Home

Light Transport in Tissue Single scattering approximation From Chapter 3, the first order scattering and transmission functions for uniform normal incidence on a slab of thickness $\tau$ are given by Equations (3.30)-(3.34) $$R(1,-\mu)={a\mu \,p(1,-\mu)\over1+\mu}\left[1-\exp(-\tau/\mu-\tau)\right]$$ (5.1) $$T(1,\mu)={a\mu \,p(1,\mu)\over1-\mu}\left[\exp(-\tau)-\exp(-\tau/\mu)\right]$$ $$T(1,1)=(a\tau p(1,1)+1)\exp(-\tau)$$ where a is the albedo, $\tau$ is the optical thickness of the sample, $\mu$ is the cosine of the angle that light exiting with the normal to the sample, and $p(1,\mu)$ is the phase function. Equations (5.1) have been multiplied by $2\pi$ to remove the integration over azimuthal angles and divided by $2\mu$ to remove the factor of $2\mu$ that was included to satisfy the star multiplication algebra. The redistribution function $h(\mu_0,\mu)$ can be replaced by the phase function because the cosine of the angle of incidence is unity (see Equation (3.24). Henceforth the phase function $p(1,\mu)$ will be written $p(\mu)$. It is assumed that the phase function is independent of azimuthal angle (implicit in the multiplication by $2\pi$ above) and that $p(\mu)$ is a complete description of the phase function. No azimuthal dependence was observed in any experiments. Equations (5.1) are exact for uniform normal illumination and are a good approximation only when the width of the incident beam is much larger than the thickness of the slab. Since typical beam diameters are about 1.0mm and the tissue samples used are approximately 0.020-0.100mm in thickness this assumption is reasonable. If the exponentials in Equations (5.1) are expanded in a Taylor series then $$R(1,-\mu)=a\tau\, p(-\mu)\left[1-{\tau(1+\mu)\over 2\mu} + \cdots\right]$$ (5.2) $$T(1,\mu)=a\tau\, p(\mu)\left[1-{\tau(1+\mu)\over 2\mu} + \cdots\right]$$ $$T(1,1)=(a\tau\, p(1,1)+1)\left[1-\tau+{\tau^2\over2}-\cdots\right]$$ The factor of $a\tau$ indicates that the amount of scattered light is directly proportional to the product of the optical thickness and the albedo (the fraction of light scattered to the total amount of light scattered and absorbed). Alternatively, $a\tau=\mu_s d$ (d is the sample thickness) indicates that the reflected and transmitted light is directly proportional to the amount of light scattered. If the phase function is isotropic, $p(\mu,\mu')=1/4\pi$, then reflection and transmission are equal and independent of the angle of exitance. These equations show the direct correspondence between the phase function and the reflected and transmitted light for very thin slabs $(t\ll1)$. Unfortunately, there are always angles $\mu$ such that $\tau/\mu$ is not small and the above approximation is invalid. These angles correspond to reflection or transmission at angles nearly parallel to the slab. If the multiplicative factor required to convert reflection and transmission into a phase function value are denoted by cR and cT respectively then $$p(-\mu)=c_R R(1-\mu) \qquad\mathrm{and}\qquad p(\mu)=c_T T(1,\mu)$$ (5.3) where $$c_R={1+\mu\over a\mu}\left[1-\exp(-\tau/\mu-\tau)\right]^{-1}$$ (5.4) $$c_T={1-\mu\over a\mu}\left[\exp(-\tau)-\exp(-\tau/\mu)\right]^{-1}$$ Equations (5.4) are plotted in Figure 5.1 for an optical thickness and albedo of unity. The correction factor is largest for light exiting at grazing angles and the measurement error at these angles will be magnified accordingly. Consequently, the phase function data at these angles will be least reliable. However, the correction factor is nearly constant for angles up to 30$^\circ$ from the normal and at these angles the light exiting the slab is nearly proportional to the phase function.

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