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.
 |
INTRODUCTION |
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.
| |
MATERIALS AND METHODS |
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.
View larger version (13K):
[in this window]
[in a new window]
|
Fig. 1.
Stimulus protocols used in experiments. In protocol
1, stimulation was given 100 times at a basic cycle length (BCL)
before switching to a new BCL. Four BCL were studied: 400, 700, 1,000, and 1,300 ms. A typical sequence of BCL was 400, 1,300, 1,000, 700, 400, 1,000, 1,300, 700, 1,000, 400, 700, 1,300, and 400. Duration of
every action potential was measured. In protocol 2, BCL1
(old BCL) and BCL2 (new BCL) were chosen from 400, 700, 1,000, or 1,300 ms. The BCL1 train always had 20 intervals. The BCL2 train ranged from
0 to 10 intervals. Finally, a stimulus was given at variable cycle
length (VCL) ranging from 150 to 1,000 ms for the purpose of
restitution curve measurement. n, Stimulus number.
|
|
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.
View larger version (34K):
[in this window]
[in a new window]
|
Fig. 2.
Schematic diagrams showing method of analysis of restitution curve
evolution data. A: in stimulus protocol 2 (Fig. 1), 0-5 and 10 extrastimuli with coupling
interval of new BCL were given after a train of 20 stimuli with
coupling interval of old BCL. Restitution curves shifted with
increasing numbers of extrastimuli (numbers in brackets). Each curve
was fit with a monoexponential equation. B: the 3 independent parameters describing the exponential restitution curves
were plotted against the number of extrastimuli. A schematic plot for
parameter is shown here with its linear regression curve. The
linear fit produced a slope and an intercept value. C and
D: the intercept and slope values found in B were
plotted against old BCL and BCL, respectively, ( )
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.
|
|
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
|
(1)
|
|
(2)
|
where K is the cumulative shift,
K0 = k0,
K1 = k0 + k1, etc., or
|
(3)
|
T is the new BCL and,
K
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
|
(4)
|
such that decreasing APD gave positive d values. From
the assumptions, it was also true that
|
(5)
|
We analyzed the dynamics of Eq. 4 for the case of
kn described by the exponentially decreasing
function
|
(6)
|
where D is a parameter of the model. The case of
constant shift per stimulus, kn = k0, was also analyzed. This was a special case
of Eq. 6 where D was equal to 1.
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 |
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.
View larger version (31K):
[in this window]
[in a new window]
|
Fig. 4.
The 3 different patterns of diastolic interval (DI), action
potential duration (APD) pair alignment onto new BCL lines. Evolution
of (DI,APD) pairs in a Purkinje fiber (PF) preparation are shown during
the 100-beat transition from a BCL of 1,300 to 400 ms (A),
1,000 to 1,300 ms (B), and 400 to 1,000 ms (C).
The diagonal line (BCL line) indicates the line where the sum of DI and
APD equals the new BCL value. The 1st through 100th DI,APD pairs after
the rate changes are shown. The steady-state DI,APD pair at the old BCL
is not shown. (See text for further discussion.)
|
|
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.
View larger version (17K):
[in this window]
[in a new window]
|
Fig. 5.
Dependence of (DI,APD) transition pattern on change of
BCL. BCL values of 400, 700, 1,000 and 1,300 ms are abbreviated to 4, 7, 10, and 13. The old BCL is shown on left and the new BCL
on right in each panel. The lines range in thickness from 1 to 6 and indicate the number of transitions between a particular pair
of BCL that showed a particular pattern. For example, the thick line
connecting 4 on the left with 13 on the right in the "Alignment after
1 beat" category indicates that there were 6 transitions (all 6 tissues) from a BCL of 400 to a BCL of 1,300 that showed alignment
after 1 beat. Oscillation was seen in 3 transitions, all in Purkinje,
alignment after 1 beat was seen in 30 transitions (18 for a BCL
increase, 12 for a BCL decrease), and direct alignment was seen in 34 transitions (14 for a BCL increase, 20 for a BCL decrease).
|
|
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.
View larger version (34K):
[in this window]
[in a new window]
|
Fig. 6.
APD evolution in a canine endocardial (Endo) preparation
stimulated 100 times each at 4 different BCL (ms). The sequence of BCL
is shown at top. APD did not reach an asymptotic value
within the 100-stimulus train duration in the majority of transitions.
Each transition was fit with a monoexponential curve for analysis (fits
not shown). The 4 horizontal lines indicate approximate values that APD
approached as determined by eye when pacing at the 4 different BCL.
|
|
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.
View this table:
[in this window]
[in a new window]
|
Table 1.
