# Single-Beat Estimation of End-Systolic Pressure-Volume Relation in Humans

## A New Method With the Potential for Noninvasive Application

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## Abstract

*Background* The end-systolic pressure-volume relation (ESPVR) provides a useful measure of contractile function. However, the need to acquire multiple cardiac cycles at varying loads limits its applicability. We therefore developed and tested a novel single-beat estimation method that is based on normalized human time-varying elastance curves [E_{N}(t_{N})].

*Methods and Results* Pressure-volume (PV) data were measured by conductance catheter in 87 patients with normal or myopathic hearts. Time-varying elastance curves were generated from 72 PV loops (52 patients) and normalized both by amplitude and time to peak amplitude. The resulting E_{N}(t_{N}) curves were remarkably consistent despite variations in underlying cardiac disease, contractility, loading, and heart rate, with minimal interloop variance during the first 25% to 35% of contraction. On the basis of this finding and assuming ESPVR linearity and constant volume-intercept, ESPVRs were estimated from one beat with the use of PV data measured at normalized time (t_{N}) and end systole (t_{max}) to predict intercept: V_{o(SB)}=[E_{N}(t_{N})×P(t_{max})×V(t_{N})−P(t_{N})×V(t_{max})]/[E_{N}(t_{N})×P(t_{max})−P(t_{N})] and slope E_{max(SB)}=P_{es}/[V_{es}−V_{o(SB)}]. Single-beat estimates were highly correlated with measured ESPVR values obtained by standard multiple-beat analysis (including data from 35 additional patients). E_{max(SB)} accurately reflected acute inotropic change and was influenced little by loading. The new estimation method also predicted measured ESPVRs better than prior techniques and was applicable to noninvasive analysis.

*Conclusions* ESPVRs can be reliably estimated in humans from single cardiac cycles by a new method that has a potential for noninvasive application.

The ESPVR is a valuable measure of ventricular systolic function for both clinical and experimental evaluations.^{1} ^{2} ^{3} ^{4} ^{5} ^{6} ^{7} ^{8} Acute ESPVR shifts primarily reflect inotropic change, while the slope (E_{es}) measures end-systolic chamber stiffness, an important determinant of the influence of vascular loading on systolic pressure and flow generation.^{3} The ESPVR evolved from a linear time-varying elastance [E(t)] model of contraction in which the left ventricle was considered an elastic structure that stiffened and relaxed along a predictable time course during the cardiac cycle.^{1} ^{9} Initial studies by Suga, Sagawa, and colleagues^{1} ^{9} supported this model by showing that the normalized E(t) curve (normalized both by peak and time to peak amplitude) of normal isolated canine hearts was fairly independent of loading conditions, contractile state, and heart rate. Subsequent studies revealed limitations to the simplified theory, such as ESPVR curvilinearity^{10} ^{11} ^{12} and dependence on ejection history (afterload).^{13} ^{14} ^{15} ^{16} However, the most marked deviations from the simple linear model were manifest when extremes of LV loading were contrasted, such as between isovolumic and ejecting beats.^{15} ^{16} Yet vascular loads in intact patients are far more constrained, leaving open the possibility that the normalized E(t) curve in humans might be less load-dependent and more consistent among patients than such studies might suggest.

One practical implication of a consistent normalized E(t) curve in humans is that the ESPVR could be estimated from a single beat. This is clearly advantageous over complex traditional methods of generating ESPVRs from multiple variably loaded cardiac cycles.^{2} ^{3} ^{5} Methods for single-beat ESPVR estimation have been previously proposed,^{17} ^{18} ^{19} ^{20} largely based on arbitrary mathematical curve fitting of LV pressure data measured during isovolumic contraction and relaxation of an ejecting beat. These methods have not been previously tested in abnormal hearts and require measurement of high-fidelity LV pressure. In contrast, estimates derived from the normalized E(t) curve are physiologically based and have the potential for noninvasive assessment.

Accordingly, the present investigation had four principal aims: (1) to test whether the normalized E(t) curve in humans is consistent among patients despite marked differences in underlying myocardial disease condition, contractile state, heart rate, and vascular loading; (2) to determine if a standard normalized E(t) curve could be applied to yield reliable ESPVR estimates from single-beat data; (3) to compare the new approach to existing single-beat estimation methods; and (4) to test the feasibility of noninvasive application of the new method.

