## Abstract

In current practice, empirical parameters such as the monoexponential time constant τ or the logistic model time constant τ_{L} are used to quantitate isovolumic relaxation. Previous work indicates that τ and τ_{L} are load dependent. A load-independent index of isovolumic pressure decline (LIIIVPD) does not exist. In this study, we derive and validate a LIIIVPD. Recently, we have derived and validated a kinematic model of isovolumic pressure decay (IVPD), where IVPD is accurately predicted by the solution to an equation of motion parameterized by stiffness (*E*_{k}), relaxation (τ_{c}), and pressure asymptote (P_{∞}) parameters. In this study, we use this kinematic model to predict, derive, and validate the load-independent index *M*_{LIIIVPD}. We predict that the plot of lumped recoil effects [*E*_{k}·(P*_{max} − P_{∞})] versus resistance effects [τ_{c}·(dP/d*t*_{min})], defined by a set of load-varying IVPD contours, where P*_{max} is maximum pressure and dP/d*t*_{min} is the minimum first derivative of pressure, yields a linear relation with a constant (i.e., load independent) slope *M*_{LIIIVPD}. To validate the load independence, we analyzed an average of 107 IVPD contours in 25 subjects (2,669 beats total) undergoing diagnostic catheterization. For the group as a whole, we found the *E*_{k}·(P*_{max} − P_{∞}) versus τ_{c}·(dP/d*t*_{min}) relation to be highly linear, with the average slope *M*_{LIIIVPD} = 1.107 ± 0.044 and the average *r*^{2} = 0.993 ± 0.006. For all subjects, *M*_{LIIIVPD} was found to be linearly correlated to the subject averaged τ (*r*^{2} = 0.65), τ_{L}(*r*^{2} = 0.50), and dP/d*t*_{min} (*r*^{2} = 0.63), as well as to ejection fraction (*r*^{2} = 0.52). We conclude that *M*_{LIIIVPD} is a LIIIVPD because it is load independent and correlates with conventional IVPD parameters. Further validation of *M*_{LIIIVPD} in selected pathophysiological settings is warranted.

- isovolumic relaxation
- mathematical modeling
- hemodynamics

in current practice, τ and τ_{L}, the exponential and logistic time constants of isovolumic relaxation, are the parameters by which isovolumic pressure decay (IVPD) is characterized (18). However, the IVPD contour has been found to be sensitive to both intrinsic relaxation properties and extrinsic load (1, 3a, 13, 15, 17, 24, 25, 32–34, 40). Indeed, the load dependence of τ is well established (1, 13, 15, 17, 32, 40), and therefore the variation in τ between subjects may be the result of intrinsic chamber property differences or may be caused by extrinsic load effects. Thus a load-independent index of isovolumic pressure decline (LIIIVPD) that overcomes the limitations of τ would be advantageous.

Recent work by Chung and Kovács (6) has demonstrated that IVPD is determined in a mathematically precise manner by the interplay of stiffness and damping/relaxation forces. The relative contribution of stiffness and relaxation to IVPD is characterized by a stiffness parameter *E*_{k} (1/s) and a damping or relaxation parameter τ_{c} (1/s^{2}). These parameters may be extracted from in vivo IVPD contours by inverting the governing differential equation and applying quantitative techniques. In addition, when compared with the monoexponential (τ) and logistic (τ_{L}) models, the τ_{c}, *E*_{k} model-predicted pressures and pressure derivatives provide a superior fit to IVPD. Although τ and τ_{L} are typically used to assess chamber relaxation (3a, 18, 24, 25), Chung and Kovács's work predicts that the τ_{c} and *E*_{k} effects jointly determine the physiology of IVPD. Furthermore, Chung and Kovács showed that τ and τ_{L} could be algebraically determined from the ratio of τ_{c} and *E*_{k} (τ, τ_{L} ∝ τ_{c}/*E*_{k}). Thus the stiffness and relaxation effects may be extracted from τ, or the stiffness and relaxation may be recombined to yield a lumped IVPD decay constant (τ or τ_{L}). It is important to note, however, that τ_{c} and *E*_{k} may be combined in various algebraic ways to yield novel lumped IVPD parameters.

While the Chung's model successfully unifies the previously unrelated monoexponential and logistic models of IVPD in a parametric limit sense, τ_{c} and *E*_{k}, just like τ and τ_{L}, are individually load dependent. We hypothesize that certain algebraic combinations of τ_{c} and *E*_{k} provide a lumped IVPD parameter that is not subject to the load dependence of τ, τ_{L}, τ_{c}, and *E*_{k}. Guided by previous methods and results applied to transmitral blood flow (E wave) (35), we derive a novel algebraic relation between τ_{c} and *E*_{k} that, unlike τ, τ_{L}, τ_{c}, *E*_{k}, and other indexes of IVPD, is independent of load. This relation defines the LIIIVPD.

