## Abstract

The simplified Bernoulli equation relates fluid convective energy derived from flow velocities to a pressure gradient and is commonly used in clinical echocardiography to determine pressure differences across stenotic orifices. Its application to pulmonary venous flow has not been described in humans. Twelve patients undergoing cardiac surgery had simultaneous high-fidelity pulmonary venous and left atrial pressure measurements and pulmonary venous pulsed Doppler echocardiography performed. Convective gradients for the systolic (S), diastolic (D), and atrial reversal (AR) phases of pulmonary venous flow were determined using the simplified Bernoulli equation and correlated with measured actual pressure differences. A linear relationship was observed between the convective (*y*) and actual (*x*) pressure differences for the S (*y* = 0.23*x* + 0.0074, *r* = 0.82) and D (*y* = 0.22*x* + 0.092, *r* = 0.81) waves, but not for the AR wave (*y* = 0.030*x* + 0.13, r = 0.10). Numerical modeling resulted in similar slopes for the S (*y* = 0.200*x* − 0.127, *r* = 0.97), D (*y* = 0.247*x* − 0.354, *r*= 0.99), and AR (*y* = 0.087*x* − 0.083,*r* = 0.96) waves. Consistent with numerical modeling, the convective term strongly correlates with but significantly underestimates actual gradient because of large inertial forces.

- pulmonary veins
- echocardiography
- fluid dynamics
- numerical modeling

the assessment of pulmonary venous flow by either transesophageal echocardiography or transthoracic echocardiography, in combination with transmitral flow, is valuable in the evaluation of left ventricular (LV) diastolic function (5, 9). Although previous numerical models have been developed to describe the physiological determinants of pulmonary venous (PV) flow (7, 17), little is known about the relationship between the pressure and velocity characteristics of PV waves in humans.

Several investigators have shown a strong relationship between PV waveform velocity characteristics, such as phase duration (1) or peak velocities (15), and cardiac pathology. Others have attempted to correlate the pulsed Doppler characteristics of the PV wave to different left atrial (LA) physiological variables, including ejection fraction, mean pressure, relaxation, minimum volume, and pulmonary arterial capillary wedge pressure (3, 10, 12,13). Rossvoll and Hatle (14) have shown that characteristics of the pulmonary atrial reversal (AR) wave, compared with the mitral inflow A wave, can predict LV end-diastolic pressure. Recently, with the use of a dog model, Appleton (2) was able to demonstrate a qualitative relationship between the PV-LA pressure difference and PV pulsed Doppler velocities. He suggests that PV velocities are related to PV-LA pressure differences and that flow through the PV into the LA was predominately convective (2). Conversely, analytic work predicts that PV flow in humans should be predominately inertial, given the length and width of the PV near the orifice to the LA (17).

Hence, if further clinical applications of the pressure and velocity characteristics of the PV waves are to be developed, then the relationship between the two, particularly in humans, needs to be further defined. Therefore, the purpose of this study was to determine whether PV velocities are related to PV-LA pressure gradients and to evaluate and quantify the relative contribution of PV inertial and convective forces. To achieve this goal, we related simultaneous PV and LA pressure data with pulsed Doppler PV velocities from patients undergoing cardiac surgery and validated the results with the predictions of a previously verified numerical model of the cardiovascular system.

## METHODS

### Patient Population

After prior approval by our Institutional Review Board, written informed consent was obtained from 12 patients (7 males, mean age 63.8 ± 8.8 yr) before they underwent first-time cardiac surgery requiring cardiopulmonary bypass. Preoperative LV ejection fraction (EF) was normal (EF > 50%) in seven, moderately depressed (EF = 35–40%) in four, and severely depressed (EF < 25%) in one. All were in sinus rhythm. Surgical procedures performed included isolated coronary artery bypass grafting (CABG) in eight, CABG with septal myomectomy in one, CABG with LV infarct exclusion surgery in one, mitral valve replacement in one, and an aortic valve replacement in one. Intraoperative transesophageal echocardiography findings are summarized in Table 1. In addition to the results described, one patient had transesophageal echocardiography evidence of mild right ventricular systolic dysfunction, and one had evidence of pulmonary hypertension.

