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Am J Physiol Heart Circ Physiol 294: H1589-H1596, 2008. First published January 25, 2008; doi:10.1152/ajpheart.00990.2007
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Physical determinants of left ventricular isovolumic pressure decline: model prediction with in vivo validation

Charles S. Chung and Sándor J. Kovács

Department of Physics, Washington University, St. Louis; and Cardiovascular Biophysics Laboratory, Cardiovascular Division, Washington University School of Medicine, St. Louis, Missouri

Submitted 27 August 2007 ; accepted in final form 22 January 2008


   ABSTRACT
TOP
 ABSTRACT
METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
 REFERENCES
 
The rapid decline in pressure during isovolumic relaxation (IVR) is traditionally fit algebraically via two empiric indexes: {tau}, the time constant of IVR, or {tau}L, a logistic time constant. Although these indexes are used for in vivo diastolic function characterization of the same physiological process, their characterization of IVR in the pressure phase plane is strikingly different, and no smooth and continuous transformation between them exists. To avoid the parametric discontinuity between {tau} and {tau}L and more fully characterize isovolumic relaxation in mechanistic terms, we modeled ventricular IVR kinematically, employing a traditional, lumped relaxation (resistive) and a novel elastic parameter. The model predicts IVR pressure as a function of time as the solution of d2P/dt2 + (1/µ)dP/dt + EkP = 0, where µ (ms) is a relaxation rate (resistance) similar to {tau} or {tau}L and Ek (1/s2) is an elastic (stiffness) parameter (per unit mass). Validation involved analysis of 310 beats (10 consecutive beats for 31 subjects). This model fit the IVR data as well as or better than {tau} or {tau}L in all cases (average root mean squared error for dP/dt vs. t: 29 mmHg/s for model and 35 and 65 mmHg/s for {tau} and {tau}L, respectively). The solution naturally encompasses {tau} and {tau}L as parametric limits, and good correlation between {tau} and 1/µEk ({tau} = 1.15/µEk – 11.85; r2 = 0.96) indicates that isovolumic pressure decline is determined jointly by elastic (Ek) and resistive (1/µ) parameters. We conclude that pressure decline during IVR is incompletely characterized by resistance (i.e., {tau} and {tau}L) alone but is determined jointly by elastic (Ek) and resistive (1/µ) mechanisms.

relaxation; stiffness; hemodynamics; diastole; mechanics; pressure phase plane; isovolumic relaxation

IN VIVO ANALYSIS of isovolumic relaxation (IVR) in mammalian hearts has traditionally relied on {tau}, the time constant of IVR, to characterize pressure decline. Typically, the rate of pressure decline as a function of time is assumed to be proportional to pressure itself and is therefore a solution to

Formula 1A(1A)
or

Formula 1B(1B)
where Po is a constant and P{infty} is the pressure asymptote (32). A geometrically convenient way to determine {tau} is to plot Eq. 1A in the time derivative of pressure (dP/dt) vs. time-varying pressure [P(t)] (pressure phase plane, PPP), where it inscribes a straight line (19, 33) with a slope of –1/{tau}, and fit it to the IVR portion of the loop inscribed by P(t) for the cardiac cycle (Fig. 1A). However, a straight line fit to the IVR portion of the loop may not always be physiological, since curvilinear IVR segments exist (Fig. 1B).


Figure 1
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  		Fig. 1. Pressure phase plane (PPP) trajectories focusing on the isovolumic relaxation (IVR) period. The full PPP loop for 1 cardiac cycle, which is inscribed clockwise, is also shown (insets). A: normal relaxation generates a linear IVR segment after (moving clockwise) peak negative time derivative of pressure (dP/dtmin), allowing fit via monoexponential decay (solid line, {tau} = 42 ms). B: example of a curvilinear IVR PPP segment conventionally fit after dP/dtmin using a logistic time constant (dotted line, {tau}L = 34 ms). C: example of an intermediate PPP IVR trajectory that is not well fit by either monoexponential (solid line, {tau} = 33 ms) or logistic formulation (dotted line, {tau}L = 20 ms), indicating the need for an improved model of IVR to describe this process. See text for details.

 
An alternative empiric constant has been proposed to fit these frequently occurring "curved" IVR segments of PPP trajectories (19). Like {tau}, the logistic time constant {tau}L provides an empirical fit, where pressure as a function of time obeys a relationship in which the rate of pressure decline is proportional to the square of the pressure and is given by

Formula 2A(2A)
or

Formula 2B(2B)
where PA is a constant and PB is the pressure asymptote (19). Unlike the monoexponential expression (Eq. 1), which can only generate a straight line (linear) relation in the PPP, the logistic relation is quadratic in P, and in the PPP, it can only generate, and therefore best fit, curvilinear PPP IVR contours (Fig. 1B). Therefore, two parameters are required ({tau} and {tau}L) to fit the range of isovolumic pressure decay shapes encountered physiologically.

Both of these measures of IVR have been used in characterization of diastolic heart failure (11, 12, 31, 35). On the molecular level, both {tau} and {tau}L have been shown to correlate with "relaxation" as defined by deactivation events such as cross-bridge cycling, calcium handing, or lusitropism (5, 20). However, although both {tau} and {tau}L have been found to be related to deactivation events, neither can fully characterize the full range of IVR PPP trajectories encountered. Furthermore, although transition between curvilinear and linear IVR PPP segments in the same heart is physiologically permitted, some segments cannot be well fit by either {tau} or {tau}L (Fig. 1C). Although particular IVR segments in the PPP may be curved, suggesting that {tau}L should be best suited for its characterization, the example in Fig. 1C shows that neither {tau} nor {tau}L can adequately characterize these physiologically allowable IVR segments. Indeed, because these types of PPP IVR segments exist, and because neither a mathematical nor physiological link between {tau} and {tau}L has been proposed, we have proposed a kinematic model-based solution of the isovolumic pressure decay problem.

