Home

Light Transport in Tissue Details of the iteration procedure The inverse method uses an N-dimensional minimization algorithm called AMOEBA [49]. This algorithm is based on the downhill simplex method of [45]. The function to be minimized is $$F(a',\tau',g')=\vert R_{\mathrm{diffuse}}^{delta-E}-R_{\math... ...m{coll}}^{\delta-E}-T_{\mathrm{diffuse}}^{\mathrm{meas}} \vert$$ (6.1) where Rdiffusedelta-E and $T_{\mathrm{coll}}^{\delta-E}$ are the diffuse reflection and total transmission as calculated with the delta-Eddington model. Only two measurements are used because either the collimated transmission Tcollmathrmmeas is known and consequently the delta-Eddington optical depth $\tau'$ is known $$\tau=-\ln\left({T_{\mathrm{coll}}^{mathrm{meas}}\over(1-r_s)^2}\right) \qquad\mathrm{and}\qquad \tau'=(1-af)\tau$$ (6.2) or gHG is known (from which g' can be derived). Thus the function $F(a',\tau',g')$ in Equation (6.1) is really a function of only two variables. The total light transmitted in the delta-Eddington model is $$T_{\mathrm{coll}}^{\delta-E}=T_{\mathrm{diffuse}}^{\delta-E}+(1-r_s)^2\exp(-\tau')$$ (6.3) where is the diffuse transmission given by Equation (4.106). It should be emphasized that the delta-Eddington collimated transmission is not equal to the measured collimated transmission (except in the special case when f=0 and $\tau'=\tau$). In particular $$T_{\mathrm{coll}}^{\delta-E}=(1-r_s)^2\exp(-\tau') \qquad\ma... ...uad T_{\mathrm{diffuse}}^{\mathrm{meas}}=(1-r_s)^2\exp(-\tau')$$ (6.4) The delta-Eddington collimated transmission is not physically observable, but merely a mathematical finesse to improve the accuracy of the approximation by treating some fraction of highly forward scattered light as `collimated' light. A Henyey-Greenstein shape may be imposed on the phase function (to second order in Legendre polynomials--see Section 1.3.3) by varying gHG instead of g'. This places a restriction on the relation between g' and f. This has the added advantage of removing f as an unknown parameter. Currently no evidence for the shape of a phase function at wavelengths other than 633nm exists, and this restriction should be made with caution. If the Henyey-Greenstein phase function is assumed then g' and f are calculated according to Equations (1.4) and (1.5). $$f=(1-\beta)g_{\mathrm{HG}}^2 \qquad\mathrm{and}\qquad g'={g_... ...HG}}(1-g_{\mathrm{HG}})\over{1\over1-\beta}-g_{\mathrm{HG}}^2}$$ (6.5) Finally, it is possible to vary g' independently of f, and omit calculating a and $\tau$ from the final values a' and $\tau'$ (which requires knowledge of f), but such model dependent parameters have not been found useful. The minimization routine amoeba assumes the range over which parameters may be varied is $-\infty$ to $\infty$. Unfortunately, the anisotropy and albedo have fixed ranges. This is remedied by transforming the albedo and anisotropy into a ``calculation space.'' As acalc varies from $-\infty$ to $\infty$ in the calculation space, the actual albedo a varies from 0 to 1. The same transformation is made on the anisotropy, since all tissues measured heretofore have had positive anisotropies. The transformation function is $$x_{calc}={2x-1\over x(1-x)}$$ (6.6) where x represents either gHG or a depending on the need. If negative anisotropies are included, then the transformation function for gHG must be altered to vary from -1 to 1 as the calculation parameter varies from $-\infty$ to $\infty$. Flowcharts for algorithms based on either known gHG or known are shown in Figures 6.4 and 6.5. These illustrate the changes of variables which take place during each iteration of the method. Starting values for an initial guess of a', $\tau'$, and g' maybe obtained in a few different ways. First, Kubelka-Munk optical properties may be calculated from the reflection and transmission and these may be converted to transport optical properties. This turns out to be not any better than just starting at fixed intermediate values of the optical properties (a'=0.5, $\tau'=1$, and g'=0.2). However, when calculating optical properties for a series of different wavelengths, excellent starting values to use for the next wavelength are the optical properties of the previous wavelength.

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