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Light Transport in Tissue Index mismatch, no incident diffuse light If the indices of refraction are mismatched and no diffuse light is incident on the slab, then boundary condition (4.22) becomes $$\int_{2\pi\,\,\mu\ge0} L_d(\mathbf{r},\hat\mathbf{s}) (\hat\... ...t\hat\mathbf{z})\,d\mathbf{\omega} \qquad\mathrm{at}\qquad z=0$$ (4.36) where $r(-\hat\mathbf{z}\cdot\hat\mathbf{s})$ is the reflection coefficient given by the Fresnel Equation (4.9). This equation states that the average downward radiance equals the reflected upward radiance. The Fresnel reflection is an even function of $\mu$, so $r(\mu)=r(-\mu)$. The R.H.S. of Equation (4.36) may be expanded using Equation (4.14) $$\int_{2\pi\,\,\mu\le0} L_d(\mathbf{r},\hat\mathbf{s}) (-\hat\mathbf{s}\cdot\hat\mathbf{z}) r(\hat\mathbf{z}\cdot\hat\mathbf{s}) \,d\mathbf{\omega} {1\over4\pi}\int_{2\pi\,\,\mu\le0} \varphi_d(\mathbf{r}) (-\hat\m... ...{s}\cdot\hat\mathbf{z}) r(\hat\mathbf{z}\cdot\hat\mathbf{s}) \,d\mathbf{\omega}$$ = $${3\over4\pi}\int_{2\pi\,\,\mu\le0} (\mathbf{F}_d(\mathbf{r})\cdot... ...{s}\cdot\hat\mathbf{z}) r(\hat\mathbf{z}\cdot\hat\mathbf{s}) \,d\mathbf{\omega}$$ + (4.37) Since the first term on the R.H.S. of Equation (4.37) is independent of the azimuthal angle, it becomes $${1\over4\pi}\int_{2\pi\,\,\mu\le0} \varphi_d(\mathbf{r}) (-\... ... =-{1\over2}\varphi_d(\mathbf{r})\int_{-1}^0r(\mu)(-\mu)\,d\mu$$ (4.38) Decomposing the flux into tangential $\hat\mathbf{t}$ and perpendicular $\hat\mathbf{z}$ components $$\mathbf{F}_d(\mathbf{r})=F_{dt}(\mathbf{r})\hat\mathbf{t}+F_{dn}(\mathbf{r})\hat\mathbf{z}$$ (4.39) allows simplification of the second term on the R.H.S. of Equation (4.37). The integral of the tangential component is zero due to the azimuthal independence $$\int_{2\pi\,\,\mu\le0} (\mathbf{F}_{dt}(\hat\mathbf{t}\cdot\... ...\cos\varphi\,d\varphi\int_{-1}^0r(\mu)\mu(1-\mu^2)^{1/2}\,d\mu$$ (4.40) The integral of the normal component of the diffuse radiant flux is $${3\over4\pi}\int_{2\pi\,\,\mu\le0} \mathbf{F}_{dn}(-\hat\mat... ...d\mathbf{\omega} =-{3\over2}F_{dn}\int_{-1}^0r(\mu)\mu^2\,d\mu$$ (4.41) Define the reflection coefficients R1 and R2 as $${R_1\over2}=\int_0^1r(\mu)\mu\,d\mu -\int_{-1}^0 r(\mu)\mu\,d\mu$$ = $$\qquad\mathrm{and}\qquad {R_2\over3}=\int_0^1r(\mu)\mu^2\,d\mu \int_{-1}^0 r(\mu)\mu^2\,d\mu$$ = (4.42) where the factors of 1/2 and 1/3 are included to ensure normalization. The symmetry of the reflection $r(\mu)=r(-\mu)$ has been used to relate the integrals in (4.42). Tables for R1 and R2 may be found in Appendix B as a function of the index of refraction ratio between the two media. Equation (4.41), definitions (4.42), and $\mathbf{F}_{\mit dn}(\mathbf{r})=mathbf{F}_d(\mathbf{r})\cdot\hat\mathbf{z}$ allow Equation (4.37) to be written $$\int_{2\pi\,\,\mu\le0} L_d(\mathbf{r},\hat\mathbf{s}) r(\hat... ...{r}) -{R_2\over2}(\mathbf{F}_d(\mathbf{r})\cdot\hat\mathbf{z})$$ (4.43) Equations (4.43) and (4.24) reduce the boundary condition (4.36) to $${1\over4} \varphi_d(\mathbf{r}) +{1\over2}(\mathbf{F}_d(\mat... ...{r}) -{R_2\over2}(\mathbf{F}_d(\mathbf{r})\cdot\hat\mathbf{z})$$ (4.44) Substituting Equation (4.25) into (4.44) and simplifying yields the following mixed inhomogeneous boundary condition for the diffuse radiance $$\varphi_d(\mathbf{r})-A_{\mathrm{top}}h{\partial \varphi_d(\... ... z}=-A_{\mathrm{top}}Q(\mathbf{r}) \qquad\mathrm{at}\qquad z=0$$ (4.45) where $${2\over3\mu_{tr}'},$$ h = $${1+R_2\over1-R_1}\qquad\mathrm{, and}\qquad$$ Atop = $$3hg'\mu_s'\pi F_0(r)\exp(-\mu_t'z/\mu_0)\mu_0(1-r_s)$$ Q(r) = (4.46) R1 and R2 are evaluated for the index of refraction ratio between the slab and the medium above the slab. The coefficient Atop may also be found by using the polynomial approximation (B.51) given in Appendix B. The boundary condition for light at the bottom boundary (located at z=d) is $$\int_{2\pi\,\,\mu\le0} L_d(\mathbf{r},\hat\mathbf{s}) (-\hat... ...t\hat\mathbf{z})\,d\mathbf{\omega} \qquad\mathrm{at}\qquad z=d$$ (4.47) Since $$\int_{2\pi\,\,\mu\ge0} L_d(\mathbf{r},\hat\mathbf{s}) r(\hat... ...{r}) +{R_2\over2}(\mathbf{F}_d(\mathbf{r})\cdot\hat\mathbf{z})$$ (4.48) Equation (4.29) reduces Equation (4.47) to $$\varphi_d(\mathbf{r})+A_{\mathrm{bottom}}h{\partial\varphi_d... ...}=A_{\mathrm{bottom}}Q(\mathbf{r}) \qquad\mathrm{at}\qquad z=d$$ (4.49) The constants h and Q(r) are defined in Equation (4.46) and Abottom is identical to Atop except that R1 and R2 are calculated using the ratio of the index of refraction of the medium beneath the slab to that above the slab. Boundary conditions (4.45) and (4.48) are appropriate for a scattering medium embedded in a non-scattering environment with a different index of refraction.

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