the ability to quantify the left ventricular (LV) relaxation rate in normal and pathological conditions is important in investigating myocardial pump function. Despite advances in noninvasive assessment of relaxation (7), invasive measurement of the ventricular relaxation rate during isovolumic relaxation remains the “gold standard.” Such invasive parameters include the first derivative of LV pressure with respect to time during isovolumic relaxation (dP/dt) and the time constant of isovolumic relaxation (τ).

The time course of the fall in LV pressure during isovolumic relaxation has been modeled using a monoexponential function with three parameters, described in Eq. 1as follows

where P(t) is LV pressure as a function of time (in mmHg), t is time (in ms), P_∞ is the asymptote to which LV pressure declines (in mmHg), and P₀ is LV pressure (in mmHg) at peak negative dP/dt (wheret = 0 ms). Measured pressure data during isovolumetric pressure decay is fitted to this model to obtain an estimation of τ (in ms).

Initially, Eq. 1 was linearized to avoid difficult calculation. Weiss et al. (1, 13) further simplified the situation by assuming a zero asymptote, yielding the reduction of a three-parameter monoexponential model to a two-parameter model

The assumption of a zero asymptote allows linearization by taking the natural logarithm of both sides of Eq. 2, from which linear regression analysis can be used to determine the least mean squared error (MSE) solution for τ and P₀. Although useful physiological insight has been gained from this approach, a disadvantage of the logarithmic transformation is to give undue weighting to data points (and noise) at low pressures.

A refinement of Weiss' log transformation approach is to substitute the differentiated monoexponential function back into Eq. 1, which allows linearization without the assumption of a zero asymptote (9). Again, linear regression analysis can be used to obtain an estimate of τ, but the differentiation process is very sensitive to noise in the signal.

The improved performance of contemporary computer hardware and software allows direct solution of Eq. 1 using nonlinear least squares parameter estimation techniques, most commonly the Levenberg-Marquardt method. This nonlinear technique allows for an accurate estimation of P₀ and τ both with and without the assumption of a zero asymptote. The use of this nonlinear technique for calculation of τ was initially validated by Bernardi et al. (1).

With nonlinear techniques widely available, they have largely superceded both the log transform and differentiation methods for solving Eq. 1. Nevertheless, the issue of choosing a two- or three-parameter model is still an open question because conflicting results have been reported when using the three-parameter model (free moving asymptote) versus the two-parameter model (zero pressure asymptote assumed) for calculation of τ. A key issue is the trade off between accuracy of fit to the observed data (which should be better with three parameters) and the confidence intervals of the derived parameters (which may worsen with three parameters if there is significant collinearity between them). The objective of this study, therefore, was to use high-fidelity animal data and numerical simulations to gain more insight into the reliability of the estimated relaxation constant when assuming either a zero or free moving pressure asymptote.

METHODS

LV pressure decay data obtained from both an animal experiment and Monte Carlo simulations were analyzed. τ was determined using the nonlinear Levenberg-Marquardt technique both with (two-parameter exponential model; LM₂) and without (three-parameter exponential model; LM₃) the assumption of a zero pressure asymptote. The differences between the two approaches were then compared for goodness of fit to the pressure curves and the confidence intervals of the estimated τ.

Animal experiment.

