## Journal of the American College of Cardiology

# Noninvasive estimation of transmitral pressure drop across the normal mitral valve in humans: importance of convective and inertial forces during left ventricular filling

## Author + information

- Received February 23, 2000.
- Revision received June 1, 2000.
- Accepted July 25, 2000.
- Published online November 15, 2000.

## Author Information

- Michael S. Firstenberg, MD
^{a}, - Pieter M. Vandervoort, MD
^{a}, - Neil L. Greenberg, PhD
^{a}, - Nicholas G. Smedira, MD, FACC
^{b}, - Patrick M. McCarthy, MD, FACC
^{b}, - Mario J. Garcia, MD, FACC
^{a}and - James D. Thomas, MD, FACC
^{a},* (thomasj{at}ccf.org)

- ↵*Reprint requests and correspondence: Dr. James D. Thomas, Department of Cardiology, Desk F-15, Cleveland Clinic Foundation, 9500 Euclid Avenue, Cleveland, Ohio 44195

## Abstract

OBJECTIVES

We hypothesized that color M-mode (CMM) images could be used to solve the Euler equation, yielding regional pressure gradients along the scanline, which could then be integrated to yield the unsteady Bernoulli equation and estimate noninvasively both the convective and inertial components of the transmitral pressure difference.

BACKGROUND

Pulsed and continuous wave Doppler velocity measurements are routinely used clinically to assess severity of stenotic and regurgitant valves. However, only the convective component of the pressure gradient is measured, thereby neglecting the contribution of inertial forces, which may be significant, particularly for nonstenotic valves. Color M-mode provides a spatiotemporal representation of flow across the mitral valve.

METHODS

In eight patients undergoing coronary artery bypass grafting, high-fidelity left atrial and ventricular pressure measurements were obtained synchronously with transmitral CMM digital recordings. The instantaneous diastolic transmitral pressure difference was computed from the M-mode spatiotemporal velocity distribution using the unsteady flow form of the Bernoulli equation and was compared to the catheter measurements.

RESULTS

From 56 beats in 16 hemodynamic stages, inclusion of the inertial term ([Δp_{I}]_{max} = 1.78 ± 1.30 mm Hg) in the noninvasive pressure difference calculation significantly increased the temporal correlation with catheter-based measurement (r = 0.35 ± 0.24 vs. 0.81 ± 0.15, p < 0.0001). It also allowed an accurate approximation of the peak pressure difference ([Δp_{C+I}]_{max} = 0.95 [Δp_{cath}]_{max} + 0.24, r = 0.96, p < 0.001, error = 0.08 ± 0.54 mm Hg).

CONCLUSIONS

Inertial forces are significant components of the maximal pressure drop across the normal mitral valve. These can be accurately estimated noninvasively using CMM recordings of transmitral flow, which should improve the understanding of diastolic filling and function of the heart.

Through the application of fluid dynamics principles, Doppler echocardiography has been widely used to evaluate cardiovascular hemodynamics, including cardiac output and the severity of valvular stenosis and regurgitation. The Bernoulli equation has been of particular utility (1,2) to calculate the pressure drop across stenotic mitral (3), aortic (4) and prosthetic valves (5). These pressure drops have

been obtained using continuous wave Doppler, which records the maximal velocity along the ultrasound beam. Use of this single velocity has meant that only the simplified version of the Bernoulli equation can be used and assumes that the entire pressure drop is due to convective acceleration of blood, neglecting losses from inertial acceleration and viscous drag. This simplification is reasonable for the restrictive orifices typical of stenotic and regurgitant lesions, but is not accurate for flow through normal mitral (6) and aortic (7) valves, where more than half of the pressure drop is typically due to inertial forces, as has been demonstrated in numerical models (8), in animals (9) and clinically (10). These relatively small pressure drops are currently measurable only by multiple simultaneous high-fidelity sensors, making them essentially unavailable in clinical cardiology.

