INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

IN ACCORDANCE WITH THE LAWS of fluid mechanics, left ventricular (LV) diastolic transmitral flow is generated by the atrioventricular pressure gradient (21, 30). In response to the rapid (~100 ms) development of the maximal pressure gradient, normal mitral valve (MV) leaflets separate widely as early rapid filling of the ventricle ensues, and the Doppler E wave is generated. Diastolic filling occurs in three phases. The first phase is early rapid filling, a phase initiated by a period of mechanical suction (dP/dV < 0, where P is pressure and V is volume) in all hearts (11), generated by the transient dominance of intracellular [titin (10)] and extracellular compartment [connective tissue matrix (4, 26)]-driven mechanical recoil over progressive myocardial relaxation. The magnitude of transmitral flow velocity can be modulated by numerous extracardiac factors, such as preload and afterload. The second phase is diastasis; if heart rate is typically <75 beats/min, this quiescent phase is characterized by an overall balance of forces (e.g., chamber wall, pericardial, thoracic, and similar) manifesting as absence of the atrioventricular pressure gradient and no change in volume. We emphasize that balance of forces during diastasis does not imply absence of forces (24). The third phase, atrial contraction, ensues when contraction of atrial myocardium pulls the MV annulus and aortic root toward the mediastinum, generating a positive atrioventricular pressure gradient, forcing blood into the ventricle (dP/dV > 0) and retrogradely into the pulmonary veins. Atrial contraction causes a simultaneous increase in LV pressure (LVP) and volume, and the relative duration of antegrade (transmitral) vs. retrograde (pulmonary vein) flow is related to atrioventricular compliance and pressure (1).

By definition, the slope (dP/dV) of the LVP-volume curve is the LV chamber stiffness (19). The diastolic pressure-volume relation is curvilinear (concave upward) in shape, and its slope (i.e., chamber stiffness) increases as the LV end-diastolic pressure (LVEDP) and volume increase. Classically, measurement of this chamber property has required catheterization-based measurement of LVP and the simultaneous change in volume.

Doppler-derived indexes of diastolic function have utilized the pattern of LV filling as seen on transmitral Doppler echocardiographic images (16). Selected indexes derived from this method include peak velocities of early rapid filling (Doppler E wave), atrial contraction (Doppler A wave), and the E-to-A ratio, as well as the acceleration time (t_acc) and deceleration time (t_dec).

Diastolic dysfunction has been associated with three types of E-wave patterns: delayed relaxation, pseudonormalized, and restrictive. Delayed relaxation is characterized by prolonged isovolumic relaxation time (IVRT), lower-than-normal E-wave and higher-than-normal A-wave amplitudes, and prolonged t_dec. The pseudonormalized pattern features increased E-wave amplitude due to elevated atrial pressure. Because this generates an increased atrioventricular pressure gradient, it "normalizes" the E-to-A ratio to >1. The restrictive pattern is characterized by short IVRT, a tall, narrow E wave with high peak velocity, short t_dec (<160 ms), and a small A wave (2, 29), and it is usually associated with a third heart sound (S3) (18). The effects of compensatory mechanisms may mask the effects of diastolic dysfunction on the LV filling pattern and indicate a lack of specificity in the ability of a transmitral flow velocity-derived index, such as the E-to-A ratio, to quantitate specific physiological attributes such as chamber compliance or stiffness. For routine interpretation of transmitral Doppler waveforms, the differentiation between the effect of relaxation and compliance has been emphasized (2). In an effort to relate measures derived from the Doppler E wave to specific physiological chamber properties, Little et al. (17) proposed a method for measurement of LV chamber stiffness. In their study in dogs (17), average LV chamber stiffness [(P/V)_avg] was defined as the pressure change (P) from minimum LVP to LVEDP divided by the change in LV volume during the same time interval. E-wave deceleration was modeled via the equation of motion (Newton's law) for an undamped harmonic oscillator. When applied to the deceleration portion of the E wave, its solution related the rate of flow deceleration directly to atrial pressure and MV area and inversely to the sum of left atrial and LV stiffness, as previously modeled (20, 28). Solution of the differential equation predicted that the deceleration portion of the E wave should be well fit by a cosine function, the argument of which contained t_dec. The final result obtained by Little et al. predicted that chamber stiffness (K_LV) could be calculated from t_dec of the Doppler E wave as follows

