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(Circulation. 1995;92:2318-2326.)
© 1995 American Heart Association, Inc.

Articles

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

Better Alternative to Exponential Time Constant

Hiromi Matsubara, MD; Miyako Takaki, PhD; Shingo Yasuhara, MD; Junichi Araki, MD; Hiroyuki Suga, MD, DMSc

From the Department of Physiology II, Okayama University Medical School, Okayama, Japan.

Correspondence to Hiromi Matsubara, MD, Department of Physiology II, Okayama University Medical School, 2-5-1 Shikata-cho, Okayama City, Okayama 700, Japan.

	Abstract

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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.

Key Words: ventricles • mechanics • diastole • diagnosis

	Introduction

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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 al⁵ 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 model¹⁰ 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

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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, PO₂, and PCO₂ 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 beatsx3 levelsx7 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 beatsx3 levelsx7 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 beatsx3 levelsx7 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

(1)

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}

(2)

where P is a nonzero asymptote, P₀ 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.

View larger version (21K): [in this window] [in a new window]   Figure 1. Schematic of pressure–time [P(t)] curves (A and E), semilogarithmic curves of P(t)-PB and P (B and F, respectively), dP/dt curves (C and G), and dP/dt–P(t) phase–plane curves (D and H) by the logistic (A through D) and monoexponential (E through H) models. A, P(t) curve by the logistic model (Equation 1) decays monotonically from P(0) (PA/2+PB) toward PB. TL is the time for P(t)-PB to decay from P(0) to its 2/(1+e) (0.54). B, Semilogarithmic curve of P(t)-PB in the logistic model is slightly concave to the origin. C, dP/dt curve in the logistic model (Equation 3) is blunt near time zero. D, dP/dt–P(t) phase–plane curve in the logistic model (Equation 5) is downward-convex. E, P(t) curve by the monoexponential model (Equation 2) decays monotonically from P0+P toward P. TE is the time for P(t)-P to decay from P(0) to its 1/e (0.33). F, Semilogarithmic curve of P(t)-P in the monoexponential model is linear. Negative reciprocal of the slope of this relation equals TE. G, dP/dt curve in the monoexponential model (Equation 4) is sharp near time zero. H, dP/dt–P(t) phase–plane curve in the monoexponential model (Equation 6) is linear. The negative reciprocal of the slope of this relation also equals TE.

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

(3)

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

(4)

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

(5)

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

(6)

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 ( P₀, 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; P₀, 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 RMS¹¹ 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

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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).

View larger version (26K): [in this window] [in a new window]   Figure 2. Plots of LV pressure [P(t)] (A and C) and residual (observed minus model) plots of the logistic (B) or monoexponential model (D) during isovolumic relaxation obtained from a representative isovolumic beat. Cutoff point is EDP+0 (see "Methods"). shows observed data points measured every 2 ms. These are the same in A and C. Solid lines show theoretically calculated curves using the logistic (A) and monoexponential (C) models. A, The logistic model (Equation 1; solid line) precisely fitted the observed P(t) curve (). In this beat, the best-fit parameters of Equation 1 were PA=132.0 mm Hg, TL=34.0 ms, and PB=0.6 mm Hg. B, Residual plots of the best-fit logistic model shown in A. Residuals were always close to zero. C, The exponential model (Equation 2; solid line) similarly fitted the observed P(t) curve (). In this beat, the best-fit parameters in Equation 2 were P0=76.3 mm Hg, TE=60.4 ms, and P=-8.2 mm Hg. D, Residual plots of the best-fit monoexponential model shown in C. Residuals were always larger than those of logistic model (B).

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 Hg². 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 Hg², ranging from 0.0026 to 0.7518 mm Hg² 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 Hg². On average, the RMS in the seven hearts was 0.6449±0.5275 mm Hg², ranging from 0.0534 to 3.0051 mm Hg² 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.

