Light Transport in Tissue

Solution of the one-dimensional diffusion equation

The one-dimensional diffusion Equation (4.75), in dimensionless units, is

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

(4.84)

where the source $S(\zeta)$ and k_d² are given by

$\displaystyle S(\zeta)$	=	$\displaystyle -3a'[1+g'(1-a')](1-r_s)\pi F_0\exp(-\zeta/\mu_0)$
$\displaystyle \kappa_d^2$	=	3(1-a')(1-g'a')	(4.85)

The solution of Equation (4.84) is the sum of a particular solution and a homogeneous solution,

$\begin{displaymath} \varphi_d(\zeta)=\varphi_d^{homo}(\zeta)+\varphi_d^{part}(\zeta) \end{displaymath}$

(4.86)

The particular solution has the form

$\displaystyle \varphi_d^{part}(\zeta)$	=	$\displaystyle c_3\exp(-\zeta/\mu_0)$
c₃	=	$\displaystyle {-3\mu_0^2\over1-\kappa_d^2\mu_0^2}a'[1+g'(1-a')](1-r_s)\pi F_0$	(4.87)

The homogeneous solution depends on the albedo of the slab. Solutions for various cases are given in the following subsections.

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