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Cardiovascular Biophysics Laboratory, Washington University School of Medicine, St. Louis, Missouri
Submitted 12 October 2005 ; accepted in final form 20 March 2006
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ABSTRACT |
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 TOP ABSTRACT METHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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peak atrioventricular (AV) gradient] to maximum viscoelastic resistive force [peak resistive force (cEpeak)] relation. To validate load independence, we analyzed E-waves recorded while load was varied via tilt table (head up, horizontal, and head down) in 16 healthy volunteers. For the group, linear regression of E-wave derived kxo vs. cEpeak yielded kxo = M (cEpeak) + B, r2 = 0.98; where M = 1.27 ± 0.09 and B = 5.69 ± 1.70. Effects of diastolic dysfunction (DD) on M were assessed by analysis of preexisting simultaneous cath-echo data in six DD vs. five control subjects. Average M for the DD group (M = 0.98 ± 0.07) was significantly lower than controls (M = 1.17 ± 0.05, P < 0.001). We conclude that M is a LIIF because it uncouples intrinsic DF (i.e., the pressure-flow relation) from extrinsic load (left ventricular end-diastolic pressure). Larger M values imply better DF in that increasing AV pressure gradient results in relatively smaller increases in peak resistive losses (cEpeak). Conversely, lower M implies that increasing AV gradient leads to larger increases in resistive losses. Further prospective validation characterizing M in well-defined pathological states is warranted. diastolic function; Doppler echocardiography; mathematical modeling
Doppler echocardiography is the preferred method for noninvasive diastolic function (DF) assessment. Doppler-derived indexes have been used to characterize DF in numerous cardiac disorders, including heart failure, myocardial infarction, hypertrophic cardiomyopathy, and hypertension (1, 37). In current practice, most DF indexes are derived by visual inspection of transmitral E- and A-waves. These shape-derived indexes include peak velocity of the E-wave (Epeak), duration of the E-wave, acceleration (AT) and deceleration (DT) times of the E-wave, and area under the E-wave [velocity-time integral (VTI)]. Additional indexes include the peak velocity of the A wave (Apeak) and the ratio of the Epeak and Apeak velocities (E/A). Most clinically relevant Doppler-derived DF indexes have proved to be load dependent in animals and humans, both in health and in disease (3, 8, 11, 14, 17, 26, 30, 32, 33, 38, 40–42, 45, 48). Several newer load-independent indexes have been empirically suggested, from velocity of propagation (Vp), to annular velocities derived from Doppler tissue imaging (E'/A', E', E/E'), but there is no consensus as to whether these indexes are truly load independent (9, 15, 24, 27, 28, 31, 47).
To quantify DF, we modeled filling in kinematic terms via the parameterized diastolic filling (PDF) formalism (18–20). The formalism is a lumped parameter, predictive rather than accommodative (23), model that characterizes transmitral flow according to the motion of a damped simple harmonic oscillator (SHO). This model characterizes transmitral blood flow velocity in terms of elastic, inertial, and damping forces. During filling, the elastic driving force generates both inertial forces, causing acceleration, and resistive (damping) forces, opposing acceleration. The three (mathematically) independent model parameters, spring constant k, damping constant c, and initial spring displacement xo, fully characterize the velocity of the SHO (i.e., E-wave velocity contour). These parameters are determined by solving the "inverse" problem of filling, using the clinical E-wave contour as the input, and the model parameters as the best-fit determined output (12). The model-predicted velocity provides an excellent fit to the Doppler E-wave contour recorded clinically (12). In accordance with first principles-based predictions and causality, the model parameters have physiological analogs that have been experimentally validated in vivo (Table 1). Specifically, the chamber stiffness (dP/dV) is linearly related to k (21); the initial peak force (kxo) that drives the oscillator is the analog of the peak atrioventricular (AV) pressure gradient generating transmitral blood flow (2); 1/2kx
is the energy (ergs) available before valve opening (20); and xo is linearly related to the volumetric load, i.e., the VTI of the E-wave (20). In addition, the PDF formalism has been extensively tested and validated in subjects with a wide range of cardiac pathologies and loads, including hypertension (22), heart failure (34), and diabetes (35).
