Cardiovascular Research Laboratory, Departments of Medicine
(Cardiology), Physiology, and Physiological Science, University of
California, Los Angeles, California 90095 USA
Scroll wave (vortex) breakup is hypothesized to underlie
ventricular fibrillation, the leading cause of sudden cardiac death. We
simulated scroll wave behaviors in a three-dimensional cardiac tissue
model, using phase I of the Luo-Rudy (LR1) action potential model. The
effects of action potential duration (APD) restitution, tissue
thickness, filament twist, and fiber rotation were studied. We found
that APD restitution is the major determinant of scroll wave behavior
and that instabilities arising from APD restitution are the main
determinants of scroll wave breakup in this cardiac model. We did not
see a "thickness-induced instability" in the LR1 model, but a
minimum thickness is required for scroll breakup in the presence of
fiber rotation. The major effect of fiber rotation is to maintain twist
in a scroll wave, promoting filament bending and thus scroll breakup.
In addition, fiber rotation induces curvature in the scroll wave, which
weakens conduction and further facilitates wave break.
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INTRODUCTION |
Numerous experiments suggest that scroll waves
(or their two-dimensional (2D) analogs, spiral waves) are a major cause
of reentrant cardiac arrhythmias (Allessie et al., 1973
;
Chen et al., 1988; Davidenko et al.,
1992; Winfree, 1994a; Gray et al., 1995, 1998; Chen
et al., 1997; Witkowski et al., 1998). In
particular, ventricular fibrillation, the leading cause of sudden
cardiac death, has been identified with spiral or scroll wave breakup, in which wavelets are continually being created and destroyed (Janse and Wit, 1989; Lee et al., 1996;
Garfinkel et al., 1997; Kim et al., 1997;
Witkowski et al., 1998; Weiss et al.,
1999; Garfinkel and Qu, 1999). If this is true,
then understanding the determinants of scroll wave stability becomes
extremely important. Studies of 2D homogeneous cardiac tissue models
have shown that the transition from a single spiral wave to spiral wave
breakup occurs when the slope of the action potential duration (APD)
restitution relation becomes sufficiently steep (Karma,
1994; Courtemanche, 1996; Qu et al.,
1999a). In three dimensions (3D), however, the situation is
much less clear. Most 3D studies of scroll waves have been carried out
in Fitzhugh-Nagumo (FHN)-type models (Keener and Tyson,
1992; Winfree, 1994b; Panfilov and
Keener, 1995; Biktashev, 1998; Gray and
Jalife, 1998; Berenfeld and Pertsov, 1999),
which grossly oversimplify cardiac electrophysiology. Several of these studies showed that scroll wave breakup can occur in tissue that is
sufficiently thick in the third dimension, while the corresponding 2D
spiral wave remains intact. However, FHN-type models lack a key
property of cardiac tissue: APD restitution, which has been shown to be
very important for the stability of reentry (Karma, 1994; Courtemanche, 1996; Qu et al.,
1999a).
In addition, real cardiac tissue contains significant preexisting
heterogeneities, such as fiber rotation and endo- to epicardial and
base-to-apex electrophysiological differences. Recently,
Panfilov and Keener (1995) showed that fiber rotation
could cause scroll wave breakup in the FHN type model. Fenton
and Karma (1998a,b) simulated the effect of fiber rotation on the stability of scroll waves, using a simplified Beeler-Reuter action potential model. Their
major conclusion was that fiber rotation created regions of highly
localized filament twist, called "twistons." These twistons migrate
toward the boundaries of the tissue; when a twiston collides with a
boundary it causes a scroll wave filament to break, producing daughter
scroll waves.
However, the connection between these results and actual wave
instabilities in cardiac tissue is uncertain, because the stability of
scroll waves is model-dependent, and the cell models used in these
studies are highly simplified compared to real cardiac cells. Therefore, it is important to carry out simulations in physiologically more realistic models. In this paper, we used a 3D tissue model with
phase I of the Luo-Rudy (LR1) action potential model (Luo and
Rudy, 1991), which contains physiologically realistic
formulations of most cardiac ionic currents. We studied how scroll wave
stability is affected by electrical restitution, tissue thickness,
initial filament twist, and fiber rotation. Our major conclusion
is that steep APD restitution remains the primary cause of wave
instability in 3D tissue, as it is in 2D, with tissue thickness,
filament twist, and fiber rotation playing ancillary roles.
| |
METHODS |
Mathematical model
The partial differential equation (PDE) for cardiac conduction
is (Panfilov and Holden, 1997)
|
(1)
|
where V is the transmembrane potential,
Cm is the membrane capacitance, and
Iion is the total ionic current density of the membrane. 
= 
/SvCm is the diffusion
tensor, where is the conductivity tensor and
Sv is the surface-to-volume ratio of the cell.
We used no flux boundary condition (Fenton and Karma,
1998b; Berenfeld and Pertsov, 1999):
· (
V) = 0, where
is the unit vector normal to the boundary. In a 3D
system, has the following matrix structure
(Panfilov and Holden, 1997; Vetter and McCulloch,
1998):
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(2)
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We used the LR1 action potential model, in which the total ionic
current is Iion = INa + Isi + IK + IK1 + IKp + Ib, where INa = 
Nam3hj(V 
ENa) is the fast inward Na+ current;
Isi = sidf(V Esi) is the slow inward current, assumed to be the
L-type Ca2+ current; IK = Kxx1(V EK) is the slow outward time-dependent K+
current; IK1 = K1K1
(V EK1) is the time-independent K+ current;
IKp = 0.0183Kp(V EKp)
is the plateau K+ current; and
Ib = 0.03921(V + 59.87) is
the total background current. m, h, j, d, f, and
x are gating variables satisfying the following type of
differential equation:
|
(3)
|
where y represents the gating variables.
y and 
y are functions of
V. The ionic concentrations are [Na]i = 18 mM, [Na]o = 140 mM, [K]i = 145 mM, and [K]o = 5.4 mM, while the intracellular Ca2+ concentration obeys
![<UP>d</UP>[<UP>Ca</UP>]<SUB><UP>i</UP></SUB>/<UP>d</UP>t=<UP>−</UP>10<SUP><UP>−</UP>4</SUP>I<SUB><UP>si</UP></SUB>+0.07(10<SUP><UP>−</UP>4</SUP>−[<UP>Ca</UP>]<SUB><UP>i</UP></SUB>).](/content/vol78/issue6/fulltext/2761/img006.gif)
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(4)
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Details of the LR1 action potential model were presented in Luo
and Rudy's paper (Luo and Rudy, 1991). By setting
[K]o = 5.4 mM, the maximum conductance of
IK and IK1 are
K = 0.282 mS/cm2 and
K1 = 0.6047 mS/cm2.
