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Cardiovascular Research Laboratory, Departments of Medicine (Cardiology), Physiology, and Physiological Science, University of California, Los Angeles, California 90095
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ABSTRACT |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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Spiral wave breakup is a proposed mechanism underlying the transition from ventricular tachycardia to fibrillation. We examined the importance of the restitution of action potential duration (APD) and of conduction velocity (CV) to the stability of spiral wave reentry in a two-dimensional sheet of simulated cardiac tissue. The Luo-Rudy ventricular action potential model was modified to eliminate its restitution properties, which are caused by deactivation or recovery from inactivation of K+, Ca2+, and Na+ currents (IK, ICa, and INa, respectively). In this model, we find that 1) restitution of ICa and INa are the main determinants of the steepness of APD restitution; 2) for promoting spiral breakup, the range of diastolic intervals over which the APD restitution slope is steep is more important than the maximum steepness; 3) CV restitution promotes spiral wave breakup independently of APD restitution; and 4) "defibrillation" of multiple spiral wave reentry is most effectively achieved by combining an antifibrillatory intervention based on altering restitution with an antitachycardia intervention. These findings suggest a novel paradigm for developing effective antiarrhythmic drugs.
fibrillation; antiarrhythmic drugs; chemical defibrillation; ventricular tachycardia
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INTRODUCTION |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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SPIRAL WAVES as the substrate of reentrant arrhythmias were first predicted theoretically (23, 47) and were later observed in several cardiac preparations (4, 9). Simulations have shown that, depending on the underlying electrophysiological properties, spiral waves can be stationary, meander, or spontaneously break up into a fibrillation-like state. The factors controlling spiral wave stability may be very relevant to clinical arrhythmias. It is well known that ventricular fibrillation (VF) develops in stages, with the first stage corresponding to polymorphic or monomorphic ventricular tachycardia (VT) lasting from several to many beats (29, 46). In electrically induced VF in the canine heart, activation mapping studies showed that VF typically begins as two reentrant wave fronts in a figure eight (4), which can be terminated by appropriately timed premature stimuli delivered during the "protective zone" (1, 19). After two to five rotations, however, the initial wave fronts break up into multiple wave fronts, and the ability of a single extrastimulus to terminate fibrillation is lost. These observations suggest that the initiation of fibrillation corresponds to the generation of one or two reentrant wave fronts, which subsequently break up to produce the multiple reentrant wave fronts that characterize fully developed fibrillation. Consequently, understanding the electrophysiological mechanisms that control the stability of spiral wave reentry may provide useful insights for defining the desirable properties of antifibrillatory drugs.
The restitution properties of the cardiac action potential duration (APD) and conduction velocity (CV) were shown to be important determinants of the stability of reentrant arrhythmias in general (6, 7, 13, 21, 36). Restitution is the property that, as the diastolic interval of a premature beat varies, the APD and CV of that beat also vary, typically decreasing with decreasing diastolic interval. When the restitution curve relating APD to the preceding diastolic interval has a steep slope (>1), reentry around an anatomic obstacle becomes subject to complex oscillations in cycle length (CL) and APD, both in experimental preparations (11, 13) and in computer simulations (7, 36). In simulations in two-dimensional (2-D) sheets of cardiac tissue, restitution characteristics were also shown to be important determinants of spiral wave stability, influencing whether a single spiral wave remains stationary, meanders, or breaks up into multiple reentrant wave fronts resembling cardiac fibrillation (6, 21). However, the effects of steepness of restitution properties on spiral wave stability are not straightforward. For example, Karma (21) found that with increasing steepness of APD restitution, spiral wave reentry became progressively unstable, leading to breakup in a 2-D sheet based on a simplified two-variable model of the cardiac action potential. In contrast, Courtemanche (6) found that speeding the kinetics of Isi (slow inward current) in the Beeler-Reuter action potential model increased the maximum slope of APD restitution but prevented spiral wave breakup. The relative importance to spiral wave stability of the maximum slope of APD restitution, the range of diastolic intervals over which the slope is steep, and the interaction with CV restitution are therefore not entirely clear. It is also not clear to what extent APD restitution properties of an isolated cardiac cell are predictive of the tissue restitution properties relevant to spiral wave stability, because diffusive (axial) currents in addition to membrane ionic currents were shown to alter APD restitution properties (26).
The goal of this study was to further clarify the role of cardiac restitution properties in spiral wave stability, in a context that could be potentially extrapolated to and tested in experimental studies. The major determinants of APD and CV restitution at the cellular level are the restitution kinetics of inward and outward currents. We used phase 1 of the Luo-Rudy model of the ventricular action potential (LR1) (27), which formulates the most important cardiac ionic currents in detail, to show how the restitution properties of the major ionic currents contribute to APD and CV restitution properties and to examine the extent to which single-cell restitution properties predict spiral wave behavior in a 2-D sheet of simulated cardiac tissue. Our findings show that single-cell restitution properties are generally, but not always, predictive of spiral wave stability. Furthermore, these findings clarify that the range of diastolic intervals for which the slope of APD restitution is steep, rather than the maximum value of the slope, is the critical determinant of spiral wave breakup. These results provide a template for predicting how the restitution properties of individual K+, Ca2+, and Na+ currents, as can be measured experimentally using appropriate voltage-clamp protocols, could be altered to influence APD and CV restitution, and hence, spiral wave stability. Assuming that cardiac restitution properties turn out to be important in the stability of clinical arrhythmias, this suggests a potentially useful strategy for evaluating the antifibrillatory potential of antiarrhythmic drugs by considering their effects on ionic current restitution properties as well as their traditional antitachycardia properties.
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METHODS |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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Mathematical Modeling
Model of electrical wave propagation. The most widely used equation simulating electrical wave propagation in cardiac tissue is a cable equation that considers cardiac tissue as a continuous system (ignoring the microscopic cell structure)
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(1) |
is
resistivity, subscripts x and
y indicate transverse and longitudinal
directions, and Sv is the
surface-to-volume ratio. In Eq. 1, we use the formulation of
Iion
(µA · cm
2)
described in the LR1 model (27), in which
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(2) |
|
(3) |
= m(V)
and 
m = m(V)
are both functions of voltage. We simulated a square sheet of cardiac
tissue with "no-flux" boundary conditions, i.e.
