Anal. Chem. 1990, 62, 1457-1464 (29) (30) (31) (32) (33) (34) (35) (36) (37)
Ranz, W. E.: Wong. J. B. Ind. Eng. Chem. 1852, 4 4 , 1371. Giilespie, G. R.; Johnstone, H. F. Chem. Eng. Prog. 1855, 51, 74-F. Mercer, T. T.; Stafford, R. G. Ann. Occup. Hyg. 1868, 12, 41. Stern, S. C.;Zeller, H. W.; Schekman, A. I. Ind. Ens. Chem. Fundam. 1862, 1, 273. Loktev, E. K.; Goldberg, V. M.; Kerber, M. L.; Akutin, M. S. Izv. Vyssh. Uchebn. Zaved., Khlm. Khlm. Teknol. 1978, 27, 723. Lundgren, D. A. J . Air. Pollut. Control Assoc. 1867, 4 , 225. Lundgren, D. A., et ai. Aerosol Measurement; University Presses of Florida: Gainsviile, FL, 1979. Rao, A. K.; Whitby, K. T. J . AerosolSci. 1878, 9 , 87-100. oodo,T.; Takashima, Y.; i-ianzawa, M. J . Chem. ~ n gJpn. . 1881, 74, 76-78.
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(38) Esmen, N. A.; Ziegler, P.; Whitfield, R. J . AerosolSci. 1878, 9 , 547.
RECEIVED for review February 1, 1990. Accepted April 12, 1990. This work was supported by the National Institutes of Health under Grant No. GM 32002 and presented at the 16th annual FACSS meeting in Chicago, IL, Oct 1-6, 1989, as paper number 71. R.G.M. was supported by a Research Career Development Award from the National Institute of Environmental Health Sciences under Grant No. ES00130.
Determination of Physiological Levels of Glucose in an Aqueous Matrix with Digitally Filtered Fourier Transform Near-Infrared Spectra Mark A. Arnold and Gary
W.Small*
Department of Chemistry, T h e University of Zowa, Iowa City, Iowa 52242
A procedure is described for the measurement of cilnicaiiy relevant concentratlons of glucose in aqueous solutions with near-infrared (NIR) absorbance spectroscopy. A glucose band centered at 4400 cm-' is used for this analysis. NIR spectra are collected over the frequency range 5000-4000 cm-l with a Fourier transform spectrometer. A narrowband-pass optical interference filter is placed in the optical path of the spectrometer to eliminate ilght outside this restricted range. This configuration provides a 2.9-fold reduction in spectral noise by utilizing the dynamic range of the detector solely for light transmitted through the filter. I n addition, a novel spectral processing scheme is described for extracting glucose concentration information from the resulting absorbance spectra. The key component of this scheme is a digital Fourier filter that removes both high-frequency nolse and low-frequency base-llne variations from the spectra. Numerlcal optimizatlon procedures are used to identify the best locatlon and wldth of a Gausslan-shaped frequency response function for this Fourier filter. A dynamic area calculation, coupled with a simple linear base-llne correction, provides an Integrated area from the processed spectra that is linearly related to glucose concentratlons over the range 1-20 mM. The linear callbration model accurately predicted glucose levels in a series of test solutions with an overall mean percent error of 2.5 % . Based on the uncertainty in the parameters defining the callbration model and the variability of the magnitudes of the integrated areas, an overall Uncertainty of 7.8 % was estimated for predicted glucose concentrations.