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To test the load independence of the predicted ratio, we altered load in healthy human volunteer subjects by changing tilt table position while recording transmitral flow. We investigated changes in traditional echo-derived indexes, changes in PDF parameters, and the load dependence of the predicted dimensionless ratio. Results of other studies have unambiguously shown that, relative to horizontal, preload increases with head-down tilt and decreases with head-up tilt (7, 11). In addition, many studies have altered preload by performing tilt table position changes (8, 11, 17, 30, 32, 33, 40, 48).
To assess the ability of slope M to differentiate between control and diastolic dysfunction (DD) states, we analyzed preexisting data from patients undergoing simultaneous catheterization-echocardiography using micromanometric (Millar) catheters. We calculated the kxo-to-cEpeak relationship from E-waves as a function of load, which was defined by beat-to-beat left ventricular (LV) end-diastolic pressure (LVEDP) respiratory variation. We determined the linearity of the kxo-to-cEpeak relationship as a function of LVEDP in this data set, as well as the ability of the slope (M) of the linear regression relationship to differentiate between DD and control states.
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Modeling physiology: decoupling the kinematics of filling from load. E-wave velocity contours are known to be load dependent, and, because the PDF formalism provides an excellent fit to all observed E-waves, the PDF parameters will vary as E-waves vary in response to changes in load.
To derive a load-independent index, we sought to decouple load (xo) from filling by employing our kinematic model of filling (the PDF model). Specifically, we note that, regardless of how E-wave contours respond to variations in load, the equation of motion (Newton's law), which includes inertial, damping, and recoil forces for the SHO, remains valid. The equation that approximates the E-wave is:
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Experiment
Data acquisition. Pulsed Doppler echocardiography was used to acquire continuous transmitral blood flow data from 16 subjects (9 men, 7 women, ages 20–30 yr), while subjects were positioned on a tilt table (Trex Medical, Danbury, CT). Doppler data were acquired in the apical four-chamber views with the sample volume gated at 1.5–2.5 mm and directed between the tips of the mitral valve leaflets orthogonal to the mitral valve plane. The subjects were healthy medical and graduate students on no prescribed medications. None of the subjects had a history of heart disease, coronary artery disease, hypertension, or diabetes. Before participation in the study, all subjects gave informed consent for participation in accordance with a protocol reviewed and approved by the Washington University Medical Center Human Studies Committee (IRB).
Doppler data were obtained with a clinical echocardiographic imaging system (Acuson Sequia 256, Mountain View, CA) equipped with a 2-MHz transducer. Heart rate was recorded simultaneously via ECG limb lead II and displayed on the E- and A-wave images. Blood pressure was monitored via a digital blood pressure cuff (Medtronic LifePak 12, Minneapolis, MN).
The data acquisition protocol consisted of initial (baseline) E- and A-wave recording with the subject supine and the tilt table in the horizontal position for 5 min, such that the heart rate was in steady state. Baseline blood pressure was also obtained. After 5 min in the horizontal position, the table, including appropriate padding and straps to ensure safety, was gradually tilted to a 90° head-down position. Since the heart/diaphragm shifted during tilt, the transducer was suitably reoriented to obtain transmitral flow. After resolution of heart rate transients, typically after several minutes, transmitral flow was recorded. After several minutes, the tilt table was returned to the horizontal position. Once heart rate and blood pressure returned to baseline levels, transmitral flow was again recorded. After 5 min horizontally, the tilt table was moved to 90° head-up position. The transducer was adjusted to account for the heart shift during tilt, and, after heart rate and blood pressure transients resolved, transmitral flow was recorded. In 2 of the 16 subjects, the persistent increase in heart rate during head-up tilt resulted in significant E- and A-wave merging. With significant merging, the E-wave deceleration portion is lost, and thus it is not possible to reliably fit merged waves via the PDF formalism. Hence for these two subjects, head-up E-waves were not analyzed, but the head-down and horizontal data were included in the overall analysis. After 5 min upright, subjects were tilted to the horizontal position, and a concluding set of transmitral flow images was recorded. All data were recorded on VHS tape for off-line analysis using a custom-editing station.
