In this paper, we describe a method used to determine the optical properties, namely, the effective scattering and absorption coefficients, employing an optimized three-dimensional-printed single integrating sphere. The paper consists of two parts, and in Part 1, the theoretical investigation of an optimized measurement and the evaluation routine are presented. Using an analytical and a numerical model for the optical characterization of the integrating sphere, errors caused by the application of a non-ideal sphere (the one with ports or baffles) were investigated. Considering this research, a procedure for the precise determination of the optical properties, based on Monte Carlo simulations of the light distribution within the sample, was developed. In Part 2, we present the experimental validation of this procedure.

Based on theoretical investigations of the light propagation within an integrating sphere, we developed an accurate method to determine the optical properties of scattering media using an integrating sphere-based setup. The method takes into account the exact sphere geometry as well as the different angular distributions of the reflected and transmitted light from the sample and the calibration standard. We tested our novelties successfully in theory with Monte Carlo simulations and in practice using a 3D printed and professionally coated integrating sphere. As a result, we were able to determine precisely the effective scattering coefficient, μ s ′ , and the absorption coefficient, μ a , between 400 nm and 1500 nm in a range of μ a = 1 e − 3 m m − 1 to 10 m m − 1 and μ s ′ = 0.2 m m − 1 to 100 m m − 1 . Usually, the accuracy was around 1% for μ s ′ and around 3% for μ a for turbid phantom media with an optical thickness τ = μ s ′ d > 1 and a transmittance signal > 0 . 1 % .

A novel approach for precise determination of the optical scattering and absorption properties of porcine tissue using an optimized integrating sphere setup was applied. Measurements on several sample types (skin, muscle, adipose tissue, bone, cartilage, brain) in the spectral range between 400 nm and 1400 nm were performed. Due to the heterogeneity of biological samples, measurements on different individual animals as well as on different sections for each sample type were carried out. For all samples, we used an index matching method to reduce surface roughness effects and to prevent dehydration. The derived absorption spectra were used to estimate the concentration of important tissue chromophores such as water, oxy- and deoxyhemoglobin, collagen and fat.

Quantitative spatial frequency domain imaging (SFDI) for separation of the absorption (${\mu }_{\mathrm{a}}$) and the reduced scattering coefficient (${\mu }_{\mathrm{s}}^{\prime }$) gained importance in biomedical imaging over the last years. It facilitates the investigation of different kinds of tissue changes and biological processes, e.g., blood perfusion, neuronal death, and brain cellular composition changes.^{1 – 3} To eliminate specular and surface reflections in SFDI, crossed linear polarizing filters are generally used.⁴ However, as shown by Wiest et al.,⁵ the use of crossed polarizers causes significant errors in ${\mu }_{\mathrm{s}}^{\prime }$ and ${\mu }_{\mathrm{a}}$ when using a nonpolarized model for solving the inverse problem. As it is shown in this work, surface scattering $R_S$ leads to an offset in the spatial frequency resolved reflectance $R_{\mathrm{SFD}}(F)$. This offset mostly affects high spatial frequencies $F$, as for those $R_{\mathrm{SFD}}(F)$ is steadily decreasing^6,7 and therefore, the ratio $\frac{R_s}{R_{\mathrm{SFD}}(F)}$ is increasing. Correcting for this offset without any more information than $R_{\mathrm{SFD}}(F)$ is hardly possible, because the subdiffusive scattering properties (e.g., phase function) and the surface scattering affect the spatial-frequency resolved reflectance in a similar way. Bassi et al.⁸ were the first to investigate the phase shift due to turbid media, but the phase information was not used for solving the inverse problem. By now, the phase information was only used for determining the topography of the sample in order to correct for changes of the reference intensity.⁹

We demonstrate optical phantom experiments on the phase function parameter $γ$ using spatial frequency domain imaging. The incorporation of two different types of scattering particles allows for control of the optical phantoms’ microscopic scattering properties. By laterally structuring areas with either $TiO2$ or $Al2O3$ scattering particles, we were able to obtain almost pure subdiffusive scattering contrast in a single optical phantom. Optical parameter mapping was then achieved using an analytical radiative transfer model revealing the microscopic structural contrast on a macroscopic field of view. As part of our study, we explain several correction and referencing techniques for high spatial frequency analysis and experimentally study the sampling depth of the subdiffusive parameter $γ$.