We have previously proposed that the complete Bernoulli equation could be quantified by analyzing color Doppler M-mode (CMM) echocardiograms (11), a spatiotemporal display of velocity along an ultrasound scan line. This approach assumes the CMM lies approximately along a streamline of flow, so that the Euler equation (a differential version of the Bernoulli equation) can be applied directly to the data. We recently validated this approach in a canine model (12), showing an improved correlation (from r = 0.15 to 0.85) when the inertial component was included. In this study, we sought to extend this approach to humans using intraoperative techniques (13). If successful, this approach could have significant utility in the assessment of diastolic function, because the driving force directly responsible for left ventricular (LV) filling is the instantaneous atrioventricular pressure difference during diastole.

## Methods

### Data acquisition

We studied 8 patients (7 male, age 60.4 ± 15.4 years) undergoing first-time coronary bypass surgery. Baseline hemodynamic parameters are summarized in Table 1. All patients had either trivial (n = 6) or mild mitral (n = 2) regurgitation. Left ventricular function was normal in two, mildly depressed in four, and severely depressed in two. The study was approved by the human subjects review committee of the Cleveland Clinic Foundation, and all patients provided informed consent. A pulmonary artery catheter and radial artery catheter were placed for routine hemodynamic monitoring. After induction of anesthesia a transesophageal echo probe was inserted to the midesophageal level.

#### High fidelity pressure measurements

The specifics of this intraoperative protocol have been previously described (14). Briefly, following a midline sternotomy and pericardiotomy, a high-fidelity dual sensor micromanometer catheter (Millar, Houston, Texas) was introduced through the right upper pulmonary vein. Under echocardiographic guidance and direct pressure waveform monitoring, the catheter was advanced into the left atrium (LA) and positioned across the mitral valve with the distal pressure sensor in the LV and the proximal sensor in the left atrium, 5 cm proximal to the catheter tip. Spatial alignment of the pressure waveforms with the CMM images was obtained by two-dimensional imaging from the basilar transesophageal window to identify the mitral annulus and guide the catheter to approximately 3 cm into the LV. Shorter distances result in artifact from mitral valve movement whereas longer distances interfered with LA pressure measurements. The catheter had been previously zeroed and calibrated according to manufacturer specifications. The pressure signals were amplified using a universal amplifier (Gould, Valley View, OHIO), digitized at 1,000 Hz using an AT-MIO-16 multifunction data acquisition card (National Instruments, Austin, Texas) and transferred for storage onto a Windows-based personal computer (Gateway, North Sioux City, South Dakota) using a customized acquisition program developed in the LabVIEW programming environment (National Instruments, Austin, Texas). In addition, this customized software generated a timing marker signal that was recorded in conjunction with the pressure waveforms and fed into the echocardiograph allowing for temporal alignment of the Doppler and the hemodynamic data during analysis.

#### Doppler ultrasound measurements

A 5 MHz multiplane transesophageal echo transducer was used with a Sonos 1500 or 2500 (Hewlett Packard, Andover, Massachusetts) equipped with digital storage and retrieval capabilities. From a midesophageal imaging window a basal four-chamber view was obtained. The M-mode cursor was aligned across the mitral valve in parallel with LV inflow using two-dimensional color Doppler flow mapping. The color Doppler velocities were obtained using an M-mode format, providing the spatiotemporal velocity distribution along the Doppler scanline from the LA to the LV throughout the cardiac cycle. The spatial resolution is approximately 0.5 mm, the temporal resolution is 5 ms, and the velocity resolution is 6.25% of the Nyquist velocity (typically about 3 cm/s). All images were stored onto magneto-optical disc for offline processing.

For each patient, 8-s simultaneous recordings of LA and LV intracardiac pressures and CMM Doppler were obtained during suspended respiration at baseline (following aortic cannulation, but prior to the institution of cardiopulmonary bypass) and during infusion of intravenous phenylephrine (titrated to a mean aortic pressure of 100 mm Hg). These conditions were chosen to reflect the wide range of physiologic conditions obtained clinically while minimizing the risk for ischemia in patients with known coronary disease. Each patient tolerated the protocol well and there were no complications.

