Vol. 280, Issue 2, H535-H545, February 2001
Electrophysiological heterogeneity and stability of reentry in
simulated cardiac tissue
Fagen
Xie,
Zhilin
Qu,
Alan
Garfinkel, and
James N.
Weiss
Departments of Medicine (Cardiology), Physiological Science and
Physiology, University of California at Los Angeles, California
90095
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ABSTRACT |
Generation of wave break is a
characteristic feature of cardiac fibrillation. In this study,
we investigated how dynamic factors and fixed electrophysiological
heterogeneity interact to promote wave break in simulated
two-dimensional cardiac tissue, by using the Luo-Rudy (LR1) ventricular
action potential model. The degree of dynamic instability of the action
potential model was controlled by varying the maximal amplitude of the
slow inward Ca2+ current to produce spiral waves in
homogeneous tissue that were either nearly stable, meandering,
hypermeandering, or in breakup regimes. Fixed electrophysiological
heterogeneity was modeled by randomly varying action potential duration
over different spatial scales to create dispersion of refractoriness.
We found that the degree of dispersion of refractoriness required to
induce wave break decreased markedly as dynamic instability of the
cardiac model increased. These findings suggest that reducing the
dynamic instability of cardiac cells by interventions, such as
decreasing the steepness of action potential duration restitution, may
still have merit as an antifibrillatory strategy.
dispersion of refractoriness; spiral wave breakup
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INTRODUCTION |
CARDIAC FIBRILLATION
is characterized by multiple waves of excitation coursing through
myocardial tissue. Because each wave has a finite lifetime
(4), new wave breaks are constantly generated during
fibrillation. Traditionally, it has been held that wave break is
produced by electrophysiological and anatomic heterogeneities in the
tissue, specifically "dispersion of refractoriness" (10, 14,
20). However, modeling studies of cardiac tissue have shown that
wave break can occur spontaneously in completely homogeneous tissue if
the electrophysiological properties of the cardiac cell model have
certain properties, such as a steeply sloped action potential duration
(APD) restitution curve (6, 17, 27, 28). This type of wave
break arises solely from the dynamics of cardiac propagation and is
related to steep APD restitution causing oscillations in wavelength
before localized wave break (27), without any requirement
for fixed heterogeneities in the tissue. Real cardiac tissue has
appropriate dynamic electrophysiological properties to permit this form
of wave break (12, 30) but is also characterized by
significant fixed regional electrophysiological differences (1,
2, 18, 25, 35), which may be further exacerbated by disease
processes. In addition, fixed heterogeneities do not always promote
wave break but can also stabilize waves by anchoring reentry (13,
16, 32).
The purpose of this study, therefore, was to investigate the interplay
between fixed and dynamic electrophysiological heterogeneities in the
generation of wave break during fibrillation. This is a critically
important issue with respect to developing new therapeutic strategies
to prevent fibrillation clinically: fixed heterogeneities are a
difficult therapeutic target, whereas dynamic properties of the cardiac
action potential, such as APD restitution steepness, can be potentially
modified by drugs.
We approached this problem in simulated two-dimensional (2-D) cardiac
tissue, by using a physiologically based cardiac ventricular action
potential model, phase 1 of the Luo-Rudy (LR1) model
(19). The LR1 model can be readily modified to produce
four distinct types of spiral wave reentry in simulated homogeneous
tissue, which have been previously characterized in simulations
(6, 7, 9, 15, 17, 22, 23, 27, 28, 31) and also observed
experimentally (4, 8). In increasing order of dynamic instability, they are stable or nearly stable spiral waves, meandering spiral waves, hypermeandering spiral waves, and spiral wave breakup. Starting with homogeneous tissue, we introduced progressively increasing degrees of fixed electrophysiological heterogeneity (dispersion of refractoriness) over a range of spatial scales and
determined the effects on various types of spiral wave behavior.
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MATERIALS AND METHODS |
Mathematical Modeling
We used a continuous ionic model to study the propagation of the
action potential in the 2-D cardiac tissue. Ignoring microscopic cell
structure, cardiac tissue can be treated as a continuous system in
which propagation in 2-D can be modeled by a partial differential equation
|
(1)
|
where V is the membrane voltage (mV),
Cm equalling 1 µF/cm2
is the membrane capacitance, D equalling 0.001 cm2/ms is the diffusion current coefficient, and
Iion is the sum of ionic currents. Our
2-D tissue consisted of 400 × 400 nodes, corresponding in
physical dimensions to an 80 mm × 80 mm square, with no-flux
boundary conditions at the edges.
