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Light Transport in Tissue Least squares fit Once the experimental data has been converted into an equivalent phase function measurement pmeas using Equation (5.13), it must be fit to a phase function. It was found that a modified Henyey-Greenstein phase function $$p_{\mathrm{m-HG}}(\cos\theta)={1\over4\pi}\left\lbrace \beta... ...\mathrm{HG}}^2-2g_{\mathrm{HG}}\cos\theta)^{3/2}}\right\rbrace$$ (5.14) accurately represented the phase function for human dermis. If the albedo for a tissue is unknown, or a correction factor in Equation (5.13) is not known precisely, then it is convenient to include an arbitrary multiplicative factor in the expression for the phase function. If this is done then $$p_{\mathrm{meas}}(\cos\theta)=\gamma p_{\mathrm{m-HG}}(\cos\theta) \qquad\qquad \gamma={1\over4\pi a}$$ The second equality $(\gamma=1/4\pi a)$ is useful only when all factors in Equation (5.13) are known. This is a poor way to measure the albedo of a material because it requires excellent absolute accuracy in the goniometric measurement rather than good relative accuracy. To fit the modified Henyey-Greenstein phase function to the measured phase function three parameters must be determined: the multiplicative factor g, the amount of light scattered isotropically $\beta$, and the anisotropy factor gHG. Figure 5.4: Raw goniophotometer data from 100$\mu$m sample of human dermis. Error bars are constant in magnitude and equal to the resolution of the A/D converter in the computer (0.005V). Figure 5.5: Corrected goniophotometer data from a 100$\mu$m sample of human dermis. Error bars have also been corrected and no longer have a constant magnitude. One of the assumptions in the modified Henyey-Greenstein phase function is that light scattered in the backwards direction (reflected light) is scattered isotropically. Figure 5.5 shows a plot of the corrected data. The light reflected is not absolutely isotropic (a flat response) but increases slightly around $\pm$180$^\circ$ due to internal reflectance of the forward peak. Subtraction of this reflected light yields a nearly constant value in the backwards direction. The fraction of light scattered isotropically ($\gamma\beta$) is determined by averaging values for $p_{\mathrm{meas}}(\theta_{\mathrm{exit}})$ in the backwards direction. Figure 5.6: Linearized phase function data from 100$\mu$m sample of human dermis showing increasing error for grazing angles (small $\cos \theta _{\mathrm {exit}}$). The reflected data provides information for $\gamma\beta$ and the transmitted data is used to find two more parameters gHG and $\gamma$. The transformation $$x=\cos\theta \qquad\mathrm{and}\qquad y=(p(\theta)-\gamma \beta)^{-2/3}$$ (5.15) reduces the modified Henyey-Greenstein equation into a linear equation of the form y=mx+b. The slope and intercept are given by $$m=-{2g_{\mathrm{HG}}\over[\gamma(1-\beta)(1-g_{\mathrm{HG}}^... ...athrm{HG}}^2\over[\gamma(1-\beta)(1-g_{\mathrm{HG}}^2)]^{2/3}}$$ (5.16) If Equation (5.15) is used to transform the data then a graph similar to Figure 5.4 is obtained. The error values were calculated with a fixed photometric error of 0.005V in the goniophotometer. The errors in $p_{\mathrm{meas}}(\theta)$ are roughly proportional to the correction factor shown in Figure 5.2. A weighted least squares fit must be used to find the slope and intercept of the best-fit line because of the widely varying errors between data points. The error in the slope ($\Delta m$) and intercept ($\Delta b$) may also be calculated [4]. The phase function parameters $\beta$ and gHG may be recovered from the slope and intercept by solving Equations (5.16). This results in the following physical expressions for gHG and $\beta$ in terms of the calculated intercept b, slope m, and product $\gamma\beta$ $$g_{\mathrm{HG}}=-{b\over m}-\sqrt{{b^2\over m^2}-1}$$ (5.17) $$\gamma=-{2(g_{\mathrm{HG}}/m)^{3/2}\over 1-g_{\mathrm{HG}}^2... ...a\beta \qquad\mathrm{and}\qquad \beta={\gamma\beta\over\gamma}$$ The errors in the values calculated for gHG and $\beta$ are found using the standard error propagation formula, $$(\Delta g_{\mathrm{HG}})^2=\left( {\partial g_{\mathrm{HG}}\... ...ft( {\partial g_{\mathrm{HG}}\over\partial b}\Delta b\right)^2$$ (5.18) Inserting Equations (5.17) leads to the following equation for the error in the anisotropic value of gHG $$\Delta g_{\mathrm{HG}} = {1\over m}\left(1+{b\over\sqrt{b^2-... ... \left((\Delta b)^2 + {b^2(\Delta m)^2\over m^2} \right)^{1/2}$$ (5.19) The error in $\gamma$ is $$\Delta\gamma = \gamma\left[{(9\Delta m)^2\over 4m^2}+\left( ... ...Delta g_{\mathrm{HG}})^2\over g_{\mathrm{HG}}^2} \right]^{1/2}$$ (5.20) The error in $\beta$ is $$\Delta \beta = \beta \sqrt{\left({\Delta(\gamma\beta)\over\g... ...\beta}\right)^2 + \left({\Delta(\gamma)\over\gamma}\right)^2}$$ (5.21) where $\Delta(\gamma\beta)$ is the standard deviation of the average of the backwards scattered light used to find $\gamma\beta$. Intra-sample variation of gHG and $\beta$ was much greater than the errors arising from the fitting process.

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