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1Departments of Medicine and 2Anesthesiology, College of Physicians and Surgeons, Columbia University, New York City, New York; and 3Department of Medicine, John Hopkins Medical Institutions, Baltimore, Maryland
Submitted 23 November 2005 ; accepted in final form 12 January 2006
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONREFERENCES |
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0 mmHg) was predicted by using the relation V0 = Vm·(0.6 – 0.006·Pm) and V30 by V30 = V0 + (Vm,n – V0)/(Pm/An) (1/Bn). The entire EDPVR of an individual heart was then predicted by forcing the curve through Vm, Pm, and the predicted V0 and V30. This technique was applied prospectively to the ex vivo human EDPVRs not used in determining optimal An and Bn values and to 36 in vivo human, 12 acute and 14 chronic canine, and 80 in vivo and ex vivo rat studies. The root-mean-square error (RMSE) in pressure between measured and predicted EDPVRs over the range of 0–40 mmHg was <3 mmHg of measured EDPVR in all settings, indicating a good predictive value of this approach. Volume-normalized EDPVRs have a common shape, despite different etiology and species. This allows the entire curve to be predicted by a new method with a potential for noninvasive application. The results are most accurate when applied to groups of hearts rather than to individual hearts.
cardiovascular disease; diagnostic techniques
Most studies of the EDPVR have been undertaken in ex vivo models of either beating or arrested hearts in which a balloon is placed in the LV chamber and volume is varied while pressure is measured (5, 12, 22). In vivo assessment of the EDPVR requires use of a means of continuous volume measurement, such as a conductance catheter or multidimensional sonomicrometry, during transient inferior vena caval occlusion (IVCO) (1, 2, 15). For human subjects, the conductance catheter technique is the only feasible volume measurement technique for this purpose. However, required use of invasive techniques to assess the EDPVR has limited its clinical utility in patients.
Until recently, measurement of the end-systolic pressure-volume relationship (ESPVR) also required continuous invasive pressure-volume measurements over a range of volumes created by IVCO. However, Sunagawa and colleagues (25) and Kass and colleagues (7, 24) suggested means of estimating the entire ESPVR from measurement of pressure-volume data from a single beat. Kass and colleagues developed and validated a noninvasive approach in patients from measurements of blood pressure (sphymometry), ventricular volumes (e.g., echocardiography, radionucleotide ventriculography), and Doppler-derived measurements of specific time intervals of the cardiac cycle (7, 24). Such approaches have the potential to overcome the obstacles that have impeded more widespread use of the pressure-volume approach in studies of ventricular mechanics in health and disease. Availability of a comparable approach for assessing the EDPVR would be another major step toward achieving this goal.
We obtained ex vivo EDPVRs from normal and diseased human hearts having a very wide range of heart sizes. We derived an algorithm for normalizing ventricular volume that yielded volume-normalized EDPVRs from all hearts that were very similar to each other. This indicated the existence of a common underlying shape of the EDPVR, independent of heart condition, which can be expressed by a simple analytical expression. This same equation was also shown to be relevant for hearts of normal and diseased animals. It is demonstrated how this analytical equation yields a strategy for predicting the whole EDPVR from a single end-diastolic pressure-volume point. Finally, this approach was validated using the data obtained from in vivo normal and failing hearts of humans, dogs, and rats.
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONREFERENCES |
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Eighty freshly harvested human hearts were obtained for the study. Hearts were from five normal subjects not suitable for transplantation, 23 transplant patients with ischemic cardiomyopathy, 30 transplant patients with idiopathic cardiomyopathy, and 22 patients with end-stage cardiomyopathy supported with an LV assist device (LVAD, TCI HeartMate, Thoratec, Pleasanton, CA, average length of support, 114 ± 86 days). All hearts were perfused with cold hypocalcemic, hyperkalemic cardioplegia solution at explant. The passive LV pressure-volume relationship of each arrested heart was measured as described previously with an intraventricular balloon (4).
In Vivo Human Experiments
Single-beat estimation of EDPVR was tested by using in vivo human data from 10 healthy old and 13 healthy young controls and from 13 patients with heart failure and a normal ejection fraction (HFNEF). The in vivo EDPVRs were constructed by conductance catheter technique using transient inferior vena caval occlusion as described in detail previously (7, 20, 24).
