# Logistic Time Constant of Isovolumic Relaxation Pressure–Time Curve in the Canine Left Ventricle

## Better Alternative to Exponential Time Constant

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## Abstract

*Background *The time constant of left ventricular (LV) relaxation derived from a monoexponential model has been widely used as an index of LV relaxation rate or lusitropism, although this model has several well-recognized problems. In the present study, we proposed a logistic model and derived a “logistic” time constant (*T*_{L}) as a better alternative to the conventional “exponential” time constant (*T*_{E}).

*Methods and Results *A total of 189 beats (147 isovolumic and 42 ejecting beats) were investigated in seven canine excised cross–circulated heart preparations. We found that the logistic model fitted much more precisely all the observed LV isovolumic relaxation pressure–time [P(t)] curves than the monoexponential model (*P*<.05). The logistic model also fitted well both the time curve of the first derivative of the observed P(t) (dP/dt) and the dP/dt–P(t) phase–plane curve. Like *T*_{E}, *T*_{L} indicated that volume loading depressed LV lusitropism and that increasing heart rate and ejection fraction augmented it. *T*_{L} was independent of the choice of cutoff point defining the end of isovolumic relaxation; *T*_{E} was dependent on that choice.

*Conclusions *We conclude that the logistic model better fits LV isovolumic relaxation P(t) than the monoexponential model in the present heart preparation. We therefore propose *T*_{L} as a better alternative to *T*_{E} for evaluating LV lusitropism.

LV diastolic dysfunction may precede its systolic dysfunction and cause congestive heart failure in certain heart diseases.^{1} ^{2} Thus, assessment of LV diastolic function is essential in basic and clinical cardiac studies. Many investigators have attempted to derive reliable indexes to evaluate LV relaxation from observed cardiac hemodynamics^{3} ^{4} and mathematical models expressing LV pressure decrease during isovolumic relaxation.^{5} ^{6}

The time constant of LV relaxation derived from a monoexponential model has been used widely as an index for evaluating LV relaxation rate or lusitropism in both experimental^{5} ^{6} ^{7} ^{8} ^{9} and clinical^{1} ^{10} ^{11} studies. Weiss et al^{5} originally determined the time constant by fitting the monoexponential model with a zero asymptote to LV pressure decrease during isovolumic relaxation after the time of peak negative value of the first derivative of LV pressure (dP/dt). Subsequent investigators added a nonzero asymptote to the monoexponential model^{7} ^{8} ^{11} ^{12} or proposed a two–sequential monoexponential model^{10} to improve the goodness of fit of the curve. Some of these investigators pointed out that the LV relaxation pressure decrease could not be characterized precisely by a monoexponential model.^{7} ^{8}

The monoexponential model and its modifications are merely empirical.^{8} ^{13} Therefore, there is no need to adhere to the monoexponential model for expressing LV pressure decrease during isovolumic relaxation. A model better than the monoexponential model has been expected.

In the present study, we proposed a logistic model as a new empirical model for LV isovolumic relaxation and investigated how well this model could express the pressure decrease observed experimentally during isovolumic relaxation after the time of peak −dP/dt. From this logistic model, we successfully derived a new “logistic” time constant (*T*_{L}) that is superior to the conventional “exponential” time constant (*T*_{E}) for evaluating LV lusitropism.

## Methods

### Surgical Preparation

Experiments were performed in the canine excised, cross–circulated (blood–perfused) heart preparation that has been used consistently in our laboratory.^{14} ^{15} Surgical procedures were described in detail elsewhere.^{16} Briefly, two mongrel dogs (body weight, 11 to 20 kg) were anesthetized with pentobarbital sodium (25 mg/kg IV) after premedication with ketamine hydrochloride (50 mg IM) and intubated in each experiment. Anesthesia was maintained by fentanyl (100 μg/h IV). Both dogs were heparinized (10 000 U IV). 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, which was the heart donor, was opened midsternally under artificial ventilation. The arterial and venous cross–circulation tubes from the support dog were cannulated into the left subclavian artery and the right ventricle through the right atrial appendage, respectively, of the donor dog. The heart-lung section was isolated from the systemic and pulmonary circulations by ligation of the descending aorta, inferior vena cava, brachiocephalic artery, superior vena cava, azygos vein, and bilateral pulmonary hili, in this order. 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. We gave diphenhydramine hydrochloride (10 mg IV) and indomethacin (5 mg IV) to the support dog to minimize occasional systemic hypotension under blood cross circulation.^{17} Our experience has shown that these doses of diphenhydramine and indomethacin help maintain the arterial blood pressure of the support dog without noticeably affecting contractile performance of cross–circulated canine hearts.