We hypothesized that the same physical and physiological principles govern IVR for all hearts; what differs from heart to heart are the parameters of the model. Specifically, mathematical modeling of elastic recoil forces, opposed by resistance (physiologically, relaxation) and inertial forces, jointly determine pressure decline during IVR. Furthermore, because this model should be able to predict (i.e., fit) all IVR PPP trajectories, straight or curvilinear, we also hypothesized that the two distinct and previously (mathematically) unrelated empirical curve-fitting parameters ({tau} and {tau}L) are in fact subject to the same physical principles (inertial, elastic, and resistive forces) such that their variation is a reflection of altered balance between inertial, elastic, and resistive mechanisms.


   METHODS
TOP
 ABSTRACT
 METHODS
RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
 REFERENCES
 
Modeling pressure decay kinematically. Deactivation necessarily plays a role in any attempt to characterize IVR, and {tau} and {tau}L have been shown to be good correlates of relaxation or deactivation (5, 20, 22, 28). The effects of the residual cross bridges or delayed calcium sequestration manifest as a resistive modulator against elastic recoil (22); thus we lump the standard deactivation parameters along with any additional viscous consequences into a resistive parameter.

End-systolic stored elastic forces that drive filling (13, 14) have been attributed to extracellular matrix, intracellular titin (8, 24), microtubules (21), and the heart's visceral pericardium (10). Such an elastic restoring force at end systole must overcome the residual forces of contraction (22). Therefore, inclusion of a restoring force, characterized by a spring constant-like, elastic parameter, is required.

If deactivation were solely responsible for the pressure drop during IVR, cross bridges would uncouple, but no sarcomere or myocyte lengthening would accompany the reduction in wall tension. The presence of wall motion, in the form of myocardial torsion during IVR (25) and chamber shape change, indicates that tissue motion accompanies relaxation/deactivation. Newton's law requires that restoring forces that generate (tissue) motion be subject to the laws of inertia, and therefore an inertial term is required in modeling IVR.

Empirical models commonly relate myocardial properties with pressure decline and other physiological factors (1, 2). LaPlace's law is used to relate wall strain (displacement) to wall stress and chamber pressure for a particular choice of chamber geometry. Thus LaPlace's law permits a change of variable from displacement (x) to chamber pressure (P) in Newton's law (per unit mass), expressing the balance among inertial, resistive, and restoring force terms (d2x/dt2 + Rdx/dt + Kx = 0), where R and K are suitable resistive and elastic parameters. This is the familiar equation of damped harmonic motion (7). With these forces in mind, the equation of motion that kinematically models IVR is

Formula 3(3)
where d2P/dt2 is the inertial term required by Newton's Law; µ is a resistive parameter maintaining the deactivation effects typically characterized via {tau} or {tau}L, producing a resistive "force," (1/µ)dP/dt; and Ek is a lumped elastic parameter characterizing the component responsible for the elastic restoring force, EkP. The fit to the IVR segment of any PPP contour, ranging from linear (overdamped) to curvilinear (underdamped) shapes is achieved by suitable variation of µ and Ek in the solution to Eq. 3 (see APPENDIX) and determining the goodness of fit using standard methods (23).

Patient data. We analyzed data from 31 subjects in the existing Cardiovascular Biophysics Laboratory Database of high-fidelity micromanometric left ventricular (LV) pressure recordings (18). Criteria for including data in this study were normal sinus rhythm, absence of valvular abnormalities (including regurgitation or stenosis), and absence of wall motion abnormalities. Washington University Medical Center Human Studies Committee (Institutional Review Board) approved informed consent was obtained before catheterization. Subject demographics are summarized in Table 1. Data sets were acquired during elective cardiac catheterization at the request of a referring cardiologist for the evaluation of suspected coronary artery disease. Data acquisition methods have been described previously (3, 4, 18, 27, 34). Briefly, high-fidelity simultaneous LV pressure-volume and aortic root pressure measurements were obtained using a 6-F multiple transducer-tipped pigtail pressure-volume conductance catheter (SPC-562 or SSD-1034; Millar Instruments, Houston, TX) amplified and calibrated via standard transducer control units (TC-510; Millar Instruments). Pressure signals were input simultaneously to clinical monitoring systems (Quinton Diagnostics, Bothell, WA or GE Healthcare, Milwaukee, WI) and a custom personal computer via a research interface (Sigma-5DF; CD Leycom, Zoetermeer, The Netherlands) at a sampling rate of 200 Hz. Conductance signals were also stored on the research interface but were not used in this study. No subjects had active ischemia during catheterization, and ejection fraction was computed from the suitably calibrated ventriculogram.


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  		Table 1. Summary of patient demographics

 
Hemodynamic data sets were converted for analysis via a custom Matlab script (Matlab 6.0; MathWorks, Natick, MA) and analyzed off-line via custom analysis software (LabVIEW 6; National Instruments, Austin, TX). For each subject, 10 continuous cardiac cycles of Millar catheter-recorded pressure data were considered to minimize respiratory or similar physiological variations. Data sets were smoothed digitally by using a five-point average to suppress small noise in the derivative (19), attenuating 50% of signal at 40 Hz and 90% above 60 Hz, followed by calculation of continuous dP/dt vs. time t from the smoothed data. For each beat, LV end-diastolic pressure (LVEDP) and peak negative dP/dt (dP/dtmin) were extracted from the PPP (3, 6, 34) or equivalent time-domain contours.