Na = 23 mS/cm2 and
si = 0.09 mS/cm2 were
fixed in the LR1 model. The original 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 scroll wave dynamics. We fixed
Na = 16 mS/cm2 as in the
later version of the Luo and Rudy model (Luo and Rudy, 1994) and K = 0.423 mS/cm2, but we varied si and
d and f of the Ca2+ channel
to investigate various types of vortex dynamics. Most of our
simulations were carried out with a twofold speedup of Ca2+
kinetics, i.e., d 
0.5d and
f 0.5f. When
si is around 0.05 mS/cm2, the
baseline APD is around 200 ms; we refer to this case as normal. We
changed si to alter the 2D spiral wave
dynamics, allowing us to investigate the effects of tissue thickness
and fiber rotation for different regimes of 2D spiral wave behavior.
The rationale for selecting the LR1 model instead of later versions
(Luo and Rudy, 1994; Jafri et al.,
1998; Chudin et al., 1999) that formulate
intracellular Ca2+ dynamics was as follows: 1) LR1 model is
computationally much cheaper; 2) the 2D behavior of the LR1 model is
clearer than that of the later versions; 3) there is slow drift in ion
concentrations in the later versions (Xu and Guevara,
1998), which introduces artifacts that are difficult to eliminate.
Numerical simulation
We implemented the model in various structures: a 1D cable, 2D
homogeneous tissue, 3D homogeneous tissue, 3D tissue with fiber rotation, and a hollow cylinder with fiber rotation. The numerical method used in most cardiac simulations is the forward Euler method (Courtemanche et al., 1993; Pollard et al.,
1993; Courtemanche, 1996; Efimov et al.,
1995; Muzikant and Henriquez, 1998), although advanced numerical methods have been developed (Quan et al.,
1998; Vigmond and Leon, 1999; Qu and
Garfinkel, 1999). We integrated Eqs. 1-4, using an operator
splitting and adaptive time step methods (Qu and Garfinkel,
1999). We split Eq. 1 into an ordinary differential equation
(ODE), which is the reaction term
|
(5)
|
and a partial differential equation, which is the diffusion term
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(6)
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ODEs for the gating variables (Eq. 3) were integrated with the
method of Rush and Larsen (1978). Equations 3 and 4 were
integrated by a first-order explicit method. Adaptive time step methods
were used in integrating Eqs. 3-5; the time step varied from 0.02 to 0.2 ms. The PDE (Eq. 6) was integrated by using a first-order explicit
method with time step that was a fraction of 0.2 ms to satisfy the
stability requirement. The space step was fixed at 0.015 cm in all
simulations. Integration of ODEs and PDE was carried out alternatively
as required by the operator splitting method. The spatial
discretization of Eq. 6 is a three-point centered difference for 1D, a
five-point centered difference for 2D, a seven-point centered
difference for 3D homogeneous tissue, and the centered difference
scheme in Fenton and Karma (1998b) for 3D with fiber
rotation. The details on the numerical stability and accuracy of the
splitting method in 2D simulation were presented previously by
Qu and Garfinkel (1999). Here we directly adopted this
method for the 3D case, using a time step of 0.025 ms to integrate
Eq. 6.
1D cable
In Eq. 1, if all of the elements of the tensor D are
set to zero except Dxx, then Eq. 1 becomes a 1D
cable equation. We set Dxx = 0.001 cm2/ms.
2D homogeneous tissue
In Eq. 1, if all of the elements of D are set to zero
except Dxx and Dyy, we
obtain the cable equation for 2D homogeneous tissue. We set
Dxx = Dyy = 0.001 cm2/ms.
3D homogeneous tissue
In Eq. 1, if the elements of D are set to zero,
except for Dxx, Dyy, and
Dzz, we obtain the equation for 3D homogeneous
tissue. We set Dxx = Dyy = 0.001 cm2/ms, and
Dzz = 0.0002 cm2/ms.
3D tissue with fiber rotation
We assume that the fibers are parallel and uniform in the
x-y plane but rotate along the z direction.
Therefore, has the following matrix structure
(Panfilov and Keener, 1995; Fenton and Karma,
1998b):
where
|
(7)
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D
is the diffusion constant along the
fiber direction, and D
is the transverse
diffusion constant. 
(z) is the twist angle along the
z direction. We used a uniform fiber rotation angle
(z) = 
z. The total angle of fiber rotation from
endocardium to epicardium is = Lz.
In this paper, we used D = 0.001 cm2/ms and D = 0.0002
cm2/ms (Fenton and Karma, 1998b).
Hollow cylinder with fiber rotation
We also simulated reentry on the surface of a hollow cylinder
with fiber rotation. In this case, the diffusion term in Eq. 6 becomes
![∇ · <A><AC>D</AC><AC>˜</AC></A>∇=[(D<SUB><UP>xx</UP></SUB>−D<SUB><UP>yy</UP></SUB>)<UP>sin</UP> 2ϕ−2D<SUB><UP>xy</UP></SUB><UP>cos</UP> 2ϕ] <FR><NU>∂</NU><DE>R<SUP>2</SUP>∂ϕ</DE></FR>](/content/vol78/issue6/fulltext/2761/img014.gif)
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(8)
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where R is the radius of the cylinder.
Dxx, Dxy,
Dyy, and Dzz are taken
from Eq. 7. A five-point centered difference scheme was used. To keep
the same spatial resolution, 
= 0.015 cm/R was
used in integrating Eq. 8.
Measuring APD restitution
We defined APD by the duration during which V > 72 mV, while DI was defined by the duration during which
V < 72 mV. APD restitution was calculated in a 1D
cable paced at one end. By slowly and progressively increasing the
pacing rate, we obtained APD restitution curves.