In Vivo Canine Experiments
Data were also obtained from 12 open chest dogs and from 14 chronically instrumented awake dogs. In the chronically instrumented dogs, data were available from each dog under normal conditions and following induction of heart failure by repeated coronary embolization (23).
For acute experiments, LV pressure (LVP) was measured by a catheter tip micromanometer (Millar, Houston, TX) placed inside the LV from the right or left carotid artery. A plastic balloon occluder was placed around the inferior vena cava for temporary occlusion (IVCO). Six ultrasound crystals were positioned in the midmyocardium (3 pairs to measure anterior-to-posterior, base-to-apex and septum-to-lateral wall distances) to estimate LV volume (LVV) based on an ellipsoidal model: LVV = 
/6(a·b·c), where a, b, and c are the principle axis dimensions.
For chronic experiments, animals were instrumented with an indwelling pressure gauge (Konigsberg), sonomicrometer crystals, and an balloon occluder around the inferior vena cava as described previously (28). LVV was obtained the same way as in acute experiments. Measurements of EDPVRs were obtained in a conscious state of the animal by transient IVCO at baseline and 4 wk after induction of heart failure by repeated, daily intracoronary microsphere embolization.
Ex Vivo and In Vivo Rat Experiments
Data were also obtained from Sprague-Dawley rats in heart failure 6 and 12 wk after coronary artery ligation (n = 58) and in 22 normal control rats. For in vivo assessment of the LV passive pressure-volume characteristics, the rats were intubated after induction of anesthesia with isoflurane, and the right carotid artery was cannulated with a Millar conductance catheter (Millar Instruments, Houston, TX). LV EDPVRs were performed by temporary IVCO. For ex vivo assessment of the LV passive pressure-volume characteristics, the heart was arrested in diastole and excised quickly. The left atrium was opened, and a thin latex balloon was attached to the end of a 10-cm length of stiff polyethylene tubing with fenestrations at the distal 3 mm of its tip. LVP was measured using a 5-Fr Millar micromanometer as volume was infused into the balloon at 0.025-ml increments.
All procedures involving human hearts were approved by the Institutional Review Board of Columbia University and the John Hopkins Medical Institutions. All animals involved in these studies received humane care in compliance with the Guide for the Care and Use of Laboratory Animals published by the National Institutes of Health (NIH Publication No. 85-23, Revised 1996). These studies were approved by the Institutional Animal Care and Use Committee of Columbia University.
Theoretical Considerations
EDPVR analysis.
Although human, canine, and rat hearts in different states (normal, failing, and post-LVAD) have very different sizes, earlier studies have shown that EDPVRs from such hearts can all generally be described by one of several nonlinear analytical expressions (e.g., EDP = 
·EDV
or EDP = ·e·EDV) with different coefficient values (17). This suggests that EDPVRs share a common underlying shape and that, if appropriately scaled using a single descriptive equation, the curves may converge. To test this hypothesis, ex vivo EDPVRs from humans were normalized (EDPVRn) in the volume dimension to account for the unstressed volume (V0, volume at which EDP is 0 mmHg) and V30, the volume at which EDP equals 30 mmHg, according to the following equation:
 | (1) |
 | (2) |
Single-beat approach to predicting EDPVR. As will be shown, data indicated the EDPVRns were similar from ex vivo human, dog, and rat hearts, and curve fitting yielded essentially the same values for An and Bn in the different species, indicating a common underlying shape of the EDPVR. From this observation, it is possible to predict the entire EDPVR from a single point on the curve as detailed forthwith.