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1, and
x = y = 0.5 k
· cm (8) to produce a planar wave CV
of 0.57 m/s, which is physiological for cardiac muscle.
The LR1 model, developed for guinea pig ventricular muscle, has an APD
of ~360 ms, longer than the APD of guinea pig or human ventricle at
37°C, which is ~200 ms. To shorten the APD, we decreased the
maximum conductance of
ICa
(
Ca)
from 0.09 to 0.07 mS/cm2 and
increased the maximum conductance of the time-dependent
K+ current
(K)
from 0.282 to 0.705 mS/cm2.
Extracellular K+ concentration was
5.4 mM. With these changes, the resting APD for 90% repolarization is
~200 ms. The maximum slope of APD restitution (~2.5), as well as
the range of diastolic intervals over which the slope exceeded 1 (30 ms), was also close to the range of values we have measured
experimentally in isolated rabbit ventricular myocytes at 35°C
(16). We use this parameter setting as our control case.
Measurement of APD and CV restitution.
To measure APD restitution in the single-cell LR1 model, we used an
S1-S2 stimulus protocol. At a basic pacing CL (S1-S1) of 1,000 ms, S2
was applied after a variable diastolic interval. The stimulus strengths
of S1 and S2 were fixed at two times threshold [stimulation
current (Isti) = 
40 µA/cm2], with
a pulse duration of 1.2 ms. APD was defined using a
threshold voltage of 72 mV, in which
V < 72 mV is defined as the
diastolic interval and V > 72
mV is considered the action potential. (72 mV is near the
voltage at which the action potential is 90% repolarized.)
Altering APD and CV restitution. Ion channel dynamics in the LR1 model uses a Hodgkin-Huxley formulation modeled by differential equations expressing a relaxation process to a steady-state value. Both the time course of relaxation and final steady-state variables are functions of voltage. The relaxation properties of K+, Ca2+ and Na+ currents (IK, ICa, and INa, respectively) determine their restitution properties. The restitutions of ionic currents are major determinants of APD and CV restitution (27, 39). To eliminate the effects of the current restitution on APD restitution, we modified their properties in the following manner. The first requirement was that the elimination of restitution of these currents have no effect on the fully rested APD (i.e., at diastolic intervals >1,000 ms). Therefore, we formulated the gating variables of IK, ICa, and INa as functions of voltage during a fully rested action potential. We then used the resulting set of gating variables corresponding to each ionic current for calculating action potentials at all (shorter) diastolic intervals. Because these functions were now solely voltage dependent, the effect of restitution of the currents on APD restitution was eliminated (Figs. 1 and 2). In the voltage-clamp mode, as might be used experimentally for assessing drug effects, this is equivalent to making the current insensitive to variations in diastolic interval (Fig. 1). This method, although phenomenological, was much more practical than attempting to modify the rate constants of a given ionic current individually, because of the marked interdependencies between different currents during the action potential.
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(4) |
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(5) |
![F<SUB><IT>d f</IT></SUB> (<IT>V</IT>) = <FENCE><AR><R><C>0.72 (<IT>V</IT> − 101.7) exp [−0.0455 (<IT>V</IT> + 4.8)], <IT>V</IT> > 1</C></R><R><C>− (0.46<IT>V</IT> + 55){1 −0.0008 exp [−0.113(<IT>V</IT> + 20)]}, −82 < <IT>V</IT> < 1</C></R><R><C>−0.57, <IT>V</IT> < −82</C></R></AR></FENCE>](/content/vol276/issue1/fulltext/H269/img008.gif)
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(6) |
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(7) |
![F<SUB><IT>mh</IT></SUB> (<IT>V</IT>) = [<IT>m</IT>′ (<IT>V</IT>)]<SUP>3</SUP> <IT>h</IT>′ (<IT>V</IT>)](/content/vol276/issue1/fulltext/H269/img011.gif)
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![<IT>m</IT>′ (<IT>V</IT>) = {−0.02 + exp[(<IT>V</IT> + 26)/17]}/{1 + exp[(<IT>V</IT> + 26)/17]}](/content/vol276/issue1/fulltext/H269/img012.gif)
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![<IT>h</IT>′ (<IT>V</IT>) = {1 − 1.25 exp[(<IT>V</IT> − 50.7)/29.4]}/{1 + exp[(<IT>V</IT> − 50.7)/29.4}}](/content/vol276/issue1/fulltext/H269/img013.gif)
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NaR)
markedly flattened the CV restitution curve (Fig.
2C). We therefore also modified
INa in another
way, to eliminate the effects of
INa restitution
on APD restitution without affecting CV restitution (NaR*)
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(8) |
max), and hence CV restitution, from being affected by the alteration of
INa.
Although the restitution of peak
I'Na was not
significantly altered, preserving normal CV restitution (Fig. 2C, NaR*), the restitution of
total charge carried by
I'Na was eliminated
(Fig. 3).
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x, where
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(9) |
is a weight: by varying from 0 to 1, the
restitution of the corresponding ionic current can be continuously
varied to any desired extent.
Chemical defibrillation.
