Vol. 279, Issue 4, H1869-H1879, October 2000
Dynamics of action potential head-tail interaction during
reentry in cardiac tissue: ionic mechanisms
Thomas J.
Hund1,
Niels
F.
Otani1, and
Yoram
Rudy1,2,3
Cardiac Bioelectricity Research and Training Center, and
Departments of 1 Biomedical Engineering, 2 Physiology
and Biophysics, and 3 Medicine, Case Western Reserve
University, Cleveland, Ohio 44106-7207
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ABSTRACT |
In a sufficiently short
reentry pathway, the excitation wave front (head) propagates into
tissue that is partially refractory (tail) from the previous action
potential (AP). We incorporate a detailed mathematical model of the
ventricular myocyte into a one-dimensional closed pathway to
investigate the effects of head-tail interaction and ion accumulation
on the dynamics of reentry. The results were the following:
1) a high degree of head-tail interaction produces
oscillations in several AP properties; 2) Ca2+-transient oscillations are in phase with AP duration
oscillations and are often of greater magnitude; 3) as the
wave front propagates around the pathway, AP properties undergo
periodic spatial oscillations that produce complicated temporal
oscillations at a single site; 4) depending on the degree of
head-tail interaction, intracellular [Na+] accumulation
during reentry either stabilizes or destabilizes reentry; and
5) elevated extracellular [K+] destabilizes
reentry by prolonging the tail of postrepolarization refractoriness.
reentry; ion accumulation; alternans; restitution
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INTRODUCTION |
REENTRY IS THE
UNDERLYING mechanism of many common cardiac arrhythmias
(11, 32, 36), which are a major cause of death and
disability. Effective diagnosis, prevention, and treatment of these
life-threatening arrhythmias require an in-depth understanding of the
reentrant action potential (AP).
During reentry in a sufficiently short pathway, the excitation wave
front (head) propagates into tissue that is still refractory (tail)
from the previous excitation. Interaction between the head and tail of
an AP is a common phenomenon in the heart. It has been observed in an
anatomically defined reentry pathway (18), along the arm
of a spiral wave (21), and in the leading circle of
functional reentry (2). Head-tail interaction is also
known to have a profound effect on the dynamics of the reentrant AP. For example, a significant degree of head-tail interaction produces oscillations in key AP properties such as AP duration (APD), conduction velocity (
), and cycle length (CL) (8, 18, 21, 34, 42). Such oscillations often precede spontaneous termination of reentry (14, 17, 32) or breakup of the reentry loop into multiple pathways resulting in fibrillation.
Theoretical studies of reentry (8, 21, 33, 42) have used
relatively simple models of the cardiac AP (4, 15, 29, 30)
that lack realistic elements such as intracellular calcium handling and
dynamic intracellular ion concentration changes during excitation.
Changes in the intracellular concentrations of ions such as
Ca2+ ([Ca2+]i) and
Na+ [Na]i) occur during rapid pacing, drug
application, and ion-channel dysfunction (35).
Importantly, intracellular ions accumulate during reentrant
tachyarrhythmias due to the rapid repetitive excitation of the cells.
These ionic changes affect the AP through various membrane currents and
modify the dynamics of reentry. In this study, we examine reentry in a
one-dimensional closed pathway using the Luo-Rudy dynamic (LRd) model
of a mammalian ventricular myocyte (28, 39, 43, 46). This
model accounts for dynamic intracellular concentration changes of
Na+, Ca2+, and K+ during
excitation. It also incorporates the effects of such changes on
transmembrane currents that determine the AP, including currents through ion channels, pumps, and exchangers. The aims of this study are
to examine: 1) the response of the reentrant AP to varying degrees of head-tail interaction and 2) the effect of ion
accumulation on this relationship.
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METHODS |
The reentry pathway.
By connecting individual LRd ventricular cell models (28, 39, 43,
46) with passive resistances to represent gap junctions (38), a ring of cells is created (33) that
represents a closed pathway in the heart (a schematic is shown in Fig.
1). Simulations were conducted for
gap-junction conductances (gj) of 0.076 and 0.285 mS. The simulated behavior of interest, namely the effect of ion
accumulation on reentry dynamics, did not depend on the particular
value of gj. Results for
gj = 0.076 mS are presented unless
otherwise stated. With this choice, we could use a shorter pathway and
significantly reduce computing time. Also,
gj = 0.076 mS produced slowed conduction,
which is a prerequisite for sustained reentry and is observed in the
border zone of a healing infarct (10) and during ischemia
(38). Computations converged for the entire range of
parameter values used with the chosen spatial and temporal
discretization steps [
x = 100 µm; t
varies from 0.005 to 1.0 ms (38)].
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Fig. 1.
