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1 Institute of Biomedical and Life Sciences, Glasgow University, Glasgow G12 8QQ, United Kingdom; 2 Center for Interdisciplinary and Complex Systems, Northeastern University, Boston, Massachusetts 02115; and 3 Medizinische Klinik der Universität Würzburg, 97080 Würzburg, Germany
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ABSTRACT |
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 TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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Decreasing the slope
of the dynamic, but not conventional, restitution curves is
antifibrillatory. Cardiac memory/accommodation underlies the
difference. We measured diastolic interval (DI) and action potential
duration (APD) in epicardial, endocardial, and Purkinje tissue from
eight dogs. Consecutive 100-stimulus trains were given to study
transitions between basic cycle lengths (BCL) ranging from 400 to 1,300 ms. (DI,APD) pairs aligned immediately on the line DI + APD = BCL (64/67) or oscillated (3/67). The shifting effect of up to 10 extrastimuli on restitution curves was also measured. These curves were
fit with the equation APD = 
+ 
exp(
DI/
), where
is asymptote, is drop, and is time constant. Linear
regression of the parameters against the number of extrastimuli showed
that premature and postmature stimuli decreased and increased and
and increased and decreased , respectively. Analysis of a
mathematical model treating memory as an exponentially decreasing shift
of restitution curves shows that oscillatory DI,APD is expected with
large 
BCL, steep restitution slope, or increased cardiac accommodation. The model explains phase shifts and suggests a common
mechanism for Purkinje and myocardial electrical alternans.
arrhythmias; dynamic restitution; short-term memory; modeling
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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SUDDEN CARDIAC DEATH claims over 300,000 lives in the United States each year (53) despite the invention of the implantable cardiac defibrillator and improvement in antiarrhythmic drugs. A better understanding of the mechanisms of ventricular tachyarrhythmias is still needed for prediction and treatment of sudden cardiac death. Modeling studies predict that a steeply (>1) sloped action potential duration (APD) restitution curve should produce breakup of spiral waves into ventricular fibrillation, whereas a shallow slope should prevent ventricular fibrillation (23, 36, 37). Studies from patients with coronary artery disease (9, 52), and some animal studies (25, 51) show that the maximum slope of the restitution curve can be much less than 1. If that were always true, ventricular fibrillation would never occur. A recent experiment designed to study this paradox demonstrated that the slope of the "dynamic" restitution curve, which is the relationship between APD and preceding diastolic interval (DI) measured during rapid pacing or during ventricular fibrillation, has a slope greater than that measured by the standard S1S2 protocol (25). The slope of the dynamic restitution curve has also been found to correlate with tachycardia stability. For example, verapamil, which was observed to convert ventricular fibrillation to ventricular tachycardia experimentally (49, 44), decreases the slope of the dynamic restitution curve, whereas procainamide, which decreases the slope of the standard restitution curve, has no effect on ventricular fibrillation (39). The difference between the two types of restitution curves is due to cardiac memory. Cardiac memory increases the effective slope of the restitution function (15). It has also been shown that spiral wave breakup can be induced in a mathematical model with flat standard restitution curves if memory is included (10). These results point to a critical role for cardiac memory in the stability and perpetuation of ventricular arrhythmias.
However, cardiac memory has not been measured systematically for
several reasons. First, memory implies that the entire past activation
history of cardiac tissue determines a single APD of interest. It is
difficult to quantify history by a single value. In contrast, APD
dependence on preceding DI, i.e., restitution, is easily quantified by
varying the coupling interval of a premature stimulus. Second, the
significance of cardiac memory in arrhythmogenesis had been equivocal
until the recent studies of dynamic restitution. Third, for most of the
history of cardiac electrophysiology APD was measured manually, and the
laborious nature of this task limited the number of APDs that could be
measured. The goal of this study was to explore methods for quantifying
and characterizing cardiac memory in concrete ways. The model of
cardiac memory we present is based on the concept that memory is the
amount by which restitution curves are shifted with stimulus basic
cycle length change (BCL). The model produces clear predictions
about the dynamic behavior of APD after a cycle length change and is
able to reproduce our experimental results as well as explain several
phenomena seen in the literature.
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MATERIALS AND METHODS |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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Experiments
Hearts were excised from eight adult beagle or mongrel dogs of either sex anesthetized with pentobarbital solution (86 mg/kg iv, Fatal-Plus; Votech Pharmaceuticals, Dearborn, MI) and placed in cool Tyrode solution (in mmol/l: 0.5 MgCl2, 0.9 NaH2PO4, 2.0 CaCl2, 137.0 NaCl, 24.0 NaHCO3, 4.0 KCl, and 5.5 glucose). All experimental procedures were conducted in accordance with guidelines set by the Institutional Animal Care and Use Committee of the Center for Research Animal Resources at Cornell University. Purkinje fibers (PF; n = 4), strips of endocardial tissue (n = 4), and strips of epicardial tissue (n = 3) dissected from either left or right ventricle were mounted in Plexiglas chambers and superfused with 37.0°C Tyrode solution gassed with 95% O2-5% CO2. The tissue was stimulated by bipolar platinum wire electrodes (interelectrode distance 1 mm) at twice late diastolic threshold intensity. Transmembrane potentials recorded by conventional microelectrode technique were digitized by AcqKnowledge software (version 3.2.6; Biopac Systems) at 1,000 Hz (resolution of 1 ms) and analyzed with a program written in the MatLab language (version 5.2; MathWorks). APD and DI were measured at 95% repolarization. If stimulus coupling intervals were short and action potentials arose before full repolarization, APD value of the truncated action potential was calculated from an extrapolation of phase 3 to the baseline. APD values were rounded to the nearest integer value. Nonlinear curve fitting was performed with SigmaPlot software (version 4.11; Jandel Scientific). Linear regression and statistical analyses were performed with StatView software (version 5.0; SAS Institute). P values < 0.05 were considered to be statistically significant. Values are expressed as means ± SD unless otherwise noted.Two stimulus protocols were used (Fig.
