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Light Transport in Tissue Phase functions When light strikes a particle with an index of refraction different from its environment, the light is refracted. The angle at which the light is bent is a function of the size and shape of the particle as well as the wavelength of the incident light and the incidence angle of the light. In general, each particle will have a different scattering profile. This scattering profile is called the phase function. This name is misleading since the scattering profile has no connection with the phase of the incident light waves and would be more appropriately called a scattering function. The phase function $p(\hat\mathbf{s},\hat\mathbf{s}')$ describes the amount of light scattered from the direction denoted by the unit vector $\hat\mathbf{s}$ into the direction $\hat\mathbf{s}'$. There are a number of ways in which the phase function may be normalized, but the most natural is that used by the astrophysicists. They treat the phase function as a probability distribution; consequently, their normalization condition requires the integral of the phase function over all angles to equal unity $$\int_{4\pi} p(\hat\mathbf{s},\hat\mathbf{s}')\,d\omega = 1$$ (1.1) where $d\omega$ is a differential solid angle in the $\hat\mathbf{s}$ direction. This condition does not permit the phase function to describe absorption of light by the particle, the phase function is a description of only the distribution of scattering by the particle. Thus $p(\hat\mathbf{s},\hat\mathbf{s}')\,d\omega$ is the probability that a photon incident from the $\hat\mathbf{s}$ direction will leave in the differential unit of solid angle in the $\hat\mathbf{s}'$ direction. The phase function will differ in general from particle to particle. For simplicity an average phase function which adequately describes the most important features of the scattering process is used. This average phase function is further constrained by assuming that the probability of scattering from one direction into another is a function only of the angle between the two directions, thus $p(\hat\mathbf{s},\hat\mathbf{s}') = p(\hat\mathbf{s}\cdot\hat\mathbf{s}') = p(\cos\theta)$. The simplest phase function is the isotropic phase function $$p(\hat\mathbf{s}\cdot\hat\mathbf{s}') = {1\over 4\pi}$$ (1.2) The factor of $1/4\pi$ results from the normalization condition (1.1) and the fact that there are $4\pi$ steradians in a complete circle. The phase function has units of sr-1. If the phase function is not isotropic, then a parameter called the average cosine of the phase function is used to describe the degree of anisotropy of the phase function. This parameter is often denoted by g and is defined as the integral over all angles of the phase function multiplied by the cosine of the angle $$g = \int_{4\pi} p(\hat\mathbf{s} \cdot \hat\mathbf{s}') (\hat\mathbf{s} \cdot \hat\mathbf{s}') \,d\omega$$ (1.3) The choice of a single scattering phase function is a compromise between realism and mathematical tractability. Jacques et al. have shown that a modified Henyey-Greenstein function describes single particle light scattering in human dermis quite well [32]. More recently Yoon et al. have found similar results for human aorta [69]. The modified Henyey-Greenstein function is $$p_{\rm m-HG}(\cos\theta) = {1\over4\pi} \left[ \beta + (1-\b... ...hrm{HG}}^2 - 2g_{\mathrm{HG}}\cos\theta\right)^{3/2} } \right]$$ (1.4) In this function, the first term $\beta$ represents the amount of light scattered isotropically. The second term is the Henyey-Greenstein function. The function is normalized such that the integral of the phase function over all solid angles is unity. When $\beta=0$ this phase function reduces to the Henyey-Greenstein phase function. A popular phase function is the Eddington phase function $$p_{\rm Eddington}(\cos\theta) = {1\over4\pi} \left[1+3g'\cos\theta \right]$$ (1.5) With this approximation the transport equation may be reduced into a diffusion equation [31,55]. Such a solution provides a qualitative picture of radiative transport in media which is not highly forward scattering. Unfortunately, the anisotropy (the average cosine of the phase function) for tissue such as dermis [32], aorta [70] and bladder [9] have values of 0.8-0.9. This suggests that the Eddington approximation would not be very good for modeling light