# The Power-Velocity Integral at the Vena Contracta

## A New Method for Direct Quantification of Regurgitant Volume Flow

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## Abstract

*Background*—Noninvasive quantification of regurgitation is limited because Doppler measures velocity, not flow. Because backscattered Doppler power is proportional to sonified blood volume, power times velocity should be proportional to flow rate. Early studies, however, suggested that this held only for laminar flow, not for regurgitant jets, in which turbulence and fluid entrainment augment scatter. We therefore hypothesized that this Doppler power principle can be applied at the proximal vena contracta, where flow is laminar before entrainment, so that the power-times-velocity integral should vary linearly with flow rate and its time integral with stroke volume (SV).

*Methods and Results*—This was tested in vitro with steady and pulsatile flow through 0.07- to 0.8-cm^{2} orifices and in 36 hemodynamic stages in vivo, replacing the left atrium with a rigid chamber and column for direct visual recording of mitral regurgitant SV (MRSV). In 12 patients, MRSV was compared with MRI mitral inflow minus aortic outflow and in 11 patients with 3D echo left ventricular ejection volume–Doppler aortic forward SV. Vena contracta power in the narrow high-velocity spectrum from a broad measuring beam was calibrated against that from a narrow reference beam of known area. Calculated and actual flow rates and SV correlated well in vitro (*r*=0.99, 0.99; error=−1.6±2.5 mL/s, −2.4±2.9 mL), in vivo (MRSV: *r*=0.98, error=0.04±0.87 mL), and in patients (MRSV: *r*=0.98, error=−2.8±4.5 mL).

*Conclusions*—The power-velocity integral at the vena contracta provides an accurate direct measurement of regurgitant flow, overcoming the limitations of existing Doppler techniques.

Valvular regurgitation, at times life-threatening, is common in virtually all heart diseases. Accurate regurgitant volumes are critical to guide therapy, especially now that valve repair is widely available and favors early intervention before functional deterioration.^{1} Current noninvasive techniques cannot measure regurgitant flow rate directly at the lesion,^{2} ^{3} ^{4} ^{5} ^{6} ^{7} which requires the product of velocity and orifice area: Velocity×area=flow rate.

Doppler measures velocity, but orifice area is unknown and often varies throughout the regurgitant flow period.^{8} ^{9}

To overcome this limitation, we developed a new approach using the backscattered acoustic power from the Doppler signal^{10} to provide the area information we need: Velocity×power∝flow rate.

Backscattered power returning to the ultrasound transducer is a nonlinear function of hematocrit, resulting from constructive and destructive interference of sound waves returning from scatterers within the Doppler sample volume.^{11} ^{12} ^{13} For blood of a given hematocrit, however, flowing through a thin disk-like sample volume of fixed height, backscattered power in the Doppler spectrum of flow velocities is linearly proportional to the blood volume of moving scatterers and therefore will be linearly proportional to the cross-sectional area (CSA) of flow within the beam^{14} ^{15} : a flow with twice the CSA, and therefore twice the volume of moving scatterers within the sample volume, will return Doppler signals with twice the power, so long as the areas all lie within the Doppler beam (Figure 1⇓). Backscattered power from nonmoving or stagnant blood within the sample volume but outside the flow CSA does not contribute to the power in the Doppler spectrum from rapidly moving blood. Backscattered power therefore has the potential to provide the CSA information we need to calculate flow rate.

Because backscattered power also depends on round-trip attenuation of sound and backscattering coefficient (backscattered power/volume, a function of hematocrit),^{12} ^{13} these power measurements must be calibrated in the same individual against power returning from a beam of known CSA that lies entirely within flow at the same depth as the pathological flow of interest: The hematocrit, backscattering coefficient, and attenuation are the same, so that any changes in Doppler power should relate to changes in the CSA of flow within the beam.^{15}

It has long been assumed that this principle holds only for laminar flow, such as that in blood vessels,^{15} and cannot be applied to regurgitant jets because, for a given flow rate, backscattered power is increased by turbulent eddies^{11} ^{13} ^{16} and fluid entrainment into the jet.^{17} ^{18} Regurgitant flow, however, is laminar, not turbulent, at the proximal vena contracta, the smallest jet CSA, where velocity is highest (Figure 2⇓).^{5} ^{6} ^{17} ^{19} Therefore, we proposed the new hypothesis that mitral regurgitant (MR) flow can be quantified directly at the vena contracta, where flow is laminar before entrainment. At this site, identified by a narrow velocity spectrum corresponding to laminar flow, total backscattered power integrated over the velocity spectrum should be linearly proportional to the vena contracta CSA: Jet CSA∝∫_{velocity} power (at velocity v) dv, or the power integral (PI).