1). In protocol 1, we measured
sequential DI and APD values after an abrupt change from one cycle
length to another. The tissue was paced 100 times at one BCL,
100 times at a second BCL, 100 times at a third BCL, and so forth,
until all 12 transitions between the 4 BCL of 400, 700, 1,000, and
1,300 ms had been covered. Each 100-stimulus train was begun
synchronized to the last stimulus of the previous train. All 1,200 APDs
were measured. In protocol 2, we measured shifts in the
restitution curve produced by multiple extrastimuli. Effects of
different numbers of premature stimuli on a given restitution curve
were studied by inserting different numbers of premature stimuli
between a train of 20 stimuli and the test stimulus given at variable
coupling intervals for measuring restitution. More specifically, the
tissue was given a 20-stimulus train (S1) at a BCL of 400, 700, 1,000, or 1,300 ms, an n-stimulus train (S2) at a BCL of 400, 700, 1,000, or 1,300 ms, and a final stimulus (S3) that was coupled to the
last stimulus of the second train by a coupling interval (S2S3) of 150, 200, 250, 300, 400, 700, or 1,000 ms. The APD produced by the last S2
stimulus of the second train and the APD produced by the final stimulus
(S3) were measured. This protocol was repeated for n of 0, 1, 2, 3, 4, 5, and 10. In both protocols, there were transitions
between various pacing cycle lengths. We called the pacing cycle length
before a transition "old" BCL and the one after a transition (such
as the S2 train in protocol 2) "new" BCL. The new BCL
became old BCL when the pacing cycle length was changed again. We
defined BCL as old BCL new BCL, so BCL was positive when
the pacing rate was accelerated.
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Stimulus protocol 1 was tested in six preparations (1 epicardial, 3 endocardial, and 2 PF). Data were studied in two ways. First, the DI preceding and the APD following each stimulus were plotted as pairs in DI,APD parameter space to assess how the pairs approached the line representing the equation DI + APD = new
BCL. Second, APD for each stimulus was plotted vs. stimulus number to
assess how APD (100-APD sequence) evolved toward the steady-state value
of the new BCL. In that analysis, APD values were fit by an exponential
function of the form APD = + exp(stimulus number/
) with three independent parameters , , and , which we called asymptote, drop, and time constant. Note that in this protocol is not the conventional time constant measured in milliseconds but has units of stimulus number, which can be converted to time by
multiplying by new BCL.
Data from stimulus protocol 2 were studied in five
preparations (2 epicardial, 1 endocardial, and 2 PF tissue). The data
analysis process is best followed by referring to Fig.
2. To characterize evolution of
restitution curves after different numbers of extrastimuli, we fit each
restitution curve (the 7 restitution curves after 0-10
extrastimuli) with an exponential function of the form APD = + exp (DI/), where is time constant (Fig.
2A). The values of a given parameter were plotted
against the number of premature stimuli and fit with linear regression
functions, resulting in a slope and intercept value for each of the
three parameters (Fig. 2B). These six slope and intercept
parameters were in turn plotted against new BCL, old BCL, or BCL.
Fig. 2C shows the intercept of plotted against old BCL;
Fig. 2D shows the slope of against BCL. These plots
were themselves fit with linear regression functions. Because the slope
should not have changed if BCL did not change, regression fits of the
slope parameter plots were forced through the origin. A fourth
parameter, 
, describing the horizontal shift of restitution curves
was also computed using the formula * ln('/), after and
were obtained from the exponential regression fitting. (The
asterisk symbol signifies multiplication here and hereafter.) '/
is the ratio of after extrastimuli to in the absence of
extrastimuli. This formula assumes a constant and merely reflects
the fact that a vertical shift of an exponential curve is equivalent to
a horizontal shift of that curve, similar to the way in which a
vertical shift of a straight line (y = ax + b) by amount c
(y = ax + b c) is equivalent to a horizontal shift of the same line by
amount c/a [y = a(x c/a) + b]. Once all of the curve fitting was completed, the
statistical significance of the regression lines of the slopes and
intercepts on BCL parameter (new, old, BCL) was determined, and if
there was no significant dependence of an exponential parameter on any
of the BCL parameters, the parameter was assumed fixed for a given
tissue and its average value was computed.
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Mathematical Model
We assumed that the restitution curve was linear, i.e., that it could be described by an equation of the form y = a * x + b, where y is APD and x is DI. This assumption was made to facilitate the drawing of mathematically exact conclusions but was justified on the basis of our experimental results from stimulus protocol 1 showing rapid, nonoscillatory adaptation of DI and APD to change in cycle length, so that there was only a narrow (and in that sense, linear) segment of restitution curve explored after the first two beats in the majority of cases. Our second assumption was that an abrupt change to a new BCL produced a change in memory described by a vertical shift of restitution function of kn ms with each new stimulus. The first shift was indexed 0 (see Fig. 3). A vertical shift of k is equivalent to a horizontal shift of k/a in a linear system, so analysis of the vertical shift case can be extrapolated to horizontal shift by constant scaling. The two assumptions gave the following iterative equations
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(1) |
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(2) |
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(3) |
1 = 0 (see Fig. 3). The subscript
n denotes the stimulus number at the new BCL,
xn is the DI preceding that stimulus, and
yn is the APD produced by that stimulus. The
steady-state DI and APD values at the old BCL were
x0 and y0, respectively.