(1)

where  is the density of blood, L is the effective, mitral "plug-flow" length, and A is the (constant) effective MV area. To validate the Doppler-predicted t_dec-K_LV relation, (P/V)_avg was measured using micromanometric LVP values. The change in volume (V) associated with P was based on an ellipsoidal model of the LV with axes corresponding to anterior-posterior, septal-lateral, and long-axis dimensions. A strong relationship was observed between the t_dec-predicted K_LV and (P/V)_avg invasively measured in eight dogs. Thus t_dec of the Doppler E wave was shown to be a noninvasive index of average chamber stiffness for E waves, for which (concave downward) the deceleration portion was well fit by a cosine function (17). Recently, using the same conceptual approach, Garcia et al. (5a) estimated LV operating stiffness from t_dec in 18 subjects undergoing open heart surgery.

For quantitative diastolic function assessment from Doppler echocardiography, a parameterized diastolic filling formalism has been developed and previously validated (6, 7, 12). This model accounts for the mechanical suction attribute of the heart (11, 26) via a damped, simple harmonic oscillator as a paradigm for filling. The Doppler E-wave contour is the analog of the solution for velocity as a function of time to the equation of motion of the oscillator. According to Newton's law, the equation of motion for the E wave is given by

(2)

where the overdot denotes differentiation with respect to time, m (in g) is inertia, c (in g/s) is the damping constant, and k (in g/s²) is the spring constant. For convenience, we set m = 1 and compute parameters per unit mass. The oscillator springs back from nonzero (x₀, in cm) initial displacement (stored elastic strain) as transmitral flow commences. With the entire Doppler E-wave contour's maximum velocity envelope (MVE) as input, a computer-based, automated, least-squares (Levenberg-Marquardt) fit of the model-predicted solution for velocity as a function of time solves the "inverse problem" of diastole and generates the three (unique) model parameters x_o, k_E, and c (6, 7, 12). These three parameters account for the amplitude, width, and rate of decay of the E wave, respectively. For the A wave (7), the damped simple harmonic oscillator model utilizes a forcing function [F(t) = F₀ sin (t)], the analog of atrial systole, and utilizes the parameters F₀ and  (the magnitude and angular frequency of the forcing function, F). This model-based image-processing (MBIP) strategy for analysis of Doppler E and A waves eliminates the need to digitize the contour by hand or determine its attributes such as maximum velocity, E-to-A ratio, t_acc, or t_dec by eye (Fig. 1).

View larger version (38K): [in this window] [in a new window]   Fig. 1.   Overview of the model-based image-processing (MBIP) method of model fitting and parameter determination with the Doppler image as input. Sequence of steps is as follows: acquire digital transmitral Doppler image (A), select diastolic interval of interest (B), determine maximum velocity envelope to be used as input for MBIP (C), and perform fit using Levenberg-Marquardt method (D). E: results, i.e., parameter values and standard deviations, including mean square error (MSE) as a measure of goodness of fit. x0, initial displacement (cm); c, damping constant (g/s); k, spring constant (g/s2), Fo, magnitude (dyn); , angular frequency (rad/s).

