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Am J Physiol Heart Circ Physiol 291: H2942-H2949, 2006. First published August 4, 2006; doi:10.1152/ajpheart.00259.2006
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Frequency-based analysis of the early rapid filling pressure-flow relation elucidates diastolic efficiency mechanisms

Yue Wu and Sándor J. Kovács

Cardiovascular Biophysics Laboratory, Washington University, St. Louis, Missouri

Submitted 13 March 2006 ; accepted in final form 25 July 2006


   ABSTRACT
TOP
 ABSTRACT
MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 GRANTS
 REFERENCES
 
Stiffness- and relaxation-based diastolic function (DF) assessment can characterize the presence, severity, and mechanism of dysfunction. Although frequency-based characterization of arterial function is routine (input impedance, characteristic impedance, arterial wave reflection), DF assessment via frequency-based methods incorporating optimization/efficiency criteria is lacking. By definition, optimal filling maximizes (E wave) volume and minimizes "loss" at constant stored elastic strain energy (which initiates mechanical, recoil-driven filling). In thermodynamic terms, optimal filling delivers all oscillatory power (rate of work) at the lowest harmonic. To assess early rapid filling optimization, simultaneous micromanometric left ventricular pressure and echocardiographic transmitral flow (Doppler E wave) were Fourier analyzed in 31 subjects. A validated kinematic filling model provided closed-form expressions for E wave contours and model parameters. Relaxation-based DF impairment is indicated by prolonged E wave deceleration time (DT). Optimization was assessed via regression between the dimensionless ratio of 2nd (Q2) and 3rd flow harmonics (Q3) to the lowest harmonic (Q1), i.e., (Q2/Q1) or (Q3/Q1) vs. DT or c, the filling model’s viscosity/damping (energy loss) parameter. Results show that DT prolongation or increased c generated increased oscillatory power at higher harmonics (Q2/Q1 = 0.00091DT + 0.09837, r = 0.70; Q3/Q1 = 0.00053DT + 0.02747, r = 0.60; Q2/Q1 = 0.00614c + 0.15527, r = 0.91; Q3/Q1 = 0.00396c + 0.05373, r = 0.87). Because ideal filling is achieved when all oscillatory power is delivered at the lowest harmonic, the observed increase in power at higher harmonics is a measure of filling inefficiency. We conclude that frequency-based analysis facilitates assessment of filling efficiency and elucidates the mechanism by which diastolic dysfunction associated with prolonged DT impairs optimal filling.

diastolic function; Fourier analysis; Doppler echocardiography; cardiac catheterization; efficiency

THE EPIDEMIOLOGIC FACTS THAT heart failure is the most frequent single diagnosis-related group for hospitalization (10, 14) and that diastolic heart failure, which comprises up to 50% of heart failure admissions, has reached epidemic proportions (14) have provided added impetus for detailed characterization of all aspects of the physiology of diastole and methods by which diastolic dysfunction can be diagnosed. Hence, investigation of how the heart works when it fills (diastolic function) is proceeding on multiple theoretical and experimental fronts spanning six orders of magnitude, involving molecular, genetic, and cellular biological as well as organ system mechanisms (14). Consensus has developed that for phenotypic characterization of diastolic function (DF) and diastolic dysfunction (DD), the most appropriate measures, in addition to filling (stroke) volume and ejection fraction, are chamber stiffness and relaxation (1, 2, 14, 17).

The concept of pathological stiffness in cardiovascular physiology is familiar in the context of properties of the arterial circulation (5, 19, 20). In addition to stiffness of the arterial tree, quantitatively characterized in terms of pressure (P) and flow (Q) (or dV/dt), frequency-based characterization of the periphery achieved by Fourier analysis of the oscillatory pressure and flow waveforms is standard (5, 19, 20, 24). Despite the successful application of Fourier methods in characterizing arterial stiffness, little effort has been made to use similar methods to study the stiffness-related mechanisms of DF or DD. To overcome this limitation, we previously introduced the concept of impedance analysis as a method for DF characterization (28). In precise analogy to frequency-based analysis of pressure and flow applied to the peripheral circulation, our method determined input impedance, characteristic impedance, and the reflection coefficient of the left ventricle (LV) by analysis of pressure and flow during early rapid filling (Doppler E wave). As anticipated from theoretical and physiological arguments based on the role of the LV as a suction pump, we found that the LV generates a negative reflection coefficient during the E wave due to the phase difference between pressure and flow. Moreover, filling in normal hearts operates very near the optimal phase angle (180°) that minimizes the reflection coefficient (28). The existence of an optimal phase angle (between P and Q) implies optimization between the dominant (1st harmonic) pressure wave and flow wave during early rapid filling. This is equivalent to maximizing filling volume at a suitably low pressure for a given amount of available stored elastic energy. In support of the hypothesis that normal filling is close to optimal, we found that subjects with DD (elevated LV end-diastolic pressure or delayed relaxation) demonstrated decreased filling efficiency in terms of a phase shift of their pressure-flow relations relative to normal by having a higher reflection coefficient phase angle, which deviates from its optimal value of 180° and causes a higher input impedance (28).