## Methods

### Patient Groups

Data were analyzed from 87 patients: 15 with normal hearts and 72 with varying forms of chronic heart disease. Patient age ranged from 38 to 80 years (mean, 58). Control subjects were referred for cardiac catheterization to assess atypical chest pain and had normal ECGs, coronary angiograms, contrast ventriculography, and echocardiograms. Nineteen patients had HCM: 4 with familial disease and 15 with acquired disease associated with chronic hypertension and presenting with congestive symptoms. Seventeen patients had dilated cardiomyopathy (DCM): 14 nonischemic and 3 ischemic. Seven additional patients had dilated hearts with reduced ejection fraction (EF) as the result of prior myocardial infarction and ventricular aneurysm (ANE). Last, 29 patients had coronary artery disease (CAD) but normal baseline ventricular function.

The patient group reflected a broad range of baseline contractility (E_{es}, 0.5 to 4.9 mm Hg/mL), EF (16% to 87%), chamber size (EDV, 47 to 496 mL), heart rate (47 to 159 min^{−1}), end-systolic pressures (84 to 205 mm Hg), and end-diastolic pressures (1 to 47 mm Hg). Hemodynamic characteristics for each group are provided in Table 1⇓. Written informed consent was obtained from all patients, and the protocols were approved by The Johns Hopkins Joint Committee on Clinical Investigation.

### Procedures

Patients were premedicated with benzodiazepam (5 to 10 mg) and diphenhydramine (25 to 50 mg). After routine coronary angiography, left ventriculography, and right heart catheterization, PV relations were determined by conductance catheter technique^{21} combined with transient inferior vena caval balloon obstruction. Details of this procedure have been previously described.^{5} ^{22} Briefly, a 7F volume catheter (Webster Labs, Millar, or Sentron) was introduced via a femoral artery, advanced to the LV apex under fluoroscopic guidance, and connected to a stimulator-microprocessor (Sigma-V, Leycom, or VCU, Cardiac Pacemakers Inc) to generate a volume signal. This signal was calibrated so that mean PV loop width matched stroke volume (SV) derived by thermodilution (cardiac output/heart rate) and EF matched that obtained by contrast ventriculography. A 2F micromanometer catheter (SPC-320 Millar) was placed within the distal lumen of the conductance catheter to measure chamber pressure.

Conventional measurement of the ESPVR from multiple PV loops was made with the use of a 7F balloon catheter (SP-9168, Cordis) to transiently obstruct inferior vena cava inflow. This resulted in transient preload reduction with little baroreflex, providing typically 10 or more beats at varying preloads. In several studies, heart rate was altered by atrial pacing, while in others, dobutamine (5 to 10 μg/kg per minute), verapamil (5 to 10 mg), esmolol (1 to 3 mg/kg bolus+0.1 to 0.3 mg/kg per minute continuous IV), or an experimental inotropic agent, OPC-18790 (5 to 10 μg/kg per minute),^{5} was infused intravenously to alter inotropic state. Data were digitalized at 200 Hz with the use of custom software and analyzed off-line.

### ESPVR Determination From Multiple Loops

The reference or “gold standard” ESPVR was derived from multiple variably preloaded cardiac cycles^{22} shown by the example in Fig 1A⇓. Points of maximal pressure/(volume−V_{o}), where V_{o} was the ESPVR volume-axis intercept, were calculated for each beat in the series by the use of an iterative method to measure V_{o}.^{3} ^{23} Data were fit by orthogonal regression^{24} yielding a multiple-beat ESPVR slope (E_{es(MB)}) and intercept (V_{o(MB)}). This regression minimizes the perpendicular distance between the data points and the regression line rather than assuming dependent/independent variable assignments.

### Normalized Elastance Curve: E_{N}(t_{N})

To derive the normalized time-varying elastance curve, PV data from 3 to 5 sequential steady-state beats were signal-averaged to yield a PV loop. Time-varying elastance [E(t)] was defined as the instantaneous ratio of P(t)/[V(t)−V_{o(MB)}]. The maximal value of E(t) [E_{max(SB)}] and the time to achieve E_{max(SB)} referenced from the R wave of the ECG (t_{max}) were both determined. The normalized E(t) function was then defined asE_|<|N|>|(t_|<|N|>|)|<|=|>|E(t_|<|N|>|)/E_|<|max(SB)|>|wheret_|<|N|>||<|=|>|t/t_|<|max|>|To combine E_{N}(t_{N}) relations, each curve was resampled at 200 equispaced intervals by linear interpolation and the results averaged for each patient and/or hemodynamic group. The mean E_{N}(t_{N}) curve was the average of data from 72 randomly selected loops from 52 patients (12 normal, 12 HCM, 9 DCM, 12 CAD, and 7 ANE), with several patients providing more than one condition (ie, altered heart rate or contractility).