The estimation of model parameters (µ and Ek) was achieved using a Levenberg-Marquardt fitting algorithm from the dP(t)/dt data (23). The start point was defined by a drop in d2P(t)/dt2 of one-half after the inflection point in dP(t)/dt, which defined Formula 3o. Data were fit until 5 ms before mitral valve opening pressure (defined by LVEDP) (3, 19). The algorithm provided parameters for µ, Ek, and Po, whereas P{infty} was calculated by minimizing the root mean squared error (RMSE) in the P(t) vs. t contour. For comparison with traditional methods of analysis, {tau} (Eq. 1) was estimated as conventionally done using data from 5 ms after dP/dtmin until 5 ms before mitral valve opening pressure as determined by LVEDP via a linear regression to the IVR portion of the PPP loop instead of a Levenberg-Marquardt algorithm with no difference in results. The logistic time constant ({tau}L) was estimated via Eq. 2, using a Levenberg-Marquardt fitting algorithm on the same data as for {tau} (23), as described by Matsubara et al. (19). All parameterization was done using LabVIEW.

Model validation. Quality of fit was assessed by calculating RMSE for model-based, monoexponential, and logistic fits for all 310 beats from 5 ms after dP/dtmin until 5 ms before mitral valve opening for consistent comparison for both the P(t) vs. t and dP(t)/dt vs. t contours (19). We also calculated correlation coefficients (r values) for both the P(t) vs. t and dP(t)/dt vs. t contours for all fits versus actual data and tested the significance of the difference of these r values after their Z transformation {Z = 1/2[ln(1 + r) – ln(1 – r)]} as described by Matsubara et al. (19).

Additional analysis was performed to test the kinematic model's ability to predict/fit data obtained under physiological load variation. LV pressure data during the Valsalva maneuver in three subjects was analyzed (Table 1). Data during the strain and release phases were analyzed for parameters as described above. In four subjects, pressure data for a nonejecting premature ventricular contraction (PVC) (Table 1), producing load variation and eliminating ventricular-arterial interactions, was analyzed. Both the PVC beat and the normal beat immediately preceding it were analyzed, while simultaneous aortic root pressure verified LV pressures did not cause ejection.

Model predictions. When the differential equation governing the monoexponential ({tau}) relationship (Eq. 1) as its solution is compared with Eq. 3, it provides mechanistic insight into the relation between {tau} and µ and/or Ek. Note that Eqs. 1 and 3 differ by the inertial term (d2P/dt2). When the recoil (EkP) and relaxation [(1/µ)dP/dt] terms numerically dominate the inertial term (d2P/dt2 {approx} 0) (27), Eq. 3 reduces to Eq. 1, requiring that {tau} = 1/µEk and being valid in the "overdamped regime" (1/µ2 – 4Ek > 0). In descriptive terms, {tau} is the e-folding time, i.e., the time required for IVR pressure to drop by a factor of 1/e (0.367879). Thus we tested the prediction that, for linear IVR segments in the PPP, {tau} will be significantly correlated with 1/µEk. The P2 term in the logistic equation is mathematically similar to the inertial term. Although no quantitative relationship can be made between {tau}L and 1/µEk, the logistic relationship was also compared statistically based on the presence of the P2 term (Eq. 2).

Neither the monoexponential nor the logistic parameter-predicted pressure declines are able to characterize data before dP/dtmin. In other words, in the PPP, these empirical choices for isovolumic pressure decay are unable to fit the U-shaped bowl of the PPP at dP/dtmin. We tested the prediction that the values predicted by either of the two empirical characterizations or our model of pressure decline would be different from the actual dP/dtmin for all beats. Predictions for dP/dtmin were made using the best fit-determined values for {tau}, {tau}L, µ, and Ek and were compared with actual dP/dtmin values. Comparisons were done via linear regression with zero intercept.


   RESULTS
TOP
 ABSTRACT
 METHODS
 RESULTS
DISCUSSION
 APPENDIX
 GRANTS
 REFERENCES
 
Model validation. The obtained parameter values for monoexponential, logistic, and kinematic model parameters are provided in Supplemental Tables (supplemental data for this article is available online at the American Journal of Physiology-Heart and Circulatory Physiology website). Large standard deviations relative to mean values in the parameters such as (µ) are indicative of beat-to-beat variation. Variation in model fit is comparable, whereas large deviation in {tau}- and {tau}L-predicted dP/dtmin values further indicates their inability to accurately fit the measured dP/dtmin.

Statistically, the model-predicted contour provided the best and most consistent fit to the IVR portion of the PPP contour (Table 2). Examining RMSE for linear versus curvilinear PPP subjects clearly indicated that neither {tau} nor {tau}L was able to accurately fit the data. Only the model fit was able to consistently indicate low error in both P vs. t and dP/dt vs. t planes consistently for all subjects, indicating it may be a more physiologically accurate model of IVR. Model fit provided the best fit (r = 0.996) to the data (P < 0.001 by paired t-test of Z transform values) compared with the monoexponential (r = 0.989) and logistic (r = 0.980) fits (310 beats), and all were statistically different.


