Light Transport in Tissue

Conservative scattering a=1, finite slab

When the albedo is unity, then the differential Equation (4.84) becomes

$\begin{displaymath} {d^2\varphi_d(\zeta)\over d\zeta^2}=S(\zeta) \end{displaymath}$

(4.97)

The homogeneous solution is

$\begin{displaymath} \varphi_d^{homo}(\zeta)=c_1+c_2\zeta c \end{displaymath}$

(4.98)

and the particular solution is given by Equation (4.87) as with a'=1. The constants c₁ and c₂ are determined using the boundary conditions (4.90) and (4.91)

$\begin{displaymath} c_1-A_{\mathrm{top}}h'c_2=-A_{\mathrm{top}}Q'(0)-c_3(1+A_{\mathrm{top}}h'/\mu_0) \end{displaymath}$

(4.99)

and

$\begin{displaymath} c_1+c_2(\tau'+A_{\mathrm{bottom}}h')=\left\lbrace A_{\mathrm... ...(1-A_{\mathrm{bottom}}h'/\mu_0)\right\rbrace\exp(-\tau'/\mu_0) \end{displaymath}$

(4.100)

Equations (4.99) and (4.100) are two linear equations in the two unknowns c₁ and c₂. These coefficients determine the homogeneous solution for a finite slab with conservative scattering.

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