The methods employed by Little et al. (17) indicated that the deceleration portion of the E wave (starting at the peak of the E wave) should be well fit by a cosine function, whereas the solution of Eq. 2 predicts that the entire E wave (starting at its onset) is well fit by the product of a sine function and a decaying exponential. It has been shown (14) that, for small values of the exponential decay term, the MBIP-predicted E-wave contour (sine wave, starting at the onset of the E wave) is equivalent to the cosine (starting at the peak of the E wave) fit to the deceleration portion of the E wave proposed by Little et al. The equivalence of sine to cosine (shifted by /2 rad) permits derivation of an exact algebraic relationship relating t_dec (and K_LV) as employed by Little et al. to k_E of the parameterized diastolic filling model. For validation, transmitral Doppler images were subjected to MBIP, and the predicted linear relationship between K_LV and k_E has been previously verified (14).

To validate the predicted linearity of the (E-wave-generated) k_E-(P/V)_avg relation, simultaneous LVP and transmitral flow data were obtained in 131 subjects exhibiting a broad range of E-wave patterns. The method also permitted determination of chamber stiffness during early rapid filling (P/V)_E and its correlation to k_E. In those having a period of diastasis, passive chamber stiffness, k_A, was determined from the A wave and compared with postdiastasis stiffness [(P/V)_PD].

Glossary

LV	Left ventricular
KLV	Chamber stiffness computed by Little et al. (17) (mmHg/cm3)
tdec	Deceleration time (s)
(P/V)avg	Average chamber stiffness (mmHg/cm3)
(P/V)E	Chamber stiffness of early rapid filling (mmHg/cm3)
(P/V)PD	Passive (postdiastasis) chamber stiffness (mmHg/cm3)
MV	Mitral valve
m	Mass (g)
c	Damping constant (g/s)
k	Spring constant (linear form; g/s2)
x	Displacement (cm)
x0	Initial displacement (displacement at t = 0; cm)
MBIP	Model-based image processing
kE	Stiffness parameter for the Doppler E wave (g/s2)
kA	Stiffness parameter for the Doppler A wave (g/s2)
TVI	Time-velocity integral
MSE	Mean square error
MVE	Maximum velocity envelope
LVP	LV pressure (mmHg)
LVEDP	LV end-diastolic pressure (mmHg)
IVRT	Isovolumic relaxation time (s)

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Clinical data acquisition. Simultaneous transmitral Doppler and high-fidelity hemodynamic (LVP) data were obtained in 131 patients undergoing elective cardiac catheterization at the request of their referring physician. Informed consent was obtained in accordance with Washington University Medical Center Human Studies Committee guidelines. All subjects were in sinus rhythm. None of the subjects had any significant valvular abnormality. Simultaneous hemodynamic and Doppler echocardiographic data were acquired before injection of iodinated contrast for selective coronary angiography and left ventriculography. Subjects were sedated with diazepam (2.5-5.0 mg iv). Left ventriculography was performed in the conventional 30° right anterior oblique view using a power injector (Mark V, MedRad) via a 7-Fr pigtail catheter (Millar, Houston, TX).

MBIP. Transmitral Doppler E and A waves were subjected to MBIP as previously described (6, 7, 12, 14) for model parameter determination (Fig. 1). We also previously showed that averaging of parameters from individual images is superior to averaging of images, with an average variation of <10% for all parameters (8, 9). Accordingly, individual beats were selected from the original transmitral digital Doppler image data set. To account for beat-to-beat variability, at least five beats per subject underwent MBIP, and the resulting parameters were averaged (8, 9).

View larger version (50K): [in this window] [in a new window]   Fig. 2.   Selected examples of in vivo Doppler transmitral flow velocity images obtained during cardiac catheterization. A: E wave having a deceleration portion well fit by a cosine function and the MBIP-predicted contour. This deceleration contour is similar to E-wave shapes considered in the canine study by Little et al. (17). B: E wave having an inflection point during the deceleration portion. Fit generated by the MBIP method is superimposed.

