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Light Transport in Tissue Summation of series In the previous section it was shown that for large N, that is, when $k_N \gg k_{\mathrm{pole}}$ the eigenvalue approaches $$k_N\rightarrow {(N-1)\pi\over\tau'}\qquad\hbox{as}\qquad N\rightarrow\infty$$ (D17) in both cases. The total diffuse radiance in the three-dimensional solution to the delta-Eddington approximation is expressed as an infinite sum, Equation (4.141) $$\varphi_d(\mathbf{r})=\sum_{n=1}^\infty {\sin(k_n\zeta+\gam... ...r h'}+{Q_0\sin(k_n\tau'+\gamma_n)\exp(-\tau')\over h'} \right]$$ As $k_N\rightarrow (N-1)\pi/\tau'$ the following limits are valid $$\lambda_N^2\rightarrow k_N^2\qquad\qquad N_N^2\rightarrow{\tau'\over2}$$ $$\sin k_N\tau'\rightarrow0\qquad\qquad \cos k_N\tau'\rightarrow(-1)^{N-1}$$ (D18) $$\sin \gamma_N\rightarrow1\qquad\qquad \cos \gamma_N\rightarrow0$$ Furthermore, as $(N-1)\pi/\tau'\rightarrow\infty$, then the limit of zN may also be found $$\lambda_N^2\rightarrow k_N^2\qquad\qquad z_N\rightarrow1$$ (D19) Substituting the limiting forms of Equations (D.18) and (D.19) into Equation (4.141), yields the following expression for the total diffuse radiance in three dimensions. $$\varphi_d(\mathbf{r}) \sum_{n=1}^N\varphi_d^n(\mathbf{r}) -{S_0\tau'^3B_N(\rho)\over45 h'}{90\over\pi^4} \sum_{n=N-1}^\infty {\cos(n\pi\zeta/\tau')\over n^4}$$ = (D20) $${Q_0\tau'B_N(\rho)\over h'} \left[1+(-1)^N\exp(-\tau')\right] {6\over\pi^2}\sum_{n=N-1}^\infty {\cos(n\pi\zeta/\tau')\over n^2}$$ - The lower index is correct because $k_N\rightarrow (N-1)\pi/\tau'$. The anisotropic surface term in the equation above converges as 1/n2. This is very slow. A better method for evaluating these sums numerically is to use $$\sum_{n=N-1}^\infty {\cos(n\pi\zeta/\tau')\over n^2}= \sum_{... ...)\over n^2}- \sum_{n=1}^{N=2} {\cos(n\pi\zeta/\tau')\over n^2}$$ (D21) $$\sum_{n=N-1}^\infty {\cos(n\pi\zeta/\tau')\over n^4}= \sum_{... ...)\over n^4}- \sum_{n=1}^{N=2} {\cos(n\pi\zeta/\tau')\over n^4}$$ (D22) The infinite sums are [17, Equation 1.443] $$\sum_{n=1}^\infty {\cos(n\pi\zeta/\tau')\over n^2}= {\pi^2\o... ...u'^2}\right] \qquad\mathrm{if}\qquad 0\le{\zeta\over\tau'}\le2$$ (D23) $$\sum_{n=1}^\infty {\cos(n\pi\zeta/\tau')\over n^4}= {\pi^4\o... ...u'^4}\right] \qquad\mathrm{if}\qquad 0\le{\zeta\over\tau'}\le2$$ (D24) A similar correction exists for the derivative of the diffuse radiance with respect to z. This derivative is needed to calculate flux densities and may be used to calculate fluence rates. In the expression below, the isotropic term converges as 1/n3 and the anisotropic term coverges as 1/n. Consequently, the numerical finesse used above is even more useful for calculating the derivative $${\partial\varphi_d(\mathbf{r})\over\partial\zeta} \sum_{n=1}^N\varphi_d^n(\mathbf{r}) -{S_0\tau'^2B_N(\rho)\over3 h'}{6\over\pi^3} \sum_{n=N-1}^\infty {\sin(n\pi\zeta/\tau')\over n^3}$$ = (D25) $${Q_0 B_N(\rho)\over h'} \left[1+(-1)^N\exp(-\tau')\right] {2\over\pi}\sum_{n=N-1}^\infty {\sin(n\pi\zeta/\tau')\over n}$$ + The correction terms may be evaluated using the following infinite sum formulas $$\sum_{n=1}^\infty {\sin(n\pi\zeta/\tau')\over n^3}= {\pi^3\o... ...u'^3}\right] \qquad\mathrm{if}\qquad 0\le{\zeta\over\tau'}\le2$$ (D26) $$\sum_{n=1}^\infty {\sin(n\pi\zeta/\tau')\over n^4}= {\pi\ove... ...tau'}\right] \qquad\mathrm{if}\qquad 0\le{\zeta\over\tau'}\le2$$ (D27) Equation (D.28) cannot be used when $\zeta = 0$. Physically, this corresponds to the top boundary. The easiest method for finding the derivative at this point is to use the boundary condition (4.79). In this case, the derivative of the diffuse radiance is given by $${\partial\varphi_d(\mathbf{r})\over\partial\zeta}= {A_{\math... ..._0\pi F_0(\rho)+\varphi_d(\mathbf{r})\over A_{\mathrm{top}}h'}$$ (D28)

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