
deviation from the resting potential (white area) is 0.7
mm in the first case, whereas in the second case it is 1.4
mm, which is twice as large. The average ratio of these
distances for cathodal vs. anodal stimuli obtained from
all animals (n ⫽ 4) was 1.9 ⫾ 0.3. Second, VC areas for
the anodal stimulus have a maximum depolarization of
⫹1.9 mV and therefore, in absolute terms, are 2.1
times more pronounced than the VA areas for the
cathodal stimulus, where the maximum hyperpolarization is only ⫺0.9 mV. The average ratio (n ⫽ 4) of the
maximum polarizations of the side virtual electrodes
for anodal vs. cathodal stimulation was 2.3 ⫾ 0.2.
To understand the reason for the asymmetry described, we conducted numerical simulations on the
basis of the active bidomain model. Figure 2 illustrates
simulated maps of the Vm that correspond to the ex-
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were placed onto a Langendorff apparatus where they were
retrogradely perfused with oxygenated modified Tyrode solution as previously described (21). Motion artifacts in optical
recordings were suppressed by 15 mM 2,3-butanedione monoxime (BDM). The voltage-sensitive dye pyridinium 4-{2[6-(dibutylamino)-2-naphthalenyl]-ethenyl}-1-(3-sulfopropyl)hydroxide (di-4-ANEPPS; 0.5 ␮M; Molecular Probes; Eugene,
OR) was added to the circulating solution within 5 min.
Hearts were stained for 40 min before the recordings were
started.
The optical mapping setup used for acquisition of the
optical signals was described previously (20, 21). Briefly, the
light produced by a 250-W quartz-tungsten-halogen DC-powered light source passed through a 520 ⫾ 45-nm excitation
filter, was reflected by a 585-nm dichroic mirror, and passed
through a 50-mm lens before it illuminated a rabbit heart.
The fluorescence emitted from the heart was collected by the
same lens, passed through the dichroic mirror, and filtered
by a long-pass filter (⬎610 nm). The light was then detected
by a l6 ⫻ 16 photodiode array (C4675; Hamamatsu).
For each subthreshold test, stimulus Vm traces (ST-Vm)
were recorded from 256 channels of the photodiode array
focused on the 5 ⫻ 5-mm window around the tip of the
test-stimulus electrode. Sampling was performed at a rate of
2,900 frames/s. Therefore, we had a space resolution of 312
␮m and a time resolution of 0.3 ms.
For improvement of the signal-to-noise ratio, we averaged
10 consecutive recordings for each stimulus polarity kept at
the same duration and current strength. Averaging resulted
in a threefold improvement in the signal-to-noise ratios. Vm
maps and ST-Vm traces were constructed from these averaged data.
Calibration was performed as previously described (5).
Briefly, we assumed resting potential to be ⫺85 mV and
action potential amplitude to be 100 mV.
To reach steady state, the heart was paced with 20 basic
pulses (with a cycle length of 350 ms) using a bipolar electrode at the base of the left ventricle far from the field of view.
After the basic drive, a test-current stimulus was applied
from a unipolar electrode (0.12-mm platinum-iridium-Tefloncoated wire) that was positioned in the middle of a 5 ⫻ 5-mm
field of view at the anterior wall of the left ventricle. The test
stimulus was delivered at a coupling interval of 300 ms from
a constant-current source (A385; WPI; Sarasota, FL). An
ECG was recorded to verify suprathreshold and subthreshold
pacing. After determination of the diastolic pacing thresholds
for cathodal and anodal pulses, only subthreshold test stimuli were applied. The subthreshold amplitude was kept at 0.1
mA less than the cathodal threshold, and the duration was
20–60 ms. The polarity of the stimuli was reversed each time
to ensure a correct comparison between anodal and cathodal
stimulations. Basic pacing stimulus strength was adjusted to
twice the diastolic threshold of excitation.
Numerical models. We guided our experiments with computer simulations based on the two-dimensional active bidomain model with Luo-Rudy phase I ion channel kinetics (17).
For the cases of passive bidomain simulations, the transmembrane current was described by Ohm’s law. The twodimensional slice of cardiac tissue was assumed to have
straight fibers and constant intracellular and extracellular
conductivities along and across the direction of the fibers.