Flow rate Q equals vena contracta area times velocity; therefore, power times velocity, integrated over the vena contracta velocity spectrum, should be proportional to regurgitant flow rate: Q∝∫_{velocity} power (at velocity v) vdv, or the power-velocity integral (PVI).

Finally, the time integral of power times velocity should be proportional to regurgitant stroke volume: RSV=∫_{time} Q (at each time t) dt, or the power-velocity-time integral (PVTI).

We tested this method in vitro; in vivo with a new model providing a direct regurgitant volume gold standard; and in initial patient applications, with calibration to provide clinically useful absolute flow values.

## Methods

### In Vitro Studies

Flow was driven from a Plexiglas left ventricle (LV)-simulating chamber (5.7-cm diameter) through an orifice 10 cm from the ultrasound transducer (typical mitral depth) to a left atrium (LA)-simulating chamber (Figure 2⇑). Circular orifice areas of 0.2, 0.5, and 0.8 cm^{2}, corresponding to mild to severe clinical lesions, were studied at 5 steady flow rates from 20 to 60 mL/s from a piston pump that minimized cavitation (modified Mark IV Power-injector, Medrad), and with 4 parabolic pulses of 20 to 70 mL from syringe injections; a 0.07-cm^{2} orifice was also studied at 20-mL/s flow. Microbubble generation by the orifice pressure drop in blood or surfactant-containing analogues was resolved by use of degassed distilled water and 19% glycerol (specific gravity, 1.043 g/cm^{3}); adding 48 600 polystyrene microspheres/mL (25.2-μm diameter; Duke Scientific) produced a backscattering coefficient equivalent to that of human blood.^{20}

### In Vivo Experimental Studies

We developed a new canine model to measure MR volume directly without flowmeters (Figure 3⇓).^{21} In adult dogs (20 to 25 kg) anesthetized with 30 mg/kg IV sodium pentobarbital and ventilated, a nondistensible LA chamber was sutured to the mitral annulus via a Dacron sewing ring, and the atrial walls were sutured tightly around it. This chamber was directly attached to a 1.0-cm-diameter column within which MR stroke volume (MRSV) produced a vertical fluid excursion with each systole that was videotaped and measured (1.3 cm=1 mL). With right heart bypass, all venous return was collected, and the oxygenated blood was pumped via a reservoir and wide-bore cannula into the LV through a 1-way valve. A total of 36 different hemodynamic stages were analyzed in 3 dogs with regurgitant orifices of 0.12 to 0.21 cm^{2} cut into the anterior leaflet, and afterload was changed by aortic clamping.

### Patient Studies

Initial clinical applications involved 23 patients with MR with a range of severity and etiologies (Table⇓), including eccentric jets, studied from a transthoracic apical approach with at least fair to good image quality. In 12 patients (age, 52±17 years; 9 male, 3 female), MRSV by Doppler power was compared with mitral inflow minus aortic outflow MRI (see below) obtained within 30 to 60 minutes of Doppler.

A second group of 11 patients (age, 65±13 years; 8 male, 3 female) had MRSV calculated from 3D echo LV ejection volume (see below) minus forward aortic stroke volume derived from LV outflow tract CSA times the integral over time of power-weighted mean velocity to account for the sampled velocity spectrum.^{22}

### Doppler Methods

In all studies, the 2.5-MHz transducer (1.8-MHz Doppler) of a Hewlett-Packard 5500 scanner was used to record velocities up to 800 cm/s with high pulse repetition frequency Doppler from a thin sample volume (0.35 cm) placed in the vena contracta to record the narrowest high-velocity spectrum just beyond the orifice. To prevent signal reduction at low flow rates, a low wall filter (200 Hz, or ≈8 cm/s) was used. Compress, reject, transmit power, receive gain, depth, and velocity range settings were kept constant in the in vitro series and within each patient and animal.