Thirty-six total patient conditions were obtained; eight were excluded from analysis because of technical factors not evident at the time of data collection. These factors included catheter malposition in one (distal pressure transducer not positioned appropriately in the PV), excessive noise after digital reconstruction of the pulsed Doppler signals in one, failure of a proximal pressure sensor in one, S-T segment changes during afterload altering maneuvers in two, and excessive atrial or ventricular ectopy during afterload altering maneuvers in three.

### Intraoperative Procedure

After routine induction of general anesthesia, median sternotomy, and pericardiotomy, high-fidelity pressure transducers (Millar Instruments, Houston, TX) were positioned in the left PV, the LA, and the LV through a small right PV incision. Before insertion, all catheters were immersed in warm saline for at least 30 min to minimize drift, and each was individually calibrated to atmospheric zero. Appropriate anatomical placement was confirmed through the use of transesophageal echocardiography and visualization of appropriate chamber-specific waveforms.

Signals were amplified with a universal amplifier (Gould, Valley View, OH) and recorded digitally through an NB-MIO-16 multifunction input/output board (National Instruments, Austin, TX) with 12-bit resolution and a sampling frequency of 1,000 Hz. The digital signals were recorded with a customized data acquisition and analysis application developed with the use of LabVIEW (National Instruments) on a standard Pentium-based personal computer running Windows 95.

Transesophageal echocardiography was performed with the use of a Hewlett-Packard Omniplane probe connected to a Sonos 1500 or 2500 echocardiograph (Hewlett-Packard, Andover, MA). For recording of PV spectral Doppler signals, the sample volume was placed in the left upper PV at 1 cm from the junction with the LA and corresponded to the location of the PV pressure transducer. The view was optimized to align the PV flow with the cursor. Pulsed Doppler audio signals were acquired and digitized at 20 kHz simultaneously with the pressure measurements by connecting the audio output of the echocardiograph to the above data-acquisition apparatus. Pulsed Doppler audio signals were processed by using a short-time Fourier analysis (20-kHz sampling frequency with 256 sample width, 128 sample shift per analysis, with the use of a Hamming window) to reconstruct spectral Doppler images and extract the PV velocity profiles (8).

For each patient, 8-s recordings of intracardiac pressure and PV pulsed Doppler velocity were obtained during suspended respiration at*1*) baseline (after aortic cannulation, but before the institution of cardiopulmonary bypass), *2*) during infusion of intravenous phenylephrine (titrated to a mean aortic pressure of 100 mmHg), and *3*) on partial-flow cardiopulmonary bypass (1–2 l/min). These conditions were chosen to obtain the widest range of physiological conditions that may be encountered during the clinical evaluation of the widest range of pathophysiological conditions, ranging from low cardiac output to hypertension.

### Mathematical Modeling

A previously described and clinically verified numerical model of the cardiovascular system (17) was used to determine the relationship between PV-LA pressure gradients and velocities under a wide range of stroke volumes. In short, our model is a closed-loop, lumped-parameter system based on 24 first-order differential equations that simulate pressure, volume, and flow throughout the heart and pulmonary and systemic vasculature. Initial model parameters, similar to those obtained and clinically verified by previous intraoperative studies, were used (17). Specifically, with regards to the PV-LA junction, a resistance of 30 g/s · cm^{4}, an inertia value of 3.0 g/cm^{4}, an initial total PV volume of 300 ml, and a compliance of 15.0 ml/mmHg were used. Total systemic volume was altered under constant LA and LV diastolic and systolic parameters to yield a modeled stroke volume that range from 25 to 80 ml. For each cardiac output tested, peak velocities and gradients were determined from the model output for the PV systolic (S wave), diastolic (D wave), and AR wave waveforms.