In Eq. 1, we use the formulation of
Iion (in µA/ cm2) described in the
LR1 model (19), in which
|
(2)
|
where INa = 
Nam3hj(V 
54.40) is the fast inward Na+ current;
Isi = sidf(V Esi) is the slow inward Ca2+
current (roughly corresponding to the L-type Ca2+ current),
where Esi = 7.7 13.0287 ln
([Ca2+]i) is the reversal potential of
calcium; [Ca2+]i is the intracellular
Ca2+ concentration; IK = Kxx1(V + 77.62) is the slow outward time-dependent K+ current;
IK1 = K1K1
(V + 87.95) is the
time-independent K+ current;
IKp = 0.0183Kp(V + 87.95)
is the plateau K+ current; and
Ib = 0.03921(V + 59.87) is
the total background currents. The ionic gating variables m, h,
d, f, and x are all governed by a Hodgkin-Huxley-type
formulation of ordinary differential equations (19).
Details of the ionic currents formulations can be found in Luo and Rudy
papers (19).
In the original LR1 model, the values (in mS/µF) of maximal
conductances of ionic currents are
Na = 23, si = 0.09, K = 0.282, and
K1 = 0.6047. With these values, the
LR1 model has an APD of ~360 ms and a very steep APD restitution
curve. We have varied some of these parameters to study the spiral wave
reentry dynamics. We set K = 0.705 and controlled the dynamic stability of spiral waves by varying
si from 0 to 0.07, as described
previously (6, 27, 28). To simulate dispersion of
refractoriness, we altered APD regionally by varying
K1 between 0.24188 and 0.60470. Most of
simulations are carried out with the normal Ca2+ kinetics
in the LR1 model. Altering si
significantly changed both APD and the slope of APD restitution curve.
In the simulation to examine the spiral reentry in inhomogeneous tissue
with flatten APD restitution without significant shortening APD
(Fig. 7), we modified these parameters as:
Na = 16, si = 0.06, and
K = 0.141. Ca2+ kinetics
were sped up by replacing the channel relaxation time 
d
and f with 0.1 · d and
0.1 · f.
Computer Simulations
Numerical simulations were performed in the isolated cell, the
one-dimensional (1-D) ring, and 2-D tissue. To simulate a single cell,
we integrated the following ordinary differential equation
|
(3)
|
where Isti is the external stimulus
current. We used a square-wave stimulus (40 µA × 2 ms)
at a constant frequency. The duration of the pulse was 2 ms, and the
strength is 40 µA/cm2, which is about two times the
threshold stimulus strength. We used the fourth-order Runge-Kutta
method to integrate Eq. 3 with a fixed time step equalling
0.01 ms.
We used a 1-D ring of tissue for measuring APD restitution (see below),
in which propagation is governed by the following partial differential
equation
|
(4)
|
Equation 4 was solved using the forward Euler method
with time step 0.01 ms and space step 0.02 cm.
For numerical simulation in 2-D tissue, the conventional Euler method
to integrate Eq. 1 is computationally tedious and costly. Therefore, we solved Eq. 1 using the well-known
operator-splitting method. We split the nonlinear operator
(Iion term) and the diffusion operator in
Eq. 1 into two terms, and then we integrated the two terms
separately and alternatively. We use an alternating-direction implicit
method to integrate the partial differential equation of the diffusion
term, and a time adaptive second-order Runge-Kutta method
(
tmin
0.01 ms and
tmax 0.1 ms) to integrate the ordinary differential equation of the reaction term with all the gating variable
equations. The time step of integration of the PDE was set to
tmax to keep all cells synchronized. The
space steps were set at dx = dy = 0.02 cm. With this approach, the integration speed increased more than
10-fold, with the relative error not exceeding 2% (26).
Full details of the numerical methods and criteria for assuring
numerical stability have been provided in detail previously
(26). Tissue size was fixed at 80 × 80 mm3 (400 × 400 nodes) in all 2-D simulations
throughout the paper. All simulations were written in FORTRAN code and
run on DEC Alpha stations.
Electrophysiological Measurements and Induction of Reentry
APD restitution refers to the relationship between APD and the
previous diastolic interval (DI). APD is measured as the portion of the
cardiac cycle (in ms) during which V > 72 mV, and DI
is measured as the portion during which V < 72 mV.
Because APD restitution in tissue is different from that in a single
cell (28, 34) due to diffusive currents, we measured APD
restitution in a 1-D ring (equivalent to a planar wave in 2-D tissue).
A unidirectional wave was initiated in the ring, and APD and DI were
measured at steady state. APD restitution was obtained by progressively
shortening the length of the ring until conduction failed.
Spiral wave reentry in 2-D tissue was initiated by two successive
perpendicular rectilinear waves. Tip trajectories of spiral waves were
traced by using the intersection point of successive contour lines of
voltage corresponding to 30 mV measured every 2 ms. The intersection
points of these successive contour lines form a tip trajectory. In the
case of wave break, the number of spiral tips was defined as the total
number of intersection points in the whole tissue.