### Power-Velocity Analysis

Digitally recorded Doppler video display intensities, nonlinearly compressed, were reconverted to their original uncompressed acoustic amplitudes and power (amplitude squared) on the basis of the acquisition compress and reject settings. Power and power times velocity were then integrated over all velocities in the narrow vena contracta velocity spectrum (Figure 4⇓) at each time point with MATLAB software (Version 5.1, MathWorks). Steady-flow power and power-velocity integrals (PI and PVI) were averaged over the recorded time samples (≈200 to 350×4.9 ms/line). Pulsatile-flow PVI was integrated over time to obtain the PVTI.

### Doppler Beam Size

The entire 1.2×2.0-cm transducer aperture produces an estimated sample volume of 3.1-mm lateral×5.2-mm elevation dimension at 10 cm based on half-maximum-power beam width.^{23} To encompass larger jets, we created a broad distal measuring beam by reducing the transducer aperture with a Tyvek (Dupont) mask (Figure 5⇓). In vitro, a 7-mm-diameter circular aperture increased beam width to 6.75 mm (circular 0.36-cm^{2} CSA); in vivo, a larger (10-mm) aperture was used to record weaker signals (beam width=5.8 mm, CSA=0.26 cm^{2}).

### Power Calibration

Calibration converts unitless power to absolute areas and accounts for the variation in backscattered power among individuals for the same blood volume due to different attenuation and backscattering coefficients.^{13} We calibrated power from the broad measuring beam encompassing the vena contracta using a narrow reference beam placed within the flow area (Figure 5⇑).^{15} ^{24} The reference beam provides the ratio between power and area because its CSA is known. We then applied this calibrating ratio to the power from the broad measuring beam to determine the CSA of flow within the beam. Ultrasound attenuation and backscattering coefficient are the same for both beams and cancel out, compensating for individual differences.

The power received from the measuring beam (PI_{meas}), with its reduced transducer aperture, must be multiplied by a correction factor (CF) to compensate for the decrease in transmitted and received power compared with the reference (ref.) beam, which uses the full aperture: or simply where the calibrating factor c_{cal}=(ref. beam CSA/PI_{ref})×CF. This CF depends only on the physical properties of the transducer and can be determined in vitro as the ratio of PI_{ref} to PI_{meas} for a small orifice (0.07 cm^{2}) encompassed by both beams (CF: 7.7 in vivo, 63 in vitro).

### Magnetic Resonance Imaging

MRSV was obtained as mitral inflow minus aortic outflow with a 1.5-T system (GE Signa). Phase contrast cine acquisitions were obtained in planes aligned with the mitral annulus and orthogonal to the mid ascending aorta.^{25} Phase velocity maps were integrated over the appropriate flow areas, integrated over time, and subtracted.

### 3D Echocardiography

In the second patient group, LV volumes were calculated with a polyhedral surfacing algorithm^{26} from reconstructed endocardial borders in 10 rotated apical views collected with a transthoracic Omniplane probe (Hewlett-Packard) with ECG and respiratory gating.

### Statistical Analysis

Doppler power results (PI, PVI, and PVTI) were compared with reference values (CSA, flow rate, and volume) by linear regression. Agreement was assessed by plotting differences against reference values (or, in patients, the mean of calculated and reference values),^{27} comparing mean differences with zero by *t* test.

## Results

### In Vitro Studies

The PI was linearly proportional to regurgitant orifice area (ROA) up to and including 0.5 cm^{2} (Figure 6A⇓, left-hand axis, *r*=0.99); the area of 0.8 cm^{2}, which is clinically extreme,^{28} was incompletely assessed with current beam size and was not included in the regression. With dual-beam calibration of PI to ROA (c_{cal}=8.9×10^{−}^{4}, right-hand axis), regression gave *y*=0.84*x*+0.01cm^{2}, SEE=0.01cm^{2}, with good agreement between calculated and actual ROA <0.8 cm^{2} (Figure⇑ 6B; mean difference=−0.04±0.03 cm^{2}, *P*<0.0001); slight underestimation was evident at 0.2 and 0.5 cm^{2}, as expected for Doppler (effective) versus anatomic orifice area.^{29} Figure 6C⇓ shows that the narrow reference beam fit within all the orifices used and therefore returned the same power regardless of orifice area and flow rate over 32 combinations studied.