Average chamber stiffness. Average chamber stiffness was defined as the ratio of the change in pressure to the change in volume [(P/V)_avg] during the time interval from minimum LVP to LVEDP, as defined by Little et al. (17). Briefly, (P)_avg was defined as the difference between minimum LVP and LVEDP for the same beats subjected to MBIP. An example of the micromanometric LVP trace from a single individual and the simultaneous transmitral Doppler image are shown in Fig. 3. The pressure portion utilized for determination of (P)_avg is shown in Fig. 3C. (V)_avg was calculated by using the sum of the time-velocity integral (TVI) for the E and A waves commencing from the time of minimum LVP to LVEDP. To obtain the corresponding change in volume, the two-dimensional echocardiographic effective MV area was multiplied by the TVI. V as a function of time is shown in Fig. 3B. The effective MV area is the echocardiographic annular area multiplied by the (dimensionless) factor R (R < 1), where R is the ratio of annular area E- and A-wave TVI to leaflet tip E- and A-wave TVI. The Doppler E-wave (MBIP)-derived parameter k_E was plotted vs. the invasively obtained (P/V)_avg for each subject. To facilitate direct comparison with the method of Little et al. (17), t_dec was measured directly, allowing determination of K_LV. This allowed experimental validation of the theoretically predicted (14) linear relationship of k_E to K_LV.

View larger version (33K): [in this window] [in a new window]   Fig. 3.   Selected Doppler image and a portion of the simultaneous micromanometric left ventricular (LV) pressure (LVP) trace used for computation of chamber stiffness (P/V, where P is pressure and V is volume) for selected phases of filling. A: simultaneous transmitral flow velocity as imaged by Doppler echocardiography. B: change in LV volume as a function of time since mitral valve opening. According to Little et al. (17), (V)avg is defined as the volume change during the minimum LVP to LV end-diastolic pressure (LVEDP) interval. (V)E and (V)PD, volume changes during the E-wave and atrial contraction portions, respectively. C: high-fidelity recording of LVP. Single diastolic interval is shown. (P)avg, rise in pressure from minimum LVP to LVEDP; (P)E, rise in pressure from minimum LVP to the end of the E wave (diastasis in this case); (P)PD, postdiastatic rise in pressure due to atrial systole; ECG, electrocardiogram.

Early rapid-filling stiffness. Stiffness during the E-wave interval (E-wave stiffness) was defined as the change in pressure divided by the change in volume [(P/V)_E] for the interval from minimum LVP to LVP at diastasis. The E-wave-derived stiffness parameter k_E was plotted vs. (P/V)_E [see Fig. 3 for graphical definition of (P)_E and (V)_E].