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) 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). This corner frequency was high enough not to blunt the original pressure signal. The balloon, primed with water without any air bubbles, was connected to a custom-made volume servo pump (Bokusui-Brown). LV volume was accurately controlled and measured with the servo pump. LV epicardial ECG was recorded with a pair of screw-in electrodes to trigger data acquisition and to identify end diastole.

Temperature of the heart in an acrylic box was monitored and maintained with heaters near 36°C (34.8°C to 37.5°C) throughout the experiment. The left atrium was electrically paced at a constant rate of 140 bpm throughout the experiment, ≈20% above a spontaneous sinus rate, to avoid arrhythmias. Systemic arterial blood pressure of the support dog, which was 108 to 136 mm Hg (124±10 mm Hg, mean±SD) throughout the experiment, served as coronary perfusion pressure of the excised heart. It was maintained stable in each experiment by slow transfusion of whole blood reserved from the heart donor dog or by infusion of dextran solution as needed. Arterial pH, Po_{2}, and Pco_{2} of the support dog were repeatedly measured and maintained within their physiological ranges with supplemental oxygen and intravenous sodium bicarbonate.

### Experimental Protocol

Experiments were performed in seven hearts. In each experiment, the following three protocols were performed: varied preload, varied heart rate in isovolumic contractions, and varied afterload in ejecting contractions. Steady state isovolumic contractions were obtained by fixing LV preload EDV at a desirable level with the volume servo pump. Steady state ejecting contractions against a desirable afterload (ejection) pressure were obtained by adjustment of stroke volume from a given EDV with the same pump.

In each steady state, 3 separate beats were sampled for analyses. Twenty–one isovolumic beats and 6 ejecting beats were sampled in each heart. A total of 189 beats consisting of 147 isovolumic and 42 ejecting beats were investigated in the seven hearts.

In protocol 1 (varied preload; n=63 beats [3 beats×3 levels×7 hearts]), LV EDV was varied to three different levels. We used isovolumic contractions at a constant heart rate (140 bpm) to exclude any influences of varied afterload and heart rate. On average, EDV and EDP ranged between 15.2±3.7 and 20.6±4.4 mL and between 1.0±3.9 and 9.3±4.6 mm Hg (mean±SD, n=21 beats), respectively.

In protocol 2 (varied heart rate; n=63 beats [3 beats×3 levels×7 hearts]), left atrial pacing rates were varied by ±20 bpm from 140 bpm. We used isovolumic contractions at a constant EDV (20.9±4.9 mL) to exclude any influences of varied preload and afterload.

In protocol 3 (varied afterload; n=63 beats [3 beats×3 levels×7 hearts]), LV ejection pressure was changed to vary EF from 0% (isovolumic contraction; n=21 beats to be added to 63+63 isovolumic beats in protocols 1 and 2) to approximately 15% (16.0±1.7%) and 30% (31.0±4.6%) at constant EDV (20.3±4.1 mL) and heart rate (140 bpm). This protocol simulates the in situ LV response to varied afterload.

### Data Analyses

LV pressure and volume data were sampled at 2-ms intervals and processed with a signal processor (model 7T18, NEC San-ei). End diastole was identified as the onset of the QRS wave of the LV epicardial ECG. In isovolumic contractions, the end point of LV relaxation was identified as the time when LV isovolumic relaxation pressure [P(t)] returned to the level of the preceding EDP. In ejecting contractions, the onset (end of ejection) and the end (onset of filling) of LV isovolumic relaxation were determined by the LV volume data. dP/dt was obtained by differentiating digitized P(t) data on a computer. To suppress a small noise in the derivative signal, raw P(t) signals were smoothed digitally by five–point, nonweighted, moving averaging on a computer.

### Mathematical Analyses

A new logistic model for LV P(t) during the isovolumic relaxation period defined above was given by

where P_{B} is a nonzero asymptote, P_{A} is an amplitude constant, t is time, and *T*_{L} is the time constant of the exponent. We designated *T*_{L} as a logistic time constant to distinguish it from the conventional time constant of the monoexponential model.