The investigation conformed to the Guide for the Care and Use of Laboratory Animals (8) published by the National Institutes of Health and was approved by the Animal Research Committee of the Cleveland Clinic Foundation. Eight healthy adult mongrel dogs of either sex weighing 29.7 ± 7.4 kg were studied. The dogs were anesthetized with 25 mg/kg intravenous pentobarbital sodium, and anesthesia was maintained throughout the experiments with additional aliquots of pentobarbital sodium. After the dogs underwent tracheal intubation, positive pressure mechanical ventilation was instituted using room air. A micromanometer catheter (Millar; Houston, TX) was introduced into the left atrium (LA) through the LA appendage and positioned across the mitral valve with the pressure sensor in the LV. LA pressure was recorded by an additional single sensor catheter. Pressure and electrocardiogram signals were digitally acquired with 1-ms/12-bit resolution using a multifunction input-output board (AT-MIO-16, National Instruments; Austin, TX) interfaced with a computer workstation (Intel 80486 PC) using customized software developed using LabView version 5.0 (National Instruments). Data acquisition was performed at baseline (for each dog experiment), during isoproterenol infusion (for 6 dog experiments), and during esmolol infusion (for 3 dog experiments). Baseline runs were initiated after allowing sufficient time for hemodynamics to stabilize before starting the experiment. Esmolol or isoproterenol medication runs were initiated after completion of a satisfactory number of baseline acquisition runs. Isoproterenol was infused at 0.025–0.4 μg · kg⁻¹ · min⁻¹intravenously, and data acquisition runs were initiated after sufficient washin time for an appropriate heart rate response and hemodynamics to stabilize. Similarly, esmolol was infused at 0.2–0.3 mg · kg⁻¹ · min⁻¹intravenously, with data acquisition after hemodynamic stabilization. For the eight dogs, 45 recordings during baseline, 12 recordings during isoproterenol infusion, and 8 recordings during esmolol infusion were registered, with each recording containing ∼7 consecutive heartbeats. We thus captured 340, 94, and 56 pressure decays at baseline and during isoproterenol and esmolol infusion, respectively. Post acquisition numerical analysis of raw pressure data was performed using another custom numerical analysis program developed in LabView. In this study, dP/dt was calculated to define the period of isovolumic relaxation as the time period between maximum negative pressure change and the first LA to LV pressure crossover. τ was determined from the pressure curves with the use of the nonlinear Levenburg-Marquardt technique both with and without the assumption of a zero pressure asymptote. With each derivation of the coefficients, a MSE value was calculated as a measure of “goodness-of-fit” of the specific model to the pressure data (Eq. 3) as follows

In this equation, (x_i,y_i) are the input data points, f(x_i;a₁… a_M) =f (X, A) is the nonlinear function (where a₁… a_M are coefficients), N the number of input data points, and ς_i the variance. To analyze the efficiency of the different models in estimating τ, the accompanying standard error for each derivation of τ was calculated. Method-dependent differences were analyzed.

Monte Carlo simulation.

One hundred instances of 125 different diastolic pressure curves were created with the use of Monte Carlo simulation in the following manner. First, an exact monoexponential curve was constructed using Eq.1 with the coefficients P₀ = 70 mmHg, τ = 60 ms, and P_∞ = 0 mmHg. By adding Gaussian noise (mean value 0 mmHg and SD 0.4 mmHg) randomly, 100 “data curves” were created from this exact monoexponential pressure decay. The simulation of one pressure curve is illustrated in Fig.1, showing the exact monoexponential curve (A), the Gaussian noise (B), and the simulated curve (C). From each of the 100 data sets, τ and P₀ were estimated using LM₂, and τ, P₀, and P_∞ were estimated using LM₃. With each derivation of the coefficients, the MSE value (Eq. 3) was calculated as a measure of goodness-of-fit of the specific model to the simulated pressure data. The estimated coefficients from the 100 data curves with the accompanying standard error were compared with the actual coefficients that produced the original pressure curve. To analyze a range of parameter values, the simulation was repeated with P₀varying from 70 to 110 mmHg in steps of 10 mmHg, τ varying from 40 to 120 ms in steps of 20 ms, and P_∞ varying from −5 to +5 mmHg in steps of 2.5 mmHg. Thus a total of 5 × 5 × 5 = 125 combinations of the coefficients are simulated, yielding the analysis of 12,500 pressure curves. The results were compared with the findings of the dog experiment.

Statistics.

All statistics were performed using SPSS version 9.0 (Chicago, IL). Values are means ± SD. Normally distributed variables, calculated using the different models, were compared using repeated-measures ANOVA. Post hoc testing was performed using either a Bonferronit-test (equal variances assumed) or a Dunnett'st-test (equal variances not assumed). Nonnormally distributed variables were compared using the nonparametric Friedman test for related variables. The level of significance was set at aP value of 0.05.