Dynamic Luo-Rudy (LRd) cell models are connected by
intercellular resistive pathways (gap junctions) with conductance
(gj) to form a closed pathway of reentry. Head
and tail of reentrant action potentials (AP) are indicated. Below the
reentry pathway a schematic of the LRd cell model is shown with various
ion channels, pumps, and exchangers represented in the model:
Na+ channel current (INa); L-type
Ca2+ current [ICa(L)];
Na+-Ca2+ exchange current
(INaCa); Na+-K+ pump
current (INaK); slow delayed-rectifier
K+ current (IKs); junctional
sarcoplasmic reticulum (JSR); and network SR (NSR). LRd model details
are provided in the literature (28, 39, 43, 46). This
model code can be downloaded from the research section of Web site
http://www.cwru.edu/med/CBRTC. See this Web site for additional
definitions of the current abbreviations.
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Once reentry has been initiated in the closed pathway, the path length
(L) is reduced by removing 10 cells in the plateau phase of
their respective APs, similar to the protocol used by Vinet and Roberge
(42). Propagation is allowed to reach a steady state,
after which 10 additional cells are removed. This process is repeated
until the pathway is sufficiently small such that oscillations in APD
are persistent (last more than 10 revolutions). After the onset of
oscillations in AP properties, cells are removed two at a time to
achieve greater spatial resolution. Close to the point of termination,
cells are removed one at a time. In the case of transient oscillations,
the system is considered to be in steady state once APD changes by less
than 1%. In the case of persistent oscillations, two different
protocols are used to determine steady state. In the first protocol,
propagation is defined to be in steady state when APD oscillations
maintain the same pattern for at least 10 revolutions ("short steady
state"). In the second protocol, steady state is declared when an
oscillatory pattern repeats itself for at least 100 revolutions
("long steady state").
The APD is the time between the AP upstroke and 90%
repolarization. The peak Ca2+ transient concentration
([Ca2+]i,peak) is defined as the maximum
value of the Ca2+ transient following the AP upstroke.
Diastolic interval (DI) is measured as the time between 90%
repolarization and the next upstroke. In the presence of APD
oscillations, DI and APD are recorded around the pathway for several
revolutions to create the APD restitution curve (APD as a function of
the previous DI). In the absence of oscillations, the restitution curve
is created by applying a premature stimulus on the tail of the
reentrant AP, which induces transient oscillations in APD and DI
(18). These data are then used to create the APD
restitution curve. CL is measured as the time between two successive
upstrokes at a particular cell. During oscillations, intermediate APD
(APDmed), DI (DImed), and
[Ca2+]i,peak
([Ca2+]i,peak,med) for a single cell are the
midpoints between the maximum and minimum values.
In certain simulations, we use a [Na+]i clamp
protocol in which [Na+]i is held at a fixed
value and prevented from accumulating as reentry continues. In another
simulation, we uncouple cells (set gj = 0) away from the AP upstroke.
The region of coupled cells and the AP wave front propagate together in
the reentry pathway during this simulation. This allows us to minimize
the effects of electrical loading on the dynamics during sustained reentry.
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RESULTS |
The regime of stable AP behavior.
Figure 2 shows APD (A),
[Ca2+]i,peak (B), and CL
(C) as functions of L for L < 150 cells. The data shown are the steady-state values recorded at each
L during the short steady-state protocol. Below a certain
L (the bifurcation point, Lcriti,
marked with an arrow in Fig. 2A), significant head-tail
interaction occurs, and the nonlinear properties of the system become
apparent as all AP properties (APD,
[Ca2+]i,peak, CL, DI, and ) display
complex oscillatory behavior (DI and are not shown). The shaded
region in Fig. 2 indicates the region of oscillatory behavior. For
L > Lcrit, reentry is stable (oscillations in AP properties are transient and subside within 10 revolutions). We begin by discussing the dependence of AP properties on
L in the stable regime.
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Fig. 2.
A: AP duration (APD); B: peak
intracellular Ca2+ transient concentration
([Ca2+]i,peak); C: cycle length
(CL) as functions of length of reentry pathway (L). These
data are the result of a protocol requiring an oscillatory pattern to
repeat itself for at least 10 revolutions before being defined as
steady state ("short steady state"). Shaded region corresponds to
where oscillations in AP properties occur. Bifurcation point
(A) (Lcrit) = 80 cells in this
protocol and is marked with an arrow. In the case of transient
oscillations (L > Lcrit), last
value observed before reducing L is plotted. For persistent
oscillations (L < Lcrit),
values during last 10 cycles before reducing L are
plotted.
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CL is the time it takes for the reentrant wave front to propagate once
around the pathway. As L becomes shorter (for
L > Lcrit), the wave front
takes less time to travel around the reentry pathway and CL decreases
(Fig. 2C). A smaller CL implies that every cell in the
reentry pathway is being stimulated at a more rapid rate, which results
in reduced APD (Fig. 2A) and increased
[Ca2+]i,peak (Fig. 2B). These
rate-dependent AP changes are manifestations of rate-adaptation
cellular processes that are well understood (43, 46).
Time dependence of the critical L for bifurcation.
For L < Lcrit, APD, DI,
[Ca2+]i,peak, CL, and begin to oscillate.