Data analysis. For each subject at each stage of tilt, five beats were captured and analyzed by model-based image processing (MBIP). The MBIP method by which PDF parameters are obtained from digitized recordings of transmitral flow has been previously described (19–21). Briefly, transmitral waves for individual beats were analyzed from still images of transmitral velocity profiles, and conventional Doppler-derived indexes were calculated. In addition, E- and A-wave contours were fit via the PDF formalism, which permits the contour of the E-wave to be expressed in a closed-form mathematical expression. Minimizing the error between the model-predicted contour and the clinical E-wave contour via the Levenberg-Marquardt algorithm provides best-fit (unique) model parameters (c, k, xo) for each E-wave. Since directly measured and model calculated values of Epeak and AT show nearly perfect agreement, Epeak was calculated directly from the model-predicted contour for the E-wave.
Additional validation in pathological cases. In addition to testing the LIIF noninvasively in healthy normal subjects via tilt table, we tested the load dependence of M in subjects with and without DD undergoing diagnostic catheterization. This additional analysis used existing data from previous studies (2, 25), utilizing the Cardiovascular Biophysics Laboratory database of simultaneous Doppler echocardiographic transmitral flow and micromanometric (Millar) catheter-derived intraventricular (LV) pressure. We note that the original intent of the Cardiovascular Biophysics Laboratory database was not explicit testing of load dependence. However, because some subjects manifested significant beat-to-beat respiratory variation of LVEDP (load) and simultaneously recorded E-wave contours, the data were suitable for determination of the kxo-to-cEpeak relationship as a function of load, i.e., varying LVEDP. We, therefore, selected subjects who had good quality E-waves as well as significant load variation (LVEDP variation > 10 mmHg) in response to the respiratory cycle. Selection criteria for inclusion in the DD group required normal sinus rhythm, no evidence of valvular disease, no active ischemia, normal ejection fraction (EF) > 60%, and elevated LVEDP (>19 mmHg). Subjects in the control group had normal sinus rhythm, no valvular disease, no active ischemia, normal EF > 60%, and normal LVEDP. Because subjects are referred for catheterization to establish the presence of coronary artery disease, variable degrees of coronary artery disease were present in both the control and DD groups. However, no subject in either group had ongoing or active ischemia. Demographics are presented in Table 2. For each subject, 25 consecutive E-waves and simultaneous LV pressures were analyzed. For each subject, good quality E-waves were selected and analyzed using the MBIP procedure described above. LVEDP was determined from the simultaneous LV pressure data.
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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In agreement with other studies (8, 11, 18, 30, 32, 33, 40), both E- and A-wave shapes varied in response to changes in load generated by changes in tilt table position. Because the PDF parameters are determined from the contour of the waves, changes in load resulted in changes in the PDF parameters. Figure 1 shows representative Doppler waves from a subject in the head-up, horizontal, and head-down position, as well as the PDF model-predicted fit to the E-wave contours. Table 3 summarizes the average values for both traditional Doppler indexes and PDF parameters under different load states i.e., tilt table positions.
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In accordance with the prediction (see METHODS) that the maximum driving force must be linearly related to peak viscous (resistive) force, a linear regression via least mean square error of cEpeak to kxo was performed. Figure 2A shows a representative kxo vs. cEpeak plot for one subject. Head-up data from two subjects were not included, but their head-down and horizontal data fit the kxo vs. cEpeak regression with high r2. Among all subjects, the average slope of the kxo vs. cEpeak plot was 1.27 (SD 0.09), and the average intercept was 5.69 (SD 1.70). The average r2 value for each subjects' linear regression was r2 = 0.95. The combined data from all subjects plotted together yielded a linear regression with r2 = 0.98 (Fig. 2B).
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Our experimental tilt table-derived results show the existence of a load-independent linear relationship between kxo and cEpeak. One issue is whether the damped-SHO model itself requires that the kxo vs. cEpeak relation must always be linear. To address this issue, we selected random values of k, xo, and c to create E-waves, with the requirement that their amplitudes and durations fall within the range of values that has been previously observed and used to fit Doppler data via the PDF formalism. If our observations were a tautology, these E-waves would show a linear relationship between k, xo, and c. The graph of kxo vs. cEpeak using such random values (within the physiological range) is shown in Fig. 3.