We compared the goodness of fit of the logistic curve and the monoexponential curve to the same P(t) curve during the same isovolumic relaxation period. We chose the following equation as the monoexponential model.^{7} ^{8} ^{11}

where P_{∞} is a nonzero asymptote, P_{0} is an amplitude constant, t is time, and *T*_{E} is the time constant of the exponent that has conventionally been used as the time constant of the monoexponential function. We called this time constant the exponential time constant. Equation 2 is a better monoexponential model^{11} ^{12} than the original monoexponential model with zero asymptote.^{5}

The P(t) curve of Equation 1 resembles that of Equation 2, as shown in Fig 1A⇓ and 1E⇓. However, the semilogarithm of Equation 1 minus P_{B} is slightly concave to the origin, as shown in Fig 1B⇓, whereas the semilogarithm of Equation 2 minus P_{∞} is theoretically linear, as shown in Fig 1F⇓. Although this difference may be small, it leads to the following substantial difference between the two models.

Differentiating Equation 1 yields dP/dt of the logistic model:

Differentiating Equation 2 yields dP/dt of the monoexponential model:

Equations 3 and 4 are shown in Fig 1C⇑ and 1G⇑, respectively. Fig 1C⇑ shows that the rate of increase in dP/dt given by Equation 3 [second derivative of P(t)] is gradually accelerated with time during the initial phase and is gradually decelerated during the later phase. In contrast, the rate of increase in dP/dt given by Equation 4 is continuously decelerated throughout isovolumic relaxation, as shown in Fig 1G⇑. As a result, the curve of Equation 3 near time zero (time of peak −dP/dt) is blunt, whereas the curve of Equation 4 near time zero is sharp. This difference between the two models underlies the following important difference.

From Equations 1 and 3, the dP/dt-P(t) phase–plane curve of the logistic model is given by

From Equations 2 and 4, the dP/dt-P(t) phase–plane curve of the monoexponential model is given by

These two models are obviously different in the phase–plane diagram. The trajectory of Equation 5 shows downward convexity, as shown in Fig 1D⇑, whereas that of Equation 6 shows linearity, as shown in Fig 1H⇑.

We obtained the best-fit set of the three parameters (P_{A}, P_{B}, and *T*_{L}) of the logistic model (Equation 1) and those ( P_{0}, P_{∞,} and *T*_{E}) of the monoexponential model (Equation 2) for each experimentally observed P(t) curve by nonlinear curve fitting on a computer. Then, we obtained theoretical dP/dt and dP/dt–P(t) phase–plane curves by substituting the best-fit sets of the three parameters into the corresponding equations ( P_{A}, P_{B}, and *T*_{L} into Equations 3 and 5; P_{0}, P_{∞}, and *T*_{E} into Equations 4 and 6). To evaluate the goodness of fit of each model, we compared the best-fit theoretical curves with the individual observed P(t), dP/dt, and dP/dt–P(t) curves.

If the logistic model (Equation 1) could express isovolumic relaxation P(t) more precisely than the monoexponential model (Equation 2), *T*_{L} in Equation 1 would be applicable as a better index of LV lusitropism than *T*_{E} in Equation 2. We compared these two time constants obtained from the P(t) data of protocols 1 through 3.

Previous investigators empirically chose various cutoff points to define the end of isovolumic relaxation in ejecting contractions.^{5} ^{7} ^{8} ^{9} ^{11} ^{12} Representative examples are (1) the time when P(t) returned to the level of the preceding EDP (cutoff point, EDP+0),^{5} ^{11} (2) the time when P(t) returned to the level of 5 mm Hg above the preceding EDP (cutoff point, EDP+5),^{7} and (3) the time when P(t) returned to the level of 10 mm Hg above the preceding EDP (cutoff point, EDP+10).^{9} ^{12} We then calculated *T*_{L} and *T*_{E} by using P(t) data up to the three different cutoff points in 14 arbitrarily selected isovolumic beats from the seven hearts. We studied whether *T*_{L} and *T*_{E} remained constant for the three different cutoff points.

### Statistical Analyses

We evaluated the goodness of fit of Equations 1 and 2 by comparing correlation coefficients between the best-fit theoretical and observed LV P(t) curves during isovolumic relaxation after the peak −dP/dt. The best-fit theoretical (Equations 3 and 4) and observed dP/dt curves and the best-fit theoretical (Equations 5 and 6) and observed dP/dt–P(t) phase–plane curves were also compared by their correlation coefficients. We tested the significance of the difference of these correlation coefficients (*r*) by a paired *t* test after their Z transformation: Z=1/2[ln(1+*r*)–ln(1–*r*)].^{18} This statistical procedure was performed in each beat for both logistic and monoexponential models.

