Forward modeling of axial trans-lumenal diffuse optical imaging with a cylindrical applicator using continuous-wave photon-illumination

The geometry of trans-lumenal diffuse optical measurement is considerably different from that of externally-applied diffuse optical imaging. In externally-applied diffuse optical imaging of breast, brain, etc, an analytic solution to the diffusion equation for a planar semi-infinite medium is often applied. This solution works accurately for planar applicator and is a good approximation for a ring applicator of considerable size. In trans-lumenal diffuse optical imaging of internal organs like the prostate, the applicator likely should have a convex surface profile for interfacing with a typically circular cross-section of the lumen. The influence of this convex applicator shape upon the photon transport is expected to cause a deviation from the solution predicted by a semi-infinite planar boundary. This interference, if available, is particularly relevant to the axial geometry in trans-lumenal diffuse optical imaging. This work investigates the analytic solution of continuous-wave photon diffusion equation for axial imaging when a cylindrical trans-lumenal applicator is used. The Green's function of the photon diffusion equation in an infinite medium geometry is expanded in cylindrical coordinates, and an image-source method is utilized to derive the analytic solution for circular concave & circular convex boundary profiles based on extrapolated boundary condition. Numerical evaluations are conducted to examine the effect of the circular boundary. Empirical solution potentially useful for calibrating the photon remission data in a circular boundary is also derived. The numerical evaluation results and the empirical solution are subject to validation against Monte Carlo simulations and experimental measurements.