in such tissues. Another possible phase function is the delta-Eddington approximation [34], $$p_{\delta-\mathrm{E}}(\cos\theta) = {1\over4\pi} \left[2f\delta(1-\cos\theta)+ (1-f)(1+3g'\cos\theta) \right]$$ (1.6) where f is the fraction of light scattered into the forward peak and g' is an asymmetry factor. As $f\rightarrow1$ the phase function becomes exactly a delta function, and as $f\rightarrow0$ the phase function reduces to the Eddington approximation. This phase function also allows reduction of the transport equation to a diffusion equation. Consequently, it is desirable to approximate the modified Henyey-Greenstein function using the delta-Eddington phase function. Joseph et al. used the delta-Eddington phase function above to approximate the Henyey-Greenstein phase function [34]. Only slight modification must be made to their derivation to arrive at an approximation for the modified Henyey-Greenstein function. Recalling that the Dirac delta function may be expanded as a sum of Legendre polynomials [44] allows expansion of the delta-Eddington phase function $$p_{\delta-\mathrm{E}}(\cos\theta) = {1\over4\pi} \left[f\su... ...1)P_n(\cos\theta)+(1-f) \big( 1+3g'P_1(\cos\theta)\big)\right]$$ (1.7) collecting like terms $$p_{\delta-\rm E}(\cos\theta) = {1\over4\pi} \left[1+3[f+g'(1-f)]P_1(\cos\theta)+5fP_2(\cos\theta) + \cdots \right]$$ (1.8) The modified Henyey-Greenstein phase function may also be expanded as a sum of Legendre polynomials using the expansion from [60] for the Henyey-Greenstein function $$p_{\rm m-HG}(\cos\theta) = {1\over4\pi} \left[ \beta P_0(\co... ...um_{n=0}^\infty (2n+1)g_{\mathrm{HG}}^nP_n(\cos\theta) \right]$$ (1.9) and collecting like terms $$p_{\rm m-HG}(\cos\theta) = {1\over4\pi} \left[1+3(1-\beta)g... ...)+5(1-\beta)g_{\mathrm{HG}}^2 P_2(\cos\theta) + \cdots \right]$$ (1.10) Now the first terms of each series are equated. The very first term (n=0) is $1/4\pi$ for both series. This results from the normalization of the phase functions. The next term (n=1) corresponds to three times the average cosine of the phase function g [31]. Thus for the delta-Eddington approximation, the average cosine of the phase function is g = f + (1-f)g' (1.11) and for the modified Henyey-Greenstein function the average cosine is $$g = (1-\beta)g_{\mathrm{HG}}$$ (1.12) The average cosine of the phase function g is a measure of how much light is scattered in the forward direction. The anisotropy can be any value between -1 and 1. If g=-1, then scattering is completely in the backwards direction; if g=1, then scattering is totally in the forwards direction; and if g=0, then scattering is isotropic. If Equations (1.11) and (1.12) are equated then a relation between the parameters of the two phase functions is obtained $$f+(1-f)g' = (1-\beta)g_{\mathrm{HG}}$$ (1.13) Proceeding in a similar manner, the second moments of the two phase functions may be equated $$f=(1-\beta)g_{\mathrm{HG}}^2$$ (1.14) And using Equation (1.13) an expression for g' may be obtained $$g' = {g_{\mathrm{HG}}(1-g_{\mathrm{HG}}) \over {1\over1-\beta}-g_{\mathrm{HG}}^2}$$ (1.15) Notice that if $g_{\mathrm{HG}}\rightarrow1$ then $g'\rightarrow0$ provided that $\beta\ne0$, otherwise $g'\rightarrow{1\over2}$. For example, measurements of the phase function of dermis at 633nm yields the modified Henyey-Greenstein parameters gHG=0.91 and $\beta=0.10$. Using Equations (1.14) and (1.15) the corresponding values for the delta-Eddington phase function may be found, i.e., f=0.75 and g'=0.29. This illustrates the way that the delta-Eddington approximation accommodates for strongly forward scattering phase functions by lumping a large portion of the scattering into the forward directed Dirac delta function and allowing the anisotropy to fall. Increased accuracy results since it is known that the diffusion approximation is poor for large values of anisotropy but is relatively good when the scattering is nearly isotropic. For completeness, the following equations relate the modified Henyey- Greenstein parameters to the delta-Eddington parameters $$g_{\mathrm{HG}}={f+(1-f)g'\over 1-\beta} \qquad{\rm and}\qquad \beta = 1-{f\over g_{\mathrm{HG}}^2}$$ (1.16) These are useful as long as f is not zero. If f is zero then no solution is possible, because the equations relating the first and second moments are no longer independent.

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