## METHODS

### Monoexponential and Logistic Models of IVPD

The monoexponential model of IVPD, first proposed by Weiss et al. (38), assumes that the time derivative of pressure decay is proportional to pressure. The governing differential equation for the monoexponential model is (1) where τ is the monoexponential time constant of IVPD and P_{∞} is the pressure asymptote. In Weiss's original formulation, P_{∞} was assumed to be 0. A convenient method for the determination of τ is to calculate the negative inverse of the slope of the IVPD in the pressure phase plane (PPP), where dP(*t*)/d*t* is plotted against P(*t*) (11, 18, 32).

While the monoexponential model defines a straight line in the PPP, the logistic IVPD model generates a curve in the PPP and therefore has been used to accommodate nonlinear IVPD PPP segments (22). The differential equation for the logistic model is (2) where τ_{L} is the logistic time-constant of IVPD and the pressure asymptote is given by the sum P_{A} + P_{B}. No simple geometric method for the determination of τ_{L} has been found. Instead, nonlinear fitting algorithms are required to extract τ_{L} from IVPD pressure contours.

### Chung Model of IVPD

In recent work, Chung et al. (6) unified the empiric monoexponential and logistic IVPD models with a general model that completely characterizes the wide range of physiologically observed IVPD PPP trajectories. Based on physiological-kinematic arguments and published experimental results (3a, 14, 19, 20, 24, 30, 33, 36, 39), Chung et al. argued that IVPD is governed by the interplay of inertial, stiffness, and relaxation forces. The relative values of these forces determine the resulting inertial force and, by Newton's law, cause small-scale tissue displacements (isovolumic torsion and chamber shape change) (33). Using Laplace's law to transform displacements to pressures, Chung et al. proposed the following differential equation to account for IVPD: (3) This equation can be solved in the underdamped regime (4*E*_{k} > τ_{c}) for pressure or for the time derivative of pressure as (4) (5) where P_{o} is the initial pressure assuming zero pressure asymptote, Ṗ_{o} is the initial time derivative of pressure, and The critically damped (4*E*_{k} = τ_{c}) and overdamped (4*E*_{k} < τ_{c}) solutions can be determined by evaluating *Eqs. 4* and *5* at ω = 0 (critically damped) or ω = *i*β (overdamped) limits. The procedure for extracting *E*_{k} and τ_{c} from an isovolumic pressure contour, the equivalent of solving the “inverse problem of IVPD,” is described in *Automated τ*_{c} *and E*_{k} *fitting*.

### Human In Vivo Hemodynamic Data

Twenty-five datasets were selected from the Cardiovascular Biophysics Laboratory database of simultaneous micromanometric catheter-recorded left ventricular (LV) pressure and echocardiographic data (21). The subjects were scheduled for an elective diagnostic cardiac catheterization at the request of their referring physicians to rule out the presence of coronary artery disease. All subjects provided informed consent before the procedure in accordance with a protocol approved by the Barnes-Jewish Hospital/Washington University Human Research Protection Office. The criteria for data selection from the database included normal sinus rhythm, normal valvular function, and no wall motion abnormalities. The group was chosen to be clinically heterogeneous, so as to test the generality of the LIIIVPD. Thus subjects with low ejection fraction (EF), significantly elevated τ and τ_{L}, and/or significantly elevated end-diastolic pressure were also included. Patient demographics are presented in Table 1.

Our method of high-fidelity, in vivo pressure-volume recording has been previously detailed (2, 7, 21) . Briefly, after arterial access and placement of a 64-cm sheath (Arrow, Reading, PA), a 6-Fr micromanometer-tipped pigtail (triple pressure transducer) pressure-volume, conductance catheter (SSD-1034, Millar Instruments, Houston, TX) was directed into the mid-LV in a retrograde fashion across the aortic valve under fluoroscopic control. The three pressure transducers were located such that the distal and middle transducers recorded LV pressure and the proximal pressure transducer recorded simultaneous aortic root pressure. Pressures were fed to the Catheterization Laboratory amplifier (Quinton Diagnostics, Bothell, WA; General Electric) at a sampling frequency of 200 or 240 Hz. The LV pressure, the LV volume from the conductance catheter, and one ECG channel were also simultaneously recorded on a disk in digital format using our multichannel physiological data acquisition system, consisting of a Pentium class computer, with 100 MB hard disk, 64 MB RAM, and NB-M10-16H digitizing board. The sampling rates were controlled using Leycom Software (Leycom Sigma-5, CardioDynamics, Rijnsburg, The Netherlands).

Load variation is achieved by performing a Valsalva maneuver during catheterization. Load dependence is further assessed by an analysis of the hemodynamic response following spontaneous or catheter-generated premature ventricular contractions (PVCs) (2, 4, 9, 37).

Following the catheterization, the EF was determined by ventriculography. The remainder of the catheterization and coronary angiography proceeded in the usual manner.