[At steady state, the restitution function is no longer shifting, and
(x0,y0) must lie at the
intersection of x + y = old BCL and the
steady-state restitution function at the old BCL.] We also defined the
difference dn in consecutive APD as
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(4) |
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(5) |
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(6) |
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Matching of Model to Experimental Data and Simulations
We explored ways of obtaining model parameters (a and b defining the restitution curve and k0 and D defining the shift) from the experimental data and identified several techniques. These methods were found to be applicable for obtaining exponential restitution curve parameters as well. The results of stimulus protocol 1 were simulated with these parameters to test the validity of the model.| |
RESULTS |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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DI,APD Pair Evolution (Stimulus Protocol 1)
When the 100 APD after BCL changes were plotted against preceding DI, three patterns in the evolution of DI and APD were seen. An example of each pattern taken from one PF experiment is shown in Fig. 4. In approximately one-half of the transitions between two BCL (34/67), all 100 (DI,APD) pairs aligned immediately on the line representing the equation DI + APD = new BCL, hereafter called the BCL line (Fig. 4B). In another group of transitions (30/67), the first (DI,APD) pair after the transition to new BCL lay away from the BCL line but the remaining 99 pairs aligned on the BCL line (Fig. 4C). Only rarely (3/67) did we see (DI,APD) pairs oscillate before aligning with the BCL line (Fig. 4A). In all transitions, once a (DI,APD) pair aligned with the BCL line, the remaining pairs moved up and to the left if pacing rate was slowed or down and to the right if pacing rate was increased.
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Figure 5 shows the relationship between
the patterns of transition and the BCL change producing them. The
thickness of the lines (thinnest = 1, thickest = 6) indicate
the number of transitions that showed that particular pattern. Figure 5
shows that direct alignment was likely when the BCL change was ±300 ms
and between longer BCL and that alignment requiring one beat was likely
when the BCL change was ±600 or 900 ms, especially when going from or
going to a BCL of 400 ms. All three instances of oscillations observed
were in a PF preparation. The direction of BCL change did not
predispose toward either direct or one-step alignment. Other than the
observation of oscillation in one PF experiment, there were no obvious
tendencies for one tissue to show one pattern over another.
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APD Evolution (Stimulus Protocol 1)
We also analyzed the temporal evolution of APD values independently of DI as a function of beat number after abrupt changes in BCL using the same data as in DI,APD Pair Evolution. Hereafter, "APD evolution" refers to APD vs. beat number. As the example of endocardium stimulated with protocol 1 in Fig. 6 typifies, in the majority of transitions, APD at each new BCL increased or decreased sharply at first and then less sharply. However, the slope of change rarely became zero, i.e., reached a clear plateau, within the 100 beats of our observation, even though we have drawn, by eye, approximate plateau values in Fig. 6. This slow but continuous evolution of APD is well established (17, 31). There were a few transitions where the difference between the two steady-state values were smaller than the measurement noise or where the pattern appeared to be biphasic (e.g., 1,300
1,000 and 700 1,000 in
Fig. 6). Transitions showing such biphasic patterns were omitted from
the curve-fitting analyses described next.
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Each transition segment from one BCL to another was curve fit with the
exponential curve APD = + exp (n/)
where n is stimulus number. The different BCL combinations
provided ~12 values of , , and for each experiment. We
studied the nine correlations between each of , , and and
each of old BCL, new BCL, and BCL. We found in all six experiments
that was highly correlated with new BCL value (correlation
coefficient range ~0.801-0.968, mean 0.883 ± 0.069;
P < 0.001 for all) and was highly correlated with
BCL (correlation coefficient range ~0.812- 0.977, mean
0.925 ± 0.0631; P < 0.0001 for all except PF
dog 2, in which P = 0.006), whereas did
not correlate with any of new BCL, old BCL, or BCL (P 
0.5 for 14 of 18 fits). Each tissue gave a similar mean value of ,
and the mean over the aggregate of 66 transitions was 25.91 ± 10.10. Table 1 lists the equations relating and to new BCL and BCL values, respectively. The values in Table
1 can be used to compute expected APD evolution curve formulas for a given BCL change.
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Restitution Curve Evolution (Stimulus Protocol 2)
For each tissue, 7-12 pairings of old BCL and new BCL were studied. In general, a rate increase (premature new BCL) decreased the asymptote and drop and increased the time constant, i.e., restitution curves became lower and flatter. A rate decrease (postmature new BCL) had the opposite effect, i.e., it increased the asymptote and drop and decreased the time constant. The exception to this rule was in epicardium. Drop magnitude did not show a direction of change dependent on BCL. Figure 7 shows an example of restitution curve evolution data obtained from epicardial tissue with BCL change from 1,300 to 400 ms. Fig. 7A corresponds to the schematic diagram in Fig. 2A, and Fig. 7B corresponds to Fig. 2B with results for five additional BCL transitions shown. Computation of a horizontal plus vertical restitution curve shift parameter in the curve fits did not produce fits better than those based on vertical shift alone. The change in shift parameters with increasing numbers of S2 was not necessarily monotonic, as can be seen from the restitution curve for two S2 in Fig. 7A and from the nonmonotonicity of the graphs in Fig. 7B. The following three paragraphs describe how well the data were fit by linear regression.