In this study, we applied frequency-based analysis to LV pressure (LVP) and flow to further characterize the efficiency-related mechanisms of the pressure-flow relation in early diastole. In analogy to methods previously utilized for characterization of arterial stiffness, although never previously applied for the characterization of DF, we analyzed the relationship between Fourier amplitudes as a function of increasing harmonics and measures of DD such as E wave deceleration time.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 MATERIALS AND METHODS
RESULTS
 DISCUSSION
 GRANTS
 REFERENCES
 
Data acquisition. Data acquisition has been previously described (28). Briefly, informed consent for participation was obtained from all subjects in accordance with Washington University Medical Center Human Studies Committee (Institutional Review Board) guidelines. Thirty-one subjects (age 56 ± 9 yr, ejection fraction 62 ± 18%, 19 men) undergoing elective diagnostic cardiac catheterization had simultaneous limb lead II of the electrocardiogram (ECG), LVP, and transmitral flow velocities recorded. Transthoracic transmitral flow was recorded via an Acuson Sequoia C256 echocardiographic imager (Mountain View, CA) equipped with a 2-MHz transducer, using pulsed Doppler echocardiography. In accordance with American Society of Echocardiography criteria (22), the sample volume (3 mm) was located at the mitral leaflet tips (using the four-chamber view) and the direction of insonification was aligned as parallel as possible to the color Doppler-determined transmitral flow direction. Baseline filters were set at the lowest setting. Micromanometric LVP contours from a Millar catheter (model SPC-560-1; Millar Instruments, Houston, TX) were fed to a catheterization laboratory amplifier (Quinton Diagnostics, Bothell, WA) for on-site monitoring, and the signal was also acquired by a Leycom multichannel control unit (model Sigma 5 DF; CD Leycom, Zoetermeer, The Netherlands). The precision of the pressure signal is 0.5 mmHg. Pressure, volume, and ECG signals were analog-to-digital converted at 200 Hz and saved to an accompanying data acquisition system. Pressure and ECG were fed into the physiological auxiliary port of the echocardiography imager for temporal alignment with transmitral flow.

Echocardiographic image analysis. Transmitral flow Q(t) (cm3/s), i.e., rate of LV volume increase, was defined as the product of Doppler E wave velocity (cm/s) and constant effective mitral valve area (cm2), i.e., velocity x area. To eliminate uncertainties related to Doppler E and A wave superposition, only subjects with a diastatic interval or minimal overlapping were considered. For each subject, three beats were selected for analysis, and the results were averaged to get time-invariant E wave parameters. Transmitral flow (E waves) and pressure contours were temporally aligned and coupled relative to a fiducial square-wave marker in the pressure channel. To achieve additional precision in temporal alignment of pressure and flow data, cardiac cycle timing events were overdetermined by utilizing both the QRS and mitral valve opening and closing features on the echo image. E wave (flow) data analysis and processing were done using an IBM personal computer running a custom-made LabVIEW 6 (National Instruments, Austin, TX) program, utilizing a previously validated model-based image processing (MBIP) method for determination of LV diastolic function (9, 13). The MBIP method models the kinematics of filling in analogy to the motion of a simple harmonic oscillator (SHO). The Levenberg-Marquardt algorithm is used to minimize the difference between the model-predicted velocity and the actual E wave contour to generate the three SHO parameters: c (damping coefficient), k (spring constant), and xo (initial displacement). These parameters uniquely characterize the rate of decay, frequency, and amplitude of the E wave. Compared with other echocardiographic indexes that only rely on one or two points of the entire E wave contour, the MBIP method utilizes nearly all the data points on the contour and concomitantly generates a quantitative measure of goodness of fit (9, 21) whose mean squared error is <2 cm/s. In addition, the three parameters have well-established physiological analogs: chamber stiffness as k (16), chamber viscoelasticity/relaxation as c (6), and volumetric load as xo (15). Related indexes include peak atrioventricular pressure gradient as kxo (3) and stored elastic strain energy to power recoil as 1/2kxo2 (18).