### Data Analysis

#### Clinical data.

For each 8-s physiological condition measured, three representative complete cycle waveforms were analyzed with the use of a customized LabVIEW data analysis application. Acceleration time (time to peak velocity), deceleration time (extrapolated time from peak velocity to zero), and peak velocities were determined for each S, D, and AR waveform of PV flow. The results of each of the three cycles measured were then averaged together to yield the velocity profiles for each patient under a specific hemodynamic condition. Because the pulsed Doppler waveforms were acquired simultaneously with the intracardiac pressures, the corresponding pressure waveforms were also analyzed (Fig. 1). Peak actual pressure gradients (ΔP_{act}), waveform acceleration times, and deceleration times were similarly determined.

#### Theoretical construct.

The unsteady Bernoulli equation for the pressure drop Δp(*t*) between two points along a streamline of flow may be written as
Equation 1where *v* is the blood velocity at the two points of interest (*v*
_{1} and *v*
_{2}); ρ is blood density (1.05 g/cm^{3}); *M* is the inertance of blood flow, a distributed term reflecting the effective mass of blood being accelerated between the two points; and*R* is a resistive term reflecting the effects of viscosity along the path, generally considered negligible and hence ignored (16). The basis for this assumption lies in the application of the Poiseuille equation (*Eq. 2
*)
Equation 2This equation describes the viscous resistance to flow (Δp_{visc}) as being a function of the viscosity of blood (μ), the peak velocity (*V*
_{max}), and the length of the column (*L*), divided by the radius squared (*r*
^{2}). Although based on steady-state laminar flow, when applied to the PV, for the range of human cardiac output (2–6 l/min), over the distance measured (5 cm between pressure transducers on the multisensor catheter), and for the range of PV diameters measured in these experiments (1.0–1.5 cm; Table 1), the contribution of the resistance term ranges from 0.006 mmHg (for a PV diameter of 1.4 cm and a cardiac output of 2 l/min) to 0.14 mmHg (for a PV diameter of 1.0 cm and a cardiac output of 6 l/min). Similarly, the initial velocity within the LA has also been shown to be negligible and is also ignored (18).

For the PV analysis, the simplified Bernoulli equation was used to calculate the convective pressure gradient (ΔP_{conv}), reflecting the first term on the right-hand side of *Eq. 1
*. Least squares linear regression analysis was used to determine the correlation between ΔP_{act} and the corresponding ΔP_{conv} for the S, D, and AR waves.

Mean inertance (*M*), as applied to the unsteady Bernoulli equation, was determined from *Eq. 1
* by subtracting the convective component derived from the pulsed Doppler velocity from the actual pressure gradient and dividing the result by the mean acceleration d*v*/d*t* of the corresponding PV pulsed Doppler wave

#### Numerical simulation.

Least squares linear regression analysis was performed on the model output data to determine a mathematical relationship between ΔP_{act} and ΔP_{conv} for the S, D, and AR waves. Pressure and velocity acceleration and deceleration times for each phase of the PV waveform were compared with the use of Student's*t*-tests with paired testing when appropriate.*P* < 0.05 was considered statistically significant.

An assumption of our numerical model is that PV inertance is a constant. To validate this hypothesis, the constant used in the numerical modeling was compared with the values obtained from the in situ results. In addition, to further validate both our numerical model and our in situ findings, a predicted model pressure gradient (ΔP_{model}) for each PV wave was determined. ΔP_{model} was determined by using the unsteady Bernoulli equation and combining the temporal and velocity characteristics of the actual pulsed Doppler with the value for *M* used in the numerical modeling. Similarly, the predicted gradients from the combined numerical modeling and pulsed Doppler data were compared with the actual corresponding PV pressure gradients by use of least squares linear regression and paired Student's *t*-tests.