Measuring spiral tip, vortex filament, and tip velocity
The location of the spiral tip in 2D was calculated by the
intersection of two successive isovoltage lines at a time interval of 1 or 2 ms, using 30 mV as the threshold for the isovoltage lines. These
tips, one for each z level, formed the filament of the 3D
scroll wave. We also used the tip positions to calculate tip velocity
and twist angle, which we defined as the difference between the
direction of tip movement at a layer and the direction of tip movement
in the endocardium (see Fig. 8).
| |
RESULTS |
As mentioned above, we and others have previously argued that
spiral wave dynamics in 2D tissue are governed largely by electrical restitution properties (Karma, 1994;
Courtemanche, 1996; Qu et al., 1999a). By
altering Ca2+ current amplitude and kinetics (Qu et
al., manuscript submitted for publication) in the LR1 model in
2D tissue, distinct spiral wave phenotypes can be generated, including
stable, weak meander, strong meander, and breakup. The strategy in this
study was to determine how the third dimension affects the
corresponding scroll wave behavior when these spiral wave phenotypes
are placed in 3D tissue. Specifically, we studied the roles of
restitution, tissue thickness, initial filament twist, and fiber
rotation, respectively.
Spiral wave behaviors in homogeneous 2D tissue
APD restitution refers to the curve relating APD to the previous
diastolic interval (DI). From detailed investigation of spiral wave
meander and breakup in the LR1 model (Qu et al., manuscript submitted for publication), we found that in 2D, chaotic
meander and spiral wave breakup are caused by a APD restitution slope greater than 1. When APD restitution slope is smaller than 1 everywhere, only stable (period 1) spiral waves or quasiperiodically
meandering spiral waves exist. Fig. 1
A shows the APD restitution
curve for the LR1 model with normal Ca2+ current kinetics,
and Fig. 1 B shows the curve for the LR1 model with
Ca2+ current kinetics sped up by a factor of 2. In both
cases, the slope of APD restitution increased as
si increased, but in Fig. 1 A,
the steepening occurred uniformly, whereas in Fig. 1 B, the steepening was restricted to short DI's, with the APD restitution slope reduced at long DI's. (Thus, in the latter case, maximum APD
restitution slope was increased, but the range of DI's with a slope
greater than 1 was reduced.) Fig. 1, C and D,
shows the corresponding spiral wave behaviors for the two cases. With
normal Ca2+ current kinetics (Fig. 1 C), the
spiral wave was stable at low si and
developed progressively more violent meander as
si increased, leading to spontaneous
breakup near si = 0.045 mS/cm2. With the Ca2+ current kinetics sped up
(Fig. 1 D), the meandering behavior was extended over a
wider range of si and was more violent
before breakup, which occurred at >0.065 mS/cm2 (similar
to the findings of Courtemanche, 1996, who used the Beeler-Reuter model). Although addressed elsewhere (Qu et al., manuscript submitted for publication), CV restitution, in
addition to APD restitution, also alters spiral wave behavior.
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FIGURE 1
APD restitution curves and 2D spiral wave phenotype as
a function of maximum Ca2+ current conductance
si in the LR1 model, with either normal
Ca2+ current kinetics (A and C), or
with kinetics sped up by a factor of 2 (B and D).
(A) APD restitution curves for
si = 0, 0.03, and 0.055 mS/cm2 with normal Ca2+ current kinetics.
(B) APD restitution curves for
si = 0.045 and 0.07 mS/cm2 with Ca2+ kinetics sped up, i.e.,
d 0.5d and f 0.5f. The dashed line in B is a redrawing of
the APD restitution curve for si = 0.055 mS/cm2 in A. The dotted lines in
A and B are reference lines of slope 1. (C) Spiral wave behaviors with normal Ca2+
current kinetics. a: stable spiral
(si = 0, j gate clamped
to 1); b-c: weak meander
(si = 0 and 0.03 mS/cm2); d: strong meander
(si = 0.04 mS/cm2);
e: breakup (si = 0.055 mS/cm2). (D) Spiral wave behaviors with
Ca2+ current kinetics sped up. a: stable spiral
(si = 0, j gate clamped
to 1); b-c: weak meander
(si = 0 and 0.03 mS/cm2); d-g: strong meander
(si = 0.04, 0.045, 0.05, or 0.06 mS/cm2). The spiral tip trajectories were all plotted on
the same spatial scale (shown in Dg). In Ce, the
tissue size is 4.8 cm × 4.8 cm. The same gray scale as shown in
Ce was used for all snapshots in subsequent figures.
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In this paper, we refer to the period 1 spiral wave in Fig. 1,
Ca and Da, as stable; the meander
shown in Fig. 1, Cb-d and Fig. 1 D, b and
c, as weak meander; the meander shown in Fig. 1
Dd-g as strong meander; and spontaneous breakup
into multiple spirals, shown in Fig. 1 Ce, as
breakup. Weak and strong meander can be either quasiperiodic
or chaotic. Note that the transition from weak meander to strong
meander and breakup corresponds to increased steepness of APD restitution.
Scroll wave dynamics in homogeneous 3D tissue
do tissue thickness
and initial filament twist cause spiral wave breakup?
Having identified appropriate parameters to control 2D spiral wave
phenotype, we next studied the corresponding scroll wave phenotypes in
homogeneous 3D tissue. First we examined the stability of a straight
scroll filament against small perturbations, to address the issue of
whether tissue thickness can by itself induce scroll filament
instability in the LR1 model, as has been seen in some simplified
models (Aranson and Bishop, 1997; Gray and Jalife, 1998; Biktashev, 1998; Nam et
al., 1998; Qu et al., 1999b).