To introduce changes in the restitution properties of ionic currents to
simulate an acute pharmacological intervention after spiral wave
reentry had already been initiated, we used Eq. 9. Starting at t = t0, increases
from 0 to 1 by
![&ggr; = 1 − exp [−(<IT>t</IT> − <IT>t</IT><SUB>0</SUB>)/&tgr;] for <IT>t</IT> ≥ <IT>t</IT><SUB>0</SUB>](/content/vol276/issue1/fulltext/H269/img016.gif)
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(10) |
is the time constant for the rate at which the drug effect on the ionic
current takes effect. We also simulated traditional antiarrhythmic drug
effects (classes I, III, and IV) by blocking the relevant ionic
current, i.e., reducing the maximum conductance of
INa,
ICa, or
IK by 20%
![<OVL><IT>G</IT></OVL><SUB><IT>x</IT></SUB>′ = <OVL><IT>G</IT></OVL><SUB><IT>x</IT></SUB> {0.8 + 0.2 exp [−(<IT>t</IT> − <IT>t</IT><SUB>0</SUB>)/&tgr;]} for <IT>t</IT> ≥ <IT>t</IT><SUB>0</SUB>](/content/vol276/issue1/fulltext/H269/img017.gif)
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(11) |
Computer Simulation
Numerical simulation of cardiac conduction in tissue requires large spatial arrays with many cells (because of the space and time constants inherent in the dynamics) and small time steps (because of the steep rate of rise of the cardiac action potential, e.g,max 
400 mV/ms in LR1). Because the conventional forward Euler method to
integrate Eq. 1 is computationally
tedious and costly, we developed a new integration method to speed
computation without losing accuracy. Specifically, using the well-known
operator-splitting method (40), we split Eq. 1 into an ordinary differential equation (ODE) and a
partial differential equation (PDE) and then integrated them separately
and alternately. We used an alternating direction implicit (ADI) method
(35) to integrate the PDE, a time-adaptive second-order Runge-Kutta
method [minimum time step
(
tmin) 
0.02 ms and maximum time step
(tmax 0.2 ms)] to integrate the ODEs, and the method of Rush and Larsen
(37) to integrate the ODEs for the gating variables like
Eq. 3. The integration
time step of the PDE was set to
tmax.
Simulations were carried out in a 9 cm × 9 cm tissue divided into
400 × 400 elements. For the integration of the single cell and
the one-dimensional cable of cardiac cells, we used fourth-order
Runge-Kutta and finite-difference methods. Simulations were carried out
on a 266-MHz DEC Alpha work station. We tested the accuracy of our
numerical method in a cable of cells by changing both the time step and
the space step (x). We set parameters as in the control case and fixed
tmax = 0.2 ms. Table 1 shows APD and CV for
tmin = 0.01 ms
and tmin = 0.02 ms, for a x from 0.01 to 0.03 cm. There was a 5-6% change in CV and a <0.2% change in APD
when we increased x from 0.01 to
0.0225 cm. There was an ~1% change in CV when
tmin was
increased from 0.01 to 0.02 ms. We also compared the accuracy and the
speed of our method to the conventional Euler method (Qu and Garfinkel,
unpublished observations), with similar results.
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RESULTS |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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Effects of Eliminating Ionic Current Restitution on APD and CV Restitution
Figure 1, A and B, shows the effects of eliminating the restitution of IK, ICa, or INa on the typical currents elicited by a voltage clamp pulse in the LR1 model of a single ventricular cell, as might be observed during an experimental drug testing protocol in an isolated ventricular myocyte. Figure 2 shows the effects on APD restitution. None of the modifications changed the APD of the fully rested cell, but they had very different effects on APD restitution. Particularly important is the effect on the slope of APD restitution, which is illustrated in Fig. 2B.Eliminating IK restitution (i.e., making IK deactivation an instantaneous function with respect to membrane voltage) increased the slope of the APD restitution curve at both short and moderate diastolic intervals (Fig. 2, A and B). CV restitution was unaffected (Fig. 2C).
Eliminating ICa restitution (i.e., making recovery from inactivation instantaneous with respect to membrane voltage) markedly decreased the slope of the APD restitution curve at moderate diastolic intervals (25-100 ms), but not at short (<25 ms) or long (>100 ms) diastolic intervals. At short diastolic intervals, a region remained in which the APD restitution slope was steeper than in the control case. This steep region is the effect of INa restitution on maximum voltage (Vmax), as noted below, which determines the extent to which ICa is activated through its voltage dependence and hence has a large effect on APD. CV restitution was not significantly affected (Fig. 2C).
Eliminating INa
restitution decreased the slope of the APD restitution curve to <1 at
short diastolic intervals (<50 ms) but had little effect at
intermediate to long diastolic intervals (>50 ms). However, the slope
of APD restitution was now <1 everywhere, although it approached very
close to 1 (reaching 0.92) at intermediate diastolic intervals. Our
analysis showed that the effect on APD restitution did not directly
result from the contribution of
INa to the
plateau currents, which is negligible in the LR1 model. Rather,
INa determines
the Vmax reached
during the action potential upstroke. The value of
Vmax in turn
strongly determines the extent to which
ICa is activated,
by virtue of its intrinsic voltage dependence. Thus, when
INa restitution
is eliminated,
Vmax is less
depressed so that
ICa is more fully
activated at short diastolic intervals, and APD is thereby preserved.
Eliminating INa
restitution also virtually eliminated CV restitution (Fig.
2C). For the special case in which
INa restitution
on APD was eliminated without altering CV restitution (NaR*; see
METHODS), the effect on APD
restitution was nearly equivalent (Fig.
2C).
In summary, eliminating restitution of the three currents individually
affected both the maximum value of the slope of APD restitution and the
range over which the slope exceeded 1. Without IK restitution,
the range was widened; without
ICa restitution, the range was decreased; without
INa restitution
the slope was <1 everywhere, but just barely so for moderate
diastolic intervals. In the case in which
INa restitution
was eliminated without altering CV restitution (NaR*), however,
the slope also approached 1 at very short diastolic intervals. To
produce an APD restitution curve with slope well below 1 everywhere, it
was necessary to eliminate restitution of both
ICa and
INa (Fig. 2,
A and
B).
Effects of Eliminating Ionic Current Restitution on Spiral Wave Stability
Reentrant spiral waves were initiated in the 2-D tissue model by two successive perpendicular rectilinear wave fronts (34). Figure 4A shows the result for the LR1 model with normal ionic current restitution properties. After initiation, the spiral wave went through several rotations before breaking up spontaneously into multiple meandering wave fronts, simulating the transition from VT to VF. Breakup was preceded by oscillations in the wavelength (product of APD and CV) in time and space along the arm of the spiral wave, which increased in amplitude until the wavelength at one point became too short to propagate. The resulting break in the arm of the spiral wave led to the formation of two new daughter spiral waves. Eventually, additional spiral waves were created by the same process, and the activation pattern took on a highly irregular appearance of multiple meandering wave fronts. Existing spiral waves were also annihilated as they ran into borders or fused with other spirals, so that the number of wave fronts changed continually. The resulting local activation patterns were highly irregular, as shown by the time series of diastolic interval, APD, and CL (Fig. 4B) obtained by monitoring intracellular potential at a fixed site in the tissue (Fig. 4C).