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  		Table 2. Averaged root mean squared error

 
Although the statistics favor our model over empirical ({tau}, {tau}L) characterizations, the PPP is the ideal arena for visual characterization and determination of the goodness of fit for any IVR pressure decay model, because the PPP coordinate system easily differentiates between various empirical fits that are more difficult to differentiate from the P(t) contour alone. Figure 2 illustrates the benefit of the model for characterization of all three approaches by comparison with Fig. 1. For all cases, the model proposed (Eq. 3) adequately fits the pressure decay, whereas the monoexponential ({tau}) and logistic ({tau}L) models have limitations based on various amounts of linearity or curvilinearity of the individual IVR portions of the PPP contour.


Figure 2
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  		Fig. 2. IVR PPP trajectories fit by our proposed model employing elastic (Ek) and relaxation/viscoelastic (µ) parameters, using the same data as in Fig. 1. Model-generated fits illustrate the advantage of the model vs. monoexponential and logistic formulations (Fig. 1). Note the ability to fit all PPP trajectories encountered and the ability to fit multiple points before dP/dtmin, as well as dP/dtmin itself. Parameter values are µ = 8 ms, Ek = 2,540 1/s2 for A, µ = 40 ms, Ek = 600 1/s2 for B, and µ = 10 ms, Ek = 2,740 1/s2 for C. See text for details.

 
In analyzing data in three subjects performing a Valsalva maneuver, significant changes in pressure were observed (Fig. 3A). During the Valsalva (Fig. 3), µ had no dependence on LVEDP (average r = 0.19), whereas {tau} (average r = 0.61) and {tau}L (average r = 0.67) were significantly related to LVEDP; {tau} and 1/µEk remained tightly correlated. The four cases of nonejecting premature ventricular contraction show that the isovolumic relaxation portion of the inscribed PPP contour differs markedly from the relaxation portion of the preceding ejecting beat contour (34) (Fig. 4). In comparing the sequence of beats plotted, it is self-evident that one would have to choose between {tau} (preceding and following beat; Fig. 4B) and {tau}L (premature contraction; Fig. 4C) to fit the relaxation portion of the respective contours.


Figure 3
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  		Fig. 3. A: time-varying left ventricular (LV) pressure [P(t)] vs. time. B: PPP plots for subject 20 during Valsalva maneuver. Pre-Valsalva (light shaded line), peak Valsalva (solid line), and post-Valsalva (dark shaded line) beats are highlighted with their kinematic model fits in B. The kinematic model fits all beats. The relaxation parameter (µ) is more consistent and less load (LV end-diastolic pressure)-dependent than either monoexponential ({tau}) or logistic ({tau}L) formulations (note the slope and slight curvature change in the peak pressure beat). See text for details.

 

Figure 4
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  		Fig. 4. A: LV P(t) vs. time for normal ({triangleup}) and nonejecting, premature ventricular contraction (PVC; {circ}) from subject 2. B: PPP plot of P(t) from A including normal beat ({triangleup}) and PVC ({circ}). C: PPP plot of only the PVC ({circ}). The preceding and following normal cardiac cycles are shown for reference. In B, note the kinematic model-generated linear fit (solid line) to the IVR portion of a normal beat (µ = 6 ms, Ek = 3,930 1/s2). In C, note that the entire PVC pressure decay segment is isovolumic and curvilinear. The kinematic model provides excellent fit, shown as a solid line (µ = 146 ms, Ek = 270 1/s2). The curvilinear (logistic) model can also provide a close fit to PVC, but not to normal beats having linear IVR segments. See text for details.

 
Comparison of model predictions. Average measured and predicted dP/dtmin values are provided in detail in the Supplemental Tables. The model-predicted versus measured dP/dtmin values were not distinguishable [model dP/dtmin = 0.99(actual dP/dtmin), r = 0.99]. Because the monoexponential fit is linear and the actual PPP is always curvilinear near dP/dtmin, the monoexponential fit systematically overestimates the magnitude of dP/dtmin by nearly 10% [predicted dP/dtmin = 1.09(actual dP/dtmin), r = 0.98]. Furthermore, the logistic fit was shown to underestimate the actual magnitude of dP/dtmin [predicted dP/dtmin = 0.88(actual dP/dtmin), r = 0.94], as would be predicted due to its curvilinearity. Utilizing floating intercepts does not significantly alter regression relations (model: slope = 1.01, r = 0.99; monoexponential: slope = 1.14, r = 0.99; logistic: slope = 0.86, r = 0.96).

Figure 5 A shows the linear regression between {tau} and 1/µEk for all 31 subjects and a separate regression for 23 subjects having linear IVR portions of the loop for which {tau} provided an excellent fit ({tau} = 1.15/µEk – 11.85, r = 0.98). As expected, data from subjects with curvilinear PPP trajectories drastically altered the relationship ({tau} = 0.70/µEk – 19.10, r = 0.40). Figure 5B shows the linear regression relation between {tau}L and 1/µEk for the 23 subjects with linear IVR segments in the PPP. Because of the dominating factors (Eq. 2 with P2 for logistic vs. Eq. 3 with d2P/dt2 for model), the regression relation continues to deviate from a linear relationship.