### Single-Beat Estimation of ESPVR

The new method for single-beat ESPVR estimation is schematically shown in Fig 1B⇑. For each steady-state cardiac cycle, t_{max} (the time from R wave to maximal P/V ratio) was determined and time coordinates transformed to normalized time t_{N} (by Equation 2). PV data were then measured at time t_{N}[P(t_{N}) and V(t_{N})] and t_{max}[P(t_{max}), V(t_{max})], with linear interpolation as needed. The ESPVR estimate was based on two assumptions: (1) that the E(t) model was linear and (2) that the volume-axis intercept [V_{o(SB)}] was constant for a given cardiac cycle. Chamber elastance at time t_{N} wasE(t_|<|N|>|)|<|=|>|P(t_|<|N|>|)/|<|[|>|V(t_|<|N|>|)|<|-|>|V_|<|o(SB)|>||<|]|>|and elastance at time t_{max} wasE(t_|<|max|>|)|<|=|>|E_|<|max(SB)|>||<|=|>|P(t_|<|max|>|)/|<|[|>|V(t_|<|max|>|)|<|-|>|V_|<|o(SB)|>||<|]|>|From Equation 1, the ratio of E(t_{N})/E_{max(SB)} equaled E_{N}(t_{N}). Thus, combining Equations 1, 3, and 4, we obtainedV_|<|o(SB)|>||<|=|>|\frac|<||<|[|>|P_|<|N|>|(t_|<|N|>|)|<|\times|>|V(t_|<|max|>|)|<|-|>|V(t_|<|N|>|)|<|\times|>|E_|<|N|>|(t_|<|N|>|)|<|]|>||>||<||<|[|>|P_|<|N|>|(t_|<|N|>|)|<|-|>|E_|<|N|>|(t_|<|N|>|)|<|]|>||>|where P_{N}(t_{N})=P(t_{N})/P(t_{max}). Once V_{o(SB)} was calculated, E_{max(SB)} was determined from Equation 4.

A modification of this method used an iterative procedure to estimate t_{max}. As noted above, t_{max} was first derived assuming V_{o}=0; however, once a V_{o(SB)} estimate was calculated, t_{max} could be redetermined at the time of maximal P/(V−V_{o(SB)}), and then Equations 3 through 5 recalculated, yielding a new V_{o(SB)}. This process could be repeated until successive V_{o(SB)} estimates were ≤1 mL from each other. We tested whether this iterative method improved the predictive accuracy of the method.

### Validation of Single-Beat ESPVR Estimation

Single-beat estimates for both E_{max(SB)} and V_{o(SB)} were compared with corresponding parameters derived from multiple-beat ESPVR analysis [E_{es(MB)} and V_{o(MB)}]. This comparison was made between the same loops used to generate the average E_{N}(t_{N}) curve as well as in a second independent set of data from 35 additional patients. This second group also reflected a broad range of cardiac diseases and included data from several patients before and after inotropic intervention. Thus, comparisons were made for a total of 126 conditions measured in 87 patients.

To determine if the new single-beat estimation accurately indexed acute contractility change induced by inotropic agents, ΔE_{max(SB)} was compared with ΔE_{es(MB)} in 21 patients who received dobutamine (n=5), OPC-18790 (n=5), esmolol (n=9), or verapamil (n=2). To test whether the single-beat E_{max(SB)} estimate was influenced by loading changes, individual cycles at varying preloads measured during IVC occlusion were subjected to single-beat analysis. Beat-to-beat variability of E_{max(SB)} and comparisons of these estimates to E_{es(MB)} were made. This analysis was performed in a random subset of 18 patients (7 normal, 3 HCM, 5 CAD, and 3 DCM).

Last, E_{max(SB)} estimates were compared with those obtained by the previously reported single-beat method of Sunagawa et al.^{17} For each steady-state beat, pressure data between EDP and dP/dt_{max} and between dP/dt_{min} and the point where pressure declined to EDP were fit by nonlinear regression to the formulaP(t)|<|=|>|0.5|<|\times|>|P_|<|iso|>||<|\times|>||<|[|>|1|<|-|>|cos(|<|\omega|>|t|<|+|>||<|\phi|>|)|<|]|>||<|+|>|EDPwhere P_{iso} was the estimated peak isovolumic pressure. For this analysis, EDP was equal to the pressure at which dP/dt exceeded 10% of dP/dt_{max}. E_{max,iso} was then calculated fromE_|<|max,iso|>||<|=|>||<|[|>|P_|<|iso|<|-|>||>|P(t_|<|max|>|)|<|]|>|/|<|[|>|EDV|<|-|>|V(t_|<|max|>|)|<|]|>|Both E_{max,iso} and E_{max(SB)} were then compared with the multiple-beat derived E_{es(MB)} value.

