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Light Transport in Tissue Index matching, no incident diffuse light If no diffuse light is incident on the top surface of the slab, then the obvious boundary condition for diffuse light in the slab at a boundary is $$L_d(\mathbf{r},\hat\mathbf{s})=0\qquad \mathrm{if}\qquad\hat\mathbf{s}\cdot\hat\mathbf{z}>0,\qquad\mathrm{at} z= 0$$ (4.21) where the vector r has the usual cylindrical coordinates $(r,z,\theta)$ and $\hat\mathbf{z}$ is directed into the slab. Condition (4.21) requires that diffuse light from all inward directions be zero at the top surface. Recalling the diffuse radiance in the diffusion approximation (Equation (4.14) $$L_d(\mathbf{r},\hat\mathbf{s}) = {1\over4\pi}\varphi_d(\mathbf{r})+{3\over4\pi}\mathbf{F}_d(\mathbf{r})\cdot\hat\mathbf{s}$$ It is evident that since both $\varphi _d(\mathbf {r})$ and Fd(r) are independent of $\hat\mathbf{s}$, both must be identically zero to satisfy condition (4.21) and the boundary condition in Equation (4.21) cannot be used. The usual choice for a boundary condition at a surface is the Marshak condition [7]. This condition requires that the diffuse radiant flux per unit area downwards at the surface equal zero. $$\int_{2\pi\,\,\mu\ge0} L_d(\mathbf{r},\hat\mathbf{s}) (\hat\... ...hat\mathbf{z})\,d\mathbf{\omega}=0 \qquad\mathrm{at}\qquad z=0$$ (4.22) The notation `` $2\pi\,\,\mu \ge 0$'' under the integral sign indicates that the integration is done over the hemisphere in which $\mu$ is positive. The positive z-direction is into the slab and the cosine angles $\mu=\hat\mathbf{s}\cdot\hat\mathbf{z}$ are all positive for directions pointing forwards or into the slab. The extra $(\hat\mathbf{s}\cdot\hat\mathbf{z})$ term is needed to project the radiance in the z-direction. Equation (4.22) may be rewritten using expansion (4.14) for the diffuse radiance $$\int_{2\pi\,\,\mu\ge0} L_d(\mathbf{r},\hat\mathbf{s}) (\hat\... ...athbf{z})(\hat\mathbf{s}\cdot\hat\mathbf{z})\,d\mathbf{\omega}$$ (4.23) Further simplification is obtained using the hemispherical Equations (C.12) and (C.16) from Appendix C, $$\int_{2\pi\,\,\mu\ge0} L_d(\mathbf{r},\hat\mathbf{s}) (\hat\... ...}) +{1\over2}(\mathbf{F}_d(\mathbf{r})\cdot\hat\mathbf{z}) = 0$$ (4.24) Recalling the relation between Fd(r) and $\varphi _d(\mathbf {r})$, Equation (4.18) $$\nabla\varphi_d(\mathbf{r})=-3\mu_{tr}'\mathbf{F}_d(\mathbf{... ...\mu_s'(1-r_s)\pi F_0(r)\exp(-\mu_t'z/\mu_0)\mu_0\hat\mathbf{z}$$ and taking the vector dot product of Equation (4.18) with $\hat\mathbf{z}$ yields upon rearrangement $$\mathbf{F}_d(\mathbf{r})\cdot\hat\mathbf{z}=-{1\over3\mu_{tr... ...mu_s'\over\mu_{tr}'}(1-r_s)\pi F_0(r)\exp(-\mu_t'z/\mu_0)\mu_0$$ (4.25) Substituting Equation (4.25) into (4.24) and simplifying yields the boundary condition for diffuse light at the upper boundary $$\varphi_d(\mathbf{r})-h{\partial \varphi_d(\mathbf{r})\over\partial z}=-Q(\mathbf{r}) \qquad\mathrm{at}\qquad z=0$$ (4.26) where (rs=0 for matched indices of refraction) $$h={2\over3\mu_{tr}'} \qquad\mathrm{and}\qquad Q(\mathbf{r})=3hg'\mu_s'\pi F_0(r)\exp(-\mu_t'z/\mu_0)\mu_0$$ (4.27) The anisotropic surface factor Q(r) results from the difference in scattering into the forward and backward hemispheres at the boundary due to anisotropic scattering. This factor is zero when scattering is isotropic (g'= 0). The boundary condition for light at the bottom boundary (located at z=d) is $$\int_{2\pi\,\,\mu\le0} L_d(\mathbf{r},\hat\mathbf{s}) (-\hat... ...hat\mathbf{z})\,d\mathbf{\omega}=0 \qquad\mathrm{at}\qquad z=d$$ (4.28) The inward normal to the slab at the bottom boundary is $-\hat\mathbf{z}$, and $(-\hat\mathbf{z}\cdot\hat\mathbf{s})$ is a positive projection angle needed to project the diffuse radiance onto the $-\hat\mathbf{z}$ axis. Since $$\int_{2\pi\,\,\mu\le0} L_d(\mathbf{r},\hat\mathbf{s}) (-\hat... ...}) -{1\over2}(\mathbf{F}_d(\mathbf{r})\cdot\hat\mathbf{z}) = 0$$ (4.29) the boundary condition (4.22) becomes $$\varphi_d(\mathbf{r})+h{\partial \varphi_d(\mathbf{r})\over\partial z}=Q(\mathbf{r}) \qquad\mathrm{at}\qquad z =d$$ (4.30) with h and Q(r) defined in Equation (4.27) above. Equations (4.26) and (4.30) are the appropriate boundary conditions for tissue embedded in a non-scattering environment. Reflection had not been considered and so these boundary conditions implicitly require the scattering media to have the same index of refraction as the tissue. Equation (4.30) is also appropriate for tissue having a black backing, since in this case light is not internally reflected (i.e., all light reaching the black rear surface is absorbed.)

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