Determination of Left Ventricular Chamber Stiffness From the Time for Deceleration of Early Left Ventricular Filling
Background A noninvasive measure of left ventricular (LV) chamber stiffness (KLV) would be clinically useful. Our theoretical analysis predicts that KLV can be calculated from the time for deceleration of LV early filling (tdec) by
where ρ=density of blood, L=effective mitral length, and A=mitral area.
Methods and Results We tested this hypothesis in eight conscious dogs instrumented for measurement of LV pressure (P) with use of a micromanometer and volume (V) with use of sonomicrometers. KLV was determined as the slope of the late diastolic portion of the LV P-V loop. KLV was varied from 0.99±0.35 to 2.58±0.92 mm Hg/mL with use of three graded doses of phenylephrine. We assumed that ρ=1.0 and that L/A=3.4. Thus, we predicted that KLV=(0.08/tdec)2 . The LV filling pattern was determined from the derivative of LV volume (dV/dt). tdec was measured from peak early filling to the end of early filling. Predicted KLV and actual KLV were closely correlated (r=.94, SEE=0.06 mm Hg/mL, P<.05). The regression line was close to the line of identity (slope=0.95, intercept=0.13 mm Hg/mL). Dobutamine did not alter the relation between tdec and KLV. tdec determined from the mitral valve flow velocity measured with Doppler echocardiography correlated well with that measured by dV/dt (r=.89, P<.01) but was 0.02 seconds longer. KLV-calculated tdec from the corrected Doppler tdec provided a good estimate of measured KLV (r=.75, SEE=0.5 mm Hg/mL, P<.01).
Conclusions LV chamber stiffness can be determined from the time for deceleration of LV early filling, which can be measured noninvasively.
The pattern of left ventricular (LV) filling determined by Doppler echocardiography or nuclear angiography is used to noninvasively evaluate LV diastolic performance. Although abnormal LV diastolic properties alter the pattern of LV filling, it has not been possible to measure LV chamber stiffness noninvasively.1
Conditions associated with increased LV stiffness are associated with a more rapid rate of deceleration of early filling and a shorter time for this deceleration. Based on theoretical analyses, Thomas et al2 3 and Flachskampf et al4 have predicted that the rate of early flow deceleration should vary directly with atrial pressure and mitral valve area and inversely with the combined stiffness of the left atrium (LA) and LV. We recently observed that during the time of early filling deceleration, LA pressure is relatively constant.5 Thus, during this period the apparent stiffness of the LA is very low. Our previous theoretical analysis5 predicted that the early filling deceleration time (tdec) should be proportional to the inverse square root of LV stiffness or 1/. Our observations in conscious dogs during the progressive development of pacing-induced heart failure were consistent with this prediction.5 This suggests that the chamber stiffness of the LV could be calculated from the time for early filling deceleration (tdec). This has practical clinical importance, since tdec can be measured noninvasively with the use of Doppler echocardiography.
Before tdec can be used to measure KLV, several issues remain. First, the proportionality constant between tdec and KLV must be evaluated. Our initial theoretical analysis predicted that
However, in this derivation, the numerical constant () depends on the time during flow deceleration that the integral of dV/dt is evaluated. If KLV is to be calculated from tdec, the proportionality constant must be known exactly. In “Appendix 1,” we provide a new analysis that avoids this problem. Second, the ability of tdec to predict KLV must be evaluated prospectively. Finally, tdec in our experimental studies is measured from analysis of the derivative of LV volume (dV/dt). In clinical studies, tdec is determined with the use of Doppler mitral valve flow velocity. Thus, tdec measured with the use of these two techniques must be compared. This study was undertaken to address these issues in conscious dogs.
Healthy mongrel dogs weighing between 24 and 37 kg were instrumented with use of the technique that we have described previously.6 7 Anesthesia was induced with xylazine (2.0 mg/kg IM) and sodium pentobarbital (6 mg/kg IV) and maintained with halothane (1% to 2%). The pericardium was opened through a left thoracotomy. Micromanometer pressure transducers (Konigsberg Instruments) and polyvinyl catheters for transducer calibration (1.1 mm ID) were inserted into the LV through an LV apical stab wound and into the LA through the LA appendage. Three pairs of ultrasonic crystals (5 MHz) were implanted in the endocardium of the LV to measure the anterior-posterior, septal-lateral, and base-apex dimensions.8 All wires and tubing were exteriorized through the posterior neck.