Figure 5
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  		Fig. 5. Comparison of monoexponential ({tau}) and logistic ({tau}L) parameters to kinematic model-generated 1/µEk. A: linear regression for {tau} vs. 1/µEk for linear (filled symbols; solid line) and all IVR PPP trajectories (filled and open symbols; shaded line) are shown. B: linear regression for {tau}L vs. 1/µEk for linear IVR PPP segments (filled symbols; solid line) and all IVR segments (filled and open symbols; shaded line). Note the highly linear correlation for 23 subjects in A and B (230 beats; filled symbols), indicating that elastic (Ek) and resistive (µ) parameters together are jointly related to {tau} and {tau}L. Including data for the subjects with curvilinear IVR PPP segments (310 beats; filled and open symbols) alters the regression slightly (from solid regression line to shaded regression line) for both {tau} and {tau}L. See text for details.

 

   DISCUSSION
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
APPENDIX
 GRANTS
 REFERENCES
 
Although {tau}, traditionally referred to as the time constant of IVR, has been applied as an index of diastolic function and also has been correlated with deactivation events (5, 22), there are limitations to its application. The logistic time constant was proposed as a solution to the problem of pressure decay shapes that are not well fit by a monoexponential (19). The P(t) vs. t contours can generally be well approximated by either {tau} or {tau}L but have limitations, since neither can adequately model the full range of PPP contours encountered. Also, although µ and Ek differ for each of the isovolumic pressure decay portions of the three contours, our proposed single kinematic model (Eq. 3) is the only method that can continuously accommodate all of the observed data in the setting of this load perturbation and provides a unifying physics- and physiology-based mechanistic model for characterizing isovolumic pressure decay in ejecting or nonejecting beats. Furthermore, no clear relationship exists between the original monoexponential characterization and the logistic constant, even though both are supposed to characterize the same physiological IVR event. This may be especially apparent for IVR PPP segments that fall "in between" the monoexponential and logistic fits (Fig. 1C) or where curvature alters (PVC; Fig. 4).

Because LV pressure follows an oscillatory course between systole and diastole, its second derivative (i.e., curvature, d2P/dt2) necessarily changes sign from negative to positive. This requires that d2P/dt2 go through zero and that dP/dtmin exist. However, the {tau}- and {tau}L-based fits accommodate data only after dP/dtmin (19, 32) and therefore characterize only a limited part of the IVR period or give physical meaning for the minimum value (dP/dtmin), manifesting as the bottom of the U-shaped "bowl" in the PPP.

Peak negative dP/dt is a required component of the kinematic model incorporating inertia, elasticity, and resistance (Eq. 3), and the model provides insight into why dP/dtmin always occurs as part of IVR and the factors that determine pressure decline both before and after dP/dtmin. The model (Eq. 3) is motivated by the existence of elastic elements that generate a restoring force during IVR (8). To the extent that these assumptions are justified, the force balance between the elastic and resistive forces predicts that dP/dtmin must exist. After the inertial effects of ejection and subsequent aortic valve closure, pressure decline accelerates until dP/dtmin is reached, when the restoring force (EkP) is exactly balanced by resistive forces [(1/µ)dP/dt] and, as expected, the inertial term is small (d2P/dt2 {approx} 0). Beyond dP/dtmin, restoring forces persist and pressure continues to decline, while resistive effects continue to slow the rate of pressure decline.

The mechanistic insights afforded by this model are apparent in both its derivation and interpretation. Inertia, from tissue motion and torsion (25); inclusion of resistance, from the deactivation events traditionally thought to be characterized by {tau} and {tau}L (5, 20, 22, 28); and elasticity, generated recoil force from the stiff, springlike intra- and extracellular elements such as titin, extracellular matrix, and visceral pericardium (8, 10, 13, 21, 24): these three terms constitute three physiologically and physically required force-generating mechanisms that contribute to pressure decline during IVR. Characterization of the elastic and resistive mechanisms of IVR are significant, given that resistance (or traditionally relaxation, via {tau}) has been offered as a major cause of "diastolic heart failure," concluding that abnormal "relaxation" is significant based on {tau} being viewed as a "pure" measure of relaxation (35). However, prior findings indicate that chamber stiffness is also altered (12, 31) in diastolic heart failure. By testing a hypothesis motivated by physical mechanisms (17), we believe a more robust characterization of IVR mechanisms can be provided.

This approach elucidates the physiological mechanisms underlying the traditional empirical ({tau}, {tau}L) indexes of IVR. Previous work also has shown high correlation between {tau} and alternate kinematic elastic parameters along with windkessel parameters, which have compliant properties (4). We note the good linear relationship (r2 = 0.96) that exists between {tau} and 1/µEk (Fig. 5A) in subjects with linear IVR PPP portions, indicating how {tau} is related to elasticity (Ek). Furthermore, the model solution indicates that the resistance term (or deactivation, driven by calcium cycling, cross-bridge deactivation, or other) is the dominant mechanism during IVR, manifesting as linear IVR segments in the PPP. The logistic approach via {tau}L is the kinematic opposite, where elastic restoring forces exceed resistive forces (generating curved PPP segment). As Fig. 5A indicates, elastic restoring force dominates resistance and inertia, resulting in curvilinearity in the IVR segment of the PPP.

The logistic time constant also has some correlation with our viscoelastic parameter 1/µEk (Fig. 5B), indicating that it is related to both resistance and elasticity. These comparisons and results (Figs. 2, 4, 5, and 6) indicate that the two historically independent and mathematically unrelated empirical parameters ({tau} or {tau}L) comprise the elastic (Ek) dominance and resistive (relaxation, 1/µ) dominance parametric limits of our model. Increasing restoring force relative to resistive force alters the PPP contour to become more curvilinear, increasing dP/dtmin and shifting the pressure at which it occurs, whereas increasing resistive force µ (relative to restoring force) creates a more linear PPP segment and causes an opposite shift in dP/dtmin (Fig. 6). What appeared to be two disparate, unrelated characterizations of isovolumic pressure decline are now fully encompassed and mechanistically (i.e., kinematically) explained by a single causality-based, kinematic modeling paradigm.