Temporal synchronization accuracy. Because of the rapid increase in pressure at mitral valve closing, i.e., the start of isovolumic contraction, the timing of mitral valve closing (end of A wave) can be easily discerned from P(t) and is synchronized to the end of the A wave. The intraobserver variation for alignment of flow to pressure in a given cardiac cycle was determined by comparison of the precision of repeated alignments for the entire set of pressure/flow data over a time interval of two or more weeks. For data sets (E wave durations) with a typical duration of 250–300 ms, the accuracy was within 10 ms, yielding an intraobserver variation of <5% for pressure-flow alignment for the same beats.

Data processing. As previously described ( 28 ), simultaneously acquired and digitized pressure and flow data were subjected to Fourier transform (FT) by the following equations:

Formula 1(1)
where N is the number of data points of digitized P(t) and Q(t), i.e., P(n) and Q(n), respectively, from mitral valve opening to the end of the E wave at which the mitral valve closes; {omega}m = 2{pi}mfs/N; m, n = 0, 1, ..., N – 1; and fs is the data acquisition sampling rate, which is 200 Hz in our data acquisition setting. Traditionally, harmonic frequencies are defined as fm = {omega}m/2{pi}. Specifically, f1 is called the fundamental frequency and is defined as fs/N.

Prediction and analysis. Considering the concept of work in mathematical terms {defined as: W = {int}[P(t) –PLA(t)]·Q(t)dt, where P(t) is LVP, PLA(t) is left atrial pressure (LAP), and Q(t) is transmitral flow rate}, it can be shown that optimal efficiency is achieved when all work is delivered at the fundamental (n = 1) frequency, with the approximation that the direct current DC components of P(t) and PLA(t) are indistinguishable (see DISCUSSION). Thus, for a ventricle using stored elastic strain energy to initiate recoil-driven filling, it follows that the optimal state is achieved if all oscillatory components reside at a single (1st) harmonic because of the orthogonality property of the sine function. Hence, whether an oscillatory pressure component at a certain frequency {omega}n, [P({omega}n)sin({omega}nt)], does effective work on an oscillatory flow component at another frequency {omega}m, [Q({omega}m)sin({omega}mt)] depends on their relative frequencies, i.e., the effective work is P({omega}n)·Q({omega}m)·{delta}m,n, where {delta}m,n is defined as

Formula 2(2)

To achieve optimal filling, we predict that higher harmonics (n > 1) should be negligible or close to zero so that the largest possible contribution of the oscillatory pressure and flow component resides at the lowest frequency, therefore maximizing {int}[P(t) – PLA(t)]·Q(t)dt for the lowest value of n. The FT of a sinusoid at a fixed frequency yields a single component at that frequency. FT of an oscillatory waveform that deviates from a pure sine wave will generate components of higher frequency than the fundamental, typically with diminishing amplitudes as a function of increasing frequency. The amplitude of each component can be used to compute the amount of oscillatory power at that frequency (19). To evaluate the contribution of the nondominant components to the oscillatory power, we normalize the harmonic power relative to the 1st harmonic and define the harmonic power factor in the form of Qn/Q1 and Pn/P1, where Pn and Qn are the nth harmonic amplitudes of pressure and flow, respectively, and P1 and Q1 are the 1st harmonic amplitudes of the pressure and flow waveforms, respectively.

By determining harmonic power factors above the fundamental (n > 1), we can determine efficiency by regressing Qn/Q1 vs. E wave deceleration time or the kinematic model-derived damping constant c. Hence, we can assess whether a Doppler E wave is indeed optimized in terms of its dependence on oscillatory power.


