Eccentric lumen stenosis

Eccentric lumen stenosis concerns a narrowed vessel lumen whose geometry breaks the symmetry of a circular, concentric reference model. Noncircular shape and displacement away from the reference centre are distinct geometric features; neither alone identifies the composition of the surrounding tissue. The practical issue is that one observed diameter may not characterize the remaining lumen area.

For an original noncircular model, take an elliptical lumen with semiaxes $a$ and $b$. Its area and area-equivalent lumen diameter areIf full principal diameters are $D_1=2a$ and $D_2=2b$, then $D_{\mathrm{eq}}=\sqrt{D_1{D}_2}$. This circle has the same area, not necessarily the same flow resistance, as the ellipse.

The width of the ellipse projected onto an axis at angle $\theta$ to its major axis isWith $a=2$ and $b=1\mathrm{mm}$, principal projected widths range from $4$ to $2\mathrm{mm}$. Relative to a circular $4\mathrm{mm}$ reference, those two widths would suggest diameter reductions $0\%$ and $50\%$, although the model's true area reduction is fixed at $50\%$. The area-equivalent diameter is about $2.828\mathrm{mm}$.

This ideal projection calculation explains why observation direction matters. It is not a complete model of a contrast-filled, curved, moving vessel. An observed image also depends on viewing geometry, overlap, resolution and the boundary-reconstruction method. Judkins' primary coronary-arteriography work used selective catheterization with serial and cine recording, providing an apparatus for observing the contrast-filled arterial route.^[ref]

Diameter versus area stenosis conversions that square one diameter ratio assume a circular model. They should not be applied automatically to a single width of an eccentric or noncircular lumen. Two principal diameters can determine the area of the stated ellipse, but a general irregular outline requires more shape information. Shape, percentage narrowing and the physiological consequence of vascular stenosis remain separate questions.

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.