Ronald A. Kahn and Ivan S. Salgo

The Bernoulli Equation

Echocardiographic techniques may be used to estimate intracardiac and intravascular gradients and flows. Newton’s conservation of energy states that the energy within a closed system must remain the same. If blood passes through an area of stenosis, the potential energy (as represented by a high pressure) must be converted into kinetic energy, as seen as high blood flow velocities. Additionally, some energy will be expended for blood acceleration and deceleration in pulsatile systems. Finally, some energy will be lost as heat by the viscous forces generated by friction.

These relationships have been described by Bernoulli as:

(Equation 4-1)

The first term represents the kinetic energy expenditure that results in acceleration of blood over the obstruction. The second term of the equation represents unsteady acceleration and deceleration of pumping blood. These are “inertial” terms. The final term represents kinetic energy loss due to viscous friction. The Bernoulli equation uses the assumption that blood is incompressible. During clinical application, the energy expended due to the cyclic acceleration and deceleration, as well as the energy loss due to viscous forces, are negligible and may be ignored, leaving just the first term. Since both velocities are squared, and v₂ is significantly larger than v_{1 (}v₂ >> v₁₎, v₁ may be ignored as well. Thus, the equation may be simplified to: p₁ − p₂ = 0.5 ρ().

For clinical echocardiography, the simplified Bernoulli equation may be modified further to convert SI units (Pascals, kg/m²) to mm Hg. Since 1 mm Hg is equal to 133.3 Pa:

(Equation 4-2)

Determination of Intravascular Pressures

The Bernoulli equation may be used to estimate intracardiac pressures. By measuring retrograde flow velocity through a regurgitant valve, transvalvular pressures during systole may be determined, allowing for estimation of intracardiac pressures. For example, the maximum velocity of the tricuspid regurgitation (TR) jet reflects the pressure drop across a regurgitant tricuspid valve (P_TR), which is the systolic pressure difference between the right ventricle (RV) and right atrium (RA). RV systolic pressure can be obtained by adding an estimated or measured RA pressure (RAP) to the systolic pressure gradient across the tricuspid valve during systole. Assuming there is some degree of tricuspid regurgitation (TR), the peak blood flow velocity during systole across the tricuspid valve may be measured using continuous wave Doppler (CWD), as described in Chapter 3. The pressure gradient across the tricuspid valve may be calculated using the simplified Bernoulli equation (Equation 4-2). If this value is added to the estimated or measured RA pressure, the RV end-systolic pressure (RVESP) may be measured. In the absence of right ventricular outflow tract obstruction, pulmonary artery (PA) systolic pressure will be the same as RVESP. For example,

Similarly, pulmonary regurgitation (PR) velocity represents the diastolic pressure difference between the PA and the RV. Therefore, PA diastolic pressure = RV end-diastolic pressure (RVEDP) + 4 × (PR end-diastolic velocity) ², assuming that RVEDP is equal to RAP (estimated or measured). Similarly, mitral regurgitation (MR) jet velocity represents the systolic pressure difference between the left ventricle (LV) and the left atrium (LA). In patients without LVOT obstruction or aortic stenosis, systolic blood pressure (SBP) is essentially equal to LV systolic pressure; therefore, LAP is equal to SBP − 4 · × (MR) ². Finally, aortic regurgitation (AR) velocity reflects the diastolic pressure gradient between the aorta and the LV. In summary:

Stevenson compared six different echocardiographic techniques to measure PA pressure. ¹ When compared with direct measurements, some of these techniques yielded highly accurate correlations (r = 0.97), but they were not applicable in all patients.

Doppler Measurements of Flow: Velocity-Time Integral (VTI)

In addition to measuring gradients, the measurements of blood flow velocity may be used to estimate flow within a given structure. The derivative of a function is the slope of the curve at a given point, while an integral of a function is the area under the curve between two points along its x-axis. Given an equation that would describe distance transversed, the time derivative or slope at any given point would represent its velocity; the time derivative of the velocity at any given point would be its acceleration ( Fig. 4-1). Similarly, given a graph of acceleration versus time, the integral would yield a velocity measurement; the integral of a velocity-versus-time graph would yield a distance traversed. A CWD velocity profile is a display of velocity versus time. If one integrates this velocity profile between two time points (i.e., calculates the area under the curve), the distance traversed of “a region of blood” flowing during this period may be estimated. Since flow velocity is not constant throughout a flow cycle, all of the flow velocities during the entire ejection period is integrated to measure “distance traversed” of this “region of blood.” This integration of flow velocities in a given period of time is called the velocity-time integral (VTI) and yields length or distance (in centimeters [cm] or meters [m]). When flows at a particular location along the LVOT or aorta are required (i.e., spacial specificity is necessary), PWD should be used, provided the velocity does not exceed the PW Nyquist limit. High pulse repetition frequency (HPRF) may also be used. Multiple “ROI” points may be seen indicating range ambiguity of the cursor. If the velocities are too high, it is not possible to localize the jet without assumptions.

VTI may be used to calculate flow. The cross-sectional area (CSA) for a circular orifice such as the LVOT is:

(Equation 4-3)

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