Studies were begun after full recovery from instrumentation (10 days to 2 weeks after surgery). The LV and the LA catheters were connected to pressure transducers (Sratham p23Db, Gould) calibrated with a mercury manometer. The signal from the micromanometer was adjusted to match that of the catheter. The LA micromanometer was adjusted to match LA and LV pressures at the end of long periods of diastasis.
The analog signals were recorded on an eight-channel oscillograph (Astro-Med), digitized with an on-line analog-to-digital converter (Data Translation Devices) at 200 Hz, and stored on a floppy disk memory system by use of a 386 computer system. Each data acquisition period lasted for 12 seconds, spanning several respiratory cycles.
Data were recorded with unsedated animals lying quietly in a sling. Control data were acquired after full recovery from the surgical instrumentation. In eight animals, LV chamber stiffness (KLV) was varied with the use of three graded doses of phenylephrine (approximately 2, 4, and 6 μg/kg per minute). In a separate group of seven dogs, the effect of increasing contractility and speeding the rate of relaxation was assessed by dobutamine infusion (approximately 5 μg/kg per minute).
Immediately after the acquisition of the micromanometer/ultrasonic crystal data and during similar heart rates, image-directed, pulsed-wave spectral Doppler tracings of mitral valve inflow were performed from the LV apex with the use of a Hewlett-Packard model Sonos 1500 ultrasound imaging system fitted with a dual-frequency 3.5/2.7-MHz transducer, an S-VHS video recorder, and an optical digital disk recorder in six of the animals during phenylephrine infusion. A small sample volume was placed at the mitral valve leaflet tips, and the transducer position was adjusted to align the cursor as close to perpendicular to the mitral valve annulus as possible and to maximize flow velocity and minimize spectral dispersion. Tracings were made at a 100-mm/s sweep speed, and several multiple-beat, digital cineloops were recorded for analysis.
At the conclusion of the studies, the animals were given an overdose of pentobarbital and the hearts were examined to confirm the proper positioning of the instrumentation.
Data Processing and Analysis
The stored digitized data were analyzed by computer algorithm developed in our laboratory. Hemodynamic values in each dog were obtained by averaging the data obtained during the steady-state recording spanning several respiratory cycles. End diastole was defined as the relative minimum of LV pressure after the A wave. If this was not clearly apparent, the peak R wave of the surface ECG was used to indicate end diastole. End ejection was defined as the time of minimum dP/dt. The LV volume was calculated as a general ellipsoid using the equation VLV=(π/6) · DAP · DSL · DLA, where DAP, DSL, and DLA are the anterior-posterior, septal-lateral, and long-axis dimensions. This method of volume calculation gives a consistent measure of LV volume (r>.97, SEE<2 mL) despite changes in LV loading conditions and chamber configuration.8 9 10 11
Ventricular filling patterns were measured with use of the time derivative of LV volume (dV/dt).6 7 The characteristics of these patterns were evaluated by determining the maximum rates of early diastolic LV filling (peak E) and atrial filling (peak A). The deceleration time of early diastolic LV filling (tdec) was defined as the time interval between maximum rate of early diastolic LV filling and the zero intercept of the deceleration slope, as previously described.5 When atrial filling occurred before early diastolic LV filling decelerated to zero, the slope was linearly extrapolated to the zero line to obtain tdec. The deceleration rate of early diastolic LV filling was calculated as peak E divided by tdec (Fig 1⇓).
The average LV chamber stiffness during diastole was obtained by dividing the change of the pressure from the time of minimum LV pressure to end-diastolic pressure (ΔPLV) by the change of the volume during this period.
