## 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

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; Ca^{2+} 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).

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.

## 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, , and 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*)/d*t*, 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}}).

#### 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*)/*h*
_{1}]
}) − *H* ⋅ (1 − exp{−[(*t* −*G*)/*h*
_{2}]
}). 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 [*m*
_{1},*m*
_{2},*h*
_{1}, and*h*
_{2} 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*)/d*t*, is given by*H* ⋅ (*m*
_{1}/*h*
_{1} ⋅ [(*t*−*G*)/*h*
_{1}]^{m1}
^{−1} ⋅ exp{−[(*t*−*G*)/*h*
_{1}]^{m1}}) −*H* ⋅ (*m*
_{2}/*h*
_{2} ⋅ [(*t*−*G*)/*h*
_{2}]^{m2}
^{−1} ⋅ exp{−[(*t*−*G*)/*h*
_{2}]^{m2}}), with the same degrees of freedom as W(*t*). dW(*t*)/d*t*= 0 maximizes W(*t*) to its theoretical peak pressure. When the second derivative of W(*t*), d^{2}W(*t*)/d*t*
^{2}, is equal to 0, dW(*t*)/d*t*is maximized or minimized to the theoretical peak positive and negative first derivative of P (±dP/d*t*). Because we cannot solve dW(*t*)/d*t*= 0 and d^{2}W(*t*)/d*t*
^{2}= 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*)/d*t*, are six, the same as those of dW(*t*)/d*t*.

#### Curve fitting by hybrid Weibull function.

We obtained the best-fit set of the six parameters (*m*
_{1},*m*
_{2},*h*
_{1},*h*
_{2},*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*)/d*t*. We calculated correlation coefficients between each P(*t*) curve and the best-fit W(*t*) and between each dP(*t*)/d*t*curve and dW(*t*)/d*t*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*)/d*t*fit to the entire P(*t*) and dP(*t*)/d*t*curves by the correlation coefficients.

The peak pressure and peak ±dP/d*t*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/d*t*, time to end-systolic pressure, and time to peak ±dP/d*t*.

#### 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*)/d*t*. We calculated correlation coefficients (*r*) between each P(*t*) curve and the best-fit L(*t*) and between each dP(*t*)/d*t*curve and dL(*t*)/d*t*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).

## 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*)/d*t*curves calculated from the best-fit W(*t*) curves in Fig. 3,*A* and*B*, closely fitted the dP(*t*)/d*t*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.

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*)/d*t*and dP(*t*)/d*t*curves were 0.999024 ± 0.000460.

The theoretically calculated values for end-systolic pressure, time to end-systolic pressure, peak ±dP/d*t*, and time to peak ±dP/d*t* 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*).

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 (*m*
_{1},*m*
_{2},*h*
_{1},*h*
_{2}) 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.5
*A*) and between correlation coefficients of dW(*t*)/d*t*and those of dL(*t*)/d*t*(Fig. 5
*B*) in all 345 beats in the 23 hearts. The correlation coefficients of W(*t*) and dW(*t*)/d*t*were always larger than those of L(*t*) and dL(*t*)/d*t*.

Figure 5 also shows the relationships between RMS values of W(*t*) and those of L(*t*) (Fig.5
*C*) and between RMS values of the dW(*t*)/d*t*and those of dL(*t*)/d*t*(Fig. 5
*D*) in all 345 beats. The RMS values of W(*t*) and dW(*t*)/d*t*were always smaller than those of L(*t*) and dL(*t*)/d*t*. 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*).

## 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 (*m*
_{1},*m*
_{2},*h*
_{1}, and*h*
_{2}) in W(*t*) (seemethods) are essential to express the entire time course of any LV isovolumic pressure. dW(*t*)/d*t*also fitted dP(*t*)/d*t*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 Ca^{2+} 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, Ca^{2+}-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 ∼10^{19} in 100 g of myocardium (150 μmol/kg) (2). The amount of Ca^{2+} released for the CB attachment is of a comparable order of magnitude (50–100 μmol/kg, 2 orders of magnitude greater than free Ca^{2+} or Ca^{2+} transient) (4, 10, 25, 32). As described in the introduction, all Ca^{2+} 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.

## Acknowledgments

The authors greatly thank Kimikazu Hosokawa for the care of the experimental animals and Dr. Tad W. Taylor for linguistic advice on our manuscript.

## Footnotes

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).

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.

- Copyright © 1999 the American Physiological Society