Gary W. Small
Center for Intelligent Chemical Instrumentation
Department of Chemistry
Ohio University
Athens, OH 45701
Mark A. Arnold
Department of Chemistry
Iowa Advanced Technology Laboratories
University of Iowa
Iowa City, IA 52242
The potential use of near-infrared (near-IR) spectroscopy for noninvasive glucose measurements has attracted significant recent attention [1]. The principles behind this measurement are (1) to allow near-IR light to penetrate a region of body tissue and thereby excite vibrations in the constituent molecules; (2) to measure the amount of light absorbed as a function of wavelength; and (3) to use the resulting data to construct a calibration model that relates the spectral information to the concentration of blood glucose. The generation of this calibration model requires the measurement of reference glucose concentration values during the spectral data acquisition, typically through the collection of blood samples and the use of a conventional clinical glucose analyzer. The calibration model can be used subsequently to predict unknown glucose concentrations.
The construction of a successful calibration model requires the extraction of glucose-dependent information from the spectral background produced by the body tissue. Inspections of spectra collected from concentrated glucose solutions reveal significant glucose absorption bands centered at wavelengths of 1.67, 2.13, 2.27, and 2.33 µm. These bands arise from combination and overtone molecular vibrations associated with C-H and O-H bonds of the glucose moleule. The principal near-IR absorbers in tissue are water, proteins, and fat. Each is present in significantly greater quantity than glucose, and the spectral signals arising from each of these species overlap with one or more of the glucose absorption bands.
In addition, at wavelengths where the tissue is absorbing strongly, the precision of the optical measurement is degraded because the reduced amount of light escaping from the tissue does not produce a signal that is sufficiently greater than the intrinsic measurement noise. Furthermore, if the device used to couple the optical measurement to the tissue applies pressure, physical changes in the tissue such as dislocation of the fat can lead to a lack of reproducibility in the optical measurement. An additional problem caused by the presence of water is extreme temperature sensitivity in the background spectral response. The positions of the water absorption bands shift to shorter wavelengths with increasing temperature.
Since the water background is non-zero even between the absorption peaks, the glucose bands lie on top of this temperature-sensitive background.
Thus, temperature variations can cause apparent changes in the shapes of the glucose bands.
The presence of overlapping spectral signatures dictates that the optical measurement must be made over multiple wavelengths. Thus, the raw form of the data arising from the noninvasive measurement is a set of light intensity values collected over a series of spectral resolution elements.
These intensities represent either the light intensity transmitted through a section of tissue or the intensity reflected from the tissue, depending on the design of the interface between the tissue and spectrometer. The light intensities are a function of the spectrometer itself (e.g., the light source intensity, detector sensitivity, etc.), the scattering properties of the tissue, and the chemical composition of the tissue as manifested in the absorption of light at characteristic wavelengths.
If the intensity measurement is made over p spectral resolution elements, each spectrum is a 1 ´ p vector. A set of n spectra can be assembled into an n ´ p spectral data matrix, X. The columns of X are independent variables that can be used in the development of a calibration model relating spectral intensities to glucose concentration. This model can be represented as
(1)
where ci is the glucose concentration for sample i, the xi,j are the p spectral intensities measured for sample i, and ei is the error associated with the model. Development of the model requires an n ´ 1 vector of reference glucose concentrations for use in defining the model parameters.
For a transmission measurement, the relationship between light intensity and the concentrations of the absorbing species is described by the Beer-Lambert law. The light intensity can be expressed as
(2)
where the Ij term specifies the light intensity measured at resolution element j in the absence of absorbing species, q denotes the number of absorbing species, Îk,j is an intrinsic property of species k termed its absorptivity, b is the optical path length or thickness of the tissue sample, and ci,k is the concentration of species k in sample i. The form of (2) is based upon several key assumptions: (1) the species do not interact in a manner that causes a change of Îk,j with concentration; (2) the values of Îk,j are constant across the wavelengths that comprise the resolution element; (3) the Îk,j are independent of experimental parameters such as temperature; (4) the ci,k are low enough to make intermolecular interactions negligible; and (5) b is constant from measurement to measurement.
If the values of Ij can be measured, (2) can be linearized easily to
(3)
where ai,j is termed the absorbance of sample i at resolution element j.
This transformation allows the construction of calibration models of the form
(4)
where the bj parameters are estimated with the use of conventional multiple linear regression methods. The use of data at multiple resolution elements allows the subtraction of the components on the right side of (3) that represent the interferences.
Three problems complicate this simplified treatment of the data. First, the values of Ij are difficult to measure in the context of the noninvasive analysis. In practice, one of several strategies is used. In some cases, approximate values of Ij are measured through the use of a reference sample. Alternately, a constant of Ij = 1 is assumed for all j, and the log transform is performed anyway, or the xi,j are simply used directly to build the calibration model. The latter approach exploits the Maclaurin series expansion of 10-x » 1 - x.
