The aim in this chapter is to explain the theory underlying some quantitative and semiquantitative methods used in perioperative TEE to estimate cardiac output, to evaluate the orifice size in valve lesions, and to estimate pressures and pressure gradients. Most quantitative methods involve simple algebra, geometry, and physics; however, it is often difficult to obtain accurate data for the calculations used in these methods, so their results should always be interpreted in the light of other echocardiographic and clinical information. In this chapter, the emphasis is on basic principles, grounded, where possible, in concepts likely to be familiar to readers and therefore easy to remember. The theory presented here should inform the use of formulas commonly employed in echocardiography; their clinical application is discussed in those chapters that deal with the relevant clinical situations. These formulas are summarized in two tables: Table 21-1 (dealing with calculations based on estimations of flow) and Table 21-2 (dealing with estimations of pressure gradients, pressure acceleration, and pressure deceleration). Reference to these tables while reading the text will assist in following the development of the concepts presented in this chapter and in understanding the interrelationships among the equations.

TABLE 21-2

Equations Involving the Simplified Bernoulli Equation, Pressure Acceleration, and Pressure Deceleration

Number	Equation	Comment
21-18	<SPAN role=presentation tabIndex=0 id=MathJax-Element-20-Frame class=MathJax style="POSITION: relative" data-mathml='P1−P2=QR’>P1−P2=QRP1−P2=QR P1−P2=QR P1−P2=QR P 1 − P 2 = QR	Pressure (P), flow (Q), and resistance (R): Analogous to Ohm’s law in physics. Laminar flow is assumed. If Q = 0 or R = 0, P 1 = P 2 .
21-19	<SPAN role=presentation tabIndex=0 id=MathJax-Element-21-Frame class=MathJax style="POSITION: relative" data-mathml='ΔP=4(V22−V12)’>ΔP=4(V22−V21)ΔP=4(V22−V21) ΔP=4(V22−V12) ΔP=4(V22−V21) Δ P = 4 ( V 2 2 − V 1 2 )	The simplified Bernoulli equation (an approximation): Use Equation 21-19 if V 1 > 1.5 m/sec. Velocity (V) must be in m/sec if Δ P is to be in mm Hg. Applies to V max and V mean (but see text for details).
21-20	<SPAN role=presentation tabIndex=0 id=MathJax-Element-22-Frame class=MathJax style="POSITION: relative" data-mathml='ΔP=4V2′>ΔP=4V2ΔP=4V2 ΔP=4V2 ΔP=4V2 Δ P = 4 V 2
21-21	<SPAN role=presentation tabIndex=0 id=MathJax-Element-23-Frame class=MathJax style="POSITION: relative" data-mathml='ΔPmean=2.4Vmax2′>ΔPmean=2.4V2maxΔPmean=2.4V2max ΔPmean=2.4Vmax2 ΔPmean=2.4V2max Δ P mean = 2.4 V max 2	Mean gradient in aortic stenosis.
21-22 a	<SPAN role=presentation tabIndex=0 id=MathJax-Element-24-Frame class=MathJax style="POSITION: relative" data-mathml='PRV−systolic=PRA+ΔPRV−RA’>PRV−systolic=PRA+ΔPRV−RAPRV−systolic=PRA+ΔPRV−RA PRV−systolic=PRA+ΔPRV−RA PRV−systolic=PRA+ΔPRV−RA P RV − systolic = P RA + Δ P RV − RA	RV systolic pressure from the gradient across a regurgitant TV. Also gives pulmonary arterial systolic pressure if PV is normal.
21-23 (and see 19-1,19-2)	<SPAN role=presentation tabIndex=0 id=MathJax-Element-25-Frame class=MathJax style="POSITION: relative" data-mathml='PPA−systolic=PRV−systolic+ΔPPA−RV’>PPA−systolic=PRV−systolic+ΔPPA−RVPPA−systolic=PRV−systolic+ΔPPA−RV PPA−systolic=PRV−systolic+ΔPPA−RV PPA−systolic=PRV−systolic+ΔPPA−RV P PA − systolic = P RV − systolic + Δ P PA − RV	Pulmonary arterial systolic pressure in the presence of pulmonary stenosis (uncommon). Δ P PA−RV is obtained from the jet through PV.
21-24	<SPAN role=presentation tabIndex=0 id=MathJax-Element-26-Frame class=MathJax style="POSITION: relative" data-mathml='PPA−mean=PRA+ΔPPA−RV−early diastolic’>PPA−mean=PRA+ΔPPA−RV−early diastolicPPA−mean=PRA+ΔPPA−RV−early diastolic PPA−mean=PRA+ΔPPA−RV−early diastolic PPA−mean=PRA+ΔPPA−RV−early diastolic P PA − mean = P RA + Δ P PA − RV − early diastolic	Pulmonary arterial mean pressure using the gradient across a regurgitant PV, calculated using the peak (early) diastolic regurgitant V through the PV. RV pressure is assumed equal to RA pressure in diastole.