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In epicardium (n = 2; pooled data), dependence of the
restitution parameters on number of stimuli at new BCL was
statistically significant for 8 of 14 transitions for the asymptote,
whereas drop and time constant value dependence were statistically
significant for 0 and 2 of 14 transitions, respectively. However, for
time constant, the direction of change was similar to the other tissues (increasing with premature new BCL, decreasing with postmature BCL) for
11 of 14 transitions. This result is interpreted to mean that the
correct trend for time constant change was present but was not enough
to overcome the experimental noise. The rate of change of asymptote and
time constant over number of extrastimuli (i.e., slope of and slope
of in a Fig. 2B-type plot) depended on BCL with
P = 0.16 and 0.0001, respectively (i.e., the slope of
these slopes in a Fig. 2D-type plot was different from zero with the given P values).
In endocardium (n = 1), dependence of the restitution parameters on number of stimuli at new BCL was statistically significant for four, six, and four of seven transitions for the asymptote, drop, and time constant values. The direction of change for the restitution parameters was similar to the other tissues (lower and flatter restitution curves with premature new BCL, higher and steeper curves with postmature new BCL), with six, six, and seven of seven transitions being consistent with the general trend for asymptote, drop, and time constant, respectively. The rate of change of asymptote, drop, and time constant over number of extrastimuli depended on new BCL with P = 0.0007, 0.002, and 0.0003, respectively (i.e., the slope of these slopes in a Fig. 2C-type plot was significantly different from zero).
In PF (n = 2; pooled data), dependence of the
restitution parameters on number of stimuli at new BCL was
statistically significant for 13, 5, and 2 of 20 transitions for the
asymptote, drop, and time constant values, respectively. The small
number of significant linear fits for asymptote and drop was due to the
presence of oscillations of those parameters depending on whether the
number of stimuli at the new BCL was even or odd. Nevertheless, if the data were fit with straight lines, the direction of change for the
restitution parameters was similar to the other tissues (lower and
flatter restitution curves with premature new BCL, higher and steeper
curves with postmature new BCL), with 18, 17, and 17 of 20 transitions
being consistent with the general trend for asymptote, drop, and time
constant, respectively. The rate of change of asymptote, drop, and time
constant over number of extrastimuli depended on BCL with
P = 0.0001, 0.0005, and 0.12, respectively.
Table 2 summarizes the formulas that
describe how restitution curves evolve when BCL is changed from one
value to another in the different ventricular tissue types. The
formulas were obtained from the curve fits just described and can be
used for restitution curve prediction.
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Dynamics of Linear Restitution Model
We analyze here the dynamics of the model described in MATERIALS AND METHODS.Case of exponentially decreasing vertical shift.
From Eqs. 1 and 2 we obtain
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(7) |
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(7`) |
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(8) |
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(9) |
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(10) |
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(11) |
![d<SUB>n</SUB>=c<SUP>n</SUP> ∗ [d<SUB>0</SUB>+k<SUB>0</SUB>/(c−D)]−D<SUP>n</SUP> ∗ [k<SUB>0</SUB>/(c−D)]](/content/vol282/issue4/fulltext/H1534/img013.gif)
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(11`) |
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(12a) |
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(12b) |
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(13) |
D). Large-amplitude oscillations would be expected when this term is large
because of a large change of BCL (large d0),
restitution slope (large d0), or vertical shift
of restitution (large k0). The attenuation of
oscillations is dependent on c, with a time constant equal
to the reciprocal of ln c , i.e., large c
would produce longer duration oscillations, whereas c
close to 0 would produce immediate attenuation of the oscillations.
Case of constant k.
When D = 1, kn = k0 for all n and
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(14) |
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(15) |
k/(1 + a). This means that
the slopes of the APD evolution curves should never become zero but
asymptotically approach the constant value k/(1 + a) after the initial oscillatory phase has passed.
Therefore, the constant k case is a good model for the
behavior of the APD evolution curves in which APD continues to decrease
or increase indefinitely, i.e., fails to reach a plateau, as observed
in some BCL changes (Fig. 6). We also note that in the constant
k case, the transients of dn diminish
at a rate of cn [d0 + k/(1 c)].
Case of horizontal + vertical shift.
The case of horizontal shift is easily extrapolated from the vertical
shift case in the linear restitution model. A downward shift of
k coupled with a leftward shift of 
is equivalent to a
downward shift of k a * and
Eq. 11 becomes
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(16) |
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n is described by
exponentially decreasing function
0 * Ln where L is
a model parameter.
Obtaining Model Parameters from Experimental Data
Ideally, one would like to obtain parameters for the dynamic model quickly without having to apply time-consuming stimulus protocols to every new patient or tissue specimen. The two categories of parameters that need to be found are those that describe the restitution function and those that describe the vertical shift.Finding parameters for linear restitution model.
In the linear restitution model, there are four parameters,
a and b, which describe the linear restitution
curve, and k0 and D, which describe
the vertical shift of the restitution curve. In theory, one could use
any four random data points from the APD evolution data to solve four
equations with four unknown values. For example, for the constant
k case, where D = 1, y1 = ax1 + b, y2 = ax2 + b k,
and y3 = ax3 + b 2k can be solved to give a = (y1 + y3 2y2)/(x1 + x3 2x2),
b = y1 ax1, and k = (y1 ax1) (y2 ax2).
In practice, however, unless the data cover a wide range of
x or y values, for example, by oscillating, the differences between x or y values are small and
comparable to the order of experimental noise. This makes accurate
calculations of the parameters using randomly selected data points
impossible. Instead, we use the fact that
(x0,y0), the steady-state
DI,APD values at old BCL, and
(x1,y1), the DI,APD
values associated with the first stimulus at the new BCL, lie on the
same restitution curve f(x) (see Fig. 3). This
gives us values for a and b.
