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Light Transport in Tissue Analytic Method One method of generating a random number $\xi$ with a specified distribution $p(\xi)$ is to create a random event for the variable $\xi$ such that the random event falls with frequency p(x)dx in the interval $(\xi,\xi+d\xi)$. This method requires the normalization of the probability density function over the interval (a,b) $$\int_a^b p(\xi)\,d\xi = 1$$ (A1) This is done by choosing a random number R uniformly distributed in the interval [0,1] and requiring $$R=\int_a^\xi p(\xi')\,d\xi'$$ (A2) [6,56]. Note that $R(\xi)$ represents the cumulative probability distribution function for $p(\xi')$. In the variable stepsize Monte Carlo method, the stepsize is randomly generated based on the probability that the photon will interact in a given distance. If the unnormalized probability density function for the distance $\Delta s$ is $$p(\Delta s) = \exp(-\Delta s)$$ (A3) then the normalized probability density function over the interval $(0,\tau)$ $$p(\Delta s) = {\exp(-\Delta s)\over 1-\exp(-\tau)}$$ (A4) When the probability density function $p(\Delta s)$ is substituted into (A.2), a generating function for $\Delta s$ in obtained $$\Delta s = -\ln\left[1-R(1-\exp(-\tau))\right]$$ (A5) If the random variable $\xi$ is distributed over the interval $(0,\infty)$ then the appropriate generating function is $$\Delta s = -\ln(1-R)$$ (A6) Since R is a random number uniformly distributed between zero and one, so is (1-R). If R' is a random number uniformly distributed between zero and one, then it may be substituted for (1-R) and and Equation (A.6) may be simplified to $$\Delta s = -\ln R'$$ (A7) A normalized phase function describes the probability density function for the azimuthal and longitudinal angles for a photon when it is scattered. If the phase function has no azimuthal dependence, then the azimuthal angle $\varphi$ is uniformly distributed between 0 and $2\pi$, and may be generated by multiplying a pseudo-random number R uniformly distributed over the interval [0,1] by $2\pi$ $$\varphi = 2\pi R$$ (A8) The probability density function for the longitudinal angle $\theta$ between the current photon direction and the scattered photon direction is found by integrating the phase function over all azimuthal angles $p(\cos\theta)$. For example, the probability density function for an isotropic distribution is $$p(\cos\theta)={1\over2}$$ (A9) Substituting Equation (A.9) into Equation (A.2) yields the following generating function for cosine of the longitudinal angle $\theta$ $$\cos\theta=2R-1$$ (A10) The probability density function corresponding to the Henyey-Greenstein phase function is $$p(\cos\theta)={1\over2}{1-g_{\mathrm{HG}}^2\over (1+g_{\mathrm{HG}}^2-2g_{\mathrm{HG}}\cos\theta)^{3/2}}$$ (A11) The generating function for this distribution obtained using Equation (A.2) [68] is $$\cos\theta = {1\over 2g_{\mathrm{HG}}}\left\lbrace 1+g_{\ma... ...over1-g_{\mathrm{HG}}+2g_{\mathrm{HG}} R}\right] \right\rbrace$$ (A12) This equation should not be used for isotropic scattering--Equation (A.10) should be used in that case. The probability density function for the modified Henyey-Greenstein phase function is $$\cos\theta = {1\over 2}\left\lbrace \beta+{(1-\beta)(1-g_{\... ...mathrm{HG}}^2-2g_{\mathrm{HG}}\cos\theta)^{3/2}} \right\rbrace$$ (A13) To generate a longitudinal angle with this distribution, two random numbers (R0 and R) uniformly distributed between zero and one are needed. In this distribution, light is either scattered isotropically or anisotropically. The first random number is used to determine which type of scattering occurs. The fraction of light scattered isotropically ($\beta$) is compared with the first random number, if $\beta<R_0$ then the photon is scattered isotropically according to Equation (A.10), otherwise the photon is scattered using the generating function given by Equation (A.12).

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