Passive chamber stiffness. Passive chamber stiffness was determined for subjects having a period of diastasis. Presence of diastasis indicated a period of no (or insignificant) transmitral flow, i.e., abolition of the atrioventricular pressure gradient with concomitant equilibration of residual atrioventricular forces. Passive chamber stiffness after diastasis was defined by the ratio of change in pressure to change in volume due to atrial systole [(P/V)_PD; Fig. 3]. In analogy to the relationship of k_E to E-wave-determined stiffness, we sought to validate the prediction that the A-wave-derived parameter k_A is linearly related to the passive chamber stiffness. Accordingly, the relationship of k_A to (P/V)_PD was compared. Linear least-squares best fit and corresponding r value were determined using DeltaGraph for echo-derived vs. invasively derived definitions of stiffness for the above-defined average, early rapid-filling, and passive (postdiastasis) phases of filling.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Group attributes. LV ejection fraction (LVEF) as determined by ventriculography for the entire group was normal: 70 ± 12% (SD). Additional characteristics of the sample include gender (85 men and 46 women), age (53 ± 10 yr), heart rate (68 ± 12 beats/min), and LVEDP (17 ± 5 mmHg). MBIP-derived fits of the predicted velocity contours to the clinical Doppler E-wave images showed excellent agreement. For all (n = 131) subjects, the average mean squared error (MSE) was 7.23 × 10⁴ for the E wave and 6.14 × 10⁴ for the A wave. The ability to compute a measure of goodness of fit in the form of MSE is an inherent advantage of the MBIP approach. Figure 2 illustrates model-predicted transmitral flow velocity superimposed onto selected clinical E and A waves. Least-squares linear best fit and correlation coefficient in this human study were determined in analogy to the study by Little et al. (17) in their canine model. In addition, the method and the data set permitted noninvasive prediction and invasive validation of stiffness during selected phases of diastole. Specifically, the Doppler-derived early rapid-filling stiffness, k_E, could be compared with (P/V)_E, and (postdiastasis) chamber stiffness, k_A, could be compared with (P/V)_PD. In analogy to Little and Downes (16), Fig. 4 displays the data for the echocardiographically determined parameter k_E and its relation to the simultaneous, invasively determined average stiffness: K_LV = (P/V)_avg. Linear regression, with least MSE best fit (n = 131), yielded k_E = 1,402 · (P/V)_avg + 59.2 with strong correlation (r = 0.84). The E-wave stiffness parameter k_E is compared with K_LV of Little and Downes as determined by t_dec of the Doppler E wave in Fig. 5. Linear regression yielded k_E = 1.110 · (A/L) · K_LV + 55.8 (r = 0.85). Figure 6 depicts k_E vs. (P/V)_E. Linear regression yielded k_E = 229.0 · (P/V)_E + 112 (r = 0.80). Figure 7 depicts the Doppler A-wave-derived parameter k_A vs. (P/V)_PD. Linear regression yields k_A = 1,640 · (P/V)_PD  8.40 (r = 0.89). Figure 8 combines Figs. 4, 6, and 7 and indicates the following order: slope of the linear regression relation for early stiffness < slope for average stiffness < slope for postdiastasis stiffness. This is consistent with the concave-upward shape of the LV diastolic pressure-volume curve.

View larger version (27K): [in this window] [in a new window]   Fig. 4.   MBIP-generated, Doppler E-wave-derived parameter (kE) vs. invasively determined average chamber stiffness [(P/V)avg] for 131 subjects. Best fit was provided by kE = 1,402 · (P/V)avg + 59.2 (r = 0.84).

View larger version (26K): [in this window] [in a new window]   Fig. 5.   MBIP-generated kE vs. invasively determined chamber stiffness (KLV) based on deceleration time (tdec) of the Doppler E-wave for 131 subjects. Best fit was provided by kE = 1.110 · (A/L) · KLV + 55.8, where A is (constant) effective mitral valve area, L is effective mitral "plug-flow" length, and  is density of blood (r = 0.85).

View larger version (28K): [in this window] [in a new window]   Fig. 6.   MBIP-generated kE vs. invasively determined early rapid filling chamber stiffness [(P/V)E] for 131 subjects. Best fit was provided by kE = 220.9 · (P/V)E + 113 (r = 0.80).

View larger version (30K): [in this window] [in a new window]   Fig. 7.   MBIP-generated, Doppler A-wave-derived parameter (kA) vs. invasively determined passive chamber stiffness [(P/V)PD] for 113 subjects with diastasis. Best fit was provided by kA = 1,640 · (P/V)PD  8.40 (r = 0.89).

View larger version (31K): [in this window] [in a new window]   Fig. 8.   MBIP-generated, Doppler-derived parameters vs. invasively determined chamber stiffness for kE vs. (P/V)avg (), kE vs. (P/V)E (), and kA vs. (P/V)PD (). Overall, average stiffness of the ventricle is considerably greater during atrial contraction than during early filling, and average stiffness (which lumps the 2 phases together) falls between early and late stiffness values.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