We also analyzed residuals of Equations 1 and 2 and compared RMS^{11} between them by an *F* test for clearer demonstration of the difference in the goodness of fit. RMS is calculated as residual sum of squares divided by the residual degrees of freedom.

*T*_{L} and *T*_{E} calculated in protocol 1 were plotted against EDP, and simple linear regression analysis was done. Changes in *T*_{L} and *T*_{E} in protocols 2 and 3 were compared by repeated–measures ANOVA.^{9} The constancy of *T*_{L} and *T*_{E} calculated for the three different cutoff points of isovolumic relaxation was also compared by repeated–measures ANOVA. Data were expressed as mean±SD.

We considered the results statistically significant when *P*<.05.

## Results

### Results of Curve Fittings

Fig 2A⇓ shows a representative observed LV P(t) curve during isovolumic relaxation and the best-fit logistic P(t) curve (Equation 1) obtained in protocol 3. The logistic P(t) curve almost completely fitted the observed P(t) curve. Between the best-fit logistic and the observed P(t) curves, *r*=.9999. In all other hearts, logistic P(t) curves also closely fitted observed P(t) curves. On average, in the seven hearts, *r*=.9998±.0003, ranging from *r*=.9972 to *r*=1.0000 in all 189 beats (147 isovolumic and 42 ejecting beats).

Fig 2B⇑ shows residual (observed minus model) plots of the same best-fit logistic curve (Equation 1) shown in Fig 2A⇑. Residuals were always very close to zero. RMS of this beat was 0.053 mm Hg^{2}. In all other hearts, residuals of the logistic model were also very close to zero. The average RMS in the seven hearts was 0.1056±0.1260 mm Hg^{2}, ranging from 0.0026 to 0.7518 mm Hg^{2} in the 189 beats.

Fig 2C⇑ shows the same observed isovolumic P(t) curve shown in Fig 2A⇑ and the best-fit monoexponential P(t) curve (Equation 2). The monoexponential P(t) curve also fitted the observed P(t) curve well. Between the best-fit monoexponential and observed P(t) curves, *r*=.9991, which is a little lower than that of the logistic function (Fig 2A⇑). On average, in the seven hearts, *r*=.9988±.0006, ranging from *r*=.9961 to *r*=.9998 in the 189 beats. However, these correlation coefficients were significantly (*P*<.05) smaller than those described above for the logistic model.

Fig 2D⇑ shows residual plots of the same best-fit monoexponential model (Equation 2) shown in Fig 2C⇑. Although residuals were reasonably small, they were always larger than those of the logistic model (Fig 2B⇑). RMS of this beat was 0.6744 mm Hg^{2}. On average, the RMS in the seven hearts was 0.6449±0.5275 mm Hg^{2}, ranging from 0.0534 to 3.0051 mm Hg^{2} in the 189 beats. These RMSs were significantly (*P*<.05) larger than those of the logistic model. Therefore, the goodness of fit to the relaxation P(t) curve by the logistic model was always better than by the monoexponential model.

Fig 3A⇓ shows the calculated dP/dt (Equation 3) curve with the best-fit parameters obtained for the logistic model (Equation 1) from the same P(t) data shown in Fig 2A⇑. This curve closely fitted the observed dP/dt curve. Between the calculated and observed dP/dt curves, *r*=.9969. In all other hearts, calculated dP/dt curves also closely fitted the corresponding observed dP/dt curves. On average, in the seven hearts, *r*=.9946±.0033, ranging from *r*=.9866 to *r*=.9989 in the 189 beats.

Fig 3C⇑ shows the calculated dP/dt (Equation 4) curve with the best-fit parameters obtained for the monoexponential model (Equation 2) from the same P(t) data shown in Fig 2A⇑. This curve was obviously much sharper than the observed dP/dt curve near the peak −dP/dt. Between the calculated and observed dP/dt curves, *r*=.9619. On average, in the seven hearts, *r*=.9529±.0218, ranging from *r*=.8578 to *r*=.9866 in the 189 beats. These correlation coefficients were significantly (*P*<.01) smaller than those between the theoretical curves by the logistic model and the observed dP/dt curves in all beats. Therefore, the goodness of fit to the observed dP/dt curve by the logistic model was always better than by the monoexponential model.