The time constant of the isovolumic fall in LV pressure was determined by fitting the steady-state data from end ejection to mitral valve opening to the equation P=PAe−t/T+PB, where t is the time from end ejection, T is the exponential time constant of relaxation, and PA and PB are constants determined by the data. The time derivatives of LV pressure and volume were calculated with use of the five-point gaussian technique.12
Analyses of the Doppler tracings were performed off-line by a single observer who was blinded to the results of micromanometer/ultrasonic crystal data analyses. Tracings were reloaded into the Hewlett-Packard Sonos 1500 ultrasound system from the optical-digital disk and analyzed with use of software provided with the system. The three best tracings were analyzed, and the results were averaged. Data were excluded if the mitral flow pattern was not adequately defined to measure tdec. Early diastolic flow deceleration time was measured as the time from the peak early filling velocity to termination of early filling. In tracings in which low velocity filtration of Doppler signals or the onset of late (atrial) filling obscured the termination of early diastolic flow, the flow velocity slope was extrapolated to the baseline.
Time Course of LA Pressure
We evaluated the time course of LA pressure during the time of early filling deceleration by dividing this period into four quarters.
Changes in the variables with use of three doses of phenylephrine and dobutamine were assessed using repeated-measures ANOVA. If significant differences were present, paired comparisons between values at control and values after injection of phenylephrine and dobutamine were performed with use of the Student-Newman-Keuls test. A probability level of <.05 was accepted as significant. Values are expressed as mean±SD.
Effects of Phenylephrine on Hemodynamics
The effects of alterations in load with the three graded doses of phenylephrine on hemodynamic parameters are shown in Table 1⇓. Phenylephrine did not significantly change heart rate. The graded doses of phenylephrine produced a progressive increase in LV end-diastolic pressure from the control of 8.3±3.8 to 15.1±6.0, 18.6±5.2, and 25.2±6.1 mm Hg (P<.05); LV end-systolic pressure from the control of 96.1±10.6 to 116.5±18.0, 126.6±14.8, and 147.8±14.1 mm Hg (P<.05); minimal LV pressure from the control of −1.3±3.3 to 1.1±4.4, 2.3±3.3, and 6.5±4.5 mm Hg (P<.05); and mean LA pressure from the control of 2.8±3.2 to 7.7±5.8, 11.3±5.0, and 19.5±6.0 mm Hg (P<.05). LV end-diastolic volume increased from the control of 40.6±14.0 to 44.1±14.5, 45.9±15.4, and 47.1±15.2 mL (P<.05), as did LV end-systolic volume from the control of 27.1±10.7 to 31.9±12.1, 33.4±12.2, and 36.3±12.7 mL (P<.05). However, stroke volume was almost unchanged until the second phenylephrine infusion; it then significantly decreased from the control of 13.5±3.7 to 10.8±4.4 mL (P<.05). The maximum early diastolic LA-LV gradient increased after the third phenylephrine infusion (from 4.6±1.2 to 8.6±2.3 mm Hg, P<.05), as did maximal early rapid filling (from 99±37 to 153±75 mL/s, P<.05). The tdec shortened progressively from 87±17 to 52±10 ms (P<.05).
Prediction of LV Chamber Stiffness (KLV)
A typical example of the LV pressure-volume loops, LV and LA pressures, and dV/dt recorded during the three graded doses of phenylephrine is shown in Fig 1⇑. Measured LV chamber stiffness progressively increased from 1.03±0.32 at control, reaching 2.62±0.87 mm Hg/mL at the highest dose (P<.05) (Table 1⇑). Our theoretical analysis predicts that the time for deceleration of early filling (tdec) is given by
where ρ=density of blood, L=effective mitral length, and A=mitral area (see “Appendix 1”). We assumed that ρ=1.0 g/cm3 and that L/A=3.4. Thus, we predicted that KLV=(0.08/tdec)2. Predicted KLV and measured KLV were closely correlated (r=.94, SEE=0.06, P<.01) (Fig 2⇓). The regression line was close to the line of identity (slope=0.95±0.06, intercept=0.13±0.11 mm Hg/mL).
