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Editorials: Editorial

Bernoulli: Still Simple, Still Useful

Sniecinski, Roman M. MD

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doi: 10.1213/ANE.0000000000000490

In 1978, Hatle et al.1 measured blood velocities across stenotic mitral valves (MVs) using a “maximal frequency estimator” in 10 patients concomitantly undergoing cardiac catheterization. By making some simple assumptions, they calculated the pressure drop using the Bernoulli equation and found excellent agreement with the values directly obtained via the catheter pressure recordings. Since that time, “pressure gradient = 4V2” has become one of the most utilized calculations in echocardiography. In this issue of Anesthesia & Analgesia, Trzcinka et al.2 investigate the use of the simplified Bernoulli equation in a very unique clinical situation.

In actuality, the “standard” Bernoulli equation is itself a simplification of a set of mathematical expressions used in fluid dynamics known as the Navier-Stokes equations (Fig. 1). They describe the relationship between flow rate and pressure gradient for incompressible fluids. These are second-order partial differential equations that are nonlinear and actually have no exact analytical solution. Applying the full Navier-Stokes equations without making any assumptions whatsoever (i.e., not assuming steady flow along a 2-dimensional axis) requires application of advanced computational fluid dynamic modeling, something impractical for clinical use. As such, all derivations of the Bernoulli equation make some assumptions about the flow through an orifice; the simplified version just happens to make the most (Table 1). Perhaps incredibly, these assumptions seem to hold true the majority of the time. In an in vitro model of the left ventricle, use of the simplified Bernoulli equation to calculate pressure drops across a 0.93-cm2 orifice yielded results almost identical to those determined via computational fluid dynamics requiring billions of calculations on a supercomputer.3

Figure 1
Figure 1:
Derivation of the Bernoulli equation. The Navier-Stokes equations describe motion along any coordinate system. Note that (A) is written in vector format and would actually consist of 3 equations in scalar format for a Cartesian coordinate system (i.e., 3 dimensions of x, y, z).JOURNAL/asag/04.02/00000539-201412000-00005/math_5MM1/v/2017-07-21T020537Z/r/image-tiffis a substantive derivative that would be applied to all coordinates in the system, thus accounting for a change in quantity as a particle moves through different regions of the system. If viscous forces are negligible (note the dropout of “μ”), Euhler’s equation of motion (B) is derived. The Bernoulli equation (C) simplifies the vector considerations by focusing on 2 points assumed to lie along the same streamline. The integral term at the beginning of the unsteady Bernoulli equation considers flow acceleration (or deceleration). When flow is steady, this term becomes zero. ρ = density, g = gravitational acceleration, p = pressure, μ = viscosity, V = velocity, z = height, ▽ is the “del” vector operator that represents the derivatives of each components of the velocity vector JOURNAL/asag/04.02/00000539-201412000-00005/math_5MM2/v/2017-07-21T020537Z/r/image-tiff.14
Table 1
Table 1:
Assumptions Made by the Simplified Bernoulli Equation

An important, although not necessarily obvious, inference of the simplified Bernoulli equation is that blood velocity achieved across multiple orifices into the same cardiac chamber should be equal since the pressure differential across them would be identical. Rewritten, the Bernoulli equation is

, indicating that the pressure differential is the only variable influencing velocity. Multiple in vitro studies performed in the 1980s have supported this.4 Relevant to the MV, there are congenital anomalies involving tissue bridges which create the so-called double-orifice MV.5 This configuration is also found with the edge-to-edge technique of MV repair, commonly referred to as the “Alfieri stitch,” first reported in the mid-1990s.6 The double orifice is created by sewing together opposing scallops of the anterior and posterior leaflets. More recently, the development of a percutaneous system (the MitraClip®) to accomplish this feat has increased interest in this unusual valve morphology.7

Elevated transmitral pressure gradients are known to occur following creation of the double-orifice MV whether done surgically or percutaneously.8,9 Unlike obtaining gradients through traditional MV repairs, however, the intraoperative echocardiographer is presented with a bit of a dilemma: with 2 (or more) orifices, where should the Doppler cursor be placed? It has been suggested, and theoretically supported by the simplified Bernoulli equation, that either orifice would yield similar, if not identical, results.10 Interestingly, some investigators advocate obtaining a gradient through the larger orifice,11 others use the smaller orifice,9 and yet others have stated “neither pulsed or continuous wave Doppler mitral velocity have been validated for use with a double-orifice valve after percutaneous MV repair.”12

The study by Trzcinka et al.2 sheds some light on this issue by applying the simplified Bernoulli equation to the double-orifice MV in vivo. In a set of 15 patients who underwent an Alfieri stitch repair of the MV, they recorded the transvalvular flow through each orifice and carefully planimetered the area of each opening using 3-dimensional transesophageal echocardiography. Not surprisingly, the orifices on either side of the Alfieri stitch were asymmetrical in area. Contrary to what the simplified Bernoulli equation would suggest, however, there was also a difference in velocities, and thus calculated gradients, as measured by continuous wave Doppler. At the same time, there was no relationship between the size differential of the 2 orifices and the mean gradient obtained through each. While differences in individual echocardiographer’s acquisition and measurement techniques can certainly contribute to these results, clearly some assumptions of the simplified Bernoulli equation did not hold true. This is not necessarily surprising, however, if one considers that the different orifices of a mechanical bileaflet valve have been shown to have different pressure drops.13 If pieces of plastic can alter the flow characteristics through an orifice, surely bunches of tissue have the potential to do the same. Unfortunately, accounting for these altered streamlines and including viscous forces requires an advanced degree in biofluid mechanics.

The good news for the busy clinician in all of this is that statistically significant findings are often clinically irrelevant ones. Trzcinka et al.2 found the mean pressure difference between the 2 orifices to be only about 1 mm Hg. Given that transmitral flow is heavily influenced by heart rate and rhythm, volume status, as well as left ventricular compliance characteristics, a 1-mm Hg difference is not likely to alter decision making. As pointed out by the authors, placing the Doppler cursor through either orifice is likely to yield clinically acceptable results. While intraoperative echocardiographers should keep the assumptions made by the Bernoulli equation in mind, these caveats should not dissuade them from applying this simple but useful tool.


Name: Roman M. Sniecinski, MD.

Contribution: This author wrote the manuscript.

Attestation: Roman M. Sniecinski approved the final manuscript.

Conflicts of Interest: Roman M. Sniecinski received research funding from Covidien and received research funding from Grifols.

This manuscript was handled by: Martin J. London, MD.


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