### Data analysis

#### Calculation of the invasive transmitral pressure differences

The instantaneous transmitral pressure difference (Δp_{cath}[t]) was calculated from the direct pressure measurements in the LA and LV throughout the diastolic time interval. From the instantaneous pressure difference, we measured the peak pressure difference ([Δp_{cath}]_{max}) and the time from pressure crossover to peak pressure difference (t _{[Δpcath]max}) during early filling.

#### Calculation of the noninvasive transmitral pressure differences

*Hydrodynamic background.* The hydrodynamic theory normal mitral valve flow and the noninvasive assessment of transmitral gradients have been discussed in detail (11). Briefly, three-dimensional flow in the cardiac cavities is governed by the Navier–Stokes equations for incompressible fluid. If we consider flow along a streamline, the Navier–Stokes equations can be rewritten as the Euler equation describing the local pressure (p) and velocity (v) relationship (equation 1). _{C}); the second component is the inertial term (Δp_{I}), also called the acceleration component of the total pressure difference across the valve—a term that requires knowing the length of the column of blood being accelerated, and a term not available with pulsed Doppler. For restrictive orifices, the proximal velocity v_{1} in the convective component is low relative to v_{2}, which represents the maximal velocity across the area of restricted flow. Therefore, the convective term can be further simplified to equation 3, the simplified Bernoulli equation, which is routinely used during echocardiographic studies to calculate the pressure drop across stenosed valves or estimate right ventricular systolic pressure from the tricuspid regurgitant velocity profile. ^{3}) is approximately 4, yielding the familiar clinical echocardiographic equation Δp = 4v^{2}. Although these equations also contain a resistive term to account for the viscous effects of the flow, previous research has demonstrated these forces to be negligible and hence they are ignored when applied to intracardiac flow (15).

##### Color Doppler M-mode analysis

Assuming that the Doppler M-mode cursor closely approximates an inflow streamline, the color-coded Doppler velocity map displays a spatiotemporal velocity distribution v(s,t) along an inflow streamline from the LA to the LV throughout the diastolic time interval. Custom software has been developed in the LabVIEW scientific programming environment (National Instruments, Austin, Texas) to extract and calibrate the velocity information in the color Doppler images. Regions of aliased velocities (velocities exceeding the Nyquist velocity) can be identified by the user and unwrapped using an unaliasing algorithm to abstract the true velocities. These regions then underwent filtering with a 3-pixel by 3-pixel median filter. From the color Doppler M-mode images the maximum velocity profile was extracted over time and used to compute the convective component of the pressure difference throughout the diastolic time interval, Δp_{C}[t], using the simplified Bernoulli equation (equation 3). The instantaneous contribution of the inertial component was calculated as the product of blood density ρ, and the integral of the temporal acceleration term (∂v/∂t) along the ultrasound scanline from the LA (s_{LA}) to the LV (s_{LV}). The distances along the scanline were 5 cm, and consistent with the catheter placement and measurements. Once the mitral annulus was identified from the CMM image, the proximal point was 2 cm into the LA whereas the distal point was 3 cm into the LV. The noninvasive Doppler-derived transmitral pressure difference, Δp_{C+I}[t], was subsequently calculated as the sum of convective and inertial components using the unsteady form of the Bernoulli equation (equation 2)(16).

### Statistics

All statistics were performed using Systat 9.0 for Windows (SPSS, Inc, Chicago, Illinois). The Doppler-derived gradients from both the simplified Bernoulli equation (Δp_{C}[t]) and the unsteady Bernoulli equation (Δp_{C+I}[t]) were compared with the invasively measured pressure gradients (Δp_{cath}[t]) using linear regression analysis. The correlation between the invasive and noninvasive pressure differences was calculated over the time interval between the invasively obtained LA–LV pressure crossover at the beginning and end of diastole, including both the early- (E) and late (A)-filling waves. Correlation coefficients were compared after Fisher z-transformation by a paired *t* test. The magnitude (Δp_{max}) and timing (t_{(Δp)max}) of the maximum pressure gradient during early diastolic filling obtained by the Doppler-derived methods were compared with the invasive measurements using paired *t* test and linear regression analysis. The differences in magnitude (ε) and timing (Δt) of the maximum pressure gradient during early filling between the noninvasive pressure measurements and the catheter measurements were also calculated as follows:

## Results

### Doppler-derived pressure gradients using the simplified bernoulli equation

Figure 1 (a and b) shows an example of the data acquisition as described in the methods section. The instantaneous pressure difference (Δp_{C}[t]) was calculated using the simplified Bernoulli equation to yield the pressure drop due to convection. Figure 2 shows an example of the transmitral pressure difference in a representative cardiac cycle. The peak pressure difference ([Δp_{C}]_{max}) (y) during early diastole was significantly underestimated relative to the invasive measurements ([Δp_{cath}]_{max}) (x) with y = 0.31x + 0.69, r = 0.72, ε_{C} = −1.70 ± 1.39 mm Hg (Figs. 3 and 4). ⇓⇓ The pressure gradients calculated by the simplified Bernoulli equation also showed a systematic time lag compared with the invasive pressure measurements (Fig. 2). The time difference between the peak catheter pressure and the peak pressure measurements by the simplified Bernoulli equation (Δt_{C}) was 51.2 ± 16.5 ms. With the simplified Bernoulli equation the mid-diastolic gradient reversal was never seen and the overall correlation with catheter measurements throughout all of diastole was poor (r = 0.35 ± 0.24).

### Doppler-derived pressure differences using the unsteady bernoulli equation

Figure 2 also compares the invasive pressure measurements and the Doppler-derived pressure differences using the unsteady Bernoulli equation. The peak transmitral pressure differences during early diastole calculated by the unsteady Bernoulli equation ([Δp_{C+I}]_{max}) (y) approximate closely the invasive pressure differences ([Δp_{cath}]_{max}) (x) with y = 0.95x + 0.24, r = 0.96, ε_{C+I}= 0.08 ± 0.54 mm Hg (Figs. 3 and 4), significantly better (p < 0.0001) than for the simplified Bernoulli equation. The timing of the peak pressure difference calculated by the unsteady Bernoulli equation also closely coincides with the invasive peak pressure difference with Δt_{C+I} = 9.7 ± 10.9 ms, significantly (p < 0.0001) less than the temporal mismatch when only the convective term was used. Including the inertial term in the Doppler estimate showed a very good overall temporal correlation with invasive pressure measurements during diastole with r = 0.81 ± 0.15. This increase in the correlation coefficient from r = 0.35 ± 0.24 was highly significant (p < 0.001) by paired *t* testing following Fisher’s z-transformation of the individual coefficients.

## Discussion

Pressure gradients in the circulatory system and across cardiac valves are the result of a combination of convective, inertial and viscous forces (18). For normal intracardiac flow, viscous forces are very low, except in the immediate vicinity of the cardiac walls, and can be neglected. Several investigators (3–19,20) have shown that for flow through restrictive orifices, the convective forces (Δp_{C}) are dominant, and (for constant flow) Δp_{C} ≈ 1/D^{4}, where D is the orifice diameter (21). This concept has been clinically validated to calculate pressure gradients across stenotic valves noninvasively from Doppler velocities using the simplified Bernoulli equation, which only includes the convective term. However, previous in vitro (19) and theoretical (11) work has shown that as the orifice diameter increases, the convective component decreases dramatically and the inertial contribution (Δp_{I}) to the pressure gradient becomes more important. Several investigators have suggested the importance of inertial forces and how Doppler echo significantly underestimates transmitral pressure gradients by its inability to measure them (9,10). Despite this importance, inertial forces have never been assessed for flow across the normal mitral valve in humans. Beyond providing the first direct measurements of this inertial component in humans, this study also validates a clinically applicable method to quantify these gradients in an entirely noninvasive manner.