Effect of Dobutamine
Dobutamine increased the heart rate from 108±13 to 121±11 beats per minute (P<.05) and LV dP/dtmax from 2023±121 to 2645±447 mm Hg/s (P<.05) and increased the rate of LV relaxation as indicated by a decrease in the time constant of LV pressure fall from 27.5±3.1 to 24.1±2.1 ms (P<.05) (Table 2⇓). Dobutamine did not alter tdec or KLV. The relation between predicted and measured KLV was not significantly altered by dobutamine (Fig 3⇓).
Measurement by Doppler
tdec measured from dV/dt and by Doppler determination of mitral flow were well correlated (r=.89, P<.01) (Fig 4⇓). The slope of the regression line was close to unity (1.0±0.1). However, there was an offset of 0.02 seconds. We corrected for this offset and predicted KLV from the Doppler measured tdec as
This provided a good estimate of measured KLV (Fig 5⇓), although there is more scatter than when tdec is measured from dV/dt.
Time Course of LA Pressure
The time course of LA pressure during filling deceleration is shown in Fig 7⇓. During all conditions, LA pressure did not significantly change during the first half of filling deceleration. At the end of the period, LA pressure increased. During control and the low dose of phenylephrine, there was no change three-quarters of the way through the deceleration period. There was an increase in LA pressure at this time at the two higher doses of phenylephrine.
The clinical evaluation of LV diastolic function has been a difficult challenge.1 Despite much effort, there has not been a noninvasive method to measure LV diastolic stiffness. In this study, we confirmed the theoretical prediction that the time for deceleration of early diastolic filling is determined by LV chamber stiffness. Thus, in conscious dogs, KLV can be approximated as (0.08 s/tdec)2 mm Hg/mL. In addition, we found that tdec can be measured by Doppler echocardiography. These results suggest a noninvasive method of clinically determining KLV.
As blood leaves the LA during early diastolic filling its pressure falls, and as LV relaxation and elastic recoil are completed, LV pressure begins to rise with the increase in LV volume.6 13 These effects decrease and then reverse the LA-LV pressure gradient. This decelerates and then stops the initial rapid flow into the ventricle. The magnitude of the fall in LA pressure and rise in LV pressure in early diastole depends both on the volume of the blood leaving the LA and entering the LV and on the stiffness of the LV and LA.2 3 14 However, during the time of early flow deceleration, there is rapid flow into the LA from the pulmonary veins.15 16 Thus, as we have observed previously,5 LA pressure initially remains relatively constant during early flow deceleration; this indicates that LA stiffness should not have an important influence on the deceleration of early filling.
Our theoretical analysis (see “Appendix 1”) predicts that the early filling deceleration time should be proportional to the inverse of the square root of LV stiffness or
This is analogous to the oscillation time for a spring, which is proportional to the square root of the spring’s stiffness constant. Furthermore, our analysis predicts that the proportionality constant between tdec and 1/ is
where ρ is density of blood (≈1 g/cm3), L is the effective length, and A is the effective area of blood moving through the mitral orifice. We evaluated the proportionality constant as follows: Flachskampf et al17 showed that L, the effective length of the mitral flow orifice, is approximately three times the diameter of the orifice plus the length of the leaflets. Assuming a canine mitral area of 2 cm2 and leaflet length of 2 cm, this yields L/A=3.4. Using ρ=1.0 g/cm3 and 1333 dyne/cm2=1 mm Hg results in
In patients with normal mitral functional area of 4 cm2 and mitral length of 3 cm,18 the constant would be similar, 0.07 seconds.
We evaluated this theoretical prediction that KLV can be estimated from tdec by altering LV stiffness by increasing LV afterload with phenylephrine. This increased LV diastolic volume causing the LV to operate on a steeper portion of its curvilinear diastolic P-V relation. Over the range of KLV we studied (1 to 4 mm Hg/mL), tdec measured from LV filling curve (dV/dt) provided an excellent estimate of KLV.
