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Light Transport in Tissue Corrections for internal reflection and refraction The tissue samples were held between two glass slides and submerged in saline. Because of the differing indices of refraction, corrections for the reflectance and refraction at the saline-glass-tissue interface were made. The index of refraction of glass (nglass) was measured to be 1.54. The assumed value for index of refraction of 0.9% saline solution ( nsaline) was 1.33. The index of refraction for tissue ( ntissue) was based on the generalization that the index of refraction of a tissue varies linearly between 1.33 to 1.5 for water contents between 100 percent and 0 percent [2]. The index of refraction would vary from 1.38 to 1.36 as water content varies from 70 to 85 percent, and so ntissue was chosen to be 1.37. The specular reflection of the incident beam is given by Equation (B.21) $$r(\theta)={r_1+r_2-2r_1r_2\over 1-r_1r_2}$$ (5.7) with $\theta=0^\circ$ and where r1 is the Fresnel reflection for light passing normal to saline-glass interface and r2 is the coefficient for light passing from glass to tissue. Using Equation (B.29) to find r1 and r2 yields r1=0.0054, r2=0.0034, and $r_s=r(0^\circ)=0.0088$ or a specular reflectance of about 0.9%. The raw data was subjected to a series of calculation steps to achieve a description of the light that exited the tissue at a given angle $\theta_{\mathrm{exit}}$ as opposed to the light that was observed at a given angle $\theta_{\mathrm{obs}}$. 1. The raw data, recorded as Volts (V) but representing collected power in Watts, was normalized by the direct beam measurement ( Vdirect) to obtain the collected power relative to a one Watt incident beam. Division by (1-rs) corrected for the specular reflectance from the front glass slide as the incident beam entered the tissue $$P(\theta_{\mathrm{obs}})={V(\theta_{\mathrm{obs}})\over V_{\mathrm{direct}}(1-r_s)} \qquad \mathit{in Watts}$$ (5.8) 2. The collected power was divided by the solid angle of collection of the optical fiber bundle ($\omega$). The solid angle is $\omega=4\pi(A_d/4\pi R_g^2)$ steradians where Ad was the collection area of the fiber bundle and Rg was the radius of the goniophotometer arm. This calculation yielded the observed radiant intensity $A(\theta_{\mathrm{obs}})$ $$A(\theta_{\mathrm{obs}})={P(\theta_{\mathrm{obs}})\over \omega} \qquad \mathit{in Watts/steradian}$$ (5.9) 3. The observed radiant intensity was corrected for the refraction at the tissue-glass-saline interfaces which caused the solid angle to expand as light exited the tissue (the n2-Law see Appendix B). $$B(\theta_{\mathrm{obs}})=A_(\theta_{\mathrm{obs}}) {\cos(\th... ...2_{\mathrm{tissue}}} \qquad \hbox{\textit{in Watts/steradian}}$$ (5.10) where $\theta_{\mathrm{exit}}$ is the angle at which light exits the tissue before refraction $$n_{\mathrm{tissue}}\sin(\theta_{\mathrm{exit}}) = n_{\mathrm{saline}}\sin(\theta_{\mathrm{obs}})$$ (5.11) 4. The value $B(\theta_{\mathrm{obs}})$ was corrected for Fresnel reflection at the tissue-glass and glass-saline interfaces, which allowed only a fraction, $1-r(\theta)$, of the light to escape and reach the detector $$I(\theta_{\mathrm{obs}})={B(\theta_{\mathrm{obs}}) \over 1-r(\theta_{\mathrm{exit}}) }$$ (5.12) where $r(\theta_{\mathrm{exit}})$ is determined using Equation (5.7) with r1 equal to the Fresnel reflection for light passing from the tissue to the glass slide and r2 equal to the reflection for light passing from the glass slide to the saline solution. The value $I(\theta_{\mathrm{obs}})$ was then attributed to the true angle of exitance from the tissue, $\theta_{\mathrm{exit}}$ as opposed to the observed angle, $\theta_{\mathrm{obs}}$, in consideration of the refraction at the tissue-glass and glass-saline interfaces. 5. Finally, modified correction factors c'R=acR and c'T=acT were applied to the reflected and transmitted light respectively. The modified correction factors permitted analysis of the data without knowledge of the albedo characterizing the tissue by allowing the albedo to be lumped with other unknown calibration factors in a multiplicative constant g described Section 5.4.2. Combining the corrections into one equation yields $$a\,p(\theta_{\mathrm{exit}})= {c'_T V(\theta_{\mathrm{obs}})... ...}(1-r(\theta_{\mathrm{exit}}))} \qquad \hbox{\textit{in 1/sr}}$$ (5.13) where cT' should be replaced with cR' for reflected angles. The most significant correction factors are c'R and c'T. The other corrections are relatively small and only become significant at oblique angles. Figure 5.4 shows the raw goniometric data as a function of the angle measured with the goniophotometer and Figure 5.5 shows the corrected data as a function of the angle that light leaves the tissue before being refracted. The goniophotometer resolution is 0.005V which corresponds to an intensity of 0.01W/sr. This is determined by the A/D conversion unit in the computer. The background noise was comparable to the resolution of the goniophotometer. The error bars in Figure 5.4 have a constant magnitude. In Figure 5.5 the errors in the phase function $p_{\mathrm{meas}}(\theta_{\mathrm{exit}})$ differ because the correction factor depends on the angle. Data in the ranges $90\pm15^\circ$ and $-90\pm15^\circ$ in Figure 5.5 are absent because light exiting the tissue at these angles is totally internally reflected ( $\theta_{\mathrm{critical}}$ is about 75$^\circ$).

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