Beyond Bernoulli: Improving the Accuracy and Precision of Noninvasive Estimation of Peak Pressure Drops

Abstract

Background: Transvalvular peak pressure drops are routinely assessed noninvasively by echocardiography using the Bernoulli principle. However, the Bernoulli principle relies on several approximations that may not be appropriate, including that the majority of the pressure drop is because of the spatial acceleration of the blood flow, and the ejection jet is a single streamline (single peak velocity value).

Methods and results: We assessed the accuracy of the Bernoulli principle to estimate the peak pressure drop at the aortic valve using 3-dimensional cardiovascular magnetic resonance flow data in 32 subjects. Reference pressure drops were computed from the flow field, accounting for the principles of physics (ie, the Navier-Stokes equations). Analysis of the pressure components confirmed that the spatial acceleration of the blood jet through the valve is most significant (accounting for 99% of the total drop in stenotic subjects). However, the Bernoulli formulation demonstrated a consistent overestimation of the transvalvular pressure (average of 54%, range 5%-136%) resulting from the use of a single peak velocity value, which neglects the velocity distribution across the aortic valve plane. This assumption was a source of uncontrolled variability.

Conclusions: The application of the Bernoulli formulation results in a clinically significant overestimation of peak pressure drops because of approximation of blood flow as a single streamline. A corrected formulation that accounts for the cross-sectional profile of the blood flow is proposed and adapted to both cardiovascular magnetic resonance and echocardiographic data.

Keywords: Bernoulli principle; biomarker; blood pressure; hemodynamics; stenosis; valve.

Figure 1.

Left, Schematics of the velocity field at the vena contracta (VC) acquired during systole with continuous Doppler (1D encoded velocity value, top) and 4D flow cardiovascular magnetic resonance (CMR; 3D-encoded 2-dimensional velocity field, bottom). Right, Definition of the anatomic regions to compute the TPD from the left ventricular outflow tract (LVOT; plane 1) to the VC (plane 2). Two other anatomic regions are defined for the Material B in the Data Supplement, the ascending aorta (AA) from the VC to the brachiocephalic artery (plane 3) and the descending aorta (DA) from the left subclavian artery (plane 4) to a plane at the same height of the aortic valve plane (plane 5).

Figure 2.

Mathematical formulations to compute a pressure drop. Compliant models can be added in the formulations labeled with (*), but this is not applied in this work. SAW indicates simplified advective WERP; SB, simplified Bernoulli; and WERP, work–energy relative pressure.

Figure 3.

Instantaneous transvalvular pressure drop (TPD) and its components computed for group I (n=20) and group II (n=12) using work–energy relative pressure (WERP) formulation. Each line with range illustrates the mean±SD of the distribution.

Figure 4.

Instantaneous transvalvular pressure drop (TPD; mean±SD values during systolic frames) estimated for groups I and II using work–energy relative pressure (WERP; left), simplified advective WERP (SAW; center), and simplified Bernoulli (SB; right) formulations.

Figure 5.

Linear regression between the reference mean transvalvular pressure drop (TPD) from 4D flow cardiovascular magnetic resonance (CMR) data using the work–energy relative pressure (WERP) formulation against the mean TPD estimated using the simplified Bernoulli (SB; black) and simplified advective WERP (SAW; gray) formulations in the 2 groups of patients. Case-specific values for subjects in group I (circles) and group II (squares), regressions for the estimation methods (solid lines), and identity line (dashed gray line).

Figure 6.

Velocity magnitude distribution from 4D flow data: in plane visualization (left) and 3D surface plot (right) in representative control (top) and stenotic (bottom) patients. The deviation from a flat profile in these 2 examples causes a simplified Bernoulli (SB) overestimation of a 20% and 136%, respectively.