Our theoretical analysis predicts that tdec should be determined by KLV but not by the rate of LV relaxation, contractility, or heart rate. Consistent with this prediction, we found that although dobutamine increased LV contractility and the rate of LV relaxation, it did not alter the ability of tdec to predict KLV (Fig 3⇑).
We determined LV stiffness from the slope of mid and late diastolic portions of the LV pressure loop, beginning at the time of minimum LV pressure, when LV relaxation is nearing completion. This period spans the time of flow deceleration. Thus, the KLV we determined indicates the functional LV chamber stiffness during the time of flow deceleration and may differ from the passive or end-diastolic stiffness. The LV diastolic pressure-volume relation is exponential in shape, with increasing slope (ie, stiffness) with increasing volume. Thus, the increase in average stiffness (KLV) we measured by infusion of phenylephrine resulted from the increase in LV volume.
Can KLV be calculated from tdec when the LV stiffness is altered by pathological conditions? We previously observed that tdec, measured from dV/dt, and 1/ were linearly related as KLV increased during the development of pacing-induced heart failure.5 We reanalyzed this data, as shown in Fig 6⇓. The predicted and measured KLV are similar when KLV is varied by phenylephrine or during the induction of pacing-induced heart failure and not altered by dobutamine. In each animal, measured and predicted KLV were linearly related (r=.78±.07, SEE=0.5±0.2 mL, P<.05 in each animal). This suggests that KLV can be estimated by tdec during pathological conditions.
We used endocardial diameter gauges to measure LV volume. This technique has been extensively validated in past studies and accurately reflects LV volume under a wide variety of normal and pathological conditions.8 9 10 11 In patients, Doppler measurements of the mitral valve flow velocity are used to assess LV filling patterns. This technique has the advantage that it is noninvasive and can be repeated serially. In this study, we found that tdec measured from the Doppler mitral flow tracing correlated very well with tdec measured from dV/dt. However, tdec from the Doppler flow velocity tracing was on the average about 20 ms longer than tdec measured from dV/dt. This could be due to an underestimation of tdec by the ultrasonically measured dV/dt. However, there are several possible reasons why Doppler might overestimate tdec. First, the measurement of the Doppler signal was made with use of the outside edge of the Doppler envelope, which is the most clearly discernible. Second, Doppler peak filling velocities are not determined solely by LV chamber diastolic properties.5 The Doppler velocity profile at a single point in the inflow tract is influenced both by the LV volume change (dV/dt) and by the propagation of the inflow wave past the sample point. Third, the mitral leaflets come together during flow deceleration. The resulting decrease in mitral valve cross-sectional area would tend to delay the fall in the flow velocity, producing a longer tdec. All of these factors may contribute to the longer tdec determined by Doppler. Finally, obtaining high-quality Doppler recordings of mitral valve velocities is technically more difficult in instrumented dogs than clinically in humans.
The theoretical analysis depends on the simplifying assumption that LA pressure is relatively constant during flow deceleration. We observed previously that this is correct during the first three quarters of the period of filing deceleration.5 However, by the end of the period of tdec, LA pressure increases (see Fig 7⇓). With the higher doses of phenylephrine, LA pressure was constant only through the first half of the filling deceleration period. Increases in LA pressure during filling deceleration would be a source of potential error in the calculation of KLV from tdec. “Appendix 2” contains an evaluation of the magnitude of this error. Under control conditions, the assumption of a constant LA pressure produced less than a 5% overestimation of KLV. Under the worst case, during high-dose phenylephrine, the assumption produced up to a 14% overestimation of KLV.
Our data suggest that Doppler-derived tdec may have to be corrected by subtracting 0.02 seconds in order to predict KLV. Taking into account the size of the mitral apparatus in patients, this results in the formula for patients
The wider scatter in the Doppler-derived data may result partially from the technical difficulty in obtaining Doppler mitral valve recordings in the instrumented animals. Even if this problem were avoided, the estimation of KLV from Doppler tdec would not be accurate enough to detect small changes but should be able to distinguish changes of the order of 1 mm Hg/mL.
