## Abstract

Average left ventricular (LV) chamber stiffness (ΔP_{avg}/ΔV_{avg}) is an important diastolic function index. An E-wave-based determination of ΔP_{avg}/ΔV_{avg} (Little WC, Ohno M, Kitzman DW, Thomas JD, Cheng CP. *Circulation* 92: 1933–1939, 1995) predicted that deceleration time (DT) determines stiffness as follows: ΔP_{avg}/ΔV_{avg} = *N*(π/DT)^{2} (where *N* is constant), which implies that if the DTs of two LVs are indistinguishable, their stiffness is indistinguishable as well. We observed that LVs with indistinguishable DTs may have markedly different ΔP_{avg}/ΔV_{avg} values determined by simultaneous echocardiography-catheterization. To elucidate the mechanism by which LVs with indistinguishable DTs manifest distinguishable chamber stiffness, we use a validated, kinematic E-wave model (Kovács SJ, Barzilai B, Perez JE. *Am J Physiol Heart Circ Physiol* 252: H178–H187, 1987) with stiffness (*k*) and relaxation/viscoelasticity (*c*) parameters. Because the predicted linear relation between *k* and ΔP_{avg}/ΔV_{avg} has been validated, we reexpress the DT-stiffness (ΔP_{avg}/ΔV_{avg}) relation of Little et al. as follows: DT_{k} ≈ . Using the kinematic model, we derive the general DT-chamber stiffness/viscoelasticity relation as follows: DT_{k,c} = (where *c* and *k* are determined directly from the E-wave), which reduces to DT_{k} when *c* ≪ *k*. Validation involved analysis of 400 E-waves by determination of five-beat averaged *k* and *c* from 80 subjects undergoing simultaneous echocardiography-catheterization. Clinical E-wave DTs were compared with model-predicted DT_{k} and DT_{k,c}. Clinical DT was better predicted by stiffness and relaxation/viscoelasticity (*r*^{2} = 0.84, DT vs. DT_{k,c}) jointly rather than by stiffness alone (*r*^{2} = 0.60, DT vs. DT_{k}). Thus LVs can have indistinguishable DTs but significantly different ΔP_{avg}/ΔV_{avg} if chamber relaxation/viscoelasticity differs. We conclude that DT is a function of both chamber stiffness and chamber relaxation viscoelasticity. Quantitative diastolic function assessment warrants consideration of simultaneous stiffness and relaxation/viscoelastic effects.

- echocardiography
- mathematical modeling
- diastole

the instantaneous slope of the left ventricular (LV) pressure-volume relation (dP/dV) defines chamber stiffness and serves as one of the two main parameters (the other is relaxation) by which diastolic function is conventionally quantitated (15, 34, 45, 46). Because the instantaneous slope dP/dV varies throughout the cardiac cycle and increases with increasing end-diastolic volume (EDV), many investigators (10, 25, 26, 28, 32, 42) have estimated chamber stiffness by defining ratios of pressure differences to volume differences over relevant portions of filling. For example, average chamber stiffness (ΔP_{avg}/ΔV_{avg}) = (P_{ed} − P_{min})/(V_{ed} − V_{min}) (where P_{ed} and V_{ed} represent end-diastolic pressure and volume and P_{min} and V_{min} represent minimum pressure and volume) is a reasonable approximation for the average operating LV stiffness. Determination of ΔP_{avg}/ΔV_{avg} via high-fidelity, micromanometric pressure recording defines the invasive “gold standard.” Because ΔP_{avg}/ΔV_{avg} is a “relative,” rather than an “absolute,” index, the requirement to use invasive methodology for its determination is not absolute. Accordingly, noninvasive methods for estimation of LV stiffness have been proposed (9, 10, 21, 25, 26, 28, 35). Little et al. (26), as well as other groups (10, 28), experimentally validated a strong correlation between the E-wave deceleration time (DT) and the invasively determined LV hemodynamic operating stiffness (ΔP_{avg}/ΔV_{avg}). Specifically, Little et al. used a kinematic approach to derive chamber stiffness, *K*_{LV}, related to DT as (1) where ρ is the density of blood, *L* is the effective mitral plug flow length, and *A*_{MV} is the constant effective mitral valve area (MVA). *Equation 1*, which was validated in eight dogs, shows a strong linear relation between DT-predicted *K*_{LV} and measured ΔP_{avg}/ΔV_{avg}. Further validation of *Eq. 1* for humans undergoing open-heart surgery and humans undergoing catheterization was achieved by Garcia et al. (10) and Marino et al. (28), respectively.

Importantly, the relation between DT and stiffness specified by *Eq. 1* implies that two LVs with E-waves having indistinguishable DTs should have indistinguishable LV stiffness (ΔP_{avg}/ΔV_{avg}), if it is assumed that chamber volumes are similar and that the constant lumped coefficient in *Eq. 1* is the same for both. Therefore, subjects with similar MVA, blood density, and chamber volumes and indistinguishable DTs by Doppler echocardiography, but significantly different ΔP_{avg}/ΔV_{avg} by catheterization, would suggest that although *Eq. 1* is an excellent correlate of chamber stiffness, it is incomplete.