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(17) |
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(18) |
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(19) |
![=y<SUB>2</SUB>−[a+b ∗ (new BCL<IT>−y</IT><SUB>1</SUB>)]](/content/vol282/issue4/fulltext/H1534/img024.gif)
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(20) |
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(21) |
![=d<SUB>0</SUB>/(1−c)+k<SUB>0</SUB>/[(1−c)(1−D)]](/content/vol282/issue4/fulltext/H1534/img026.gif)
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(22) |
![D=1−k<SUB>0</SUB>/[(y<SUB>0</SUB>−y<SUB>n</SUB>)(1−c)−d<SUB>0</SUB>]](/content/vol282/issue4/fulltext/H1534/img027.gif)
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(23) |
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(24) |
Finding parameters for exponential restitution model.
Two of the three parameters for a monoexponential restitution curve can
be obtained similarly to Eqs. 17 and 18 from the
first two values of APD evolution data. For example
![&bgr;=(y<SUB>0</SUB>−y<SUB>1</SUB>)/[exp(−<IT>x</IT><SUB>0</SUB><IT>/&tgr;</IT>)<IT>−</IT>exp(−<IT>x</IT><SUB>1</SUB><IT>/&tgr;</IT>)]](/content/vol282/issue4/fulltext/H1534/img029.gif)
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(25) |
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(26) |
, either from a table of normal values or by using an S1S2 protocol
to obtain one more DI,APD value to establish the restitution function.
The parameter k0 is obtained as in Eq. 19
![k<SUB>0</SUB>=[&agr;+&bgr; ∗ exp(−<IT>x</IT><SUB>2</SUB><IT>/&tgr;</IT>)]<IT>−y</IT><SUB>2</SUB>](/content/vol282/issue4/fulltext/H1534/img031.gif)
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(27) |
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(28) |
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(29) |
![D=1−k<SUB>0</SUB>/[&agr;+&bgr; ∗ exp(−<IT>x<SUB>n</SUB>/&tgr;</IT>)<IT>−y<SUB>n</SUB></IT>]](/content/vol282/issue4/fulltext/H1534/img035.gif)
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(30) |
yn. In summary, two of the three parameters defining the exponential restitution curve and the two parameters defining the shift of restitution with each stimulus can be obtained from the first three and last values of APD evolution data
y0, y1,
y2, and yn, where
n is large enough for APD to have reached a plateau value.
Special case of restitution curve reconstruction from oscillatory
data.
We discovered a crude visual method by which a nonlinear restitution
curve and early shift values could be reconstructed simultaneously in
the special case where DI,APD pairs oscillate several times, thereby
providing a wide range of DI points. In this method, (DI1, APD1), (DI2, APD2 k), (DI3, APD3 2k), ..., [DIn,
APDn (n 1) * k] are plotted for the n oscillating
(DI,APD) pairs for various values of k until the curve
produced by connecting the points resembles an exponential restitution
curve. For example, the first five DI,APD pairs measured from the BCL
1,300 to 400 ms data of a PF experiment (Fig. 4A) were found
to produce the smoothest restitution curve when the five pairs were
plotted with a k value of 5 ms. The restitution
curve found for this data set using this method was
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(31) |
Cross-Validation of Model by DI,APD Evolution Simulation
We reconstructed restitution curve evolution for the 60 experimental runs of stimulus protocol 1 with Eqs. 25-30 and from Table 2. We found that in practice,
was unrealistically large if the difference between
y0 and y1 was small
(
5). This was true of 25 of 60 runs. Furthermore, if the difference
between y1 and y2 was
small, the k0 value was unrealistically small or
of the wrong sign. This was true of a further nine runs.
Reconstructions of DI,APD pair evolution for the remaining 26 runs
produced results quantitatively and qualitatively similar to the
original experiments, especially for the cases in which the behavior
was simple with immediate or almost immediate alignment of DI,APD pairs
on the BCL line. We show in Fig.
8A the results of simulation
of the most complex behavior, i.e., the oscillatory DI,APD evolution of
the PF going from BCL of 1,300 to 400 ms shown in Fig. 2A. The main difference between the simulation and the original data was in
the position of the third, fourth, and fifth points relative to the
first and second points. In the experiment, the APDs of those points
are higher.
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We hypothesized that the conclusions drawn from the analysis of the
linear restitution model (Eq. 11) would hold to a first approximation for the exponential restitution case, i.e., that a
greater change of BCL and steeper restitution slope (greater and
smaller ) would lead to a larger amplitude of oscillation and that a
steeper restitution slope would lead to a longer duration of
oscillation. This is shown to be true in Fig. 8, B and
C, in which a small and large restitution slope are
compared. A comparison of Fig. 8A vs. 8B and
8C also shows that a small k0
is responsible for a lower position of the first and second APD values
relative to the subsequent values.