Second, since each of the n samples requires a corresponding reference glucose value, it is impractical for n to be large. Thus, n is often <<p, with the result that the solution of (4) is underdetermined.
This limits the number of independent variables that can be used in the model and necessitates the use of selection procedures to determine which variables should be used.
Third, the characteristics of the noninvasive measurement typically violate some or all of the assumptions used in the derivation of (2). For example, unless temperature is tightly controlled, shifts in the water absorption bands impact (2) as changes in the Îk,j values of water. Similarly, a highly precise interface between the spectrometer and tissue measuring site is required to ensure that b remains constant from measurement to measurement.
To help overcome these problems, a combination of three approaches can be used. First, preprocessing methods can be employed to remove artifacts from the measured intensity values or the computed ai,j . For example, digital filters can be used to remove features from the intensity data on the basis of width [2]. In this way, wide features such as baseline artifacts or narrow features such as superimposed noise can be eliminated before the construction of the calibration model is attempted.
A second strategy is the use of latent variable methods to extract information from the xi,j or ai,j for the purpose of reducing the number of independent variables. The two most widely used methods are principal component analysis (PCA) and partial least-squares (PLS) [3]. Each computes a new set of spectral variables based on linear combinations of the original variables. For example, ti,1, the value of the first latent variable for sample i, is computed as
(5)
where the aj are weights that describe the manner in which each original variable contributes to the new latent variable. Although (5) is written with xi,j, the actual values used can be the preprocessed xi,j or the ai,j. Taken together, the set of aj values is termed a “loading vector”. The second latent variable is constructed by computing a second loading vector, typically one that is orthogonal to the first. A maximum of p latent variables can be computed, although the number used in practice is typically much less than p.
The PCA and PLS methods differ in the manner in which the loading vectors are computed. The PLS method extracts loading vectors that explain the covariance between the spectral variables and the reference analyte concentrations, while the PCA technique computes loadings that simply explain the variance in the spectral variables. The goal of both calculations is to “enrich” the new latent variables in information regarding the absorbing species and to remove redundant information. Once the latent variables are computed, a calibration model of the type described by (4) is most often used, with the values of the new variables replacing the ai,j in the equation. The number of latent variables to use, as well as which of the original p variables to include in the calculation of the latent variables, represent optimization issues that must be addressed.
Recent work in our laboratories has focused on the application of genetic algorithms to these optimization problems [4-5].
A third approach is to use formal nonlinear modeling techniques to attempt to encode the relationships described by (2). The most flexible implementation of this strategy is the use of artificial neural networks (ANN) [6]. The ANN is a flexible mathematical model that can be used to approximate both linear and nonlinear functions. Inputs to the model can be the original or preprocessed xi,j, the ai,j, or the latent variables computed by PCA or PLS. The model parameters are termed weights, and are determined by iterative optimization techniques.
For any of these data handling methods to be effective in building a robust calibration model for glucose, the optical measurement must be made with the greatest precision possible, and the interface between the human subject and the spectrometer must allow a reproducible measurement to be made. Furthermore, the experimental protocols must be devised to ensure that there are no correlations between glucose concentrations and time-dependent data artifacts. Finally, a sufficient quantity of data must be collected to ensure that the computed calibration models can be adequately tested. These latter two considerations are especially important. As increasingly sophisticated data handling methods are used, the possibility of fortuitous results and chance correlations also increases. Only through carefully designed protocols can these chance occurrences be avoided.
References
1. H. M. Heise, “Near-infrared Spectrometry for in vivo Glucose Sensing,” in Biosensors in the Body: Continuous In Vivo Monitoring, D. M. Fraser, Ed., Wiley, Chichester, UK (1997).
2. G. W. Small, M. A. Arnold, and L. A. Marquardt, “Strategies for Coupling Digital Filtering with Partial Least-Squares Regression: Application to the Determination of Glucose in Plasma by Fourier Transform Near-Infrared Spectroscopy,” Anal. Chem. 65, 3279-3289 (1993).
3. H. Martens and T. Næs, Multivariate Calibration, Wiley, Chichester, UK (1989).
4. R. E. Shaffer, G. W. Small, and M. A. Arnold, “Genetic Algorithm-Based Protocol for Coupling Digital Filtering and Partial Least-Squares Regression: Application to the Near-Infrared Analysis of Glucose in Biological Matrices,” Anal. Chem. 68, 2663-2675 (1996).
5. A. S. Bangalore, R. E. Shaffer, G. W. Small, and M. A. Arnold, “Genetic Algorithm-Based Method for Selecting Wavelengths and Model Size for Use with Partial Least-Squares Regression: Application to Near-Infrared Spectroscopy,” Anal. Chem. 68, 4200-4212 (1996).
6. K. Jagemann, C. Fischbacher, K. Danzer, U. A. Müller, and B. Mertes, “Application of Near-Infrared Spectrosccopy for Non-Invasive Determination of Blood/Tissue Glucose Using Neural Networks,” Zeitschrif. für Physikalische Chemie, Bd. 191, 179-190 (1995).