Our work was motivated by a specific example (Fig. 1) from two subjects with similar EDVs undergoing elective diagnostic catheterization in whom simultaneous micromanometric LV pressure (Millar) and transmitral flow (E waves) were recorded. For both subjects, catheterization-determined stiffness (ΔP_{avg}/ΔV_{avg}) and an E-wave-derived kinematic LV stiffness analog (*k*) (21, 25) were calculated for several consecutive beats (see methods). Although the DTs in Fig. 1 are indistinguishable, all the invasive and noninvasive stiffness indexes were significantly different between the two subjects (see Table 1).

To resolve the “indistinguishable DT but distinguishable measured stiffness (ΔP_{avg}/ΔV_{avg})” dilemma, we investigated the functional relation between DT and ΔP_{avg}/ΔV_{avg}. We address the dilemma conceptually and provide experimental data in support of our model-predicted hypothesis that DT is determined jointly by chamber stiffness and chamber relaxation/viscoelasticity.

## METHODS

#### Overview.

We employ a validated kinematic 1987 model for filling, the parameterized diastolic filling (PDF) formalism, which accommodates relaxation/viscoelastic effects via a “damping” parameter (*c*) (18). We derive a general expression for stiffness that depends on measured DT and PDF relaxation/viscosity (*c*; see *Eq. 5*). Because the general expression for DT is not solely dependent on *k*, two subjects with indistinguishable DT need not have equivalent stiffness. We validate our derived general expression for DT as a function of stiffness and relaxation/viscoelasticity by analysis of 400 E waves recorded from 80 subjects undergoing simultaneous echocardiography-catheterization.

#### Inclusion criteria.

Eighty subjects were selected from an existing Cardiovascular Biophysics Laboratory database of simultaneous Doppler echocardiographic transmitral flow recordings and micromanometric catheter-derived LV pressures obtained during diagnostic cardiac catheterization (2, 25). All subjects underwent elective cardiac catheterization at the request of their referring physician. Before data acquisition and cardiac catheterization, all subjects provided signed, informed consent to participate in the study, which was approved by the Washington University Medical Center Human Studies Committee (Institutional Review Board). Selection criteria from the database required that subjects have normal sinus rhythm, normal valvular function, and clearly identifiable E and A waves. Subjects with significant merging of E and A waves were excluded. In 19 of the 80 subjects, ejection fraction was <55%; in 35 of the 80 subjects, LV end-diastolic pressure was ≥19 mmHg. Demographic information for the group is summarized in Table 2.

#### Simultaneous acquisition of echocardiography and high-fidelity pressure data.

Our methodology for simultaneous micromanometric LV pressure-transmitral flow acquisition is described elsewhere (2, 25) and briefly reviewed in appendix a.

#### Doppler analysis methods.

DT was measured manually as defined by standard criteria (1) as the base of the triangle approximating the deceleration portion of the E wave (Fig. 1).

For each beat, the PDF formalism was also used to fit the E-wave contour (11, 18). The PDF formalism is a lumped-parameter, predictive, rather than accommodative, model (24) that characterizes transmitral flow according to damped simple harmonic oscillatory (SHO) motion in terms of elastic, inertial, and damping/viscoelastic forces (see appendix a for review of the PDF formalism).

As in earlier work (25), diastolic volume differences (ΔV) were calculated by multiplication of MVA by the velocity-time integral (VTI) over the relevant portion of the diastolic period. For estimation of the effective (constant) MVA, the ventriculography-determined (calibrated) stroke volume was divided by the average total VTI (E-wave VTI + A-wave VTI).

#### Calculation of LV hemodynamic operating stiffness.

Two subjects with similar EDV from the group (Fig. 1) and clear diastasis intervals between E and A waves were chosen for a preliminary analysis. For each subject, DT and ΔP_{avg}/ΔV_{avg} were measured in 7–10 consecutive beats. In addition, E waves were fit by the PDF formalism for each beat. Beats from each subject with indistinguishable DT were grouped together. To test whether LV stiffness values were indistinguishable in subjects with indistinguishable DT, hemodynamically determined operating stiffness values, as well as PDF model-derived LV stiffness (*k*), were compared between beats with indistinguishable DT. In addition to comparison of mean values between subjects for stiffness parameters and DT, all measured values (DT, hemodynamic stiffness, and PDF stiffness) for each beat from *subject 1* (9 beats) were compared with measured values from *subject 2* (7 beats) by ANOVA using Microsoft Excel (Microsoft, Redmond, WA) see Table 1.