The parameters for the cardiac tissue were the same as those
used by Roth (24). The boundaries were sealed for the intracellular space and grounded (zero potential) for the extracellular space. Different-polarity STSs of 0.8 ⫻ threshold
strength and 60 ms duration were applied from a point-size
source to the center of a 2 ⫻ 2-cm square domain. The space
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Fig. 1. Experimental virtual electrode polarizations
(VEPs) at different times during subthreshold stimulation (STS). Optically recorded and averaged
maps of transmembrane potentials (Vm) were obtained from a 5 ⫻ 5-mm area of anterior epicardium
of rabbit heart for unipolar subthreshold current
anodal and cathodal stimuli (⫾0.5 mA, 20 ms).
Sepulveda’s dog-bone pattern of VEPs for anodal (A)
and cathodal (B) stimuli grows in size with time and
becomes asymmetric; central depolarized area for
cathodal stimulus at 10 ms is larger than the corresponding hyperpolarized area for anodal stimulus.
Vm varies from ⫺90 mV (in blue) to ⫺80 mV (in red);
⫺85 mV (in white) represents the resting state.
one might expect, the ST-Vm curves from the two sites are
symmetric with respect to the stimulus polarity. They
reflect the passive VEP dynamics. First the transmembrane VEPs in the central and the side areas grow rapidly, reaching a peak by 4–5 ms after the application of
the pulse. Later, as the side VEP areas move away from
the center, the corresponding polarizations taken from
the sites at some distance along the direction of the fibers
decrease to a lower value (in the example shown, to zero).
The second peak of the voltage traces after the withdrawal of the pulse indicates the charge diffusion and the
quick elimination of the dog-bone pattern.
As mentioned, if the tissue were completely passive
and linear, then the reversal of the stimulus polarity
would simply invert the VEP pattern, and depolarized
areas and responses would become hyperpolarized and
vice versa, which is shown in Fig. 3. However, experimental traces of ST-Vm on Fig. 4 clearly demonstrate
that for the cathodal stimulus, the central VC area
grows to a larger extent than the corresponding VA
Fig. 2. VEPs at different times during
STS obtained from active two-dimensional bidomain simulations with LuoRudy ion channel kinetics. Images for
anodal (A) and cathodal (B) stimuli
(⫾60 ␮A/cm, 20 ms) show the growth of
the VEP with time as well as the slight
asymmetry of the VEP with respect to
the stimulus polarity. Vm variations
are within the range of ⫾1 mV; ⫺84.5
mV represents the resting state (in
white).
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perimental maps from Fig. 1. Similar dynamics of
growing VEPs can be observed. Transmembrane VEPs
reach maximum values at nearly 4 ms after the stimulus application. At 10 ms, the pattern is asymmetric
with respect to the stimulus polarity, which is similar
to the experimental observations: namely, the central
hyperpolarized area for the anodal pulse is slightly
smaller than the depolarized area for the cathodal
pulse, and the smallest distance from the pacing point
to the zero-deflection line is 0.6 mm in the first case
and 0.8 mm in the second case. In addition, the maximum depolarization at the VCs for the anodal stimulation is almost twice as high (⫹0.34 mV) compared
with the maximum hyperpolarization at the VAs for
the cathodal stimulation (⫺0.18 mV).
To analyze the asymmetry in detail, we considered
ST-Vm traces from two different spots of oppositely polarized areas of VEP for both anodal and cathodal stimuli.
Figure 3 demonstrates such traces for bidomain simulations assuming a passive and linear tissue response. As
SUBTHRESHOLD VIRTUAL ELECTRODES
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Fig. 3. Simulated VEP and subthreshold stimulation
induced by Vm (ST-Vm) for the passive bidomain model.
A: VEP patterns for cathodal and anodal stimuli of the
same amplitude at 2 ms after the stimulus application.
B: time courses of the Vm changes from the two points of
virtual cathode (VC; in red) and virtual anode (VA; in
blue) indicated by boxed areas in A. For the passive
model, the VEP at any time is reversed with the reversal of the stimulus polarity.
Nonlinearity in inward rectifier current causes dogbone-shaped asymmetry. We analyzed the contributions of different ionic currents into the formation of
the ST-Vm traces at the two sites of interest. Figure 6A
shows the time course of the changes of the major
currents during stimulation. This image illustrates
that the total transmembrane current is affected primarily by the inward rectifier potassium current (IK1),
which is known to be responsible for the maintenance
of the resting potential. According to our simulations,
for cathodal STS, due to rather small deviations of Vm
from the resting value, the IK1 plays the major role for
almost the entire central depolarized dog-bone-shaped
Fig. 4. Experimental VEP and ST-Vm. A: optical maps
of VEP patterns for cathodal and anodal stimuli of the
same amplitude at 2 ms after the stimulus application.