Figure 6
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  		Fig. 6. Comparison of the effect of altering recoil force via elastic force (Ek) or resistive force via relaxation (µ) in the kinematic model-predicted IVR portion of the PPP contour. A: increasing stiffness from Ek = 1,042 1/s2 (solid line) to 1,500 1/s2 (shaded line) shifts the trajectory downward (increase in dP/dtmin) and to the left, generating smaller values for {tau}. B: increasing the rate of relaxation by lowering µ from 16 ms (solid line) to 10 ms (shaded line) shifts the trajectory upward and to the right, causing dP/dtmin to decrease in magnitude and occur at a higher P; the IVR trajectory becomes more linear, and {tau} increases. Varying model parameters allows fitting of all PPP shapes. See text for details.

 
Model considerations and limitations. Our approach may be subjected to the criticism that two model parameters, µ and Ek, can always provide a better curve fit to data than a single parameter such as {tau} or {tau}L. Although two free parameters is in general always better than one free parameter when performing conventional curve fitting to data points (such as comparing linear vs. quadratic vs. cubic spline fitting methods), the necessity for two parameters was dictated by modeling the physics and physiology in elastic and resistive terms, in addition to the fact that two parameters, {tau} and {tau}L, are already used to characterize IVR. Because a linear equation of motion has a unique invertible solution (15), it provides a set of best-fit model parameters by which specific, testable predictions (dP/dtmin) can be made. Although we minimized mean squared error via Levenberg-Marquardt (19, 23), the analysis is best viewed as a solution of the "inverse problem" (i.e., model parameter determination from pressure data) rather than simple curve fitting, because the model is based on a set of physical, physiological, and kinematic hypotheses that apply to all hearts (17).

Equation 3 is a linear second-order differential equation, but in modeling the physiology it is always possible to consider alternative (and more complicated) formulations of elastic or resistive forces. The resistive force term could be modeled nonlinearly [for example, as P(dP/dt) or (1/µ)(dP/dt)2 (16)], but such a choice provides a poor match with previous findings regarding {tau} or {tau}L (5, 19, 32). Although ventricular operating stiffness (dP/dV, where V is volume) and elastance [P/(V – Vo)] change significantly and nonlinearly during the entire cardiac cycle (26) our concern is with the IVR segment only, where we assume the simplest linear pressure-displacement relationship. This is justified, because a key elastic component (titin) of the tissue is known to have a time-independent, linear force-length relationship (8) and function, as originally predicted from requirements for diastolic suction (14), that is, as a linear, bidirectional spring. Alternatively, one may view this formulation as a linear approximation to nonlinear phenomena confined to a limited physiological domain, i.e., IVR. This is analogous to using maximum elastance (Emax) to characterize (i.e., linearize) the end-systolic pressure-volume relationship. Accordingly, we treat Ek and µ as constants when the chamber is isovolumic (4), a choice that is further supported by a validated model of early, rapid filling (14).

However, we recognize that it may be advantageous to modify Eq. 3 by altering the resistance term. Specifically, if one wished to explicitly include residual cross-bridge interactions in µ (22) (i.e., rate of cross-bridge uncoupling), a time-varying coefficient for the dP/dt term could be considered. Alternatively, relaxation could be modeled as force decay, leaving µ as a pure viscosity constant and including relaxation as a separate time-varying term, acting as a "forcing function" on the right side of the equation. Although the exact algebraic form for such a forcing function is unknown [linear or exponential decline would be expected for cross bridges (3, 5)], there also might be a problem with parameter uniqueness in solving the inverse problem. However, such a formulation might allow for better characterization of certain isometric states (34).

A complete, detailed physical description of IVR requires models containing detailed three-dimensional fiber orientation, electrical activation, electromechanical coupling, development of (tensorial) stresses, and inclusion of sarcomeric and cross-bridge dynamics. These have been developed (9) but were not intended for human in vivo data as input to compute model parameters as output. A more complete analysis utilizing pharmacological intervention in animal models would clearly improve our detailed understanding of the mechanisms and physiological meaning of our model's parameters. However, because of the excellent agreement between model-predicted and experimentally measured pressure decay and peak negative dP/dt, including in the setting of load variation and PVCs, our lumped approach of including elastic restoring force opposed by inertia and resistance as a paradigm for modeling IVR (22) is reasonable.

Physiologically, our model only considers inertia, resistance (e.g., deactivation or relaxation events), and elastic components (e.g., stiffness) of the ventricle. Additional physiological mechanisms have not been considered, such as the contribution of aortic blood momentum (29), which may produce PPP trajectories that have a concave downward feature.

We provided robust examples but not quantitative relationships for load dependence via Valsalva and nonejecting PVC, since they were the only available load-varying phenomena in our data set. Although the relative load independence during Valsalva is reassuring, a more complete, systematic evaluation of load dependence is required before the load dependence or load independence of the parameters can be determined. Specific interventions (inotropes, vasodilators, and IVC occlusion, among others) are required to further characterize the role of elastic and resistive forces, the load (in)dependence of the proposed indexes, and their use in characterization of genetically modified laboratory animals and ultimate clinical utility.