### Hemodynamic Analysis

Hemodynamic data were analyzed using custom-automated Matlab programs (Matlab 6.0; MathWorks, Natick, MA). For each subject, the time derivatives of pressure, left ventricular end-diastolic pressure (LVEDP), mitral valve opening (MVO) time, maximum and minimum pressure and pressure derivatives (P_{max}, P_{min}, dP/d*t*_{min}, and dP/d*t*_{max}), and IVPD inflection point were determined by automated Matlab scripts. LVEDP was defined by the LV pressure at the ECG R-wave peak. The MVO time was determined as the time point where the decaying pressure contour was closest to the LVEDP of the subsequent filling beat (5, 16, 23, 27).

#### Automated τ and τ_{L} fitting.

The PPP was used to determine τ for each beat in each subject. The least-squares determined slope of the dP/d*t* versus P plot over the interval between 5 ms after dP/d*t*_{min} and 5 ms before the estimated MVO time was equal to −1/τ (11, 18, 32). τ_{L} was determined for each beat according to the methods described by Matsubara et al. (22), using a customized Levenberg-Marquardt algorithm (31). Automated Matlab scripts were used for both τ and τ_{L} determination.

#### Automated τ_{c} and E_{k} fitting.

For each IVPD contour in each subject, τ_{c}, *E*_{k}, P_{o}, and Ṗ_{o} were extracted via *Eq. 5* from dP/d*t* versus *t* data using a Levenberg-Marquardt fitting algorithm to the dP/d*t* data (6). The start point for the fitting was determined from the inflection point in the IVPD pressure contour preceding dP/d*t*_{min}, and the end point for fitting was taken to be 5 ms before the LVEDP-estimated MVO time. Once τ_{c}, *E*_{k}, P_{o}, and Ṗ_{o} were found, P_{∞} was determined via *Eq. 4* using the Levenberg-Marquardt algorithm with the four other parameters held constant. The large size of the dataset for each subject, which always includes spurious (noisy) data due to ectopy, patient cough, motion, etc., justified an automated screening procedure to exclude nonphysiological data. This was accomplished by determining the root mean square error (RMSE) between the model fit dP/d*t* and the raw dP/d*t* data for all beats for a given subject. IVPD contours, which are nonphysiological and noisy, generate high dP/d*t* RMSE values compared with acceptable physiological dP/d*t* data. Accordingly, we discarded beats having the largest 50th percentile of RMSE values. This approach ensured that only physiological (smooth) data were included in the final analysis and provided the additional advantage of being automated, thereby minimizing observer bias in beat selection.

### The Load-Independent Index of IVPD

#### Derivation of LIIIVPD.

Recently, using a harmonic oscillator as the kinematic analog of early rapid filling (20), we have derived and validated a load-independent index of diastolic function (LIIDF) (35). Because the governing differential equations for early rapid filling and IVPD are similar, we approached the derivation of the LIIIVPD in a manner analogous to the LIIDF derivation. We provide mathematical details in the appendix.

In the LIIDF work, we found that the peak force driving transmitral flow was linearly related to the peak force opposing transmitral flow. We showed that the slope of this linear relation was constant in the face of beat-to-beat load variation and therefore was load independent. Chung's kinematic model of IVPD, similar to the previously validated model of transmitral flow (20, 35), was characterized by damped harmonic motion. Thus, in this study, by analogy, the effective peak elastic (recoil) forces that drive pressure decline during isovolumic relaxation were predicted to be linearly related to the peak resistive forces that oppose cross-bridge uncoupling and pressure decline. The model's equivalent of peak elastic recoil force is given by *E*_{k}·(P*_{max} − P_{∞}), whereas the peak resistive force is given by τ_{c}·(dP/d*t*_{min}). It is important to note that P*_{max} is the model-predicted maximum pressure, which occurs before aortic valve closure. Thus the Chung model of IVPD predicts the following linear relationship among a set of load-varying IVPD contours: (6) where *M*_{LIIIVPD} and *B*_{LIIIVPD} are constants and the magnitude of dP/d*t*_{min} is used. See appendix for mathematical details. It is important to note that each individual IVPD contour has unique values for *E*_{k}·(P*_{max} − P_{∞}) and τ_{c}·(dP/d*t*_{min}). Thus each IVPD contour defines a single point in the *E*_{k}·(P*_{max} − P_{∞}) versus the τ_{c}·(dP/d*t*_{min}) plane. Whereas τ_{c}, *E*_{k}, P_{∞}, P*_{max}, and dP/d*t*_{min} may change as a result of beat-to-beat load variation, the slope *M*_{LIIIVPD} is predicted to remain constant in the face of load variation and thus is the predicted LIIIVPD.

