Light Transport in Tissue

Index matching, diffuse light incident

When the light incident on a slab is diffuse, either in addition to the collimated incidence or as the sole light source, it is included in the boundary conditions. The diffuse radiance $L_{di}(\mathbf{r},\hat\mathbf{s})$ is assumed isotropic (Lambertian) and might be generated experimentally with an integrating sphere. Since the net diffuse radiant flux downwards equals the net flux of the incident diffuse radiance.

$\begin{displaymath} \int_{2\pi\,\,\mu\ge0} L_d(\mathbf{r},\hat\mathbf{s}) (\hat\... ...t\hat\mathbf{z})\,d\mathbf{\omega} \qquad\mathrm{at}\qquad z=0 \end{displaymath}$

(4.31)

Equation (4.31) may be simplified using Equation (4.24) and using the isotropy of the incident diffuse radiance $L_{di}(\mathbf{r},\hat\mathbf{s})$

$\begin{displaymath} {1\over4} \varphi_d(\mathbf{r}) +{1\over2}(\mathbf{F}_d(\mathbf{r})\cdot\hat\mathbf{z})=\pi L_{di}(\mathbf{r},\hat\mathbf{s}) \end{displaymath}$

(4.32)

Substituting the expression for $\mathbf{F}_d(\mathbf{r}) \cdot\hat\mathbf{z}$ from Equation (4.25) and simplifying yields

$\begin{displaymath} \varphi_d(\mathbf{r})-h{\partial \varphi_d(\mathbf{r})\over\... ... L_{di}(\mathbf{r},\hat\mathbf{s}) \qquad\mathrm{at}\qquad z=0 \end{displaymath}$

(4.33)

where

$\begin{displaymath} h={2\over3\mu_{tr}'} \qquad\mathrm{and}\qquad Q(\mathbf{r})=3hg'\mu_s'\pi F_0(r)\exp(-\mu_t'z/\mu_0)\mu_0 \end{displaymath}$

(4.34)

The anisotropic surface factor Q(r) accounts for the difference in scattering into the forward and backward hemispheres. The surface factor Q(r) is zero when scattering is isotropic (g'=0).

For completeness, when diffuse light is incident on the bottom surface of the slab then the appropriate boundary condition becomes

$\begin{displaymath} \varphi_d(\mathbf{r})+h{\partial \varphi_d(\mathbf{r})\over\... ... L_{di}(\mathbf{r},\hat\mathbf{s}) \qquad\mathrm{at}\qquad z=d \end{displaymath}$

(4.35)

where h and Q(r) are defined in Equation (4.34). Both Equations (4.33) and (4.35) assume that the scattering medium is adjacent to non scattering media. Furthermore, the non-scattering media must have the same index of refraction as the scattering media. If either of these conditions is not satisfied then these are not physically appropriate. Consequently, these boundary conditions are not particularly useful for solving multi-layered problems since adjacent tissues may have different optical properties and will both probably scatter light.

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