The role of tissue thickness
To study the stability of an initially straight scroll filament
against small perturbations, we lined up identical spiral waves,
stacked along the z direction to form a vertically straight scroll wave, and then gave a random perturbation to this state, i.e.,
V(x, y, z, 0) = V0(x, y, z, 0) + 10
mV* (random 0.5), in which V0(x,
y, z, 0) is the 2D spiral wave solution and random is a
random number uniformly distributed in [0, 1]. We define a quantity,
![s(t)=<FR><NU>1</NU><DE>L<SUB><UP>x</UP></SUB>L<SUB><UP>y</UP></SUB>L<SUB><UP>z</UP></SUB></DE></FR> <LIM><OP>∭</OP><LL><UP>xyz</UP></LL></LIM>[V(x, y, z, t)−<A><AC>V</AC><AC>&cjs1171;</AC></A>(x, y, z, t)]<SUP>2</SUP><UP>d</UP>x <UP>d</UP>y <UP>d</UP>z,](/content/vol78/issue6/fulltext/2761/img016.gif)
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(9)
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to measure the growth rate of this perturbation, where
(x, y, z, t) = (1/Lz)
zV(x, y, z, t)dz, and
Lx, Ly, and
Lz are the dimensions of the tissue. Note that
s(t) is the integral over all space of the distance between
the actual value of V at a point (x, y, z) and
the value of V at that (x, y) point averaged
over the z axis. Thus, if s(t) decays to zero,
then the straight vortex filament is stable; otherwise it is unstable.
Stable, weak meander, and strong meander regimes. Fig.
2 compares s(t) for scroll
waves corresponding to stable, weak meander, and strong meander regimes
of 2D spiral waves (Fig. 1 D, a, c, and e,
respectively) in 9-mm-thick tissue. The initial perturbations became
progressively damped over time, demonstrating that for these parameter
ranges, scroll waves were stable against small perturbations. We also
performed simulations in 15-mm-thick tissue, as well as other parameter
regions in Fig. 1 D, and obtained similar results. In
addition, using the LR1 model with the original Ca2+
current kinetics, the comparable parameter regions shown in Fig. 1
Ca-d gave similar results.
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FIGURE 2
Stability of scroll filaments in homogeneous 3D tissue
to small perturbations. s(t) versus time for the scroll
waves in the 2D stable (A), weak meander (B), and
strong meander (C) regimes shown in Fig. 1 D. (A)
si = 0, j = 1; (B)
si = 0.035 mS/cm2; (C)
si = 0.045 mS/cm2. The tissue
size was 3 cm × 3 cm × 0.9 cm in A and
B, 4.8 cm × 4.8 cm × 0.9 cm in C.
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Several previous studies (Aranson and Bishop, 1997;
Gray and Jalife, 1998; Biktashev, 1998;
Nam et al., 1998; Qu et al., 1999b) in
excitable media with two-variable cell models have found that an
instability can occur when the tissue thickness exceeds a certain critical value. In this instability, a small perturbation applied to a
straight scroll wave leads to a helical scroll wave or to scroll wave
breakup. We recently demonstrated that such instabilities are diffusion
induced (Qu et al., 1999b) and model dependent. In the
present model, we did not observe these instabilities. In summary,
straight filaments were all stable in these parameter regimes, and no
scroll breakup was found. Thus our findings indicate that a conversion
from 2D no-breakup behavior to 3D breakup behavior, caused by tissue
thickness, does not occur with the LR1 action potential model.
Breakup regime. To examine how breakup is affected by the
presence of the third dimension, we implemented the LR1 model in the 2D
breakup regime shown in Fig. 1 Ce and characterized the effects of tissue thickness on the filament stability. When the tissue
was thinner than a critical value (~3 mm), the filaments were stable
against small random perturbations, i.e., s(t) finally decayed to zero (Fig. 3 A),
even though the number of scrolls and their movements were chaotic.
Thus chaotic motion in 2D was synchronized in the third dimension in
thin tissue. When the tissue was thicker than the critical value,
filament break occurred, resulting in transmural reentry. The system
then displayed "fully developed turbulence" in 3D (Fig. 3
B). When the tissue thickness was only slightly greater than
the critical value, filaments became unstable, and filament bending and
twisting occurred. These findings demonstrate that the chaotic motions
of the 2D spiral wave can be synchronized in the z axis when
the tissue is thin, but synchronization is lost when the thickness
exceeds a critical length. (For discussions of spatiotemporal chaos
synchronization and its relation to system size, see Heagy et
al. (1995), Hu et al. (1995), Ding and
Yang (1997), Yang et al. (1998), and Qu
et al. (1999b).
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FIGURE 3
Effect of tissue thickness on filament stability in the
breakup regime. s(t) versus time for scroll breakup in 2D
spiral wave breakup regime, comparing 0.15-cm-thick tissue
(A) and 0.9-cm-thick tissue (B). The insets show
the scroll filaments at 200 ms (A) and 750 ms
(B). Parameters are the same as in Fig. 1 Ce.
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The role of initial filament twist
In the simulations described above, identical 2D spiral waves were
stacked to form a straight vortex filament. To determine whether
initial filament twist induces scroll breakup, we stacked up spiral
waves from a 2D simulation at different phases, forming a twisted
scroll vortex. Then we put this twisted scroll wave into 3D homogeneous
tissue to study its behavior.
Stable and weak meander regimes. When parameters were such
that the 2D spiral wave was in the stable or weak meander regimes, an
initially twisted scroll wave became untwisted, finally forming a
scroll wave with a straight filament, as shown in Fig.
4. The initial twist angles were over
300° from the top to bottom (i.e., "epicardial" to
"endocardial") surfaces in a 9-mm-thick tissue. This corresponded
to a time lag between the top and bottom surfaces of 36 ms for the
stable spiral wave regime and 48 ms for the weak meander regime, i.e.,
about one rotation period. In both cases, the twisted filament became
straight with time. Even with an initial twist angle of 720° (about
two rotations) in the stable spiral wave regime, the scroll wave
finally untwisted. Consistent with previous studies of twisted scroll
waves in excitable media (Biktashev et al., 1994), we
found that a twisted scroll wave rotated faster than an untwisted one.
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FIGURE 4
Straightening of initially twisted filaments in
homogeneous 3D tissue, for the 2D stable (A) and weak
meander (B) regimes (corresponding to Fig. 1 D, a
and e). The scroll filaments at 10 and 2000 ms are shown as
indicated, along with the continuously traced scroll tip trajectories
on the bottom "endocardial" surface. (A)
si = 0, j = 1. (B)
si = 0.035 mS/cm2. The time
lags in initial conditions from the top to the bottom surfaces were 36 ms in A and 48 ms in B. Tissue size was 3 cm × 3 cm × 0.9 cm.
