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Light Transport in Tissue Internal reflection The possibility of internal reflection occurs when the photon is propagated across an index of refraction discontinuity. Typically, reflection will only occur at the boundaries of the medium. The probability that a photon will be reflected is determined by the Fresnel reflection $R(\theta_i)$ $$R(\theta_i)={1\over2}\left[ {\sin^2(\theta_i-\theta_t)\over\... ...an^2(\theta_i-\theta_t)\over\tan^2(\theta_i+\theta_t)} \right]$$ (2.13) where $\theta_i=\cos^{-1}\mu_z$ is the angle of incidence on the boundary and the angle of transmission $\theta_t$ is given by Snell's law $$n_i\sin \theta_i = n_t\sin\theta_t$$ (2.14) where ni and nt are the indices of refraction of the medium from which the photon is incident and transmitted, respectively. A random number $\xi$ uniformly distributed between zero and one is used to decide whether the photon is reflected or transmitted. If $\xi < R(\theta_i)$ then the photon is reflected, otherwise the photon is transmitted. The details of how the photon is reflected depends upon the variance reduction technique used in the Monte Carlo method. Figure 2.5: Geometry of photon reflection at an interface. In the slab geometry shown only the z-coordinate and $\mu _z$ direction angle change. Once a photon leaves the tissue a new photon is initialized, except when photon weighting is used. If the photon is internally reflected, then the position and direction of the photon are adjusted accordingly. For a slab with thickness t, the exiting photon position is obtained by computing the position assuming transmission and changing only the z component of the photon coordinates $$z' = \cases{ -z &if $z<0$,\cr 2\tau-z) &if $z>\tau$.\cr}$$ (2.15) The change in photon direction is $$\mu_z'= -\mu_z$$ (2.16) and both $\mu_x$ and $\mu_y$ remain unchanged (Figure 2.5). When photon weighting is used then the photon may be both reflected and transmitted. If the old weight is w, then the new weight of the transmitted photon is $w'=w(1-R(\theta_i))$. The reflected photon's position and direction are calculated as above and the new weight of the photon is given by $w'=wR(\theta_i)$. When a glass slide placed on the surface of the tissue, creating a tissue-glass-air interface, the situation is slightly more complicated than for a tissue-air interface. When no weighting is used then reflection coefficients are calculated for each interface and the photon is propagated until it is reflected by or transmitted through the glass slide. For weighted photons, the photon should be split into two photons at each interface--one that is transmitted and one that is reflected. These photons in turn would be propagated and split as necessary until all photons are terminated. This has the advantage of creating many photons with small weights near the surface, which is a region of interest. The disadvantage is that this is awkward to implement. A simpler method is to no longer treat the photon as weighted, and let the whole weight of the photon either be reflected or transmitted at all interfaces.

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