Fig 3B⇑ shows that the calculated dP/dt–P(t) phase–plane curve (Equation 5) with the best-fit parameters also closely fitted the observed phase–plane curve. Between the calculated and observed curves, *r*=.9961. In all other hearts, the calculated phase–plane curves also closely fitted the observed phase–plane curves. On average, in the seven hearts, *r*=.9938±.0037, ranging from *r*=.9852 to *r*=.9992 in the 189 beats.

Fig 3D⇑ shows that the calculated phase–plane curve with Equation 6 was linear, whereas the observed phase–plane curve was obviously curvilinear. Between the calculated and observed curves, *r*=.9619. On average, in the seven hearts, *r*=.9556±.0211, ranging from *r*=.8884 to *r*=.9888 in the 189 beats. These correlation coefficients were significantly (*P*<.01) smaller than those between the calculated (Equation 5) and observed phase–plane curves in all beats. Therefore, the goodness of fit to the observed phase–plane curve by the logistic model was always better than by the monoexponential model.

All these results indicated that our logistic model expressed the LV P(t) decrease during isovolumic relaxation more precisely than the conventional monoexponential model.

### Comparison of Time Constants for Evaluating LV Relaxation

Fig 4A⇓ shows the relations of *T*_{L} and *T*_{E} to EDP in isovolumic contractions (protocol 1). *T*_{L} increased with EDP elevated by increasing EDV (*T*_{L}=0.5901 EDP+24.9940, *r*=.7060, *P*<.001, n=63 isovolumic beats). *T*_{E} also increased with EDP (*T*_{E}=1.1855EDP+40.7060, *r*=.7124, *P*<.001, n=the same 63 beats). Although *T*_{L} was always smaller than *T*_{E} at any LV EDP, both time constants increased similarly with volume loading; percent changes in the two time constants with volume loading were not significantly different (*P*>.29).

Fig 4B⇑ shows the relations of *T*_{L} and *T*_{E} to heart rate (protocol 2). *T*_{L} decreased significantly with increasing heart rate (*P*<.01). *T*_{E} also decreased significantly with increasing heart rate (*P*<.05). Although *T*_{L} was always smaller than *T*_{E} at any heart rate, both time constants decreased similarly with increasing heart rate; percent changes in the two time constants with increasing heart rate were not significantly different (*P*>.39).

Fig 4C⇑ shows the relations of *T*_{L} and *T*_{E} to EF (protocol 3). *T*_{L} significantly decreased with increasing EF (*P*<.01). *T*_{E} also significantly decreased with increasing EF (*P*<.01). Although *T*_{L} was always smaller than *T*_{E} at any LV EF, both time constants decreased similarly with increasing EF; percent changes in the two time constants with increasing EF were not significantly different (*P*>.30).

These results indicated that both *T*_{L} and *T*_{E} were equivalent in evaluating LV lusitropism.

### Effects of Isovolumic Relaxation Cutoff Points on *T*_{L} and *T*_{E}

Fig 5A⇓ and 5C⇓ show the same observed phase–plane curves as shown in Fig 3B⇑ and 3D⇑ with the three different cutoff points (EDP+0, EDP+5, and EDP+10; see “Methods”). Fig 5B⇓ and 5D⇓ are close-ups of the terminal parts of the two curves. The logistic model curves (Equation 5) with the three different cutoff points were almost superimposable even in the close-up. Consequently, *T*_{L} was virtually constant, regardless of the changed LV isovolumic relaxation cutoff point: *T*_{L}=34.0 ms for EDP+0, *T*_{L}=34.4 ms for EDP+5, and *T*_{L}=34.3 ms for EDP+10.

In contrast, the slope of the monoexponential model line (not curve; Equation 6) obviously decreased by advancing the LV isovolumic relaxation cutoff point, as shown in Fig 5C⇑ and 5D⇑. Because the reciprocal of the slope of these theoretical lines indicates *T*_{E} (see Equation 6),^{7} ^{8} *T*_{E} was gradually increased with the advancing isovolumic relaxation cutoff point; *T*_{E}=60.4 ms for EDP+0, *T*_{E}=72.8 ms for EDP+5, and *T*_{E}=83.9 ms for EDP+10 even in the same beat.

Similar results were obtained in the other 13 beats. On average, *T*_{L} was almost constant (*P*>.81), regardless of the isovolumic relaxation cutoff points, as shown in Fig 6⇓. However, *T*_{E} was significantly (*P*<.01) increased with the advancing isovolumic relaxation cutoff point, as shown in Fig 6⇓.