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Strong meander regime. In contrast to the stable and weak
meander 2D regimes, a twisted scroll wave in the strong meander regime
broke up if it was sufficiently twisted (Fig.
5). When the time lag from the top to the
bottom surface was small, the scroll wave remained intact. No filament
bending or break occurred, and the twisted filament eventually
untwisted, becoming straight. At a critical time lag, however, filament
bending occurred, leading to filament break. The break formed a new,
less twisted filament plus a scroll ring (Fig. 5 A). The
scroll ring finally shrank and disappeared, and the new filament
eventually became untwisted. When the top-to-bottom time lag was even
larger, filament bending led to very complex scroll breakup patterns
(Fig. 5, B and C). We are not sure whether scroll
breakup in this case was a sustained motion or a transient process,
because of the limited computer run time. However, in thinner tissue,
breakup was transient, and finally only straight filaments were left in
the tissue (Fig. 6).
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FIGURE 5
Filament break induced by initial filament twist in the
2D strong meander regime (corresponding to Fig. 1 D,
si = 0.045 mS/cm2).
(A) Time lag between the top and bottom surfaces (set in the
initial conditions) of 60 ms. A scroll ring forms and then disappears.
(B) Time lag, 72 ms. Complex filament breakup occurs.
(C) Snapshots of scroll waves in B, showing the
waves on the top "epicardial" surface, as well as on the
forward-facing vertical surfaces. Below them are same waves on the
bottom "endocardial" surface. The tissue size was 4.8 cm × 4.8 cm × 0.9 cm. The timing of each panel is indicated.
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FIGURE 6
Scroll breakup due to an initially twisted filament in
the 2D strong meander regime is transient. s(t) versus time
for the case shown in for Fig. 5 B, except that the tissue
thickness has been reduced to 0.45 cm. Insets show filaments at 50, 100, and 900 ms.
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We also simulated scroll waves close to the parameter boundary at which
spiral wave breakup begins (Fig. 7). In
this case, breakup did not occur if the time lag from top to bottom was
small but did occur if the time lag was increased beyond a critical value. However, this breakup was not induced by filament bending. Instead, a scroll ring formed as a result of conduction block in the
scroll wave arm, which was unrelated to filament collision. When the
twist was large, the scroll ring expanded and collided with the tissue
boundary, producing complex scroll breakup.
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FIGURE 7
Scroll ring formation in the absence of filament
collision, for the 2D strong meander regime. The strong meander regime
was set very near the boundary for 2D breakup, with
si = 0.048 mS/cm2, and
d 0.6d, f 0.6f. (A) Time lag between the top and bottom
surfaces (set in the initial conditions) of 48 ms. A small scroll ring
forms independently without filament bending/collision and subsequently
disappears. (B) Time lag, 72 ms. A larger scroll ring
appears, leading to complex filament breakup. Tissue size was 4.8 cm × 4.8 cm × 0.9 cm.
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Scroll wave behavior in inhomogeneous 3D tissuedoes fiber
rotation induce scroll wave breakup?
In real cardiac tissue, conduction is anisotropic, being faster
parallel to fiber direction and slower transversely. Fiber direction
also rotates from epicardium to endocardium through the ventricular
wall, producing rotational anisotropy. To study 3D tissue with fiber
rotation, we adjusted the diffusion coefficient to slow conduction
velocity in one axis (transverse axis) while maintaining normal
conduction velocity in the other (parallel axis). By convention, the
parallel axis defines the fiber direction. Fiber direction was then
rotated at a uniform rate along the z axis between the top
("epicardial") and bottom ("endocardial") surfaces. We began
with a scroll wave with a straight filament, composed of stacked-up
identical spiral waves from 2D homogeneous tissue.
Stable and weak meander regimes
Fig. 8, A and
B, shows the tip trajectories on the "epicardial" and
"endocardial" surfaces for a stable 2D spiral wave (corresponding to Fig. 1 Da) as the tissue thickness (A) and
degree of fiber rotation (B) were progressively increased.
The scroll wave remained stable for any thickness and rotation rate up
to 48°/mm. At 48°/mm, however, the filament began to drift. The
drift still existed when we reduced z to 0.0075 cm. We
are uncertain whether this instability was real or was due to numerical
inaccuracy.
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FIGURE 8
Effect of fiber rotation rate on filament bending in
3D tissue, for the 2D stable spiral regime (corresponding to Fig. 1
Da). Tip trajectories from the bottom "endocardial"
surface (dashed lines) and the top "epicardial" surface
(solid lines) are superimposed, for different tissue
thickness (Lz) and fiber rotation rate ().
(A) = 12°/mm, Lz = 1.5, 3, 6, and 9 mm, respectively. (B) Lz = 3 mm, = 12, 24, 36, and 48°/mm. (C) A plot of the
angular difference () of propagation directions of the spiral
tips in a given layer (depth z) relative to the endocardium
(illustrated in the inset). The maximum differences occur at
3-6 mm depth. (D) A plot of the distance between the scroll
tip in a given layer (depth z) and the scroll tip on the
endocardium, projected in the x-y plane, versus
t and z.
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Fig. 8 C shows a space-time plot of the filament twist angle
from "endocardium" to "epicardium" in a 9-mm-thick tissue. The twist angle was measured as the angle between the direction in which
the tip was moving in a given layer and the direction in which the tip
moved on the endocardial surface, as illustrated in the insets between
Fig. 8 C and Fig. 8 D. The maximum twist occurred
between 3 and 6 mm along the z axis, reaching nearly 80°,
which was greater than the fiber rotation angle between 3 and 6 mm. In
one period, the maximum twist angle occurred twice and the filament
untwisted twice. Fig. 8 D shows the distance, projected in
the x-y plane, between the tip at various depths in the
tissue and the tip on the "endocardial" surface. The maximum also
occurred between 3 and 6 mm. Interestingly, the period of the scroll
wave did not change with the fiber rotation rate, except for extremely
large (>48°/mm) fiber rotation rates, at which the scroll wave
rotated a little bit more slowly. In this regime, the cycle length for
this scroll was always 35 ms. This differs from the rotation period of
a twisted scroll wave in homogeneous tissue, which runs faster than a
straight one (Biktashev et al., 1994).