   RESULTS
TOP
 ABSTRACT
 MATERIALS AND METHODS
 RESULTS
DISCUSSION
 GRANTS
 REFERENCES
 
A typical synchronized pressure and flow tracing as a function of time is shown in Fig. 1A. The Fourier analysis results for the P and Q waveforms of early rapid filling, i.e., amplitudes of DC, 1st to 7th harmonics are shown in Fig. 1B. There is no physiological reason to consider frequency content beyond 10 Hz (beyond the 3rd harmonic) for the E wave; therefore, we set the cutoff frequency at 10 Hz (28). For clarity, in Fig. 1, C and D, we plotted the harmonic components of P and Q from the 1st to 7th harmonics. As anticipated (28), we observed that the lower the harmonic, the higher the amplitude, and from the 5th and greater harmonics, the amplitude contributed <5% of the total oscillatory amplitude. Accordingly, the 5th and greater harmonics were deemed negligible. E waves with longer DTs were associated with broadening of the oscillatory amplitude spectrum at higher harmonics, as shown in Fig. 2. The relationship between increased values for the damping parameter c and the associated manifestation of more oscillatory power at higher harmonics is shown in Fig. 3. Availability of the Fourier components permits quantitation of the oscillatory power ratio, defined as power at different harmonics: W({omega}n) = P({omega}n)·Q({omega}n). The relative total power at higher harmonics in terms of the ratio of the 2nd and 3rd harmonic power to the 1st, i.e., (P2·Q2)/(P1·Q1) and (P3·Q3)/(P1·Q1) are shown in Figs. 4 and 5. Using the average value of P2/P1, P3/P1, Q2/Q1, and Q3/Q1 in Table 1, we calculated that an ideal E wave is 12% more efficient than the whole subject set (see DISCUSSION).


Figure 1
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  		Fig. 1. A: typical simultaneously acquired, digitized waveforms of left ventricular pressure (LVP) and transmitral flow as a function of time for 1 cardiac cycle having a duration of 900 ms, equivalent to a heart rate of 66 beats/min. The E wave contour is obtained via model-based image processing of the Doppler E wave image. Only the ~300-ms interval from mitral valve opening (MVO) to diastasis is subjected to Fourier analysis. FT, Fourier transform; MVC, mitral valve closing. B: FT results for the ~300-ms early rapid filling segment shown in A are shown as (normalized) Fourier amplitude vs. frequency (f1–f7). Note the rapid decline in amplitude for increasing harmonics. The decrease of flow harmonic amplitude at higher harmonics relative to the first is faster than that of LVP. P, pressure; Q, flow. C: LVP and the first 7 harmonic components plotted on the same scale. D: transmitral E wave and the first 7 harmonic components. See text for details.

 

Figure 2
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  		Fig. 2. The relation between the 2nd and 3rd oscillatory flow harmonics relative to the fundamental (Q1) and E wave deceleration time (DT). Longer DT indicates worse diastolic function. Q1, Q2, and Q3 are the E wave Fourier amplitudes at the 1st (fundamental), 2nd, and 3rd harmonics, respectively. Note r values and monotonic increase in amplitude ratios with prolongation of DT, indicative of less efficient filling. See text for details.

 

Figure 3
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  		Fig. 3. The relation between the 2nd and 3rd oscillatory flow harmonics relative to the fundamental (Q1) and kinematic filling model-derived damping constant c. We expected relationships similar to those of Fig 2. Larger values of c account for longer DT and indicate worse diastolic function. Q1, Q2, and Q3 are the E wave Fourier amplitudes at the 1st (fundamental), 2nd, and 3rd harmonics, respectively. Note monotonic increase in amplitude ratios with increased values for c, indicative of less efficient filling. Note r values in Fig. 3 are higher than those in Fig. 2. See text for details.

 

Figure 4
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  		Fig. 4. The power ratio defined by P·Q at the 2nd and 3rd harmonics relative to the 1st harmonic is plotted as a function of DT. Relatively greater power at higher harmonics (2nd and 3rd) is positively correlated with DT and indicates decreased efficiency. The regression relations are (P2·Q2)/(P1·Q1) = 0.01213 + 0.00094DT, r = 0.58, and (P3·Q3)/(P1·Q1) = –0.01233 + 0.00038DT, r = 0.54. See text for details.

 

Figure 5
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  		Fig. 5. The power ratio defined by P·Q at the 2nd and 3rd harmonics relative to the 1st harmonic is plotted as a function of the kinematic filling model-derived damping constant c. Relatively greater power at higher harmonics (2nd and 3rd) is positively correlated with c and indicates decreased efficiency. The regression relations are (P2·Q2)/(P1·Q1) = 0.0751 + 0.0062c, r = 0.74, and (P3·Q3)/(P1·Q1) = 0.0150 + 0.0024c, r = 0.66. Note r values in Fig. 5 are higher than those in Fig. 4. See text for details.

