Vol. 277, Issue 5, H1940-H1945, November 1999
Weibull distribution function for cardiac contraction:
integrative analysis
Junichi
Araki1,
Hiromi
Matsubara1,2,
Juichiro
Shimizu1,
Takeshi
Mikane1,3,
Satoshi
Mohri1,2,
Ju
Mizuno1,3,
Miyako
Takaki4,
Tohru
Ohe2,
Masahisa
Hirakawa3, and
Hiroyuki
Suga1
Departments of 1 Physiology II,
2 Cardiovascular Medicine, and
3 Anesthesiology and
Resuscitology, Okayama University Medical School, Okayama 700-8558; and
4 Department of Physiology II,
Nara University Medical School, Kashihara, Nara 634-8521, Japan
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ABSTRACT |
The Weibull
distribution is widely used to analyze the cumulative loss of
performance, i.e., breakdown, of a complex system in systems
engineering. We found for the first time that the difference curve of
two Weibull distribution functions almost identically fitted the
isovolumically contracting left ventricular (LV) pressure-time curve
[P(t)] in all 345 beats
(3 beats at each of 5 volumes in 23 canine hearts;
r = 0.999953 ± 0.000027; mean ± SD). The first derivative of the difference curve also closely
fitted the first derivative of the
P(t) curve. These results suggest
the possibility that the LV isovolumic
P(t) curve may be characterized by
two counteracting cumulative breakdown systems. Of these, the first breakdown system causes a gradual pressure rise and the second breakdown system causes a gradual pressure fall. This Weibull-function model of the heart seems to give a new systems engineering or integrative physiological view of the logic underlying LV isovolumic pressure generation.
systems engineering; weakest-link principle; ventricular pressure; curve fitting; integrative physiology
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INTRODUCTION |
THE REDUCTIONISTIC ADVANCE in
molecular and cellular biology has deepened our understanding of
individual elements and components of complex living systems (1, 5, 6, 17a). The logic of the interplay of molecules, cells, and organs, i.e.,
how molecular performances are integrated into cellular and organic
performance, is expected to be understood by the accumulation of
reductionistic understanding.
In cardiac physiology, molecular biological techniques have
characterized physicochemical properties of cross-bridge (CB) cycling
as the elementary step of myocardial contraction (11, 16, 17, 29).
However, this reductionistic approach is insufficient to reveal the
logic linking all the individual CB cycles in the contracting left
ventricle (LV). The spatiotemporal relationships among countless CB
cycles in LV contraction are complicated;
Ca2+ diffusion is a physically
probabilistic event and plays an important role in the
excitation-contraction coupling prerequisite to CB cycling (9, 11, 16,
17, 19, 26, 27, 29, 31). Therefore, an integrative physiological view
of LV contraction based on the known molecular events remains to be elucidated.
We searched for macroscopic knowledge of LV contraction as a
clue to give us an integrative physiological view of the logic underlying LV contraction. As the first step, we have already decomposed the LV isovolumic pressure-time curve
[P(t)] into a pressure-rising curve and a pressure-falling curve (Fig.
1) and reported (18) that the LV isovolumic
P(t) curve was well fitted by the
"hybrid logistic function
[L(t)]" as a
difference of two logistic functions. We also showed that the same
L(t) was applicable to both isolated
and in situ papillary muscles (22, 24).
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Fig. 1.
Schematic illustration of decomposition of left ventricular (LV)
isovolumically developed pressure-time curve. We decomposed LV
isovolumically developed pressure-time curve (solid line) into
pressure-rising (dotted line) and pressure-falling curves (dashed
line).
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In this study, we report a more interesting mathematical function for
the LV P(t) curve than our previous
L(t). By trial and error, we found
that the difference curve of two Weibull distribution functions (7, 14,
21, 23) better fitted all the LV isovolumic P(t) curves despite fewer degrees of
freedom than the L(t) (18). Although
curve fitting is empirical, the mathematical property of the Weibull
distribution would give a new integrative physiological insight into
the logic underlying LV isovolumic pressure generation.
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METHODS |
Surgical preparation.