RESULTS

Animal experiment.

Table 1 summarizes the results of the determinations of τ in the dog experiment using the nonlinear Levenburg-Marquardt method with a zero pressure asymptote (LM₂) and a nonzero moving asymptote (LM₃) for the baseline measurements and during infusion of isoproterenol and esmolol. The mean values for τ are the result of averaging the calculated τ values obtained from LV pressure recordings using LM₂ and LM₃. Method-dependent differences were observed. At baseline, τ values obtained with LM₂ were consistently shorter than the values obtained with LM₃(P < 0.001), whereas during either isoproterenol or esmolol infusion, τ values obtained with LM₂ were consistently higher than the values obtained with LM₃. LM₃ most closely fits the pressure decays, as reflected by the significant difference in MSE between original and fitted pressure decays at baseline and during isoproterenol and esmolol infusion (P < 0.001). In contrast, however, the standard error of the estimate was significantly higher when τ was calculated using LM₃ compared with LM₂ at baseline as well as during isoproterenol and esmolol infusion (P < 0.001). LM₃ showed that P_∞ significantly increases during isoproterenol and esmolol infusion compared with baseline values. For both LM₂ and LM₃, τ decreased with isoproterenol and increased with esmolol infusion (P < 0.001). However, the relative change compared with baseline values for both isoproterenol and esmolol infusion was different when using LM₂ compared with LM₃. For LM₂, the infusion of isoproterenol resulted in a decrease of τ of 26%, whereas for LM₃ the decrease of τ was 42%. With the use of LM₂, the infusion of esmolol resulted in an increase of τ of 67%, whereas LM₃ resulted in an increase of τ of 30%.

Monte Carlo simulation.

Table 2 shows the results from a representative 2 of 125 pressure simulations (τ = 60 ms, P₀ = 70 mmHg, and P_∞ = 0 mmHg, and τ = 60 ms, P₀ = 70 mmHg, and P_∞ = −2.5 mmHg). In these and all other simulations, the standard errors of the τ estimates were significantly smaller for the zero asymptote model (LM₂) compared with the moving asymptote model (LM₃) (P < 0.001).

First, τ, P₀ and P_∞ were determined from 100 pressure decays created with Monte-Carlo simulation starting from a monoexponential curve using the Levenburg-Marquardt technique with a fixed zero asymptote (LM₂) and a free-moving asymptote (LM₃). The results are shown in Table 2. Both methods had a comparable MSE. For each method, the mean values of the calculated coefficients were a good approximation of the exact coefficients. However, the estimate of τ calculated using LM₃ had a larger SE (P < 0.001). Figure2A shows the regression between the calculated τ and the calculated P₀ for both LM₂ and LM₃. From this graph, it is obvious that, for this particular simulation, the nonlinear method with fixed zero asymptote (LM₂) approach is the better one because this approach provides the smallest confidence interval on the estimated coefficients.

A second Monte-Carlo simulation was done starting from a monoexponential curve with a negative pressure asymptote (P_∞ = −2.5 mmHg). Again τ, P₀, and P_∞ were estimated using LM₂ and LM₃ (cf. Table 2). In this second simulation, LM₃ had the smallest MSE (P < 0.001). Moreover, the mean values of the calculated coefficients, estimated using LM₃, were good approximations of the exact coefficients. The standard error on the estimated values was comparable to the standard error accompanying this method in the first simulation. By analogy with the first simulation, using LM₂ resulted in a significantly lower standard error for τ. However, LM₂significantly underestimated the exact values of τ, as illustrated in Fig. 2B, which shows the regression between the calculated τ and the calculated P₀ for both LM₂ and LM₃. Simulation of pressure curves with a positive instead of a negative pressure asymptote revealed an overestimation instead of an underestimation of τ when using the fixed zero asymptote approach.