The method for determining LV hemodynamic operating stiffness is described elsewhere (25). Briefly, average chamber stiffness and early rapid filling stiffness were calculated from ratios of LV pressure changes to volume changes over appropriate time intervals. In accordance with the method of Little et al. (26), average chamber stiffness was defined as ΔP_{avg}/ΔV_{avg} = (P_{ed} − P_{min})/(V_{ed} − V_{min}). Early rapid filling stiffness, the effective E-wave LV stiffness, was defined as the ratio of the change in pressure to the change in volume from minimum LV pressure to diastasis LV pressure: ΔP_{E}/ΔV_{E} = (P_{diastasis} − P_{min})/(V_{diastasis} − V_{min}).

#### General expression for DT.

Using a kinematic approach, in 1995 Little et al. (26) modeled E-wave deceleration, starting at the E-wave peak (E_{peak}), via Newton's law (F = *ma*). The inertial force was opposed solely by elastic ventricular forces, which were proportional to ventricular stiffness. This kinematic undamped model of filling for fitting the DT portion of the Doppler E wave is based on the following equation: d^{2}*v/*d*t*^{2} = *K*_{LV}^{2} () (where is E-wave velocity), which is mathematically equivalent to an undamped SHO having the following equation of motion (per unit mass): (2) where *x* is displacement (cm), *k* is stiffness per unit mass (g/s^{2}), and is the second time derivative of displacement [or acceleration (cm/s^{2})].

To account for the role of the LV as a mechanical suction pump in early diastole and to predict E-wave contours, the PDF formalism utilizes a damped SHO. The equation of motion (per unit mass) is (3) where *c* is the damping constant (g/s), *ẋ* is the velocity (cm/s), and *x*, *k*, and are defined as for *Eq. 2*. Apart from the choice of initial conditions for the two models, *c* is the primary difference between the PDF formalism and the model of Little et al. (26). Thus, in deriving a PDF formalism-based equivalent to the period of oscillation-DT relation (*Eq. 1*), we expect the PDF formalism-derived formula to be a function of stiffness (*k*) and relaxation/viscosity (*c*), as shown in appendix a (*Eq. A1*).

To find the PDF model-predicted stiffness-DT relation, we derive a closed-form expression for DT in terms of PDF parameters. We calculate DT from *Eq. A1* by taking the difference between the time at which the velocity crosses the origin [E-wave duration (E_{dur})] and the time of peak velocity [acceleration time (AT)]. We simplify the result via a Taylor series expansion (see appendix b) in the limit 2*k* > *c*^{2}. The simplified expression becomes (4) where DT_{k,c} is the DT predicted by a “stiffness-and-viscoelasticity” model of filling. *Equation 4* can be inverted to have form similar to that of *Eq. 1*, thereby expressing stiffness as a function of DT and *c*: (5) *Equation 1* is recovered in the *c* = 0 limit.

#### Comparative analysis between DT_{k} and DT_{k,c}.

Lisauskas et al. (25) showed for a large sample size (*n* = 131) that the E-wave-derived kinematic stiffness *k* is strongly linearly correlated with catheterization-determined ΔP_{avg}/ΔV_{avg}. Kovács et al. (21) showed that *k* is equal to a constant multiple of the Little et al. (26) DT-determined stiffness for low *c*-valued E waves. Thus, instead of calculating ΔP_{avg}/ΔV_{avg} (as we did for the preliminary work presented in Fig. 1) for all 80 subjects in the comparative analysis, we used PDF-derived *k* as a surrogate for invasively determined stiffness. To ensure the validity of using *k*, instead of ΔP_{avg}/ΔV_{avg}, as a surrogate, we once again determined the correlation between *k* and ΔP_{avg}/ΔV_{avg} in a subgroup of our patients (*n* = 20) using the methods of Lisauskas et al. Consistent with results of Lisauskas et al., a strong linear correlation (*r* = 0.76) between *k* and ΔP_{avg}/ΔV_{avg} was observed for the subgroup of subjects (data not shown).

By incorporation of the established linear relation between *k* and ΔP_{avg}/ΔV_{avg} (21, 25), the prediction between DT and kinematic stiffness of Little et al. (26) in *Eq. 1* becomes (6) where DT_{k} is DT predicted by a “stiffness-only” model of filling. *Equation 6*, derived from Little et al. (26) (*Eq. 1*), reflects the *c* = 0 limit of *Eq. 4*, derived via the PDF formalism. For the comparative analysis, we used the PDF formalism to analyze 400 E waves. To account for physiological variation, *k*, *c*, and *x*_{0} were computed for five consecutive beats for each of the 80 subjects and averaged for each subject (12). In addition, clinical DT was measured by standard triangle approximation methods (1) for each beat, and the five-beat average DT for each subject was calculated.

For computation of DT_{k,c} and DT_{k}, stiffness (*k*) and relaxation/viscosity (*c*) were substituted into *Eqs. 4* and *6*. Clinically measured DT was compared with the model-predicted DT_{k,c} and DT_{k} by least-mean-square regression. The difference between measured and predicted DT values was analyzed by paired *t*-test (Microsoft Excel, Microsoft).