The procedure starts with measurement of EDP and EDV on a single beat (Pm, Vm, respectively). Based on Eq. 1, normalized measured volume (Vm,n) is defined as follows:
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 | (4) |
 | (5) |
 | (6) |
 | (7) |
 | (8) |
and to force the curve through the measured point on the EDPVR and the predicted V0 (Eq. 8) and V30 (Eq. 5) according to the following:
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 | (10) |
and from Eqs. 9 and 10; and 4) use these estimates of and to specify the entire EDPVR by:
 | (11) |
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Results are expressed as means ± SD. Root-mean-square error (RMSE) between measured and predicted pressures over the entire EDPVR was calculated for each heart as [
(Pm – Pp)2]1/2 with the summation performed over each measured pressure-volume point. One-way ANOVA was used to detect significant differences in RMSE values among the different groups of hearts (ex vivo and in vivo human hearts, acute dog experiments, chronic dog experiments, and ex vivo and in vivo rat hearts). Dunn's test was used for post hoc multiple comparison. Analyses were performed using commercially available software (SPSS 11.5, Chicago, IL).
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONREFERENCES |
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Representative examples of EDPVRs measured from ex vivo human hearts are shown in Fig. 2A. As shown in these examples, EDPVRs from different types of hearts (normal, idiopathic dilated cardiomyopathy, ischemic cardiomyopathy, post-LVAD) spanned very wide ranges of volumes. Some hearts reached filling pressures of 30 mmHg at as little as 80 ml, and others reached this same pressure at more than 300 ml. These same EDPVRs are shown in Fig. 2B after volume was normalized according to Eq. 3. These curves reveal a high degree of concordance between hearts in these different states.
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15 mmHg selected for the test (i.e., these are Pm and Vm for Eqs. 9 and 10). The dotted line shows the curve fit from the raw data using a nonlinear analytic expression (EDP = P0 + ·EDV). The solid line shows the EDPVR predicted from Eqs. 9, 10, and 11. The average RMSE between measured and predicted EDPVRs from the 40 hearts used for the test was 1.74 ± 1.2 mmHg. Figure 4, A and B, shows examples for which the RMSE between measured and predicted EDPVRs was close to the average for the overall group; these results, which are indicative of the average results, reveal a good predictive capability of the approach. Note in these two examples that the predicted V30 point is very close to the measured value. Figure 4C shows an example with a very low RMSE value (0.51 mmHg, among the best predictions), whereas Fig. 4D shows an example with one of the highest RMSE values (5.70 mmHg, among the worst predictions). Interestingly, we did not find any differences in the predictive capability of this approach among the different types of hearts studied (i.e., chronic heart failure, post-LVAD, or normal control hearts).
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10, 15, and 20 mmHg). Figure 6 shows that when a measured pressure-volume point having an EDP above 15 mmHg is used to predict the EDPVR, the prediction is better than when it is predicted by a point having an EDP of 10 mmHg or less (P < 0.05 10 mmHg vs. 15 and 20 mmHg).
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EDPVRs were measured from 36 human hearts in vivo by using the conductance catheter technique of volume measurement and transient IVCO to reduce preload. The EDP-EDV from the starting loop (i.e., before IVCO) were the designated Pm and Vm values for EDPVR prediction. Overall, the average RMSE obtained from the 36 EDPVRs was 2.99 ± 1.72 mmHg. Four examples (two average, one above average, and one of the worst examples) are shown in Fig. 7, A–D, respectively. The example showing the worst predictive value (Fig. 7D) was from a patient with heart failure with normal ejection fraction (HFNEF) in which the EDPVR did not have an asymptote with the x-axis, which was considered to be because of high right-sided filling pressures and pericardial mediate interventricular interactions. Another means of assessing the accuracy of the prediction was to compare measured and predicted pressures at an intermediate volume, e.g., at a volume yielding 15 or 20 mmHg, V15 and V20, respectively. For these in vivo human hearts, measured versus predicted V15 was 113 ± 27 vs. 110 ± 27 ml, and for V20 these numbers were 120 ± 28 vs. 115 ± 29, differences of only 3–5 ml or less than a 5% error.
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To explore the overall general applicability of this approach, data from acute and chronic in vivo dog hearts and in vivo and ex vivo rat hearts were also explored. Plots comparable to those of Fig. 3 revealed similar behavior in each of these settings and, remarkably, the parameter values for An and Bn were nearly identical to those identified in the ex vivo human hearts (Table 1). Also shown in Table 1, the means ± SD RMSE values between measured and predicted EDPVRs and the regression coefficients for the relationship between measured and predicted V30 values for all groups were also, in general, comparable among these different settings so that prospective EDPVR predictions (Fig. 8) were generally excellent. The RMSE values for in vivo human and chronic dog hearts were slightly higher than those attained in the other settings, but the values were still relatively small, indicating good prospective predictive capability of this approach.