### Color doppler flow mapping

Color Doppler flow mapping provides a convenient two-dimensional color-coded map of the velocity distribution within the cardiac chambers and across valves (22). These color Doppler velocity maps are usually interpreted in a qualitative or semiquantitative way to detect the presence or absence of abnormal flows such as valvular regurgitation (23) or shunts (24–26). However, a few applications exploit the actual velocities digitally encoded in the map and most have focused on quantification of valvular regurgitation from the acceleration or momentum of flow proximal to the regurgitant orifice using various techniques (27–30). Quantitative analysis is commercially available to calculate cardiac output automatically by spatiotemporal integration of the color Doppler velocity profile across the LV outflow tract throughout the LV ejection phase (31). More recently, the change in velocity distributions as flow enters into the LV (i.e., wavefront propagation) has demonstrated that this parameter is a useful preload independent index of diastolic filling (14) and can estimate LV filling pressures when combined with other Doppler indices (32).

### Results of the current study

This study is the first to apply complex image processing techniques (calculation of spatial and temporal partial derivatives) and fundamental fluid dynamics (Euler equation) to analyze diastolic transmitral pressure gradients in humans using color Doppler M-mode echocardiography. The spatial and temporal partial derivatives were calculated from these velocities and the Euler equation was applied to yield the local pressure gradient along the scanline. Because the Euler equation is a differential expression of the Bernoulli equation (18), these data were integrated from the atrium to the ventricle to yield the time course of the transmitral pressure difference including both convective and inertial losses. Because these reference points are spatially fixed, this technique allows for determining a pressure gradient along a streamline of laminar flow (approximated by an ultrasound scanline) that is independent of intrinsic cardiac motion during early filling, such as the known backward motion of the LA (33). These pressure drops agreed closely in timing and magnitude with those recorded by a high-fidelity catheter. In contrast, pressure drops estimated by the simplified (convection only) Bernoulli equation showed a significant (∼50%) underestimation and time lag. Although the clinical significance and determinants of this underestimation are unknown, previous in vitro modeling suggest the magnitude is independent of the total transmitral gradient but related to the geometry of the mitral apparatus (34). Assuming that the Doppler scanline closely coincides with a mitral inflow streamline, the inertial component of ventricular filling can be calculated by spatial integration of flow acceleration and combined with convective forces to estimate the transmitral pressure difference. Including the inertial forces provides much more accurate estimates of the peak gradients during early filling, almost entirely correcting the systematic time lag and revealing frequently the previously missed mid-diastolic pressure difference reversal. This pressure reversal was observed in 5 of the 8 patients and in 7 of the 10 stages in these patients (average magnitude: 0.15 ± 0.08 mm Hg). Although the determinants of this reversal could not be addressed in our study, application of our methods may provide future clues as to their significance. These data are consistent with our earlier canine study, which showed improvement in the catheter-Doppler correlation of transmitral pressure drop from r_{C} = 0.15 ± 0.23 (simplified Bernoulli) to r_{C+I} = 0.85 ± 0.08 (unsteady Bernoulli, p < 0.001), with corresponding improvements in the accuracy of the magnitude (ε_{C} = 0.91 ± 0.32 mm Hg vs. ε_{C+I} = 0.01 ± 0.24 mm Hg) and timing (Δt_{C} = 36 ± 17 ms vs. Δt_{C+I}= 2 ± 10 ms) of the peak pressure drop with the inclusion of the inertial component (p < 0.001 for improvement in both parameters).

### Study limitations

The application of the Euler equation to color Doppler M-mode data assumes that LV inflow is laminar and that the ultrasound scanline closely coincides with an inflow streamline. Fortunately, numerical modeling (35,36) and magnetic resonance phase encoding studies have shown that flow is largely laminar in early filling through a normal mitral valve (37). For mitral stenosis or prosthetic mitral valves, inflow is turbulent and the assumption of laminar flow is no longer valid. However, because these orifices are more restrictive, the simplified Bernoulli equation is applicable (3,5,20).