Conclusions. Left ventricular isovolumic pressure decline is incompletely characterized by the traditional relaxation indexes {tau} and {tau}L. We have elucidated the physiology of isovolumic pressure relaxation/decay (IVR) by considering the forces (recoil, resistance, inertia) that determine it. The model explains the circumstance that determines peak negative dP/dt and unites the previously disjointed curve fits provided by {tau} and {tau}L under a single, easily understood physical/physiological prediction-based parametric limit mechanism. We found that recoil forces opposed by relaxation-related resistive and inertial effects determine isovolumic pressure decline.


   APPENDIX
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
GRANTS
 REFERENCES
 
Solutions to Eq. 3

Equation 3 is a linear differential equation characteristic of damped oscillatory motion. It has unique solutions depending on the relative magnitudes of the relaxation (µ) and elastic (Ek) parameters. The three cases are 1) 1/µ2 < 4Ek ("underdamped" kinematics):

Formula A1(A1)
2) 1/µ2 = 4Ek ("critically damped" kinematics):

Formula A2(A2)
and 3) 1/µ2 > 4Ek ("overdamped" kinematics):

Formula A3(A3)
where

Formula A4(A4)
and

Formula A5(A5)
where Po and Formula A5o are the pressure and dP/dt at t = 0 and P{infty} is a constant.


   GRANTS
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
REFERENCES
 
This work was supported in part by American Heart Association Heartland Affiliate Award 0610042Z (to C. S. Chung), the Whitaker Foundation, National Heart, Lung, and Blood Institute Grants HL-54179 and HL-04023 (to S. J. Kovács), the Barnes-Jewish Hospital Foundation, and the Alan A. and Edith L. Wolff Charitable Trust.


   ACKNOWLEDGMENTS
 
The assistance of the Barnes-Jewish Hospital cardiac catheterization laboratory staff and discussion with Leonid Shmuylovich and Marc Sherman are gratefully acknowledged.

Present address of C. S. Chung: Department of Molecular and Cellular Biology, University of Arizona, Tucson, AZ.


   FOOTNOTES
 

Address for reprint requests and other correspondence: S. J. Kovács, Cardiovascular Biophysics Laboratory, Washington Univ. Medical Center, 660 South Euclid Ave. Box 8086, St. Louis, MO 63110 (e-mail: sjk{at}wuphys.wustl.edu)

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.


   REFERENCES
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 GRANTS
 REFERENCES
 