#### Calculation of LIIIVPD.

The load-independent index of IVPD, *M*_{LIIIVPD}, is determined by an analysis of a set of load-varying IVPD contours from one subject. In Fig. 1, the steps for analyzing one IVPD contour to yield a single point in the plot of *E*_{k}·(P*_{max} − P_{∞}) versus τ_{c}·(dP/d*t*_{min}) are summarized. First, using the methods presented in *Automated τ*_{c} *and E*_{k} *fitting*, the Chung IVPD model parameters were extracted from the IVPD contour of interest. This process yielded unique τ_{c}, *E*_{k} , P_{∞}, P_{o}, and Ṗ_{o} parameter values for each beat. P*_{max} was next determined for the IVPD contour of interest via *Eq. A2* by using the calculated *t*_{P*max} in *Eq. 4* (see appendix for details). Subsequently, the dP/d*t*_{min} was determined directly from the data. Finally, the products *E*_{k}·(P*_{max} − P_{∞}) and τ_{c}·(dP/d*t*_{min}) were calculated for each beat, and these values were plotted in the *E*_{k}·(P*_{max} − P_{∞}) versus the τ_{c}·(dP/d*t*_{min}) plane.

These steps were repeated for every IVPD contour for every subject, thus yielding a set of points in the plot of *E*_{k}·(P*_{max} − P_{∞}) versus τ_{c}·(dP/d*t*_{min}). In accordance with the predicted linear relationship in *Eq. 6*, a linear regression between all points in the *E*_{k}·(P*_{max} − P_{∞}) versus τ_{c}·(dP/d*t*_{min}) plane yields the slope *M*_{LIIIVPD}, the intercept *B*_{LIIIVPD}, and the Pearson correlation coefficient *r*^{2}. Since the IVPD contour shape varied from beat to beat, the location of the point inscribed in the *E*_{k}·(P*_{max} − P_{∞}) versus τ_{c}·(dP/d*t*_{min}) plane varied as well. However, *Eq. 6* predicted that for a given subject, all beats should fall on the same linear regression line, with constant slope *M*_{LIIIVPD}. Thus the slope *M*_{LIIIVPD} was the predicted load-independent index of IVPD.

#### Validation of M_{LIIIVPD}.

For each of the 25 subjects in the current study, *M*_{LIIIVPD}, *B*_{LIIIVPD}, and *r*^{2} were calculated as described above via a custom Matlab script. The load independence of *M*_{LIIIVPD} was assessed by the proximity of *r*^{2} to 1. A low *r*^{2} value would imply that *M*_{LIIIVPD} was load dependent.

To ensure that the value of *M*_{LIIIVPD} reflected the clinical and physiological information, individual *M*_{LIIIVPD} and *B*_{LIIIVPD} values for each subject were correlated with average EF, τ, τ_{L}, and dP/d*t*_{min} values from each subject.

## RESULTS

### Hemodynamic Analysis and Application of Chung IVPD Model

An average of 284 beats were fit to the Chung IVPD model for each subject. The average RMSE (before the exclusion of any beats) between the model fit and the dP/d*t* versus the *t* contour was 19 mmHg/s (an average 2% deviation between model and measured contour). As described in methods, nonphysiological beats with high RMSE were excluded. Specifically, for each subject, the beats with RMSE above the subject-specific RMSE average value were excluded from further analysis, resulting in the analysis of an average of 107 beats per subject. Across all the subjects, the average values and standard deviation for τ_{c}, *E*_{k}, P_{o}, Ṗ_{o}, and P_{∞} were 89 1/s, 1,500 1/s^{2}, 105 mmHg, −1,040 mmHg/s, and 1.70 mmHg, respectively. The number of beats included in the analysis, as well as the P_{max}, dP/d*t*_{min}, τ, and τ_{L} average values, standard deviations, and ranges for each subject are shown in Table 2.

### Physiological Variation in IVPD Pressure Parameters

The IVPD contour in the PPP varied significantly with load, and therefore within any given subject, a significant beat-to-beat variation was observed in τ, τ_{L}, P_{max}, and dP/d*t*_{min}. A significant variation was seen primarily in beats following PVCs or Valsalva maneuvers. Indeed, the large range of values presented in Table 2 reflects the significant hemodynamic variation between perturbed (PVC and Valsalva) and normal physiology. As reflected by the Table 2 standard deviations, the steady-state hemodynamic variation was also significant in the subjects studied.

### Determination of Predicted Load Independence

Despite the wide variation in τ, τ_{L}, and dP/d*t*_{min} within each subject, and in accordance with the prediction (*Eq. 6*), the linear least-squares regression between *E*_{k}·(P*_{max} − P_{∞}) and τ_{c}·(dP/d*t*_{min}) yielded strong linear relationships for each subject. Figure 2 shows the *E*_{k}·(P*_{max} − P_{∞}) and τ_{c}·(dP/d*t*_{min}) plot for *subject 5*. Below the plot, the IVPD portion of the PPP is shown for three individual beats acquired in the subject. The leftmost contour shows a post-PVC beat having the highest τ value among all the heart cycles. The middle contour is a beat with the median τ value, and the rightmost contour is the beat with minimum τ value. The gray curve shows the Chung IVPD model fit to the PPP contour, whereas the light gray line shows the linear τ fit. Notice that each individual beat inscribes a particular point in the *E*_{k}·(P*_{max} − P_{∞}) versus the τ_{c}·(dP/d*t*_{min}) plot, and these individual beats are colinear with the rest of the 71 beats measured in this particular subject. Thus, despite significant changes in τ between these three individual beats, the *M*_{LIIIVPD} defined by these beats remains the same and is consistent with the *M*_{LIIIVPD} defined by all the analyzed beats in this subject.