### E_{max(SB)} Estimation From Aortic Pressure Data

By selecting t_{N} to occur at the onset of cardiac ejection, then V(t_{N})=EDV, and P(t_{N})=AoP_{DIA}, the aortic diastolic pressure. The end-systolic pressure, or P(t_{max}), was estimated from arterial systolic pressure byP(t_|<|max|>|)|<|=|>|P_|<|es|>||<|=|>|(0.9|<|\times|>|AoP_|<|SYS|>|)as previously validated,^{25} while V(t_{max}) was approximated by ESV. Thus, from arterial systolic and diastolic pressures, and values for ESV and EDV, V_{o(SB)} could also be estimated byV_|<|o(SB)|<|-|>|AOP|>||<|=|>|\frac|<|AoP_|<|DIA|>||<|\times|>|ESV/(0.9|<|\times|>|AoP_|<|SYS|>|)|<|-|>|EDV|<|\times|>|E_|<|N|>|(t_|<|N|>|)|>||<|(AoP_|<|DIA|>|/(0.9|<|\times|>|AoP_|<|SYS|>|)|<|-|>|E_|<|N|>|(t_|<|N|>|)|>|andE_|<|max(SB)|<|-|>|AOP|>||<|=|>|(0.9|<|\times|>|AoP_|<|SYS|>|)/(ESV|<|-|>|V_|<|o(SB)|>|)For this analysis, t_{max} was the time between the R wave of the ECG and the time when arterial pressure declined by 10% of peak. t_{N} was the normalized time at the upstroke of arterial pressure (ie, diastolic pressure). E_{N}(t_{N}) was determined at this time from the averaged E_{N}(t_{N}) curve, and this value was then inserted into Equation 9. Comparisons were made between E_{max(SB)} estimated from arterial pressure analysis and the corresponding estimate based on PV data (Equation 4) in 24 randomly selected patients.

### Statistics

Data are presented as mean±SD. Single-beat estimates were compared with multiple-beat measures of resting ESPVR slope and intercept by linear regression. Similar analysis was used to test the accuracy of changes in E_{max(SB)} induced by inotropic intervention and for the comparison of PV loop and arterial pressure-derived estimates. Load dependence was tested with the use of a repeated-measures ANOVA and coefficient of variation.

## Results

### Normalized Elastance Curve: E_{N}(t_{N})

Fig 2⇓ displays averaged E_{N}(t_{N}) curves (solid line ±1 SD−dashed line) for each patient group. There was relatively little variance about the mean for each E_{N}(t_{N}) curve among the groups, particularly early in contraction (before the first inflection point). Furthermore, the mean E_{N}(t_{N}) curves were quite similar between groups despite marked differences in disease condition, afterload, preload, and contractility. This is highlighted in Fig 2H⇓, in which the mean curves are superimposed. E_{N}(t_{N}) curves before and after positive inotropic stimulation (Fig 2F⇓, E_{es} from 1.5±0.5 to 2.7±0.9) or atrial pacing (Fig 2G⇓, heart rate at 76±10, 99±9, and 130±12 min^{−1}) were virtually identical. A combined averaged E_{N}(t_{N}) curve based on all these data is shown in Fig 3⇓. Again, the least variance occurred in early contraction (isovolumic phase), while the greatest variability was during late relaxation. This is similar to findings reported from isolated canine heart studies.^{1} ^{9}

### Single-Beat ESPVR Estimation

On the basis of the consistency of E_{N}(t_{N}), we then tested the single-beat ESPVR estimation method. While Equations 4 and 5 yielded V_{o(SB)} and E_{max(SB)} estimates at any time t_{N}, it was preferable to select t_{N} when the physiological variance of E_{N}(t_{N}) was small, ie, in early contraction (also see Fig 3⇑). Equation 5 also revealed V_{o(SB)} to be a hyperbolic function of E_{N}(t_{N}) with a vertical asymptote at E_{N}(t_{N})=P_{N}(t_{N}). If t_{N} was selected near this time, small errors could lead to instability of the V_{o(SB)} estimate. An optimal t_{N} could be theoretically deduced to minimize both physiological and mathematical variances. As demonstrated in the “Appendix,” this value was between 0.25 and 0.35.

Fig 4A⇓ displays scatterplots of single-beat E_{max(SB)} and V_{o(SB)} estimates versus corresponding values from multiple-beat ESPVRs (E_{es(MB)} and V_{o(MB)}). The single-beat estimates were the average of results using t_{N}=0.25, 0.30, and 0.35. Both sets of parameters were highly correlated, with regressions given by E_{es(MB)}=1.01×E_{max(SB)}+0.03, *r*=.903, SEE=0.42; and V_{o(MB)}=0.87×V_{o(SB)}−1.4, *r*=.91, SEE=27.3 (both *P*<.001). These slopes and intercepts were not significantly different from unity and zero, respectively. The lower plots display differences between single- and multiple-beat ESPVR slope and intercept estimates. There was a tendency for the E_{max(SB)} to underestimate E_{es(MB)} (*r*=.46, *P*<.001), whereas this was not observed for V_{o(SB)} (*r*=.1, *P*=NS).

