# Noninvasive Assessment of the Ventricular Relaxation Time Constant (τ) in Humans by Doppler Echocardiography

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## Abstract

*Background* The time constant of ventricular relaxation (τ) is a quantitative measure of diastolic performance requiring intraventricular pressure recording. This study validates in humans an equation relating τ to left ventricular pressure at peak −dP/dt (P_{0}), pressure at mitral valve opening (P_{MV}), and isovolumic relaxation time (IVRT_{inv}). The clinically obtainable parameters peak systolic blood pressure (P_{s}), mean left atrial pressure (P_{LA}), and Doppler-derived IVRT (IVRT_{Dopp}) are then substituted into this equation to obtain τ_{Dopp} noninvasively.

*Methods and Results* High-fidelity left atrial and left ventricular pressure recordings with simultaneous Doppler by transesophageal echocardiography were obtained from 11 patients during cardiac surgery. Direct curve fitting to the left ventricular pressure trace by Levenberg-Marquardt regression assuming a zero asymptote generated τ_{LM}, the “gold standard” against which τ_{calc} {IVRT_{inv}/[ln(P_{0})−ln(P_{MV})]} and τ_{Dopp} {IVRT_{Dopp}/[ln(P_{s})−ln(P_{LA})]} were compared. For 123 cycles analyzed in 18 hemodynamic states, mean τ_{LM} was 53.8±12.9 ms. τ_{calc} (51.5±11 ms) correlated closely with this standard (*r*=.87, SEE=5.5 ms). Noninvasive τ_{Dopp} (43.8±11 ms) underestimated τ_{LM} but exhibited close linear correlation (n=88, *r*=.75, SEE=7.5 ms). Substituting P_{LA}=10 mm Hg into the equation yielded τ_{10} (48.7±15 ms), which also closely correlated with the standard (*r*=.62, SEE=11.6 ms).

*Conclusions* The previously obtained analytical expression relating IVRT, invasive pressures, and τ is valid in humans. Furthermore, a more clinically obtainable, noninvasive method of obtaining τ also closely predicts this important measure of diastolic function.

Ventricular diastolic dysfunction implies the requirement for elevated filling pressures to maintain cardiac output, resulting ultimately in congestive cardiac failure. Normal ventricular performance during early diastole is largely dependent on the process of active myocyte relaxation. Invasive ventricular pressure and volume data have been widely used in the quantification of such diastolic performance.^{1} ^{2} ^{3} Intraventricular pressure decay during the IVRT, before the onset of filling, follows an approximately exponential curve^{4} numerically characterized by its maximum negative slope (peak −dP/dt) and its τ (see Fig 1⇓). The requirement for high-fidelity intraventricular pressure recording to estimate τ and peak −dP/dt has logistically limited the clinical utility of these parameters in the quantification of diastolic function.

Isovolumic relaxation time duration can be obtained invasively and by several noninvasive modalities, including phonocardiography and M-mode and Doppler echocardiography.^{5} This isolated parameter has been shown to vary significantly in disease states associated with diastolic dysfunction.^{6} ^{7} However, IVRT duration represents the physiological summation of diastolic myocardial function and the degree of preload compensation.^{8} ^{9} Consequently, attempts have been made to derive or infer τ and peak −dP/dt from IVRT duration and other noninvasive parameters, such as the downslope of the mitral regurgitation Doppler profile^{10} ^{11} ^{12} . This technique is limited to patients with mitral regurgitant jets in whom high-quality Doppler spectra can be obtained. A more universally applicable method for such quantification in the routine clinical setting remains to be described and validated.

## Methods

### Background Mathematical Correlations

Instantaneous P_{V}, P_{0}, and t derived from the invasive LV pressure trace are related by the monoexponential equationP_|<|V|>||<|=|>|P_|<|0|>|\mathit|<|e|>|^|<||<|-|>|t/|<|\tau|>||>||<|+|>|bwith a time constant τ.^{4} Curve fitting can be achieved by use of the Levenberg-Marquardt nonlinear least-squares parameter estimation technique.^{13} A basic assumption may be made regarding the theoretical asymptote (*b*) of the pressure-decay curve. Clearly, ongoing diastolic filling dictates that such an asymptote will never actually be reached. In a canine model with complete occlusion of the mitral valve allowing ongoing relaxation, Yellin et al^{14} determined the absolute asymptote of LV pressure decay to be −7.3±3.3 mm Hg. They went on to show that the simplified assumption of a zero asymptote (*b*=0) generated similar values for τ to the true nonzero asymptote. Thus, for clinical purposes, our study has assumed a zero asymptote (*b*=0).