INTRODUCTION Considerable effort has been placed on the development of a reliable glucose sensor capable of serving as the active sensing element for an implantable artificial pancreas (1-4). To be considered for this application, the sensor must be able to measure glucose selectively and continuously under in vivo conditions for at least 1year. Although glucose sensors have been successfuly developed for short-term continuous monitoring (5-7), none of these devices is suitable for long-term
in vivo applications. The primary limitation has been a fundamental lack of compatibility between the sensor and the site of implantation (8). The issue of biocompatibility has been recognized for many years as the ultimate barrier to the successful development of an implantable glucose sensor (9, 10). An alternative approach is to monitor in vivo glucose levels by a noninvasive spectroscopic measurement. In concept, noninvasive blood glucose measurements can be made by transmitting a selected band of radiation through a vascular region of the body and calculating the clinically relevant glucose concentration from the resulting transmission spectrum. With such a noninvasive approach, biocompatibility issues are irrelevant because nothing is in direct contact with the sample. In addition, a noninvasive approach is reagentless and, therefore, is not limited by the stability or consumption of reagents. These attributes make a noninvasive blood glucose sensing scheme ideal for the continuous measurements required for the artificial pancreas. Such a sensing scheme would also be attractive for short-term continuous measurements such as during emergency treatment of extreme hyperglycemia. A noninvasive monitor would also be preferred for daily patient measurements at home where a simple and painless technique is desirable, especially for measurements on children. Because neither chemical reagents nor physical separations can be used in a noninvasive approach, the primary concern is selectivity. Overall, the spectroscopy alone must provide sufficient selectivity for the measurement. The challenge is to establish a procedure that accurately predicts the concentration of glucose at clinically relevant levels in the complex matrix of whole blood from the information in a transmission spectrum. Results from two recently reported investigations suggest that vibrational spectroscopy can be used for this measurement. Zeller and co-workers (11)have concluded that a single frequency in the midinfrared region is suitable for selective glucose measurements. Their preliminary results indicate that measurements at 1040 cm-' will provide selective information over proteins, carbohydrates associated with proteins, hemoglobin, urea, and lipids. Heise and co-workers (12),on the
0003-2700/90/0362-1457$02.50100 1990 American Chemical Society
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Flgure 1. Near-infrared absotbance spectra for whole blood (solid line) and 0.1 M, pH 7.2 phosphate buffer (dashed line).
other hand, have proposed a multivariate calibration method based on the partial least-squares algorithm. In their work, a multivariate calibration model is constructed with midinfrared absorbance spectra over the spectral range 1500-750 cm-l. Their model is based on a sample population composed of anticoagulated blood from 150 randomly selected patients in a general hospital. Although these measurements were performed on discrete, isolated blood samples, the success of this model suggests that a multivariate approach can provide suitable selectivity for continuous noninvasive measurements. We are evaluating the feasibility of using near-infrared (NIR) spectroscopy as the basis for noninvasive blood glucose measurements. Our initial efforts have centered around establishing a procedure for measuring clinically relevant concentrations of glucose in an aqueous matrix. NIR spectra of a pH 7.2 phosphate buffer and a human blood sample are presented in Figure 1. The major features in both spectra over the frequency range 10000-4000 cm-' (1000-2500 nm) are those of water. Clearly, the large amount of water in blood, coupled with the strong NIR absorption characteristics of water, make water the primary interference for glucose measurements in blood. Dull and Giangiacomo (13) have shown that glucose in water can be quantified with NIR spectroscopy. The detection limit of their system was only 167 mM, however, and their ability to measure lower concentrations was limited by the strong absorption of water. Normal levels of glucose in human blood range from 3.5 to 6.1 mM (14).Before attempts to measure glucose in whole blood can begin, we must first demonstrate the ability to measure 3 mM glucose in a relatively simple aqueous matrix. Success will require a 50-fold improvement in the detection limit reported by Dull and Giangiacomo. In this paper, we describe a procedure for measuring glucose in the 1-20 mM concentration range in an aqueous matrix with NIR absorption spectroscopy. The key components of this procedure are (1)the incorporation of a band-pass filter in the optical path of a Fourier transform NIR spectrometer and (2) the use of a novel Fourier filtering procedure in the spectral processing scheme. The band-pass filter significantly enhances the sensitivity of the detector for the spectral region of interest. Fourier filtering effectively removes both high- and low-frequency noise from the raw absorbance spectra. Numerical optimization procedures have been used to identify the best frequency response function for this Fourier filtering step. Overall, a linear calibration model has been obtained by relating the integrated area under a selected region of the processed absorbance spectrum with the corresponding glucose concentration.