All experiments were performed according to the "Guiding Principles
for the Care and Use of Animals in the Field of Physiological Sciences" approved by the Council of the Physiological Society of
Japan. Experiments were performed in the excised, cross-circulated (blood perfused) canine heart preparation that has consistently been
used in our laboratory. Surgical procedures were described in detail
elsewhere (3, 18, 20). Briefly, two mongrel dogs (body wt 5-24 kg)
were anesthetized with pentobarbital sodium (25 mg/kg iv) after
premedication with ketamine hydrochloride (10 mg/kg im) and
artificially ventilated. Both dogs were heparinized (10,000 U/dog). The
larger dog was used as the metabolic supporter; the common carotid
arteries and right external jugular vein were cannulated and connected
to the arterial and venous cross-circulation tubes, respectively.
The chest of the smaller dog, the heart donor, was opened midsternally.
The arterial and venous cross-circulation tubes from the support dog
were cannulated into the left subclavian artery and the right ventricle
via the right atrial appendage, respectively, of the donor dog. The
heart-lung section was isolated from the systemic and pulmonary
circulation by ligating the descending aorta, inferior vena cava,
brachiocephalic artery, superior vena cava, azygos vein, and bilateral
pulmonary hili sequentially. The beating heart, supported by cross
circulation, was then excised from the chest. Coronary perfusion of the
excised heart was never interrupted during the preparation.
The left atrium was opened, and all the LV chordae tendineae were cut.
A thin latex balloon (unstressed volume ~50 ml) mounted on a rigid
connector was fitted into the LV, and the connector was secured at the
mitral annulus. LV pressure was measured with a miniature pressure
gauge (model P-7, Konigsberg Instruments, Pasadena, CA) placed inside
the apical end of the balloon, processed with a DC strain amplifier,
and low-pass filtered at a corner frequency of 100 Hz (model 6M76, NEC
San-ei, Tokyo, Japan). This corner frequency was high enough not to
distort the original P(t) signal.
The balloon, primed with water without any air bubbles, was connected
to a custom-made volume servo-pump (Air-Brown, Tokyo, Japan). The
servo-pump enabled us to control and measure LV isovolumic volume
accurately. LV epicardial electrocardiogram (ECG) was recorded with a
pair of screw-in electrodes to trigger data acquisition and to identify
the onset of contraction.
The temperature of the heart in an acrylic box was monitored and
maintained with heaters near 36°C (35.7-37.5°C) throughout the experiment. The left atrium was electrically paced at 137 ± 12 (mean ± SD) beats/min, ~20% above the spontaneous sinus rate, to
avoid arrhythmias. The systemic arterial blood pressure of the support
dog, which was 120 ± 12 mmHg, served as the coronary perfusion
pressure of the excised heart; it was maintained stable in each
experiment by slowly transfusing whole blood reserved from the heart
donor dog or by infusing dextran solution as needed. Arterial pH,
PO2, and
PCO2 of the support dog were
repeatedly measured and maintained within physiological ranges with
supplemental oxygen and intravenous sodium bicarbonate as needed.
Data sampling.
We performed the experiments in a total of 23 hearts. In each
experiment, steady-state isovolumic contractions were produced at five
different LV volumes by changing LV volume between 11.8 and 44.0 ml/100
g LV with the volume servo-pump. Three consecutive steady-state
isovolumic beats were sampled for analyses at each fixed LV volume. We
investigated a total of 345 beats (= 23 hearts × 5 volumes × 3 beats). LV pressure and volume were sampled at 2-ms intervals
and processed with a signal processor (7T18, NEC San-ei). The onset of
contraction was identified as the rise of the QRS wave of the LV
epicardial ECG. The end of contraction was identified as the time when
LV pressure returned to the end-diastolic pressure level. We analyzed
the developed LV P(t) curve from the onset to the end of LV contraction. On the average, end-diastolic pressure was 0.8 ± 6.9 mmHg. The first derivative of LV
P(t), dP(t)/dt,
was obtained by digitally differentiating the LV
P(t). To suppress a small noise in
the derivative signal, raw pressure signals were first smoothed
digitally by five-point, nonweighted moving average.
Weibull distribution function.
In systems engineering, the Weibull distribution is widely used to
characterize the time course of the cumulative breakdown of a complex
system (7, 14, 21, 23). The Weibull distribution characterizes the
performance of the entire system regardless of its actual structure and
constitution. The Weibull distribution function is given by
H · (1 
exp{[(t G)/h]m}),
where H is the total number of
breakable subsystems in a given complex system,
m is the shape parameter,
h is the scale parameter, and
G is the location parameter of the
function curve (7, 12-14, 21, 23, 30). These parameters change the
shape and height of the Weibull distribution curve considerably;
m and
h are the only two parameters
essential to determine the time course of the Weibull distribution
curve (Fig. 2).