Similar to the stable regime, in the 2D weak meander regime
(corresponding to Fig. 1 C, c and d,
and 1 Dc), a single scroll wave always remained intact. No
breakup was observed, even at a fiber rotation rate up to 30°/mm in
9-mm-thick tissue. To further substantiate the robustness of these
findings, we also changed APD restitution by speeding up the
Ca2+ kinetics (Xu and Guevara, 1998) by a
factor of 10, i.e., d 0.1d and
f 0.1f. With this speedup, the slope
of APD restitution was largely reduced and produced a 2D spiral wave
with weak meander for si = 0.06 mS/cm2. We did not see scroll breakup with a fiber rotation
rate of 18°/mm in 9-mm-thick 3D tissue (in the physiological range).
Strong meander regime
In contrast to the stable or weak meander regimes, in the 2D
strong meander regime, the resulting scroll wave behavior changed. For
small degrees of fiber rotation, a scroll wave remained intact, with no
filament bending or breakup. However, when the fiber rotation rate was
large, the scroll wave broke up. Fig. 9
shows the scroll filament in a 9-mm-thick tissue at a fiber rotation
rate of 18°/mm. In the progression from the initially straight
filament to scroll breakup, several distinct processes occurred:
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FIGURE 9
Filament breakup in 3D tissue with fiber rotation, for
the 2D strong meander regime (corresponding to Fig. 1 De).
Fiber rotation rate = 18°/mm. Tissue size: 4.8 cm × 4.8 cm × 0.9 cm. The time after initiation of the scroll wave is
indicated above each panel.
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First, filament bending occurred (t = 50
ms). The bending filament collided with itself to form a small scroll
ring (t = 54 ms). This scroll ring shrank and
disappeared; the bent filament straightened (t = 70 ms).
Later, another scroll ring formed from the bent filament without
collision (t = 400-430 ms). This scroll ring expanded
and broke at the boundaries (t = 430 ms), while the
bent filament straightened without forming new scroll rings or
colliding with the boundary (t = 430 ms).
Still later, a large scroll ring formed as the filament bent and
collided with itself (t = 570-600 ms); this scroll
ring collided with one of the boundaries, producing breakup.
These processes occurred repeatedly and finally led to a pattern of
complex scroll breakup in the tissue (t = 800 and 850 ms).
Fig. 10 shows snapshots of
the scroll waves corresponding to Fig. 9. Note that the spiral waves on
the "endocardial" and "epicardial" surfaces became more and
more desynchronized with time. Fenton and Karma
(1998a,b)
showed that filament unbending or the first filament break occurred
when the bending filament collided with a boundary. We did not see this
in our simulation in Fig. 9, but we did see this occur when the tissue
was thinner. The filament behaviors in Fig. 9 for the tissue with fiber
rotation were similar to the filament behaviors in Figs. 5 and 7 for
twisted scroll waves in homogeneous tissue, except that breakup in the
tissue with fiber rotation was sustained, while in homogeneous tissue it may have been transient because the scroll waves may finally have
all untwisted.
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FIGURE 10
Snapshots for the simulation shown in Fig. 9,
illustrating scroll breakup in 3D tissue with fiber rotation, for the
2D strong meander regime. Panels show the waves on the top
"epicardial" surface, as well as on the forward-facing vertical
surfaces. Below them are the same waves on the bottom "endocardial"
surface.
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After scroll breakup occurred, the average APD, average DI, and the
average CL were shorter than in the case with no breakup. Fig.
11 shows the distribution of DI, APD,
and CL for 2D strong meander (Fig. 1 De) and 3D scroll
breakup (Fig. 9). The values for 3D were obtained after scroll wave
breakup in the fully developed turbulent regime. Note that the curves
from the 3D case were all shifted to the left with respect to the
values in 2D in which the single spiral wave remained intact, with
average values in 3D versus 2D: 19 versus 35 ms for DI; 52 versus 69 ms
for APD; and 71 versus 104 ms for CL. Thus the average DI, APD, and CL in 3D were shorter by 45%, 25%, and 30%, respectively, because of
scroll breakup.
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FIGURE 11
Probability distributions of average DI
(A), APD (B), and CL (C) in the 2D
strong meander regime. Solid lines: for the broken scroll
waves in 3D tissue with fiber rotation shown in Figs. 9 and 10.
Broken lines: for the 2D spiral wave shown in Fig. 1
De (dashed lines). The action potential parameters are
identical in the two cases. DI, APD, and CL were recorded at
every other grid point for a duration of 1 s in 3D and for 3 s in 2D.
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When we explored the parameter space more fully, we found that whether
a scroll wave broke up or not depended on three factors: si, or equivalently, the APD restitution
properties (which determine the 2D spiral wave dynamics), the fiber
rotation rate, and tissue thickness, as summarized in Fig.
12. Fig. 12 A shows the
relationship between the critical fiber rotation rate c
at which scroll breakup occurred and si
in a 9-mm-thick tissue. For si < 0.04 mS/cm2, scroll breakup did not occur, even at very
large fiber rotation rates. At si = 0.04 mS/cm2, breakup was not observed until exceeded
25°/mm. For si > 0.04 mS/cm2, c progressively decreased as
si increased and was close to zero at
si = 0.06 mS/cm2, at
which 2D spiral waves are close to their breakup regime.
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FIGURE 12
Relationships between 3D scroll behavior and
si (controlling APD restitution
steepness), fiber rotation rate , and tissue thickness
Lz.  , No breakup occurred;  , breakup
occurred. We connected the full circles by dotted lines to distinguish
the no-breakup and breakup regions; i.e., below the line is the
no-breakup region, while above is the breakup region. (A)
-si parameter space. (B)
Lz-si parameter space.
(C) -Lz parameter space.
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Fig. 12 B shows the relationship between critical tissue
thickness Lc at which scroll breakup occurs and
si, for a fixed fiber rotation rate
= 15°/mm. For si 
0.04 mS/cm2, scroll breakup did not occur in tissue up to 15 mm
thick. For si between 0.045 and 0.06 mS/cm2, the Lc was ~4-6 mm.
