## Ib

Theoretically, stroke volume can be calculated at any valve annulus.96,97,100-102 In clinical practice, however, this is not always possible (e.g., it is difficult to obtain an accurate diameter of the pulmonary artery in every patient). Because the measurement of annular radius is squared in the computation of area, it is the most important source of error of Doppler stroke-volume analyses. Stroke-volume analysis through the mitral annulus is cumbersome; it is uncertain whether the mitral annulus is best described as a circle or an ellipse, and the cross-sectional area of the annulus probably changes slightly during diastole. Calculations using the tricuspid annulus are hampered by similar problems. Despite these limitations, measurements of stroke volume through the various cardiac valves are clinically useful and can be used to calculate pulmonary-to-systemic shunt ratios, regurgitant volumes,103-110 and orifice areas of stenotic valves by the continuity equation111-115 (see below).

### The Bernoulli Equation

An important application of Doppler echocardiography is the calculation of pressure gradients within the cardiovascular system using a modification of the Bernoulli equation.116-118 This theorem states that the pressure drop across a discrete stenosis in the heart or vasculature occurs because of energy loss due to three processes: (1) acceleration of blood through the orifice (convective acceleration), (2) inertial forces (flow acceleration), and (3) resistance to flow at the interfaces between blood and the orifice (viscous friction).119- Therefore, the pressure drop across any orifice can be calculated as the sum of these three variables (Fig. 13-41). In most clinical situations, the contribution of inertial forces and viscous friction are minimal and can be discounted. Since convective acceleration is determined by velocity, the pressure gradient can be calculated from the velocities of blood proximal to and at the level of an orifice as gradient = 4[(orifice velocity)2 - (proximal velocity)2]. If the blood velocity proximal to the stenosis is low (<1.0 m/s), this term can be ignored as well. The resulting modified equation states that the pressure gradient across a discrete orifice is equal to four times the square of the peak velocity (V) through the stenosis (PG = 4^/2)116-119

Figure 13-41: The modified Bernoulli equation. Pressure drop across a small orifice can be estimated as four times the square of the peak velocity (if the proximal velocity is less than 1 m/s). V1 and P1 = proximal velocity and pressure; V2 and P2 = distal velocity and pressure. (Modified from Pearlman AS. Technique of Doppler and color flow Doppler in the evaluation of cardiac disorders and function. In: Schlant RC, Alexander RW, eds. The Heart, Arteries, and Veins, 8th ed. New York: McGraw-Hill; 1994:2229, with permission.)

Figure 13-41: The modified Bernoulli equation. Pressure drop across a small orifice can be estimated as four times the square of the peak velocity (if the proximal velocity is less than 1 m/s). V1 and P1 = proximal velocity and pressure; V2 and P2 = distal velocity and pressure. (Modified from Pearlman AS. Technique of Doppler and color flow Doppler in the evaluation of cardiac disorders and function. In: Schlant RC, Alexander RW, eds. The Heart, Arteries, and Veins, 8th ed. New York: McGraw-Hill; 1994:2229, with permission.)

The modified Bernoulli equation can be used to calculate pressure gradients across any flow-limiting orifice and has been validated against invasive measurements.116-122 The method was originally applied to aortic, mitral, and pulmonic stenosis, but further uses have been identified. If at least trivial valvular regurgitation is present, systolic gradients across the tricuspid and end-diastolic gradients across the pulmonic valve can be calculated.123,124 If the RV diastolic pressure is known (or estimated as the right atrial or central venous pressure), peak RV and pulmonary artery pressure (assuming pulmonic stenosis is absent) can be computed as follows125 126:

Peak pulmonary artery pressure = 4(TR velocity)2

End-diastolic pulmonary artery pressure (PAD) also can be calculated: PAD = 4(end-diastolic pulmonary regurgitation velocity)*

In the presence of mitral regurgitation, a variety of calculations can be made. With measurement of peak systolic arterial pressure, systolic left atrial pressure can be estimated127:

Left atrial systolic pressure = systolic blood pressure

Further, the acceleration of the MR jet can be used to estimate LV systolic dP/dt.128 Thus, from the Bernoulli equation, the LA-to-LV pressure gradients at regurgitant velocities of 1 and 3 m/s are 4 and 36 mmHg, respectively. Therefore, dP/dt can be calculated as 32 mmHg divided by the time (in seconds) required for the mitral regurgitant jet to accelerate from 1 to 3 m/s. In the case of ventricular septal defects or aortopulmonary shunts, measurements of the peak systolic arterial pressure and the peak Doppler velocity across the defect allows calculation of the right ventricular (or pulmonary arterial) systolic pressure.