H specifies the amplitude of the
curve, G determines the onset time of
the curve, and 1 exp{[(t G)/h]m}
gives a probability distribution function of unity amplitude. The first
derivative is given by
H · (m/h · [(t G)/h]m 
1 · exp{[(t G)/h]m}).
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Fig. 2.
Weibull distribution curves change with varying parameters.
A: effects of changing
m (shape parameter).
H (no. of breakable subsystems),
h (scale parameter), and
G are constant, and
m changes from 0.5 to 6. Here
G is location parameter and determines
onset time of curve. As m increases,
curve becomes steeper. B: effects of
changing h.
H, m,
and G are constant, and
h changes from 50 to 200. As
h increases, slope of curve decreases
and takes a longer time to reach a steady value.
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Hybrid Weibull function.
We named the difference of two Weibull distribution functions
a "hybrid Weibull" function
[W(t)]:
W(t) = H · (1 exp{[(t G)/h1]m1}) H · (1 exp{[(t G)/h2]m2}). The first and second terms on the right-hand side correspond to the
pressure-rising and -falling component curves, respectively (Fig. 1).
Parameters G and
H are common to both terms, because we
assumed that each curve starts simultaneously from zero pressure and
reaches the same absolute pressure. G
corrects a small time lag between the onsets of data sampling and
observed pressure rise. H determines
the amplitudes of both curves. Consequently, only four parameters
[m1,
m2,
h1, and
h2 in
W(t)] are essential to express
any LV isovolumic P(t) curve. The
degrees of freedom are six in W(t).
The first derivative of W(t),
dW(t)/dt,
is given by
H · (m1/h1 · [(t G)/h1]m11 · exp{[(t G)/h1]m1}) H · (m2/h2 · [(t G)/h2]m21 · exp{[(t G)/h2]m2}),
with the same degrees of freedom as
W(t).
dW(t)/dt = 0 maximizes W(t) to its
theoretical peak pressure. When the second derivative of
W(t),
d2W(t)/dt2,
is equal to 0, dW(t)/dt
is maximized or minimized to the theoretical peak positive and negative
first derivative of P (±dP/dt).
Because we cannot solve
dW(t)/dt = 0 and
d2W(t)/dt2 = 0 analytically, we solved them numerically using Mathematica Enhanced
(version 2.2; Wolfram Research, Champaign, IL) on a Macintosh computer.
Hybrid logistic function.
We have previously demonstrated that the hybrid logistic function
[L(t)], as the
difference of two logistic functions, could fit well the LV isovolumic
P(t) curve (18) as well as the
isometric papillary muscle force curve (22, 24).
L(t) is given by
a/{1 + exp[(4 · b/a) · (t c)]} d/{1 + exp[(4 · e/d) · (t f)]} + g; a,
b, c,
d, e,
f, and
g are parameters. The degrees of
freedom are seven, greater by one than those of
W(t), although those of the first
derivative of L(t),
dL(t)/dt,
are six, the same as those of
dW(t)/dt.
Curve fitting by hybrid Weibull function.
We obtained the best-fit set of the six parameters
(m1,
m2,
h1,
h2,
H,
G) of the
W(t) curve for each observed
isovolumic LV P(t) curve by the
least-squares method on the computer. These best-fit parameter values
were adopted as the calculated parameters of
dW(t)/dt.
We calculated correlation coefficients between each P(t) curve and the best-fit
W(t) and between each
dP(t)/dt
curve and
dW(t)/dt
with the same best-fit parameters. These curve fittings were performed
for all 345 curves in the 23 hearts. We evaluated goodness of
W(t) and
dW(t)/dt
fit to the entire P(t) and
dP(t)/dt curves by the correlation coefficients.
The peak pressure and peak ±dP/dt
characterize the LV contraction. We also evaluated goodness of
W(t) fitting to the
P(t) curve by simple linear
regression between theoretically calculated and observed values for
each end-systolic pressure, peak
±dP/dt, time to end-systolic
pressure, and time to peak ±dP/dt.
Comparison of goodness of fit between the hybrid Weibull function
and hybrid logistic function.