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RESULTS |
Creating Spiral Wave Phenotypes and Dispersion of Refractoriness
To create different spiral wave phenotypes in the LR1 action
potential model, we altered the maximal conductance of the slow inward
current (si) (27). As shown
in Fig. 1, a spiral wave initiated in
homogeneous, isotropic 2-D tissue was nearly stable when
si = 0 (Fig. 1A),
exhibited quasiperiodic meander when si = 0.030 (Fig. 1B), and
chaotic hypermeander when si = 0.049 (Fig. 1C). Beyond
si = 0.055, spontaneous spiral wave
breakup occurred (not shown). These different behaviors corresponded to increases in the steepness of the APD restitution slope, as has been
shown previously (6, 17, 27, 28).
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Fig. 1.
Spiral wave dynamics as a function of the maximal
conductance of the slow inward current
(si) when action potential duration (APD)
is modulated by changing the time-dependent K+ conductance
(K1) in homogeneous isotropic
two-dimensional (2-D) tissue (80 × 80 mm). A-C:
nearly stable (si = 0), meander
(si = 0.030), and hypermeander
(si = 0.049) regimes, respectively.
Spiral wave behavior is shown both for the maximum (top row)
and minimum (bottom row) values of
K1 (0.60470 and 0.21488) used to modify
APD in subsequent simulations. Left: voltage snapshots at
steady state (2 s after initiation of the spiral). White represents
depolarized tissue at the wave front, and black represents repolarized
tissue at the wave back. Middle: trajectory of the spiral
tip. Right: Poincaré plots of successive cycle lengths
(CL) of the spiral.
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To create dispersion of refractoriness, we increased APD regionally by
reducing the maximal conductance of the
K1 from its control
value of 0.60470. We chose to alter K1
instead of K because prolonging APD by
reducing K markedly increased the slope
of APD restitution to the extent where the spiral wave behavior transitioned to a dynamically less stable phenotype. In contrast, the
effects of reducing K1 on APD restitution
slope and spiral wave stability were much less dramatic. For example,
in the case of meander (si = 0.030),
a 26% increase in APD produced by decreasing K caused the maximal slope of APD
restitution to increase from 0.91 to 1.81, whereas the same increase in
APD by modifying K1 increased the maximum
slope from only 0.91 to 0.95.
Figure 2 summarizes the effects of
reducing K1 on the single cell APD and
its tissue restitution curve for the three values of
si corresponding to nearly stable,
meandering, and hypermeandering spiral waves. For all values of
si , reducing
K1 prolonged APD and slightly increased
the slope of APD restitution. In addition, resting membrane potential
was depolarized by 5 mV at the smallest value of
K1. Despite these changes, however, the
phenotype of the spiral wave remained in the same general category at
both extremes of K1 values, as
illustrated by the tip trajectories and Poincaré plots in Fig. 1,
A-C. This permitted regional variation of APD without
changing the general category of spiral wave phenotype so that
preexisting APD dispersion and dynamic instability could be varied
independently.
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Fig. 2.
APD and APD restitution as a function of
K1 for the different spiral wave
regimes corresponding to si = 0 (nearly stable), si = 0.030 (meander), si = 0.049 (hypermeander). A: APD and the relative change in APD
(APD, in %change) versus K1, for a
planar wave paced at a CL of 500 ms. B-D: APD
restitution curves for the different values of
si. Solid and dashed lines represent
si = 0.60470 and 0.24188, respectively. For reference, the dotted line has a slope of 1.
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In 2-D tissue (80 mm × 80 mm), we introduced the
K1-induced dispersion of refractoriness
over variable spatial scales (x) ranging from 1 to 20 mm
as follows: the tissue was divided into different regions of size
x2, and then the value of
K1 for each region was randomly selected from a range [K1min,
K1max], where
K1max was fixed at 0.60470, the
original value in the LR1 model. Thus the strength of the fixed
electrophysiological heterogeneity can be defined as
K1 = K1min K1min. We arbitrarily chose a random
rather than an ordered dispersion of refractoriness because this is the
most general case, recognizing that a random pattern is not necessarily
the most relevant to specific physiological or pathophysiological processes.
In the tissue, the regional effect on APD (and hence dispersion of
refractoriness) depended on both x and
K1, as shown in Fig.