*M*_{LIIIVPD}, *B*_{LIIIVPD}, and *r*^{2} values for each subject are presented in Table 3. All subjects showed highly linear *E*_{k}·(P*_{max} − P_{∞}) versus τ_{c}·(dP/d*t*_{min}) relations, with the average *r*^{2} value equal to 0.993. Across all subjects, *M*_{LIIIVPD} varied between 1.06 and 1.169 and *B*_{LIIIVPD} varied between 10,620 and 45,990 mmHg/s. As a further test of the consequences of the automated beat selection process, in 14 subjects whose data were “clean” in the physiological sense, using all beats (including the excluded high RMSE beats) in the calculation of *M*_{LIIIVPD} changed the value of *M*_{LIIIVPD} by <5%. Thus, for physiologically “clean” data, the beat exclusion criteria have a minimal impact on *M*_{LIIIVPD}. To ensure consistency in the analysis, the same RMSE-based beat exclusion criteria were applied to all subjects.

### Clinical Correlations with M_{LIIIVPD}

*M*_{LIIIVPD} values in each subject showed a strong linear correlation with the average EF (*r*^{2} = 0.52), τ (*r*^{2} = 0.65), τ_{L} (*r*^{2} = 0.50), and dP/d*t*_{min} (*r*^{2} = 0.63) values from each subject. High *M*_{LIIIVPD} values were associated with lower τ and τ_{L} values, higher EF values, and more negative dP/d*t*_{min} values (see Fig. 3). The intercept *B*_{LIIIVPD} values from each subject did not show a significant correlation to the average EF (*r*^{2} = 0.22), τ (*r*^{2} = 0.38), τ_{L} (*r*^{2} = 0.55), or dP/d*t*_{min} (*r*^{2} = 0.31) values.

## DISCUSSION

The time course of IVPD has been shown to depend on both intrinsic ventricular parameters and extrinsic load effects (3, 3a, 13, 15, 17, 24, 30, 32, 36, 38, 40), and, therefore, despite some initial claims of load independence, both τ and τ_{L} have proven to be load dependent. In this study we use the recently proposed Chung model of IVPD (6) to derive and validate a novel lumped load-independent index of IVPD, *M*_{LIIIVPD}.

### Known Load Dependence of τ and τ_{L}

In the initial studies describing τ, investigators noted insignificant changes in τ associated with volume loading or heart rate variation and concluded τ to be independent of systolic stress and end-systolic fiber length (12, 38). However, these initial studies were not performed in the intact heart, and later studies in both anesthetized and conscious dogs found τ to be significantly dependent on systolic load (13, 17, 32). In addition, τ was shown to be dependent on volume loading and end-diastolic pressure (32), pharmacologically generated increases in contractility and load (13, 17), and the timing of acute afterload perturbations (15). τ_{L} was suggested as an alternative to τ, and studies in human heart failure subjects, as well as in isolated canine hearts, suggested that τ_{L} is less load dependent than τ (25, 34). However, the application of τ_{L} may not be appropriate when the PPP trajectory is highly linear (6). Furthermore, a study in patients undergoing cardiac surgery found that the load variation due to leg lift significantly affected both τ and τ_{L} in a select group of patients (10).

### LIIIVPD Findings

#### Physical interpretation of M_{LIIIVPD}.

The Chung IVPD model uses a linear damped harmonic oscillator as the kinematic analog for chamber behavior during IVPD (6). The rapid decay of wall stress and associated strain directly generates a pressure decay through Laplace's law, and therefore *Eq. 3* is written in terms of pressure and pressure derivatives. However, if we consider IVPD kinematically and in terms of displacements, *Eq. 3* becomes a balance of inertial, resistive, and elastic forces. IVPD is governed by an elastic, mechanical recoil component that drives down the pressure, as well as a viscoelastic resistive component related to cross-bridge uncoupling that modulates pressure decay. These elastic and resistive components of IVPD have peak forces associated with them, and the LIIIVPD is the slope of the peak elastic force driving pressure decay versus peak resistive force opposing pressure decay. Thus *Eq. 6* predicts that for an individual subject, the peak driving force and peak resistive force maintain a constant relation, with a constant slope *M*_{LIIIVPD} in the face of load variation. As load changes, the level of peak elastic force required to drive down the pressure during isovolumic relaxation also changes. However, because the Chung IVPD model is linear, increased elastic (recoil) forces will be accompanied by increased peak resistive (viscoelastic) forces. A chamber where increased peak elastic (recoil) forces are accompanied by relatively large increases in peak resistive forces will have a low *M*_{LIIIVPD} value, whereas increased elastic peak forces accompanied by relatively small increases in peak resistive forces will have a higher *M*_{LIIIVPD} value. Thus *M*_{LIIIVPD} is an index of isovolumic relaxation that is related to the efficiency with which the relaxing ventricle adapts to changes in load.

