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Light Transport in Tissue The transport equation The transport equation describes the behavior of light in a slab [8] $$(\hat\mathbf{s}\cdot\nabla)L(mathbf{r},\hat\mathbf{s}) = -\m... ...thbf{s},\hat\mathbf{s}') L(mathbf{r},\hat\mathbf{s})\,d\omega'$$ (1.17) Here the integral is over all solid angles and $d\omega'$ is the differential solid angle in the direction $\hat\mathbf{s}'$. Notice that the radiance is a function of five variables: three in the r vector and two in the $\hat\mathbf{s}$ unit vector. (Since $\hat\mathbf{s}$ is a unit vector the magnitude is fixed and consequently one degree of freedom has been removed.) The left hand side of the transport equation describes the rate of change of the intensity at the point indicated by r in the direction $\hat\mathbf{s}$. This rate of change is equal to the intensity lost due to absorption and scattering (the first term on the R.H.S.) plus the intensity gained through light scattering from all other directions into the direction $\hat\mathbf{s}$ (the last term on the R.H.S.). The assumptions implicit in the transport equation are those mentioned in the assumptions Section 1.2 above. The first of these is that the medium is assumed to be homogeneous. This means that any variation in the scattering and absorption of the medium must be on length scales much smaller than the depth of the slab. Another more questionable assumption, from a tissue optics standpoint, is that each particle is sufficiently isolated that its scattering pattern is independent of all other particles. This is known as the far field approximation in geometrical optics, and is clearly violated for typical tissues because the scattering and absorbing particles are in contact with one another. A related assumption is that scattering by all particles may be described by a single function known as the phase function. This means that there exists an ensemble average scattering pattern for all the scattering centers in the medium. Yet another assumption is that the intensity distribution is assumed to be in a steady state, which is valid if the light is incident for longer than a few nanoseconds. Finally, it is assumed that there are no light sources in the medium.

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