Light Transport in Tissue

Dimensionless form of the diffusion equation

All optical properties heretofore have been expressed in terms of dimensional parameters. This is useful for emphasizing the underlying physics of the derivation of the boundary conditions and the diffusion equation. However this notation is cumbersome and simplification is possible with the use of non-dimensional quantities. The three non-dimensional constants are the modified albedo a', the modified optical depth $\tau'$ , and the modified anisotropy factor g'. This section expresses $\tau'$ and a' in terms of the absorption coefficient $\mu_a$ , the scattering coefficient $\mu_s$ , the delta-Eddington parameter f, and the tissue thickness d.

The usual definitions for the optical depth $\tau$ and albedo a are independent of the phase function characterizing the medium.

$\begin{displaymath} a={\mu_s\over\mu_s+\mu_a} \qquad\mathrm{and}\qquad \tau=(\mu_a+\mu_s)d \end{displaymath}$

(4.72)

However, the modified optical depth $\tau'$ and modified albedo a' are specific to the delta-Eddington approximation and depend on the delta-Eddington phase function. These modified quantities depend on reduced scattering coefficients based on the delta-Eddington phase function:

$\begin{displaymath} P_{delta-E}(\cos\theta)={1\over4\pi}\left\lbrace 2f\delta(1-\cos\theta)+(1-f)(1+3g'\cos\theta)\right\rbrace \end{displaymath}$

(4.73)

The reduced scattering coefficient is defined as $\mu_s'=\mu_s(1-f)$ (Section 4.1). The modified albedo and modified optical depth are

$\begin{displaymath} a'={\mu_s'\over\mu_s'+\mu_a}={(1-f)a\over1-af} \qquad\mathrm{and}\qquad \tau'=(\mu_a+\mu_s)d=(1-af)\tau \end{displaymath}$

(4.74)

The inverse equations are

$\begin{displaymath} a={a'\over1-f+a'f} \qquad\mathrm{and}\qquad\tau=\left(1+{a'f\over1-f}\right)\tau' \end{displaymath}$

The diffusion Equation is (D.21),

$\begin{displaymath} \nabla^2\varphi_d(\mathbf{r})-3\mu_{tr}'\mu_a\varphi_d(\mathbf{r})=S(\mathbf{r}) \end{displaymath}$

(4.75)

and the source function S(r) is

$\begin{displaymath} S(\mathbf{r}) = -3\mu_s'(\mu_{tr}'+\mu_t'g')(1-r_s)\pi F_0(r)\exp(-\mu_t' z/\mu_0) \end{displaymath}$

(4.76)

The boundary conditions for mismatched indices of refraction between the slab and its non-scattering environment are given by Equation (4.45) for the top surface

$\begin{displaymath} \varphi_d(\mathbf{r})-A_{\mathrm{top}}h{\partial \varphi_d(\... ... z}=-A_{\mathrm{top}}Q(\mathbf{r}) \qquad\mathrm{at}\qquad z=0 \end{displaymath}$

and Equation (4.49) for the bottom surface

$\begin{displaymath} \varphi_d(\mathbf{r})+A_{\mathrm{bottom}}h{\partial\varphi_d... ...}=A_{\mathrm{bottom}}Q(\mathbf{r}) \qquad\mathrm{at}\qquad z=d \end{displaymath}$

The parameter h and the function Q(r) are defined by Equation (4.46).

$\begin{displaymath} h={2\over3\mu_{tr}'} \qquad\mathrm{and}\qquad Q(\mathbf{r})=3hg'\mu_s'\pi F_0(r)\exp(-\mu_t'z/\mu_0)(1-r_s) \end{displaymath}$

The variables A_top and A_bottom incorporate internal reflection of light and depend only on the index of refraction of the slab (Section 4.2.3).

Converting Equation (4.75) to non-dimensional quantities, requires replacement of the cylindrical coordinates z and r by non-dimensional variables $\zeta=z(\mu_s'+\mu_a)$ and $\rho=r(\mu_s'+\mu_a)$ . The cylindrically symmetric for of the diffusion Equation (4.75) becomes

$\begin{displaymath} {\partial^2\varphi_d(\mathbf{r})\over\partial\zeta^2} + {1\o... ...al\rho^2} - 3(1-a')(1-a'g')\varphi_d(\mathbf{r})=S(\mathbf{r}) \end{displaymath}$

(4.77)

The source term (4.76) is then

$\begin{displaymath} S(\mathbf{r})=3(1-r_s)a'[1+g'(1-a')]\pi F_0(\rho)\exp(-\zeta/\mu_0) \end{displaymath}$

(4.78)

The boundary conditions remain the same, with h replaced by h' and Q(r) by Q'(r)

$\begin{displaymath} \varphi_d(\mathbf{r})-A_{\mathrm{top}}h'{\partial \varphi_d(... ...-A_{\mathrm{top}}Q'(\mathbf{r})\qquad\mathrm{at}\qquad \zeta=0 \end{displaymath}$

(4.79)

for the top surface and

$\begin{displaymath} \varphi_d(\mathbf{r})+A_{\mathrm{bottom}}h'{\partial \varphi... ...m}}Q'(\mathbf{r})\qquad\mathrm{at}\qquad \zeta=d(\mu_a+\mu_s') \end{displaymath}$

(4.80)

for the bottom surface. The parameters h' and Q'(r) are given by

$\begin{displaymath} h'={2\over3(1-g'a')} \qquad\mathrm{and}\qquad Q'(\mathbf{r})=3h'g'a'\pi F_0(\rho)\exp(-\zeta/\mu_0)\mu_0(1-r_s) \end{displaymath}$

(4.81)

Rearranging Equation (4.18) to express the diffuse radiant flux in terms of the average diffuse radiance d yields the following equation

$\begin{displaymath} \mathbf{F}_d(\mathbf{r})=-{1\over3\mu_{tr}'}\nabla\varphi_d(... ...tr}'}(1-r_s)\pi F_0(\rho)\exp(-\zeta/\mu_0)\mu_0\hat\mathbf{z} \end{displaymath}$

(4.82)

In dimensionless parameters this is

$\begin{displaymath} \mathbf{F}_d(\mathbf{r})=-{h'\over2}{\partial\varphi_d(\math... ...r_s)\pi F_0(\rho)\mu_0\exp(-\zeta/\mu_0) \right]\hat\mathbf{z} \end{displaymath}$

(4.83)

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