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The area-equivalent lumen diameter is the diameter of a circle having the same area as a measured lumen. For positive area $A$,It converts an area observation into a length scale without asserting that the actual lumen is circular. The definition applies to an irregular outline as well as to an ideal geometric shape.

For an ellipse with full principal diameters $D_1,D_2$, area is $A=\pi D_1{D}_2/4$, soAn original ellipse with diameters $4$ and $2\mathrm{mm}$ has area $2\pi \mathrm{m}{\mathrm{m}}^2$ and area-equivalent diameter $2\sqrt{2}\simeq 2.828\mathrm{mm}$. The arithmetic mean of its two diameters is $3\mathrm{mm}$, which would define a circle of larger area.

Using an equivalent diameter in a circular area-ratio formula is valid by construction:Using an arbitrary single projected width instead is not. Eccentric lumen stenosis can show different widths in different views while retaining the same area. Primary selective coronary arteriography used serial and cine recording to observe the contrast-filled arterial route; interpreting such images still requires the chosen projection and reconstruction model.^[ref]

The equivalent diameter also differs from hydraulic diameter, $D_h=4A/P_w$, which includes wetted perimeter $P_w$. Two equal-area outlines can have different perimeters and flow resistance. An area-equivalent circle is therefore not automatically an equivalent flow model.

For small area uncertainty, differentiation gives $\mathrm{d}{D}_A/D_A=\frac{1}{2}\mathrm{d}A/A$. A $4\%$ relative standard uncertainty in area gives approximately $2\%$ in the derived diameter if other inputs are fixed. This propagates area uncertainty; it does not correct an incorrectly segmented outline or a mismatched reference region.

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 reductionsSince $A=\pi d^2/4$, the conversion isMultiply 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 $\pi (2)(4)/4=2\pi \mathrm{m}{\mathrm{m}}^2$, 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 $\mathrm{d}{s}_A/\mathrm{d}{s}_d=2(1-s_d)$. 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.^[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.