21-25 (and see 19-3,19-4)	<SPAN role=presentation tabIndex=0 id=MathJax-Element-27-Frame class=MathJax style="POSITION: relative" data-mathml='PPA−end diastolic=PRA+ΔPPA−RV−end diastolic’>PPA−end diastolic=PRA+ΔPPA−RV−end diastolicPPA−end diastolic=PRA+ΔPPA−RV−end diastolic PPA−end diastolic=PRA+ΔPPA−RV−end diastolic PPA−end diastolic=PRA+ΔPPA−RV−end diastolic P PA − end diastolic = P RA + Δ P PA − RV − end diastolic	Pulmonary arterial end-diastolic pressure: End-diastolic regurgitant V through the PV is taken from a shoulder, which can be identified late in diastole (as it can with aortic regurgitation)–(see Figure 21-6 ): this is used to calculate Δ P PA−RV−end diastolic . RV pressure is assumed equal to RA pressure in diastole.
21-26 (and see 19-5,19-6)	<SPAN role=presentation tabIndex=0 id=MathJax-Element-28-Frame class=MathJax style="POSITION: relative" data-mathml='PLA=PLV−systolic−ΔPLV−LA’>PLA=PLV−systolic−ΔPLV−LAPLA=PLV−systolic−ΔPLV−LA PLA=PLV−systolic−ΔPLV−LA PLA=PLV−systolic−ΔPLV−LA P LA = P LV − systolic − Δ P LV − LA	LA pressure from the gradient across a regurgitant MV: This involves subtracting a large number from another very large number to obtain a small number, and so it is susceptible to error.
21-27	<SPAN role=presentation tabIndex=0 id=MathJax-Element-29-Frame class=MathJax style="POSITION: relative" data-mathml='PLA=PRA+ΔPLA−RA’>PLA=PRA+ΔPLA−RAPLA=PRA+ΔPLA−RA PLA=PRA+ΔPLA−RA PLA=PRA+ΔPLA−RA P LA = P RA + Δ P LA − RA	LA pressure from the gradient across an ASD: With TEE, it is difficult to align the Doppler signal with the flow across an ASD, and any error in estimation of V is squared.
21-28	<SPAN role=presentation tabIndex=0 id=MathJax-Element-30-Frame class=MathJax style="POSITION: relative" data-mathml='LVEDP=Paorta−diastolic−ΔPaorta−LV’>LVEDP=Paorta−diastolic−ΔPaorta−LVLVEDP=Paorta−diastolic−ΔPaorta−LV LVEDP=Paorta−diastolic−ΔPaorta−LV LVEDP=Paorta−diastolic−ΔPaorta−LV LVEDP = P aorta − diastolic − Δ P aorta − LV	LV end-diastolic pressure from the gradient across a regurgitant AV: Δ P aorta−diastolic is usually obtained from a peripheral arterial line. V for estimating Δ P aorta−LV with Equation 21-20 is taken at the end of diastole. In practice, peripheral arterial pressure may not equal central aortic pressure in the presence of aortic regurgitation; there may be considerable error inherent in subtracting a large number from a large number to obtain a small number, and it can be difficult to obtain a satisfactory view for estimating the V of the aortic regurgitant jet.
21-29 (and 7-11)	<SPAN role=presentation tabIndex=0 id=MathJax-Element-31-Frame class=MathJax style="POSITION: relative" data-mathml='dP/dt=32mm Hgt’>dP/dt=32mm HgtdP/dt=32mm Hgt dP/dt=32mm Hgt dP/dt=32mm Hgt dP / dt = 32 mm Hg t	LV systolic function from flow acceleration in a mitral regurgitant jet: t is the time for the flow V or a mitral regurgitant jet to increase from 1 to 3 m/sec. Values >1200 mm Hg/sec are normal and <800 mm Hg/sec indicate severe dysfunction. Alternatively, t can be used directly, equivalent values being <26 msec and >40 msec, respectively.
21-30	<SPAN role=presentation tabIndex=0 id=MathJax-Element-32-Frame class=MathJax style="POSITION: relative" data-mathml='VP2=0.7VP’>VP2=0.7VPVP2=0.7VP VP2=0.7VP VP2=0.7VP V P 2 = 0.7 V P	Basis of pressure half-time (Pt 1/2 ) estimation: Pt 1/2 is the time for the jet V to fall from any given value ( V P —usually V max is chosen) to 0.7 V P . Pt 1/2 is inversely related to orifice size, so it increases as mitral stenosis becomes more severe and decreases as aortic regurgitation becomes more severe. In mitral stenosis, Pt 1/2 is inaccurate in the presence of AV block, with mechanical valves and with the compliance changes that follow valvuloplasty; it shortens with increased cardiac output and any cause of elevated left ventricular end-diastolic pressure, including more than mild aortic regurgitation (but severe aortic regurgitation may impeded the opening of the anterior leaflet of the MV and artificially prolong Pt 1/2 ). In aortic regurgitation, Pt 1/2 < 200 msec indicates a severe lesion. Pt 1/2 is shortened by increased systemic vascular resistance and reduced LV compliance and is unreliable in the presence of MV disease.
21-31 (and see 19-5,19-6)	<SPAN role=presentation tabIndex=0 id=MathJax-Element-33-Frame class=MathJax style="POSITION: relative" data-mathml='CSAMV=220Pt1/2′>CSAMV=220Pt1/2CSAMV=220Pt1/2 CSAMV=220Pt1/2 CSAMV=220Pt1/2 CSA MV = 220 Pt 1 / 2	Orifice area in mitral stenosis using Pt 1/2 : Pt 1/2 > 220 equates to CSA MV < 1 cm 2 and severe stenosis.

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