Thomas et al^{15} demonstrated in a canine model that IVRT correlates with P_{0} and P_{MV}. Furthermore, at mitral valve opening (ie, when t=IVRT and P_{V}=P_{MV}), Equation 1 becomes P_{MV}=P_{0}*e*^{−IVRT/τ}. Taking the logarithm of this equation and rearranging yields Equation 2, relating τ to P_{0}, P_{MV}, and IVRT. In their model, τ_{calc} correlated well with τ obtained by direct curve fitting to the ventricular pressure trace.|<|\tau|>|_|<|calc|>||<|=|>|IVRT_|<|INV|>|/|<|[|>|ln(P_|<|0|>|)|<|-|>|ln(P_|<|MV|>|)|<|]|>|In the same preparation, Thomas et al proceeded to examine this mathematical model, substituting the clinically obtainable P_{s} for the invasively acquired P_{0}. This, as expected, yielded a systematic underestimation of τ, because P_{s} is systematically higher than P_{0}. Notwithstanding this offset, the relationship between the direct-fitted and calculated τ was linear, with a high degree of correlation and a regression slope of near unity.

### Purpose of This Study

Invasive intraventricular pressure traces were obtained, and direct Levenberg-Marquardt curve fitting generated τ_{LM}, considered to be the “gold standard” for this study. The purpose of this study was then to validate the mathematical framework that would allow derivation of this time constant from first invasive and then noninvasive parameters. To demonstrate the rational basis of Equation 2, we set out to show that τ_{LM} was related to the invasive parameters IVRT_{inv}, P_{0}, and P_{MV.} Then τ_{calc}, derived from Equation 2, was compared with the standard τ_{LM} to validate this equation in humans. Having shown that the fundamental tenet of the equation is valid, we generated a noninvasive approximation, τ_{Dopp}, calculated from P_{s}, P_{LA}, and IVRT_{Dopp} via Equation 3:|<|\tau|>|_|<|Dopp|>||<|=|>|IVRT_|<|Dopp|>|/|<|[|>|ln(P_|<|s|>|)|<|-|>|ln(P_|<|LA|>|)|<|]|>|This was also compared with τ_{LM} to determine the reliability of this clinically obtainable parameter. In clinical echocardiography, right atrial pressure is routinely assumed to be 10 mm Hg to allow calculation of right ventricular systolic pressure from the tricuspid regurgitation Doppler spectrum. We made a similar assumption and tested Equation 3 with a presumed P_{LA} of 10 mm Hg to test a fully noninvasive parameter that may be used in outpatient echocardiography practice.

### Data Acquisition

An integrated system for simultaneous acquisition of physiological and ultrasound data during cardiac surgery has been developed in our institution and reported previously.^{16} Pressure recordings were obtained with high-fidelity pressure catheters (Millar Instruments). Before insertion, the catheters were immersed in saline to minimize “drift” and then calibrated relative to atmospheric pressure. Pressure signals were amplified with a universal amplifier (Gould). Up to four channels (including peripheral arterial pressure) and an ECG were recorded simultaneously. Amplified signals were then digitized with an NB-MIO-16 multifunction input/output board (National Instruments). All signals were digitized with 12-bit resolution and a sampling frequency of 1000 Hz.

Doppler signals were recorded via a Hewlett-Packard Omniplane probe connected to a Sonos 1500 Echocardiograph (Hewlett-Packard). Doppler spectral display images were frozen and stored to optical disk in 1.5-second frames by use of the digital storage and retrieval system. Velocity profiles were extracted with 5-ms temporal resolution from the images by use of a proprietary tagged image file format reader. The spectral Doppler information was synchronized with the physiological waveforms with a computer-generated marker signal, which is placed on the image through the auxiliary physiological input of the echocardiograph machine and stored with the digital images on optical disk.