These results demonstrate the dependency of *T*_{E} on the choice of the isovolumic relaxation cutoff point. In contrast, the present results demonstrate the independence of *T*_{L} on the choice of cutoff point. In this respect, *T*_{L} is superior to *T*_{E}.

## Discussion

### Problems in the Monoexponential Model

The monoexponential model of LV P(t) decrease during isovolumic relaxation has commonly been used in both experimental^{5} ^{6} ^{7} ^{8} ^{9} and clinical^{1} ^{10} ^{11} studies since Weiss et al^{5} proposed it in 1976. On the basis of this model, LV relaxation rate or lusitropism has conventionally been characterized by *T*_{E}. The goodness of fit of the monoexponential model (Equation 2) to the experimentally observed P(t) decrease during isovolumic relaxation appears acceptable, as shown in Fig 2C⇑. Therefore, it seems reasonable for previous investigators to have believed that the P(t) decrease during isovolumic relaxation could be expressed reasonably by the monoexponential function.^{5} ^{6}

If the P(t) decrease during isovolumic relaxation could be expressed precisely by a monoexponential function, the semilogarithm of the pressure decrease above baseline (P_{∞} in Equation 2) should be linear, as shown in Fig 1F⇑. The semilogarithm of the observed P(t) decrease during isovolumic relaxation has been recognized to be slightly concave to the origin, even after baseline correction.^{8} This deviation from linearity, although small, is recognized as the basic problem of the monoexponential model,^{8} but this model is still used as an acceptable approximation.^{7} ^{8}

However, our results indicated that the monoexponential model curve of dP/dt (Equation 4) near the peak −dP/dt to be much sharper than the experimentally observed curve, as shown in Fig 3C⇑. Previous investigators recognized this problem and reported that the monoexponential fit was poor during the initial 10 to 20 ms after the peak −dP/dt.^{6} ^{19} ^{20} Although they claimed this tendency to be due to the flexibility of the aortic valve of in vivo hearts in ejecting contractions,^{6} ^{20} our results showed the same tendency even in isovolumic contractions, as illustrated in Fig 3C⇑.

The force [F(t)] decline phase–plane curve in an isometric twitch of the cat papillary muscle is curved but not linear even after the peak rate of F(t) decline.^{21} This result also indicates that F(t) decline in the papillary muscle could not be characterized as a monoexponential function because the phase–plane curve of the monoexponential model should be linear.^{21} In the whole-heart preparation in the present study, the observed P(t) decrease phase–plane curve also curved against our expectation of the linearity on the basis of the monoexponential model (Equation 6; Figs 3D⇑ and 5C⇑).

Taken together, these results indicate that the monoexponential model cannot precisely characterize the LV isovolumic relaxation P(t) decrease, even though the monoexponential curve resembles the experimentally observed LV P(t) decreasing curve.

Several previous investigators also recognized the limitation of the monoexponential model.^{7} ^{8} ^{10} ^{13} ^{21} Some proposed the other methods to analyze the LV lusitropism.^{10} ^{13} ^{21}

Rousseau et al^{10} divided the isovolumic relaxation period into the early and late phases and then fitted the two phases to two different monoexponential functions. Although they claimed that the goodness of fit of this two–sequential monoexponential model was higher than that of the monoexponential model, this model provided a discontinuous LV P(t) curve. Therefore, both dP/dt and phase–plane curves derived from this model are also discontinuous and cannot precisely fit the observed dP/dt and phase–plane curves.

Mirsky^{13} suggested a calculation method of relaxation half-time and time constant based on polynomial fitting to the LV P(t) during isovolumic relaxation. However, he did not put any direct theoretical meaning on the coefficients of the polynomial terms.

Finally, Sys and Brutsaert^{21} calculated a time constant from the terminal isometric twitch F(t) decline curve of the cat papillary muscle because they had noticed that the phase–plane curve near the end of F(t) decline tended to be linear. However, they did not define the time when the phase–plane curve started to become linear during isometric relaxation. In addition, their method will not be applicable for ejecting contractions in the whole LV because the phase–plane curve near the end of isovolumic relaxation corresponding to their terminal F(t) decline is usually unobtainable as a result of LV filling in ejecting contractions.

### Advantages of the Logistic Model

Because none of the previously proposed models could sufficiently express the LV P(t) decrease, we proposed the logistic model as a new model for LV P(t) decrease during isovolumic relaxation. The logistic function has been used in many fields of bioscience to express various curves^{22} ^{23} ^{24} and has the following attractive features.