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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES
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History of Research in Cardiac Memory and Restitution
APD is known to be a function both of immediately preceding DI or coupling interval and of previous activation history (12, 13). In general, APD of a premature activation is shorter when the DI or coupling interval is shorter (30), a dependence that has come to be called electrical restitution (1) after its similarity to mechanical restitution (4). APD is also dependent on heart rate, usually being longer at slow heart rates (20, 33, 47) as seen in prolongation of QT intervals at slow heart rates on the surface electrocardiogram. This dependence of APD and refractory period on activation history (32) is also manifest in the downward shift of the restitution curve at fast heart rates (2, 5) and has come to be called cardiac memory (22, 8) because the tissue tends to have longer APD than expected from DI if previous APD were long, and vice versa, as though remembering its previous APD. The function of memory is to optimize the ratio of diastolic filling time to systolic ejection time.Suggestion for New Memory Terminology
The shift of restitution curve with change to new cycle lengths is an effect of memory. When switching to a new pacing rate, the restitution curve does not jump immediately to the new restitution curve but is somewhere between the old and the new. Therefore, one might say that the curve still remembers and lingers near the old curve. In that sense, more memory implies smaller or slower shifts to the new state. This is the sense used when stating that ventricular muscle has more memory than Purkinje tissue, because a premature stimulus changes the refractory period less in ventricular muscle than in the Purkinje system (28). In other words, more memory means less memory effect, and this is confusing. We therefore suggest use of another term, accommodation (27), as an inverse concept of memory. It is short compared with other terms used historically, such as cumulative effect (17), information retention (14), and cycle length-independent change (12). Greater restitution curve shifts and greater dynamic restitution curve slope could be attributed to greater accommodation. This term would abolish the need for circumlocutions such as "declining memory effect" to describe the shift of restitution curves after rate changes. It would also be useful in differentiating short-term memory, such as we analyzed in this study, from changes in T wave polarity induced by ventricular pacing that persist for hours to weeks after resumption of normal atrioventricular conduction, a phenomenon that is also referred to as cardiac memory (6, 40, 41).Methods of Cardiac Memory/Accommodation Quantification
The standard restitution curve is always sufficient for explaining changes in APD after one premature stimulus, because restitution is defined and measured as the relationship between one premature stimulus coupling interval or DI and the following APD at a particular pacing rate. It is measured directly, and no assumptions are made. However, when there is more than one premature stimulus, APD values deviate more and more from that predicted from the restitution curve because of the effects of accommodation. There is no universally accepted way of quantifying accommodation. One can normalize steady-state restitution curves (5, 8) and regard the normalization factor as a measure of memory. This normalization factor has been used to model the memory effect and its role in spiral wave breakup (10), although normalization with the steady-state restitution curves is predicated on making assumptions about the time course of restitution change between two steady states. Two theoretical models of memory have treated it as a variable that gets incremented with every action potential but also decays with time (16, 34), i.e., memory increases when there are many action potentials per unit time and decreases with long DI. The latter model successfully reproduces qualitative aspects of complex alternans phenomena seen in experiments that cannot be simulated by iteration of a standard restitution curve. The model presented in this study does not attempt to characterize accommodation by an independent variable as in the two theoretical studies. It is a simple phenomenological model in which accommodation is characterized by the initial restitution curve shift value and its decay with additional extrastimuli.Experimental Results
The results of the protocol 1 demonstrated two things. First, as shown by other investigators (17, 31), APD frequently does not equilibrate to a new steady-state value within 100 beats of a change to new BCL. Instead, the APD continues changing beyond 100 beats, or if the change is between two relatively long BCL APD can increase then decrease, or decrease then increase. This means that it is sometimes difficult to define steady-state APD, or for that matter, restitution function, for a given BCL. Second, oscillation of DI,APD pairs in the DI,APD space before alignment on the DI + APD = new BCL line was rare. This was to be expected. For example, Saito et al. (42) noted that the maximum BCL producing oscillations in canine ventricular tissue was <400 ms, whereas the minimum BCL used in this study was 400 ms. Oscillation in the present study was seen only in PF when the new BCL was very short. The mathematical model provided three conditions for oscillation of DI,APD evolution to occur, all of which were satisfied in the PF that showed oscillation. In the three oscillations that did occur, the pattern of DI,APD pairs oscillating back and forth on a single line for many beats, followed by alignment on the BCL line as described by Vick (48), was not observed. In simulations such as Fig. 8A, however, we were able to reproduce similar behavior, which leads us to believe that such patterns can occur experimentally.The results of the protocol 2 showed that the parameters of the monoexponential curve fits to restitution could be treated as changing linearly with increasing numbers of premature stimuli up to 10 stimuli. As might be expected, the changes were proportional to difference between new and old BCL. In general, faster pacing flattened and lowered restitution curves whereas slower pacing sharpened and raised restitution curves, as seen in cat papillary muscle (2) and canine PF (8). Epicardium was an exception in that drop magnitude did not show a direction of change related to BCL. It was also found that computation of a horizontal + vertical restitution curve shift parameter in the curve fits produced fits no better than those based on vertical shift alone.
Earlier studies studying the shift of restitution curves produced by multiple S2 using finer DI spacing showed that the slope of the early part (short DI) of the restitution curve was sharper for the second S2 (restitution curve after 1 premature stimulus) than for the first S2 (the standard restitution curve) (24, 51). A reappraisal of the plots of the drop parameter against number of premature stimuli confirms the earlier results in that the drop parameter is frequently larger for the second S2 in endocardium and Purkinje (not shown), usually in the transition from a large BCL to a BCL of 400 ms. However, the results of the present study (transient increase of slope followed by flattening of restitution curve slope over 10 premature S2) do contradict the earlier results seen in Purkinje (increased slope maintained over 4 premature S2; Ref. 51). This may be due to the larger number of prematures applied in the current study or to the difference between refractory period and APD restitution, but it is most likely due to the fact that new BCL was set at the minimum in the earlier study (effective refractory period of the last S1 beat + 10 ms). This interpretation is more likely because of the known steeper slope of dynamic restitution at shorter BCL (25), although dynamic and standard restitution cannot be compared directly.
We never saw biphasic APD restitution curves (11, 21, 24, 50) in which the ventricular myocardial restitution curve has a local maximum at short DI. The action potentials at this peak have been referred to as supernormal premature action potentials (3) and are believed to reflect potentiation of the calcium current. The prevalence of biphasic restitution curves as opposed to monotonically increasing restitution curves is not known. The probable reason we did not see obvious APD peaks, assuming some had been present, was the coarse resolution measurements of restitution curves.