 

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  		Table 1. Study group characteristics

 

   DISCUSSION
TOP
 ABSTRACT
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
GRANTS
 REFERENCES
 
Consideration of the heart as a pump indicates that it is ~15–18% efficient in mechanical terms when the area of the pressure-volume loop (external work per cardiac cycle) is compared with the potential (chemical) work (arteriovenous O2 difference) extracted from its arterial supply (20). Although several nonlinear and one linear model of diastolic filling have been proposed (15, 21), no previous work addresses the efficiency of filling (diastole) by itself, in either mechanical or thermodynamic terms. Because the kinematic model and other nonlinear models (21) of filling accurately predict E wave contours, the modeling paradigm itself provides a method by which filling efficiency can be characterized. The essential feature of kinematic modeling of filling is the requirement that the model account for the role of the LV as a mechanical suction pump (dP/dV < 0 at mitral valve opening and briefly thereafter). Filling is viewed as being analogous to a spring (stored, end-systolic elastic strain in tissue) recoiling at mitral valve opening, overcoming the effects of inertia (blood/tissue) and resistance (viscous effects) (9, 13). This paradigm yields a linear, invertible filling model that obeys the kinematics of a damped SHO and exploits its parameters. The model determines unique physiological parameters (i.e., solves the "inverse problem" of filling) by using the contour of the clinical Doppler transmitral E wave as input (9, 13). The physics of SHO motion and the availability of solutions for velocity (time derivative of displacement) as a function of time in closed algebraic form naturally lead to a frequency-based analysis of filling. An idealized, undamped spring has a natural (fundamental) frequency. If the motion is damped (i.e., presence of viscosity), Fourier analysis of the velocity necessarily includes higher oscillatory components (harmonics) at integer multiples of the fundamental frequency. In the absence of damping, all the oscillatory energy is delivered at the fundamental frequency (1st harmonic). In the presence of increasing amounts of damping, the 2nd and higher harmonic components become manifest, and a greater portion of the total oscillatory (potential) energy is distributed among the higher harmonic components. Accordingly, by both kinematic and thermodynamic criteria, an optimal E wave generated by the rules of SHO motion and driven by a fixed amount of stored elastic energy (1/2kxo2 in kinematic model terminology), maximizes filling volume in the time available at physiological (low) pressures. In the language of kinematic modeling (SHO motion), such an ideal E wave corresponds to undamped (lossless) motion, equivalent to a negligible value for the damping constant (i.e., c = 0). From the perspective of optimization of efficiency (or maximization of work) in frequency terms, the existence of oscillatory power at frequencies higher than the fundamental is the sine qua non of inefficiency, because the atrioventricular pressure gradient is dominated by the 1st harmonic, and sinusoidal functions are orthogonal. The causal, physiological consequences are that a given harmonic of the pressure waveform does work only on the same harmonic of the flow waveform (E wave; see below for mathematical details) (19, 20). The concept of power optimization can be appreciated by comparing an ideal E wave and a damped E wave having the same filling volume. The ideal E wave Qi(t) is expressed as Qi({omega}1)·sin({omega}1t), and the damped E wave Qd(t) is expressed as Qd({omega}1)·sin({omega}1t) + Qd({omega}2)·sin({omega}2t) + Qd({omega}3)·sin({omega}3t) + ..., where Qi({omega}1), Qd({omega}1), Qd({omega}2), and Qd({omega}3) are the harmonic components of each. Qi({omega}1)= Qd({omega}1) + Qd({omega}2) + Qd({omega}3), since the filling volume is the same. The amount of work to generate the ideal E wave is Wi = P1·Qi({omega}1). To generate the damped E wave, the actual amount of work required is Wactual = P1·Qd({omega}1) + P2·Qd({omega}2) + P3·Qd({omega}3). In quantitative terms the optimization can be expressed as the difference between ideal work and actual work, divided by actual work, i.e., power optimization {eta} = (Wi – Wactual)/Wactual. With the data from Table 1, the power optimization {eta} computed according to the above expression is 12% for an ideal E wave compared with the average actual E waves for the entire group, i.e., for the same amount of initial energy available for filling (1/2kxo2 in terms of parameterized diastolic filling), an ideal or optimized E wave delivers 12% more PQ power.