We obtained the best-fit set of the seven parameters
(a-g) of the
L(t) curve as well for each
isovolumic LV P(t) curve by the
least-squares method. These best-fit parameter values were adopted as
the calculated parameters of
dL(t)/dt.
We calculated correlation coefficients
(r) between each
P(t) curve and the best-fit L(t) and between each
dP(t)/dt
curve and
dL(t)/dt
with the best-fit six parameters. These curve fittings were performed
for all 345 curves in 23 hearts.
We evaluated the goodness of fit of the
W(t) with the
L(t) by comparing
r between the best-fit theoretical and
observed LV P(t) curves in a total
of 345 beats. We also compared the goodness of fit between
W(t) and
L(t) in the 345 beats by residual
mean squares (RMS) (28). RMS takes into account the difference of the
number of function parameters, being calculated as the residual sum of
squares divided by the residual degrees of freedom (the number of
points analyzed minus the number of variables in the function).
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RESULTS |
Goodness of fit of hybrid Weibull function with varied preload.
Figure 3,
A and
B, shows two
W(t) curves best fitted to the
P(t) curves at the smallest and
largest LV volumes in the 23 experiments. They are virtually
superimposed; it is impossible to distinguish the observed data points
from the fitted curves at this magnification. The correlation
coefficients were 0.999954 and 0.999928, respectively. The best-fit
W(t) curve was decomposed into the
two components, namely, the pressure-rising and -falling curves (Fig.
1). Figure 3, C and
D, shows that the
dW(t)/dt
curves calculated from the best-fit
W(t) curves in Fig. 3,
A and
B, closely fitted the
dP(t)/dt curves calculated from P(t) curves
in Fig. 3, A and
B; the correlation coefficients were
0.999247 and 0.998755. The calculated derivative curves were also
closely fitted to the derivative curves of the observed pressure. The
first derivative curves of the pressure-rising and -falling curves were
also drawn.
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Fig. 3.
Representative hybrid Weibull function
[W(t)] curves best
fitted to 2 observed isovolumically developed pressure-time curves of
canine LV (A and
B) and their first derivative curves
(C and
D) at LV volumes of 11.8 (A and
C) and 44.0 (B and
D) ml/100 g LV. In
A and
B, dotted curves represent
experimental pressure values sampled at 2-ms intervals at respective LV
volumes, solid curves represent best-fit
W(t), and short-dashed and
long-dashed curves represent pressure-rising and pressure-falling
components, respectively. In C and
D, dotted curves represent first
derivative values obtained by digitally differentiating observed
pressure-time values in A and
B, solid curves represent first
derivative curves of W(t) with
best-fit parameters, and short-dashed and long-dashed curves represent
first derivatives of pressure-rising and pressure-falling components,
respectively.
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All 345 curves in the 23 hearts yielded essentially the same results.
The correlation coefficients between all the
W(t) and P(t) curves were 0.999953 ± 0.000027 (mean ± SD). The correlation coefficients between all the
dW(t)/dt
and
dP(t)/dt
curves were 0.999024 ± 0.000460.
The theoretically calculated values for end-systolic pressure, time to
end-systolic pressure, peak
±dP/dt, and time to peak ±dP/dt of all 345 beats showed a
very highly linear correlation with observed values. All regression
lines were close to the identity lines (Fig.
4,
A-F).
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Fig. 4.
Relationship between observed and calculated values of indexes under
varied preloads (n = 345 beats).
Regression line is indicated by solid line, and line of identity is
indicated by dashed line in each panel.
A: relationship between observed and
calculated end-systolic pressures with high correlation
[y = 1.0001x + 0.0986, correlation
coefficient (r) = 0.9998, P < 0.01].
B: relationship between observed and
calculated peak positive first derivative of pressure
(dP/dt) values with high correlation
(y = 1.0240x 23.4573, r = 0.9978, P < 0.01).
C: relationship between observed and
calculated peak negative dP/dt values
with high correlation (y = 0.9518x 15.5234, r = 0.9974, P < 0.01).
D: relationship between observed and
calculated time to end-systolic-pressure values with high correlation
(y = 0.9889x + 1.9726, r = 0.9959, P < 0.01).
E: relationship between observed and
calculated time to peak positive dP/dt
values with high correlation (y = 0.9578x + 1.1663, r = 0.9696, P < 0.01).
F: relationship between observed and
calculated time to peak negative dP/dt
values with high correlation (y = 0.9880x + 3.0094, r = 0.9946, P < 0.01).
