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Light Transport in Tissue Mechanics of photon propagation A photon is uniquely described by five variables: three spatial coordinates for the position and two directional angles for the direction of travel. Initially, cylindrical coordinates were used for the photon's position and two angle variables relative to the line between the photon and the origin were used for its direction. Cylindrical coordinates were chosen for symmetry reasons. The primary disadvantage of this description was that the angle variables changed with each photon step, even when the photon continued travelling in the same direction. Another less critical problem was that this geometry has several special trigonometric cases, which complicates the mechanics of photon propagation. Figure 2.4: Monte Carlo coordinate system. Figure A shows the Cartesian coordinate system and how the photon's direction cosines are specified. Figure B shows how $\theta$ and $\varphi$ are specified when a photon is scattered. Alternatively, Carter and Everett have described the photon's spatial position with Cartesian coordinates and the direction of travel with three direction cosines [6]. The required formulas for propagation are simpler, and the angle variables describing photon direction do not change unless the photon's direction changes. The direction cosines are specified by taking the cosine of the angle that the photon's direction makes with each axis. These are specified by $\mu_x$, $\mu_y$, and $\mu _z$ corresponding to each of the x-, y-, and z-axes respectively (Figure 2.4A). For a photon located at (x,y,z) travelling a distance $\Delta s$ in the direction $(\mu_x,\mu_y,\mu_z)$, the new coordinates (x',y',z') are given by $$x+\mu_x\Delta s$$ x'= $$y+\mu_y\Delta s$$ y'= (2.10) $$z+\mu_z\Delta s$$ z'= If a photon is scattered at an angle $(\theta,\varphi)$ from the direction $(\mu_x,\mu_y,\mu_z)$ in which it is travelling (Figure 2.4B), then the new direction $(\mu_x',\mu_y',\mu_z')$ is specified by $$\mu_x' = {\sin\theta \over \sqrt{1-\mu_z^2} } (\mu_x\mu_z \cos\varphi-\mu_y\sin\varphi)+\mu_x\cos\theta$$ $$\mu_y' = {\sin\theta \over \sqrt{1-\mu_z^2} } (\mu_y\mu_z \cos\varphi+\mu_x\sin\varphi)+\mu_y\cos\theta$$ (2.11) $$\mu_z' = -\sin\theta\cos\varphi\sqrt{1-\mu_z^2} + \mu_z\cos\theta$$ If the angle is too close to the normal (say $\vert\mu_z\vert>0.99999$), the following formulas should be used $$\mu_x' = \sin\theta\cos\varphi$$ $$\mu_y' = \sin\theta\sin\varphi$$ (2.12) $$\mu_z' = \mu_z / \vert\mu_z\vert \cos\varphi$$ to obtain the new photon directions.

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