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Light Transport in Tissue

Conclusions

This chapter solved the diffusion equation for mismatched boundary conditions. Comparison of the approximate delta-Eddington solutions with adding-doubling calculations indicates that delta-Eddington approximation works well for calculations of reflection and transmission. It works moderately well for many fluence rate calculations, but should not be used to calculate fluence rates in tissues with high albedos and mismatched boundary conditions.

Table: Percent errors in the delta-Eddington fluence rates. Both sets of data assume that $g_{\protect\mathrm{HG}}=0.875$, a=0.99, and an optical depth of 4.0. The unmatched data corresponds to an air-glass-tissue-glass-air medium which has indices of refraction 1.0/1.5/1.4/1.5/1.0. Delta-Eddington is more accurate for matched boundaries.

	Matched			Unmatched
Depth	Exact	Diffusion	Error	Exact	Diffusion	Error
0.000	1.307	1.297	-0.7	2.866	2.191	-23.6
0.125	1.364	1.314	-3.7	2.872	2.210	-23.0
0.250	1.401	1.329	-5.1	2.878	2.228	-22.6
0.375	1.429	1.343	-6.0	2.883	2.244	-22.2
0.500	1.453	1.355	-6.7	2.889	2.259	-21.8
0.625	1.473	1.366	-7.3	2.894	2.273	-21.5
0.750	1.490	1.375	-7.7	2.899	2.284	-21.2
0.875	1.504	1.382	-8.1	2.904	2.295	-21.0
1.000	1.516	1.388	-8.4	2.908	2.304	-20.8
1.125	1.526	1.393	-8.7	2.912	2.312	-20.6
1.250	1.533	1.397	-8.9	2.916	2.319	-20.5
1.375	1.538	1.399	-9.1	2.920	2.325	-20.4
1.500	1.542	1.400	-9.2	2.924	2.329	-20.3
1.625	1.543	1.400	-9.3	2.927	2.333	-20.3
1.750	1.543	1.398	-9.4	2.930	2.335	-20.3
1.875	1.541	1.396	-9.4	2.933	2.337	-20.3
2.000	1.537	1.392	-9.4	2.936	2.337	-20.4
2.125	1.532	1.388	-9.4	2.938	2.337	-20.5
2.250	1.524	1.382	-9.3	2.941	2.335	-20.6
2.375	1.515	1.376	-9.2	2.943	2.333	-20.7
2.500	1.505	1.368	-9.1	2.945	2.330	-20.9
2.625	1.492	1.360	-8.9	2.947	2.326	-21.1
2.750	1.478	1.351	-8.6	2.950	2.321	-21.3
2.875	1.462	1.341	-8.3	2.952	2.315	-21.6
3.000	1.444	1.330	-7.9	2.955	2.309	-21.9
3.125	1.424	1.318	-7.4	2.957	2.302	-22.2
3.250	1.402	1.305	-6.9	2.960	2.294	-22.5
3.375	1.377	1.292	-6.2	2.963	2.286	-22.9
3.500	1.349	1.278	-5.3	2.967	2.277	-23.3
3.625	1.318	1.263	-4.2	2.971	2.267	-23.7
3.750	1.281	1.248	-2.6	2.975	2.257	-24.1
3.875	1.236	1.232	-0.3	2.980	2.246	-24.6
4.000	1.162	1.215	4.6	2.985	2.234	-25.2

Figure: Comparison of delta-Eddington (solid lines) and adding-doubling (squares) fluence rates. The optical properties are $g_{\protect\mathrm{HG}}=0.875$, a=0.99, and an optical depth of 4.0. The upper curves for an air-glass-tissue-glass-air medium have indices of refraction 1.0/1.5/1.4/1.5/1.0.

Figure: Comparison of delta-Eddington (solid lines) and adding-doubling (squares) fluence rates. The optical properties are $g_{\protect\mathrm{HG}}=0.875$, $\tau =4$, and an air-glass-tissue-glass-air medium with indices of refractions 1.0/1.5/1.4/1.5/1.0. Differences between the two methods decrease with decreasing albedo.

Figure: Comparison of delta-Eddington (solid lines) and adding-doubling (squares) fluence rates. Both sets of lines assume a=0.99, $\tau =4$, and an air-glass-tissue-glass-air medium with indices of refraction 1.0/1.5/1.4/1.5/1.0.

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