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These results indicate that W(t)
could well characterize the entire time course, the parts, and the
first derivative of the LV P(t)
curve regardless of preload and evidently needs only four parameters
(m1,
m2,
h1,
h2) to express
the entire time course of a given
P(t) curve.
Comparison of goodness of fit between hybrid Weibull function and
hybrid logistic function.
Figure 5 shows the relationships between
correlation coefficients of W(t) and
those of L(t) (Fig.
5A) and between correlation coefficients of
dW(t)/dt
and those of
dL(t)/dt
(Fig. 5B) in all 345 beats in the 23 hearts. The correlation coefficients of
W(t) and
dW(t)/dt
were always larger than those of
L(t) and
dL(t)/dt.
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Fig. 5.
Relationships between correlation coefficients of
W(t) and those of hybrid logistic
function [L(t)]
(A), between correlation
coefficients of first derivative of
W(t)
[dW(t)/dt]
and those of L(t)
[dL(t)/dt]
(B), between residual mean squares
(RMS) of W(t) and those of
L(t)
(C), and between RMS of
dW(t)/dt
and those of
dL(t)/dt
(D) in all 345 beats in all 23 hearts. P(t) is LV isovolumic
pressure-time curve, and
dP(t)/dt
is first derivative of P(t). Lines
of identity are indicated by solid lines.
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Figure 5 also shows the relationships between RMS values of
W(t) and those of
L(t) (Fig.
5C) and between RMS values of the dW(t)/dt
and those of
dL(t)/dt
(Fig. 5D) in all 345 beats. The RMS
values of W(t) and
dW(t)/dt
were always smaller than those of
L(t) and
dL(t)/dt.
These results evidently indicate that
W(t) could always better fit LV
P(t) curves than
L(t) despite a smaller number of
parameters of W(t).
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DISCUSSION |
The present results show that our proposed
W(t) fits the observed LV isovolumic
P(t) curve excellently at any
preload in the canine heart. The
W(t) with only six parameters
expresses almost completely any of the LV isovolumic
P(t) curves we studied. Only four
parameters (m1,
m2,
h1, and
h2) in
W(t) (see
METHODS) are essential to express
the entire time course of any LV isovolumic pressure.
dW(t)/dt
also fitted
dP(t)/dt
well. Therefore, we would consider the possibility that the LV
isovolumic P(t) curve may have
Weibull function characteristics in both pressure-rising and -falling phases.
Systems engineering view.
Given a complex system consisting of multiple elements that are working
together, if the system stops its performance (breakdown) even when
only one of the elements stops its performance, their performances (but
not structures) are considered to link in series and hence the system
is called a series-link system (12, 13). Conversely, if
the system does not stop its performance until all the elements stop
their performances, these performances are considered to link in
parallel and the system is called a parallel-link system. There are
intermediate types of systems between the series-link and parallel-link systems.
The probability that the system would stop its performance within a
given period is highest in the series-link system, lowest in the
parallel-link system, and intermediate in the other types of systems.
Therefore, the cumulative breakdown distribution of the series-link
system, i.e., the Weibull distribution, saturates fastest of all the
cumulative breakdown distributions.
In systems engineering, many complex systems consisting of multiple
elements are known to be series-link systems (23). Therefore, the
Weibull distribution has been widely used to characterize the
cumulative distribution of the time-dependent breakdown of the complex
system. Because the LV could be considered as a complex system, it
seems reasonable that LV performance was characterized by the Weibull distribution.
Physiological significance.
Our present study has revealed that the LV isovolumic pressure curve
highly resembles the W(t), i.e., the
difference of two Weibull functions. This suggests that both the
pressure-rising and -falling component curves may enable the LV to
raise and lower its pressure as fast as possible because of the
advantage of the series-link system described in
Systems engineering view.
This logic in the LV pressure generation may provide the most
beneficial strategy of contraction to the LV as a pressure generator or
compression pump.
Integrative view.
The Weibull properties of the LV
P(t) allow us to view cardiac
contraction in a new, integrative manner. When a function with a small
number of parameters accurately fits the
P(t) curve, we must examine the
possibility that the mathematical property of the function and the
physiological property of the P(t)
curve have something in common with each other.