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It is pertinent to ask about the sensitivity of the predicted curves to potential inaccuracies in measurement of pressures and volumes, particularly for the clinical scenario when both parameters may be measured noninvasively. Of course, any technique for measuring the EDPVR is equally sensitive to the methods used to measure both pressure and volume. Such an analysis is relatively straightforward because of the simple analytical nature of the prediction algorithm, therefore, for the sake of brevity, only two important examples involving errors in Pm variation are considered in this section; a similar analysis of error in Vm yields similar findings. Figure 9 shows EDPVRs predicted from a common Vm of 100 ml and Pm values of 16 mmHg along with Pm values of 14 and 18 mmHg and 13 and 19 mmHg (i.e., 16 ± 2 and 16 ± 3 mmHg). As seen, the impact is relatively minor if one were interested in determining ventricular capacitance. For example, consider an arbitrary clinical application in which one desired to define capacitance as the volume at a pressure of either 10, 15, or 20 mmHg (V10, V15, V20, respectively). In this example, the 6-mmHg range of possible errors of specifying Pm would result in only an 6 ml range of predicted V10, V15, or V20, which amounts to a range of approximately ±3%.
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Another important aspect of the prediction algorithm is the estimation of V0 using Eq. 8, which can be associated with significant uncertainty and inaccuracy especially as the volume of the heart increase as in systolic heart failure. Figure 9C shows a hypothetical example with an assumed Vm of 200 ml at a Pm of 10 mmHg. Equation 9 predicts a V0 of 108, but based on Fig. 1, V0 could exhibit considerable variability. The three graphs of Fig. 9C show EDPVR predictions with V0 values of 75, 100, and 125 ml. Despite this rather large variation in the assumed V0 value, the impact on predicted EDPVR is relatively mild.
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONREFERENCES |
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The proposed single-beat EDPVR estimation method is based on the premise that the end-diastolic pressure-volume relation can generally be described by common nonlinear analytical expressions with coefficient values that are reasonably well linked with the size of the heart (17). This suggests that overall these EDPVRs share a common underlying shape. The present results are intended to facilitate the use of the EDPVR in clinical and research settings where assessment of changes in passive ventricular properties are important, such as in certain disease states and following therapeutic interventions. The prospective prediction of EDPVRs measured in vivo using the conductance catheter technique combined with transient inferior vena caval occlusion serve as important validation of the approach.
The low RMSE values between measured and predicted EDPVRs indicate good predictive accuracy of the approach. The accuracy was greatest when the pressure of the single measured end-diastolic pressure-volume point existed within a pressure range over 15 mmHg. Specifically with regard to the clinical situation, this single-beat approach provided a predicted EDPVR that was on average within 3 mmHg of that measured in humans using the conductance technique.
Taken to the extreme, the underlying assumption that the EDPVR of all hearts share a common underlying shape regardless of species or disease state would lead to the conclusions that passive myocardial properties and chamber geometry change in only one way and that any two hearts that have a single common end-diastolic pressure-volume point would have identical EDPVRs. However, consider even the extreme case shown in Fig. 4D. In this example, measured and predicted values of and are considerably different. Despite such deviations, which reflect expected variations in material properties and geometry, the differences between the actual and predicted EDPVRs are relatively small. The average results indicate that the expected deviation between predicted and measured EDPVR curves is typically <3 mmHg. Nevertheless, this means that results obtained using the proposed approach would be most accurate when applied to groups of hearts (where such deviations would average out), rather than to individual hearts.
To illustrate this point, consider the example shown in Fig. 10. Little et al. (16) provided group-average end-diastolic pressure-volume data in Table 1 of their publication. These data were obtained in an experimental canine preparation in which volumes were measured by sonomicrometry and pressure measured by a solid-state transducer over a range of conditions achieved by phenylephrine administration. These data, which are rarely provided in published papers, were plotted and the EDPVR predicted using the proposed approach. This completely independent data suggest that the proposed approach is reasonable. We propose that one potentially important use of this approach is in a research setting in which group-averaged EDPVRs from different groups are to be estimated and compared.