3 for the case of a planar wave paced at
a cycle length of 500 ms, for the three cases of
si. With
K1min = 0.24188 (corresponding
to K1 = 0.36282), the left column illustrates that for small values of x, the minimum and
maximum APD were nearly the same due to the strong smoothing effects of diffusion currents. As the spatial scale increased, regional
differences in APD increased, saturating at around x = 9 mm. At this point, the minimum and maximum APD corresponded to
their respective values for tissue with homogenous
K1. Although for x > 9 mm, the APDs in the center of different regions were not affected
by diffusion currents, the APD values along the boundary of these
regions were still influenced by diffusion currents. Figure 3
illustrates how K1 affected the
difference between the minimum and maximum APD at the two extremes of
x = 1 mm (middle) and 20 mm
(right). For the small spatial scale (x = 1.0 mm), both minimum and maximum APD, and the difference between them,
increased with increasing K1. For
x = 20.0 mm, however, only the maximum APD
increased. The minimum APD remained constant due to the fixed value of
K1, and the inability of diffusion
currents to affect APD except at the boundaries of a region.
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Fig. 3.
Effect of spatial scale (x) on the
dispersion of APD. A: nearly stable spiral wave regime
(si = 0). B: meander
regime (si = 0.030). C:
hypermeander regime (si = 0.049).
Left: maximum (APDmax) and minimum
(APDmin) in the tissue during planar wave conduction at a
CL of 500 ms, versus x from 0 to 20 mm.
K1max = 0.6047 and
K1min = 0.24188. Middle
and right columns: APDmax and APDmin
versus the strength of electrophysiological heterogeneity
(K1 = K1max K1min) for x = 1 mm (middle ) and 20 mm (right).
K1max = 0.6047 and
K1min was varied.
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Effects of Dispersion of Refractoriness on Spiral Wave Behavior
Figures 4-8 show how the
dispersion of refractoriness, introduced into simulated 2-D tissue (80 mm × 80 mm) by altering K1 as
described above, affected the behavior of spiral waves. The key issue
is the extent to which fixed dispersion of refractoriness promotes
breakup of reentrant waves that, in electrophysiologically homogeneous
tissue, would have remained intact as single spiral waves.
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Fig. 4.
Effect on spiral wave behavior of moderate fixed
electrophysiological heterogeneity
(K1 = 0.18141) imposed over a large
spatial scale (x 20 mm). A: nearly stable
spiral wave regime (si = 0).
B: meander regime (si = 0.030). C: hypermeander regime
(si = 0.049). Row a:
regional variation of APD (absolute on the left and relative
APD in %change on the right) over the surface of the
tissue. Rows b-e: voltage snapshots at t = 100, 500, 1,000, and 2,000 ms after initiation of a spiral wave in either
the region of shortest APD (left column of each pair in
A-C) or longest APD (right column of each
pair in A-C). Row f: corresponding tip
trajectories for the single spiral waves or tip number for the spiral
wave breakup in C. Row g: corresponding
Poincaré plots of successive cycle lengths. Only the spiral wave
initiated in the shortest APD region in C (in the
hypermeander regime, si = 0.049)
broke up into a multispiral fibrillation-like state.
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Fig. 5.
Effect on spiral wave behavior of large fixed
electrophysiological heterogeneity
(K1 = 0.36282) imposed over a large
spatial scale (x = 20 mm). Same as Fig. 4 except
that K1 was increased to 0.36282. See
Fig. 4 legend (rows a-e) for details. The
dispersion of APD over the surface (row a) is greater than
that seen in Fig. 4, and new wave break occurred in all spiral wave
regimes when the spiral wave was initiated in the region of shortest
APD, as well as in the region of long APD for the hypermeander regime
(C).
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Fig. 6.
Effect on spiral wave behavior of large fixed electrophysiological
heterogeneity (K1 = 0.36282)
imposed over a large spatial scale (x = 20 mm).
Traces correspond to A-C in Fig. 5. A:
nearly stable spiral wave regime
(si = 0). B: meander
regime (si = 0.030).
C: hypermeander regime
(si = 0.049). Rows 1 and 2 correspond to the initiation of the spiral wave
in the region of shortest APD and in the longest APD, respectively.
Left column shows averaged membrane potential of the 20 × 20 mm region with the shortest APD (top trace) and
longest APD (bottom trace); the middle column
shows the corresponding FFT spectra (linear and log scales); and the
right column shows the corresponding Poincaré plots of
successive cycle lengths. Note that spectra which appear single-peaked
in linear scale appear broadband in log scale.
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Fig. 7.
Effect on spiral wave behavior of fixed electrophysiological
heterogeneity (K1 = 0.18141)
imposed over a large spatial scale (x = 40 mm), with
the modification of LR1 model (see MATERIALS AND METHODS).
A: APD restitution curves for
K1 = 0.6047 (solid line) and 0.42329 (dashed line), respectively. For reference, the dotted line has a slope
of 1. B: regional variation of APD over the surface of the
tissue. C: voltage snapshots at t = 100, 1,000, and 2,000 ms after initiation of a spiral wave in the longest APD
region. D: same as C except with initiation of a
spiral wave in the shortest APD region.