View larger version (35K): [in this window] [in a new window]   Figure 3. Plots of dP/dt vs time (A and C) and dP/dt vs instantaneous LV pressure [P(t)] (phase–plane diagram; B and D) during isovolumic relaxation obtained from the same beat shown in Fig 2. shows observed data points measured every 2 ms; each solid line shows theoretically calculated curve using the logistic (A and B) or monoexponential (C and D) models. A, The theoretically calculated best-fit dP/dt–time curve (Equation 3) (PA=132.0 mm Hg, TL=34.0 ms, and PB=0.6 mm Hg) precisely fitted the observed dP/dt–time curve. B, The theoretically calculated best-fit phase–plane curve (Equation 5) precisely fitted the observed phase–plane curve. C, The theoretically calculated best-fit dP/dt–time curve (Equation 4) (P0=76.3 mm Hg, TE=60.4 ms, and P=-8.2 mm Hg) was obviously different from the observed dP/dt–time curve, especially near the peak –dP/dt. D, The theoretically calculated best-fit phase–plane line (not curve; Equation 6) was also obviously different from the observed phase–plane curve.

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).

View larger version (18K): [in this window] [in a new window]   Figure 4. Plots showing the relations of logistic time constant (TL) and exponential time constant (TE) to EDP (A), heart rate (B), and EF (C). A, Correlations between LV EDP (x axis) and time constants TL and TE (y axis) in 63 isovolumic contractions. Both TL () and TE () were significantly (P<.001) increased with elevating EDP. Regression lines of TL and TE are shown by the solid and dashed lines, respectively. B, Relations between heart rate (x axis) and TL and TE (y axis) in 21 isovolumic contractions (mean±SD). Both TL () and TE () were significantly (P<.05) decreased with increasing heart rate. C, Relations between LV EF (x axis) and TL and TE (y axis) in 21 ejecting contractions (mean±SD). Both TL () and TE () were significantly (P<.01) decreased with increasing EF.

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.

View larger version (39K): [in this window] [in a new window]   Figure 5. Representative plots of observed dP/dt vs instantaneous LV P(t) (phase–plane diagram) obtained from the same beat demonstrated in Fig 2. Observed data points from the time of peak –dP/dt to the time when P(t) reached the three different relaxation pressure levels—ie, the same as the preceding EDP (EDP+0), 5 mm Hg above the preceding EDP (EDP+5), and 10 mm Hg above the preceding EDP (EDP+10)—are presented by ++, +, and , respectively. B and D are close-ups of rectangular areas in A and C, respectively. Solid, dashed, and dotted lines indicate best–fit curves in the logistic model (Equation 5) in A and B and best–fit lines in the monoexponential model (Equation 6) in C and D for the three cutoff points. A, Three best–fit curves in the logistic model for the three cutoff points were virtually identical. B, Three best–fit curves in the logistic model were almost superimposable even in the close–up. C, Three best–fit lines in the monoexponential model for the three cutoff points were markedly different even in the same beat. The slopes of these lines decreased and TE increased with advancing cutoff point. D, More details are explicit in the close–up.

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.

View larger version (16K): [in this window] [in a new window]   Figure 6. Plot showing the summarized relations of the logistic time constant (TL) and exponential time constant (TE) (y axis) and three different cutoff points (EDP+0, EDP+5, and EDP+10; x axis) in 14 isovolumic contractions (mean±SD). TL was constant regardless of varied cutoff points, whereas TE was significantly (P<.01) increased with advancing cutoff point.

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

Top Abstract Introduction Methods Results Discussion References

 
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 al⁵ 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 al¹⁰ 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.

Mirsky¹³ 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 Brutsaert²¹ 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 (P₀, 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 dogs⁷ 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

	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.

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

(7)

Similarly, substituting t=T_L into Equation 1 yields

(8)

From Equations 7 and 8,

(9)

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 P₀>0 and T_E>0.

Substituting t=0 into Equation 2 yields

(10)

Similarly, substituting t=T_E into Equation 2 gives

(11)

From Equations 10 and 11,

(12)

Therefore, T_E is the time for P(t)-P to decay from P(0) to its 1/e (0.33).

Received February 7, 1995; revision received April 19, 1995; accepted May 22, 1995.

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