Core Doppler Math: Modified Bernoulli, VTI, Stroke Volume & the Continuity Equation

Why Doppler Math Matters

Measurement Techniques, Maneuvers, and Sonographic Views make up 25% of the AE content outline, and the calculations in this section are the mathematical foundation for nearly every quantitative hemodynamic assessment tested: pressure gradients, valve areas, stroke volume, and cardiac output. Hotspot and calculation-style items commonly present a traced Doppler envelope or a set of measured diameters and ask the candidate to apply one of these formulas directly, so accurate recall of each constant matters as much as conceptual understanding. Master these four relationships and the advanced calculations in Section 6.5, plus the valve-severity grading in Chapters 7-8, become straightforward substitutions into formulas you already know.

The Modified Bernoulli Equation

The full Bernoulli equation relates pressure and velocity across a flow convergence but includes terms for flow acceleration, viscous friction, and gravitational potential. In clinical echocardiography those terms are negligible for the high-velocity jets crossing stenotic or regurgitant orifices, so the equation simplifies to the modified Bernoulli equation:

ΔP = 4V²

where ΔP is the peak (or instantaneous) pressure gradient in mmHg and V is the peak (or instantaneous) jet velocity in m/s measured by CW Doppler. A 4 m/s aortic jet therefore corresponds to a 64 mmHg peak gradient (4 × 4² = 64 mmHg).

The simplified form assumes the proximal (upstream) velocity is low enough (<1 m/s) to be ignored. When the proximal velocity is elevated — a high-output state, a serial stenosis, or a small LVOT — the full modified equation must be used instead: ΔP = 4(V2² − V1²), where V2 is the distal (stenotic) velocity and V1 is the proximal velocity. Ignoring an elevated proximal velocity overestimates the gradient. Accurate gradients also require the Doppler beam to be as parallel to flow as possible — because velocity is squared in the equation, a beam-alignment error produces an even larger error in the calculated pressure gradient.

Velocity-Time Integral (VTI)

The velocity-time integral is the area traced under the spectral Doppler envelope for one cardiac cycle, measured in centimeters. Physically, VTI represents the "stroke distance" — how far a column of blood travels past the sample site with each beat. VTI is obtained by tracing the modal (outer-edge) velocity of the PW or CW spectral waveform and is the building block for stroke volume, cardiac output, the continuity equation, and — later in this guide — the dimensionless index and the Qp/Qs shunt ratio.

Stroke Volume and Cardiac Output

Because flow through a cylindrical column equals cross-sectional area times the distance the blood travels, stroke volume (SV) at any site is:

SV = CSA × VTI

For a circular structure such as the LVOT, cross-sectional area is calculated from the measured diameter (D, in cm):

CSA = π(D/2)² = 0.785 × D²

So the standard LVOT-based stroke volume is SV = 0.785 × (LVOT diameter)² × LVOT VTI, in mL. Multiplying stroke volume by heart rate gives cardiac output:

CO = SV × HR (mL/min, divided by 1000 for L/min)

Cardiac index (CI) further divides CO by body surface area. Typical adult resting values: LVOT diameter ≈1.8–2.2 cm, LVOT VTI ≈18–22 cm, SV ≈60–100 mL, CO ≈4–8 L/min, CI ≈2.5–4.0 L/min/m².

The Continuity Equation

The continuity equation applies conservation of mass to two sites in series along the same flow stream (no shunt or regurgitation between them): flow proximal to a stenotic valve equals flow through the stenotic orifice during the same ejection period, so CSA₁ × VTI₁ = CSA₂ × VTI₂, or equivalently A₂ = A₁ × V₁ / V₂ using peak velocities in place of VTIs. Applied to aortic stenosis:

AVA = (CSA_LVOT × VTI_LVOT) / VTI_AV

using the LVOT as the known reference site and the aortic valve as the unknown orifice. The velocity-based shortcut (A₂ = A₁V₁/V₂) is faster but slightly less accurate since it ignores the shape of the flow envelope. A related, LVOT-diameter-independent shortcut is the dimensionless index (DVI) — VTI_LVOT ÷ VTI_AV, or equivalently V_LVOT ÷ V_AV — useful when the LVOT diameter is difficult to measure reliably; severity cutoffs for AVA and DVI are covered with aortic stenosis in Chapter 7.1.

Worked example: LVOT diameter 2.0 cm, LVOT VTI 24 cm, AV VTI 120 cm.

1. CSA_LVOT = 0.785 × 2.0² = 3.14 cm²
2. SV = 3.14 × 24 = 75.4 mL (near-normal)
3. AVA = (3.14 × 24) / 120 = 75.4 / 120 = 0.63 cm² — in the severe aortic stenosis range

Formula Reference Table

These relationships recur throughout the exam's Measurement Techniques domain and are prerequisites for the advanced hemodynamic calculations in Section 6.5.