Analysis with the iterative method to update t_{max} and thus V_{o(SB)} calculations (see “Methods”) yielded virtually the same results, with slightly lower slopes and greater scatter: (E_{es(MB)}=0.97×E_{max(SB)}+0.12, *r*=.88, SEE=0.46; and V_{o(MB)}=0.80×V_{o(SB)}−1.8, *r*=.90, SEE=28.3 (both *P*<.001). The E_{max(SB)}−E_{es(MB)} difference plot also did not change, revealing a slight underestimation by the single-beat method at high elastances (slope=−0.197, versus −0.20 without iteration). Thus, despite its potential theoretic benefit, the iteration procedure did not improve ESPVR estimation, suggesting that small errors about t_{max} [and thus P_{(tmax)} and V_{(tmax)}] were not a major source of error in the estimation process.

Table 2⇓ provides data for analyses using t_{N} values other than those between 0.25 and 0.35. As predicted in the “Appendix,” these other ranges of t_{N} yielded greater variance, with somewhat lower correlation coefficients, and higher SEE for the single- and multiple-beat ESPVR parameter regressions.

### Sensitivity of Single-Beat Estimate to Inotropic Change

Fig 5⇓ compares single-beat versus multiple-beat estimates of E_{max} change due to inotropic interventions. Fig 5A⇓ displays example pressure-volume data with dashed lines corresponding to multiple-beat ESPVRs. The drug intervention primarily induced a change in the relation slope. The solid lines are corresponding single-beat ESPVR estimates based on the resting (high) preload beat and were similar to the multiple-beat ESPVR. Fig 5B⇓ shows group results for ΔE_{max(SB)} versus ΔE_{es(MB)}. There was a strong direct correlation (*r*=.92, *P*<.0001), with a slope not statistically different from 1.0 and intercept not different from zero.

### Sensitivity of Single-Beat Estimate to Preload Change

Fig 6⇓ shows the results of tests for the influence of loading change on E_{max(SB)} estimation. E_{max(SB)} was determined from PV loops at initial (high) and reduced (after IVC obstruction) preload volume from a single multiple-beat ESPVR. Fig 6A⇓ displays example PV loop data, with beats at high and low EDV highlighted in bold. The two sets of single-beat ESPVR predictions from these two beats are displayed by a dashed line and were very similar to the multiple-beat ESPVR (solid line). Fig 6B⇓ displays time plots of the same data. The lower tracing shows the single-beat E_{max(SB)} estimate derived from each beat in the series, revealing high reproducibility despite marked changes in chamber loading. The solid line is the multiple-beat derived E_{es(MB)}.

For the group data, the average coefficient of variation (100×SD/mean) of repeated single-beat estimates at varying preloads was 9.4±4%. Comparisons of E_{es(MB)} to E_{max(SB)} at high and low EDV in all 18 patients (Fig 6C⇑) revealed no significant differences between mean values (2.31+1.12, 2.36+0.99, and 2.28+1.09 mm Hg/mL), and values in individual subjects were also generally similar.

### Comparison of New Single-Beat Estimation Technique to Prior Methods

Fig 7A⇓ displays a comparison of E_{max(SB)} determined by the new method to the single-beat E_{max,iso} estimation based on the method of Sunagawa et al.^{17} Both estimates are plot versus E_{es(MB)}. In 36 of 126 loops (28%), Equation 6 predicted P_{iso} (isovolumic pressure) that was lower than P_{es,} resulting in a nonsensical (negative) ESPVR. Interestingly, this occurred more often in failing hearts (HCM, DCM, ANE) than in controls (38% versus 12.8%, *P*=.002 by Fisher's exact test). Data from the remaining 88 loops are shown in Fig 7A⇓. E_{max,iso} generally overestimated E_{es(MB)} (regression slope of 0.53, intercept of 0.73 mm Hg/mL), in comparison to a slope and offset of 1.01 and 0.008 for the new method (*P*<.001 by ANCOVA), and displayed a larger variance (SEE=0.57 versus 0.41 for E_{N}(t_{N}) method).

Another often-used and simpler E_{es} estimation is the ratio of P_{es}/V_{es}, assuming V_{o}=0. Fig 7B⇑ displays this ratio versus measured E_{es(MB)}. There were marked discrepancies between actual E_{es(MB)} and the P_{es}/V_{es} ratio, particularly at low values, with overestimation of E_{es(MB)} by P_{es}/V_{es} at higher values.