Images and physiological traces were then analyzed off-line by customized software implemented in LabVIEW (National Instruments). Specific algorithms were designed to calculate the following parameters: (1) P_{0}, previously shown to be a close approximation to pressure at aortic valve closure^{9} ; (2) P_{MV} and P_{LA}; (3) automated exponential curve fitting to the LV pressure trace during IVRT, yielding τ_{LM} via the Levenberg-Marquardt technique; (4) P_{s}; (5) IVRT_{inv}, the time interval from P_{0} to P_{MV} (see Fig 1⇑); (6) cycle length; and (7) IVRT_{Dopp} (see Fig 1⇑). A pulsed-wave Doppler cursor is placed in the area of the anterior mitral valve leaflet to capture an LVOT envelope and the mitral inflow profile. The interval from the aortic valve artifact at the end of the LVOT envelope to the mitral valve artifact at the beginning of the mitral E wave was considered to be IVRT_{Dopp.}

### Patient Population

After approval of the protocol by our Institutional Review Board, data were acquired from patients undergoing routine cardiac surgery after they had given written informed consent. Measurements were taken both before and after cardiopulmonary bypass when possible to achieve a wide variety of loading and inotropic states.

### Statistical Methods

All variables are summarized as mean±SD and range. Comparison of continuous variables was performed with Student's *t* test and univariate linear regression and was expressed as correlation coefficient, probability value of the regression, and regression formula. The relationships of the parametric variable IVRT_{inv} to P_{0}, P_{MV}, and τ_{LM} were evaluated by linear regression analysis. With τ_{LM} as the standard, the merit of the analytically derived parameters invasive τ_{calc} and Doppler-derived τ_{Dopp} was assessed by univariate regression. The SEE is presented as a measure of accuracy of the techniques examined compared with the standard. Bland and Altman plots^{17} were used to examine for systematic error of the techniques tested.

All invasive measurements of pressure and IVRT_{inv} were automated with the customized software. To examine for interobserver variability, the measurement of IVRT_{Dopp} was performed by two observers blinded to each other's results and to the values of τ_{LM} in 50 cardiac cycles. The paired results were analyzed with linear regression and paired Student's *t* test. Beat-to-beat variability was analyzed by intraclass correlation with ANOVA,^{18} expressed as an intraclass correlation coefficient, *r*. Values close to 1.0 represent minimal beat-to-beat variability.

## Results

Eleven patients 44 to 76 years old (mean, 59 years) were studied in a total of 18 hemodynamic states. Seven of these patients were undergoing routine coronary bypass surgery with normal LV size and systolic function; two, mitral repair surgery for severe regurgitation; and two, LV assist device implantation for severe systolic dysfunction. The major hemodynamic and Doppler parameters are summarized in the Table⇓. Of 123 cardiac cycles analyzed, 35 had suboptimal Doppler tracings and thus were excluded from the noninvasive comparison. Cycle length was 703±142 ms; P_{s}, 109±36 mm Hg; and P_{0}, 68±31 mm Hg. P_{MV} (10.5±5.7 mm Hg) was significantly higher than P_{LA} (8.8±3.7 mm Hg, *P*<.0001).

### Relationships Between IVRT_{inv}, IVRT_{Dopp}, τ_{LM}, and Ventricular Pressures

IVRT_{inv} (103±39 ms) was significantly shorter than IVRT_{Dopp} (115±36 ms, *P*<.001). There was close linear correlation between the paired values (n=88, *r*=.9, *P*<.0001). As in the canine model,^{15} there was a linear relation between IVRT_{inv} and P_{0} (n=123, *r*=.51, *P*<.0001) and between IVRT_{inv} and P_{MV} (n=123, *r*=.61, *P*<.0001). IVRT_{inv} showed linear correlation with the direct curve-fitted τ_{LM} (n=123, *r*=.65, *P*<.0001) (see Fig 2⇓).

Interobserver variability for the measurement of IVRT_{Dopp} was assessed in 50 cardiac cycles. The difference in mean IVRT_{Dopp} between the two observers was 2±11 ms (*P*=NS). There was close linear correlation between the two sets of paired measurements (*r*=.96, *y*=0.94*x*+8.6).