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Fig. 8.
Effect on spiral wave behavior of large fixed
electrophysiological heterogeneity
(K1 = 0.36282) imposed over a
small spatial scale (x = 4 mm). A: nearly
stable spiral wave regime (si = 0).
B: meander regime (si = 0.030). C: hypermeander regime
(si = 0.049). Row a:
regional variation of APD over the surface of the tissue, which was
reduced by the smaller spatial scale compared with Fig. 5. Rows
b-e. voltage snapshots at t = 100, 500, 1,000, and
2,000 ms after initiation of a spiral wave. Row f:
corresponding tip trajectories for the single spiral waves or tip
number for the spiral wave breakup in C. Row g:
corresponding Poincaré plots of successive cycle lengths.
Only the spiral wave initiated in C (in the hypermeander
regime, si = 0.049) broke up into a
multispiral fibrillation-like state.
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Moderate dispersion of refractoriness, large spatial scale.
Figure 4 illustrates the case of a moderate degree of
electrophysiological heterogeneity
(K1 = 0.18141) imposed over a large spatial scale (x = 20 mm) for values of
si = 0 (Fig. 4A), 0.030 (Fig. 4B), and 0.049 (Fig. 4C) corresponding to
the nearly stable, meandering, and hypermeandering regimes,
respectively. Row a of Fig. 4 shows the regional variation
in APD for a planar wave propagating across the tissue, at a pacing
cycle length of 500 ms. Regional APD variation, corresponding
approximately to dispersion of refractoriness, ranged from 10-18%
in the three cases, being greatest for
si = 0 and smallest for
si = 0.049. Rows b-e show
membrane voltage snapshots at 100, 500, 1,000, and 2,000 ms,
respectively, after initiation of spiral wave activity. For each value
of si, spiral waves were initiated either
in the region with the shortest APD (left panel in each pair in
A-C) or in the region with the longest APD
(right panel in each pair in A-C). Row
f in Fig. 4 shows the tip trajectories of the spiral waves if they
remained intact or, when new wave break occurred, the number of spiral
wave tips. Row g shows the corresponding Poincaré plots of successive cycle lengths. Three main
observations are apparent. First, in cases in which the spiral wave
remained intact, its tip motion became more irregular than in
homogenous tissue (compare with Fig. 1) due to the random gradients in
electrophysiological properties. Second, as the inherent dynamic
instability of the spiral wave increased (i.e., at
si = 0.049 corresponding to the
hypermeander regime), wave break was more easily induced by the fixed
electrophysiological heterogeneities, creating new spiral waves. Third,
in the latter case, whether new wave break occurred depended on where
the spiral wave was initiated. If initiated in the region of longest
APD, the spiral wave remained intact; when initiated in the region of
short APD, however, the spiral wave broke up.
Large dispersion of refractoriness, large spatial scale.
To determine how sensitive these findings were to parameter values, we
explored other regions in parameter space. Figure
5 shows the case in which the fixed
electrophysiological heterogeneity was further enhanced by increasing
K1 to 0.36282, imposed over the same
large spatial scale (x = 20 mm) as in Fig. 4. The
dispersion of APD during a planar wave increased to nearly 30% for
si = 0 and 20% for
si = 0.049 (Fig. 5A). As
shown in rows b-e in Fig. 5, when spiral wave
reentry was initiated in a region of short APD, spontaneous wave break
now occurred at all three values of si.
For the largest value of si = 0.049 (corresponding to the hypermeander regime) wave break occurred even
when the spiral wave was initiated in the region of longest APD.
For the six sets of data shown in Fig. 5, Fig.
6 shows the average intracellular
membrane potential over the 20 × 20 mm2 region, Fast
Fourier Transform (FFT) spectra, and Poincaré plots of successive
cycle lengths during spiral wave reentry, in both the regions of
shortest APD and longest APD. In the nearly stable spiral wave regime
(si = 0 in Fig. 6A), even
though initiation of the spiral wave in the region of short APD caused
wave break and multiple spiral waves to form (Fig. 6A, 1),
the regional cycle lengths quickly settled into periodicity, and after
the transient, no further new wave breaks occurred. Regions with short
APD had short cycle lengths, and regions with long APD had longer cycle lengths, with minimal quasiperiodic modulation. The FFT spectra of the
average voltage in each region had one dominant peak, but at different
frequencies in the two regions, reflecting the stable excitation in the
corresponding region. In contrast, when the spiral wave was initiated
in the region of long APD (Fig. 6A, 2), the spiral wave
remained intact and the FFT spectra and the cycle lengths in regions of
long and short APD were identical.
