= Flow-convergence angle correction
\b[<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> $\Omega$, then
$$
Q=\Omega r^2v_a,\qquad \frac{Q}{Q_{\rm hemisphere}}=\frac{\Omega}{2\pi}.
$$
The 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.<Recommendations for Noninvasive Evaluation of Native Valvular Regurgitation by William A. Zoghbi and others>[Section II.B.6.c, p. 313, Shape of PISA]{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 $\beta$ has
$$
\Omega=\int_0^{2\pi}\int_0^\beta\sin\theta\,d\theta\,d\varphi
=2\pi(1-\cos\beta).
$$
For $\beta=60\,{}^\circ$, 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 $0.18096\,\mathrm{cm^2}$. 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.<Recommendations for Noninvasive Evaluation of Native Valvular Regurgitation by William A. Zoghbi and others>[p. 313, constrained convergence and limits of reporting a corrected estimate]{ref}
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