Modeling Results and Implications
Mathematical analysis of the dynamics where restitution curves were modeled as linear functions provided the conditions under which DI,APD oscillations would be expected after a rate change. Two factors determine whether oscillations are detectable or not, amplitude and duration of oscillation. Both the amplitude and duration have to be large for oscillation to be visible. For example, if amplitude were large but attenuation occurs in one beat, no oscillation would be seen. Oscillation is also missed if attenuation occurs slowly but amplitude of oscillation is smaller than the experimental noise. Therefore, oscillation is predicted to be visible when the restitution curve slope is steep, the BCL change is large, and vertical shift is large. Of these three criteria, the first is well known mathematically to produce APD alternans. The other two criteria are presented here for the first time.Lepeschkin (29) hypothesized that electrocardiographic
alternans at fast heart rates was due to APD dependence on preceding DI, coupled with DI alternans during alternans. Subsequently, some
investigators confirmed this hypothesis whereas others found otherwise
(7, 45). It was later argued that the conflicting experimental results arose from the fact that Lepeschkin's theory was applicable to Purkinje tissue but not to ventricular myocardium in
that intracellular calcium concentration was also a factor contributing
to APD (46). For example, Saito et al. (42)
compared electrical alternans produced in both tissues. In their study, memory was accounted for by plotting the difference between two sequential APD values (y) against the difference between
APD values obtained by projection of sequential DI values onto the restitution curves (yrtt), i.e., the
difference between APD in the absence of a memory effect. In PF, they
found y = 1.02yrtt 1.6, whereas in myocardium, y 
2yrtt, i.e., PF data gave a slope of 1 and
myocardium a slope of 2. In terms of the linear restitution model
developed here, yrtt = c * dn1.
From Eq. 11 we obtain
|
(32) |
y vs.
yrtt should give y = yrtt + µ, where µ is some positive value. That
is, the model predicts a slope of 1. If
k0 * Dn1
is close to 0 [n < 4 in Saito et al.
(42)] then the model result matches the results of Saito
et al. in PF and APD values during alternans are indeed a result of
restitution plus effects of accommodation. What then does one make of
the slope = 2 results in myocardium? Is our model of accommodation
inapplicable to myocardium? Figure 7 in the report by Saito et al.
(42) gives a clue. The initial shift
k0 appears to be large in ventricular myocardium
compared with later shifts, judging from the large space between the
restitution curve and later points. Because d1
is greater than d
y vs. yrtt > 1. In other words, if the term
k0 * Dn1
is comparable to or greater than yrtt, then
one can indeed get a slope of 2 or even higher. Furthermore, yrtt is seen to be small compared with the PF
case, making it more likely for the term
k0 * Dn1
to outweigh the yrtt term. In other words,
our model shows that it is still possible to explain ventricular
myocardial APD by using the concepts of restitution and accommodation
as in PF and that the particular argument used by Saito et al. cannot
be used to diminish the role of electrical restitution and memory in
determining APD in myocardial tissue. That said, the model does not
negate the importance of intracellular calcium concentration in
determination of myocardial APD. Changes to intracellular calcium
concentration affect APD restitution (24) and APD
alternans (19, 26, 43). We conjecture that the calcium
concentration determines APD restitution and accommodation and that the
two can be measured and modeled decoupled from calcium concentration
without consequence because they already encode the calcium
concentration information.
A study by Hirata et al. (18) reviewed some experimental
literature and made the point that APD alternans could be of two types,
even or odd. The odd type refers to alternans where the odd numbered
beats (beat 1 being the first APD at the shortened cycle length) have
the longer APD. They noted that the odd type was seen more commonly in
myocardium than in Purkinje and that the even type was usually seen
with a large change in BCL. If there were no accommodation at all, one
would expect to see only even-type alternans. A shift in the phase can
be hypothesized to occur in two ways: one is the presence of a biphasic
restitution curve, and the other is to take into account the
accommodation effect. From Eq. 9,
d1 = a * d0 + k0 is true. The sign of
d0 is always positive for a shortening of BCL.
However, the sign of d1 can be either positive
or negative depending on the relative sizes of the various parameters.
If d1 is negative, even-type alternans follows.
The sharper slope a of PF restitution is in agreement with this
expected result. This equation also explains why even-type alternans is
more likely with a larger change of BCL; d0 is
larger in that case. If restitution slope a is smaller as in
ventricular myocardium and k0 is large,
d1 can be positive, thereby shifting the phase
from even to odd-type alternans.
Cross-Validation of Model
Finding model parameters from the experimental data was not as simple as expected from theory. This was due mainly to the relatively large size of experimental noise compared with the small changes of APD during APD evolution. Nevertheless, we proposed several methods for obtaining model parameters requiring the measurement of only the first few DI,APD values and the 100th DI,APD value after a rate change. These parameters could be applied toward construction of either a linear or exponential restitution curve model. Simulation of the simple behavior (immediate or almost immediate alignment) produced realistic results. Simulation of the most complex behavior, oscillatory DI,APD in PF, was similar to experiment quantitatively and qualitatively except in one respect. This was in the positions of the early data pairs relative to the first two data pairs. The most likely cause of this discrepancy was the assumption of linear change in asymptote and drop parameters with increasing stimulus number. In the current experiments, the asymptote and drop parameters of the restitution curves showed small oscillations in PF after BCL change to very rapid rates, but this behavior was not captured in the linear regression analyses of the parameters over stimulus number. We had measured restitution curves after 0-5 and 10 S2 stimuli, which meant that there were only seven data points per parameter that could be fitted by regression analysis. This was enough for linear regression but not enough for exponential curve fitting.Limitations
Our model assumed that the restitution curve was linear and that the slope of that curve was fixed, although the results of stimulus protocol 2 showed that drop and time constant also evolved. These assumptions were justified by the fact that, in the absence of oscillations, only a small part of the restitution curve was explored by the dynamics of the tissues. However, at shorter BCL, where alternans is obligatory, these assumptions will no longer hold. It did appear, nevertheless, that conclusions drawn from the linear restitution model about the conditions producing longer or larger alternans were applicable to simulations using exponential restitution curves.We reduced the dynamics of shift parameter k to two cases, that of a constant k and that of an exponentially decreasing k, and limited our analyses to those cases. However, accommodation very likely involves several mechanisms at the cellular level, each with a different time scale. Therefore, models with more complicated dynamics of k, such as where k is a sum of a constant and exponentially decreasing function, or where there are two or more time constants to the decrease of k may provide a better fit to experimental data, especially if it is of such duration that it spans the multiple time scales.
