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Light Transport in Tissue

Approximations for the Boundary Coefficient A

In Section 4.3 a constant A characterizing the boundary conditions for the diffusion approximation was defined. It depends on the Fresnel moments R₁ and R₂,

$$A={1+R_2\over1-R_1}$$ (B46)

Values for A may be calculated three different ways:

1.

numerical integration of Equations (B.10)-(B.12),

2.

Keijzer's approximation, or

3.

Star's approximation.

Substituting Equations (B.34) and (B.35) into Equation (B.46) gives,

$$A^{\mathrm{Keijzer}}= {1+R_2^{\mathrm{Keijzer}}\over1-R_1^{\... ...{1+\mu_c^3+(1-\mu_c^3)R_F(0)\over 1-\mu_c^2-(1-\mu_c^2)R_F(0)}$$ (B47)

which reduces to

$$A^{\mathrm{Keijzer}}={n(n^2+1)\over2}+\left(n-{1\over n}\right)\sqrt{n^2-1}$$ (B48)

Similarly using Equations (B.30) and (B.31) yields

$$A^{\mathrm{Star}}= {1+R_2^{\mathrm{Star}}\over1-R_1^{\mathrm... ...r}}} ={1+\mu_c^3\over 1-\mu_c^2}={1-\mu_c+\mu_c^2\over1-\mu_c}$$ (B49)

$$A^{\mathrm{Star}}=n^2+{(n^2-1)^{3/2}\over n}$$ (B50)

Exact values for A obtained by calculating R₁ and R₂ numerically are presented in Table B.6. These values were fitted to a cubic polynomial to find

A^cubic(n)=-0.13755n³+4.3390n²-4.90466n+1.68960 (B51)

Table B.6 shows a comparison of the various approximations for A. The errors resulting from the cubic approximation are an order of magnitude smaller than those from the Star and Keijzer approximations.

Table B.5: Difference between polynomial approximations (B.43) to (B.45) of the moments of the Fresnel reflection and the exact values. The approximations for R₁ and R₂ are much better than the approximation for R₀.

ni/nt	$\Delta R_0$	$\Delta R_1$	$\Delta R_2$
1.000	-0.115	-0.011	0.015
1.100	0.065	0.006	-0.009
1.200	0.019	0.005	-0.004
1.300	-0.016	-0.001	0.002
1.400	-0.029	-0.004	0.005
1.500	-0.023	-0.005	0.003
1.600	-0.005	-0.002	0.000
1.700	0.015	0.002	-0.003
1.800	0.024	0.004	-0.004
1.900	0.012	0.003	-0.002

Table B.6: The boundary condition A and errors in the cubic polynomial, the Keijzer and the Star approximations. The cubic approximation is much better for nearly all cases.

ni/nt	Aexact	$\Delta A^{\mathrm{cubic}}$	$\Delta A^{\mathrm{Keijzer}}$	$\Delta A^{\mathrm{Star}}$
1.00	1.000	0.010	0.000	0.000
1.10	1.353	0.015	-0.050	-0.056
1.20	1.810	0.011	-0.103	-0.683
1.30	2.346	0.004	-0.157	-0.131
1.40	2.955	0.000	-0.211	-0.632
1.50	3.636	-0.001	-0.237	-0.454
1.60	4.388	0.003	-0.322	-0.778
1.70	5.213	0.008	-0.378	-0.610
1.80	6.113	0.016	-0.434	-1.010
1.90	7.089	0.023	-0.490	-1.260

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