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Eliminating restitution of either IK or INa did not prevent spiral wave breakup (Fig. 5, A and C) or the highly irregular fluctuations in intracellular potential and beat-to-beat intervals (Fig. 6, A and C). In contrast, eliminating restitution of ICa did prevent spiral wave breakup (Fig. 5B). The spiral wave was not stationary but meandered chaotically, as illustrated by the trajectory of the spiral wave tip in Fig. 5B and in the record of intracellular potential and beat-to-beat intervals recorded at a fixed site in the tissue in Fig. 6B. Although dominated by the quasiperiodic motion, the fine structure of the meander was chaotic (unpublished observations).
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In contrast to the modifications to IK and ICa, the modifications to INa affected CV restitution as well as APD restitution (Fig. 2). The changes in CV restitution appeared to play an important role in spiral wave stability. When spiral wave breakup was prevented by eliminating ICa restitution (Fig. 5B), the additional elimination of INa restitution restored spiral wave breakup (data not shown but similar to the effect of INa elimination alone shown in Fig. 5C). To further delineate the role of changes in APD versus CV restitution in causing spiral wave breakup in this case, we modified INa so that its effects on APD restitution were retained but CV restitution was unaffected (see METHODS). In this case, spiral wave breakup was completely prevented (Fig. 5D), and the degree of meander was even less prominent (although still quasiperiodic and mildly chaotic) than when restitution of ICa was eliminated (Fig. 6D).
These results illustrate that both CV and APD restitution play important roles in spiral wave stability. Spiral wave breakup was promoted either by flattening the slope of CV restitution (Figs. 2C and 5C) or by increasing the slope of APD restitution (to >1) over a wide range of diastolic intervals (Figs. 2A and 5A). In the latter case, it was the range over which the slope was steep (>1) rather than the maximum steepness of APD restitution that was critical: for the case in which ICa restitution was eliminated, the maximum value of the slope actually increased at very short diastolic intervals (Fig. 2B), yet spiral wave breakup was prevented because of the shallow slope (<1) over the remaining wide range of diastolic intervals (Fig. 5B).
In all of the above cases in which IK, ICa or INa restitution was modified individually, a region of steep slope (closely approaching or exceeding 1) in the single-cell APD restitution curve remained. To make the slope shallow (much less than 1) everywhere required eliminating restitution of both ICa and INa. In this case (using the modified INa that did not alter CV restitution), spiral wave breakup was also prevented. Although a small degree of quasiperiodic meander of the spiral wave remained (Figs. 5E and 6E), the quasiperiodic meander was no longer chaotic, representing a qualitative change (i.e., a bifurcation point) in the behavior of the spiral wave (unpublished observations).
These results show that by appropriately modifying restitution properties of cardiac ionic currents, it is possible to suppress both wavelength and CL oscillations to make reentrant spiral waves more stable, preventing, in a model, the transition from VT to the VF-like state.
Effects of Blocking Ionic Currents on Spiral Wave Stability
Elimination of restitution of ionic currents is easily achieved in a computer model, but pharmacological tools to achieve the same effects on ionic currents in real cardiac tissue are not necessarily available. Therefore, it is useful to consider how less selectively targeted pharmacological interventions affect APD and CV restitution and spiral wave stability. For example, because ICa relaxation is the major factor regulating the steep region of APD restitution in the LR1 model, simply reducing the absolute magnitude of ICa, without specifically modifying its relaxation properties (a class IV antiarrhythmic drug effect), might be predicted to lessen the steepness of APD restitution. This is because the variation in ICa magnitude with diastolic interval will be smaller relative to other ionic currents during the plateau and have a lesser effect on APD. To test this strategy, we examined individually the effects of decreasing the magnitudes of IK, ICa, and INa by 50% (analogous to class III, IV, and I antiarrhythmic drug effects, respectively) without otherwise affecting their relaxation or other kinetic properties. Figure 7, A-C, shows the effects on the APD and CV restitution curves. Reducing IK by 50% (i.e., by decreasingK
from 0.705 to 0.3525 mS/cm2)
increased the steepness of APD restitution over a broad range (Fig. 7,
A and
B) by preferentially prolonging APD
at long diastolic intervals and did not affect CV restitution (Fig.
7C). The initiated spiral wave still
broke up (Fig. 7F), consistent with
the increased steepness of the APD restitution over a wide range of
diastolic intervals. Reducing
ICa by 50%
(i.e., by decreasing
Ca in Eq. 6 from 0.07 to 0.035 mS/cm2) decreased the range of
diastolic intervals over which APD restitution was steep by reducing
the steepness at moderate to long diastolic intervals (>25 ms),
shortened APD at long diastolic intervals (Fig. 7,
A and
B), and had no effect on the
steepness of CV restitution (Fig.
7C). Blocking
ICa prevented
spiral wave breakup, producing a single chaotically meandering spiral
wave (Fig. 7E). Reducing INa by 50%
(i.e., by decreasing
Na
in Eq. 4 from 23 to 11.5 mS/cm2) also decreased the range
of diastolic intervals over which APD restitution was steep by reducing
the steepness of APD restitution slightly at short to moderate
diastolic intervals (<100 ms) but had no effect on APD at long
diastolic intervals. It also decreased the magnitude and
slope of CV restitution. Similar to the situation in which restitution
of INa was
selectively eliminated, the effects on CV restitution on promoting
spiral wave instability outweighed its stabilizing effects on APD
restitution, so that spiral breakup still occurred (Fig.
7D).
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With the caveat that no use-dependent properties were incorporated in these simulations, these results suggest that pharmacological agents that block ICa (class IV drugs) were more effective at stabilizing spiral wave reentry than IK or INa blockers (class III or I drugs) in this model.