The quest for Doppler-derived physiological indexes of diastolic function has remained an area of active investigation (22). Progress has been made through mathematical modeling of diastolic physiology, which provided quantitative insight into how Doppler E-wave contours are determined by the transmitral atrioventricular pressure gradient (15). In turn, how the atrioventricular pressure gradient depended on numerous physiological variables, such as atrial stiffness and capacitance as well as MV inertance and chamber properties, has been also described in detail (19, 21, 24, 27, 30). One inherent limitation of the (nonlinear) physiological modeling approaches is their inability to use the clinical Doppler contour as input to generate unique model parameters as output (23). Our approach circumvents this limitation by approximating the filling process by the use of a linear model for filling. This requires the lumping of all physiological determinants of transmitral flow for the E wave into three parameters to account for its amplitude (x_o), width (k_E), and rate of decay (c). The goodness of the model-predicted fit to in vivo contours underscores the utility of the approach as a practical tool for analysis and for facilitation of clinical decisions by accommodating the full range of Doppler flow velocity patterns encountered in vivo (13, 25).

As a result of the pioneering work of Little et al. (17) in a canine model, additional progress toward physiological interpretation of E-wave attributes has been achieved by their theoretical derivation and validation of a relationship between E-wave t_dec and chamber stiffness. On the basis of their work, we previously established a direct linear relationship between the E-wave-determined parameter k_E and their t_dec-derived chamber stiffness K_LV (14). A minor difference between the study of Little et al. and this study is that the entire E wave is used as input to the model, instead of only the t_dec (2 points of the entire contour). This has the advantage of using more information to fit the model-predicted contour, thereby providing a more powerful analysis. On the other hand, it has the shortcoming of requiring computation that cannot be simply performed by inspection of the Doppler contour. Testing the analogous (k_E-average stiffness relation) hypothesis in humans, extending it to include E-wave patterns having an inflection point in the deceleration portion, and being able to consider the early rapid-filling and the late, postdiastatic (passive) stiffness phases of diastole represent the focus of the present in vivo work. One key result (Fig. 4) relates invasive and noninvasive determination of (average) chamber stiffness in vivo in humans (n = 131). Figure 4 is the analog of the experimentally determined relationship observed in a canine model (n = 8) by Little et al. (17). In agreement with the canine study, a linear correlation was found.

The MBIP strategy eliminates the need to quantify the transmitral Doppler contour or determine its attributes (e.g., t_acc, t_dec, E-to-A ratio) "by eye." Figure 2 illustrates typical E-wave images demonstrating excellent agreement between model prediction and the clinical velocity contour. Furthermore, the predicted linear correlation between the noninvasively determined parameter k_E and the invasively determined (P/V)_avg, between k_E and K_LV, between k_E and early rapid-filling stiffness, and between k_A and postdiastatic stiffness has been validated in a large in vivo sample having a broad distribution of Doppler flow velocity patterns.

Although a strong correlation is observed indicating that k_E is a clinically viable noninvasive estimate of average chamber stiffness and that k_A is an excellent estimate of passive chamber stiffness, some limitations remain.

Limitations. The limitations encountered in this study fall into two categories: methodological and conceptual.

Conclusion. This study validates the model-predicted linear relationship between the MBIP-determined, E-wave-based k_E and the simultaneous, invasively obtained (P/V)_avg. Our results in humans are in concert with the work of Little et al. (17) in a canine model. We found the relationship to be k_E = 1,402 · (P/V)_avg + 59.2 with good correlation (r = 0.84). Such validation has not been previously performed in a large sample (n = 131) of human subjects having normal LVEF. The model-predicted linear relationship of k_E to (P/V)_E was also validated and observed to be k_E = 229.0 · (P/V)_E + 112 with good correlation (r = 0.80). In addition, the prediction that postdiastasis A-wave-based k_A is linearly related to (P/V)_PD was validated: k_A = 1,640 · (P/V)_PD  8.40 with excellent correlation (r = 0.89). These results establish new tools by which average, early rapid-filling, and passive (postdiastasis) chamber stiffness may be noninvasively attained on a beat-by-beat basis in vivo. The findings underscore the role that all hearts play as mechanical suction pumps in early diastole and strengthen our understanding of the relationship between Doppler echo-derived parameters and their in vivo physiological analogs.