For the meandering regime (si = 0.030 in Fig. 6B), initiation of spiral activity in the
short APD region led to partially disordered reentry. The spiral wave
remained intact in the short APD region, but wave break occurred in the
longer APD regions, which were unable to sustain 1:1 conduction due to
their longer refractory periods. As a result, the Poincaré plot
from the short APD region showed quasiperiodic meander, whereas that of
the long APD region showed disorder (Fig. 6B, 1). The
averaged regional intracellular membrane potential in the long APD
region was also irregular, resembling polymorphic ventricular
tachycardia or fibrillation. This case demonstrates that a mildly
meandering spiral wave in a localized region with a short cycle length
can produce irregular fibrillation-like activity in surrounding regions
with longer refractory periods due to conduction block, which is one of
the recently proposed mechanisms of fibrillation (3, 24,
33). Again, the FFT spectra in the two regions are different.
The FFT spectra in the short APD region displayed multiple peaks
reflecting a quasiperiodic meandering reentry. The FFT spectra in the
long APD region had only one dominant peak identical to the peak
frequency in Fig. 6B, 2. On the other hand, if the spiral
wave was initiated in a region of long APD, the spiral remained intact,
no wave break occurred in the surrounding areas (Fig. 6B,
2), and the FFT spectra in both regions had the identical dominant peak.
For the hypermeander regime (si = 0.049 in Fig. 6C), initiation of the spiral wave in either
the short or long APD region led to complex patterns of wave break. The
average intracellular membrane potential in both long and short APD
regions were irregular and fibrillation-like, and the cycle length
Poincaré plots were highly disordered reflecting fully developed
spatiotemporal chaos (Fig. 6C, 1-2). The FFT spectra in
all cases had a dominant peak at different frequencies reflecting the
averaged cycle length of excitation in that region.
As noted in Figs. 2-6, APD became very short when
si was reduced to low values. To
determine whether the failure of fixed heterogeneity to cause spiral
wave breakup at low si values (Fig. 5,
A and B) was due to the shorter APD rather than
increased dynamic stability and the shorter APD by low
si, we examined a different
modification of the LR1 model that flattened the slope of APD
restitution without shortening APD significantly, as described previously (12). As shown in Fig.
7, under these conditions, similar
results as in Fig. 5, A and B, were obtained.
When the spiral reentry was initiated in the shorter APD region (Fig.
7D), wave break occurred in the longer APD regions, but the
spiral wave in the shortest APD region always remained intact. However, no wave break occurred in the whole tissue when the spiral reentry was
initiated in the longer APD region (Fig. 7C).
Large dispersion of refractoriness, decreased spatial scale.
As the spatial scale decreased, dispersion of APD also decreased due to
the strong smoothing effects of diffusive currents. To illustrate the
effects on spiral wave reentry, Fig. 8
shows the consequences of reducing x from 20 to 4 mm for
the same large degree of electrophysiological heterogeneity as in Fig.
5 (K1 = 0.36282). Dispersion of
APD for a planar wave (cycle length of 500 ms) decreased from
20-30% to 13-22% for the three values of
si = 0, 0.030, and 0.049. Because of
the smoothing effects of diffusion current, comparable results were
obtained whether the spiral wave was initiated in a region of short APD
or long APD.
For the nearly stable regime (si = 0 in Fig. 8A), the spiral wave remained intact (Fig. 8A,
b-e), but its tip meandered due to the random
electrophysiological gradient (Fig. 8A, f). The
Poincaré plot of successive cycle lengths (Fig. 8A, g)
showed a ring instead of a single point as in homogeneous tissue.
For the meander regime (si = 0.030 in Fig. 8B), the spiral wave also remained intact (Fig.
8B, b-e), and its tip meander was more irregular than
in homogeneous tissue (Fig. 8B, f). The Poincaré plots of cycle lengths displayed chaotic characteristics (Fig. 8B, g) instead of simple quasiperiodic meander as in
homogeneous tissue.
For the hypermeander regime (si = 0.049 in Fig. 8C), however, the initiated spiral wave broke
up into a fully fibrillation-like state after 1,500 ms (Fig. 8C,
b-e). The spiral tip number (Fig. 8C, f)
increased to 15-30 in the fully developed fibrillation-like state.
At this point, the Poincaré plot of cycle lengths was highly
disordered (Fig. 8B, g).
Generally, for each spiral wave phenotype, similar outcomes were
obtained using a number of different initial random heterogeneity patterns. Conversely, if the spiral wave phenotype was varied while
using the same initial random pattern, the findings were also comparable.
Thus, comparing Figs. 5 and 8, a smaller spatial scale of APD
dispersion made wave break less likely and decreased the incidence of
spiral wave breakup and development of a fibrillation-like state.
