Regurgitant volume ($V_R$)

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 givesA 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,The technical recommendations describe this area-times-integrated-velocity approach and complementary volumetric methods.^[ref]

For stipulated $A_{\mathrm{eff}}=0.20\mathrm{c}{\mathrm{m}}^2$ and VTI 150 cm, the estimate is $30\mathrm{c}{\mathrm{m}}^3=30$ 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.

The effective regurgitant orifice area is a flow-equivalent area for a leaking heart valve: the instantaneous backward volume flow rate $Q_R(t)$ divided by the corresponding jet speed $u_j(t)$,It characterizes the effective stream passing through the leak. It need not equal the anatomical opening traced on an image, because flow contracts and its velocity is not uniform across every geometric surface. The vena contracta and the upstream convergence pattern provide complementary flow information.^[ref]

The proximal isovelocity surface area method illustrates how an area can be inferred without drawing the hole itself. Color Doppler identifies an upstream surface at a selected aliasing speed $v_a$. In an ideal hemispherical convergence model of radius $r$, conservation of mass givesContinuous-wave Doppler echocardiography supplies the jet speed. The convergence radius and jet speed must refer to matching times. The hemisphere is an assumption about flow geometry, not a property guaranteed by the instrument.^[ref]

For an original model calculation, let $r=0.8$ cm, $v_a=30$ cm/s and $u_j=500$ cm/s. Then $Q_R\simeq 120.64$ mL/s and $A_{\mathrm{eff}}\simeq 0.2413\mathrm{c}{\mathrm{m}}^2$. At otherwise fixed inputs a ten-percent radius overestimate multiplies the inferred area by $1.1^2=1.21$. The radius-squared dependence explains why a small geometric measurement error can matter substantially.

An instantaneous effective area and an integrated regurgitant volume answer different questions:If the effective area is constant, this becomes $V_R=A_{\mathrm{eff}}\mathrm{VTI}$ using the velocity-time integral. A stipulated VTI of 150 cm with the calculated constant area gives 36.19 mL. A single peak area generally cannot be pulled outside the integral when the leak changes during systole. Alternatively, $V_R/\mathrm{VTI}$ defines a velocity-weighted mean effective area over that interval. The technical report explicitly distinguishes instantaneous convergence measurements from temporally varying regurgitation.^[ref]

The proximal isovelocity surface area method estimates instantaneous flow towards a leaking heart valve from the upstream convergence field. Color Doppler supplies a contour at a chosen aliasing speed $v_a$. If that contour represents a hemisphere of radius $r$, flow approximately normal to it giveswhere the matched jet speed $u_j$ comes from continuous-wave Doppler echocardiography. The inferred effective regurgitant orifice area is a flow-equivalent area, not the area of the colored region or necessarily the anatomical hole.^[ref]

The physical idea is conservation of mass. More generally, $Q={\int }_S\mathbf{u}\cdot \mathbf{n}dS$. Replacing that integral by one speed times a surface area requires an appropriate velocity component and geometry. Doppler ultrasound measures a component along the beam, so an apparent constant-color contour is not automatically a measured three-dimensional surface of constant normal speed.

For an original hemispherical example, let $r=0.6$ cm and $v_a=40$ cm/s. The modeled backward flow is $90.48$ mL/s. With matched $u_j=500$ cm/s, the effective area is $0.1810\mathrm{c}{\mathrm{m}}^2$. If this area remains constant and the velocity-time integral is 140 cm, the modeled regurgitant volume is 25.33 mL. These are stipulated inputs, not severity thresholds.

Geometry and timing can each change that inference. In an ideal radial model restricted to a known solid angle $\Omega$, $Q=\Omega r^2{v}_a$; a hemisphere uses $\Omega =2\pi$. Thus a stipulated $\Omega =\pi$ gives half the flow at the same radius and speed. This illustrates flow-convergence angle correction; a real nonhemispherical or confined flow needs a justified geometric model rather than an arbitrary numerical adjustment.

Likewise, a radius observed at one instant must be paired with the jet speed at that instant. A peak effective area multiplied by a full-cycle velocity integral can overstate transported volume when the leak occupies only part of systole. The technical report explicitly discusses matching times, nonplanar convergence and changing regurgitant duration.^[ref] The apparatus observes an aliased velocity pattern and a jet spectrum; the flow and area are derived through the stated model.

Flow-convergence angle correction replaces the hemispherical collection area assumed by the proximal isovelocity surface area method with an area justified by the actual convergence geometry. If flow is radial and has constant normal speed $v_a$ on a spherical surface of radius $r$ spanning solid angle $\Omega$, thenThe correction concerns a flow surface, not simply the apparent opening of a heart valve in one image. The technical recommendations explicitly distinguish a nonplanar valve surface, persistent flow constraints and nonhemispherical contours.^[ref]

An original geometry comparison shows why a planar angle is insufficient without a three-dimensional model. A spherical cap within a cone of half-angle $\beta$ hasFor $\beta =60{}^{\circ }$, this gives $\Omega =\pi$, or half a hemisphere. A different model that retains a fraction of the azimuthal extent of a hemisphere gives a fraction proportional to that azimuthal angle. A measured angle must therefore be identified as a polar cone angle, an azimuthal sector or another geometric quantity before selecting a formula.

Using the stipulated spherical-cap model with $r=0.6$ cm and $v_a=40$ cm/s gives $Q=45.24$ mL/s, whereas a full hemisphere at the same radius and speed gives 90.48 mL/s. With a matched jet speed of 500 cm/s, the corresponding effective regurgitant orifice area estimates are $0.09048$ and $0.18096\mathrm{c}{\mathrm{m}}^2$. These numbers illustrate geometry, not a correction prescribed for an observed valve.

The velocity measurement adds a separate limitation. Color Doppler measures a beam-directed velocity component; a constant-color boundary need not be a sphere of constant normal flow speed. Changing the assumed collection area cannot repair an incorrectly measured radius, a nonuniform flow field or a jet velocity acquired at a different time.

The apparatus therefore provides an image and a velocity spectrum, while the corrected flow remains a model-based inference. The primary report discusses changing the aliasing contour to reduce constraint and states that a persistently unsuitable geometry may not support a simple angle correction.^[ref]