### Invasive τ_{calc} Versus Direct Curve-Fitted τ_{LM}

Direct curve fitting by use of the Levenberg-Marquardt technique yielded τ_{LM} (*x*) of 53.8±12.9 ms. This was slightly longer than τ_{calc} (*y*), 51.5±11.0 ms, calculated from the invasive data by Equation 2 (*P*<.001). Linear regression analysis showed a high degree of correlation (n=123, *r*=.87, *P*<.0001, *y*=0.74*x*+11.7) (Fig 3A⇓). The mean±SD of the differences of the paired values was −2.1±6.4 ms, indicating a small systematic underestimation of τ with Equation 2. Bland and Altman analysis (Fig 3B⇓) did not indicate a tendency toward exaggerated error at either end of the data range. The SEE for τ_{calc} was 5.5 ms, representing a ±10.6% predictive error of this technique.

### Noninvasive τ_{Dopp} Versus Direct Curve-Fitted τ_{LM}

In the 88 cycles for which acceptable Doppler tracings were available, the noninvasive Doppler-derived τ_{Dopp} (*y*, 43.8±11.4 ms) was shorter than τ_{LM} (*x*, 54.5±13.9 ms, *P*<.00001). There was, however, a high degree of linear correlation between these parameters (n=88, *r*=.75, *P*<.0001, *y*=0.6*x*+10.3), Fig 4A⇓. Bland and Altman analysis of systematic error showed a systematic underestimation of τ by Equation 3, with the mean±SD of the differences of the paired values being −10.6±9.2 ms (Fig 4B⇓). The SEE for τ_{Dopp} was 7.5 ms, representing a predictive error of ±14% for this estimation.

The further assumption that P_{LA} could be approximated to 10 mm Hg was used to generate τ_{10}. These values for τ_{10} (48.7±14.7 ms) were also significantly shorter than the standard, but they showed a good correlation with τ_{LM} (n=88, *r*=.62, *P*<.0001, *y*=0.65*x*+13). The SEE of this technique was 11.6 ms, representing a ±20% predictive error of this technique.

## Discussion

Diastolic dysfunction, although clinically important, remains difficult to quantify by noninvasive techniques. Doppler echocardiographic assessment of mitral and pulmonary vein inflow patterns has been used extensively to estimate LV diastolic performance.^{19} ^{20} ^{21} Unfortunately, most Doppler parameters derived from such velocity profiles are nonspecific for individual physiological variables. Such qualitative data define pathophysiological constellations, with numerous possible combinations of perturbations contributing to the overall appearance.^{22} Specifically, the qualitative staging of diastolic dysfunction into “delayed relaxation,” “pseudonormal,” and “restrictive” patterns is confounded by the degree of preload compensation.^{23}

The τ and peak −dP/dt derived from invasive techniques are well established as important clinical and research tools.^{1} ^{2} ^{3} ^{4} Whereas peak −dP/dt is affected by loading conditions, τ is largely preload independent.^{24} This “gold standard” parameter shortens with β-adrenergic stimulation and prolongs with age, reperfusion states, and β-blockade.^{25} Noninvasive determination of peak −dP/dt and τ from the Doppler profile of mitral regurgitation has proved to be feasible and accurate.^{10} ^{11} ^{12} The ventriculoatrial gradient was obtained via the Bernoulli equation, and peak −dP/dt was estimated from the first differential of the pressure-time curve and τ from the slope of the natural logarithm of the pressure-versus-time curve. Although it does not represent true isovolumia, this period does represent the decay of LV pressure after active systole.

Isovolumic relaxation time is defined as the period from aortic valve closure (dicrotic notch on the invasive LV pressure trace) to mitral valve opening before filling begins.^{26} Weisfeldt et al^{9} demonstrated a systematic delay from the dicrotic notch to P_{0} (pressure at peak −dP/dt) of ≈20 ms. Thus, the period from P_{0} to mitral valve opening remains on the exponential pressure decay curve and has been used widely and in our study as an obtainable IVRT period.^{4} ^{9} ^{15} Doppler-derived IVRT measures the onset of the interval from the aortic valve artifact.^{5} That IVRT_{Dopp} in our study is longer than IVRT_{inv} is therefore expected and logical.

Simple measurement of IVRT has been proposed for the assessment of diastolic function.^{7} The confounding effects of preload on this parameter^{15} have limited its usefulness. Indeed, in a canine preparation,^{27} IVRT changed in parallel with τ in conditions of varying inotropy but not with varying preload. Mathematical modeling^{15} has provided a mechanism whereby τ can be related to IVRT in combination with simple hemodynamic parameters, such as P_{0} and P_{MV}. Canine preparations have then validated that these relationships indeed accurately predict the experimental physiology.

