= Diameter versus area stenosis
\b[Diameter versus area stenosis] distinguishes two geometric ways to express <vascular stenosis> relative to a stated reference lumen. A percentage reduction in diameter is generally not the same percentage reduction in area.
For a circular narrowed lumen with diameter $d$ and a circular reference with diameter $d_0$, define the fractional reductions
$$
s_d=1-\frac d{d_0},\qquad s_A=1-\frac{A}{A_0}.
$$
Since $A=\pi d^2/4$, the conversion is
$$
s_A=1-(1-s_d)^2=2s_d-s_d^2,\qquad
s_d=1-\sqrt{1-s_A}.
$$
Multiply a fraction by 100 to report a percentage. A diameter change from $4$ to $2\,\mathrm{mm}$ gives $s_d=0.50$ and $s_A=0.75$. Conversely, a $50\%$ area reduction corresponds to about $29.3\%$ diameter reduction in this circular model. Converting a geometric definition is not the same thing as converting clinical grading systems.
<Eccentric lumen stenosis> shows why the shape assumption matters. Suppose the reference is a circle of diameter $4\,\mathrm{mm}$, while the narrowed lumen is an ellipse with full axes $2$ and $4\,\mathrm{mm}$. Its area is $\pi(2)(4)/4=2\pi\,\mathrm{mm^2}$, half the reference area. A single view measuring the $2\,\mathrm{mm}$ axis would show a $50\%$ diameter reduction, but treating that view as a circular lumen would incorrectly infer a $75\%$ area reduction.
The reference also matters. For the same measured narrow diameter $2\,\mathrm{mm}$, references $4$ and $5\,\mathrm{mm}$ give diameter reductions $50\%$ and $60\%$. A report should therefore identify the measurement plane or projection, the reference segment, the geometric convention and the imaging method.
For a small uncertainty in $s_d$, the circular-model sensitivity is $\mathrm ds_A/\mathrm ds_d=2(1-s_d)$. Uncertainty in the reference and narrow diameters, including shared measurement errors, must be propagated through their ratio rather than treating the percentage as a directly measured primitive quantity.
<Ultrasound> can supply vessel images and flow-related observations using a probe and coupling gel.<Carotid Duplex by the Society for Vascular Surgery>[Test description and probe procedure]{ref} A geometric stenosis estimate and a flow observation remain distinct measurements. Even a correctly converted lumen area does not determine the pressure loss or tissue perfusion without an appropriate model of the vessel and circulation.
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