We deliberately selected our minimum BCL of pacing to be 400 ms to avoid alternation of steady-state APD. If we had selected a shorter minimum BCL with the likelihood of alternating steady-state APD value, two restitution curves would have had to be measured for a given BCL (35, 38). Consequently, experimental design prevented us from seeing oscillatory DI,APD behavior. Future experiments will focus on analysis of accommodation at short BCL, now that a framework for analyzing accommodation at longer BCL has been established.
We did not use a large number of animals for this study for several reasons. Canine restitution data is ubiquitous in the literature, especially at the relatively long BCL that we presented here. Our goal was to collect enough data to validate our mathematical model of accommodation and thus present a new way of quantifying and analyzing restitution data. The ultimate utility of the model is expected to be in its application to the computation of dynamic restitution at short BCL, to assess the role of accommodation at the short cycle lengths present in ventricular tachycardia and fibrillation.
Summary
We began with the concept that memory is the amount by which restitution curves are shifted with pacing rate change. We measured these shifts experimentally in ventricular endocardium, epicardium, and PF tissues and constructed a simple phenomenological model of memory amenable to rigorous mathematical analysis. We assessed ways of obtaining model parameters from minimum subsets of the data and finally tested the model predictions against the data. The close match of the simulations to the data suggest that memory/accommodation for a particular cycle length change can be modeled to a first approximation as a vertical shift of restitution curve. The model also produced clear predictions about the dynamic behavior of APD after an abrupt cycle length change that was able to explain several phenomena seen in the literature.| |
ACKNOWLEDGEMENTS |
|---|
The authors thank Dr. Robert F. Gilmour, Jr. at Cornell University for the extensive use of his laboratory and for encouragement and support and Mark L. Riccio for critical assistance with data collection and transfer.
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FOOTNOTES |
|---|
First published November 29, 2001;10.1152/ajpheart.00351.2001
M. A. Watanabe was supported by National Heart, Lung, and Blood Institute National Research Service Award F32-HL-10308-01 and by the Cardiology Department of the Royal Infirmary of Glasgow. M. L. Koller was supported by a research fellowship grant from the Deutsche Forschungsgemeinschaft (Ko 1782/1-1).
Address for reprint requests and other correspondence: M. A. Watanabe, Inst. of Biomedical and Life Sciences, West Medical Bldg., Univ. of Glasgow, Glasgow G12 8QQ, UK (E-mail: mwata001{at}udcf.gla.ac.uk).
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 27 April 2001; accepted in final form 19 November 2001.
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REFERENCES |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
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J. F. Huizar, M. D. Warren, A. G. Shvedko, J. Kalifa, J. Moreno, S. Mironov, J. Jalife, and A. V. Zaitsev Three distinct phases of VF during global ischemia in the isolated blood-perfused pig heart Am J Physiol Heart Circ Physiol, September 1, 2007; 293(3): H1617 - H1628. [Abstract] [Full Text] [PDF]
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A. M. Pitruzzello, W. Krassowska, and S. F. Idriss Spatial heterogeneity of the restitution portrait in rabbit epicardium Am J Physiol Heart Circ Physiol, March 1, 2007; 292(3): H1568 - H1578. [Abstract] [Full Text] [PDF]
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A. Baher, Z. Qu, A. Hayatdavoudi, S. T. Lamp, M.-J. Yang, F. Xie, S. Turner, A. Garfinkel, and J. N. Weiss Short-term cardiac memory and mother rotor fibrillation Am J Physiol Heart Circ Physiol, January 1, 2007; 292(1): H180 - H189. [Abstract] [Full Text] [PDF]
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B. C. Knollmann, T. Schober, A. O. Petersen, S. G. Sirenko, and M. R. Franz Action potential characterization in intact mouse heart: steady-state cycle length dependence and electrical restitution Am J Physiol Heart Circ Physiol, January 1, 2007; 292(1): H614 - H621. [Abstract] [Full Text] [PDF]
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 R. D. Berger Electrical Restitution Hysteresis: Good Memory or Delayed Response? Circ. Res., March 19, 2004; 94(5): 567 - 569. [Full Text] [PDF]
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R. Wu and A. Patwardhan Restitution of Action Potential Duration During Sequential Changes in Diastolic Intervals Shows Multimodal Behavior Circ. Res., March 19, 2004; 94(5): 634 - 641. [Abstract] [Full Text] [PDF]
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 M. L. Walker, X. Wan, G. E. Kirsch, and D. S. Rosenbaum Hysteresis Effect Implicates Calcium Cycling as a Mechanism of Repolarization Alternans Circulation, November 25, 2003; 108(21): 2704 - 2709. [Abstract] [Full Text] [PDF]
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)
along with values found from stimulus runs that studied other
combinations of old and new BCL (
). Linear regression
on these data produced further slope and intercept values.
