Light Transport in Tissue

Derivation of the adding-doubling method

This derivation follows Plass et al. [47] with the terms representing internal sources omitted for clarity. Define T_nm and R_nm as the transmission and reflection operators for light incident on side n and exiting side $\mu$ . Homogeneous tissues have no preferred direction and so the matrices are equal T_nm=T_mn and R_nm=R_mn (The matrices are also symmetric.) Let the vector $L_0^+(\mu)$ denote the radiance incident from on side 0 of the slab 01, and $L_1^-(\mu)$ denote the radiance incident on side 1. Similarly define L₀^- and L₁⁺ as the radiance exiting the slab from sides 0 and 1 respectively (Figure 3.1B). The downward radiance from side 1 is the sum of the transmitted incident radiance from side 0 and the reflected radiance from side 1,

L₁⁺=T₀₁L₀⁺ +R₁₀L₁^-

(3.10)

The upward radiance from side 0 is the transmitted radiance from side 1 and the reflected radiance from side 0

L₁^-=R₀₁L₀⁺ +T₁₀L₁^-

(3.11)

Analogous formulas apply to a layer 12

L₂⁺ =T₁₂L₁⁺ +R₂₁L₂^-

(3.12)

L₁^- =R₁₂L₁⁺ + T₂₁L₂^-

(3.13)

Juxtaposition of layers 01 and 12 yields a combined layer 02. The equations relating the radiances exiting from the top and bottom of this slab are

L₂⁺ =T₀₂L₀⁺ +R₂₀L₂^-

(3.14)

L₀^- =R₀₂L₀⁺ + T₂₀L₂^-

(3.15)

Presumably, the reflection and transmission operators for the 01 and 12 layers are known. The reflection and transmission operators for the 02 layer are needed in terms of those for the 01 and 12 layers. To do this Equation (3.10) is multiplied by R₁₂ from the left and added to Equation (3.13). Since the terms containing L₁⁺ cancel

(E-R₁₂R₁₀) L₁^- =R₁₂T₀₁L₀⁺ + T₂₁L₂^-

(3.16)

Multiplying Equation (3.16) on the left by (E-R₁₂R₁₀)^-1 yields

L₁^- = (E-R₁₂R₁₀)^-1 (R₁₂T₀₁L₀⁺ + T₂₁L₂^-)

(3.17)

This equation expresses the upward radiance at the interface between two layers. An equation for the downward mid-layer radiance can be obtained similarly using Equations (3.13) and (3.10)

L₁⁺ = (E-R₁₀R₁₂)^-1 (T₀₁L₀⁺ + R₁₀T₂₁L₂^-)

(3.18)

Substituting Equation (3.18) into Equation (3.12) yields

$\begin{displaymath} L_2^+ = \left[T_{12}(E-R_{10}R_{12})^{-1}T_{01} \right]L_0^+... ...ft[T_{12}(E-R_{10}R_{12})^{-1}R_{10}T_{21}+R_{21} \right]L_2^- \end{displaymath}$

(3.19)

Comparing this equation with Equation (3.14) indicates that

T₀₂ = T₁₂(E-R₁₀R₁₂)^-1T₀₁

(3.20)

R₂₀ = T₁₂(E-R₁₀R₁₂)^-1 R₁₀T₂₁+R₂₁

(3.21)

Similarly Equation (3.11) can be substituted into Equation (3.13) to obtain

T₂₀ = T₁₀(E-R₁₂R₁₀)^-1T₂₁

(3.22)

R₀₂ = T₁₀(E-R₁₂R₁₀)^-1 R₁₂T₀₁+R₀₁

(3.23)

Equations (3.20)-(3.23) define the reflection and transmission operators for a layer comprised of two individual layers in terms of reflection and transmission operators for each layer.

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