Comparison of Fluence Rates

This section compares the delta-Eddington fluence rates with exact values obtained from the adding-doubling method. The radiances were calculated using Equations (3.17) and (3.18) and the fluence rates were obtained by integrating the radiances over all $4\pi$ solid angles. The delta-Eddington approximation consistently underestimates the fluence rate found with the adding-doubling method.

Table 4.1: Delta-Eddington errors for an index matched medium. The $\Delta R$ and $\Delta T$ values are the differences between the delta-Eddington and the exact values.

g	a	$\tau$	$\Delta R$	$100{\Delta R\over R}$	$\Delta T$	$100{\Delta T\over T}$
0	0.6	1	0.010	7.5	0.011	2.3
0	0.6	2	0.011	7.0	0.007	3.3
0	0.6	4	0.010	6.5	-0.000	-0.7
0	0.6	8	0.010	6.5	-0.000	-18.1
0	0.9	1	0.005	1.9	0.010	1.6
0	0.9	2	0.009	2.5	0.011	3.2
0	0.9	4	0.011	2.7	0.004	3.4
0	0.9	8	0.011	2.7	-0.000	-2.5
0	0.99	1	-0.002	-0.6	0.004	0.6
0	0.99	2	-0.002	-0.4	0.006	1.2
0	0.99	4	0.001	0.2	0.004	1.4
0	0.99	8	0.003	0.5	0.001	1.0
0.5	0.6	1	0.007	13.3	0.006	1.0
0.5	0.6	2	0.007	11.2	0.004	1.5
0.5	0.6	4	0.007	9.6	-0.001	-0.8
0.5	0.6	8	0.007	9.5	-0.001	-14.8
0.5	0.9	1	0.006	4.7	0.005	0.7
0.5	0.9	2	0.008	4.1	0.011	2.0
0.5	0.9	4	0.011	4.0	0.008	3.4
0.5	0.9	8	0.011	3.9	0.000	0.0
0.5	0.99	1	0.001	0.9	0.000	0.0
0.5	0.99	2	-0.001	-0.2	0.004	0.7
0.5	0.99	4	-0.000	-0.0	0.006	1.4
0.5	0.99	8	0.004	0.6	0.003	1.4
0.875	0.6	1	0.002	20.6	-0.001	-0.1
0.875	0.6	2	0.001	12.2	-0.002	-0.5
0.875	0.6	4	0.001	6.1	-0.002	-1.4
0.875	0.6	8	0.001	5.5	-0.000	-2.3
0.875	0.9	1	0.004	17.1	-0.002	-0.2
0.875	0.9	2	0.006	15.1	-0.001	-0.2
0.875	0.9	4	0.008	11.5	0.002	0.3
0.875	0.9	8	0.007	8.6	0.003	1.4
0.875	0.99	1	0.004	12.6	-0.004	-0.4
0.875	0.99	2	0.006	8.9	-0.005	-0.5
0.875	0.99	4	0.007	4.9	-0.003	-0.3
0.875	0.99	8	0.005	2.0	0.006	0.9

Table 4.2: Delta-Eddington errors for an air-glass-tissue-glass-air medium. The $\Delta R$ and $\Delta T$ values are the differences between the delta-Eddington and exact values.

g	a	$\tau$	$\Delta R$	$100{\Delta R\over R}$	$\Delta T$	$100{\Delta T\over T}$
0	0.6	1	0.022	20.4	0.018	4.5
0	0.6	2	0.017	15.7	0.005	3.0
0	0.6	4	0.016	14.6	-0.002	-5.2
0	0.6	8	0.016	14.6	-0	-27.8
0	0.9	1	0.028	12	0.022	4.3
0	0.9	2	0.029	10.8	0.012	4.1
0	0.9	4	0.027	9.5	0	0.2
0	0.9	8	0.026	9.2	-0.001	-8.7
0	0.99	1	0.004	1.3	-0.002	-0.3
0	0.99	2	0.012	2.7	-0.004	-0.8
0	0.99	4	0.016	3	-0.005	-1.6
0	0.99	8	0.015	2.4	-0.003	-1.8
0.5	0.6	1	0.017	21.6	0.013	2.8
0.5	0.6	2	0.012	16.5	0.004	1.8
0.5	0.6	4	0.011	15.4	-0.002	-3.6
0.5	0.6	8	0.011	15.4	-0.001	-21.0
0.5	0.9	1	0.025	15.3	0.021	3.5
0.5	0.9	2	0.025	13.6	0.017	4.2
0.5	0.9	4	0.023	12.3	0.003	1.7
0.5	0.9	8	0.022	11.7	-0.002	-6.0
0.5	0.99	1	0.003	1.3	-0.001	-0.1
0.5	0.99	2	0.008	2.4	0	0.0
0.5	0.99	4	0.016	3.8	-0.003	-0.7
0.5	0.99	8	0.017	3.5	-0.003	-1.4
0.875	0.6	1	0.006	10.2	-0.003	-0.5
0.875	0.6	2	0.003	6.7	-0.005	-1.4
0.875	0.6	4	0.002	4.5	-0.003	-2.3
0.875	0.6	8	0.002	4.9	-0.0	-1.9
0.875	0.9	1	0.019	21.1	0.002	0.3
0.875	0.9	2	0.020	21.1	0.003	0.5
0.875	0.9	4	0.014	15.6	0.006	1.5
0.875	0.9	8	0.011	13.2	0.003	1.9
0.875	0.99	1	0.012	9.1	-0.009	-1.0
0.875	0.99	2	0.014	8.2	-0.006	-0.8
0.875	0.99	4	0.012	5.2	0.005	0.8
0.875	0.99	8	0.015	5.2	0.012	2.5

In Figure 4.1 the boundary conditions are varied to determine how the index of the refraction of the medium affects fluence calculations. The delta-Eddington approximation works best for index matched conditions, because no approximation must be made to account for total internal reflection of light at the boundary.

The total fluence is the sum of collimated and diffuse fluences. The collimated fluence dominates when the albedo is small. This explains Figure 4.2 in which the delta-Eddington approximation is better for low than for high albedos. Figure 4.3 illustrates that the delta-Eddington approximation is better for isotropic scattering than anisotropic scattering. In this graph, the Henyey-Greenstein phase function is used with g_HG=0.875. The equivalent delta-Eddington parameters (g'=0.47, f=0.77) can be calculated using Equations (1.4) and (1.5). The effective thickness $\tau'$ of the sample is 0.97. Consequently, the diffusion region is not reached in the strongly forward scattering media in four mean free paths $(\tau=4)$

Light Transport in Tissue

Comparison of Fluence Rates