Light Transport in Tissue

Explicit Expressions for $\varphi _d(\mathbf {r})$

The solution to the diffusion equation is given by Equation (4.123). Substituting the Green's function (4.124) from the previous subsection into the volume integral on the R.H.S. of Equation (4.123) yields

$\begin{displaymath} \int_{\mathrm{volume}} G(\mathbf{r};\mathbf{r}')S(\mathbf{r}... ..._n)\over N_n^2} {z_n\over k_n^2+1}{B_n(\rho)\over\lambda_n^2} \end{displaymath}$

(4.132)

where z_n is given by

z_n	=	$\displaystyle \sin\gamma_n[1+\exp(-\tau')(k_n\sin k_n\tau'-\cos k_n\tau')]$	(4.133)
		$\displaystyle \qquad\qquad \cos\gamma_n[k_n+\exp(-\tau')(\sin k_n\tau'+k_n\cos k_n\tau')]$

and the radial term $B_n(\rho)$ is defined as

$\displaystyle B_n(\rho)$	=	$\displaystyle K_0(\lambda_n\rho)\int_0^\rho \pi\varphi F_0(\rho') I_0(\lambda_n\rho')(\lambda_n\rho')\,d(\lambda_n\rho')$	(4.134)
	+	$\displaystyle \qquad\qquad I_0(\lambda_n\rho)\int_\rho^\infty\pi F_0(\rho') K_0(\lambda_n\rho')(\lambda_n\rho')\,d(\lambda_n\rho')$

The radial term depends on the source irradiance. If the source represents a beam of finite width ( $\rho_0$ ) with constant irradiance then [50]

$\begin{displaymath} B_n(\rho)=\cases{\pi F_0[1-\lambda_n\rho_0I_0(\lambda_n\rho)... ..._0K_0(\lambda_n\rho)I_1(\lambda_n\rho_0)]&if $\rho\ge\rho_0$.} \end{displaymath}$

(4.135)

The special case of constant uniform irradiance is achieved by letting $\rho_0\rightarrow\infty$ . Since $K_1(\lambda_n\rho_0)\rightarrow 0$ as $\rho_0\rightarrow0$ , Equation (4.135) becomes

$\begin{displaymath} B_n(\rho)=\pi F_0 \end{displaymath}$

(4.136)

The radial term for an impulse (delta function) located at the origin is

$\begin{displaymath} B_n(\rho)=\pi F_0 K_0(\lambda_n\rho) \end{displaymath}$

(4.137)

A beam with a Gaussian irradiance profile having a e^-2 radius of $\rho_0$ requires that the radial term $B_n(\rho)$ be calculated numerically using Equation (4.135) with

$\begin{displaymath} \pi F_0(\rho)=\pi F_0\sqrt{{8\over\pi}}{\exp(-2\rho^2/\rho_0^2)\over\rho_0} \end{displaymath}$

(4.138)

The surface integrals in Equation (4.123) describe the contribution from the top surface due to reflected light,

$\begin{displaymath} \int_{\zeta'=0} G(\mathbf{r};\mathbf{r}')Q'(\mathbf{r}')\,dS... ...k_n\zeta+\gamma_n)\sin\gamma_nB_n(\rho)\over N_n^2\lambda_n^2} \end{displaymath}$

(4.139)

the contribution from the bottom integral is

$\begin{displaymath} \int_{\zeta'=\tau'} G(\mathbf{r};\mathbf{r}')Q'(\mathbf{r}')... ...k_n\tau'+\gamma_n)B_n(\rho)\exp(-\tau')\over N_n^2\lambda_n^2} \end{displaymath}$

(4.140)

Collecting Equations (4.132), (4.139), and (4.140) yields

$\begin{displaymath} \varphi_d(\mathbf{r})=\sum_{n=1}^\infty {\sin(k_n\zeta+\gam... ...r h'}+{Q_0\sin(k_n\tau'+\gamma_n)\exp(-\tau')\over h'} \right] \end{displaymath}$

(4.141)

The derivative of $\varphi _d(\mathbf {r})$ with respect to $\zeta$ is

$\begin{displaymath} {\partial\varphi_d(\mathbf{r})\over\partial\zeta}=\sum_{n=1}... ...r h'}+{Q_0\sin(k_n\tau'+\gamma_n)\exp(-\tau')\over h'} \right] \end{displaymath}$

(4.142)

The derivative of $\varphi _d(\mathbf {r})$ with respect to $\rho$ is

$\begin{displaymath} {\partial\varphi_d(\mathbf{r})\over\partial\rho}=\sum_{n=1}^... ...r h'}+{Q_0\sin(k_n\tau'+\gamma_n)\exp(-\tau')\over h'} \right] \end{displaymath}$

(4.143)

Numerical summations of these series are detailed in Appendix D.

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