Figure 9 summarizes the critical
boundaries of spiral wave breakup in the
x-K1 parameter space for
the nearly stable, meander, and hypermeander regimes
(si = 0, 0.030, and 0.049, respectively). Above these critical lines, spontaneous wave break creating multiple spiral waves occurred. Two features are clear in Fig.
9. First, for a fixed value of si, the
critical degree of heterogeneity required to induce wave break
decreased dramatically as spatial scale increased. Therefore, the
smaller the spatial scale, the stronger the electrophysiological
heterogeneity needed to cause spiral wave breakup. Second, the strength
of electrophysiological heterogeneity needed to induce spontaneous wave
break decreased as the spiral wave regime was more dynamically
unstable. Therefore, a hypermeandering spiral wave (e.g.,
si = 0.049) was much more likely to
break up in heterogeneous tissue than a nearly stable spiral
wave (e.g., si = 0).
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Fig. 9.
Critical boundaries for spiral wave breakup in
x-K1 parameter
space for the nearly stable (si = 0.000), meander (si = 0.030), and
hypermeander (si = 0.049) spiral
wave regimes. Below the respective lines, the spiral waves remained
intact, whereas above the lines, breakup could occur.
|
|
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DISCUSSION |
Generation of wave break is a characteristic feature of cardiac
fibrillation (4). In this study, we investigated the
interplay between dynamic factors (30) and fixed
electrophysiological heterogeneity (14, 20) in causing
wave break in simulated 2-D cardiac tissue. Our main conclusions can be
summarized as follows.
First, even without significant dynamic instability (i.e., the nearly
stable spiral wave regime), wave break occurred if fixed electrophysiological heterogeneity was sufficiently large. This effect
was independent of APD because similar results were obtained for
shallow APD restitution slope whether APD markedly shortened (Fig. 4)
or remained normal (Fig. 7). For both cases, however, the ensuing
activity was locally periodic rather than disordered as in
fibrillation. In addition, new wave break was transient, and the system
reached a new steady state characterized by multiple spirals but no new
wave break or new spiral wave formation. This finding suggests that
irregular fibrillation-like activity is impossible if APD restitution
slope is shallow throughout the tissue. This observation is consistent
with experimental observations that the maximal APD restitution slope
typically exceeds one in the majority of animal and human studies
(12, 30).
Second, with increasing dynamic instability (meander and hypermeander
regimes), less fixed electrophysiological heterogeneity was required to
produce wave break and spiral wave breakup, and the ensuing local
activity became highly disordered and aperiodic, resembling
fibrillation. A particularly interesting case is meander (Fig.
5B and 6B, 1), in which a spiral wave initiated
in a region of short APD remained intact locally, exhibiting a strongly
periodic component in its FFT spectrum [i.e., a dominant frequency
(3, 24, 33)]. Because of its short cycle length, however,
adjoining regions with long APD could not sustain 1:1 conduction and
developed complex patterns of conduction block, which has been termed
"fibrillatory conduction" by Jalife and co-workers (3, 24,
33). The FFT spectra in the long APD area (Fig. 6B,
1) also showed a dominant frequency, but also other lower
frequencies of significant power, consistent with local conduction
block. Moreover, the activation intervals in the long APD areas were
highly irregular, resembling fibrillation. This case may be relevant to
recent experimental observations (3, 24, 33), suggesting
that atrial and ventricular fibrillation may be caused by one or a few
dominant rotors (scroll waves), with most of the apparent irregularity
resulting from "fibrillatory conduction" rather than dynamic
instability. However, the results with hypermeander (Fig. 5C
and 6C) indicate that a dominant frequency in the FFT
spectra is not sufficient to exclude other mechanisms. In the latter
case, the FFT spectra also showed a clear dominant frequency, despite
the lack of any stable rotors in the tissue (compare Fig. 6B,
1, and Fig. 6C, bottom trace). Our present
study does not address the other findings, namely domains with
different dominant frequencies and optical maps of activation
sequences, which also support the latter hypothesis (3, 24,
33).
Finally, the spatial scale of the fixed electrophysiological
heterogeneity played a large role in determining the actual degree of
dispersion of refractoriness in the tissue due to the smoothing effect
of diffusion currents at small spatial scales. Fixed
electrophysiological heterogeneity occurring over a large spatial scale
was more generally effective at promoting wave break. However, at large
spatial scales, the tendency for wave break to occur depended on the
region in which reentry was initiated. Only when reentry was initiated
in regions with short APD, corresponding to a short refractory period, did wave break leading to fibrillation occur. When reentry was initiated in regions of long APD (long refractory period), the spiral
wave remained intact. This may be clinically relevant to the much
higher incidence of inducible monomorphic (stable) ventricular tachycardia in coronary artery disease than in cardiomyopathies from
other causes, because the spatial scale of regions with abnormal electrical properties (due to infarction) is typically larger in the
former case (29).