Figure 4 provides a theoretical plot of the peak driving force versus the peak resistive force. The area below the diagonal is theoretically restricted from occurring with normal physiology, because if peak resistive forces exceeded peak driving forces, then pressure decay is prohibited. Figure 4 also demonstrates that a line with an *M*_{LIIIVPD} < 1 eventually crosses over to the nonphysiological domain with an extended load variation. Thus, when compared with subjects with an *M*_{LIIIVPD} > 1, subjects with an *M*_{LIIIVPD} < 1 will have a less efficient relaxation-related beat-to-beat response to load variation and possess a limited regime of allowed load variation compared with subjects with an *M*_{LIIIVPD} > 1.

#### Clinical validation of the index.

The physics-based derivation presented in the appendix, as well as in the results presented in Fig. 2 and Table 3, demonstrates that for each subject, *M*_{LIIIVPD} remains constant in the face of load variation. In this manner, the load independence of *M*_{LIIIVPD} is both derived and validated. However, to be clinically useful, *M*_{LIIIVPD} must be related to real physiology and must have the potential to differentiate between normal physiology and pathophysiology. To assess clinical validity, *M*_{LIIIVPD} was compared with the average values of accepted, conventional measures of isovolumic relaxation.

The results of Fig. 3 must be interpreted with care. The current study consisted of subjects with a wide range in traditional isovolumic relaxation parameters: 33 < τ < 147 ms, 14 < τ_{L} < 45 ms, and −718 < dP/d*t*_{min} < −1,875 mmHg/s. Thus, despite the fact that τ, τ_{L}, and dP/d*t*_{min} are known to be load dependent, it is unlikely that the load effects alone account for the large range of variation observed among the subjects. It is therefore reasonable to conclude that in the current study, the subjects with longer τ and τ_{L} and less negative values for dP/d*t*_{min} have relaxation-related dysfunction, whereas the subjects with shorter τ and τ_{L} and more negative dP/d*t*_{min} have characteristics of normal function. Thus the correlations in Fig. 3 between *M*_{LIIIVPD} and τ, τ_{L}, and dP/d*t*_{min} demonstrate that the values of *M*_{LIIIVPD}, although confined to a narrow range among the subjects studied, convey clinical information.

We note, however, that correlating conventional indexes of isovolumic relaxation and *M*_{LIIIVPD} would have limited value if the subjects chosen were more homogeneous and possessed near-normal average τ, τ_{L}, or dP/d*t*_{min} values. Indeed τ, τ_{L}, and dP/d*t*_{min} are load dependent, and therefore modest variations in these parameters can be the result of load effects. By including subjects with widely different τ, τ_{L}, and dP/d*t*_{min} values, we ensure that the subjects have significant intrinsic relaxation-related differences among them. In this manner we minimize the effects of load in the particular subject group and therefore can conclude that the trends observed in Fig. 3 support our view that *M*_{LIIIVPD} correlates with the intrinsic relaxation-related function. Higher values of *M*_{LIIIVPD} correlate with improved function (more negative dP/d*t*_{min}, and lower τ and τ_{L}), whereas lower values of *M*_{LIIIVPD} correlate to dysfunction (more positive dP/d*t*_{min}, and prolonged τ and τ_{L}).

#### Connection to previous work.

The derivation of *M*_{LIIIVPD} is similar to the previously validated LIIDF (35). The key physiological difference is that the current work deals with the physiology of IVPD rather than transmitral flow and early rapid filling. Thus the relevant time scales, as well as initial conditions, are different between the current study and previous LIIDF-related work. However, because both IVPD and transmitral flow are governed by lumped forces that can be accurately modeled kinematically, a similar derivation can be exploited in determining a load-independent index of either isovolumic relaxation or transmitral flow. The fact that the same type of conceptual and mathematical modeling works when applied to different physiology problems underscores the multiscale power of kinematic modeling.