  1. Arts T, Bovendeerd PHM, Prinzen FW, Reneman RS. Relation between left ventricular cavity pressure and volume and systolic fiber stress and strain in the wall. Biophys J 59: 93–102, 1991.[Web of Science][Medline]
  2. Campbell KB, Taheri H, Kirkpatrick RD, Burton T, Hunter WC. Similarities between dynamic elastance of left ventricular chamber and papillary muscle of rabbit heart. Am J Physiol Heart Circ Physiol 264: H1926–H1941, 1993.[Abstract/Free Full Text]
  3. Chung CS, Ajo DM, Kovács SJ. The isovolumic pressure to early rapid filling decay rate relation: model-based derivation and validation via simultaneous catheterization-echocardiography. J Appl Physiol 100: 528–534, 2006.[Abstract/Free Full Text]
  4. Chung CS, Strunc A, Oliver R, Kovács SJ. The diastolic ventricular-vascular stiffness and relaxation relation: elucidation of coupling via pressure phase-plane derived indexes. Am J Physiol Heart Circ Physiol 291: H12415–H12423, 2006.
  5. Comacho SA, Brandes R, Figueredo VM, Weiner MW. Ca2+ transient decline and myocardial relaxation are slowed during low flow ischemia in rat hearts. J Clin Invest 93: 951–957, 1994.[Web of Science][Medline]
  6. Eucker SA, Lisauskas JB, Singh J, Kovács SJ Jr. Phase plane analysis of left ventricular hemodynamics. J Appl Physiol 90: 2238–2244, 2001.[Abstract/Free Full Text]
  7. Feynman RP, Leighton RB, Sands M. The Feynman Lectures on Physics. Reading, MA: Addison-Wesley, 1963.
  8. Helms M, Trombitás K, Granzier H. Titin develops restoring force in rat cardiac myocytes. Circ Res 79: 619–626, 1996.[Abstract/Free Full Text]
  9. Hunter PJ, Borg TK. Integration from proteins to organs: the physiome project. Nat Rev Mol Cell Biol 4: 237–243, 2003.[CrossRef][Web of Science][Medline]
  10. Jobsis PD, Ashikaga H, Wen H, Rothstein EC, Horvath KA, McVeigh ER, Balaban RS. The visceral pericardium: macromolecular structure and contribution to passive mechanical properties of the left ventricle. Am J Physiol Heart Circ Physiol 293: H3379–H3387, 2007.[Abstract/Free Full Text]
  11. Kass DA, Bronzwaer JGF, Paulus WJ. What mechanisms underlie diastolic dysfunction in heart failure? Circ Res 94: 1533–1542, 2004.[Abstract/Free Full Text]
  12. Katz AM, Zile MR. New molecular mechanisms in diastolic heart failure. Circulation 113: 1922–1925, 2006.[Free Full Text]
  13. Katz LN. The role played by the ventricular relaxation process in filling the ventricle. Am J Physiol 95: 542–553, 1930.[Free Full Text]
  14. Kovács SJ Jr, Barzilai B, Pérez J. Evaluation of diastolic function with Doppler echocardiography: the PDF formalism. Am J Physiol Heart Circ Physiol 252: H178–H187, 1987.[Abstract/Free Full Text]
  15. Kovács SJ Jr, McQueen DM, Peskin CS. Modelling cardiac fluid dynamics and diastolic function. Philos Trans R Soc Lond A Math Phys Sci 359: 1299–1314, 2001.[CrossRef]
  16. Kovács SJ Jr, Meisner JS, Yellin EL. Modeling of diastole. Cardiol Clin 18: 459–487, 2000.[CrossRef][Medline]
  17. Lipton P. Testing hypotheses: prediction and prejudice. Science 304: 219–221, 2005.[CrossRef][Web of Science]
  18. Lisauskas JB, Singh J, Bowman AW, Kovács SJ Jr. Chamber properties from transmitral flow: prediction and validation of average and passive left ventricular stiffness. J Appl Physiol 91: 154–162, 2001.[Abstract/Free Full Text]
  19. Matsubara H, Takaki M, Yasuhara S, Arki J, Suga H. Logistic time constant of isovolumic relaxation pressure-time curve in the canine left ventricle. Circulation 92: 2318–2326, 1995.[Abstract/Free Full Text]
  20. Mizuno J, Araki J, Mikane T, Mohri S, Imaoka T, Matsubara H, Okuyama H, Kurihara S, Ohe T, Hirakawa M, Suga H. Logistic time constant of isometric relaxation force curve of ferret ventricular papillary muscle: reliable index of lusitropism. Jpn J Physiol 50: 479–487, 2000.[CrossRef][Web of Science][Medline]
  21. Nishimura S, Nagai S, Katoh M, Yamashita H, Saeki Y, Okada JI, Hisada T, Nagai R, Sugiura S. Microtubules modulate the stiffness of cardiomyocytes against shear stress. Circ Res 98: 81–87, 2006.[Abstract/Free Full Text]
  22. Opitz CA, Kulke M, Leake MC, Neagoe C, Hinssen Hajjar RJ, Linke WA. Damped elastic recoil of the titin spring in myofibrils of human myocardium. Proc Natl Acad Sci USA 100: 12688–12693, 2003.[Abstract/Free Full Text]
  23. Press WH. Numerical Recipes: The Art of Scientific Computing. New York: Cambridge Univ. Press, 1986.
  24. Robinson TF, Factor SM, Sonnenblick EH. The heart as a suction pump. Sci Am 254: 84–91, 1986.[Web of Science][Medline]
  25. Rosen BD, Gerber BL, Edvardsen T, Castillo E, Amado LC, Nasir K, Kraitchman DL, Osman NF, Bluemke DA, Lima JAC. Late systolic onset of regional LV relaxation demonstrated in three dimensional space by MRI tissue tagging. Am J Physiol Heart Circ Physiol 287: H1740–H1746, 2004.[Abstract/Free Full Text]
  26. Sagawa K, Maughan L, Suga H, Sunagawa K. Cardiac Contraction and the Pressure-Volume Relationship. New York: Oxford Univ. Press, 1988.
  27. Shmuylovich L, Kovács SJ. A load-independent index of diastolic filling: model-based derivation with in-vivo validation in control and diastolic dysfunction subjects. J Appl Physiol 101: 92–101, 2006.[Abstract/Free Full Text]
  28. Solomon SB, Nikolic SD, Frater RWM, Yellin EL. Contraction-relaxation coupling: determination of the onset of diastole. Am J Physiol Heart Circ Physiol 277: H23–H27, 1999.[Abstract/Free Full Text]
  29. Sugawara M, Uchida K, Kondoh Y, Magosaki N, Niki K, Jones SJH, Sugimachi M, Sunagawa K. Aortic blood momentum—the more the better for the ejecting heart in vivo? Cardiovasc Res 33: 433–446, 1997.[Abstract/Free Full Text]
  30. Templeton GH, Nardizzi LR. Elastic and viscous stiffness of the canine left ventricle. J Appl Physiol 36: 123–127, 1974.[Free Full Text]
  31. Van Heereback L, Borbély A, Niessen HWM, Bronzwaer JGF, van der Velden J, Steinen GJM, Linke WA, Laarman GJ, Paulus WJ. Myocardial structure and function differ in systolic and diastolic heart failure. Circulation 113: 1966–1973, 2006.[Abstract/Free Full Text]
  32. Weiss J, Frederiksen JW, Weisfeldt ML. Hemodynamics determinant of the time course of fall in canine left ventricular pressure. J Clin Invest 58: 751–760, 1976.[Web of Science][Medline]
  33. Yellin EL, Hori M, Yoran C, Sonnenblick EH, Gabbay S, Frater RW. Left ventricular relaxation in the filling and nonfilling intact canine heart. Am J Physiol Heart Circ Physiol 250: H620–H629, 1986.[Abstract/Free Full Text]
  34. Zhang W, Chung CS, Kovács SJ. Derivation and left ventricular pressure phase plane based validation of a time dependent isometric crossbridge attachment model. Cardiovasc Eng 6: 132–144, 2006.[CrossRef][Medline]
  35. Zile MR, Baicu CF, Gaasch WH. Diastolic heart failure—abnormalities in active relaxation and passive stiffness of the left ventricle. N Engl J Med 350: 1953–1959, 2004.[Abstract/Free Full Text]



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L. Shmuylovich and S. J. Kovacs
Stiffness and relaxation components of the exponential and logistic time constants may be used to derive a load-independent index of isovolumic pressure decay
Am J Physiol Heart Circ Physiol, December 1, 2008; 295(6): H2551 - H2559.
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