Inflow through a normal mitral valve is not directed precisely along a Doppler scanline but typically is directed slightly laterally, following the curvature of the LV wall and leading to vortex formation from the anterior mitral leaflet (38). Consistent with clinical practice, care was taken to position the Doppler cursor through the middle of the inflow path as close to an inflow streamline as possible. Finite element modeling of transmitral flow has shown that for minor misalignment up to 15° or displacements up to a quarter of the mitral valve diameter, the correlation between the actual and reconstructed pressure difference exceeds r = 0.99 without systematic over- or underestimation (39).

Application of the Euler equation also assumes negligible transmitral viscous forces. Although these forces may be important for prosthetic valve flow (40), viscous forces appear to be important only in the immediate proximity of the valve leaflets or LV wall. Falsetti et al. (41) neglected viscous contributions in their analysis of regional LV pressure differences, using an argument that the Reynolds number for blood flow (the ratio of inertial to viscous forces) that viscous forces are approximately 1/4,000 of inertial forces.

Although our methods involve complex offline processing of digitally stored color Doppler M-mode images with ever-increasing processing speed, these algorithms could easily be incorporated within the analysis package of contemporary echocardiographs. This would allow fully noninvasive acquisition of transmitral pressure differences online, similar to the recently reported algorithms to automate calculation of cardiac output (42).

### Future directions and clinical applications

This study has demonstrated the overall importance of the inertial term in the transmitral pressure drop. Previous work has demonstrated the possibility of quantifying ventricular compliance and the relaxation time constant from characteristics of the mitral E-wave, both techniques that rely on approximations of the inertial term (43,44). Inclusion of the actual inertial term should only enhance these and other analyses of diastolic function.

The basic concept utilized to calculate transmitral pressure differences can also be extended to obtain regional intraventricular pressure differences anywhere along a streamline of laminar flow. For example, intraventricular pressure gradients between left ventricular base and apex, similar to the direct pressure measurements obtained by Courtois et al. (45) and Falsetti et al. (41), could be determined noninvasively.

This study has quantified the relative contribution of convective and inertial forces to the pressure difference across the nonrestrictive mitral valve, validating a noninvasive approach to calculate pressure differences using color Doppler M-mode echocardiography. This approach has been developed within a general theoretical framework that should be applicable to other situations of nonrestrictive flow within the heart and vasculature. This technique should provide new quantitative data of physiologic interest and create a novel approach to the complex phenomena underlying LV diastolic filling.

## Footnotes

☆ Supported in part by Grant 93-13880 from the American Heart Association, Greenfield, Texas (JT), a Grant-in-Aid from the American Heart Association Northeast Ohio Affiliate (PV), Grant 1R01HL56688, National Heart Lung and Blood Institute, Bethesda, Maryland (JT), and Grant NCC9-60, National Aeronautics and Space Administration, Houston, Texas (JT).

- Abbreviations
- ∂v/∂t
- partial derivative of velocity with respect to time
- ∂v/∂s
- partial derivative of velocity with respect to space (LA to LV)
- ∂p/∂s
- partial derivative of pressure with respect to space
- Δp
_{cath} - instantaneous transmitral pressure difference based on catheter measurement
- Δp
_{C} - convective component of the instantaneous transmitral pressure difference (using the simplified Bernoulli equation)
- Δp
_{I} - inertial component of the instantaneous transmitral pressure difference
- Δp
_{C+I} - instantaneous transmitral pressure difference derived from Doppler measurements (including both the convective and inertial components of the unsteady Bernoulli equation)
- CMM
- color Doppler M-mode
- LA
- left atrium
- LV
- left ventricle
- s
_{LV}or s_{LA} - velocity sample depth within the left ventricle or left atrium
- v
_{LV}[t] or v_{LA}[t] - velocity profile at the location of a sample volume within the left ventricle or within the left atrium

- Received February 23, 2000.
- Revision received June 1, 2000.
- Accepted July 25, 2000.

- American College of Cardiology

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