First, the semilogarithmic LV P(t) curve expressed by the logistic function is slightly concave to the origin (Fig 1B⇑) even after the baseline (P_{B}) correction, although the logistic curve per se (not semilogarithmic curve) resembles the monoexponential curve in Fig 1A⇑ and 1E⇑. This small difference between the two models allows the logistic function to express the observed LV P(t) decrease more precisely during isovolumic relaxation than the monoexponential function.

Second, the first derivative of the logistic function (Equation 3) demonstrates a blunt rise near the peak −dP/dt, as shown in Figs 1C⇑ and 3A⇑. This feature allows the logistic function to express the observed dP/dt curve precisely during isovolumic relaxation.

Third, these differences between the logistic model P(t) and dP/dt curves lead to the nonlinear phase–plane curve of the logistic model, as shown in Figs 1D⇑ and 3B⇑. This feature is the basis for the logistic function to express the observed dP/dt–P(t) phase–plane curve precisely.

Thus, the present results have shown that the logistic function expresses the observed LV P(t) decrease, dP/dt curve, and dP/dt–P(t) phase–plane curve more precisely during isovolumic relaxation than the monoexponential function. The present results have thus demonstrated the superiority of our newly proposed logistic model over the conventional monoexponential model.

Our logistic function (Equation 1) has the same number of parameters as the monoexponential function (Equation 2). These parameters (P_{A}, *T*_{L}, and P_{B}) have similar theoretical meanings as the corresponding parameters of the monoexponential function (P_{0}, T_{E}, and P_{∞}, respectively; see the “Appendix”). Therefore, *T*_{L} in our logistic model may be called the logistic time constant and can serve as a more reliable and advantageous alternative to the conventional exponential time constant.

*T*_{L} as an Index for Evaluating LV Isovolumic Relaxation Rate

The present results have shown that our newly proposed *T*_{L} is always smaller than the conventional *T*_{E}. *T*_{L} indicates the time when LV P(t)−P_{B} reaches 2/(1+e) (about 0.54) of LV P(t) at the peak −dP/dt (see the “Appendix”). In contrast, *T*_{E} indicates the time when LV P(t)−P_{∞} reaches 1/e (about 0.33) of LV P(t) at the peak −dP/dt (see the “Appendix”). Therefore, it is reasonable that *T*_{L} was always smaller than *T*_{E}, as shown in Figs 4⇑ and 6⇑.

The present study reconfirmed that *T*_{E} changes with volume loading (or increasing EDP), increasing heart rate, and decreasing afterload, as expected from the previous results.^{5} ^{7} ^{9} *T*_{L} also behaves like *T*_{E} under these conditions (Fig 4⇑). In this respect alone, both *T*_{L} and *T*_{E} appear to be useful for evaluating the LV lusitropism.

However, we must conclude that *T*_{E} is an insufficient index for evaluating the LV lusitropism. In ejecting contractions, isovolumic relaxation is terminated by the onset of LV filling. To identify this, simultaneous measurements of LV and left atrial pressures are needed,^{6} ^{8} but measuring left atrial pressure is difficult, especially in clinical settings. Therefore, previous investigators had to use various isovolumic relaxation cutoff points for the monoexponential model,^{5} ^{7} ^{9} ^{11} ^{12} and there has been no consensus on the best cutoff point.

We tested the three representative cutoff points in the monoexponential model, all of which had been used in previous studies.^{5} ^{7} ^{9} ^{11} ^{12} We have demonstrated that *T*_{E} significantly changes with the three different cutoff points even in the same beat (Fig 6⇑). Therefore, it seems meaningless to indiscriminately compare *T*_{E} values among studies using different cutoff points in the monoexponential model. Worse than this is the possibility that changes in *T*_{E} caused by different cutoff points mask real changes in the relaxation rate.

Despite some limitations, *T*_{E} could still be used by choosing a certain isovolumic relaxation cutoff point. However, it is difficult to fix the cutoff point for different ejecting contractions because the onset of ventricular filling varies among beats and hearts. In contrast to *T*_{E}, we have found that *T*_{L} is insensitive to the choice of isovolumic relaxation cutoff point. This is a definite advantage of our proposed *T*_{L} over *T*_{E}.

Accordingly, we conclude that *T*_{L} is a more reliable time constant for evaluating LV relaxation rate or lusitropism than *T*_{E}. In any respect, *T*_{E} has no advantages over *T*_{L}.

