3.2. Experimental Validation

Measurements were performed with a custom-made MDSFR device (Fig. 1). A fiber with a 300-μm core and a fiber with a 600-μm core (Optran WF, Diamond Kimberlit B.V., Almere, the Netherlands) with numerical apertures of 0.22 were connected to a halogen light source (Ocean Optics, HL-2000-FHSA) using a bifurcated fiber. The tips of the measurement fibers were polished at an angle of 15 deg, to minimize internal reflection from the fiber tip. Shutters were placed to enable separate illumination by each measurement fiber. Each fiber was connected to a separate spectrometer (Avantes ADC1000-USB). Data acquisition was performed using a custom-written LabVIEW program and analyzed using a custom-written Matlab program.

(MD)SFR setup. Two fibers with diameters of 300 and 600μm, respectively, and NAs of 0.22 were used for the measurements. Shutters were placed to enable separate illumination by each measurement fiber and each fiber was connected to a separate spectrometer. Figure reproduced from Ref. ⁹.

We prepared two sets of phantoms with Intralipid-20% (Fresenius Kabi) and varying concentrations of Evans Blue (Sigma-Aldrich). One set was diluted with water to an Intralipid-20% volume fraction of 1:16, the second set to a volume fraction of 1:25, resulting in μs′ values between approximately 7 and 25cm−1 over the full spectrum. Even though higher μs′ values can occur in tissue, we had to stay at these lower volume fractions to prevent dependent scattering within the Intralipid-20% dilutions since for dependent scattering the phase function can currently not be predicted and, therefore, we would not know what the value of psb would be to compare our results to. We prepared an Evans Blue stock solution of 5 g/l and determined its absorption spectrum using a transmission measurement through 1 cm of the stock solution diluted to a volume fraction of 0.4:80.4. Based on the absorption spectrum, we prepared samples with Intralipid 20% and Evans Blue to obtain μa = [1.0; 2.0; 3.1; 5.1; 7.7; 10.2; 15.1; 20.1] cm−1 at 605 nm (the peak of the measured absorption spectrum). We chose these values for the absorption coefficient since, in tissue, the maximum absorption coefficient of blood between 500 and 600 nm is expected to be between 3 and 15cm−1, which corresponds to blood volume fractions of 1% to 5%.

We compared the obtained values for μs′ to the measured μs′ of Michels et al.³⁶. We calculated reference values for psb using Mie theory and the size distribution for Intralipid-20% as supplied by Michels et al.:

where fIL is equal to 4.151·103nm−1.³⁶ It is noteworthy that the size distribution for Intralipid-20% is not a fractal distribution. Based on this size distribution, we calculated the phase function for a discrete distribution of 10 diameters, from 25 to 750 nm, in steps of 50 nm. Following the approach of Michels et al., we used the wavelength-dependent refractive index of water for nmed and the wavelength-dependent refractive index of soy oil for npar using the Cauchy equation:

with Iwater=1.311, Isoy=1.451 and the same values of J and K for both water and soy oil of J=1.154·104 and K=−1.132·109.^36,37

The number of counts obtained from the spectrometer was corrected for the nonlinearity of the detector.³⁸ Next, the number of counts (Isample) was converted to an absolute reflectance using:

where Iback was a measurement performed in a black container with water, to include both the dark current of the spectrometer and internal reflections at the fiber tip. Iref was a measurement performed on Intralipid-20% diluted with water to a volume fraction of 1:20. Rref was the absolute reflectance of the 1:20 Intralipid dilution, obtained using the Fresnel reflection method of Zhang et al.³⁹

The measured spectra were fitted using the model of Post et al.^25,26 and minimizing the chi-squared value of the fit. In Eq. (2), we set the refractive index to 1.33 (the refractive index of water), which makes A equal to 1.0311 for Eq. (4). To reduce the number of fit parameters, μs′ was modeled as μs′=a(λ/λ0)−b, with λ0=500nm, and μa was modeled as the product of the volume fraction of Evans Blue (φEB) and the absorption spectrum of Evans Blue obtained from the transmission measurement (μa,EB).

In MDSFR, the spectra measured by both fibers are fitted simultaneously, resulting in a single set of optical properties for both fibers. To investigate the influence of the parametrization of psb on the fit results with two fibers, the analysis was performed once using the parametrization of psb and once where a value of psb was fitted per wavelength. First, the measurements with both fiber diameters were fitted simultaneously. Next, we investigated whether it was possible to accurately extract optical properties from a measurement with only a single fiber, using the parametrization of psb. For each fit, we calculated the confidence intervals based on the method proposed by Amelink et al.⁴⁰