Radiative transport in the diffusion approximation: An extension for highly absorbing media and small source-detector separations

The diffusion approximation to the Boltzmann transport equation is commonly used to analyze data obtained from biomedical optical diagnostic techniques. Unfortunately, this approximation has significant limitations to accurately predict radiative transport in turbid media, which constrains its applicability to highly scattering systems. Here we extend the diffusion approximation in both stationary and frequency-domain cases using an approach initially formulated independently by Prahl [Ph.D. thesis, University of Texas at Austin, 1988 (unpublished)] and Star [in Dosimetry of Laser Radiation in Medicine and Biology, edited by G. J. Müller and D. H. Sliney (SPIE, Bellingham, WA, 1989), pp. 146–154; in Optical-Thermal Response of Laser-Irradiated Tissue, edited by A. J. Welch and M. J. C. van Gemert (Plenum, New York, 1995), pp. 131–206]. The solution is presented in the stationary case for infinite media with a collimated source of finite size exhibiting spherical symmetry. The solution is compared to results given by standard diffusion theory as well as to measurements made in turbid phantoms with reduced single scattering albedos $a^{\prime }$ ranging from 0.248 to 0.997. Unlike the conventional diffusion approximation, the approach presented here provides accurate descriptions of optical dosimetry in both low and high scattering media. Moreover, it accurately describes the transition from the highly anisotropic light distributions present close to collimated sources to the nearly isotropic light distribution present in the far field. It is postulated that the ability to measure the transition between this near and far field behavior and predict it within a single theoretical framework may allow the separation of the single scattering anisotropy g from the reduced scattering coefficient ${\mu }_s^{\prime }.$ The generalized formulation of diffusion theory presented here may enable the quantitative application of present optical diagnostic techniques to turbid systems which are more highly absorbing and allow these systems to be probed using smaller source-detector separations.