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Light Transport in Tissue Uniqueness of inverse procedure The iteration method implicitly assumes that a unique combination of the albedo a, the optical depth $\tau$, and the anisotropy gHG will be determined by a specified set of reflection and transmission measurements. Consider the case when the optical thickness is held constant. Increasing the albedo increases reflection and decreases transmission. Decreasing the anisotropy also increases the reflection and decreases the transmission, and so the anisotropy and the albedo are not obviously independent functions of the reflection and transmission. Figure 6.1: Flowchart for inversion process. The method involves guessing the optical properties of a tissue, calculating the reflection and transmission for these properties, comparing the calculated with the measured reflection and transmission, and repeating this process until the calculated and measured transmission match. In Figure 6.2 the dependence of the total transmission and the diffuse reflection on the anisotropy and albedo is shown. The grid was computed with the collimated transmission fixed at ten percent ( Tcollmathrmmeas=0.1) and the boundaries of the sample are matched with the outside. The intersection of the appropriate measured diffuse reflection and measured total transmission grid lines determines a unique albedo and anisotropy. Diffuse reflectances above about 0.5 are physically impossible for the assumed boundary conditions and assumed collimated transmission. The albedo and anisotropy are most sensitive to errors in the magnitudes of the diffuse reflection and total transmission measurements when the magnitudes of both these measurements are small (e.g., R=0.1, T=0.2). Now let gHG be held constant: in this case the two parameters which may be varied are the albedo a and the optical depth $\tau$. Increasing the albedo increases the reflection and decreases transmission. Since increasing the optical depth also increases reflection and decreases transmission, the optical depth is again not clearly independent of the albedo. Figure 6.3 shows how T (total transmission) and R+T (diffuse reflection plus total transmission) depend on the albedo and optical thickness for the case when gHG=0 and the sample boundaries are matched with the environment. This graph is typical of other anisotropies and boundary conditions. Measured values for R and T (or equivalently, T and R+T) determine a unique intersection point whose ordinate is a simple function of the albedo a and whose abscissa is the optical depth $\tau$. The axes in Figure 6.3 were chosen to linearize the T and the R+T contour curves. Figure: Typical diffuse reflectance (R) and total transmission (T) grid for a fixed collimated transmission (Tc=0.1). For every pair of measurements R and T, the R-T grid intersection defines a unique pair of optical properties for the albedo a and the anisotropy $g_{\protect\mathrm{HG}}$ for that tissue. Figure: Typical diffuse reflectance (R) and total transmission plus diffuse reflectance (R+T) grid for a fixed anisotropy $(g_{\protect\mathrm{HG}}=0)$. For every pair of measurements R and T, the T/R+T grid intersection defines a unique pair of optical properties for the albedo a and the optical depth $\tau$ for that tissue.

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