### Single-Beat Estimation of E_{es} From Aortic Pressure

Prior analysis (“Appendix” and Table 2⇑) indicated that a t_{N} of 0.25 to 0.35 was optimal for the single-beat ESPVR estimation. This timing was very close to that for aortic valve opening (t_{N} at onset of ejection averaged 0.258±0.035, n=18), supporting the feasibility of using Equations 9 and 10 to estimate ESPVR parameters from arterial pressure and ESV, EDV data. Fig 8⇓ displays the results of this analysis, plotting E_{max(SB)} calculated from Equation 10 versus the estimate measured from PV loop data (Equation 4, the latter assessed at t_{N}=0.25). These estimates were nearly identical, with the regression given by *y*=0.993*x*−0.004, *r*=.982, *P*<.0001, SEE=0.19 mm Hg/mL.

## Discussion

This is the first study to compare normalized time-varying elastance curves [E_{N}(t_{N})] of human left ventricles reflecting a broad spectrum of underlying cardiac diseases and/or operating conditions. We found that the E_{N}(t_{N}) curves were surprisingly similar among hearts, particularly during early contraction. This finding supported the new method of single-beat ESPVR estimation, which displayed sensitivity to contractile change but was little influenced by preload change, was an improvement over prior methods, and could be adapted for noninvasive analysis. These results should enhance the applicability of ESPVR analysis to clinical medicine for assessing acute and chronic changes in contractile function due to therapeutic interventions and for predicting hemodynamic responses to therapy.^{3}

### Assumptions and Implications of Consistent E_{N}(t_{N})

The new single-beat ESPVR estimation method was based on the time-varying elastance model of cardiac contraction, assuming that there was a consistent E_{N}(t_{N}), independent of heart rate, loading conditions, or contractility that could be scaled by both amplitude (E_{max}) and time (t_{max}) for a given ventricle. This also implied linear elastance relations (including the ESPVR) that intersected at a common volume intercept. These are admittedly oversimplifications, as studies have revealed that ESPVRs can be nonlinear,^{10} ^{11} ^{12} can be influenced by afterload,^{13} ^{14} ^{15} ^{16} ^{17} ^{18} ^{26} ^{27} and that V_{o} may change during contraction.^{27} Load-dependence of E_{max} is thought to be due to shortening deactivation,^{28} ^{29} internal resistance,^{26} and length-dependent myofilament calcium sensitivity.^{30} ^{31} Thus, it would seem somewhat surprising that E_{N}(t_{N}) curves from patients with widely varying cardiac diseases and structural adaptations would be so consistent or that the single-beat estimation would yield such reasonable predictions. However, by normalizing the time-varying elastance curve by time as well as amplitude, effects of ejection on t_{max}, such as recently analyzed by Burkhoff et al,^{15} were less influential. The resulting E_{N}(t_{N}) curves, particularly in the early phase, probably reflected myocardial properties related to actin-myosin interaction and the kinetics of stiffness generation. Differences in myofibril protein isoforms and protein kinetics among species might alter E_{N}(t_{N}), so one should not assume that the human E_{N}(t_{N}) curve directly applies to other animals.

Another potential explanation for the consistent E_{N}(t_{N}) curves in patients is that human vascular impedance load is more constrained than that typically used in isolated heart and muscle studies. Unlike muscle preparations in which shortening is abruptly released after a brief isometric contraction^{28} or in isolated hearts in which ejecting and isovolumic contractions are often contrasted,^{15} real impedance varies much less among humans even with various diseases, thereby limiting effects of ejection history on t_{max} and E_{max}. There was more variance in E_{N}(t_{N}) during relaxation, consistent with an enhanced influence of load in this period. Nonetheless, the curves were similar during the initial isovolumic contraction phase, so that during this period, a simple time-varying elastance model became an adequate descriptor of contraction. This finding is supported by prior animal data from Little and Freeman,^{27} who found that the normalized E(t) curve accurately predicted the early isovolumic pressure rise. Consistency of the isovolumic phase of E_{N}(t_{N}) also formed the basis for relations between the derivative of LV pressure and EDV^{32} and between chamber contraction and relaxation times.^{33} One caveat is that E_{N}(t_{N}) in the present study was derived from beats with an isovolumic contraction phase, and this curve should not be extended to hearts with significant mitral regurgitation without additional testing. These tests could not be made with the conductance catheter, since substantial mitral regurgitation can induce large signal artifacts.