Chemical Defibrillation
An important question is whether altering APD and CV restitution after the VF-like state is established can restore periodic behavior and convert VF to VT. To test this idea, we initiated a spiral wave with the LR1 model and allowed the VF-like state to develop. After 2 s, we then used Eqs. 9-11 with a of
1 s to introduce the modified
IK,
ICa, or
INa in which
restitution was eliminated, either alone or in combination with class
I, III, or IV antiarrhythmic drug effects. When either
IK or
INa restitution was eliminated, the fibrillation-like state persisted for a variety of
different initial conditions (Fig. 8,
A and
C). When
ICa restitution was eliminated, the multiple and variable number of wave fronts coalesced into several meandering spiral waves, whose number
remained constant (Fig. 8B). We
simulated class I, III, or IV antiarrhythmic drug effects by reducing
the magnitude of
INa,
IK, or
ICa,
respectively, by 20%, without altering their kinetic properties. None
of these interventions changed the qualitative behavior of the
fibrillation-like state (Fig. 8,
D-F), either alone or in
combination with the additional elimination of
INa or
IK restitution
(data not shown). However, when a class III antiarrhythmic drug
intervention (but not class I or IV interventions) was combined with
elimination of
ICa restitution, all wave fronts were extinguished after several rotations, as they
encountered refractory tissue from a wave back (Fig. 8,
G-I). Thus, this combined
"antifibrillatory" plus "antitachycardia" intervention was
successful at "defibrillating" the tissue to a quiescent (but
excitable) state.
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DISCUSSION |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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A number of studies have investigated spiral wave meander and breakup in 2-D and three-dimensional (3-D) simulations (8, 33) and in tissue experiments (9, 17, 34). However, only Karma's (21) and Courtemanche's (6) 2-D simulations explicitly addressed the role of restitution in these phenomena. We extended these investigations by examining directly the effects of restitution properties of the major currents on APD and CV restitution as well as on spiral wave behavior, providing a guide for pharmacological manipulations that can be tested experimentally. Although our method of eliminating the restitution of individual ionic currents was phenomenologically based, it nevertheless readily permits an explicit description of how the altered current would behave during a typical voltage-clamp protocol used in an experimental drug screening protocol applied to an isolated cardiac myocyte (Fig. 1). Importantly, the approach can be refined as more complete descriptions of the cardiac action potential (in ventricular as well as atrial tissues) are developed.
Although a number of important limitations must be considered (see Limitations), the major conclusions arising from these simulations are that in this model 1) the restitution properties of INa and ICa are the main determinants of the steep portions of the APD restitution curve, whereas IK restitution plays a lesser role; 2) steep APD restitution promotes spiral wave meander and breakup [for the latter, the range of diastolic intervals over which the slope of APD restitution is steep (>1) is more important than the maximum steepness]; 3) eliminating restitution of ICa is more effective than eliminating INa or IK restitution for preventing spiral wave breakup in this model; 4) eliminating INa restitution, which flattens CV restitution, promotes spiral wave breakup independently of APD restitution; and 5) among nine interventions tested in this model, "defibrillation" of multiple spiral wave reentry required combining an antifibrillatory intervention based on altering restitution properties (to convert VF to VT) with an antitachycardia intervention (to eliminate VT) based on blocking an ionic conductance (particularly a class III antiarrhythmic drug effect).
Cellular determinants of APD and CV restitution. In the LR1 single-cell model, restitution of INa and ICa is the major determinant of the steep portions of the APD restitution curve (Fig. 2). This is consistent with experimental findings in intact cardiac tissue that INa and ICa blockers in general reduce the slope of APD restitution (5, 41), because reducing the magnitude of the current decreases its influence on APD restitution independently of any direct effect on APD restitution properties per se, as illustrated by our simulations with the LR1 model in Fig. 7. Eliminating restitution of IK in the LR1 model also changed the steepness of the APD restitution curve, but in the opposite, i.e., steeper, direction (Fig. 2), also consistent with experimental observations (24, 25). This effect has usually been attributed to the reverse use dependence property of these drugs (18), i.e., preferential current block at long diastolic intervals, which would increase the slope of APD restitution. However, the results from the LR1 model suggest another explanation, namely that APD restitution at short diastolic intervals is dominated by restitution of INa and ICa, whereas IK restitution only assumes importance at longer diastolic intervals.
In contrast to the multiple factors influencing APD restitution, CV restitution in normally polarized tissue as simulated in this study is primarily determined by recovery from inactivation of INa. Consistent with these results, use-dependent Na+-current blockers such as lidocaine were shown experimentally to reduce the slope of CV restitution (11). However, in depolarized or partially uncoupled tissue in which INa is largely inactivated and CV depends on ICa to support the action potential upstroke (such as in the setting of ischemia), ICa and IK relaxation processes would assume greater importance (39).Mechanism by which restitution properties destabilize spiral wave
reentry.
The mechanism by which a steep restitution curve causes instabilities
was appreciated previously in studies investigating responses of
myocardium to pacing (31, 42, 44) and in reentry around an anatomic
obstacle (7, 11, 13, 36). We believe that the same basic mechanism,
diagrammed in Fig. 9, also applies to
spiral wave reentry in 2-D. Figure 9A
illustrates an APD restitution curve with slope <1. For a stationary
spiral wave with constant CL as defined by the equality CL = APD diastolic interval, the CL can be represented on the
restitution graph by a line with a slope of 1 (dashed line). A
stationary spiral wave will have an APD and diastolic interval
corresponding to the intersection of the dashed line and the APD
restitution curve. If a perturbation (e.g., a premature stimulus) is
applied to shorten the diastolic interval to the point labeled
a, the next APD will fall on the restitution curve at point b,
producing the next diastolic interval at point
c, etc. With iteration, the slope <1 ensures that the APD and diastolic interval converge back to the stable equilibrium at
the intersection point. In contrast, if the slope of the APD restitution curve is >1, as shown in Fig.
