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Light Transport in Tissue

Tabulated values

This section verifies the implementation of the doubling method and gives four digit reflection and transmission results for various albedos and optical thicknesses. The results from this chapter are summarized in Tables 3.1-3.6. The first four tables are calculations for slabs with matched boundary conditions. The last two tables give reflection and transmission values for a air-slide-tissue-slide-air sandwich.

Tables 3.1 and 3.2 give reflection and transmission for light normally incident on slabs of various optical depths and albedos. Isotropic scattering was assumed for these tables. Values for reflection and transmission for various optical depths $\tau=2^{-5}$ to $\tau=2^5$ and $\tau=\infty$ are identical to those tabulated by van de Hulst [60]. When the medium is conservative (a=1), large optical depths ( $\tau \sim 2^{14}$) are required before the slab becomes effectively semi-infinite. The adding-doubling method is subject to anomalous absorption arising from quadrature errors when optical depths become this large. The diffusion approximation was used to calculate reflection and transmission for conservative scattering and optical depths larger than $\tau=2^{10}$, because the diffusion approximation is more accurate in the diffusion region (large optical depths). This ensured that all entries in these tables were accurate to 0.01%.

Tables 3.3 and 3.4 give reflection and transmission for anisotropic scattering with a Henyey-Greenstein phase function. Many values in this table are also tabulated by van de Hulst ($\tau=2^0$ to $\tau=2^4$ and $\tau=\infty$) [61]. The values in Tables 3.3 and 3.4 are identical to those of van de Hulst, thus verifying the implementation for anisotropic scattering.

Correct implementation of the boundary conditions is verified by comparison with Giovanelli [16] who calculated reflection from a semi-infinite slab bounded by glass slides. The indices of refraction were $n_{\rm slab}=1.333$, $n_{\rm glass}=1.532$, and $n_{\rm outside}=1.0$. Giovanelli states that the fourth digit in his values is questionable.

	a=0.4	a=0.8	a=0.95	a=0.99
Giovanelli	.0858	.2072	.4155	.6541
Doubling	.0859	.2075	.4160	.6547

Once both anisotropy and the boundary conditions were verified, Tables 3.5 and 3.6 were calculated. These tables give reflection and transmission for a air-glass-tissue-glass-air sandwich. These values have not been tabulated elsewhere, and will serve as ``truth'' for evaluation of the delta-Eddington approximation.

S. A. Prahl."Light Transport in Tissue," PhD thesis, University of Texas at Austin, 1988.