### The Continuity Equation

Although transvalvular pressure gradients can be calculated from CW Doppler recordings using the modified Bernoulli equation, gradients sometimes can be misleading in the evaluation of valvular stenosis. The transvalvular gradient is determined by both the size of the stenotic orifice and the stroke volume traversing it. Severe aortic stenosis and accompanying LV systolic dysfunction may produce a low transvalvular gradient despite a small valve area, while coexistent aortic regurgitation may result in a large gradient with only mild aortic stenosis. The calculation of orifice area by Doppler echocardiography employs the continuity equation, which is derived from the law of the conservation of mass and states that the product of cross-sectional area and velocity is constant in a closed system of flow129 (Fig. 13-42). Thus, in the case of aortic stenosis, the product of the area and velocity of the left LV outflow tract equals the product of the area and velocity of the aortic valve orifice. Annulus diameter and integrated velocity measurements are derived by the standard volumetric approach, while the velocity across the stenotic orifice is derived by CW Doppler. The equation is then solved for the valve area.111-116

Figure 13-42: The continuity equation. In a closed system (top) with constant flow, Qi = Q2. Therefore, Aj x Vj must equal A2 x V^. Determination of any three of the variables allows calculation of the fourth. Clinically (bottom), the area of the left ventricular outflow tract (LVOT) can be estimated and used to determine aortic valve area. (From Hagan AD, DeMaria AN. Clinical Applications of Two-Dimensional Echocardiography and Cardiac Doppler. Boston: Little, Brown; 1989, with permission.)

Figure 13-42: The continuity equation. In a closed system (top) with constant flow, Qi = Q2. Therefore, Aj x Vj must equal A2 x V^. Determination of any three of the variables allows calculation of the fourth. Clinically (bottom), the area of the left ventricular outflow tract (LVOT) can be estimated and used to determine aortic valve area. (From Hagan AD, DeMaria AN. Clinical Applications of Two-Dimensional Echocardiography and Cardiac Doppler. Boston: Little, Brown; 1989, with permission.)

The continuity equation is simple and the constituent factors are readily measured, but a number of potential errors can occur. The most common pitfall is an inaccurate estimation of the cross-sectional area proximal to the stenosis. In addition, it is essential that blood velocity proximal to a stenosis be measured outside the area of flow acceleration. Finally, the continuity equation actually solves for the area of the vena contracta, which is usually just distal to the stenotic orifice. Although this area is very similar to the area of the stenotic orifice, occasional discrepancies occur.

### Determinants of the Size of Flow Disturbances

Although CFD yields primarily qualitative information, it is unique in its ability to provide measurements of the size of flow disturbances. It is logical that the size of a turbulent jet should correlate with the volume of blood contained within the flow disturbance. Regardless of the lesion, however, the area of turbulence recorded by CFD has multiple determinants.130-134 The volume of flow present in the disturbance is, of course, a major factor in its size. The pressure gradient operative in any flow disturbance is also an important determinant of the spatial distribution or "spray area" of turbulence.134 In addition, the size of a flow disturbance is influenced by the orifice through which flow occurs as well as the size and compliance of the receiving chamber.130-136 Finally, a number of technical factors can influence jet size as imaged by CFD, including instrument gain, the angle of incidence of the interrogating beam, the frequency and pulse repetition rate of the transducer, and the temporal sampling rate.!37 Therefore, measurements derived from the size of the turbulent jet recorded by color Doppler, are at best semiquantitative and should not be expected to correlate with the volume of blood contained in the flow disturbance.

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