Fourier analysis of the pressure waveform during early rapid filling for normal subjects reveals that most of the oscillatory power resides at the 1st (fundamental) harmonic. In contrast, for subjects with E waves of "delayed relaxation" pattern (DT > 220 ms), a smaller component of the waveform resides at the 1st harmonic. Figure 4 shows that as DT is lengthened, the ratio of P2·Q2 and P3·Q3 to P1·Q1 rises. When viewed in terms of flow, Fig. 3 shows that the ratio of Q2 and Q3 to Q1 also rises as c, the model-derived viscoelastic/damping constant, rises. The higher r values achieved in the regression relations of Fig. 3 compared with those achieved in Fig. 2 are explained by the ability of the parameter c to accurately determine the curvilinear deceleration portion of the E wave, whereas the DT used in Fig. 2 is determined by a straight line drawn to the curvilinear deceleration portion (2, 8), which usually underestimates the actual deceleration time. This also is the reason why the r values in Fig. 5 are higher than in Fig. 4.

This property of redistribution of the relative Fourier amplitudes makes filling less efficient in hearts having greater DT or greater c. In our previous work (28), we characterized the hydraulic input impedance and reflection coefficient associated with the filling process. We found that the reflection coefficient R*, a complex number defined as the ratio of the reflected wave to the incident wave in terms of amplitude ratio and phase difference, can be expressed as

Formula 3(3)
where ZC is characteristic impedance and Z1 is input impedance at the 1st harmonic (19, 20, 28). Reflection is optimized with a phase angle very near 180° since input impedance is minimized, i.e., the same pressure gradient generates greater flow (28). The reflection optimization with a phase angle very near 180° occurs in subjects with shorter DT (DT < 180 ms vs. DT > 180 ms) (28). Achievement of the optimal value for the reflection coefficient would require no damping during the E wave, i.e., an E wave having a perfect sinusoidal shape.

In previous work, we demonstrated the feasibility of applying Fourier analysis to the LV pressure contour during the E wave, where the pressure at E wave onset and termination numerically differs. This discontinuity between the beginning and the end of the cyclic pressure signal introduces a small change in the spectrum, especially at high frequencies. This discontinuity accounts for the higher input impedances observed at higher harmonics, i.e., Z3 > Z2 > Z1 (27), where Zn is defined as Zn = P({omega}n)/Q({omega}n). However, the dominant component of the pressure oscillatory power resides at the fundamental harmonic. Even in the presence of the discontinuity between beginning and end of the pressure signal, previous work has shown that the mean squared error between the raw (discontinuous) pressure signal compared with a reconstructed pressure signal using only the first three harmonics is only 1.5% (28); hence, the presence of the discontinuity has no discernible effect on the results. Moreover, the largest discrepancy between the raw pressure signal and the reconstructed first three harmonics is at the beginning and end of the transform interval corresponding to the early and late parts of the E wave, which have small flow. This results in a negligible difference to the calculation of overall work during the flow interval. To optimally utilize the available oscillatory pressure, the response of flow at the very same frequency is desired. Ideally, the transmitral flow contour should have the form of a perfect sine wave. However, the observed pressure has less oscillatory power at the 1st fundamental harmonic compared with ideal flow, as shown in Fig. 1A. Recall that it is the pressure gradient between the left atrium (LA) and LV that drives transmitral flow (7, 11, 26). It is possible that Fourier analysis of the gradient, rather than LVP, would cause a systematic offset for the energy at the fundamental frequency and more closely relate flow harmonics to pressure gradient harmonics. Based on the known features that characterize the temporal behavior of the AV gradient (7, 11, 26) and constrained by the availability of only the LVP during diagnostic catheterization, rather than both LVP and LAP, and the knowledge that during the E wave the atrium is passive and is not a harmonic source, the expression for P-Q work is as follows. For mathematical simplicity, we have used the continuous Fourier transform. The external work during the E wave is