Last, it should be reiterated that ESPVR linearity was implicit in the present method, whereas these relations can be curvilinear.^{10} ^{11} ^{12} Thus, V_{o(SB)} may have underestimated the true “V_{o}” (ie, volume intercept at zero pressure) had it been directly measurable, and this probably explains apparent “negative” V_{o} values that were often derived. V_{o(SB)} values did correlate well with V_{o(MB)}, which reflected ESPVR behavior in the physiological loading range. This is significant, since V_{o(MB)} was derived from many beats (average >10) with a net 25 and 50 mm Hg change in P_{es}. The similarity between V_{o(SB)} and V_{o(MB)} is important, since it suggests that the time course of elastance for cycles within the physiological loading range can predict local “linear” ESPVR behavior in that range. This similarly also supports the notion that nonlinear physiology rather than volume-catheter signal artifacts during IVC obstruction most likely explained negative V_{o(MB)} values, since V_{o(SB)} would not be influenced by these artifacts. ESPVR nonlinearity is rarely observed clinically, and we could not test whether beats taken from a different, steeper portion of the ESPVR would yield higher E_{max(SB)} estimates. Nonetheless, we found that the behavior of the ventricle could be predicted over a physiologically relevant loading range from a single beat.

### Comparison to Other Single-Beat Estimation Methods

Investigators have been driven by the desire to simplify the loading procedures required to measure the ESPVR and avoid the need for continuous LV volume data through the use of single-beat E_{max} estimation methods. The simplest, most frequently used method is to ignore V_{o} and report the ratio of P_{es}/V_{es}.^{32} ^{33} This is inadequate (as shown in Fig 7B⇑), since there are marked differences in V_{o}, with often large nonzero intercepts in patients with infarction or dilated cardiomyopathy^{5} ^{7} ^{8} and apparent negative values in patients with high contractility and presumably nonlinear ESPVRs.

An alternative method first proposed by Igarashi and Suga^{18} used abrupt occlusion of the aorta during isovolumic contraction to determine peak isovolumetric pressure. The ejecting portion of the PV loop was constructed by integrating aortic flow and synchronizing the result with LV pressure. E_{max} was the slope of the tangent to this curve anchored at the peak isovolumic PV point. Sunagawa et al^{17} modified this method by showing that peak isovolumic pressure could be mathematically predicted by curve-fitting isovolumic data from an ejecting beat to a cosine function. Takeuchi et al^{20} tested the method in 16 patients with normal EF and found reasonable correlations to E_{es} derived from three variably loaded beats, with an SEE of 1.2 mm Hg/mL/m^{2}. There was no assessment of the accuracy for indexing inotropic change, nor were hearts with significant LV dysfunction evaluated. Some of these limitations were addressed in an animal study,^{19} but again only in normal hearts.

We found several weaknesses in this method. First, it yielded nonsensical results in 28% of the cases (P_{es}<P_{iso}, ie, negative E_{max}), this being three times more common in patients with LV disease. Second, the method generally overestimated directly measured values. Thus, the cosine fit would not appear to apply to all hearts, in particular those with chronic chamber dysfunction. Reliance on data during relaxation (which is more sensitive to load) as well as contraction may contribute to this limitation. The new method yielded more reliable estimates over the full range of heart conditions, used a physiologically based model [E_{N}(t_{N})] rather than arbitrary curve fits, and could be used without need for left ventricular pressure data.

### Noninvasive ESPVR Estimation

Noninvasive applicability of the new single-beat ESPVR estimation method would certainly be an attractive feature. In addition to predicting V_{o} and E_{max}, noninvasive assessment of ventricular-arterial interaction could be determined by coupling E_{max} to estimates of arterial elastance (E_{a}).^{7} ^{8} ^{24} ^{34} Chamber volumes for the estimation might be obtained by two-dimensional or three-dimensional echocardiography, magnetic resonance imaging, or nuclear ventriculography, whereas arterial cuff pressures could provide diastolic and end-systolic pressures. Timing of the onset of ejection relative to end systole could be obtained noninvasively by Doppler echocardiography or from arterial pulse tracings generated by tonometry.^{35} ^{36} The present study showed only that with accurate measurements at the designated time points, noninvasive data could substitute, and thus a noninvasive approach seems feasible. However, there are admittedly considerable errors in noninvasive volume and pressure measurements, and it remains to be determined whether such errors prove to be critical limitations for ESPVR estimation.

## Appendix

Equation 3 from “Methods” predicts V_{O} of the ESPVR from the relationV_|<|O(SB)|>||<|=|>|\frac|<|P_|<|N|>|(t_|<|N|>|)|<|\times|>|V_|<|es|>||<|-|>|V(t_|<|N|>|)|<|\times|>|E_|<|N|>|(t_|<|N|>|)|>||<|P_|<|N|>|(t_|<|N|>|)|<|-|>|E_|<|N|>|(t_|<|N|>|)|>|where P_{N}(t_{N})=P(t_{N})/P_{es}. This provides a prediction of V_{o} for any value of t_{N} and E_{N}(t_{N}). However, the E_{N}(t_{N}) curve obtained from measured pooled clinical data (Fig 3⇑) displayed physiological and statistical deviation that varied with the cardiac cycle phase. Choices of E_{N}(t_{N}) at times with the least variance would likely yield more reliable and accurate estimates. Fig 9⇓ displays a plot of the standard deviation (SD) of the E_{N}(t_{N}) curve as a function of t_{N}, ranging between 0 and 1.0. SD was used as a measure of the instantaneous variance [dE_{N}(t_{N})] as a function of t_{N}. The variance was highest near late ejection and zero (by definition) at t_{N}=1.0.