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It should be noted that k, c, and xo are, from a mathematical standpoint, independent parameters. There is therefore no a priori correlation between k, c, xo, or any combination of the parameters that can be predicted. These parameters become physiologically coupled, and their magnitudes are constrained, once they are fit to clinical data (the E-wave). Once determined by fitting to actual E-waves, several correlations can be seen, such as Epeak to kxo, k to cEpeak, and c to kxo. However, kxo to cEpeak is the strongest observed correlation and is the only one with a dimensionless slope amenable to a simple physiological interpretation.
Effect of DD on M: Results From Available Data
Using existing simultaneous cath-echo data, we analyzed the kxo vs. cEpeak relationship in subjects with significant respiratory (LVEDP) variation. The results, including statistics, are summarized in Fig. 4 and Table 5. The kxo vs. cEpeak relation was highly linear for both normal (average r2 = 0.96 ± 0.02) and DD groups (average r2 = 0.90 ± 0.05). The average slope for the normal group was M = 1.17 ± 0.05, and the average value for the DD group was M = 0.98 ± 0.07 (P < 0.001 by ANOVA). Additionally, the slope intercept of the normal cath-echo group was B = 6.69 ± 0.91, whereas the slope intercept of the DD cath-echo group was B = 10.67 ± 3.35 (P = 0.03 by ANOVA). The difference in M between the cath-echo DD subjects and the healthy tilt table subjects showed stronger statistical significance (P < 0.00001) compared with the difference in M between the cath-echo normal subjects and the healthy tilt table subjects (P = 0.02). The difference in intercept B is also statistically significant between DD subjects and tilt table subjects (P < 0.0001). There was no statistically significant difference, however, between the intercept of the normal cath-echo subjects and the tilt table subjects.
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DISCUSSION |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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A major challenge facing interpretation of echo-derived DF indexes is that the indexes in current clinical use (Epeak, AT, DT, E/A, VTI) are load dependent (3, 8, 11, 14, 17, 26, 32, 33, 40–42, 44, 45, 48). Despite this uncertainty, it is often assumed that changes observed in the indexes represent changes in pathological processes. Studies have clearly established, however, that traditional transmitral flow-derived diastolic indexes are load dependent, and, therefore, changes in DF indexes could be due to load variability rather than to pathology.
Using blood pressure cuffs to noninvasively reduce preload in 12 normal subjects, Triulzi et al. (44), for example, found Epeak, E/A, VTI, DT, and AT to change significantly with preload and concluded that preload reduction in normal subjects produces a "pattern that mimics changes in left ventricular diastolic dysfunction." By altering preload with postural changes, Downes et al. (8) found that, although E/A changed, it did not change enough for normal values to migrate into the abnormal range and concluded that "simple changes in venous return do not ‘normalize’ an abnormal pattern, nor do they ‘abnormalize’ a normal pattern." Kmetzo et al. (17), however, found that 80° head-up tilt produced abnormal relaxation patterns in 22 normal heart-healthy volunteers.
Newer load-independent Doppler-derived indexes of DF have been proposed (Vp, E', E/E'), but all have been shown to have load dependence. E' has been noted to be load independent in patients with chronic ischemic syndrome (47) but has shown variability with changing preload in normal subjects (30, 31), in animals (15), and in patients undergoing hemodialysis (28). Vp has shown load independence and strong correlation to the time constant of isovolumic relaxation in patients undergoing cardiac surgery (9) but proven to be load dependent in normal volunteers (30). Even E/E', which many studies find to be load independent (25, 28, 30), has proved in animal experiments to be preload dependent (15).
Despite studies that have shown DF indexes to be load dependent, no studies have derived diastolic indexes that compensate for load. Recent studies (13) have shown, however, that intrinsic properties of the ventricle may correlate better with load-induced changes in the Doppler indexes rather than the Doppler indexes themselves. For example, Tanabe et al. (43) found that the change in E/A following a decrease in preload correlated strongly with an invasively derived intrinsic property of the ventricle.