9B, the small perturbation in
diastolic interval is unstable and becomes amplified on iteration, eventually reaching a diastolic interval shorter than the refractory period. This results in a wave break along the spiral wave arm, initiating spiral wave breakup. This contrasts to reentry in a ring (7,
11, 13, 36), in which wave break simply terminates the arrhythmia.
|
Limitations. In this study, we have described the effects of altered ionic current restitution on spiral wave stability as being mediated through APD and CV restitution. However, it could be argued that the effects on spiral wave stability are directly caused by altered ionic current restitution, and that the effects on APD and CV restitution are epiphenomena. We do not believe this to be the case, for several reasons. First, insofar as we have been able to determine, there is complete agreement between the effects of ionic current modifications on APD and CV restitution with their effects on spiral wave stability. For example, increasing the steepness of APD restitution by eliminating IK restitution, by decreasing IK (without altering its restitution properties), and by increasing ICa (also without altering its restitution) all had the same effect of increasing the slope of APD restitution, and all destabilized spiral wave reentry. Second, a similar relationship between APD restitution steepness and spiral wave breakup was previously established in other cardiac models in which individual ionic currents are either not specifically formulated or are formulated differently, such as Karma's two-variable model (21) or the Beeler-Reuter model (6). The common link to spiral wave stability with our study is APD restitution steepness, rather than alterations to restitution properties of specific ionic currents. Third, there is a clear dynamic mechanism to explain how APD restitution steepness causes destabilization of spiral wave reentry (Fig. 9), whereas the same is not true for relating ionic current restitution properties to spiral wave stability (except through their effects on APD and CV restitution).
In addition, several important caveats must be recognized in evaluating the physiological relevance of these simulations to arrhythmias in the real heart. These caveats primarily relate to two issues, the completeness and physiological accuracy of the cellular action potential model and the validity of extrapolating findings in a simulated homogeneous 2-D sheet of cardiac tissue to real cardiac tissue, which is 3-D, anisotropic, and both anatomically and electrophysiologically heterogeneous. Limitations of the LR1 model include unphysiologically slow kinetics of ICa, the incomplete description of individual time-dependent K+ currents [the rapid K+ current (IKr), the slow K+ current (IKs), and the transient outward current (Ito)], and the lack of detailed intracellular Ca2+ dynamics. We also found it necessary to make adjustments to the LR1 model to shorten the APD to a physiologically realistic value, which simulated the features of our experimentally measured ventricular APD restitution curves (16). The limitation on computational speed was one important consideration in using the LR1 model rather than a more detailed model such as the phase 3 formulation of the Luo-Rudy model (LR3) (28, 48), which is less tractable from a computational standpoint. However, we recently confirmed that eliminating ionic current restitution in these more detailed models has effects on APD and CV restitution qualitatively similar to that in the LR1 model. For example, eliminating restitution of the various IK components in the LR3 model (IKr and IKs) results in increasing the range, and steepness of APD restitution is steep (>1), similar to the LR1 model (unpublished observations). The potential effects of Ito relaxation were not studied, because Ito has not been formulated in these models. Intracellular Ca2+ dynamics also may have important effects on cardiac restitution properties (38). The increase in intracellular Ca2+ during excitation affects a variety of ionic currents influencing APD, including ICa (through Ca2+-induced inactivation), the Na+/Ca2+ exchange current, and Ca2+-activated nonselective cation and Cl currents (45). At
short diastolic intervals, Ca2+
release from the sarcoplasmic reticulum decreases and may influence APD
less prominently than at long diastolic intervals (27). Pretreatment of
cardiac tissue with agents that inhibit
Ca2+ release by the sarcoplasmic
reticulum has been reported to affect APD restitution (38), although in
isolated rabbit ventricular myocytes studied at 35°C, we found that
eliminating the intracellular Ca2+
transient had little effect on the steepness of APD restitution (16).
Because the goal of the present study was to examine the effects of
eliminating ionic current restitution on APD and CV restitution
properties, there was an important practical reason for using an action
potential model that did not incorporate detailed intracellular
Ca2+ dynamics. Specifically, this
avoided the confounding effects of intracellular
Ca2+ dynamics on ionic current
restitution (especially
ICa), which would have made it impossible to predict the properties of the altered
currents under voltage-clamp conditions relevant to drug screening
protocols. With more advanced cardiac models incorporating intracellular Ca2+ dynamics,
however, it will be possible in future studies to evaluate the
influence of intracellular Ca2+
dynamics on restitution and spiral wave behavior.
In this study, we assessed the steepness of APD restitution using an
S1-S2 stimulation protocol applied to a single simulated cardiac cell.
Several factors make it difficult to relate the restitution properties
of the single cardiac cell quantitatively to the restitution properties
during spiral wave reentry. First, the memory feature (32) of APD means
that there is no unique relationship between APD and the previous
diastolic interval, even at the single-cell level. Because the history
of previous excitation is important, it remains to be determined what
is the most accurate method for assessing restitution with a pacing
protocol, to accurately reflect restitution properties occurring during spiral wave reentry. Second, restitution in a single cell differs from
restitution in coupled cells in cardiac tissue, because diffusive (axial) currents between adjacent cells become important, especially at
short diastolic intervals (26). In 2-D and 3-D tissue, the curvature of
the wave front further modulates density of diffusive currents, thereby
influencing APD and CV in a complex curvature-dependent manner. In
reality, the slope >1 criterion as a measure predicting spiral wave
stability really refers to the restitution properties during spiral
wave reentry and not to the restitution properties of the single cell.
An example of a discrepancy between single-cell and tissue APD
restitution properties was encountered in the example in which
restitution of
INa was
eliminated (NaR). Although the maximal slope of APD restitution
in the single cell approached very close to 1 (Fig. 2,
middle), it did not actually exceed
1 anywhere. If the mechanism of spiral wave instability illustrated in
Fig. 9 is correct, the failure of the APD restitution slope to exceed 1 should have prevented spiral wave breakup by dampening oscillations in
APD and diastolic interval along the spiral wave arm. This paradox was
resolved, however, when we examined APD restitution for the NaR
case in a one-dimensional cable of cardiac cells. In contrast to the
single cell, the slope of APD restitution in the ring did exceed 1 at
short diastolic intervals (Fig. 2B, inset in middle panel).