The trade off between the magnitude of the variance on the estimated coefficients versus under/overestimation of the exact values is demonstrated in Fig. 3 at P₀ = 70 mmHg and reference values of τ = 40 ms (A), 80 ms (B), and 120 ms (C). The plots are illustrating the method-dependent sensitivity of the estimates (τ ± SD) for variations in P_∞ between 0 and −5 mmHg. In the case of a zero asymptote, both methods estimate τ well because τ do not significantly differ from the reference values (P > 0.05). LM₂ had the smallest standard deviation compared with LM₃ (P < 0.001). However, with increasing absolute values of the pressure asymptote, the values obtained using LM₂ were moving away from the exact values. Independent of the magnitude of the pressure asymptote, LM₂ kept the smallest confidence interval (P < 0.001).

To evaluate the efficiency of the two different estimators of τ, on the basis of LM₂ and LM₃, respectively, a MSE value was calculated as MSE = (Variance + Bias²) for each estimator (14). This MSE was similar to the variance on an estimated coefficient except that it was measured around the true target rather than around the (possibly biased) mean of the estimator. Formally, we can compare two estimators by calculating the relative efficiency (RE) as the proportion of the two MSE values. The RE values comparing LM₂ and LM₃ for estimating τ are plotted in Fig. 4 for pressure asymptotes varying from −5 to +5 mmHg and τ values ranging from 40 to 120 ms. Values for RE > 1 indicated a superior estimate of τ when using LM₃. Thus, in case of a zero pressure asymptote, LM₂ is always closer at estimating τ. However, with an increasing positive or negative pressure asymptote and decreasing τ, LM₃ becomes superior.

DISCUSSION

In this study, we exclusively used nonlinear techniques for estimation of τ from pressure decays using a monoexponential model:1) the nonlinear Levenberg-Marquardt method with a fixed zero pressure asymptote (LM₂), and 2) the Levenberg-Marquardt method with a variable pressure asymptote (LM₃). The two methods were chosen to observe for any divergence in the resultant τ.

Overall, both methods determined comparable values of τ for the data collected. The average values of the MSE when using LM₃were smaller compared with the values obtained when using LM₂. The smaller MSE reflects a superior goodness-of-fit of LM₃ of modeling the experimental data compared with LM₂. The inferior fitting when assuming a fixed zero pressure asymptote not only provokes a larger MSE but also has an important consequence on the estimated τ values: whereas LM₃ always provides an unbiased estimation of τ, LM₂ results in a biased estimation of τ when analyzing pressure decays with a nonzero pressure asymptote. The Monte Carlo simulation showed that in the case of a negative (positive) pressure asymptote, using LM₂ leads to a significant underestimation (overestimation) of the exact coefficients. In the animal experiment, under baseline conditions, the τ values obtained with LM₂were smaller compared with the values obtained using LM₃. In contrast, during either isoproterenol or esmolol infusion, τ values obtained with LM₃ were smaller compared with the values obtained using LM₂. According to the Monte Carlo simulations, this suggests a negative pressure asymptote and a significant underestimation of τ with LM₂ for baseline conditions and a positive pressure asymptote and a significant overestimation of τ for isoproterenol and esmolol. This was indeed confirmed by the values of the calculated pressure asymptotes using LM₃ (cf. Table 1). Figure 5shows the relationship between the actual pressure asymptote and under/overestimation of the τ values using LM₂ in more detail. An excellent correlation (r² = 0.92, P < 0.001) was observed between the difference in τ calculated with LM₂ and LM₃ and the pressure asymptote as calculated using LM₃ for the animal data under baseline and during isoproterenol and esmolol infusion. This correlation explains the differences in relative change of τ during drug infusion when using LM₂ compared with LM₃.