Formulas for computing action potential duration evolution curve
parameters from pacing cycle length values
|
|
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.
View larger version (25K):
[in this window]
[in a new window]
|
Fig. 7.
Restitution function evolution due to cardiac memory.
Example of restitution curve evolution data obtained from epicardial
(Epi) tissue. A: the conventional restitution curve measured
at BCL of 1,300 ms is shown at top. As increasing numbers of
premature stimuli (1-10) with BCL of 400 ms were
given, the restitution curve shifted downwards. B: the
restitution curves in A were fit with monoexponential
curves. The asymptotes of those curves were plotted against the number
of premature stimuli (). Plots of asymptotes are shown
for other BCL changes as well. A symbol key of 4/7 indicates a BCL
change from 400 to 700 ms.
|
|
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.
View this table:
[in this window]
[in a new window]
|
Table 2.
Formulas for computing slopes and intercepts of parameters describing
restitution functions from BCL values
|
|
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
|
(7)
|
Likewise
|
(7`)
|
Subtracting Eq. 7 from Eq. 7' gives us
|
(8)
|
or using the definition in Eq. 4
|
(9)
|
Simply to eliminate multiple parentheses, let
|
(10)
|
Iterating and expanding Eq. 9 gives
|
(11)
|
or
![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)
|
(11`)
|
In the limit of n increasing to infinity, if both
|
(12a)
|
and
|
(12b)
|
then dn converges to 0
|
(13)
|
The physiological interpretation of Eq. 13 is that
slope of the APD evolution curve (not of restitution) gradually
decreases to zero and APD achieves a steady-state value after a BCL
change. Equation 11' shows that for large n,
dn diminishes as the nth power of the
larger of D and c. When n is
small, oscillatory APD behavior is possible because c < 0, and cn changes sign depending on whether
n is odd or even. The amplitude of the DI,APD oscillations
are determined by d0 + k0/(c 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
|
(14)
|
and for convergence criterion (Eq. 12a)
|
(15)
|
The physiological interpretation of Eq. 14 is that
oscillations of APD will increase in amplitude (which in real tissue
would eventually result in a stimulus failing to excite the tissue
because of long refractory period) if the slope of the linear
restitution curve is >1, dampen if the slope is >1, and stay at a
fixed amplitude if the slope is exactly 1. The physiological
interpretation of Eq. 15 is that for large n, APD
continues to decrease or increase forever after a BCL change, by the
fixed amount 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
|
(16)
|
where leftward shift 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.
|
(17)
|
|
(18)
|
k0 is the difference between the standard
restitution curve and the restitution curve after one extrastimulus.
Therefore
|
(19)
|
![=y<SUB>2</SUB>−[a+b ∗ (new BCL<IT>−y</IT><SUB>1</SUB>)]](/content/vol282/issue4/fulltext/H1534/img024.gif)
|
(20)
|
Finally
|
(21)
|
which for large n
![=d<SUB>0</SUB>/(1−c)+k<SUB>0</SUB>/[(1−c)(1−D)]](/content/vol282/issue4/fulltext/H1534/img026.gif)
|
(22)
|
under convergence criteria (Eqs. 12a and 12b). Solving this for D gives
![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)
|
(23)
|
In summary, a, b, k0,
and D 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.
If yn does not reach a plateau value, we assume
a constant k case. With Eq. 15, k can
be computed as
|
(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)
|
(25)
|
|
(26)
|
It is necessary to obtain the third parameter, in this example
, 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)
|
(27)
|
To obtain D, we use
|
(28)
|
and
|
(29)
|
for very large n. The last two equations solved for
D give
![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)
|
(30)
|
When n is large, y values are not changing
much, so xn can be substituted by new BCL 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
|
(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.
View larger version (28K):
[in this window]
[in a new window]
|
Fig. 8.
Cross-validation of mathematical model by simulation of
DI,APD evolution. Top panels: DI,APD evolution in the DI,APD
plane. Bottom panels: APD evolution as a function of beat
number. A: the oscillatory DI, APD evolution of the Purkinje
fiber shown in Fig. 2A was simulated using model parameters
extracted from the data as described in the text. The parameters were
old BC = 1,300, new BCL = 400, = 393, = 172, = 129, k0 = 5.2, and
D = 0.89. B: a smaller restitution slope
dampened oscillation amplitude and duration. C: a greater
restitution slope increased oscillation amplitude and duration.
|
|
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.
| |
DISCUSSION |
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)
|
This means that a plot of 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
by
k0 and d2 is greater than
d
by
k0 * D, if both
d and
d are positive, one can easily get a
slope of 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. (1
TOP
ABSTRACT
INTRODUCTION