## RESULTS

For illustrative purposes, representative E-waves from two subjects with indistinguishable DTs and similar MVA and chamber volumes are presented in Fig. 1. Invasively derived stiffness (ΔP_{avg}/ΔV_{avg} and ΔP_{E}/ΔV_{E}), as well as PDF-determined stiffness (*k*), was calculated for seven beats in *subject 2* and 10 beats in *subject 1*, (see Table 1). Although DT values were indistinguishable between the two subjects (*P* = 0.89), invasively derived and kinematically derived stiffness values were significantly different (*P* < 0.001, ANOVA).

The predicted relation between DT, *k*, and *c* (*Eq. 4*) was compared with the established (inverted) relation of Little et al (26) in *Eq. 6*. In Fig. 2, the raw data of measured DT are plotted against model-predicted DT_{k,c} using *Eq. 4* and the model of Little et al. (*Eq. 6*). Least-mean-square regression yields DT = 1.15DT_{k,c} + 0.01 (*r*^{2} = 0.84) vs. *Eq. 1* (DT = 1.48DT_{k} + 0.03, *r*^{2} = 0.60). (DT − DT_{k}), the average difference between the measured DT and DT predicted by the model of Little et al. (26), was 0.082 (SD 0.022) s, whereas (DT − DT_{k,c}), the average difference between measured DT and DT predicted by the model of Kovács et al. (18), was 0.036 (SD 0.031) s. Paired *t*-test analysis of the difference between measured and predicted DT (DT − DT_{k,c} vs. DT − DT_{k}) at a significance level of α = 0.05 yields *t* = 32 and *p* ≈ 10^{−47} for a two-tailed test.

## DISCUSSION

Capitalizing on the “suction pump” (dP/dV < 0 at mitral valve opening) feature of all ventricles (4, 16, 39, 41, 43), in 1987 Kovács et al. (18) showed that damped, SHO motion (the PDF formalism) can causally explain (predict) the observed contours of E waves starting from mitral valve opening. The oscillator was driven by a bidirectional, linear spring, the biological analog of which was unknown, although the extracellular matrix was hypothesized to contribute to the recoil process in early diastole (39). In 1996, Helmes et al. (13) showed that titin develops restoring force in rat cardiac myocytes; i.e., myocytes can “push” when they relax. Importantly, the force-displacement relation of titin (and its isoforms) was experimentally found to be that of a linear, bidirectional spring, in accordance with the 1987 PDF formalism kinematic prediction. In 1995, Little et al. (26) independently posited that, starting from E_{peak}, the deceleration portion can be modeled kinematically as an undamped oscillation (i.e., a cosine function). Expressed differently, the independent derivation of Little et al., based on chamber stiffness as the mechanism that opposes inertia to decelerate the E wave, is a special (*c* = 0) limit of the solution to the suction-initiated transmitral flow problem, solved via the PDF formalism (18). It is reassuring that totally different and independent lines of physiological argument, invoking different initial conditions, lead, via Newton's law, to essentially the same mathematical expression for the E-wave contour.

The model of Little et al. (26) was used to derive *Eq. 6*, which relates DT to ventricular chamber stiffness. Although the approach of Little et al. provides an excellent approximation to the deceleration portions of many E-wave contours, for deceleration portions of E waves exhibiting an inflection point or the “delayed-relaxation” pattern, a cosine function has limited applicability. In addition, a cosine (quarter wavelength, “concave-down”) model cannot fit an E-wave deceleration contour with an inflection point (concave-down changing to “concave-up”). Thus the cosine model works best for ventricles where relaxation/viscoelastic effects are negligible relative to stiffness.

To determine why two subjects with indistinguishable DTs and chamber volume can have different catheterization-determined values of chamber stiffness, we used the PDF formalism to derive the general expression between DT, ventricular stiffness, and relaxation/viscoelasticity (*Eq. 4*). The relaxation/viscoelastic effects during filling declare themselves in the damped oscillatory features of the velocity contour, transforming a pure sine wave (low damping)-shaped E wave to an E wave well fit by a sine wave modulated by a damped exponential, as frequently encountered clinically in the delayed-relaxation pattern.

The viscoelastic parameter *c* resolves the dilemma of two subjects with indistinguishable DTs, MVAs, and EDVs having distinguishable values for invasively determined chamber stiffness (Fig. 1). Importantly, although two E waves, approximated as triangles, can have indistinguishable numeric values for DT, the actual shape and amplitude of the deceleration portion of these E waves can be different. The specific contour of the E wave, in conjunction with DT, is important, because it reflects underlying physical properties of the ventricle (5). The PDF model fits amplitude, shape, and duration of the deceleration portion and provides the information needed to uniquely determine stiffness (*k*) and relaxation/viscosity (*c*) parameters. Since *Eq. 4* does not require a one-to-one correspondence between *k* and DT, it is possible for two LVs to have indistinguishable DT but distinguishable *k*, as long as *c* is different. The plot of stiffness parameter *k* vs. relaxation parameter *c* in Fig. 3, shows how *k* and *c* must change in order for DT to remain constant.