This example illustrates that in cases in which the slope of APD
restitution approaches close to 1 in the single-cell model, tissue
measurements of APD restitution may be required to predict spiral wave
stability. Nevertheless, despite this "gray zone," our study
suggests that interventions that markedly alter the steepness of the
single-cell APD restitution slope to values considerably less than or
greater than 1 remain highly accurate for predicting the spiral wave
behavior. This is an important point for potential drug screening
experiments to predict antifibrillatory efficacy from restitution measurements.
The second major caveat is that our simulations were based on a
homogeneous, isotropically conducting 2-D medium in which the cell
model had a steep (slope > 1) APD restitution curve. We have not yet
studied how anisotropy, electrophysiological heterogeneity, and
anatomic obstacles affect the validity of these conclusions or whether
conclusions about spiral wave behavior in 2-D have direct relevance to
scroll wave behavior in 3-D tissue. For example, Fenton and Karma (12)
recently found in 3-D simulations using a simplified cardiac model that
the rotation of fiber orientation from endocardium to epicardium may
induce breakup of reentrant scroll waves by inducing filament twist.
Also, some investigators have questioned whether APD restitution
properties during spiral wave reentry in the real heart are
sufficiently steep to produce breakup at all, although there are many
examples in the literature in which APD restitution has been found to
have a slope >1 (~50% in our survey of the literature), in both
animal (2, 10, 44) and human (14, 30) studies. The ability to
extrapolate our simulation results to fibrillation in real cardiac
tissue will require this issue to be further clarified by experimental studies. It has also been hypothesized that cardiac fibrillation in 2-D
models may be irrelevant to real cardiac fibrillation, which has been
postulated to require 3-D tissue to develop under physiological
conditions. However, this hypothesis appears to be at odds with the
experimental documentation of sustained fibrillation in relatively thin
cardiac preparations, such as the full-thickness (5-9 mm) porcine
right ventricle (22), thin (2-4 mm) right ventricular sheets in
the presence of drugs that shorten action potential duration (15), and
the thin-walled atria (20).
With respect to the issue of heterogeneity, it is well established that
tissue heterogeneity promotes the initiation of functional reentry.
However, this does not necessarily imply that once spiral wave reentry
has been initiated, tissue heterogeneity continues to have a
destabilizing effect. In both experimental and clinical settings,
monomorphic VT compatible with a stationary spiral wave reentry is
virtually never seen in normal hearts, only in diseased hearts in which
heterogeneity caused by an infarct or other process has occurred. In
contrast, in healthy (less heterogeneous) hearts, functional reentry
compatible with unstable spiral wave reentry is more difficult to
induce but can invariably be initiated with a sufficiently aggressive
stimulation protocol. Once initiated in the healthy heart, however,
this type of functional reentry is never stable if sustained and always
degenerates to VF through a mechanism consistent with spiral breakup
(4, 19). Also, both theoretical and experimental studies documented
that spiral wave stability can be enhanced by tissue heterogeneities,
which tend to anchor the cores of spiral waves by creating local
source-sink mismatches (34). From a therapeutic standpoint, tissue
heterogeneity is a very difficult target, whereas cardiac restitution
properties should be predictably alterable by pharmacological
interventions. Therefore, if it can be demonstrated that spiral wave
stability is primarily controlled by cellular restitution
characteristics, the hope for developing effective antifibrillatory
drug therapy is indeed promising.
Clinical implications for antifibrillatory drug therapy. If the hypothesis is correct that cardiac fibrillation arises from a single or double reentrant wave front that subsequently breaks up into multiple reentrant wave fronts, then understanding the factors controlling the stability of reentrant wave fronts in cardiac tissue is critical for developing effective antifibrillatory therapy. To the extent that spiral wave reentry in simulated cardiac tissue provides a realistic model for fibrillation, our study suggests that drugs that alter APD and CV restitution by modulating ionic current restitution properties may markedly influence the tendency for spiral waves to break up and cause a fibrillation-like state. The traditional classification of antiarrhythmic drugs is based on their effects on APD, CV, and individual ionic currents and was devised largely to characterize their antitachycardic effects, because tachycardia is much better understood than fibrillation. Our results provide a preliminary framework for understanding how these drugs, through their effects on APD and CV restitution, may affect the tendency to fibrillation. Our modeling (Fig. 8) suggests that an ideal antiarrhythmic drug should have both antitachycardic and antifibrillatory efficacy. Subject to the various caveats discussed above, we suggest that a drug or drug combination that flattens APD but not CV restitution, in addition to an antitachycardic action, would have the ideal profile. Drugs with a favorable antitachycardia profile, but an unfavorable antifibrillation profile, could potentially contribute to proarrhythmic effects. An example might be class III antiarrhythmic drugs with reverse use dependence, which are predicted to be profibrillatory by steepening APD restitution (Fig. 7) (24). Some drugs fitting this profile have been associated with the excess mortality from proarrhythmia in clinical trials (3, 43). Given the generally disappointing results of recent clinical antiarrhythmic drug trials, a rational framework to better understand the antifibrillatory as well as antitachycardia properties of antiarrhythmic drugs is clearly needed.
| |
ACKNOWLEDGEMENTS |
|---|
The authors thank Peng-Sheng Chen, Hrayr Karagueuzian, and Boris Kogan for many helpful discussions.
| |
FOOTNOTES |
|---|
This work was supported by National Institutes of Health Specialized Center of Research in Sudden Cardiac Death P50-HL-52319, by a Fellowship from the American Heart Association, Greater Los Angeles Affiliate (to Z. Qu), and by the Laubisch and Kawata Endowments.
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests: A. Garfinkel, Dept. of Medicine (Cardiology), UCLA School of Medicine, 47-123 CHS, Los Angeles, CA 90095-1679.
Received 29 April 1998; accepted in final form 9 September 1998.