### Limitations

The lumped parameter, kinematic approach presented uses a linear differential equation with constant (i.e., time invariant) coefficients (Newton's Law) (*Eq. 3*) to model events that others have modeled using time-varying ventricular properties. Past work by Nudelman et al. (29) has compared and shown a superb agreement between a time-invariant linear kinematic model and time-varying nonlinear models of transmitral flow. Although the form of the differential equation for transmitral flow and IVPD is the same (Newton's Law), numerical experiments comparing the kinematic model and models with time-varying coefficients for IVPD have not been carried out. Additional validation of the constant coefficient, kinematic modeling used here would require that the averaged time-varying parameters agree with the constant coefficient lumped parameters. A potential limitation of the current work is the absence of experimental data to validate a time-invariant analysis in this manner. However, the ability to solve the “inverse problem” using in vivo data as input and unique parameter values as output, and the close agreement between the in vivo pressure data and the time invariant, kinematic model predicted contours, supports the conclusions of the current study.

In the derivation of the LIIIVPD, the model-based P*_{max}, determined by algebraically solving for the maximum value of *Eq. 4*, was employed. It may seem inappropriate, however, to apply an isovolumic model to intervals before aortic valve closure. Indeed, P*_{max} will not accurately reflect the actual P_{max} value, because with intact physiology, the isovolumic condition is broken at P_{max}. However, using P*_{max} as a value in the idealized limit where the chamber is isovolumic does not violate any physical principles or introduce any tautological conditions. In fact, the use of P*_{max} is analogous to the methods applied in the Doppler-echo-derived LIIIDF (35). Furthermore, if one uses the actual P_{max} value instead of P*_{max}, so that the *y*-coordinate in the regression that defines *M*_{LIIIVPD} becomes *E*_{k}·(P_{max} − P_{∞}) instead of *E*_{k}·(P*_{max} − P_{∞}), then the plots for each subject remain strongly linear but the linear correlations between *M*_{LIIIVPD} and τ (*r*^{2} = 0.44), τ_{L} (*r*^{2} = 0.41), dP/d*t*_{min} (*r*^{2} = 0.46), and EF (*r*^{2} = 0.27) decrease relative to those seen in Fig. 3. Because *M*_{LIIIVPD} is derived to be an index of isovolumic relaxation, it is reasonable to only use parameters derived from isovolumic data. Thus the use of P*_{max}, the maximum pressure in the chamber assuming that the chamber remains isovolumic from the pressure maximum down to mitral valve opening, is preferred.

A further limitation may be the concern that *M*_{LIIIVPD} does not present information beyond what the average τ or τ_{L} value already provides and therefore serves as merely a surrogate for τ. Although this is reasonable, we note that *M*_{LIIIVPD}, τ, and τ_{L} are all determined by τ_{c} and *E*_{k}. In other words, all three indexes are surrogates for how stiffness and relaxation combine to determine IVPD. Because all three indexes represent lumped parameters of IVPD and are therefore measuring the same physiological event, it is reasonable to expect these parameters to correlate with each other and to vary between subjects with different physiology. However, although each index provides information regarding IVPD, *M*_{LIIIVPD} is the only one that is derived with load independence in mind and is validated to be load independent.

The strong correlations presented in Fig. 3 may suggest that the inversion of these correlations would allow one to extract *M*_{LIIIVPD} for an individual subject by measuring τ, τ_{L}, or dP/d*t*_{min} alone. Thus it may appear that the load-independent *M*_{LIIIVPD} index could be derived from load-dependent parameters. It is important to note, however, that Fig. 3 consists of correlations among subjects with overt differences in τ, τ_{L}, and dP/d*t*_{min}. Although a general trend is observed between *M*_{LIIIVPD} and relaxation-related function, a direct causal link that is appropriate for extreme and moderate values of τ, τ_{L}, and dP/d*t*_{min} cannot be determined from Fig. 3. Thus an inversion of the correlations in Fig. 3 is not justified.

Additionally, the use of a nonhomogeneous subject group may be seen as inappropriate because the significant physiological difference between the subjects may confound the results of the work. Had the *E*_{k}·(P*_{max} − P_{∞}) versus τ_{c}·(dP/d*t*_{min}) relation shown to be nonlinear in some subset of the subjects, then the nonhomogeneous nature of the subject group could have been invoked to explain why *M*_{LIIIVPD} is load independent in some cases but load dependent in others. However, because all subjects showed a strongly conserved *M*_{LIIIVPD} slope in the face of load variation, the nonhomogeneous nature of the subject group serves to underscore the validity and general applicability of the method.

Although the number of beats analyzed is substantial (2,669), the number of subjects is modest, and therefore no claims regarding the range of normal or abnormal values for *M*_{LIIIVPD} are made. Although the range of variation of *M*_{LIIIVPD} is narrow in the absolute numerical sense, these values correlated with the underlying physiology as evidenced by Fig. 3. Although conventional indexes of isovolumic relaxation such as τ and τ_{L} may exhibit a larger numerical range of variation across subjects, much of it is due to load variation. It is therefore not surprising that a load-independent index exhibits a narrower numerical range of variation than load-dependent indexes such as τ and τ_{L}. Thus the benefit of a load-independent index of IVPD carries with it the cost of a numerically smaller range of variation.