Apart from solving the “indistinguishable DT-distinguishable chamber stiffness” dilemma, this work highlights the fact that the utilization of DT as an index of diastolic function requires the decomposition of DT into stiffness and relaxation/viscoelastic components.

#### Diastolic function and chamber material properties.

The simplest model for the material properties of the chamber assumes that the tissue is a linear elastic material. This approach lumps complex cellular and extracellular interactions into a single stiffness parameter that relates stress to strain (pressure differences to volume differences). Although some models treat tissue as purely elastic, many studies have shown that tissue is viscoelastic (8, 14, 17, 22, 36, 42, 47). The connection between a simple elastic model and a viscoelastic model for the LV is subtle but was elucidated in the pioneering work of Templeton and Nardizzi (42). They dynamically filled canine ventricles with a sinusoidally varying volume and measured the resulting chamber pressure changes as well as the operating chamber stiffness. They found that the peak of the chamber pressure perturbation was offset (in time) from the peak of the driving volume perturbation, indicating viscous effects. Importantly, any offset introduced by the coupling of regional pressure gradients to regional ventricular dimension (6, 23, 31) cannot account for the observed offset, since volume changes were measured precisely from calibrated driving piston motion, and not from ventricular dimensions. To account for viscous effects, Templeton and Nardizzi used a three-component (stiffness, damping, and inertia) linear model for the pressure contour. Inversion of the model allowed determination of the elastic stiffness and the “viscous stiffness” for each ventricle from the pressure-volume data. Templeton and Nardizzi showed that ventricular stiffness could be divided into elastic components, measured by instantaneous dP/dV measurements, and viscous components, measured by phase differences between hemodynamic pressures and volumes. Thus ventricular stiffness is best approximated by elastic dP/dV stiffness only if viscous effects are negligible.

Several studies, however, have shown viscous effects to be significant in animal models and in humans, particularly in those with pathology. For example, Rankin et al. (36) found that a viscoelastic, rather than a purely elastic, model of the ventricle was required to fit observed stress-strain data obtained in open-chest dogs. In an extension of these results to humans, Hess et al. (14) showed that, for patients with myocardial hypertrophy, a model that included viscous and elastic parameters provided a better fit to ventricular stress-strain relations than a simple elastic model. Hess et al. concluded that “it is important for the assessment of diastolic myocardial stiffness to evaluate the viscous influences during filling, because the simple elastic constants reflect a composite of elastic and viscous forces and may be misleading, especially in patients with myocardial hypertrophy.”

In the present study, we sought to integrate the experimental insights that show that ventricular tissue has significant viscous components with the well-established stiffness equation of Little et al. (26), which assumed viscous effects to be negligible. Through mathematical modeling of the physiology, we provide mechanistic insight into how the heart works when it fills by showing that DT can be decomposed into “elastic stiffness” and viscous stiffness or, in our terms, stiffness and relaxation/viscoelasticity components (7) where *a* is a multiplicative constant and ΔP/ΔV and *c* represent stiffness and viscosity/relaxation components, respectively. Consideration of DT as a purely elastic parameter neglects the second viscous term in *Eq. 7* and may lead to the prediction that subjects with indistinguishable DT must have indistinguishable catheterization-determined chamber stiffness. Additionally, the damping term in *Eqs. 4* and *7* provides a more accurate estimate of the clinically measured DT than a model with no damping (*Eq. 6*).

#### Classical physiology-to-clinical cardiology connection.

In current practice, the elastic stiffness component (due to collagen, titin, and a myriad of other factors) of Templeton and Nardizzi (42) is routinely measured (ΔP/ΔV) and the viscous stiffness component is referred to as viscosity/relaxation effects (15). General consensus has developed that diastolic dysfunction is associated with pathophysiology related to stiffness and relaxation (15, 34, 45, 46).

#### Significance of relaxation/viscoelasticity.

Relaxation is intuitively appealing as a parameter that determines diastolic function, and it is a term that has clinical and physiological interpretations. From a physiological perspective, it includes viscoelastic effects and involves processes related to the reuptake of Ca^{2+} after cross-bridge cycling and force generation, intracellular components including microtubules, and actin-titin interactions (15, 17, 22). Clinically, impaired relaxation can be characterized by two dominant phenomena: *1*) prolonged time constant (τ), determined by catheterization, or prolonged isovolumic relaxation time, determined by echocardiography, often indicates impaired Ca^{2+} handling and poor relaxation of the ventricle during isovolumic relaxation, and *2*) an E wave with prolonged DT is referred to as having the delayed-relaxation pattern. Although these measurements differ mechanistically, the delayed-relaxation pattern and a prolonged τ jointly indicate that the ventricle is operating in a regime of impaired relaxation/increased viscoelasticity necessarily associated with diastolic dysfunction (15, 34, 45, 46).