In Fig. 2, corresponding to the short steady-state protocol, Lcrit = 80 cells. However, if reentry in an
80-cell pathway continues for another 400 revolutions, the stable
beat-to-beat APD alternans (2:2 stimulus-response ratio) observed after
50 revolutions disappears. In Fig. 3
(shown on a scale of only 70 cells to expand the region of oscillatory
behavior), the long steady-state protocol is implemented. In this
figure, Lcrit = 64 cells compared with 80 cells in Fig. 2. Note in Fig. 3 the division of the oscillatory region
into subregions with distinct oscillatory patterns. (These patterns are
discussed in Figs. 7 and 8.) With models that do not account for
dynamic changes in intracellular ion concentrations (e.g., Beeler-Reuter; see Ref. 4), AP properties begin
oscillating at a well-defined (fixed) Lcrit that
does not decrease with time (8, 21, 34, 42). The LRd model
used in this study accounts for dynamic intracellular ion concentration
changes that occur when cells are stimulated rapidly during reentry.
With these physiological processes, Lcrit shows
strong time dependence (compare Figs. 2 and 3).
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Fig. 3.
AP oscillatory region in greater detail: APD
(A); [Ca2+]i,peak (B);
and CL (C) as functions of L during a protocol
that requires an oscillatory pattern to repeat itself for at least 100 revolutions before steady state is declared ("long steady state").
This protocol produces oscillations in AP properties at
L = 64 cells. Regions of periodic and quasi-periodic
behavior are labeled in A.
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Time dependence of oscillations for a relatively large L (weak
head-tail interaction) and ionic mechanism.
Insight into the time-dependent processes responsible for damping of
oscillations and reduction of Lcrit is provided
in Figs. 4 and
5. Figure 4, left, shows
APD restitution curves during reentry in a pathway with
L = 80 cells. Figure 4, right, shows
corresponding APD values recorded at every cell in the pathway during
one revolution of reentry. Oscillations in APD persist after 150 revolutions (Fig. 4A, right). The corresponding
APD restitution curve (Fig. 4A, left) has a slope
(m) whose maximum value (mmax) is
>1, which agrees with theoretical and experimental observations that
mmax 
1 for oscillations in APD to occur
(8, 18, 21, 24). After 950 revolutions (Fig.
4B), the oscillations in APD cease (right), and
mmax < 1 (left). In addition,
[Na+]i (not shown) accumulates from 18 to 21 mM, in agreement with experimental findings that
[Na+]i increases by about 30% during rapid
pacing (7). Furthermore, APDmed decreases from
75.6 to 71.8 ms, and DImed increases from 26.4 to 30.4 ms
over the course of 800 revolutions.
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Fig. 4.
Time dependence of oscillations in a relatively large
pathway (L = 80 cells). APD restitution curves
(left) and APD along reentry pathway for one revolution
(right) are calculated after 150 revolutions (A),
950 revolutions (B), and 980 revolutions with intracellular
[Na+] ([Na+]i) reset to value
after 150 revolutions (C). A line with slope (m) = maximal
slope value (mmax) is shown adjacent to each
restitution curve (left).
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Fig. 5.
Ionic mechanism of time dependence of APD restitution
curve and Lcrit: APs (A);
INaCa (B);
INaK (C), IKs
(D), and ICa(L) (E). Data
were recorded after 150 revolutions (solid line) when
oscillations were present and after 950 revolutions (dashed
line) when APD was constant from beat to beat (same conditions as
in Fig. 4, A and B, respectively).
[Na+]i increased from 18 mM after 150 revolutions to 21 mM after 950 revolutions (not shown). Arrow in
E indicates larger ICa(L)
corresponding to shorter APD in A. Vm, membrane potential.
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[Na+]i accumulation dampens oscillations
after 950 revolutions through the following mechanism. The point about
which APD and DI oscillate (the operating point, discussed further in
Fig. 10) is determined by the intersection of the line APD = 
DI + CL and the APD restitution curve (31).
[Na+]i accumulation decreases APD (shifts the
restitution curve downward, Fig. 4B) relative to control
(Fig. 4A). Assuming CL remains constant (a good assumption
because CL changes are very small relative to changes in APD and DI), a
downward displacement of the restitution curve (decrease in
APDmed) shifts the operating point to a larger DI where
mmax < 1, and oscillations disappear.
Evidence for this is given in Fig. 4C, where
[Na+]i during reentry after 950 revolutions
is reset to its value after 150 revolutions. When
[Na+]i is decreased in such a manner,
APDmed increases from 71.8 to 76.2 ms, and
DImed decreases from 30.4 to 26.7 ms. Consequently, mmax > 1 (left), and
oscillations in APD resume (right).
To provide insight into the ionic mechanism by which APD decreases with
time and oscillations cease, we compare APs (Fig. 5A) and
ionic currents (Fig. 5, B-E) observed at a
single cell after 150 revolutions (solid line) to those
observed after 950 revolutions (dashed line). Accumulation
of [Na+]i after 950 revolutions increases the
reverse-mode activity (Na+ extrusion with
3Na+-1Ca2+ stoichiometry) of the
Na+-Ca2+ exchanger INaCa
(Fig. 5B). This results in an increase of an outward
repolarizing current. Similarly, elevated
[Na+]i increases the repolarizing
Na+-K+ pump current
(INaK), which has a
3Na+-2K+ stoichiometry (Fig. 5C). As
a consequence of this increase in the total repolarizing current
secondary to [Na+]i accumulation,
APDmed decreases (downward shift of the APD restitution curve).