In contrast to the superior MSE and the unbiased estimation of τ, a drawback of LM₃ is the larger standard error on the estimated coefficients. This is primarily the consequence of the increased degree of freedom and error propagation in the algorithm for determining τ when using a free moving asymptote. The trade off between the magnitude of the standard error on the estimates (smaller when using LM₂) and the closer fit that guarantees an unbiased estimation (when using LM₃) can be evaluated quantitatively by calculation of a MSE that combines the bias and standard error on the estimates. This calculation demonstrates for the simulated data that in the case of a zero pressure asymptote, LM₂ always has a closer estimate of τ. However, with an increasing positive or negative pressure asymptote and decreasing τ, LM₃ becomes superior. This is also observed in the animal data. During drug infusion, the pressure asymptotes are small, and, consequently, the bias on the estimates is small. The τ values obtained using either LM₂ or LM₃ did not significantly differ. In contrast, the standard error of the estimates was significantly smaller when using LM₂. At baseline, a significant negative pressure asymptote was found and, as expected, the bias on the estimates was large. τ values obtained using either LM₂ or LM₃ were significantly different. The standard error of the estimates remained higher when using LM₃ compared with LM₂. With the use of the results of Table 1, we calculated the mean relative efficiency for the different groups. For the baseline data, the relative efficiency was 20.70, indicating that LM₃ provides the most reliable estimate. During isoproterenol and esmolol infusion, the mean relative efficiency became 0.22 and 0.85, which indicates that LM₂provides the most reliable estimate.

The problem of choosing a two (fixed zero pressure asymptote assumed)- or three (free moving asymptote assumed)-parameter model is still a matter of debate because conflicting results are reported. Several authors (1, 5, 11, 12) have demonstrated the use of a variable asymptote to be a more rigorous and physiologically rational method of modeling LV pressure decline during the isovolumic relaxation period. Bernardi and associates (1) demonstrated that the Levenberg-Marquardt algorithm with a variable asymptote is a most accurate method for modeling LV pressure decline during the isovolumic relaxation periods. Martin and colleagues (5) demonstrated that a variable asymptote method of determining τ was more sensitive to β-adrenergic blockade or stimulation than to drugs that altered cardiac loading conditions.

On the other hand, Yellin and colleagues (16) demonstrated that τ determined from an exponential model using a fixed asymptote method is comparable with τ determined from an exponential model using a measured or best-fit asymptote. Yellin and colleagues (16) further concluded that as long as it is consistently used in the same study, τ resulting from any method provides useful information related to diastolic function. Also, Kettunen and colleagues (4) advocate the use of a fixed zero asymptote method for practical clinical use to determine τ. This recommendation is based on their observation that τ determined using a fixed asymptote method is comparable with τ determined using a variable asymptote method with the exception of conditions of β-adrenergic blockade or stimulation. The zero asymptote method was advocated on the basis of a less complicated mathematical algorithm for most practical clinical purposes. Yamakado et al. (15) calculated τ with and without a pressure asymptote to investigate the influence of age on ventricular relaxation. No significant differences between the different approaches were observed. In contrast, Davis et al. (3) obtained opposite conclusions when analyzing ventricular relaxation rate using the zero or nonzero asymptote model.

Despite closer fitting of the pressure curves when using LM₃ and despite the bias accompanying LM₂, τ estimates with LM₂ may show better correlations with other physiological parameters than LM₃. This is demonstrated for the baseline animal data in Fig. 6, showing the regression between maximum negative dP/dt and τ when using LM₂ (A) and LM₃(B). Although LM₃ better fit the pressure decays, the better correlation was obtained using LM₂(r² = 0.51 vs.r² = 0.45). We speculate that this phenomenon is due to the biased estimates of τ with LM₂for the analysis of nonzero asymptote pressure decays. This leads to over- or underestimation of τ and thus to a broader range of τ values, automatically enhancing the correlation.

In previous studies, linear methods were used to determine τ. We also calculated τ values under baseline conditions with linear regression, assuming a fixed zero pressure asymptote as proposed by Weiss et al. (13) and Nagueh et al. (7). The results were compared with the results obtained using the Levenburg-Marquardt technique with zero pressure asymptote (LM₂). The τ values obtained using LM₂ were slightly, although significantly, larger (63.85 ± 17.12 vs. 67.83 ± 15.47 ms,P < 0.001) compared with the linear method. LM₂ also provided closer fits to the pressure decays, as reflected by the smaller MSE (2.59 ± 2.06 vs. 0.90 ± 0.75,P < 0.001).