Through mathematical modeling and experimental validation, we have advocated the use of a lumped damping parameter *c* to account for all viscous effects during filling, including prolonged τ effects and viscous effects of the tissue and blood. In two studies, one animal and one human, *c* was significantly higher in E waves acquired from diabetic than from nondiabetic control hearts (7, 38). Higher *c* values implied that the diabetic hearts had dynamic force relations during diastole that differ from diastolic force relations in normal controls, and these different relations reliably generated distinguishable transmitral velocity profiles between the two groups. Furthermore, a recent study using simultaneous echocardiographic-catheterization data predicted and validated a significant linear correlation between the E-wave-derived *c* and 1/τ, the invasively derived time constant of isovolumic relaxation (5).

#### Relaxation/viscosity effects on diastolic filling.

The main difference between the two models presented in Fig. 2 resides in the relative significance of relaxation/viscosity effects on filling. When relaxation/viscosity effects are small, *Eqs. 4* and *6* predict the same value for DT, and the two models are virtually equivalent (8). In clinical situations where E waves have a “restrictive pattern,” i.e., tall and narrow waves with short DT, the waves necessarily have high PDF *k* values, but that does not imply that *c* = 0. Nevertheless, in such cases, the value of *c*/2*k* in *Eq. 4* is much smaller than the value of , and DT predicted by *Eq. 4* is very close to DT predicted by the method of Little et al. (26) in *Eq. 6*. In other words, for patients with restrictive E waves, use of DT to estimate stiffness by the equation of Little et al. will not introduce a significant error. However, for patients with a delayed-relaxation (long-DT) E wave, *c* is significant, and *c*/2*k* cannot be ignored. In this case, use of the equation of Little et al. to estimate stiffness from DT alone will significantly underestimate LV stiffness, as can be seen by setting *c* = 0 in *Eq. 5*.

An interesting kinematic consequence of the presence of relaxation/viscoelastic effects relates to the atrioventricular pressure gradient. If damping were absent, the elastic recoil of the ventricle would be completely converted to fluid motion, and the atrioventricular pressure gradient would vanish when blood acceleration is zero, i.e., at E_{peak}. However, if viscosity is present, zero atrioventricular pressure gradient implies that the damping forces opposing flow equal the inertial force. The PDF model predicts that the atrioventricular pressure gradient vanishes not at E_{peak} but, rather, at a time interval DT–AT after E_{peak} (appendix c); e.g., it is phase shifted. In the *c* = 0 limit, DT − AT = 0, and the zero atrioventricular pressure gradient occurs, as expected, at E_{peak}. This analysis is similar to the work of Templeton and Nardizzi (42), because it predicts that viscous effects will introduce a phase shift between the pressure gradient and the resulting flow. Previous work utilizing frequency-based (Fourier) analysis of in vivo pressure-volume data has also characterized such a phase difference during the E wave.

#### Limitations of the study.

It is well known that many ventricular diastolic properties are nonlinear, and nonlinear models of filling have advantages over linear models, such as the PDF model employed in the present study, e.g., an ability to more completely characterize complex physiology. However, the more complex models, having many model parameters, cannot be inverted to generate a unique set of parameters determined by using the clinical data as input (33). Furthermore, previous work from our group has demonstrated that the PDF model-generated fit to the clinical E-wave contour was numerically indistinguishable from fits provided by well-established, complex, nonlinear models (33). A key advantage of using the simplest model that can be validated by experiment resides in its ability to directly (and uniquely) characterize in vivo clinically recorded E-wave contours in terms of three easily understood, lumped kinematic parameters. The associated limitation is the inability to determine the selective effects of individual parameters (of the nonlinear models) on the E-wave contour.

Another potential limitation may be that *k*, *c*, and DT are derived from the E wave. Therefore, one may suspect that *k* and *c* are algebraically related, and, therefore, one of the variables in *Eq. 6* can be eliminated, resulting in a one-to-one correspondence between *k* and DT. Although physiology constrains the range of observed values for *c* and *k*, we note that *k* and *c* are mathematically independent. Apart from the restriction that the wave shape is underdamped (4*k* > *c*^{2}) and that the parameters are nonnegative, there is no explicitly known functional relation between the two parameters.

Although a highly linear relation between *k* and ΔP_{avg}/ΔV_{avg} has been demonstrated in a large sample (*n* = 131) by Lisauskas et al. (25) and independently repeated for a smaller sample (*n* = 20) in the present study, one may surmise, in Fig. 2, that a more appropriate comparison would be to use *Eq. 7*, utilizing ΔP_{avg}/ΔV_{avg} from catheterization data for all 80 patients, instead of *Eq. 4*, and *k* as a surrogate for ΔP_{avg}/ΔV_{avg}. However, our use of *k*, instead of ΔP_{avg}/ΔV_{avg}, would be inappropriate only if it resulted in a tautology, in the sense of not being able to differentiate between the ability of alternate expressions (with and without viscoelasticity) to fit the same data (Fig. 2). If viscoelasticity were not an important determinant of DT and the E-wave contour is purely the result of elastic and inertial forces, then Fig. 2 would yield indistinguishable plots for DT vs. DT_{k} and DT vs. DT_{k,c}. The two equations are equivalent when resistive forces are absent, and previous work (21) has shown that if viscous effects are negligible, then *k* is (within a multiplicative constant) equal to *K*_{LV}, as proposed by the model of Little et al. (26). In contrast, if viscoelastic/resistive forces play a discernible role in determining the E-wave contour, then we anticipate a better fit to the data with DT_{k,c} than with DT_{k}, as shown in Fig. 2. Thus using the stiffness parameter *k*, rather than ΔP_{avg}/ΔV_{avg}, for comparative analysis is reasonable and appropriate.