The slow component of the repolarizing delayed rectifier K+
current (IKs) (Fig. 5D) is reduced
after 950 revolutions and therefore does not contribute to APD
shortening with time. The reduction of IKs is
the result of an increase in DI secondary to shortening of APD, which
allows IKs more time to deactivate after its
activation. The depolarizing L-type Ca2+ current
[ICa(L)] (Fig. 5E) does
not contribute to the downward shift of the restitution curve either.
This is apparent in Fig. 5E, which shows after 950 revolutions a larger ICa(L) (arrow)
during the plateau of a shorter AP (Fig. 5A).
It is important to note that in addition to
[Na+]i accumulation, there is also
[Ca2+]i accumulation
([Ca2+]i,peak,med increases from 2.24 to 2.62 µM after 800 revolutions, not shown). When
[Na+]i accumulation is prevented,
[Ca2+]i does not accumulate either, making it
difficult to state whether [Na+]i
accumulation leads to [Ca2+]i accumulation or
vice versa. However, Ca2+-sensitive currents in the model
[IKs,
ICa(L)], and forward-mode INaCa do not contribute to the downward shift of
the restitution curve. Furthermore, resetting
[Na+]i is sufficient to reverse the
time-dependent change in dynamics. These results suggest that
[Na+]i accumulation is primarily responsible
for the downward shift of the restitution curve with time.
Temporal oscillations in AP properties at a single site.
Once bifurcation occurs (L < Lcrit), oscillations in AP properties measured
at a single cell (the temporal domain) may assume a number of
complicated patterns. Figure 6,
left, shows temporal oscillations in APD (Fig.
6A), DI (Fig. 6B), and CL (Fig. 6C). The theoretical oscillatory patterns agree with experimental
measurements (Fig. 6, right) conducted by Frame and Simson
(18) in the canine tricuspid orifice ring as well as with
theoretical studies performed in other ring models (8, 21,
42). Importantly, the maximal percent changes in APD and DI
(33% and 153%, respectively) during oscillations are much greater
than the maximal percent change in CL (6%). Furthermore, APD and DI
oscillations are 90° out of phase relative to the CL oscillations
(i.e., APD and DI oscillations are maximal where CL oscillations are
minimal, as indicated by arrows in Fig. 6).
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Fig. 6.
Temporal oscillations in APD (A), diastolic
interval (DI) (B), and CL (C) recorded at a
single site in a 154-cell reentry pathway. Quasi-periodic temporal
behavior is observed that agrees with experimental recordings shown at
right (18). Maximum degree of CL oscillations
(indicated with arrow in C) occurs where
oscillations in APD and DI are at a minimum (arrows,
A and B, respectively). Conductance
gj = 0.285 mS was used in these
simulations.
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Examples of temporal oscillations in APD from the short steady-state
(Fig. 2) and the long steady-state (Fig. 3) protocols are shown in Fig.
7 and may be classified as either
periodic or quasi-periodic. Only APD (left) and
[Ca2+]i,peak (right) are shown,
but other AP properties (DI, , and CL) display a similar behavior. A
periodic 2:2 behavior (stable beat-to-beat alternans) appears in Fig.
7A, and a 5:5 periodicity (pattern repeats itself every five
beats) is shown in Fig. 7B. These oscillatory patterns agree
with patterns observed experimentally (Fig. 7, A and
B, bottom) (14, 18). An example of
quasi-periodic behavior, which is characterized by a pattern that
repeats itself only approximately, is shown in Fig. 7C.
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Fig. 7.
Different temporal oscillatory behavior observed at a
single site during reentry. APD (left) and
[Ca2+]i,peak (right) oscillations
were recorded from pathways with three different L values:
A: beat-to-beat AP alternans in a 70-cell reentry pathway;
B: 5:5 periodic behavior (pattern repeats every five beats)
in a 61-cell pathway [*APA (APD = 77 ms
and [Ca2+]i,peak = 3.4 µM) and
APB (APD = 76.8 ms and
[Ca2+]i,peak = 3.6 µM)]; and
C: quasi-periodic behavior in a 60-cell pathway. Short
steady-state protocol is used in A; long steady-state
protocol is used in B and C. Experimental CL
oscillations are shown for comparison in A (18)
and B (14). In these simulations
gj = 0.076 mS was used.
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The 2:2 pattern in Fig. 7A is an example of periodic
behavior observed during the short steady-state protocol (Fig. 2). The 5:5 pattern in Fig. 7B comes from the "periodic window"
observed during the long steady-state protocol (Fig. 3A).
Figure 7C (also from the long steady-state protocol)
contains a quasi-periodic pattern from the "quasi-periodic region"
located to the left (smaller L) of the periodic window in
Fig. 3A. As L is reduced, the oscillatory behavior passes through a quasi-periodic regime that is interrupted by a small window of periodicity at L = 61 cells (Fig.
