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Light Transport in Tissue Non-conservative scattering (a'<1), finite slab The homogeneous solution is (when the albedo is not equal to one) $$\varphi_d^{homo}(\zeta)=c_1\exp(\kappa_d\zeta)+c_2\exp(-\kappa_d\zeta)$$ (4.88) where c1 and c2 depend on the boundary conditions. The complete solution is $$\varphi_d^{part}(\zeta)=\varphi_d^{homo}(\zeta)+\varphi_d^{p... ...p(\kappa_d\zeta)+c_2\exp(-\kappa_d\zeta)+c_3\exp(-\zeta/\mu_0)$$ (4.89) Where c3 is given by Equation (4.87). The boundary conditions at the top surface and bottom surfaces are given by Equations (4.79) and (4.80) $$\varphi_d(\zeta)-A_{\mathrm{top}}h'{d \varphi_d(\zeta)\over ... ...} = -A_{\mathrm{top}}Q'(\zeta)\qquad\mathrm{at}\qquad \zeta=0$$ (4.90) $$\varphi_d(\zeta)+A_{\mathrm{bottom}}h'{d \varphi_d(\zeta)\ov... ..._{\mathrm{bottom}}Q'(\zeta)\qquad\mathrm{at}\qquad \zeta=\tau'$$ (4.91) The parameter h' and $Q'(\zeta)$ are given by Equation (4.81) $$h'={2\over3(1-g'a')} \qquad\mathrm{and}\qquad Q'(\zeta)=3h'g'a'\pi F_0(\rho)\exp(-\zeta/\mu_0) \mu_0(1-r_s)$$ (4.92) Substituting Equation (4.89) into boundary condition (4.90) yields $$c_1(1-A_{\mathrm{top}}h'\kappa_d)+c_2(1+A_{\mathrm{top}}h'\k... ...rm{top}}Q'(0)-c_3\left[1+{A_{\mathrm{top}}h'\over\mu_0}\right]$$ (4.93) Substituting Equation (4.89) into boundary condition (4.91) yields $$\exp(-\kappa_d\tau')\Big[ c_1(1+A_{\mathrm{bottom}}h'\kappa_d) c_2(1-A_{\mathrm{bottom}}h'\kappa_d) \Big] =$$ + (4.94) $$\exp(-\tau'/\mu_0) \left\lbrace A_{\mathrm{bottom}}Q'(0)-c_3\left[1-{A_{\mathrm{bottom}}h'\over\mu_0}\right] \right\rbrace$$ Equations (4.93) and (4.95) are two linear equations with constant coefficients in the two unknowns c1 and c2. These equations are easily solved using determinants.

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