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In practice, the proposed scheme for predicting the EDPVR can be accomplished noninvasively. Methods for measurement of LV volume include magnetic resonance imaging, three-dimensional echocardiography, and radionuclide ventriculography (3, 29, 30). Measurement of pulmonary vein velocity by Doppler-echocardiographic techniques provides a means of estimating LV EDP (19).
Study Limitations
The correlation between predicted and measured points was excellent in all ex vivo setups and in vivo acute dogs and rats, measured with sonomicrometry and conductance techniques, respectively. These experimental settings offer ideal conditions in which confounding factors (e.g., controlling respiration) are minimized, and the ability to vary volumes and pressures over a broad range is possible. The good, but slightly lower, correlation obtained from in vivo chronic conscious dog and human data may reflect the effects of respiration and possibly effects of interventricular interactions and pericardial constraints, which may result in an EDPVR that does not have an asymptote with the volume axis (as in Fig. 7D). However, these factors did not impair the ability of the overall approach to provide good predictions in a majority of cases. Such factors could become more important in disease states where right ventricular overload is more prominent and thus limit the utility of the present approach.
The values of An and Bn were obtained from normal and dilated cardiomyopathic hearts, which may limit the applicability of the approach to other disease states. Nevertheless, the approach appeared to apply well in vivo to patients even with heart failure and normal ejection fraction due to hypertensive and/or idiopathic hypertrophic cardiomyopathies. This again speaks to the general applicability of the underlying observation that the EDPVR generally has a common shape.
The mathematical form of the equation used to describe the EDPVR (P = V) was chosen because it was the only one of the commonly used equations that lends itself to the analytical solution permitting the current prediction. However, it is currently more common that an exponential equation be used to describe the EDPVR (e.g., P = eV). Thus the values of and generated using the present approach will differ from those now commonly appearing in the literature. For example, typical values for and obtained with the present approach would be 1.6 x 10–11 mmHg and 6.0, respectively, for normal human hearts and 4.1 x 10–9 mmHg and 5.8, respectively, for normal dog hearts.
From an analytical perspective, the prediction also depends on a degree of difference between the Pm and 30 mmHg. Thus, for Pm between 27 and 32 mmHg, the currect predictions break down. Future efforts could also resolve this limitation.
Finally, further consideration should be given to the effects of right-sided filling pressures, pericardial pressures, and ventricular interdependence. This is especially important in the data generated by caval occlusion in which these pressures are changing and can influence the resulting EDPVR. Whereas such effects can be eliminated in studies performed ex vivo (i.e., by completely unloading the RV), these effects are difficult (sometimes impossible) to discern in vivo without the use of special instrumentation (e.g., pericardial pressure measurements). Thus, even when measured invasively with a conductance catheter or by sonomicrometry, interpretation of an EDPVR measured in vivo during caval occlusion is subject to these same limitations.
In addition to potentially addressing these limitations, future efforts could possibly improve the proposed approach. The cornerstone of such efforts should be aimed at obtaining a larger amount of invasive data from human and animal hearts over a range of normal and pathophysiological states. Such data could be used to refine the parameters employed in the prediction method. Among the parameters that could possibly be refined would be those used in the prediction of V0 in different settings and an attempt to understand the degree to which actually allowing disease-specific differences in the shape of the EDPVR could improve predictive accuracy.
In conclusion, the LV end-diastolic pressure-volume relationship can be reasonably estimated from a single pressure-volume point, and the predicted relationships are generally well correlated with directly measured data. The results obtained are most accurate when applied to groups of hearts rather than to individual hearts. The potential for noninvasive application is particularly appealing and is complementary to the recently proposed approach to single beat estimation of the ESPVR.
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FOOTNOTES |
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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.
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REFERENCES |
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Vm (0.6 – 0.006·Pm), where Vm and Pm are measured volume and pressure points, respectively.
), V30 on the best-fit EDPVR; circled cross (
), predicted V30. A and B: EDPVR prediction with a root-mean-square error (RMSE) value around the average; C: EDPVR prediction with an RMSE value above average; D: EDPVR prediction with a very high RMSE.