### Study Limitations

The excised heart preparation used in the present study was controlled with a volume servo pump. We can produce any desirable contraction by independently controlling several variables (ie, EDV, stroke volume, and ejection pressure). Although ejecting contractions in this preparation resemble those of in vivo hearts, they may not be exactly the same because afterloading conditions affect relaxation.^{9} For this reason, the present results may not be directly extrapolated beyond our study conditions. However, the problems of the monoexponential model have been reported both experimentally in closed-chest dogs^{7} and clinically.^{10} Therefore, we expect that the logistic model would be applicable to in vivo hearts to circumvent the problems of the monoexponential model. This warrants in vivo heart studies of *T*_{L}.

Although the logistic model has shown excellent curve fitting to an LV P(t) decrease during isovolumic relaxation, it is still an empirical model like other previous models, including the monoexponential model. It remains unknown whether LV relaxation has the mechanism of a logistic nature.

### Conclusions

We conclude that our newly proposed logistic model provides a better curve fit to the LV pressure decrease during isovolumic relaxation in both isovolumic and ejecting contractions. *T*_{L}, one of the three fitting parameters of our logistic model, can more reliably characterize the rate of LV relaxation or lusitropism than *T*_{E}. We therefore propose *T*_{L} as a better alternative than *T*_{E}.

## Selected Abbreviations and Acronyms

bpm | = | beats per minute |

EDP | = | end-diastolic pressure |

EDV | = | end-diastolic volume |

EF | = | ejection fraction |

LV | = | left ventricular |

RMS | = | residual mean square |

## Appendix A

The following derivations explain the meaning of *T*_{L} in the logistic model. P(t) in Equation 1 decays monotonically toward asymptote P_{B} when P_{A}>0 and *T*_{L}>0.

Substituting t=0 into Equation 1 yields

Similarly, substituting t=*T*_{L} into Equation 1 yields

From Equations 7 and 8,

Therefore, *T*_{L} is the time for P(t)−P_{B} to decay from P(0) to its 2/(1+e) (≈0.54).

The following derivations explain the meaning of *T*_{E} in the monoexponential model. P(t) in Equation 2 decays monotonically toward asymptote P_{∞} when P_{0}>0 and *T*_{E}>0.

Substituting t=0 into Equation 2 yields

Similarly, substituting t=*T*_{E} into Equation 2 gives

From Equations 10 and 11,

Therefore, *T*_{E} is the time for P(t)−P_{∞} to decay from P(0) to its 1/e (≈0.33).

## Acknowledgments

This study was partly supported by Grants-in-Aid for Scientific Research (05221224, 05305007, 06213226, and 06770494) from the Ministry of Education, Science and Culture; Research Grants for Cardiovascular Diseases (7C-2) and on Aging and Health from the Ministry of Health and Welfare; Joint Research Grants Utilizing Scientific and Technological Potential in the Region (1995) from the Science and Technology Agency; and a Research Grant from the Terumo Life Science Foundation, all from Japan. We thank Profs Y. Seino, Department of Pediatrics from which Dr Yasuhara was on leave, and M. Hirakawa, Department of Anesthesiology and Resuscitology from which Dr Araki was on leave, for partial financial support of the experiments. We also thank Prof T. Tsuji for generously lending us his departmental 7T18. We thank Drs K. Kohno, Department of Anesthesiology and Resuscitology, H. Yamaguchi, Department of Cardiovascular Surgery, and J. Shimizu, Tokyo Medical College, the Department of Internal Medicine II, for surgical assistance.

- Received February 7, 1995.
- Revision received April 19, 1995.
- Accepted May 22, 1995.

- Copyright © 1995 by American Heart Association

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- Logistic Time Constant of Isovolumic Relaxation Pressure–Time Curve in the Canine Left VentricleHiromi Matsubara, Miyako Takaki, Shingo Yasuhara, Junichi Araki and Hiroyuki SugaCirculation. 1995;92:2318-2326, published online before print October 15, 1995http://dx.doi.org/10.1161/01.CIR.92.8.2318
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- Logistic Time Constant of Isovolumic Relaxation Pressure–Time Curve in the Canine Left VentricleHiromi Matsubara, Miyako Takaki, Shingo Yasuhara, Junichi Araki and Hiroyuki SugaCirculation. 1995;92:2318-2326, published online before print October 15, 1995http://dx.doi.org/10.1161/01.CIR.92.8.2318

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