It is generally accepted that the isovolumic relaxation period of the LV pressure curve is well approximated by the monoexponential decay model described by Eq. 1 (1, 3-5, 9, 11-13,15, 16) except in case of a postextrasystolic LV isovolumic pressure decay (2) and dilated cardiomyopathy (10). Alternative models have been proposed for fitting the isovolumic pressure decay (6, 10). Recently, Senzaki et al. (10) reported an improvement of quantitative analyses when using the following more complex hybrid logistic model ofEq. 4

The hybrid logistic model provided more consistent data fits, especially in dilated cardiomyopathy, when a nonlinear relationship between dP/dt and P was observed. We also fitted this model to the pressure decays of the animal study. In accordance with the results reported by Senzaki et al. (10), the τ values obtained using this model (44.29 ± 5.38 ms) were significantly smaller compared with the values obtained with the other methods. Also, the mean pressure asymptote remained positive (1.82 ± 2.79 mmHg). Assuming an exponential model, the physical meaning of the τ value is the time needed for the pressure to decrease to 37% of its initial value. In contrast, when assuming a hybrid logistic model, the physical meaning of τ is the time needed for the pressure to decrease to 54% of its initial value. Therefore, the hybrid logistic function provides τ values of another magnitude compared with the monoexponential function. Thus comparing values obtained using the different models is difficult. The MSE (0.45 ± 0.35) was significantly (P < 0.001) larger compared with the values obtained using LM₃ (0.17 ± 0.33). Thus, in this animal experiment, the monoexponential model provides the closer fit. The standard error of the estimated τ (1.58 ± 1.60 ms) was, however, smaller (P < 0.001) compared with the standard error when using LM₃ (3.44 ± 1.85 ms). Therefore, the hybrid logistic method might be a valuable alternative for LM₃, especially for pressure decays with a nonlinear relationship between dP/dt and P.

In conclusion, in this study, Monte Carlo simulations of monoexponential pressure decays provided a reference, allowing an objective comparison of different methods for estimation of the relaxation constant τ of LV pressure fall. Comparison of the experimental data with the results of the Monte Carlo simulations demonstrated a trade off between the nonlinear Levenburg-Marquardt fixed zero approach on one hand and the nonlinear Levenburg-Marquardt method with a free moving asymptote on the other hand. The latter method closer fits the pressure curve and has the advantage of producing an extra coefficient (P_∞) with potential diagnostic information. On the other hand, this method suffers from larger standard errors on the estimated coefficients. The nonlinear Levenburg-Marquardt method with fixed zero asymptote produces values of τ within a narrow confidence interval. However, in case of a nonzero negative (positive) pressure asymptote, this method significantly underestimates (overestimates) the real values. Quantitative evaluation of the trade off between bias (when using LM₂) and the magnitude of the standard error on the estimates (larger when using LM₃) demonstrates that, in case of a zero pressure asymptote, LM₂ always has a closer estimate of τ. However, LM₃ becomes superior with an increasing pressure asymptote (both positive or negative) and decreasing τ.

FOOTNOTES

• S. De Mey was a recipient of Grant IWT-971096 from the Flemish Institute for the Promotion of Scientific-Technological Research in the Industry. This study was also supported in part by National Aeronautics and Space Administration Grant NCC 9-60 (to J. D. Thomas) and by National Heart, Lung, and Blood Institute Grant R01-HL-56688-01A1 (to J. D. Thomas).

• Address for reprint requests and other correspondence: S. De Mey, Hydraulics Laboratory, St.-Pietersnieuwstraat 41, 9000 Gent, Belgium (E-mail: stefaan.[email protected]rug.ac.be).

• The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.