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Light Transport in Tissue Fixed stepsize method Figure 2.1: Flowchart of Monte Carlo with fixed propagation stepsize. The small stepsize $\Delta s$ required causes most of the computation time is spent in the inner (dashed) loop. Since the statistics are only changed in the ``Update Absorption'' and ``Update Reflection and Transmission'' boxes this method is inefficient. The simplest Monte Carlo method propagates each photon with small, fixed incremental stepsizes. A flowchart describing the process is shown in Figure 2.1. Three questions arise: ``What should the stepsize $\Delta s$ be?'', ``What is the probability of the photon being scattered?'' and, ``What is the probability of the photon being absorbed?'' The fixed stepsize $\Delta s$ chosen must be small relative to the average mean free path of a photon in the tissue. The mean free path is the reciprocal of the total attenuation coefficient. $$\Delta s \ll {1\over\mu_a+\mu_s}$$ (2.1) If $\Delta s$ is too small the photon will rarely interact with the tissue and the Monte Carlo method will be inefficient. It was found that a stepsize $\Delta s$ of one-tenth of a mean free path yielded reasonable results. The probability of absorption of a photon travelling a distance $\Delta s$ is given by Beer's Law $$P\{\mathrm{absorption}\} = 1-e^{-\mu_a\Delta s}$$ (2.2) Expanding the exponential in a Taylor series, and letting $\mu_a\Delta s\rightarrow0$, shows that the probability of photon absorption is $$P\{\mathrm{absorption}\}\approx\mu_a\Delta s$$ (2.3) Similarly, the probability that the photon will scatter in this distance is $$P\{\mathrm{scattering}\} = 1-e^{-\mu_s\Delta s}$$ (2.4) and for short pathlengths $\Delta s$ $$P\{\mathrm{scattering}\}\approx\mu_s\Delta s$$ (2.5) Assuming three disjoint events (1) absorption, (2) scattering, and (3) no interaction between the tissue and the photon, the sum of the three events equals unity. This implicitly assumes that the photon cannot be scattered and absorbed in the same propagation step. To determine if a photon is scattered or absorbed, a random number $\xi$ uniformly distributed between zero and one is generated and compared with the probability of absorption. If, $$\xi<P\{\mathrm{absorption}\}$$ (2.6) then the photon is absorbed and a new photon is launched. If $\xi$ is between $$P\{\mathrm{absorption}\}\le \xi<P\{\mathrm{absorption}\}+P\{\mathrm{scattering}\}$$ (2.7) then the photon is scattered and a new photon direction is chosen based on the phase function for the medium. If the photon is neither scattered nor absorbed then the photon has propagated the distance $\Delta s$ without interaction. One advantage of this method is that it is simple to implement. Moreover, since the stepsize is fixed, each step corresponds to a given length of time, and a time history can be generated. The primary disadvantage is that this technique is slow: the photon must be moved an average of $(\mu_a+\mu_s)/\Delta s$ times before it is either absorbed or scattered. The number of propagation steps required becomes prohibitive for highly scattering media, since the total distance travelled before absorption is large compared with the stepsize.

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