An example of an optical recording of Vm is also shown
(bottom). B: time courses of the Vm changes from the
two points of VC (in red) and VA (in blue) indicated by
boxed area in A. VEP patterns are asymmetric with
respect to reversal of the stimulus polarity, with a
larger area of depolarization for the cathodal stimulus
compared with the corresponding area of hyperpolarization for the anodal stimulus. A slight negative trend
in the resting potential is due to slow residual repolarization from the previous beat.
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area for the anodal stimulus thus suppressing the
opposite negative response from the two neighboring
VA regions and creating the ST-Vm asymmetry. Active
bidomain simulations of the Luo-Rudy model (Fig. 5)
are in good agreement with the experiment. One can
see a substantially higher level of the absolute polarization in the VC area for the anodal stimulus compared to the VA area for the cathodal stimulus. From
these observations, we can conclude that cardiac tissue
at diastole cannot be considered completely passive.
Nonlinear ionic mechanisms significantly modulate
VEP responses to STSs in a stimulus polarity-dependent manner.
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region with perhaps the nonsignificant exception of a
tiny area around the tip of the electrode. This observation excludes a possible explanation of the asymmetry
as being created by the influence of the undeveloped
excitation in the case of cathodal stimulation.
Because the IK1 in the Luo-Rudy model is time independent, i.e., its strength is determined solely by the
Vm, the discrepancies between cathodal and anodal
stimulation must be explained by the dependence of
the IK1 conductance on Vm.
Figure 6B shows that IK1 conductance is in general
increased by hyperpolarization and decreased by depolarization, although not by the same absolute value.
Because the length constant of the tissue is inversely
proportional to the square root of the transmembrane
conductance (23), it must be larger for depolarized
regions compared with hyperpolarized regions. So the
larger central depolarized area for the cathodal pulse
in comparison with the central hyperpolarized area for
the anodal pulse can be explained by the changing of
the space constant during polarization of the tissue.
The same effect of the IK1 conductance alterations over
the region of the VEP leads to stronger VC areas for
anodal stimulation compared with the VA areas for
cathodal stimulation.
The results obtained in our experiments with guinea
pig hearts were similar to those obtained for rabbit
hearts; i.e., we observed the same dynamics of VEP
Fig. 6. Ionic currents and inward rectifier potassium current (IK1) conductivities during STS. Changes in major
currents (left) are shown at the two
points of VC (in red) and VA (in blue)
during cathodal (A) and anodal (B)
STS. IK1 is the primary current in both
cases and behaves asymmetrically
with respect to the reversal of the stimulus polarity. Time courses of IK1 conductivity (GK1; right) are provided at
the same two points of VA (in red) and
VA (in blue) (indicated by boxed areas)
and the site of the stimulation (in
black). GK1 changes unequally for
cathodal vs. anodal STS.
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Fig. 5. Simulated VEPs and ST-Vm for active bidomain
model. A: VEP patterns are illustrated for cathodal and
anodal stimuli of the same amplitude at 2 ms after the
stimulus application. B: time courses of the Vm changes
from the two points of VC (in red) and VA (in blue)
indicated by boxed area in A. VEP patterns are asymmetric with respect to reversal of the stimulus polarity,
with larger area of depolarization for the cathodal stimulus compared to the corresponding area of hyperpolarization for the anodal stimulus. These results are in
agreement with experimental recordings shown Fig. 4.
SUBTHRESHOLD VIRTUAL ELECTRODES
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Fig. 7. VEP for weak cathodal stimulus applied during
plateau phase of the action potential: experiment and
active bidomain simulation. A: optical maps of VEP at
10 ms of the ⫺0.5-mA, 15-ms stimulus that was applied
at a 50-ms coupling interval. B: same VEP obtained
from active bidomain simulations is shown. For the
same stimulus strength, the resulting VEP for plateauphase stimulation was larger compared to the stimulation at resting state (see Fig. 1). Vm values are shown as
deviations from normal action potential values.