A further limitation involves the calculation of ΔP_{E}/ΔV_{E} (Table 1). ΔP_{E}/ΔV_{E} is calculated over a shorter duration (by about a factor of 2) than ΔP_{avg}/ΔV_{avg} and, as such, is more sensitive to exact temporal alignment of the pressure and echo waveform than ΔP_{avg}/ΔV_{avg}. This sensitivity manifests in Table 1 as a larger standard deviation in ΔP_{E}/ΔV_{E} values, which contributes to the higher *P* value. ΔP_{avg}/ΔV_{avg}, however, is more robust and is the measure also used by Little et al. (26). Importantly, the more significant (*P* = 0.0015) difference between the two subjects in Fig. 1 is in ΔP_{avg}/ΔV_{avg} values. This difference in ΔP_{avg}/ΔV_{avg} is intended to serve as motivation for the conceptual basis of the present study and as a basis for the experimental validation for the 80 subjects shown in Fig. 2.

Another minor limitation is the use of E-wave VTIs, instead of conductance catheter data, for ΔV determination. Indeed, MVA has been shown to vary during diastole (3), and conductance catheter volumes may avoid the error introduced by multiplying the VTI by a constant effective MVA. However, use of constant effective MVA is inherent in the derivation of the expression by Little et al. (26) in *Eq. 1*. Specifically, when ΔV is expressed as VTI·MVA, MVA cancels from *Eq. 1*, leaving only ΔP and VTI: (8)

Finally, left atrial properties, particularly left atrial stiffness, may play a role in determining DT. Although some studies suggest that DT is affected jointly by left ventricular and atrial stiffness, during the E wave the atrium is a conduit and is passive. Recently, Marino et al. (27) measured atrial stiffness during diastole in relation to ventricular stiffness and concluded that its role was minor and did not significantly affect DT.

In conclusion, different LVs with the same duration of E-wave DT and similar chamber volume can have different catheterization-determined values for chamber stiffness (ΔP_{avg}/ΔV_{avg}). Model-based analysis of E waves provides unique values for chamber stiffness (*k* ∝ ΔP_{avg}/ΔV_{avg}) and chamber viscoelasticity (*c*). For E waves that are very nearly symmetrically shaped (AT ≅ DT) about the E wave peak, DT and chamber stiffness are related as follows: . Once asymmetry is present and AT ≠ DT, the E-wave deceleration portion manifests an inflection point and lengthens. The general expression for DT applicable to all E waves depends on chamber stiffness but also requires inclusion of chamber relaxation/viscoelastic effects as follows: .

We conclude that quantitative diastolic function assessment warrants consideration of viscoelastic effects in addition to those of stiffness, because E-wave DT is determined by both. The DT vs. stiffness-viscoelasticity relation described above is a general finding applicable to all ventricles and is most significant for ventricles with relaxation abnormalities.

## APPENDIX A

#### Simultaneous acquisition of echocardiographic and high-fidelity pressure data.

Briefly, after appropriate sterile skin preparation and drape, local anesthesia (1% xylocaine) was administered, and percutaneous right or left femoral arterial access was obtained using a valved sheath (6-F, Arrow, Reading, PA). A 6-F micromanometer-tipped pigtail pressure-volume (conductance) catheter (model SPC 562, Millar Instruments, Houston, TX) was directed into the mid-LV in a retrograde fashion across the aortic valve under fluoroscopic control. Before insertion, the manometer-tipped catheter was calibrated against hydrostatic “zero” pressure by submersion just below the surface of a 37°C normal saline bath. It was balanced using a transducer control unit (model TC-510, Millar Instruments). The ventricular pressures were fed to the catheterization laboratory amplifier (Quinton Diagnostics, Bothell, WA) and output simultaneously into the auxiliary input port of a Doppler imaging system (Acuson, Mountain View, CA) and into a digital converter connected to a customized personal computer. With the subject supine, apical four-chamber views were obtained by the sonographer, 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. To synchronize the hemodynamic and Doppler data, a fiducial marker in the form of a square-wave signal was fed from the catheter transducer control unit to the echocardiographic imager and the personal computer. Approximately 25–50 beats of continuous, simultaneous transmitral Doppler and LV pressure signals were recorded on the imager's magneto-optical disk. Images of individual beats were captured from the disk for offline analysis using custom image-processing software.

