= Regurgitant volume
{title2=$V_R$}
\b[<Regurgitant volume>] is the amount of <blood> transported backward through a leaking <heart valve> over a specified interval, usually a <cardiac cycle>. In <mitral regurgitation>, it is backward transport from the <left ventricle> towards the <left atrium> during ventricular <systole>. It measures an integrated volume rather than a peak <velocity> or one still image of the jet.
For a fixed measurement surface with normal pointing in the backward direction, the instantaneous backward flux is the surface integral of the corresponding positive normal <velocity>. Integrating over time gives
$$
V_R=\int_{t_0}^{t_1}\int_A [\boldsymbol u\cdot\boldsymbol n]_+\,dA\,dt,
\qquad [x]_+=\max(x,0).
$$
A moving surface instead requires velocity relative to that surface. This convention separates material transported through a boundary from the motion of the boundary itself.
An imaging estimator combines an <effective regurgitant orifice area> with a <velocity-time integral>. If the effective area is approximately constant over the interval,
$$
V_R\simeq A_{\rm eff}\int u(t)\,dt=A_{\rm eff}\,\mathrm{VTI}.
$$
The technical recommendations describe this area-times-integrated-velocity approach and complementary volumetric methods.<Recommendations for Noninvasive Evaluation of Native Valvular Regurgitation by William A. Zoghbi and others>[Sections II.B.6 and III, effective orifice area and regurgitant volume]{ref}
For stipulated $A_{\rm eff}=0.20\,\mathrm{cm^2}$ and VTI 150 cm, the estimate is $30\,\mathrm{cm^3}=30$ mL. Equal peak velocities do not ensure equal VTI: two triangular velocity pulses of height 5 m/s and durations 0.10 and 0.20 s have integrals 25 and 50 cm. At the same effective area, their modeled backward volumes are 5 and 10 mL. These are illustrative pulses, not observed valve measurements.
A changing effective area cannot generally be pulled out of the time integral. A peak-area measurement multiplied by the full VTI can therefore answer a different question from integrating the changing area and velocity together. Independently, the ventricular balance $V_R=V_T-V_F$ requires compatible total and forward <stroke volumes>. The <regurgitant fraction> then divides $V_R$ by the relevant total volume; it does not replace the underlying amount measurement.
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