In contrast, Hasegawa et al. (13) observed Epeak (ml/s) volumetric flow to be linearly related to the peak AV pressure gradient (r2 = 0.94) in accordance with the Bernoulli relation relating pressure gradient to flow velocity. Although load was not explicitly altered, the observed change in peak AV gradient is consistent with a variable load. Interestingly, the slope of the observed relation is highly linear and is nearly indistinguishable between the control and heart failure groups. Because the slope of the Epeak vs. peak AV gradient relation discussed by Hasegawa et al. remains linear and does not change in the setting of heart failure, although load may have changed, in our view it reflects the applicability of the Bernoulli equation rather than fulfilling the criterion for a LIIF.
Applying the PDF Model
The PDF model helps elucidate the connection between the experimentally determined Epeak vs. peak AV gradient relation by Hasegawa et al. (13) and the present study. In agreement with Hasegawa et al., a plot (not shown) of Epeak vs. kxo (the peak AV gradient equivalent) using our data also generates a linear relation. The slope has units of seconds per kilogram; the linear regression relation is Epeak = 0.017(kxo) + 0.39, r2 = 0.78. The fact that an approximately linear relation is observed is predictable in part from the nonsteady Bernoulli relation, which relates pressure gradient to flow velocity. Moreover, the Epeak vs. kxo relation does not explicitly account for the effect of damping/viscosity as can be achieved by inclusion of the parameter c.
A stronger and more meaningful correlation is achieved by the kxo (peak AV gradient kxo) vs. cEpeak relation, which attains r2 = 0.98 compared with the r2 = 0.78 for the Epeak vs. peak AV gradient (kxo) relation. In addition to attaining a stronger correlation, the slope of the kxo (peak AV gradient kxo) vs. cEpeak relation is dimensionless. It complements the Hasegawa et al. (13) observation by explicitly including the effect of damping. A high value for cEpeak implies a large cEpeak must be generated to achieve the observed flow. This higher resistive force indicates that conversion of (maximum AV gradient) potential energy into kinetic energy (flow) is less efficient. The PDF (viscoelastic) parameter c is important because it has been shown to be significantly different from normal in pathophysiological states with delayed relaxation patterns such as diabetes (6, 35). A higher value of the parameter c in diabetic (human and rat) hearts implies that diabetic hearts, compared with normal hearts, are less efficient in being able to convert potential energy (pressure gradient) into kinetic (flow) energy. Therefore, we expect that, unlike the Epeak vs. peak AV gradient (kxo) relationship, the kxo (peak AV gradient kxo) vs. cEpeak relationship will have quantifiable differences between normal and abnormal LV function states.
Although not the primary intent of the present study, Fig. 4 (derived from cath-echo data acquired for another purpose) provides compelling evidence that both slope M and intercept B, of the kxo vs. cEpeak linear regression, change in LV DD states. The DD group had lower M values and higher B values compared with the normal cath-echo group and the healthy tilt table group. The normal cath-echo group had the same B values as the healthy tilt table group and slightly lower M values. This difference in M between the normal cath-echo subjects and the tilt table subjects may be due to the higher average age of the normal cath-echo subjects (44 yr) relative to the healthy tilt table group (25 yr).
These preliminary data, particularly because they were not acquired with the specific intent of determining load independence, strongly support the view that M and B can differentiate between normal and pathological states. Because M is the slope of a regression relation, it provides relative information regarding the efficiency with which the ventricle adapts to load. In contrast, the intercept correlates with LVEDP, an absolute rather than relative index. New (cath-echo) studies, specifically including maneuvers that vary load, are indicated to more fully elucidate the relationship between M and B, and phenotypically well-characterized DD states.
Interestingly, only certain regimes of the kxo vs. cEpeak graph can exist physiologically. Energy conservation makes it impossible for cEpeak, the peak resistive force opposing filling, to exceed kxo, the initial (and maximum) driving force initiating filling. If M is low, specifically <1, as preload increases, the kxo vs. cEpeak regression line approaches the kxo = cEpeak boundary, eventually crossing it and moving into the nonphysiological regime. A ventricle having M < 1 would, therefore, not be able to maintain filling function in the face of increased preload (Fig. 5).