Lc decreased to zero as the 2D spiral wave approached its breakup regime.
Fig. 12 C shows the relationship between c
and Lz with si
fixed at 0.045 mS/cm2. c decreases as
Lz increases.
Mechanisms of scroll breakup in 3D tissue
Using their simplified three-variable model, Fenton and
Karma (1998a,b) showed that twiston-induced filament bending can
lead to wavebreak and complex turbulent scroll behaviors in 3D tissue with rotational anisotropy due to fiber rotation. They expected that
the mechanisms of filament bending and breaking "do not require specific forms of APD restitution curves." However, using the LR1
model, we find that a steep APD restitution curve is still one of the
necessary conditions for the breakup, because neither a twisted scroll
wave in a homogeneous tissue nor a scroll wave in tissue with fiber
rotation broke up until the control parameter was increased to the 2D
strong meander regime, in which APD restitution is steep. We develop
our explanation for breakup in stages, as follows.
Tip meander in 2D
As shown in Fig. 1, in 2D tissue, transitions from weak to strong
meander to breakup occurred as the steepness of APD restitution increased. In the strong meander case, the spiral wave rotated with
large oscillations in both wavelength and excitable gap due to steep
APD restitution. This is illustrated in Fig. 13
A by snapshots of a spiral
wave in the strong meander regime (corresponding to Fig. 1
De). Large differences in wavelength (white-gray
areas) and excitable gap (black areas) along the spiral
wave are evident, which did not occur in the stable or weak meander
regimes (Fig. 13 B). In particular, the waveback in the weak
meander regime was smooth, but "bumps" developed in the waveback
during strong meander. These bumps were caused by the excitation of
spatial modes due to the APD and conduction velocity restitution, as we
have discussed previously (Qu et al., manuscript submitted for
publication). Because of the meandering instability, the tip
velocity of the spiral wave began to oscillate. Fig. 13 C
compares spiral tip velocity for 2D spiral waves in the stable, weak
meander, and strong meander regimes. When the spiral wave meandered,
its tip velocity oscillated. In the strong meander regime, this
oscillation became violent, varying from very fast (over 0.6 m/s) to
extremely slow (close to zero). The slow phases corresponded to the
spiral tip encountering a "bump" in the previous waveback (i.e.,
encountering relatively refractory tissue) (Fig. 13). With the tip in
its slow phase, however, the core area had more time to recover,
resulting in a larger excitable area (Figs. 13 A, b, c, and
f). Therefore, once the tip had completed its slow turn, it
moved into fully excitable tissue and propagated quickly and almost
linearly (Figs. 13 A, a, d, and g). In contrast,
in the weak meander regime, the tip was not slow enough in its slow
phase to create such a large excitable gap in the core area as in the
strong meander case.
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FIGURE 13
2D spiral tip movement. (A) Snapshots
(every 20 ms) of a spiral wave in the strong meander regime
(corresponding to Fig. 1 De). The panel at the far right
shows the position of the spiral tip (open circle) every 3 ms, illustrating a slow phase (closely spaced dots) and a
fast phase (widely spaced dots) of tip movement. Arrows
indicate the turning from slow phase to fast phase. (B)
Snapshots (every 30 ms) of a spiral wave in the weak meander regime
(corresponding to Fig. 1 Dc). The panel at the far right
shows much smaller differences between the fast and slow phases of tip
movement than in A. (C) Tip velocity for a stable 2D spiral
wave (for Fig. 1 Da), and for the weak and strong meander
regimes in A and B. Tissue size: 4.8 cm × 4.8 cm. To show the excitable gap oscillation, we painted it black when
V < 70 mV, which differs slightly from Fig. 1
De.
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How tip meander in 3D promotes filament bending
Now let us explain how a scroll wave with a twisted filament
breaks up in 3D homogeneous tissue. In homogeneous tissue, because we
stacked spiral waves from different phases, the excitable areas and the
tip velocities were different in each layer. The spiral waves in some
layers were in the slow phase, and others were in the fast phase. When
the scroll wave in a given layer completed its slow turn and began its
fast movement, it faced excitable tissue not only in its own layer but
also in the neighboring layers. If the gradient was large enough, then
it invaded its neighboring layers and propagated in the z
direction, causing filament bending. This is illustrated in Fig. 14
A, where we show snapshots
of different layers at t = 28 ms for the simulation in
Fig. 5 A. At layers below z = 2.4 mm (Fig.
14 Aa-c), the spiral wave was still in its slow phase
(trying to make the slow turn) and had a large excitable area in the
center. However, the spiral waves in the layers above z = 6 mm had already completed the slow turn and moved into the fast
phase (Fig. 14 Af-h). The spiral tips were thus able to
propagate downward, causing a breakthrough-like excitation (Fig. 14
Ac-e). For such breakthrough excitation to occur, there
must be large excitable gaps in the z direction, due to
different layers moving in different phases. This only occurs when the
spiral wave is in the strong meander regime and when the spiral waves
are sufficiently desynchronized by the twist of the filament. In the
weak meander regime, the differences between the slow-moving and
fast-moving phases of the tip are too small to generate large enough
excitable gaps in the z direction to cause filament bending
sufficient to produce wavebreak.
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FIGURE 14
Simultaneous snapshots of a scroll wave at different
depths, illustrating gradients in excitability along the z
axis caused by out-of-phase fast and slow tip movements, for the 2D
strong meander regime. The same gray scale as in Fig. 13 was used.
(A) t = 28 ms for homogeneous 3D tissue with the
initial filament twisted (the simulation shown in Fig. 5). (B)
t = 42 ms for 3D tissue with fiber rotation (the simulation
shown in Fig. 9). a-h are layers equally distanced from
lower "endocardial" to the top "epicardial" surface.
(C) Local maxima and minima in the z axis of the
scroll filaments, from t = 20 to 60 ms, for
B.
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In 3D tissue with fiber rotation, the mechanism of filament bending is
similar. As was shown in Fig. 8, fiber rotation itself did not add new
instability to the scroll wave dynamics unless it was extremely large.