Limitations
There are important limitations to this study. First, because of
computational limitations, we simulated 2-D rather than
three-dimensional tissue, like the real heart. Features such as
anisotropy, fiber rotation, and complex anatomic structures were not
taken into consideration. The third dimension of tissue thickness
(31) and fiber rotation (11) have both been
proposed to destabilize scroll wave reentry and promote wave break in
three dimension and may act synergistically with fixed
electrophysiological heterogeneities such as dispersion of
refractoriness. Second, to produce fixed electrophysiological
heterogeneity, we simulated dispersion of APD and refractoriness by
randomly varying K1, whereas in real ventricular tissue, dispersion of APD has chiefly been attributed to
differences in time-dependent K+ currents represented by
IK in the LR1 model (34).
Unfortunately, reducing K to prolong APD
markedly increased the slope of APD restitution, making it impossible
to maintain the same category of dynamic stability throughout the
tissue without extensively modifying other LR1 currents. Although
increasing APD by reducing K1 had modest
effects on the slope of APD restitution and also depolarized resting
membrane potential by several millivolts, the general phenotype of the
spiral wave was preserved at both extremes of
K1 values. Thus the key requirement to be
able to vary APD regionally while maintaining the same dynamic category of spiral wave phenotype throughout the tissue could still be met.
Third, to model dispersion of refractoriness, we arbitrarily chose
(from among the limitless possibilities) a random rather than an
ordered dispersion of refractoriness, distributed over different
spatial scales. A disadvantage of this approach is that it may not be
specifically relevant to any particular pathophysiological situation.
In the real ventricle, for example, there is a nonrandom transmural
dispersion of APD related to different action potential characteristics
in the endocardial, midmyocardial, and epicardial layers (1,
2). However, our goal in this study was to analyze the
interaction between fixed electrophysiological heterogeneity and
dynamic stability at a general level rather than to simulate specific
physiological or pathophysiological states. The latter is beyond the
scope of the present study. Fifth, the possibilities for creating
electrophysiological heterogeneities are infinite, and we did not
evaluate other interventions, such as modifying different currents or
altering junctional resistance regionally. In general, however,
dispersion of refractoriness is considered to be one of the most
important profibrillatory electrophysiological heterogeneities
(14, 20). Finally, although the LR1 ventricular action
potential model is physiologically based, it is not a complete model.
For example, it does not include intracellular Ca2+
dynamics, which also may be important to spiral wave stability (5). The observation that Ca2+ current
blockade in our study increased dynamic stability and therefore
inhibited wave break may seem at odds with clinical studies associating
Ca2+ channel blockade with increased cardiac mortality
(21). However, the concentrations of Ca2+
channel blockers required to flatten APD restitution significantly are
manyfold higher than concentrations used clinically.
Implications for Antifibrillatory Therapy
Despite these limitations, our observations clearly emphasize the
importance of dynamic properties of the cardiac cell in the development
of wave break, which initiates and sustains cardiac fibrillation. The
more dynamically stable the cell, the harder it is for fixed
electrophysiological heterogeneities, such as dispersion of
refractoriness, to induce wave break. In the normal human ventricle,
the dispersion of APD is of the order of 10%. In our simulated 2-D
tissue, this degree of dispersion of refractoriness required the
dynamics to be at least in the hypermeander regime to induce new wave
break when a spiral wave was initiated. Of course, the degree of
heterogeneity may be much greater in diseased hearts, the clinical
relevant target for antifibrillatory therapy. Nevertheless, our
findings suggest that strategies based on modifying cellular
electrophysiological properties to reduce dynamic instability, such as
reducing the steepness of APD restitution [the Restitution Hypothesis
(30)], may have merit in reducing the risk of cardiac fibrillation. Drugs that flatten APD restitution without concomitantly increasing fixed electrophysiological heterogeneity (e.g., by increasing APD dispersion) may have particular promise.
| |
ACKNOWLEDGEMENTS |
This work was supported by National Heart, Lung, and Blood
Institute Specialized Center of Research in Sudden Cardiac Death 1P50
HL-52319, by a Beginning Grant-in-Aid from the American Heart Association, Western States Affiliate to F. Xie and Z. Qu, and by the
Laubisch and Kawata Endowments.
| |
FOOTNOTES |
Address for reprint requests and other correspondence: F. Xie,
Dept. of Medicine (Cardiology), UCLA School of Medicine, 47-123 CHS,
Los Angeles, CA 90095-1679 (E-mail: fxie{at}mednet.ucla.edu).
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. Section 1734 solely to indicate this fact.
Received 5 July 2000; accepted in final form 15 September 2000.
| |
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TOP
ABSTRACT
INTRODUCTION