#### PDF formalism.

During filling, the elastic driving force generates inertial force, which causes acceleration, and resistive (damping) force, which opposes acceleration. The three (mathematically) independent model parameters *k* (spring constant), *c* (relaxation/viscosity/damping constant), and *x*_{0} (initial spring displacement) fully characterize the velocity of the SHO (i.e., E-wave velocity contour). These parameters were determined, for each beat, by solution of the “inverse” problem of filling, with the clinical E-wave contour used as the beat-by-beat input and the model parameters as the best-fit-determined output (11). Most clinical E-wave velocity contours are fit by solutions in the “underdamped” regimen of motion of the SHO, defined by 4*k* > *c*^{2}: (A1) with α = *c*/2 and ω = . Setting *c* = 0 yields undamped sinusoidal behavior, originating at the start of the E wave, which is algebraically equivalent to the solution of the model of Little et al. (26), yielding a cosine function that originates at E_{peak}.

The PDF model-predicted velocity (*Eq. A1*) and its “overdamped” equivalent provide an excellent fit to all clinically recorded Doppler E-wave contours (11, 18, 19). Accordingly, *Eq. A1* allows integration of the velocity to yield the E-wave VTIs.

In addition to generating accurate fits to all E-wave contours, the PDF model parameters have physiological analogs that have been experimentally validated in vivo. On the basis of a large sample (*n* = 131), Lisauskas et al. (25) showed that average LV hemodynamic operating stiffness (ΔP_{avg}/ΔV_{avg}), extracted from invasive measurements of pressure and volume differences, showed a strong linear correlation with the PDF model-derived elastic stiffness (*k*) extracted purely from the E-wave contour. Additionally, the peak-force *kx*_{0}, which drives the oscillator, is the analog of the peak atrioventricular pressure gradient generating transmitral flow (2); the slope of the *kx*_{0}-*c*E_{peak} relation obtained at variable loads has recently been shown in normal control subjects and patients with diastolic dysfunction to be a load-independent index of diastolic function (39); 1/2*kx* is the energy (ergs) available before valve opening (18), and *x*_{0} is linearly related to the volumetric load, i.e., the VTI of the E wave (18). Furthermore, the PDF formalism has been tested and validated in subjects with a wide range of cardiac pathologies and loads, including hypertension (20), heart failure (29, 37), diabetes (7, 38), and caloric restriction (30).

## APPENDIX B

The PDF formalism's expression for the Doppler E-wave contour (*t*) in terms of model parameters (per unit mass) is (B1) where α = (*c*/2) and ω = . E_{dur} is from the origin where (0) = 0 to the time at which the v(*t*) contour again crosses the abscissa. The sine has its first zero at ω*t* = π, or *t* = π/ω. Thus (B2) The AT can be found by solving for the time at which the derivative of *Eq. A1*, with respect to time, vanishes. Thus (B3) Since DT = E_{dur} − AT, we have (B4) We simplify *Eq. B4* by expanding the inverse tangent function as a series for ω/α > 1: (B5) The limit ω/α > 1 is equivalent to 2*k* > *c*^{2}, which is slightly more restrictive than the underdamped condition (4*k* > *c*^{2}) but is easily satisfied by most observed underdamped waves. Keeping the first two terms in *Eq. B5*, *Eq. B4* becomes (B6) Taylor's theorem allows any continuous function to be approximated as (B7) Applying *Eq. B7* to *f*(*x*) = (1 + *x*)^{b} about *x* = 0 (B8) Recall that α = (*c*/2) and ω = . Further simplification of *Eq. B6* is achieved by Taylor expansion of ω^{−1} (B9) Because *c*^{2}/4*k* < 1, we retain linear terms, since quadratic terms will be significantly smaller. Hence, *Eq. B6* becomes (B10) Finally, because *c*^{2}/4*k* is small, we eliminate higher-order terms, such as *c*^{2}/(8*k*^{3/2}), *c*^{3}/(8*k*^{2}), and *c*^{5}/(64*k*^{3}). Thus the general expression for DT as a function of *k* and *c* becomes (B11)

## APPENDIX C

The analog of the pressure gradient according to the PDF model is the force per unit area, expressed as ΔP = *kx*(*t*). Thus the atrioventricular pressure gradient vanishes when the displacement *x*(*t*) = 0. Accordingly, for underdamped (*c*^{2}/4*k* < 1) kinematics, the displacement as a function of time is (C1) We solve for the time when *x* = 0: (C2) Inverting *Eq. C2* for time and using *Eq. B4* yields (C3) Thus, for underdamped waves, the pressure crossover point occurs at *t* = DT after the start of the E wave.

## GRANTS

This study was supported in part by the American Heart Association Heartland Affiliate, the Whitaker Foundation, National Heart, Lung, and Blood Institute Grants HL-54179 and HL-04023, the Barnes-Jewish Hospital Foundation, and the Alan A. and Edith L. Wolff Charitable Trust.

## Acknowledgments

We gratefully acknowledge the assistance of the Barnes-Jewish Hospital cardiac catheterization laboratory staff. We thank Peggy Brown for expert echocardiographic data acquisition.

## Footnotes

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- Copyright © 2007 by the American Physiological Society