= Flow-profile mismatch in Doppler stroke-volume estimation
\b[Flow-profile mismatch in Doppler stroke-volume estimation] occurs when the velocity used in a <stroke volume> calculation does not represent the spatial average over the measured opening. The exact instantaneous <volumetric flow rate> is
$$
Q(t)=\int_A\mathbf v(\mathbf x,t)\cdot\mathbf n\,dA=A\bar v(t)
$$
for fixed area $A$. Consequently $\mathrm{SV}=A\int\bar v(t)\,dt$. Replacing $\bar v$ by a centerline velocity or fastest spectral envelope requires an additional profile assumption. The ASE report distinguishes the modal part of a pulsed Doppler spectrum from the outer continuous-wave envelope representing the fastest detected cells.<Recommendations for Quantification of Doppler Echocardiography by Miguel A. Quiñones and others>[PW Doppler and CW Doppler, printed p.169]{ref}
An original family of axisymmetric profiles makes the missing factor explicit. Within a circular opening of radius $R$, suppose
$$
v(r,t)=u(t)\left[1-\left(\frac rR\right)^n\right],\qquad n>0.
$$
Here $u(t)$ is the centerline velocity. Integrating concentric annuli gives
$$
\bar v(t)=\frac{2}{R^2}\int_0^R v(r,t)r\,dr
=\frac{n}{n+2}u(t).
$$
The <Doppler profile correction factor> is therefore $\alpha=n/(n+2)$ for this stipulated family. A parabolic profile $n=2$ has $\alpha=1/2$; a blunter profile $n=8$ has $\alpha=0.8$. Both have the same centerline velocity and vanish at the wall. These shapes are examples, not fitted cardiac profiles.
\Image[/doppler-profile-peak-to-mean.png]
{title=Equal centerline velocities with different area-weighted flow means}
For $R=1\,\mathrm{cm}$ and a centerline <velocity-time integral> of $20\,\mathrm{cm}$, uncorrected multiplication by $A=\pi\,\mathrm{cm^2}$ gives $62.8\,\mathrm{mL}$. The two model profiles instead transport approximately $31.4$ and $50.3\,\mathrm{mL}$. Identical peak data can therefore support different transported volumes when the profile differs.
Real Doppler measurements also involve beam direction, sample-volume extent and signal processing. Their spectral envelope need not coincide with the literal centerline trace used in this example. The illustration isolates the averaging issue; it is not a universal factor-of-two correction for clinical measurements. The sampling site and anatomical area must be matched as well.<Recommendations for Quantification of Doppler Echocardiography by Miguel A. Quiñones and others>[PW Doppler, p.168, matched velocity and area locations]{ref}
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