## RESULTS

### Human Studies

Figure 2 shows a typical data set demonstrating the actual pressure gradient between the PV and the LA along with the corresponding convective pressure drop derived from application of the simplified Bernoulli equation. Although the convective drop appears to track the actual gradient throughout the cardiac cycle, significant underestimation is evident and the convective term appears to lag behind the actual gradient. Table2 demonstrates the wide range in peak ΔP_{act}, velocities, and average percent contribution of the ΔP_{conv} within our patient population. The ΔP_{conv}, as calculated with the use of the simplified Bernoulli equation, was found to be significantly lower than ΔP_{act} for each of the three phases of the PV waveform (*P* < 0.01 for each ΔP_{conv} vs. ΔP_{act}). For the S and D waves, the convective term represented only 22.8 and 24.9%, respectively, of the ΔP_{act}, with the remaining pressure difference representing the predominately inertial components of the Bernoulli equation. For the AR wave, ΔP_{conv} accounted for <5% of the ΔP_{act}. Despite the wide variations in actual pressure gradients (*x*) and convective components (*y*), regression analysis revealed a linear correlation between the two for both the S (*y* = 0.23*x* + 0.0074;*r* = 0.82) and D (*y* = 0.22*x* + 0.092; *r* = 0.81) waves (Fig.3, *A* and *B*). Although the correlation for the S wave appears to be strongly influenced by several data points obtained during phenylephrine infusion, even in analysis of the data after exclusion of the two extreme data points, the resulting linear equation is still highly significant (*y* = 0.15*x* + 0.29;*r* = 0.71, *P* < 0.001). No correlation was found between the AR ΔP_{conv} and ΔP_{act}(*y* = 0.030*x* + 0.13; *r* = 0.10), which we attribute to the low overall contribution of convective energy to the ΔP_{act} (Fig. 3
*C*).

Of particular note are the waveform and pressure relationships observed in the single patient with severe mitral regurgitation (MR). Clinically, MR is associated with blunting, and even reversal, of the S wave. In our patient, two hemodynamic data sets were obtained. Under baseline conditions, ΔP_{act} was 4.21 mmHg and the S-wave velocity was 49.7 cm/s (ΔP_{conv} = 0.99 mmHg). Under partial-flow bypass, ΔP_{act} was 2.92 mmHg and the S-wave velocity was 38.1 cm/s (ΔP_{conv} = 0.58 mmHg). For baseline and partial-flow conditions, ΔP_{conv} accounted for 23.4 and 19.9% of ΔP_{act}, respectively. The consistency of these findings compared with the overall results (22.8% for the S wave) suggests the observed gradient-velocity relationship even in patients with severe MR.

Because a major component of the ΔP_{act} is accounted for by nonconvective forces, one would expect a temporal delay between the peak ΔP_{act} and the peak velocity for each of the three phases. As demonstrated in Table 3, all three phases demonstrated a delay in the time to peak velocity after the peak ΔP_{act}. In addition, because the velocity waves account for the kinetic energy, a delay in deceleration once the driving force of the pressure gradient ceases would also be expected. This is observed in the significant delay in the velocity deceleration time compared with the pressure-gradient deceleration time (Table 3). As demonstrated by the above application of the Bernoulli equation, the convective, or kinetic energy component, of the reversal wave is small; nevertheless, there is still a significant delay between the time to peak pressure and the time to peak velocity. Although there was a delay in the velocity deceleration, it was not significantly different from the pressure-gradient delay.

### Model Results

Model results of the relationship between the ΔP_{act}and ΔP_{conv} for the S, D, and AR waveforms were similar to those obtained invasively for stroke volumes that ranged from 25 to 80 ml (Fig. 4, *A* and*B*). For the S wave, a slope of *y* = 0.20 (*r* = 0.99) was obtained versus 0.23 for the in situ data. The model slope of the D wave was *y* = 0.25 (*r* = 0.99) compared with an in situ slope of 0.22, and for the AR wave, the model-derived slope was *y* = 0.087 (*r* = 0.96) versus 0.030. The relationships between the in situ data and the numerical modeling results are also demonstrated in Fig. 3, *A–C*.