3). We discuss the factors that contribute to the formation of these different patterns in the next section.
Interestingly, for a relatively low degree of head-tail interaction
(Fig. 7A) the maximal percent change in
[Ca2+]i,peak (94%, right) during
oscillations is much greater than the maximal percent change in APD
(17%, left). This difference disappears as the degree of
head-tail interaction increases (Fig. 7, B and
C). Furthermore, [Ca2+]i,peak
oscillations appear only in the presence of APD oscillations, and in
general the two are in phase with the larger Ca2+ transient
corresponding to the longer AP. An exception to this in-phase
relationship is observed during the 5:5 periodic pattern shown in Fig.
7B. The AP labeled A (left) is
slightly longer than the AP labeled B (77.0 ms compared with
76.8 ms); however, A has a smaller peak Ca2+
transient (right) than B (3.43 µM compared with
3.65 µM). Larger variations in APD and
[Ca2+]i,peak during the same 5:5 sequence are
in phase.
Spatial oscillations of AP properties.
To understand the different temporal oscillation patterns presented in
Fig. 7, it is helpful to consider how AP properties change in space
along the entire pathway (spatial domain) during reentry. Figure
8 shows CL as a function of location for
consecutive revolutions of the reentrant wave front under the same
circumstances as in Fig. 7. Importantly, Fig. 8 reveals that spatial
oscillations in CL are sinusoidal during reentry. Spatial oscillations
in APD, DI, , and [Ca2+]i,peak are also
sinusoidal (not shown). The wavelength of spatial sinusoidal variation
of AP properties is referred to as the oscillation wavelength (
).
The 2:2 temporal behavior in Fig. 7A is the result of a
spatial oscillation pattern with /L = 2/7 (Fig.
8A). In this case, two revolutions accommodate an integer
multiple of (2L = 7), which gives rise to
beat-to-beat alternans observed temporally at a single site (in Fig. 8,
values recorded at a single site for consecutive revolutions are marked
with an x).
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Fig. 8.
Spatial CL pattern along the reentry pathway during
consecutive revolutions in a 70-cell (A), 61-cell
(B), and 60-cell (C) pathway corresponding to
conditions and protocols in Fig. 7. L is marked by circles
at bottom of each figure, and wavelength () is shown above each
spatial CL oscillatory pattern. X, CL values at a single site for
consecutive revolutions.
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Figure 8 also shows spatial oscillatory patterns from the 5:5 periodic
(Fig. 8B) and quasi-periodic (Fig. 8C)
oscillatory regions in Fig. 3. In Fig. 8B,
/L = 5/3 such that 5 revolutions accommodate an
integer multiple of (5L = 3), which produces the
5:5 periodic pattern at a single site (Fig. 7B). For a
shorter L (Fig. 8C), the magnitude of CL
oscillation increases, and /L decreases to slightly less
than 5/3. Now five revolutions fit slightly more than 3, producing a
shift in the spatial oscillation pattern every revolution, which
generates the quasi-periodic pattern at a single site (Fig.
7C).
Two factors are responsible for the formation of different temporal
oscillatory patterns. First, increased head-tail interaction increases
CL oscillations and the complexity of the spatial oscillation pattern
(which accounts for the difference between Fig. 7, B and C). Second, changes with time (the difference
between Fig. 7, A and B), a phenomenon that is
discussed in the next section.
Time dependence of oscillations for relatively small L (strong
head-tail interaction).
The of the spatial oscillation pattern from a 70-cell pathway
(relatively small L) increases from 0.29L (Fig.
9A, shown previously in Fig.
8A) after 150 revolutions to 0.39L after another 760 revolutions (Fig. 9B). Simultaneously,
[Na+]i increases from 19.7 to 21.8 mM (not
shown), and the APD restitution curve shifts downward (Fig.
9B). In contrast to the decrease of mmax observed in Fig. 4,
mmax increases from 1.31 to 1.51 in response to
this downward shift. Consequently, the difference (APD) between the
maximum value of APD (APDmax) and the minimum value of APD (APDmin) increases from 12.4 to 16.3 ms, reflecting an
increase in the magnitude of oscillations. Note that the data in
Fig. 4 come from a relatively long pathway (L = 80 cells) in the presence of weak head-tail interaction. In contrast, Fig.
9 corresponds to a relatively short pathway (L = 70 cells) where there is strong head-tail interaction.
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Fig. 9.
Time dependence of oscillations in a relatively small
reentry pathway (L = 70 cells). APD restitution curves
(left) and APD along reentry pathway for 1 revolution
(right) are calculated after 100 revolutions (A)
and after another 760 revolutions (B).