In addition to this physiological variance, there was mathematical-based variability in the sensitivity of the V_{o} estimate as a function of the value of E_{N}(t_{N}). This could be shown by differentiating Equation A1 with respect to E_{N}(t_{N}):dV_|<|O|>|/dE_|<|N|>|(t_|<|N|>|)|<|=|>|\frac|<||<|-|>|V(t_|<|N|>|)|>||<||<|[|>|P_|<|N|>|(t_|<|N|>|)|<|-|>|E_|<|N|>|(t_|<|N|>|)|<|]|>||>| |<|-|>| \frac|<||<|[|>|P_|<|N|>|(t_|<|N|>|)|<|\times|>|V_|<|es|>||<|-|>|V(t_|<|N|>|)|<|\times|>|E_|<|N|>|(t_|<|N|>|)|<|]|>||>||<||<|[|>|P_|<|N|>|(t_|<|N|>|)|<|-|>|E_|<|N|>|(t_|<|N|>|)|<|]|>|^|<|2|>||>|Fig 9B⇑ displays this equation as a function of t_{N} and shows that this sensitivity is least during the early-mid phase of the cycle.

Multiplying the two curves shown in Fig 9A and 9B⇑⇑ yields a plot of dV_{O} as a function of t_{N} (9C), since dV_{O}=dE_{N}(t_{N})×dV_{O}/dE_{N}(t_{N}), ie, this indicates the variance about a V_{O} estimate as a function of the time point (t_{N}) chosen from the E_{N}(t_{N}) curve. This plot reveals a minimal variance when t_{N} ranges between 0.25 to 0.35. This result is consistent with the data presented in Table 2⇑, which shows the best correlation (ie, smallest SEE) between single-beat estimates and multiple-beat estimates for V_{O} and E_{es} when t_{N} was in this range.

Last, errors in the timing point t_{N} could result in substantial variation in the V_{O} estimate if the E_{N}(t_{N}) curve was itself rapidly changing at that particular time. This could be determined from the derivative of the E_{N}(t_{N}) curve, which is shown in Fig 9D⇑. This reveals a portion of the curve with a rapidly changing slope followed by a portion where the slope is reasonably constant. It is in this latter range (stippled area) that calculation of V_{O} was less likely to be affected by timing measurement errors. Relevant values of E_{N}(t_{N}) between t_{N} of 0.2 and 0.35 for use with Equations 3 and 4 are provided in Table 3⇓. In addition, a 12-term Fourier series fit toE_|<|N|>|(t_|<|N|>|)|<|=|>|A_|<|0|>||<|+|>||<|\sum_|<|n|<|=|>|1|>|^|<|12|>||>|A_|<|n|>|sin(nt|<|+|>||<|\phi|>|_|<|n|>|)is given in Table 4⇓.

## Selected Abbreviations and Acronyms

EDP | = | end-diastolic pressure |

EDV | = | end-diastolic volume |

ESPVR | = | end-systolic pressure-volume relation |

ESV | = | end-systolic volume |

HCM | = | hypertrophic cardiomyopathy |

IVC | = | inferior vena cava |

LV | = | left ventricular |

PV | = | pressure-volume |

## Acknowledgments

This study was supported by National Institute of Aging Grant AG-12249. Dr Kass is an Established Investigator of the American Heart Association. Dr Senzaki is a visiting research fellow from the University of Tokyo (Japan). Dr Chen is a visiting fellow from the Division of Cardiology, Veterans General Hospital–Taipei and National Yang-Ming University, ROC. The authors gratefully thank Drs W. Lowell Maughan, Chih-Tai Ting, Chun-Peng Liu, Mau-Song Chang, Peter H. Pak, Amit Nussbacher, Sigamitzu Ariˆe, and Giovanni Belotti for their valuable assistance in performing the pressure-volume studies and Dr Erez Nevo for his constructive comments and critique.

- Received March 5, 1996.
- Revision received June 11, 1996.
- Accepted June 17, 1996.

- Copyright © 1996 by American Heart Association

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- Single-Beat Estimation of End-Systolic Pressure-Volume Relation in HumansHideaki Senzaki, Chen-Huan Chen and David A. KassCirculation. 1996;94:2497-2506, originally published November 15, 1996https://doi.org/10.1161/01.CIR.94.10.2497
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