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Top
Abstract
Introduction
Methods
Results
Discussion
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L. C. Baker, R. Wolk, B.-R. Choi, S. Watkins, P. Plan, A. Shah, and G. Salama Effects of mechanical uncouplers, diacetyl monoxime, and cytochalasin-D on the electrophysiology of perfused mouse hearts Am J Physiol Heart Circ Physiol, October 1, 2004; 287(4): H1771 - H1779. [Abstract] [Full Text] [PDF]
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S. C. Hao, D. J. Christini, K. M. Stein, P. N. Jordan, S. Iwai, O. Bramwell, S. M. Markowitz, S. Mittal, and B. B. Lerman Effect of {beta}-adrenergic blockade on dynamic electrical restitution in vivo Am J Physiol Heart Circ Physiol, July 1, 2004; 287(1): H390 - H394. [Abstract] [Full Text] [PDF]
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 A. G. KLEBER and Y. RUDY Basic Mechanisms of Cardiac Impulse Propagation and Associated Arrhythmias Physiol Rev, April 1, 2004; 84(2): 431 - 488. [Abstract] [Full Text] [PDF]
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Z. Qu, H. S. Karagueuzian, A. Garfinkel, and J. N. Weiss Effects of Na+ channel and cell coupling abnormalities on vulnerability to reentry: a simulation study Am J Physiol Heart Circ Physiol, April 1, 2004; 286(4): H1310 - H1321. [Abstract] [Full Text] [PDF]
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K. H. W. J. ten Tusscher, D. Noble, P. J. Noble, and A. V. Panfilov A model for human ventricular tissue Am J Physiol Heart Circ Physiol, April 1, 2004; 286(4): H1573 - H1589. [Abstract] [Full Text] [PDF]
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F. Xie, Z. Qu, A. Garfinkel, and J. N. Weiss Electrical refractory period restitution and spiral wave reentry in simulated cardiac tissue Am J Physiol Heart Circ Physiol, July 1, 2002; 283(1): H448 - H460. [Abstract] [Full Text] [PDF]
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O. Bernus, R. Wilders, C. W. Zemlin, H. Verschelde, and A. V. Panfilov A computationally efficient electrophysiological model of human ventricular cells Am J Physiol Heart Circ Physiol, June 1, 2002; 282(6): H2296 - H2308. [Abstract] [Full Text] [PDF]
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K. J. Sampson and C. S. Henriquez Simulation and prediction of functional block in the presence of structural and ionic heterogeneity Am J Physiol Heart Circ Physiol, December 1, 2001; 281(6): H2597 - H2603. [Abstract] [Full Text] [PDF]
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M.-H. Lee, Z. Qu, G. A. Fishbein, S. T. Lamp, E. H. Chang, T. Ohara, O. Voroshilovsky, J. R. Kil, A. R. Hamzei, N. C. Wang, et al. Patterns of wave break during ventricular fibrillation in isolated swine right ventricle Am J Physiol Heart Circ Physiol, July 1, 2001; 281(1): H253 - H265. [Abstract] [Full Text] [PDF]
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F. Xie, Z. Qu, A. Garfinkel, and J. N. Weiss Effects of simulated ischemia on spiral wave stability Am J Physiol Heart Circ Physiol, April 1, 2001; 280(4): H1667 - H1673. [Abstract] [Full Text] [PDF]
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 B.-R. Choi, T. Liu, and G. Salama The Distribution of Refractory Periods Influences the Dynamics of Ventricular Fibrillation Circ. Res., March 16, 2001; 88 (5): e49 - e58. [Abstract] [Full Text] [PDF]
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F. Xie, Z. Qu, A. Garfinkel, and J. N. Weiss Electrophysiological heterogeneity and stability of reentry in simulated cardiac tissue Am J Physiol Heart Circ Physiol, February 1, 2001; 280(2): H535 - H545. [Abstract] [Full Text] [PDF]
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M. L. Koller, M. L. Riccio, and R. F. Gilmour Jr Effects of [K+]o on electrical restitution and activation dynamics during ventricular fibrillation Am J Physiol Heart Circ Physiol, December 1, 2000; 279(6): H2665 - H2672. [Abstract] [Full Text] [PDF]
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M. Yashima, T. Ohara, J.-M. Cao, Y.-H. Kim, M. C. Fishbein, W. J. Mandel, P.-S. Chen, and H. S. Karagueuzian Nicotine increases ventricular vulnerability to fibrillation in hearts with healed myocardial infarction Am J Physiol Heart Circ Physiol, June 1, 2000; 278(6): H2124 - H2133. [Abstract] [Full Text] [PDF]
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J.-M. Cao, Z. Qu, Y.-H. Kim, T.-J. Wu, A. Garfinkel, J. N. Weiss, H. S. Karagueuzian, and P.-S. Chen Spatiotemporal Heterogeneity in the Induction of Ventricular Fibrillation by Rapid Pacing : Importance of Cardiac Restitution Properties Circ. Res., June 11, 1999; 84(11): 1318 - 1331. [Abstract] [Full Text] [PDF]
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M. A. Watanabe and M. L. Koller Mathematical analysis of dynamics of cardiac memory and accommodation: theory and experiment Am J Physiol Heart Circ Physiol, April 1, 2002; 282(4): H1534 - H1547. [Abstract] [Full Text] [PDF]
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J. J. Fox, J. L. McHarg, and R. F. Gilmour Jr Ionic mechanism of electrical alternans Am J Physiol Heart Circ Physiol, February 1, 2002; 282(2): H516 - H530. [Abstract] [Full Text] [PDF]
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O. Bernus, R. Wilders, C. W. Zemlin, H. Verschelde, and A. V. Panfilov A computationally efficient electrophysiological model of human ventricular cells Am J Physiol Heart Circ Physiol, June 1, 2002; 282(6): H2296 - H2308. [Abstract] [Full Text] [PDF]
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M. Swissa, Z. Qu, T. Ohara, M.-H. Lee, S.-F. Lin, A. Garfinkel, H. S. Karagueuzian, J. N. Weiss, and P.-S. Chen Action potential duration restitution and ventricular fibrillation due to rapid focal excitation Am J Physiol Heart Circ Physiol, May 1, 2002; 282(5): H1915 - H1923. [Abstract] [Full Text] [PDF]
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), DI (
), CL (
), and
V during time interval from 0 to 5 s
after initiation of spiral wave reentry, for corresponding cases shown
in Fig. 
