Because reflectance values are not uniquely associated with a single combination of scattering and absorption properties, using any model in the inverse direction is a fundamentally ill-posed problem and requires use of wavelength-dependent relationships for and to reduce the dimensionality of the problem. In models of tissue, the spectral shape of is often based on a power law,14,15,23,27 and the absorption coefficient is often determined by a linear combination of concentrations and extinction coefficients for oxyhemoglobin and deoxyhemoglobin [ and , respectively] sometimes with the addition of a correction for a vessel-packing factor, , to account for the fact that blood is confined within blood vessels rather than uniformly distributed.29,30 Thus, the tissue parameters can be represented as follows: Display Formula
(3)where is a normalization wavelength. Here, , which is the absorption coefficient of whole blood, and are calculated as30Display Formula
(4)where is the concentration of hemoglobin in whole blood. Other contributions to the tissue absorption coefficient (e.g., beta carotene for adipose tissue in the visible, or water for any tissue in the NIR) can be added to as appropriate for the specific tissue and wavelength range. The remaining five coefficients include and , representing the scattering coefficient at a normalization wavelength, , and the scattering power-law exponent, related to average scatterer size, respectively, and , , and , representing the blood volume fraction (%), the hemoglobin oxygen saturation (%), and the average blood vessel radius, respectively. These wavelength relationships for and [Eqs. (3) and (4)] can then be substituted into any desired forward model. Measured reflectance spectra are fit to the model using a nonlinear least-squares curve fitting method, such as the Levenberg-Marquardt method, with , , , , and as the fitting coefficients. Thus, each measured spectrum can be characterized by these five physiological parameters. While these wavelength-dependent expressions for and are specific to tissue, by selecting alternative wavelength-dependent expressions, reflectance models can be used in the inverse direction to extract optical properties of any turbid medium.