Our study demonstrates that early diastolic filling deceleration time decreases as LV stiffness increases. Our observations are consistent with a theoretical analysis that predicts that KLV=(0.08/tdec)2 mm Hg/mL. Furthermore, this study suggests that Doppler measurement of tdec may be a clinically useful noninvasive method to evaluate LV chamber stiffness.
By Newton’s second law, the deceleration rate of early diastolic filling is equal to the force (F) applied to the blood divided by the mass (M) of the blood:
The force producing deceleration after peak early flow is the reverse pressure gradient across the mitral valve (PLV−PLA) multiplied by the mitral valve area (A). The mass of blood is determined by density of blood (ρ) and the volume of blood in the mitral orifice, which is given by the effective area (A) multiplied by the effective length (L).14 17 Deceleration is the negative rate of change of flow velocity (−dv/dt). Flow velocity (v) is equal to the rate of change of LV volume (dV/dt) divided by the mitral area [(dV/dt)/A]. Therefore
The chamber stiffness of the LV is defined as KLV=dPLV/dV. Since v · A=dV/dt, applying the chain rule results in
Combining Equations 3 and 4,
This linear second-order differential equation has a solution of the form y(t)=a · cos(b · t), where d2v/dt2=−a · b2 · cos(bt). If we define t=0 to be at the peak of the E wave, v(0)=E, then
Since v(t) reaches zero at t=π/2, the time for early flow deceleration (tdec) is given by
Therefore, within the accuracy of the simplifying assumptions, this analysis predicts that the time for early filling deceleration should be inversely related to the square root of LV stiffness. The proportionality constant depends on the viscosity of blood (ρ) and an anatomic factor, the ratio of the effective length to the effective area of the mitral valve apparatus. This conclusion differs from our previous derivation5 only in the numerical constant (π/2 versus ).
We evaluate the proportionality constant as follows. Flachskampf et al17 showed that L, the effective length of the mitral flow orifice, is approximately three times the diameter of the orifice plus the length of the leaflets. Assuming a canine mitral area of 2 cm2 and leaflets 2 cm long, this yields L/A=3.4. Using ρ=1.0 g/cm3 and 1333 dyne/cm2=1 mm Hg results in
Rearranging results in
The analysis in “Appendix 1” requires the assumption that PLA remains constant during flow deceleration. To determine the size of the error, this assumption produces in our conclusion that KLV can be calculated from tdec. In this analysis we do not assume that PLA remains constant.
K*LA is the ratio of the change in LA pressure to the volume that leaves the LA and enters the LV during flow deceleration. This apparent LA stiffness (K*LA) would be equivalent to the true LA stiffness if the volume entering the LV was the same as the change in LA volume. However, during flow deceleration, there is rapid flow into the LA from the pulmonary veins; thus, K*LA is not the same as true LA chamber stiffness. Equation 5 becomes
Using similar logic as in “Appendix 1,” this results in
To evaluate the error in calculated KLV introduced by assuming that PLA is constant (and K*LA is zero), we evaluated the magnitude of K*LA as ratio of the change in PLA during the period of flow deceleration (from 0 to 0.75 tdec) to the flow out of the LA during this period (−∫dVLV). K*LA is negative (ie, LA pressure increased despite flow out of the LA). During control, the absolute value of K*LA was less than 0.06 mm Hg/mL. Thus, the assumption of a constant PLA caused less than a 5% overestimation of KLV. In the worst case, during the high-dose phenylephrine infusion when there was the largest change in PLA, the absolute value of K*LA was 0.37 mm Hg/mL, introducing up to a 14% overestimation of KLV.
This study was supported in part by grants from the National Institutes of Health (HL-45258 and HL-42364) and the American Heart Association (21218710385 and 930133380). We gratefully acknowledge the expert secretarial assistance of Judy McClenny, the computer programming of Ping Tan, and the technical assistance of Mack Williams and Piper Millsaps.
Presented in part at the American Heart Association Scientific Sessions, Dallas, Tex, November 1994.
- Received February 16, 1995.
- Revision received April 10, 1995.
- Accepted April 16, 1995.
- Copyright © 1995 by American Heart Association
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