For the 0.07-, 0.2-, and 0.5-cm^{2} orifices, steady flow rates calculated from PVI correlated and agreed well with actual values (Figure 7⇓; *y*=0.95*x*+0.21 mL/s, SEE=2.5 mL/s), with a mean difference of −1.6±2.5 mL/s (*P*=0.001 versus 0); as before, there was underestimation for the 0.8-cm^{2} orifice. Similar correlations and agreement were observed for the 72 pulsatile stroke volumes studied (Figure 8⇓; *y*=0.92*x*+1.3 mL, SEE=2.6 mL, mean difference=−2.44±2.9 mL, *P*<0.0001).

The results, for example, showed that the PI remained virtually constant (1.5×10^{4} to 1.6×10^{4} in unitless values) for a constant ROA of 0.2 cm^{2} as flow rate varied from 20 to 40 and 60 mL/s; conversely, the PVI increased from 1.8×10^{6} to 3.6×10^{6} to 5.3×10^{6} in proportion to flow rate.

### In Vivo Experimental Studies

Calculated MR stroke volume correlated and agreed well with directly measured values of 4 to 21 mL (Figure 9⇓; *y*=0.98*x*+0.28 mL, *r*=0.98, SEE=0.89 mL, mean difference=0.04±0.87 mL, *P*=0.79 versus 0). Because Ref. Beam CSA was known and CF was determined in vitro, only reference and measuring beam powers needed to be measured in vivo to obtain absolute flow values.

### Patient Studies

In all patients, a satisfactory high-velocity narrow-spectrum Doppler signal could be recorded. In the primary 12 patients studied by MRI, calculated regurgitant stroke volume correlated and agreed well with MRI values (Figure 10⇓; *r*=0.98, *y*=0.70*x*+3.5 mL, SEE=2.4 mL, mean error=−3.6±5.1 mL, *P*=0.03); calculated values, in fact, lay close to the line of identity for regurgitant stroke volumes up to 40 mL (*r*=0.99, *y*=0.87*x*+1.2 mL, SEE=1.3 mL, mean error=−1.0±1.8 mL, *P*=0.13 [*P*=NS versus 0]), with mild underestimation only at higher values (ROA>0.5 cm^{2}), consistent with the mild potential underestimation due to currently limited beam size, as shown in vitro (Figures 6 to 8⇑⇑⇑). Results were comparable when the secondary patient group (MRSV from LV ejection volume minus aortic forward flow) was added (n=23; *r*=0.98, *y*=0.71*x*+3.5 mL, SEE=1.9 mL, mean error=−2.8±4.4 mL, *P*=0.01); for MRSV up to 40 mL (ROA<0.5 cm^{2}), values lay close to the line of identity (n=19, *r*=0.97, *y*=0.81*x*+2.1 mL, SEE=1.6 mL, mean error=−1.1±2.2 mL, *P*=0.05). Figure 11⇓ (top) shows an example of the high-velocity spectral recording in a patient.

## Discussion

Many noninvasive ultrasound measures of regurgitant volume are limited by indirect measurements,^{30} ^{31} multiple steps prone to error,^{7} ^{32} and limiting assumptions about regurgitant flows.^{3} ^{4} ^{33} ^{34} ^{35} To overcome these limitations, we introduced the new concept of integrating Doppler power times velocity at the proximal vena contracta, directly measuring regurgitant flow at the lesion itself. The results demonstrate that at the vena contracta, where flow is laminar before entrainment, jet CSA is linearly proportional to the Doppler PI, regurgitant flow rate to the PVI, and regurgitant stroke volume to the power-velocity time integral. A dual-beam approach compensates, in each individual studied, for attenuation and variations in hematocrit, providing clinically useful absolute values.

This approach has the major advantage of integrating the contribution of scatterers at all velocities within the vena contracta without simplifying assumptions about its shape or velocity distribution. It also simplifies measurement by obviating the need to measure CSA by 2D echo. Unlike other, single-time-point methods, power-velocity directly assesses and integrates dynamic variations in ROA and flow rate, as shown in Figure 11⇑, bottom left: a patient with a dilated LV shows the characteristic midsystolic decrease in ROA described in such patients,^{8} and this variation is automatically incorporated into the PVI to give flow rate and volume. Finally, this approach should be relatively immune to variations in the Doppler beam–to-flow angle θ: the cos θ decrease in measured velocity is canceled by a reciprocal increase in CSA relative to the beam, and any variations in attenuation with angle are dealt with by the dual-beam calibration (T.B., unpublished data, 1999).

