Light Transport in Tissue

Reflection, transmission, and fluence rates in one-dimension

The diffuse radiant flux per unit area is given by Equation (4.83)

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

Projecting the above equation into the z-direction yields

$\begin{displaymath} \mathbf{F}_d(\zeta)\cdot\hat\mathbf{z}= -{h'\over2}\left[ {d... ...3g'a'(1-r_s)\pi F_0(\rho)\mu_0\exp(-\zeta/\mu_0) \mu_0 \right] \end{displaymath}$

(4.103)

where h' is defined by Equation (4.81). This represents the net fluence at a depth $\zeta$ . The average diffuse radiance for a non-conservative finite medium is given by Equation (4.89).

$\begin{displaymath} \varphi_d^{part}(\zeta) =c_1\exp(\kappa_d\zeta)+c_2\exp(-\kappa_d\zeta)+c_3\exp(-\zeta/\mu_0) \end{displaymath}$

(4.104)

The derivative of $\varphi_d(\zeta)$ is

$\begin{displaymath} {d\varphi_d^{part}(\zeta)\over d\zeta} =\kappa_d c_1\exp(\ka... ...-\kappa_d c_2\exp(-\kappa_d\zeta) -c_3/\mu_0\exp(-\zeta/\mu_0) \end{displaymath}$

(4.105)

Substituting these expressions into Equation (4.103) and dividing by the total incident intensity yields the diffuse reflection and diffuse transmission. The expressions for the diffuse reflection and diffuse transmission are

$\begin{displaymath} R_d={-\mathbf{F}_d(\zeta)\cdot\hat\mathbf{z}\over(1-r_s)\pi ... ...hat\mathbf{z}\over(1-r_s)\pi F_0\mu_0}\bigg\vert_{\zeta=\tau'} \end{displaymath}$

(4.106)

Substituting (4.103)-(4.105) into (4.106) yields

$\begin{displaymath} R_d=-\left[{h'\over2\pi F_0\mu_0}(\kappa_dc_2-\kappa_dc_1+ c_3/\mu_0)+{3\over2}h'g'a'(1-r_s)\right] \end{displaymath}$

Similarly, the transmission is

$\begin{eqnarray*} T_d&=&{h'\over2\pi F_0\mu_0}[\kappa_dc_2\exp(-\kappa_d\tau') ... ...xp(-\tau'/\mu_0)]\\ &+&{3\over2}h'g'a'(1-r_s)\exp(-\tau'/\mu_0) \end{eqnarray*}$

The equations are the diffuse reflection and diffuse transmission for a slab illuminated uniformly by collimated light. The coefficients c₁, c₂ and c₃ are given in Section 4.4. The source term for heating, or the local volumetric absorption rate is [26]

$\begin{displaymath} \Phi(\zeta)=-{dF_d\over dz}-\mu_0{dF_{\mathrm{coll}}\over dz} \end{displaymath}$

(4.107)

The collimated flux is given by Equation (4.13)

$\begin{displaymath} \mathbf{F}_{\mathrm{coll}}(\mathbf{r})=\int_{4\pi}L_{\mathrm{coll}}(\mathbf{r},\hat\mathbf{s}')\hat\mathbf{s}'\,d\omega' \end{displaymath}$

The collimated radiance $L_{\mathrm{coll}}(\mathbf{r},\hat\mathbf{s})$ is defined by Equation (4.10), and the collimated flux is

$\begin{displaymath} F_{\mathrm{coll}}=\mu_0(1-r_s)\pi F_0 \exp(-\zeta'/\mu_0) \end{displaymath}$

Taking the derivative of F_coll above and substituting Equation (4.17) for the diffuse flux in Equation (4.107) results in

$\begin{displaymath} \Phi(\zeta)=\mu_a\left[ \varphi_c(\zeta)+\mu_0(1-r_s)\pi F_0 \exp(-\zeta/\mu_0) \right] \end{displaymath}$

(4.108)

This is the one-dimensional source function for light absorbed at a depth $\zeta$ in a slab.

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