The Weibull distribution expresses the time course of the cumulative
breakdown of the series-link systems in a complex system. Therefore,
the resemblance of the pressure-rising component to the Weibull
distribution suggests that the LV pressure rise occurs as the result of
time-dependent breakdowns of the series-link systems within the LV
wall. LV pressure is developed by cumulative attachment of CBs made of
myosin head bound to actin (15, 26). Each CB attachment occurs when
Ca2+ is bound to troponin C and
releases the troponin C inhibition of CB attachment (15, 26). Each CB
attachment can develop unitary force. The more the troponin C
inhibition is released, the more CB attachment occurs and the more
force is developed (26). The release of each troponin C inhibition
seems to correspond to a breakdown of a series link. Therefore,
Ca2+-free troponin C can be
considered an inhibiting system for LV pressure.
The Weibull character of the pressure-falling component suggests that
the LV pressure fall also occurs as time-dependent breakdowns of the
series-link systems within the LV wall. LV pressure falls by cumulative
detachment of CBs (15, 26). Each CB detachment occurs by hydrolyzing
ATP and loses unitary force (15, 26). The more CBs that are detached,
the more force is lost (26). Each CB detachment seems to correspond to
a breakdown of a series link. Therefore, attached CBs can be considered
a holding system for LV pressure. These CB-inhibiting and -holding
systems are considered to counteract with each other to develop and
maintain pressure and finally to relax in each LV contraction.
It is reasonable to assume that these inhibiting and holding
series-link systems are linked both in parallel and in series in the LV
wall. CBs are linked in parallel in each sarcomere and in series along
each myofibril across sarcomeres. The number of attached CBs maximally
amounts to ~1019 in 100 g of
myocardium (150 µmol/kg) (2). The amount of
Ca2+ released for the CB
attachment is of a comparable order of magnitude (50-100
µmol/kg, 2 orders of magnitude greater than free
Ca2+ or
Ca2+ transient) (4, 10, 25, 32).
As described in the introduction, all
Ca2+ diffusion and concentration
and CB attachment and detachment are spatiotemporarily probabilistic.
Although no one knows the logic underlying these probabilistic events
as a whole, we have found them to be integratively characterized by
W(t). However, there is still a
black box between CB cycling and LV pressure. For this reason,
W(t) might be a new strategy to
reveal the logic linking the individual CB cycles in LV contraction.
This strategy warrants further study as to whether
W(t) is applicable to myocardial and
myocyte force generation.
Limitation of the study.
The W(t) with only six parameters
expresses almost completely any of the LV isovolumic
P(t) curves we studied. However, we would not deny the possibility of the existence of better viewpoints and mathematical functions than ours.
We only studied isovolumic contractions. The Weibull-function model
would be applicable to the isovolumic contraction and relaxation
phases. However, we do not yet know how to modify the model during
ejection. This question remains to be solved.
We conclude that our proposed hybrid Weibull function
W(t) almost completely fits and
characterizes the LV isovolumic P(t) curve with only four parameters at any preload. Therefore, we propose
that the Weibull distribution function characterizes the integrative/summative performance of unitary force-generating cross-bridge cycles under physiological excitation-contraction coupling
in isovolumic LV contraction. The
W(t) would facilitate better
understanding of the logic underlying the performance of the heart as a
complex system consisting of countless cross bridges.
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ACKNOWLEDGEMENTS |
The authors greatly thank Kimikazu Hosokawa for the care of the
experimental animals and Dr. Tad W. Taylor for linguistic advice on our manuscript.
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FOOTNOTES |
This study was partly supported by Grants-in-Aid for Scientific
Research (07508003, 08670052, 09307029, 09470009, 09670053, 10770307, 10558136, 10877006) from the Ministry of Education, Science, Sports and
Culture, a Research Grant for Cardiovascular Diseases (7C-2) from the
Ministry of Health and Welfare, 1997-1998 Frontier Research Grants
for Cardiovascular System Dynamics from the Science and Technology
Agency, and research grants from the Ryobi Teien Foundation, the
Mochida Memorial Foundation, and the Nakatani Electronic Measuring
Technology Association, all of Japan.
Address for reprint requests and other correspondence: J. Araki, Dept.
of Physiology II, Okayama Univ. Medical School, 2-5-1 Shikata-cho,
Okayama, 700-8558, Japan (E-mail:
jaraki{at}med.okayama-u.ac.jp).
Received 24 December 1997; accepted in final form 25 May 1999.
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