Future studies with carefully selected patient groups will be needed to establish the clinical range for *M*_{LIIIVPD}. In addition, the variation of *M*_{LIIIVPD}, in repeated studies where subjects serve as their own controls, may provide an opportunity for the phenotypic characterization that provides the mechanistic insights regarding the effects of alternative (pharmacological, surgical, and device based) therapeutic modalities.

### Conclusions

We applied Chung's IVPD model to determine (*E*_{k} and τ_{c}) the stiffness and relaxation components of IVPD. Based on physical and physiological principles, we derive a parameter that avoids the load dependence of τ. The new index, *M*_{LIIIVPD}, is shown to be load independent in subjects with significantly different clinical profiles. Furthermore, *M*_{LIIIVPD} correlates with conventional isovolumic relaxation parameters. The determination of *M*_{LIIIVPD} in clinical subsets having specific pathophysiology is planned.

## APPENDIX

### Derivation of LIIIVPD

We follow the logical progression presented in a previous work (35). Although individual IVPD pressure contours vary as a result of load perturbations, *Eq. 3* remains valid, because the governing differential equation is obeyed independent of load. We consider this equation at the time of minimum dP/d*t*. At this time, the second derivative term vanishes, and *Eq. 3* becomes (A1) *Eq. A1* is a tautology and is therefore true for any IVPD contour. However, the values of dP/d*t*_{min} and P(*t*_{dP/dtmin}) are determined directly from clinical data and, when plotted, are expected to demonstrate some scatter. The Chung IVPD model provides an excellent fit to both the pressure and pressure derivative. Indeed, the plots of τ_{c}(dP/d*t*_{min}) versus *E*_{k} [P(*t*_{dP/dtmin}) − P_{∞}] for subjects from the current study are in close agreement with the line of unity (data not shown).

It is helpful here to consider the dP/d*t* versus the *t* contour. A plot of dP/d*t* versus *t* defined by *Eq. 5* would yield an inverted damped sine wave. Isovolumic relaxation typically ends near the inflection point of the dP/d*t* versus *t* downslope, but the model may be extended to the point where dP/d*t*_{min} crosses zero. This time point defines the Chung model-based maximum pressure value, and we therefore call this value P*_{max}. The time at which P*_{max} occurs can be found by solving for *t* in *Eq. 5* when dP/d*t* = 0: (A2) Evaluating P(*t*) in *Eq. 4* at the time found in *Eq. A2* yields P*_{max}.

Because the time between P*_{max} and dP/d*t*_{min} is short (10–20 ms) relative to the time scale of the IVPD duration, we can approximate the pressure decay between P*_{max} and (*t*_{dP/dtmin}) to be linear. The slope of this linear dependence may change from beat to beat and from subject to subject. However, the short time between P*_{max} and dP/d*t*_{min} implies that the pressure at dP/d*t*_{min} may be linearly approximated by P*_{max}: (A3) where α and β are constants. This linear relationship is valid (α = 0.74, *r*^{2} = 0.88) over all 2,671 beats across the 25 subjects from the current study, although for any individual subject or individual beat, the value of α may vary (data not shown).

When combining *Eq. A3* with *Eq. A1* and rearranging, we find (A4) where *M*_{LIIIVPD} and *B*_{LIIIVPD} are constants and the magnitude of dP/d*t*_{min} is used. It is important to note that the value of *M*_{LIIIVPD} cannot be determined directly from 1/α. As mentioned above, α in *Eq. A3* represents an ensemble of linear relations between P(*t*_{dP/dtmin}) and P*_{max}. The value of 1/α provides an average estimate for the overall value of *M*_{LIIIVPD} that any subject may possess, but a subject's specific *M*_{LIIIVPD} can only be determined by finding the linear regression between *E*_{k}·(P*_{max} − P_{∞}) and τ_{c}·(dP/d*t*_{min}).

The linear relationship between *E*_{k}·(P*_{max} − P_{∞}) and τ_{c}·(dP/d*t*_{min}) is predicted to be load independent because it is derived from equations and approximations that are general and load independent. Thus, although τ_{c}, *E*_{k}, P_{∞}, P*_{max}, and dP/d*t*_{min} may change with load, the slope *M*_{LIIIVPD} is predicted to remain constant in the face of load variation and thus is the predicted LIIIVPD.

## GRANTS

This work was supported in part by the National Institutes of Health, the Whitaker Foundation (Roslyn, VA), the Alan A. and Edith L. Wolff Charitable Trust (St Louis, MO), the American Heart Association (AHA), and the Barnes-Jewish Hospital Foundation. L. Shmuylovich holds predoctoral fellowship awards from the Heartland Affiliate of the AHA.

## Acknowledgments

We gratefully acknowledge the assistance of the staff of the cardiac catheterization laboratory at Barnes-Jewish Hospital at Washington University Medical Center and discussions with Wei Zhang.

## Footnotes

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- Copyright © 2008 by the American Physiological Society