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Light Transport in Tissue

Verification

Both Monte Carlo methods were implemented. The variable stepsize method was much faster and the fixed stepsize method was abandoned. The variable stepsize Monte Carlo method was checked to ensure that the sum of the transmission, reflection, and absorption was unity. The method was verified by comparing the Monte Carlo reflection and transmission results with known values.

Three different comparisons with exact values for testing all aspects of the Monte Carlo implementation were used. The errors shown in all the figures are standard errors (i.e., the standard deviations of the mean). These values were obtained by splitting one large Monte Carlo run of say 50,000 photons into ten runs of 5,000 each. The results of these ten runs were averaged and the standard error was computed.

Figure 2.8: Comparison of exact reflection values (filled circles) with variable stepsize Monte Carlo simulation. Scattering is anisotropic (g=0.75) and distributed according to the Henyey-Greenstein phase function. The index of refraction of the tissue equals that of its environment. The albedo is 0.9 and the thickness of the slab is two optical depths. Error bars indicate standard errors in the Monte Carlo simulation.

Figure 2.9: Comparison of exact transmission values (filled circles) with variable stepsize Monte Carlo simulation (empty squares). Scattering is anisotropic (g=0.75) and distributed according to the Henyey-Greenstein phase function. The index of refraction of the tissue equals that of its environment. The albedo is 0.9 and the thickness of the slab is two optical depths. Standard errors are smaller than the empty squares.

For an anisotropic phase function and a slab geometry of finite thickness with index matching, van de Hulst's tables served as a reference for reflection and transmission as a function of angle [1980b]. The phase function is for the Henyey-Greenstein phase function with an average cosine of 0.75. The slab was two optical depths thick, and index-matched with its environment. Light was uniformly incident normal to the slab. The average results from the Monte Carlo program with ten runs of 50,000 photons are plotted with standard errors, along with exact values from van de Hulst in Figures 2.8 and 2.9. The values for total reflection and total transmission are

Quantity	van de Hulst	Monte Carlo	std. dev.
Total Reflection	0.09739	0.0971	0.0003
Total Transmission	0.66096	0.6616	0.0005

Figure 2.10: Comparison of exact reflection values (filled squares) with Monte Carlo simulation (empty circles). The tissue is semi-infinite with an index of refraction mismatch of 1.5 to 1.0 at the tissue-air interface. The albedo is 0.9 and scattering is isotropic. Error bars indicate standard errors.

Finding exact solutions for media which are not index matched is difficult, but Giovanelli provides data for a semi-infinite slab with isotropic scattering and an index of refraction mismatch of 1.5 to 1.0 [16]. The internal reflection assumes Fresnel Reflection. The average of ten Monte Carlo runs of 5,000 photons with the values from Giovanelli are plotted in Figure 2.10. The albedo is 0.9 and light is normally incident. The values for total reflection are shown below

	Giovanelli	Monte Carlo	std. dev.
Total Reflection	0.2600	0.26079	0.00079

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