### Determination of Mean PV Inertance

Despite the wide range of peak pressure gradients and measured velocities, *M* was relatively constant for all conditions tested. In situ, *M* was 3.42 ± 2.28 g/cm^{4} for the S wave and 3.32 ± 2.67 g/cm^{4} for the D wave. Both of these were similar to the constant of 3 g/cm^{4} used in the numerical modeling [*P* = not significant (NS) for the S wave and*P* = NS for the D wave]. In addition, predicted S- and D-wave pressure gradients obtained from solving the unsteady Bernoulli equation were similar (*P* = NS by paired*t*-test) to actual pressure gradients (ΔP_{act} = 1.12ΔP_{model} − 0.67,*r* = 0.71, *P* < 0.001 for the S wave; ΔP_{act} = 0.89ΔP_{model} + 0.36,*r* = 0.77, *P* < 0.001 for the D wave). For the AR wave, *M* was 21.48 ± 17.97 g/cm^{4}, which was significantly different from the model constant (*P* < 0.05) and may reflect the actual reversal of flow rather than just acceleration.

## DISCUSSION

We have shown with both in situ and numerical modeling data that the peak pressure gradient (ΔP_{conv}) obtained from the peak pulsed Doppler velocity by use of the simplified Bernoulli equation correlates with, but consistently underestimates, the true gradient (ΔP_{act}). However, for both the S and D phases, but not the AR wave, there is a linear relationship that is preserved over a wide range of ΔP_{act}. In addition, we have for the first time in situ an estimate of *M* for the PV waves. Combining the in situ results with the numerical model strongly suggests that the concept of constant *M* for the S and D PV waves is acceptable for the physiological range of flow. This therefore allows for the calculation of the nonconvective forces utilizing the mean Doppler wave acceleration.

The poor correlation between the AR velocity and the AR ΔP_{act} can be explained by the physiology of the larger inertial contribution to the AR wave. Inertia is the pressure gradient or force required to accelerate a mass over a distance. As blood travels from the PV through the LA and into the LV, it acquires kinetic energy. At the onset of atrial contraction, the application of a force to cause reversal of flow (i.e., inertial contribution) must be greater than this kinetic energy first to decelerate the forward velocity and then further to accelerate the mass of blood in the reverse direction. The flat slope of the ΔP_{act} versus ΔP_{conv}regression indicates that for the hemodynamic conditions tested, the convective contribution of the reversal wave remained relatively constant. However, as has been previously shown, the magnitude of the AR-wave velocity is determined in part by a function of LV diastolic stiffness and the contractile properties of the LA during LA systole (11, 14). The relatively large inertial contribution to the overall pressure gradient is a complex interaction between the forces necessary not only to cause reversal of flow of the blood entering into the LA (i.e., LA contractility) but also the resistance of the LV (i.e., LV stiffness) during the late diastolic filling stage. Therefore, it is easy to appreciate the predominance of nonconvective forces that govern the AR wave and the way in which measurement of AR velocities may be more suited for estimating LV and LA function than for estimating PV-LA pressure gradients. The large*M* determined by the in situ data further supports these findings but also demonstrates the complex physiology and incomplete understanding of the determinants of the AR wave (11,12).

The Bernoulli equation (*Eq. 1
*) is fundamental to explaining the relationships between pressure gradients and flow/velocity characteristics. The Bernoulli equation can be divided into three independent components: a convective term [½ρΔ(v^{2})], which accounts for the change in kinetic energy; an inertial term (*M*d*v*/d*t*), which accounts for the pressure required to accelerate a mass of blood over a distance; and a viscous term (*R*), accounting for the resistance of the blood along the endocardium. In clinical applications, the simplified Bernoulli equation considers both the resistance and inertial terms to be negligible. The viscosity, or resistance, term is related to the proximity and duration of flow along a wall or surface. Clinically, the viscous contribution is significant for long tubes such as arteries, where the Poisseuille equation holds, but less so across orifices such as cardiac valves (16). The convective term forms the basis for clinically estimating transvalvular pressure gradients, and, despite several assumptions, it is extensively validated in clinical echocardiography for the estimation of pressure gradients from observed wave velocities and provides a valuable index for disease progression. The inertial term is not included in the simplified Bernoulli equation because it cannot be derived from pulsed Doppler because it requires spatial acceleration. The inertial component of transmitral flow has been shown to play a significant role in early diastolic LV filling, but its role in LA filling is unknown (4, 6).