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We provide the following hypothesis for the mechanism by which
[Na+]i accumulation increases oscillations
and in a relatively short pathway. As discussed previously,
[Na+]i accumulation decreases APD at every DI
(downward shift of the restitution curve; Fig. 9), shifting the
operating point to a larger DI. At very short DIs (high degree of
head-tail interaction), the APD restitution curve begins to flatten
(decreased m) as DI decreases. Consequently, a slight shift
of the operating point to a larger DI increases m and the
magnitude of oscillations. The result is an increase in APD of the
spatial APD oscillation pattern, which increases spatial gradients of
the membrane potential (Vm) along the reentry
pathway. However, electrotonic interaction between neighboring cells
limits the possible steepness of these gradients and therefore limits
APD. As [Na+]i accumulation increases
APD (Fig. 9) and the limit on Vm gradients imposed by electrotonic interaction is reached, increases (Fig. 9B). This increase in allows for APD to change from
APDmax to APDmin over a longer distance,
reducing the steepness of spatial gradients.
Support for our hypothesis is provided by the fact that if
[Na+]i is held constant during reentry, and CL oscillations do not change with time. If we allow
[Na+]i to accumulate but reduce the effect of
electrotonic interaction by reducing intercellular coupling (from 0.076 to 0 mS) away from the AP upstroke, CL oscillations increase slightly
with [Na+]i accumulation (from 1.5 to 2.4 ms)
but does not change. It is important to note that if we allow both
[Na+]i and to increase with time
(intercellular coupling increased back to 0.076 mS), then CL
oscillations increase to 3.3 ms (compared with 2.4 ms for shorter ),
suggesting that the increase of itself increases oscillations.
The increase in oscillations with [Na+]i
accumulation for a high degree of head-tail interaction is dependent on
flattening of the APD restitution curve for short DIs. Experimental
studies have shown a similar flattening of the APD restitution curve
for short DIs and have attributed it to increased latency between the
stimulus and the AP upstroke (6, 24). In our study, the restitution curve flattens because for very short DIs the
Ca2+ transient begins to decrease significantly, which is
in agreement with experimental observations (26). A
smaller peak Ca2+ transient results in reduced
Ca2+-dependent inactivation of
ICa(L) (not shown) and therefore an increase in
depolarizing current during the plateau of the AP, which acts to oppose
APD shortening.
Elevated [K+]o.
Experiments have shown that K+ accumulates in extracellular
clefts during rapid pacing (23). To investigate the effect
of extracellular K+ concentration
([K+]o) accumulation on the dynamics of
reentry, we increase [K+]o from 4.5 mM
(control) to 7 and 12 mM in a 78-cell pathway. In the control
([K+]o = 4.5 mM), oscillations during
reentry disappear within 600 revolutions. When
[K+]o is increased to 7 mM, the resting
Vm depolarizes from 83 to 76 mV, and
oscillations persist for over 1,000 revolutions despite a decrease of
APDmed from 72 to 63 ms (not shown). If
[K+]o is elevated instead to 12 mM, the
resting Vm depolarizes to 65 mV, and reentry
terminates within one revolution. Thus elevated [K+]o acts to enhance oscillations and
destabilize reentry.
| |
DISCUSSION |
Summary of findings.
Important findings of this study are: 1) a high degree of
head-tail interaction produces oscillations in AP properties and in the
Ca2+ transient; 2) the Ca2+
oscillations are generally in phase with APD oscillations;
3) the magnitude of Ca2+ oscillations is greater
than the magnitude of APD oscillations for moderate degrees of
head-tail interaction; 4) [Na+]i
accumulation shifts the APD restitution curve downward to smaller APD
values which, depending on the degree of head-tail interaction, may
either stabilize or destabilize reentry; 5) for a relatively low degree of head-tail interaction, accumulation of
[Na+]i tends to stabilize the dynamics of the
reentrant AP; 6) for a higher degree of head-tail
interaction, [Na+]i accumulation is
accompanied by increased wavelength of spatial oscillations, which
augments CL oscillations; and 7) elevated [K+]o increases oscillations and destabilizes reentry.
Changes in [Na+]i alter
the dynamics of reentry.
We find that the behavior of reentry evolves in time due to
accumulation of [Na+]i. Previous experimental
and theoretical studies have shown that the APD restitution curve
shifts downward (to smaller APD values) in response to an increase in
pacing rate (5, 46). It has been proposed that the
mechanism for this downward shift is either [Ca2+]i accumulation (5, 46) or
[K+]o accumulation (5). In this
study we observe that while [Ca2+]i
accumulates along with [Na+]i,
[Na+]i is the primary cause of APD shortening
and the downward shift of the restitution curve. This downward shift
may shift the point about which APD oscillates (the operating point) to
a less steep (Fig. 4) or more steep (Fig. 9) portion of the restitution
curve, depending on the degree of head-tail interaction. Therefore,
this shift may either suppress or augment AP oscillations.
A schematic is used in Fig. 10 to
summarize the effects of [Na+]i accumulation
on the APD restitution curve and the dynamics of reentry for low and
high degrees of head-tail interaction. The APD restitution curve in the
presence of [Na+]i elevation (dashed
line) is shifted downward relative to the control curve
(solid line). The operating point about which APD and DI
oscillate is determined by the intersection of the line APD = DI + CL (dashed-dotted line) with the restitution
curve. For a low degree of head-tail interaction
(CLA), the operating point occurs where
m = 1 in the control case but where m < 1 in the presence of elevated [Na+]i,
which dampens oscillations. For a high degree of head-tail interaction
(CLB < CLA), the
operating point in the presence of elevated
[Na+]i occurs where the slope is steeper than
in the control case, which increases the amplitude of oscillations. The
schematic in Fig. 10 illustrates the differential effect
[Na+]i accumulation has on the dynamics of
reentry depending on the degree of head-tail interaction in the
pathway.