### Previous Work

Previous studies used backscattered power to average amplitude-weighted velocities over a CSA, multiplying by area to estimate flow.^{22} ^{36} ^{37} ^{38} ^{39} Other studies used the PVI but only in low-velocity laminar flows, precluding direct assessment of regurgitation.^{24} ^{40} Continuous-wave Doppler studies of the entire regurgitant jet, including turbulent and entrained flow, demonstrated highly variable relations between regurgitant volume and signal intensity.^{41} ^{42} ^{43} The present approach, in contrast, examines only the laminar vena contracta to evaluate regurgitant flow most directly.

### Current Limitations and Future Application

Although the technique requires some understanding of flow through a restrictive orifice, this is similarly required for application of the simplified Bernoulli equation, part of routine clinical practice. This approach simplifies the measurement process because no separate 2D echo measure of CSA is required, and dynamic variations in ROA are incorporated automatically. The approach lends itself to the simplified application we envision in Figure 11⇑: First, the Doppler sample volume is placed just beyond the regurgitant orifice, which can be guided by visualizing the narrowest point of the jet by use of Doppler color flow mapping; the narrowest high-velocity signal is then optimized. This requires skills comparable to those for localizing the highest jet velocities for the simplified Bernoulli equation, which sonographers routinely do. Second, the spectrum can be analyzed automatically on board the machine itself, without the offline analyses we performed with the available system: power within the high-velocity Doppler spectrum can be directly extracted before conversion to display intensities to calculate PI, PVI, and PVTI and output regurgitant area, flow rate, and volume.

Although we had to vary transducer apertures manually to provide measuring and calibrating beams, that can be achieved more easily with electronic aperture variation. Indeed, both beams can in principle be generated simultaneously by connecting the transducer elements to 2 independent digital beam-forming processors, providing measurement and calibration in a single cardiac cycle. Also, because calibration, for a given patient and depth, relates power to the CSA of moving blood independent of velocity, the calibration beam can be formed only in diastole, with the measuring beam in systole (T.B., unpublished data, 1999).

The main limitation, then, would be that currently available beams are not wide enough to sample the most severe regurgitant jets (>0.5 cm^{2}); however, that reflects current basic transducer design, which optimizes spatial resolution and can be remedied with newer transducers under development.

It should be emphasized that although calibration corrects for attenuation and backscattering coefficient for each patient, the correction factor relating calibrating and measuring beam power on the basis of their different apertures is fundamentally a physical result of transducer design and therefore needs to be determined only by the manufacturer in vitro for each system design, not for each patient. Our results support this, because 1 aperture-correction factor provided consistent agreement between power-velocity and actual values across several animals, with similar results for the transducer apertures and correction factor used across multiple patients.

Finally, although this approach was tested in MR, it should potentially be applicable to a wide range of regurgitant, stenotic, and shunt lesions.

### Conclusions and Clinical Implications

The integral of Doppler power times velocity at the regurgitant vena contracta is a new, noninvasive approach that for the first time accurately measures regurgitant flow directly at the lesion itself. Because only information contained in the Doppler spectrum itself is required, this approach can potentially be readily automated for future routine clinical application. Such quantification would improve our evaluation of lesion severity and progression to guide patient interventions and test their ability to preserve ventricular function.

## Acknowledgments

This study was supported in part by grants HL-38176 and HL-57302, National Institutes of Health, Bethesda, Md. Dr Buck was supported by grant Bu1097/1-1, Deutsche Forschungsgemeinschaft, Bonn, Germany. The Mark IV Powerinjector was provided by Geoff A. Morris, Medrad, Inc, Indianola, Pa. We thank Randall Grimes, MD, PhD, for his invaluable fluid dynamics comments and Iain D. Wright and Arnold J. George (Biomedical Engineering) for constructing the flow model.

- Received December 31, 1999.
- Revision received March 23, 2000.
- Accepted March 29, 2000.

- Copyright © 2000 by American Heart Association

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- The Power-Velocity Integral at the Vena ContractaThomas Buck, Ronald A. Mucci, J. Luis Guerrero, Godtfred Holmvang, Mark D. Handschumacher and Robert A. LevineCirculation. 2000;102:1053-1061, originally published August 29, 2000https://doi.org/10.1161/01.CIR.102.9.1053
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