DISCUSSION
Conditions for detection of VEP patterns. Advances
in voltage-sensitive dye techniques have enabled us to
detect VEP patterns on the surface of the heart during
stimulation. However, in most of the reported results,
the stimuli have been either very strong (suprathreshold) or they have been applied during the plateau
phase of the action potential. In the first case, stronger
shocks produce higher VEP magnitudes and therefore
require less voltage resolution. If a strong shock is
delivered during diastole, the resulting VEP pattern
becomes overlapped by excitation within several milliseconds, thus preventing one from observing possible
VEP dynamics at later times. We show here that for
STS, the VEP pattern is most pronounced at 4–5 ms
from the moment of the stimulus application, which is
the time that is comparable with the resistance-capacitance time of the tissue. Then, because of charge
diffusion, the pattern slowly becomes less spatially
concentrated with the diminishing of the Vm differences. The STS VEP is asymmetrical with respect to
the polarity of the stimulus. We explained the modulation of the VEP pattern by the nonlinear influence of
the transmembrane currents, primarily the IK1. These
facts should be taken into consideration when one is
trying to experimentally obtain the ST-Vm distribution
as well as in attempts to make certain conclusions
regarding the tissue properties, for instance, such as
intracellular conductance (1), from the STS data.
In the second case of stimulation during the plateau
phase, larger and stronger VEPs are produced than if
the same amplitude of current stimulus were applied
during the resting state (Fig. 7). As we see it, the
reason for this lies in a seemingly paradoxical fact that
the total transmembrane conductance of the ion channels for the most part of the plateau phase is smaller
than that of the diastolic period (32). According to
theory, a bigger transmembrane resistance allows for
stronger expression of the differences in the anisotropies of the intracellular and extracellular spaces and
AJP-Heart Circ Physiol • VOL
therefore causes a more pronounced VEP pattern. This
explains why it has been easier to detect the VEPs for
shocks of high magnitude or shocks applied during the
plateau phase of the action potential.
Nonlinearity of Vm response to electric shock. The Vm
response to external shock plays a critical role in pacing, defibrillation, and cardioversion. It is generally a
function of the strength of the external field, the time of
its application, its polarity, and possibly other factors.
Many studies investigating the effect of stimulus polarity on Vm changes (⌬Vm) have appeared recently. In
experiments on cardiac cell strands, Cheek et al. (3)
demonstrated that negative ⌬Vm values for a strong
anodal stimulus applied during the plateau phase are
bigger than positive ⌬Vm values for the same-strength
cathodal shocks. This asymmetry was attributed to
nonlinearity in Ca2⫹ channel kinetics. Similar results
have been obtained from microelectrode recordings of
guinea pig papillary muscle plateau-phase ⌬Vm values
(33) and from isolated ventricular myocytes (4). The
main idea of these studies consists of emphasizing the
crucial role of active ion channel properties in generation of the Vm response to electric stimulation.
In this work, we show that even at the resting state
for a weak STS, the membrane response cannot be
accurately represented by a simple, passive resistancecapacitance model. Ionic modulation can lead to small
but experimentally detectable asymmetry of the response, which may be quite important if one takes into
account the all-or-none nature of the action potential
generation process.
Limitations. An optical mapping setup collects averaged fluorescence not only from the surface of the
epicardium, but also from a thin layer beneath that
surface (10, 11). Therefore, without knowing the exact
three-dimensional pattern of the Vm distribution, it is
hard to make conclusions about the reasons for the
observed asymmetry in response to stimuli of different
polarities. Aside from that, the experimental part of
our study was accomplished with the excitation-contraction uncoupler BDM, which is known to affect ionic
channel properties. Nevertheless, we believe that the
results of our computer simulations, which show good
qualitative agreement with experimentally observed
ST-Vm patterns, alleviate these limitations.
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development and polarity-dependent asymmetry of the
STS response. However, the small number of experiments with guinea pigs did not allow us to make any
significant comparisons between the species.
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The bidomain model we used was two dimensional
and assumed the simple geometry of straight fibers. In
addition, the Luo-Rudy phase I model of the cardiac
myocyte ignored some existent ionic channels as well
as the role of certain cellular structures such as sarcoplasmic reticulum. These and other factors might have
caused the discrepancies in the values of the ⌬Vm for
the simulations and the experiments. Despite these
limitations, we think that for our case, the chosen
model captured all the substantial features and was
sufficient for a qualitative description of electrophysiological responses of the heart tissue to STS.
This study was supported by the Whitaker Foundation and by
National Heart, Lung, and Blood Institute Grant HL-67322.
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