The factors that contribute to the different PV velocity phases are complex and difficult to evaluate independently. Appleton (2), in lightly sedated dogs, revealed valuable insight into the varying roles of both right and left heart function on the different phases of PV flow. Although all three phases of PV flow increased with increasing ventricular preload, the effects were largest on the AR. In addition, the duration and velocity of the reversal wave were shown to be directly correlated with mean LA pressure and inversely related to heart rate. Maximal late systolic PV pressure was shown to be a function of right ventricular systolic output. The diastolic phase correlated with the decline in LA pressure and the corresponding transmitral and LV filling characteristics. The successful application of the simplified Bernoulli equation is valuable in quantifying the relationships between PV velocities and ΔP_{act}. With these relationships, further insight into the complex physiological and pathophysiological mechanisms of PV flow may be better understood. Unfortunately, little work has been done examining the physiological determinants and force characteristics of PV flow in humans. Furthermore, extrapolation of animal studies is difficult because the ratios of convective and nonconvective forces may be different than in humans. Because the balance between convective and nonconvective forces is, as discussed, a function of orifice diameters and conduit lengths, the anatomical relationships in dogs (or other small animals) may not necessarily apply to humans.

These results are important because knowing that PV flow is relatively inertial under normal hemodynamic conditions provides further clues regarding some of the limitations of utilizing PV flow when assessing diastolic function. Although the duration and wave profile of PV flow have been shown to reflect atriopulmonary pressure gradients during atrial contraction and, indirectly, LV stiffness, its inertial nature dictates that its relationship would be related to heart rate and the magnitude of the D wave. Thus, in sinus tachycardia and in conditions of restrictive filling, when the D amplitude is greater, AR velocity magnitude may underestimate pressure gradients and thus a narrow AR width may be observed even when LV stiffness is high. A broader understanding of these convective and nonconvective relationships is critical for the application of PV-wave characteristics to the clinical assessment of ventricular function.

In conclusion, the application of pulsed Doppler echocardiography to estimate pressure gradients between two structures is well known. Its use is limited by the ability only to accurately quantify the convective or kinetic component of the Bernoulli equation. We have shown that the simplified Bernoulli equation consistently underestimates ΔP_{act} for the three major phases of the PV waveform because of the large role of the nonconvective forces with a fairly constant PV inertance *M* observed. The assessment and assumption of a constant value of *M* demonstrate that the inertial term of the Bernoulli equation plays a significant and definable role in the underestimation of the measurement of PV-LA pressure gradients from Doppler data. Despite this underestimation, and as further demonstrated with mathematical modeling, there exists a linear correlation between the ΔP_{conv} and ΔP_{act} for the systolic and diastolic phases, allowing for clinical estimation of the true pressure gradient on the basis of noninvasive parameters obtained for pulsed Doppler echocardiography.

## Acknowledgments

This study was supported in part by Grant 93-13880 from the American Heart Association (Greenfield, TX); Grant 1R01HL56688-01A1 from the National Heart, Lung, and Blood Institute (Bethesda, MD); Grant NCC9-60 from the National Aeronautics and Space Administration (Houston, TX) (all to J. D. Thomas); and Grant-in-Aid NEO-97-225-BGIA from the American Heart Association (Northeast Ohio Affiliate) (to M. J. Garcia).

## Footnotes

Address for reprint requests and other correspondence: M. J. Garcia, Dept. of Cardiology, Desk F15, The Cleveland Clinic Foundation, 9500 Euclid Ave., Cleveland, OH 44195 (E-mail:garciam{at}ccf.org).

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- Copyright © 2000 the American Physiological Society