Abstract
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cases using an approach initially formulated independently by Prahl [Ph.D. thesis, University of Texas at Austin, 1988 (unpublished)] and Star [in Dosimetry of Laser Radiation in Medicine and Biology, edited by G. J. M\uller and D. H. Sliney (SPIE, Bellingham, WA, 1989), pp. 146--154; in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 ${a}^{\ensuremath{'}}$ 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 ${\ensuremath{\mu}}_{s}^{\ensuremath{'}}.$ 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.
Highlights
The diffusion approximation for radiative transport in turbid media is used as the conceptual basis to analyze measurements made in many diagnostic biomedical laser applications
To calculate the measured irradiance in the generalized formalism, Eqs. ͑14͒, ͑15͒, and17͒ are substituted in Eqs. ͑27͒ and28͒, while for the standard diffusion approximation cdc(r)ϭ0 and Eqs. ͑22͒ and23͒ are substituted in Eqs. ͑27͒ and28͒
Even in the far fieldi.e., at large rthe measured irradiance for radial and tangential detection are noticeably different. This demonstrates that the magnitude of the radiant flux is significant and should not be neglected. At these large distances the datasymbolsand the predictions given by both standard diffusion theorydashed curvesand the generalized diffusion model presented heresolid curvesare congruent
Summary
The diffusion approximation for radiative transport in turbid media is used as the conceptual basis to analyze measurements made in many diagnostic biomedical laser applications. It has been demonstrated that data of this type can form the basis for noninvasive discrimination between normal and diseased tissues6͔ Despite these successes, the standard diffusion approximation and its generalized variant, the P1 approximation, have significant limitations7–9͔. Following an approach proposed independently by Prahl11͔ and Star12,13͔, we provide a generalized diffusion approximation to the Boltzmann transport equation for both stationary and frequency-domain cases. This generalized model explicitly includes a collimated source in the radiance approximation. We compare the solution to that given by the standard diffusion approximation and will demonstrate that the generalized approach shown here is accurate over a broader range of optical properties and at positions proximal to a collimated source
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