Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator III Synthetic study of continuous-wave photon fluence rate along unique spiral paths

Abstract

This is Part III of the work that examines photon diffusion in a scattering-dominant medium enclosed by a "concave" circular cylindrical applicator or enclosing a "convex" circular cylindrical applicator. In Part II of this work Zhang et al. [J. Opt. Soc. Am. A, 66 (2011)] predicted that, on the tissue-applicator interface of either "concave" or "convex" geometry, there exists a unique set of spiral paths, along which the steady-state photon fluence rate decays at a rate equal to that along a straight line on a planar semi-infinite interface, for the same line-of-sight source-detector distance. This phenomenon of steady-state photon diffusion is referred to as "straight-line-resembling-spiral paths" (abbreviated as "spiral paths"). This Part III study develops analytic approaches to the spiral paths associated with geometry of a large radial dimension and presents spiral paths found numerically for geometry of a small radial dimension. This Part III study also examines whether the spiral paths associated with a homogeneous medium are a good approximation for the medium containing heterogeneity. The heterogeneity is limited to an anomaly that is aligned azimuthally with the spiral paths and has either positive or negative contrast of the absorption or scattering coefficient over the background medium. For a weak-contrast anomaly the perturbation by it to the photon fluence rate along the spiral paths is found by applying a well-established perturbation analysis in cylindrical coordinates. For a strong-contrast anomaly the change by it to the photon fluence rate along the spiral paths is computed using the finite-element method. For the investigated heterogeneous-medium cases the photon fluence rate along the homogeneous-medium associated spiral paths is macroscopically indistinguishable from, and microscopically close to, that along a straight line on a planar semi-infinite interface.

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“…Equations (4.conC.spiral) and (4.conV.spiral) demonstrate that in concave or convex geometries of large radius there are spiral-paths along which the decay rate of photon fluence is identical to that along a straight line on a semi-infinite interface for the same line-of-sight source-detector distance. The existence of such spiral-paths for concave or convex geometry of smaller radial dimension is found numerically [21], as summarized in Fig. 3.…”

“…Based on a novel analytical approach to photon diffusion in the convex" geometry and "concave" geometry [18,19], we gained new insights to how trans-rectal DOT/FDOT measurements are different from DOT/FDOT measurements in other geometries. By comparing the photon fluence with respect to the line-of-sight source-detector distance in the concave, semi-infinite, and convex geometries, respectively, the effects of the medium-applicator interfacing curvature and the dimension of the curvature to the diffuse photon measurements are better understood [20,21,22]. The tissue is at the concave side of the circular cylindrical tissue-applicator interface, so the equivalent isotropic source of the physical source that illuminates into the medium is located closer to the center axis than the physical source is.…”

“…We also have from Eq. (4.conC) that when the source and detector are positioned along a set of spiral paths defined by the following relationship [20,21] k , i.e., identical to the rate when evaluated along a straight-line on a semi-infinite medium-applicator interface.…”

“…We also have from Eq. (4.conV) that when the source and detector are positioned along a set of spiral paths defined by the following relationship [20,21] (1.conC) and (1.conV). For a fixed source, a detector has three directions to move away from the source: along the azimuthal direction, along the longitudinal direction, and along the diagonal of the above two directions.…”

“…On a convex interface with large radius, Eq. (3.conV) indicates   2 ln d conV  reducing versus d as[21,22]…”