But for initial conditions of an untwisted scroll wave, fiber rotation
caused the filament to twist (Fig. 9). The scroll wave thereby began to
move in different phases in different layers, similar to the motion of
an initially twisted scroll in homogeneous tissue. Fig. 14 B
illustrates the filament bending process at t = 42 ms
for the simulation shown in Fig. 9. The spiral waves in layers below
z = 3.6 mm had completed their slow turn and moved to
the fast phase, but the spiral waves in the layers above z = 6 mm were still in their slow turn, creating a very large
excitable area in the core region. Therefore, a breakthrough could move
upward, bending the filament. Fig. 14 C shows the local maxima and minima in the z direction of the filaments for
the simulation in Fig. 9 from t = 20-60 ms. At
t = 25 ms, the filament began to bend. The minimum
decreased a little at first and then remained nearly constant, while
the maximum increased almost linearly. At t = 54 ms,
the bending filament broke, forming a scroll ring and a scroll with a
bent filament. At t = 58 ms, the filament unbended, and
the scroll ring shrank and disappeared at t = 60 ms.
That the minimum did not decrease indicated that there was no excitable
gap for the filament to propagate downward. However, the filament
propagated upward with an almost constant velocity (~0.15 m/s),
indicating that there was excitable tissue available for propagation.
How filament bending promotes scroll breakup
Once filament bending starts, there are several processes that can
lead to scroll breakup. If the tissue is not thick, the bend in the
filament can propagate to the boundary, where it may break in two
(Fenton and Karma, 1998a,b). If the tissue is thick, however, a scroll
ring may occur as a result of the bent filament colliding with itself
(Figs. 5 A and 9). The scroll ring can either expand until
it hits the boundaries to cause further filament break (Fig. 9) or
shrink and disappear (Figs. 5 A and 9), depending on the
twist rate. An important consequence of filament bending is the
invasion of the excitable tissue from the z direction, which
makes the DI shorter than it would be without this invasion. This
effectively shortens both APD and CL and thus moves the system into the
steeper range of the APD restitution curve, promoting further
restitution-induced instabilities. This shortening of CL due to
z direction reentry is a major cause of further wavebreak (Figs. 5 B and 9), following the first filament bending.
These wavebreaks occur in the scroll arm, which is similar to the
behavior of the wavebreaks in 2D spiral wave breakup. They are not due to a bent filament colliding with itself or with boundaries, but to
conduction block in the spiral arm as a direct result of steep APD
restitution. This kind of wave break in 3D can form either a scroll
ring (e.g., t = 400 ms in Fig. 9) or a U-shaped scroll filament (e.g., t = 800 ms in Fig. 9).
In addition, in homogeneous tissue, twist itself speeds up the scroll
wave (Biktashev et al., 1994), thus shortening CL and engaging a steeper portion of the APD restitution curve. If the system
is very close to 2D breakup, scroll breakup may occur because of this
speedup. This is the case shown in Fig. 7, where there was no filament
bending, but a scroll ring formed because of wavebreak in the scroll arm.
In tissue with fiber rotation, the CL of a scroll wave did not change
as in a twisted scroll wave in homogeneous tissue, but here another
factor facilitated wavebreak, namely curvature of the wave induced by
fiber rotation. To illustrate how curvature changes with fiber rotation
in a scroll wave, we simulated unidirectional conduction on the surface
of a cylinder. We set the diffusion coefficient on the cylinder as if
it were sliced out of the tissue with 3D fiber rotation. (Details of
the mathematics are in Eq. 8). Fig. 15
A shows the isopotential
lines of the wavefront and waveback. The curvatures of the wavefront
and waveback changed constantly in both space and time. Curvature of a
convex wave decreases conduction velocity but prolongs APD
(Comtois and Vinet, 1999). Therefore, despite the same
CL, APD prolongation in a curved wavefront makes conduction block
easier. In addition, the waveback curvature may differ from the
wavefront curvature at the same position. This allows conduction block
to be localized to one region of the wavefront, resulting in a scroll
ring in the tissue. Fig. 15 B shows such an episode of
wavebreak in reentry on the cylinder. Because the conduction failure
was very localized, the two spiral waves collided to heal the
wavebreak. This corresponds to the case of the scroll ring shrinking in
3D tissue. If the break involves a sufficient large area, the two
spiral waves continue to rotate, causing further wavebreaks to develop
in the tissue.
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FIGURE 15
Wavefront and waveback dynamics during unidirectional
reentry on the surface of a cylinder with fiber rotation, for the 2D
strong meander regime. (A) Isochrones of the wavefront
(top) and waveback positions (bottom) during
successive cycles around the cylinder. Note that both wavefronts and
wavebacks are curved. (B) Snapshots illustrating local
wavebreak as a result of curvature mismatch (head-to-tail interaction)
of the wavefront and waveback. At t = 140 ms, a convex
wavefront approaches from the left; at t = 170 ms,
wavebreak occurs when this wavefront encounters still-refractory tissue
from the previous wave (due to curvature of its waveback); at
t = 200 ms, the two broken wavefronts reseal.
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DISCUSSION |
Scroll wave breakup has been proposed as a paradigm for the
degeneration of tachycardia to fibrillation. In this study, we have
investigated scroll wave stability by examining how 2D spiral waves
behave when they are placed in 3D tissue under the following conditions: homogeneous 3D tissue in which the scroll wave has an
initially straight filament; homogeneous 3D tissue in which the scroll
wave has an initially twisted filament; and 3D tissue with fiber
rotation (rotational anisotropy). Four phenotypes of 2D spiral waves
were studied under these conditions: stable, weak meander, strong
meander, and breakup. Because scroll wave behavior is model dependent
(Qu et al., 1999b), we used a detailed cardiac action
potential model incorporating physiologically based rather than
phenomenologically based formulations of the key cardiac ionic currents.
From our findings, the major causes of the scroll breakup in this 3D
model can be summarized as follows: 1) Steep APD restitution causes
strong meander, thus promoting large oscillations in tip velocity that
lead to large excitable gaps in the scroll core area. 2) Either fiber
rotation or filament
TOP
ABSTRACT
INTRODUCTION