This present study is important for two reasons. First, it validates in the human clinical setting the canine work relating IVRT and P_{0} and P_{MV}.^{15} Canine models are able to artificially produce widely varying loading conditions, providing a large range of IVRT values with which to calculate τ_{calc}. By examining patients with a variety of cardiac disease states before and after cardiopulmonary bypass, we also have produced as wide a range as possible of P_{0}, P_{MV}, and IVRT values to test this model. The analytically derived τ_{calc} closely correlates with the direct curve-fitted τ_{LM} parameter. This intertechnique agreement validates Equation 2 as a model for early diastolic pressure decay in humans.

Second, and more important, we have shown that by substitution of the clinically obtainable values P_{s}, P_{LA}, and IVRT_{Dopp} into Equation 3, τ can be predicted with a reasonable degree of accuracy (SEE±14%). This finding introduces the possibility of the use of τ in the intensive care setting, where pulmonary wedge pressure is often available as a substitute for P_{LA}.

We have further simplified the formula by substituting an estimated value for P_{LA} of 10 mm Hg. This assumption closely parallels the right atrial pressure generalization used in the routine calculation of right ventricular systolic pressure from the tricuspid regurgitation Doppler profile. If such an assumption proved valid in Equation 3, the need for any invasive measure of left atrial pressure would be obviated. In our study, this assumed left atrial pressure did generate values for τ that correlated well with the gold standard. The standard error of this estimate was ≈±20%. Thus, for example, in a clinical setting in which Doppler IVRT was 100 ms and systolic blood pressure was 120 mm Hg, τ would be estimated at 40±8 ms. It is likely that in most clinical situations, this degree of accuracy would be adequate for the quantification of LV diastolic function.

The systematic underestimation of τ by Equation 3 relates to P_{s} being, by definition, greater than P_{0}. Techniques that estimate P_{0} from P_{s} by use of pressure-volume data^{28} may prove useful in correcting for this systematic error of Equation 3. The volume data required for these estimations were not recorded in this study, and it is proposed that these techniques should be tested in future prospectively acquired data sets.

### Limitations of This Study

This study was performed in the operating room with transesophageal echocardiography. The transferability of this information to the bedside with transthoracic echocardiography remains to be shown. Technically, 28% of cardiac cycles could not have IVRT assessed by Doppler. However, in no patient was it impossible to achieve at least two cycles with an IVRT measurement. Apart from P_{LA} and an assumed pressure of 10 mm Hg, several other techniques for the noninvasive left atrial pressure have been proposed.^{29} ^{30} ^{31} Incorporation of one or more of these techniques may offer some advantage in the predictive accuracy of our noninvasive measure. Finally, this group had widely varying LV systolic function and loading conditions. The techniques, however, were not assessed in patients with severe restrictive or constrictive conditions. These pathological conditions, with the most severe degrees of diastolic dysfunction, remain to be assessed with these techniques.

### Summary

τ can be closely approximated from the invasive parameters ventricular pressure at aortic valve opening, P_{MV}, and isovolumic relaxation time. Substitution of the noninvasive, more clinically obtainable parameters IVRT_{Dopp}, P_{s}, and P_{LA} (or assumed LA pressure of 10 mm Hg) into the same analytical expression also closely predicts this constant, τ, with a degree of accuracy acceptable for clinical practice.

## Selected Abbreviations and Acronyms

## Acknowledgments

This study was supported in part by grant 93/013880 from the American Heart Association.

- Received May 28, 1996.
- Revision received July 31, 1996.
- Accepted August 22, 1996.

- Copyright © 1997 by American Heart Association

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- Noninvasive Assessment of the Ventricular Relaxation Time Constant (τ) in Humans by Doppler EchocardiographyGregory M. Scalia, Neil L. Greenberg, Patrick M. McCarthy, James D. Thomas and Pieter M. VandervoortCirculation. 1997;95:151-155, published online before print January 7, 1997http://dx.doi.org/10.1161/01.CIR.95.1.151
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- Noninvasive Assessment of the Ventricular Relaxation Time Constant (τ) in Humans by Doppler EchocardiographyGregory M. Scalia, Neil L. Greenberg, Patrick M. McCarthy, James D. Thomas and Pieter M. VandervoortCirculation. 1997;95:151-155, published online before print January 7, 1997http://dx.doi.org/10.1161/01.CIR.95.1.151

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