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Light Transport in Tissue Evaluation of the method The method for measuring phase functions was evaluated using theoretical values for reflected and transmitted light from slabs of varying thicknesses. The light exiting a uniformly illuminated slab as a function of angle was calculated using the adding-doubling method. All calculations assumed an albedo of 0.98 (the measured albedo of the dermis at 632nm) and an isotropy factor of gHG=0.9 (again, close to that of dermis at 632nm). Four sets of data were calculated: matched and mismatched boundaries with $\beta=0.1$ and $\beta=0$. Each of the four cases was calculated for a series of optical depths ranging from $\tau =0.1$ to $\tau=5$. The mismatched boundary case was equal to that for tissue-glass-saline and the indices of refraction were 1.37/1.54/1.33. The reflected and transmitted light as a function of angle for $\beta=0$ and $\tau =0.1$, with matched and mismatched boundaries is shown in Figure 5.7. Both boundary conditions manifest a strong forward peak at 0$^\circ$. However, when the indices of refraction are mismatched, a backward peak at 180$^\circ$ results from internal reflection ($\sim$1%) of the forward peak. The method for obtaining the phase function from measured quantities was tested using the four sets of calculated data. The known values of $\beta=0.1$ and gHG=0.9 are compared with the fitted values using the procedure outlined in Section 5.4. The fitted parameters $\beta$ and gHG are presented in Figures 5.8 and 5.9 as a function of the optical thickness of the theoretical sample. The values of gHG for all four sets of data were identical. Figure 5.8 shows gHG for only one set of data is shown ($\beta$=0, mismatched conditions). This suggests that gHG is insensitive to changes in the isotropy factor $\beta$ and boundary condition. For sample thicknesse less than one optical depth the error in the fitted phase function is less than five percent. Thicknesses greater than one optical depth multiply scatter and cause large errors in the fitted phase function. Figure: Light leaving a uniformly illuminated slab as calculated by the adding-doubling method with a=0.98, $g_{\protect\mathrm{HG}}$=0.9, $\tau =0.1$. Matched (empty circles) and mismatched (1.37/1.54/1.3) conditions are shown. The peak at $\cos \theta _{\mathrm {exit}}=-1$ in the mismatched case arises from internal reflection of the forward peak ( $\cos \theta _{\mathrm {exit}}=1).$ Figure: Dependence of fitted anisotropy factor $g_{\protect\mathrm{HG}}$ on sample thickness. The true value of $g_{\protect\mathrm{HG}}$ for all optical depths is 0.9. The deviation of the fitted values from the true value is caused by multiple scattering. Figure 5.9: Sensitivity of $\beta$ to sample thickness and boundary conditions. The empty points correspond to fitted values with $\beta$=0.1 and the filled correspond to $\beta$=0. The squares indicate mismatched conditions and the circles correspond to matched conditions. Figure 5.9 shows the dependence of the fitted isotropy parameter $\beta$ as a function of optical thickness. The isotropy factor $\beta$ is more sensitive to thickness than gHG and errors do not become small until optical thicknesses approach 0.1. Moreover, $\beta$ is very sensitive to boundary conditions and significant errors exist for even very thin tissues with mismatched boundaries. The discrepancy occurs because internal reflection increases the path length of light in the tissue for which the single scattering model cannot account The data presented in Figure 5.9 suggests that $\beta$ cannot be measured reliably because any real experiment will have mismatched boundaries and subsequent analysis will overestimate the isotropy factor. Therefore, Figure 5.9 should be used to correct any measured value of $\beta$ to obtain the true value of $\beta$ for light scattered isotropically.

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