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|
Fig. 10.
Effect of [Na+]i accumulation
on dynamics of reentry in the presence of weak head-tail interaction
(CLA) and strong head-tail interaction
(CLB) (CLB < CLA). Control APD restitution curve (solid
line) and restitution curve in the presence of elevated
[Na+]i (dashed line) are shown.
Operating point is defined as the intersection of line APD = DI + CL (dashed-dotted line) and the restitution
curve. Circle on each restitution curve indicates where slope = 1. For an operating point located right (larger DI) of the circle, no
oscillations occur because the slope of the restitution curve is <1.
|
|
Accumulation of [Na+]i is known to promote
spontaneous activity, which may trigger an arrhythmia (12,
35). Our results illustrate how [Na+]i
accumulation may affect the time course of an arrhythmia once initiated. This effect can be stabilizing or destabilizing, depending on the degree of head-tail interaction of the circulating AP. For
relatively slow tachycardias (weak head-tail interaction), the effect
may be stabilization of reentry due to a shift of the operating point
to a less steep portion of the restitution curve. In contrast,
[Na+]i accumulation may act to destabilize
reentry during rapid tachycardias (strong head-tail interaction)
leading to either its termination or transition to fibrillation.
Longer leads to greater CL oscillations.
We observe that [Na+]i accumulation with time
produces spatial oscillation patterns with different s. The
existence of different s has been predicted by Courtemanche, Glass,
and Keener (8), who proposed a delay-difference equation
to describe the evolution of AP properties as the reentry wave front
propagates. They solved for all values (eigenmodes, Eq.
1), which yield solutions to the delay-difference equation once
bifurcation occurs
|
(1)
|
where 
is proportional to the slope of the dispersion relation
( as a function of DI) and k is the mode number. Because k can assume any positive integer value, an infinite number
of oscillatory modes can occur in the delay-difference model. The spatial oscillation patterns with different shown in Fig. 8 correspond to different eigenmodes in this equation. The implication of
the possible existence of many oscillatory modes is discussed in the
following text.
In our simulations, we observe that a longer /L promotes
CL oscillations (Fig. 9 and corresponding text). Fig.
11 presents two hypothetical spatial
oscillation patterns in 1/ (the reciprocal of conduction velocity)
to explain this behavior. The pattern in Fig. 9A has a
relatively long relative to L, whereas that in
Fig. 9B has a short . It is assumed that the magnitude of the oscillations in in Fig. 9A and B, is the
same and that the only difference is in . CL for one revolution is
given by
|
(2)
|
where x is the current position of the wave front. The
CL calculated in Fig. 9A and B, is therefore
greater than the intermediate CL (corresponding to no oscillations with
a fixed = med) by an amount equal to the area
of the shaded region. The larger shaded area in Fig.
9A implies that a spatial oscillation pattern with a longer
augments CL oscillations.
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|
Fig. 11.
Mechanism by which a spatial oscillation pattern with a
longer yields greater CL oscillations. Reciprocal of conduction
velocity, 1/, is plotted around the reentry pathway for one
revolution. Spatial oscillation patterns with long (A)
and short (B) are presented. CL in each panel is area
beneath the curve, and shaded area represents the amount by which CL is
greater than the intermediate CL value.
|
|
Experiments have shown that the existence of CL oscillations favors
spontaneous termination of reentry (14, 17, 18, 32). In
our study, we find that [Na+]i accumulation
with time increases of a spatial oscillation pattern, which in turn
augments CL oscillations (Fig. 9). This result suggests that
lengthening of with time (transition from one eigenmode to another)
could be one mechanism by which reentry spontaneously terminates.
Furthermore, it has been shown that AP oscillations in a
one-dimensional ring occur for the same rotation periods as does
spiral-wave breakup in two-dimensional models, a phenomenon that
produces a disordered fibrillatory state (21, 44). This
suggests that lengthening of with time and the consequent increase
in oscillations may promote spiral-wave breakup and could be one
mechanism for the transition from ventricular tachycardia to fibrillation.
Effect of elevated
[K+]o on the dynamics of
reentry.
A previous study from our laboratory (39) showed that
elevated [K+]o, by depolarizing the resting
membrane potential, caused prolonged postrepolarization refractoriness
as a result of prolonged recovery from inactivation of
INa following an AP. During reentry, this phenomenon acts to prolong the tail of refractoriness and reduced excitability that follow the AP; thus the degree of head-tail interaction increases despite APD shortening. In this study, we observe
that a moderate elevation of [K+]o promotes
AP oscillations, although a more severe elevation leads to termination
of reentry. The simulated values of elevated [K+]o (7 and 12 mM) are representative of
hyperkalemia during acute ischemia (39). The results
suggest that hyperkalemia may have a destabilizing effect on reentry
and may contribute to its termination or its disintegration into
multiple reentrant circuits and a fibrillatory state.
