# On the geometry dependence of differential pathlength factor for near-infrared spectroscopy. I. Steady-state with homogeneous medium

### D. Piao *et al.* (2015)

#### Summary

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 *m*_{a} = 0.01 mm^{-1} and
*m'*_{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.