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Abstract
This work analytically examines some dependences of the differential pathlength factor (DPF) for steady-state photon diffusion in a homogeneous medium on the shape, dimension, and absorption and reduced scattering coefficients of the medium. The medium geometries considered include a semi-infinite geometry, an infinite-length cylinder evaluated along the azimuthal direction, and a sphere. Steady-state photon fluence rate in the cylinder and sphere geometries is represented by a form involving the physical source, its image with respect to the associated extrapolated half-plane, and a radius-dependent term, leading to simplified formula for estimating the DPFs. With the source-detector distance and medium optical properties held fixed across all three geometries, and equal radii for the cylinder and sphere, the DPF is the greatest in the semi-infinite and the smallest in the sphere geometry. When compared to the results from finite-element method, the DPFs analytically estimated for 10 to 25 mm source–detector separations on a sphere of 50 mm radius with μa=0.01 mm(−1) and μ′s=1.0 mm(−1) are on average less than 5% different. The approximation for sphere, generally valid for a diameter≥20 times of the effective attenuation pathlength, may be useful for rapid estimation of DPFs in near-infrared spectroscopy of an infant head and for short source–detector separation.
Highlights
Near-infrared spectroscopy (NIRS)[1] is becoming an increasingly important modality for a number of investigational and clinical needs, including noninvasive functional study of neurophysiological connectivity,[2] bedside monitoring of cerebral hemodynamics,[3] and evaluating responses to cancer treatment.[4]
With the analytical procedures established for steady-state photon diffusion in an infinite-length cylinder domain that can be extended to deriving differential pathlength factor (DPF), the solution to steady-state photon diffusion in a spherical domain is developed by applying the “image” source method with the extrapolated boundary condition (EBC) and involving the modified spherical Bessel functions of the first and the second kinds
The solution is converted into a format employing the physical source and its image with respect to its associated semi-infinite geometry and a radiusdependent term accounting for the dimension of the sphere
Summary
Near-infrared spectroscopy (NIRS)[1] is becoming an increasingly important modality for a number of investigational and clinical needs, including noninvasive functional study of neurophysiological connectivity,[2] bedside monitoring of cerebral hemodynamics,[3] and evaluating responses to cancer treatment.[4]. The solution obtained for the infinite-length cylinder domain is presented to consist of two parts: the first part is associated with the “real” isotropic source that is the common approach of treating the light normally incident to the medium from the source fiber or channel, and the second part accounts for the contribution of the “image” source as a result of the medium boundary that is represented by the “real” source term scaled by a factor determined by the radius of the cylinder This solution is subsequently formatted to a form similar to that for the semi-infinite geometry except a shape-curvature-associated term that makes the solution for the cylinder domain approaching the one for semi-infinite geometry as the radius of the cylinder domain increases without bound. The sphere domain will be modeled in the spherical coordinates, where we are concerned about photon pathlength from a source at χ~ 0 1⁄4 ðr 0; θ 0; φ 0Þ to a detector at χ~ 1⁄4 ðr; θ; φÞ
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