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A minor limitation is the absence of end-diastolic volume (EDV) data as a correlate of preload alteration at different stages of tilt. EDV could not be reliably calculated, because only a four-chamber view was archived. Transthoracic echocardiographic studies have shown that EDV calculated from four-chamber views alone achieve only an r =0.61 correlation with true EDV (46). Although exact preload change with tilt alteration is not reported, many studies have altered preload using tilt table methodology (8, 11, 17, 30, 32, 33, 40, 47), and it is certain that preload is highest in head-down tilt and lowest in head-up tilt (7, 11).
A second limitation is that the PDF formalism is most applicable to E-waves having both ascending and descending portions. E- and A-waves become difficult to separate and discern when the A-wave merges with the E-wave and covers more than two-thirds of the E-wave deceleration (4), which typically occurs at heart rate >100 beats/min. Thus M values for subjects with high heart rates (>100 beats/min) were not computed. In the present study, only 2 of 16 subjects during head-up tilt had limited E-wave deceleration portions. Hence head-up data from these two subjects were not fit by the PDF formalism.
Finally, in our tilt table study, all data were acquired from heart-healthy volunteers, and so a detailed comparison of the kxo vs. cEpeak relation between normal and pathological states cannot be made. We predict, and results from analysis of previously acquired data obtained for other purposes in a modest sample size show, that the slope and intercept of the kxo vs. cEpeak relation changes (decreases) in the setting of dysfunction. The full potential of the index resides in performance of follow-up echocardiographic studies, where each person is his own control, showing alteration of E-wave morphology in response to pharmacological therapy. Whether observed E-wave changes are due to load, or due to intrinsic changes in the LV in response to therapy (LV remodeling), could be addressed via slope M and intercept B. Because no prediction regarding the existence of any LIIF or its noninvasive validation in normal subjects has been previously achieved, the proposed approach is a reasonable first step.
In conclusion, it is known that changing tilt table position alters load in a predictable manner and affects E-waves in a predictable way (3, 8, 11, 14, 17, 26, 32, 33, 38, 40–42, 44, 45, 48). Changes in load thereby affect the E-wave determined PDF parameters (k, xo, and c). The observed E-wave variation reflects the physiological mechanisms that respond to changes in load. Because the slope M of the maximum driving force vs. peak resistive force relationship remains constant in response to alteration of tilt table-induced load (Fig. 2), we conclude that M represents a LIIF in normal subjects. To determine whether M differentiates between control and DD states, we analyzed previously acquired cath-echo data. Results show M to be significantly lower in DD states (6 subjects with increased LVEDP and normal EF), compared with control states (5 subjects with normal LVEDP and normal EF). We emphasize that additional prospective studies, specifically designed to test load independence in well-defined pathophysiological subsets, are required to fully characterize and establish the utility of our proposed noninvasive LIIF.
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GRANTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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ACKNOWLEDGMENTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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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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TOPABSTRACTMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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70 years of age with congestive heart failure. Am J Cardiol 84: 82–86, 1999.[CrossRef][Web of Science][Medline]This article has been cited by other articles:
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  E.-Y. Choi, B. W. Choi, S.-A. Kim, S. J. Rhee, C. Y. Shim, Y. J. Kim, S.-M. Kang, J.-W. Ha, and N. Chung Patterns of late gadolinium enhancement are associated with ventricular stiffness in patients with advanced non-ischaemic dilated cardiomyopathy Eur J Heart Fail, June 1, 2009; 11(6): 573 - 580. [Abstract] [Full Text] [PDF]
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C. S. Chung and S. J. Kovacs Physical determinants of left ventricular isovolumic pressure decline: model prediction with in vivo validation Am J Physiol Heart Circ Physiol, April 1, 2008; 294(4): H1589 - H1596. [Abstract] [Full Text] [PDF]
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C. S. Chung, A. Strunc, R. Oliver, and S. J. Kovacs Diastolic ventricular-vascular stiffness and relaxation relation: elucidation of coupling via pressure phase plane-derived indexes Am J Physiol Heart Circ Physiol, November 1, 2006; 291(5): H2415 - H2423. [Abstract] [Full Text] [PDF]
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