Electrical and mechanical oscillations.
The LRd model computes both the AP and the Ca2+ transient.
This enables us to study oscillations in APD and
[Ca2+]i for different degrees of head-tail
interaction. Studies suggest that T wave alternans in the
electrocardiogram reflects APD oscillations and indicates an increased
risk for cardiac arrhythmia and sudden death (20, 37, 40).
Similar to APD alternans, beat-to-beat alternation in the force
generated by cardiac muscle (mechanical alternans) has also been
observed at rapid rates (25), and intracellular Ca2+ cycling causing [Ca2+]i
oscillations has been suggested as a mechanism (22, 25). Some studies have hypothesized that mechanical alternans generates electrical alternans (20, 25, 41), and others have
proposed the converse to be true (27). The hypothesis that
mechanical alternans causes electrical alternans is supported by
observations of mechanical alternans in the absence of electrical
alternans (41). In our study oscillations in APD and
[Ca2+]i,peak always appear together. However,
over a certain range of head-tail interaction, percent changes in
[Ca2+]i,peak during oscillations are much
larger than percent changes in APD, which may explain why mechanical
alternans has been observed without detection of electrical alternans.
In general, APD and the Ca2+ transient oscillate in phase
(a large Ca2+ transient corresponds to a long APD).
However, occasionally an out-of-phase relationship (a large
Ca2+ transient with a short APD) may occur (Fig.
7B).
Study limitations. The limitations of this study were the
following. The ring model in this study is used to investigate an important property of reentrant excitation, namely the interaction between the head and tail of the reentrant AP. The center (core) of the
reentry pathway in this model of "anatomical reentry"
(16) is inexcitable and represents an anatomical obstacle.
In functional forms of reentry, such as leading-circle or anisotropic
reentry (2, 10) or during spiral-wave activity
(13), the core is excitable. In the leading-circle
concept, the core is created by the collision of centripetal wavelets
generated by the reentrant wave front propagating along the smallest
possible pathway (the "leading circle") (2). Although
the nature of the core is different, important properties of
leading-circle reentry are represented in the ring model. For example,
there is no excitable gap in leading-circle reentry because of a high
degree of head-tail interaction (2), as is the case for
reentry in a small pathway in our simulations.
Another limitation of our model is the absence of heterogeneities in
cell properties [e.g., midmyocardial M cells (43)] or
tissue architecture [e.g., anisotropy due to fiber orientation (10)]. Although such inhomogeneities will affect the
oscillatory patterns observed during reentry, the principles
established here regarding the dynamics of head-tail interaction and
its time dependence can provide insight into these phenomena in the
inhomogeneous myocardium.
In the spiral-wave concept, a high degree of curvature at the tip of
the reentrant wave front creates a source-sink mismatch that leaves a
central core of unexcited but excitable tissue (13). The
absence of an unexcited but excitable core in our model is significant
because such a core influences the reentrant AP via electrotonic
interaction (3). Nevertheless, the phenomenon of head-tail
interaction explored here occurs along the arm of a spiral wave
(21). Furthermore, in many instances, a drifting spiral
wave anchors to an anatomical obstacle, such as a small artery
(9). In such situations, the distinction between
anatomical and spiral-wave reentry becomes less obvious.
The use of a one-dimensional model of reentry allowed us to use a
detailed model of the cardiac cell that accounts for dynamic changes of
ionic concentrations (e.g., [Na+]i
accumulation) over many reentrant cycles. With this approach, we could
characterize the temporal evolution of the AP and the dynamics of
reentry. The focus of this study is on head-tail interaction, AP
oscillations (alternans), and stability of reentrant activity. Other
properties of reentrant excitation, such as the ionic activity in the
core of a spiral wave or the effects of anisotropy, must be studied
using higher-dimensional models. The insights obtained from the present
study can guide such simulations where the use of a detailed cell model
during many reentry cycles still constitutes a major computational
challenge. Moreover, the time scale investigated in this study is still
relatively short. It allows for ion accumulation to occur but does not
take into account changes due to electrical remodeling (1, 19,
45) that occur over a much longer time frame. Remodeling
involves changes in ion channels, gap junctions, and tissue structure.
Many of these changes can be incorporated within the framework of the
detailed cell model used here and will be an important subject of
future modeling studies.
| |
ACKNOWLEDGEMENTS |
This study was supported by Grants R01-HL-49054 and R37-HL-33343
(to Y. Rudy) from the National Heart, Lung, and Blood Institute and by
a Whitaker Foundation Development Award.
| |
FOOTNOTES |
Address for reprint requests and other correspondence: Yoram
Rudy, Cardiac Bioelectricity Research and Training Center, 319 Wickenden Bldg., Case Western Reserve Univ., Cleveland, OH 44106-7207 (E-mail: yxr{at}po.cwru.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 29 November 1999; accepted in final form 21 April 2000.
| |
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ABSTRACT
INTRODUCTION