Formula 4(4)
where P(t) is LVP, PLA(t) is LAP, and Q(t) is transmitral flow. The limits of integration are from mitral valve opening to the end of E wave. (Without loss of generality or any effect on the total work, we can expand the limits of integration from –{infty} to +{infty} and pad with 0 beyond the E wave duration.) The pressure gradient between the LV and LA has been previously characterized via modeling (11, 26) and experimentally (7). After mitral valve opening, early rapid filling ensues and the pressure gradient reaches its maximum, then reduces to zero near the peak of the E wave, and then reverses sign to decelerate flow. The (passive) atrial pressure contour is similar in shape to the LV pressure contour, but with a small phase lag (11, 26). Hence, the contour of the AV pressure gradient is comparable in amplitude, duration, and frequency content to LVP during E wave (11). The dominant harmonics of the LVP spectrum must therefore be very similar to the dominant harmonics of the similarly shaped AV pressure gradient (11), and the dominant FT term of the AV gradient waveform will also reside at the 1st harmonic. These similarities permit approximation of the AV gradient spectrum as the LVP spectrum to the leading order. The work W is well approximated as

Formula 5(5)
where PAVR is the mean LAP. Putting the second term aside and considering the Fourier and inverse Fourier transform of P(t) and Q(t),

Formula 6A(6a)

Formula 6B(6b)
By substituting Eq. 6a into Eq. 5, we obtain

Formula 7(7)
We change the notation {omega}' to –{omega}' in the Q*({omega}') portion to get

Formula 8(8)
We bring Q*(–{omega}') and d(–{omega}') out of the {int}dt integration, since they do not explicitly depend on t:

Formula 9(9)
The {int}ei({omega} – {omega}')tdt = 2{pi}{delta}({omega} – {omega}'), because integration defines the delta function (29), therefore, we get

Formula 10(10)
By applying the property of the delta function, we calculate the second integral:

Formula 11(11)
Converting from continuous FT to discrete FT, we get

Formula 12(12)
Since the average LAP (PAVR) is the same as the DC component (average) of LVP [P*(0)] during E wave, i.e., P*(0) = PAVR, the second and third terms canceled out. Hence, the work done is

Formula 13(13)
In summary, we use P(t) – PAVR to approximate the AV gradient, i.e., we approximate LAP to first order during E wave as a constant (PAVR). Recall that during the E wave, the LA and the blood are both passive elements and do not introduce any active frequency components into the AV gradient. Hence, the expression for work can be reasonably approximated by the frequency content of LVP and passive flow, responding to the time-varying pressure.

When summarized, these arguments are in concert with previous results attained using other methods (18, 23, 28). These results provide further physical justification why an undamped sine wave-shaped E wave contour is desirable in early rapid filling.

Limitations. In concert with convention, and for simplification, we assumed a constant effective mitral valve area (cm2) by equating volumetric flow rate (cm3/s) with E wave velocity (cm/s). This has the potential effect of introducing a small, systematic shift to the amplitudes of all frequency/harmonic terms of the volume waveform. Because MRI and echo measurement of normal mitral valve area (4) show that it responds passively to the phasic nature of the pressure-flow relation, the small change affecting volume due to time variation of mitral valve area is always in phase with the pressure-flow relation and should not significantly affect the observed trend in the power spectrum as shown in Figs. 2 and 3.

In conclusion, we characterized the efficiency of LV filling via frequency-based analysis of simultaneous LVP and transmitral flow (Doppler E wave) waveforms. Relative to the amplitude of the fundamental harmonic, which contains the most (oscillatory) power, and relative to controls, subjects having the delayed relaxation pattern of transmitral flow had an increased fraction of their oscillatory power reside at higher harmonics. The presence of oscillatory power at higher frequencies is the sine qua non of decreased filling efficiency, which manifests as a decrement in E wave filling volume (relative to normal). The results underscore the potential value of filling efficiency-derived indexes as physiology-based measures of diastolic function.


   GRANTS
TOP
 ABSTRACT
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 GRANTS
REFERENCES
 
This work was supported in part by the Whitaker Foundation, National Heart, Lung, and Blood Institute Grants HL-54179 and HL-04023, the Heartland Affiliate of the American Heart Association (0310021Z), the Barnes-Jewish Hospital Foundation, and the Alan A. and Edith L. Wolff Charitable Trust.


   ACKNOWLEDGMENTS
 
We appreciate the contribution of sonographer Peggy Brown in acquiring high-quality echocardiographic data and the assistance of the staff of the Barnes-Jewish Hospital Cardiac Catheterization Laboratory in high-fidelity LVP recording.


   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
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 GRANTS
 REFERENCES
 

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