Sample-to-sample turbidity variations are a limiting factor in non-invasive optical techniques. Macroscopically, light propagation in turbid media such as biological tissue is governed by elastic scattering and absorption of the media. For example, prominent absorption features of hemoglobin can lead to spectral shape distortions in tissue fluorescence spectroscopy [1
], which subsequently confound interpretation of underlying fluorophores. Similarly, in diffuse reflectance and Raman spectroscopy, turbidity variations can cause sampling volume variations across samples [6
]. Such turbidity-induced sampling volume variations introduce additional non analyte-specific variance into subsequent data analysis. This additional variance results in increased prediction error of analyte concentrations and a calibration algorithm that is not robust.
Many researchers have developed methods to correct for spectral distortions in biological fluorescence spectroscopy in which the shape of the observed spectrum is significantly altered by the presence of absorbers such as hemoglobin [3
]. Our laboratory has utilized diffuse reflectance spectroscopy (DRS) in the development of intrinsic fluorescence spectroscopy (IFS) to correct for turbidity distortions, particularly, absorption-induced spectral distortions of the fluorescence line shape [1
]. Diffusely reflected light is the backward emission which undergoes numerous elastic scattering events before re-emerging from the tissue, and thereby provides a metric for the amount of tissue absorption and scattering present. The optical properties of a given sample at a particular wavelength can therefore be measured in situ
by monitoring the diffuse reflectance at that wavelength. Similarly, DRS can monitor optical properties at multiple wavelengths. By measuring fluorescence and diffuse reflectance at the excitation wavelength and over the fluorescence emission wavelengths using the same excitation-collection geometry, the method of IFS can be used to remove these distortions. The underlying principle is that in a turbid medium fluorescence excitation-emission undergoes similar scattering and absorption as diffuse reflectance. For IFS, the main goal is to remove spectral shape distortions, exemplified by the hemoglobin absorption peak near 420 nm.
We are developing quantitative biological Raman spectroscopy to measure analyte concentrations in the near infrared (NIR) wavelength range (~ 830–1000 nm) [15
]. In these studies, shape distortion is less of an issue because of the lack of absorbers with prominent spectral features in the NIR wavelength range [16
]. However, for quantitative analysis, turbidity-induced sampling volume variations become very significant. Consider, for example, two identical biological tissue samples containing a Raman analyte, except that the second sample has a larger scattering coefficient than the first. The increased scattering causes light to be localized in a smaller volume, with a corresponding higher efficiency for the collection lens. As a result, the size of the Raman signal in the second sample will be larger than that of the first, and the measured concentration of the Raman analyte will be different. Furthermore, since both samples have some degree of scattering, the measured Raman concentration in both samples will differ from the actual concentration.
In the Raman spectroscopy literature, some researchers have applied corrections based on direct absorption spectroscopy [17
]. Waters extended the formalism developed by Kubelka and Munk to relate the Raman spectrum to the measured diffuse reflectance as a function of either the Kubelka-Munk absorption or scattering coefficient [19
]. This method assumes that only one optical property is changing at a time. Thus, for powdered samples, where the size of the particles and therefore their scattering characteristic change little over time, the effect of absorption from a progressively darkening sample on the Raman spectrum can be sufficiently removed [20
]. However, the Kubelka-Munk formalism is not generally applicable to biological tissue because it assumes isotropic scattering, and biological tissue scattering is known to be anisotropic [22
]. Recently, a method of retrieving Raman spectra from subsurface layers in diffusely scattering media was implemented, in which Raman scattered light from surface regions laterally offset from the excitation laser spot was collected and analyzed via multivariate techniques [23
]. However, correction for sampling volume was not considered.
Under the photon migration framework developed by Wu et al.
], the same general principle that applies for IFS should hold true for Raman spectroscopy as well. The goal of this paper is to present a method which corrects measured analyte concentrations for turbidity-induced sampling volume variations. We refer to this method as intrinsic Raman spectroscopy (IRS). Starting from the photon migration theory, we review the analytical model for extracting intrinsic fluorescence (the fluorescence as it would be observed in the absence of scattering and absorption) and derive a parallel expression for the intrinsic Raman spectrum. Monte Carlo simulations are then employed to demonstrate its validity and elucidate the relationship between observed Raman and diffuse reflectance for semi-infinite samples and samples of finite dimension. An analysis of the methodology with respect to sample size and scattering anisotropy is presented. An experimental study is presented in the companion paper to validate this method and demonstrate that it can be used experimentally to correct for turbidity-induced spectral distortions and sampling volume variations.