EXPERIMENTAL SECTION Apparatus. Spectra were collected with a Nicolet 740 Fourier transform infrared (FTIR) spectrometer system (Nicolet Analytical Instruments, Madison, WI). The spectrometer was configured for operation in the NIR region by using a 75-W tungsten-halogen source, quartz beam splitter, and room-temperature PbSe detector. A multilayer optical interferencefdter (Barr Aasoc., Westford, Ma) was used to isolate the range 5000-4000 cm-' (2000-2500 nm). Solution temperatures were controlled with a VWR Model 1140refrigerated temperature bath (VWR Scientific, Chicago, IL). Temperatures were measured with a 1/32-in.diameter copper-constantan thermocouple probe (Omega Inc., Stamford, CT) in conjunction with an Omega Model 670 digital meter. This arrangement allowed temperatures to be measured with an accuracy of f O . l "C. Spectral processing was performed on a Prime 9955 interactive computer system operating in the Gerard P. Weeg Computing Center at the University of Iowa. Spectra were transferred from the Nicolet 620 computer to the Prime system by a serial communications link. All computer software was implemented in FORTRAN 77. Fourier transform computations were performed with subroutines from the IMSL software package (IMSL, Inc., Houston, TX). The numerical optimization calculations were implemented with subroutines described and listed in ref 15. Regression computations were performed with the MINITAB statistical software system (Minitab, Inc., State College, PA). Reagents. Reagent grade glucose and potassium phosphate salts were obtained from common suppliers. 5-Fluorouracilwas purchased from Sigma Chemical Co., St. Louis, MO. All solutions were prepared with reagent grade water obtained by passing house-distilled water through a Milli-Q three-house purification unit. Water was purified immediately before use. Procedures. Standard glucose solutions were prepared by diluting a stock glucose solution with the working buffer. The working buffer was a pH 7.2, 0.1 M phosphate solution with 0.044% 5-fluorouraciladded as a preservative. The stuck glucose solution had a concentration of 1.002 M and was prepared by dissolving the appropriate amount of dried glucose powder in the working buffer. Calibration standards were prepared from the stock solution immediately before use. Single-beamspectra were collected by the following procedure. Glucose standards were heated or cooled to the desired temperature. A small volume of solution was placed in a cell made from infrared quartz with a 1-mmpath length (Wilmad Glass Co., Buena, NJ). The filled cell was then positioned in the sample holder of the spectrometer. A glass jacket was positioned around the sample in the holder to control the solution temperature. A thermocouple probe was positioned in the sample solution, and the temperature was continuously recorded as the spectral data were collected. Unless noted otherwise, all spectra were collected at 20 "C. Data collection was initiated after the solution reached a constant temperature. Double-sided interferograms with 16384 points were collected based on 256 coadded scans. These interferogramswere triangularly apodized and Fourier transformed. The point spacing in the transformed spectra was 1.9 cm-'. Mertz phase correction was applied to the spectra, with the phase array used based on 200 points on each side of the interferogram centerburst. The resulting single-beam spectra were transferred to the Prime computer system where the processing steps detailed below were performed.
RESULTS AND DISCUSSION Investigation of Experimental Parameters. The successful quantitation of glucose in an aqueous matrix requires the identification of glucose spectral bands that can be isolated from those of water. As indicated in Figure 1, the NIR spectrum of water is dominated by large absorption bands with peak maxima a t 6876,5267, and 3800 cm-' (1454,1899,2632 nm). The spectral regions outside these bands were inspected for the presence of usable glucose bands. On the basis of this inspection, two NIR regions were found to contain glucose information. The region 6500-5600 cm-' (1538-1786 nm) contains three glucose bands of relatively low absorptivity. This is the region
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Figure 2. Near-Infrared spectra for glucose (solid line) and 0.1 M, pH 7.2 phosphate buffer (dashed line).
used previously by Dull and Giangiacomo (13)to determine high concentrations of glucose (3-52% w/w, approximately 167-2887 mM) in aqueous solutions. Spectra of 100.3 mM glucose in phosphate buffer collected in our laboratory, however, revealed no apparent glucose signals in this region. A second region, 4800-4200 cm-' (2083-2381nm), contains glucose bands of higher absorptivity. The solid line in Figure 2 plots a spectrum of pure glucose in KBr over the region 5400-3800cm-' (1852-2632nm).The dashed line in this f i e traces the correspondingspectrum of phosphate buffer. Both spectra have been referenced to an air background spectrum and are plotted in normalized absorbance units to facilitate a qualitative comparison. Of the available glucose absorption bands, the band centered at 4400 cm-' (2273nm) is located near a minimum in the water absorbance. In addition, this glucose band is significantly narrower than the nearby water absorption bands. This difference in band shape can be exploited in the development of a spectral processing scheme to separate glucose information from the water background. The potential of the band at 4400 cm-' was tested by inspecting the spectrum of a 100.3 mM glucose solution. The glucose signal was clearly evident in this spectrum, confirming the utility of this band. For these reasons, subsequent efforts focused on this glucose absorption band. Given that 100.3 mM represents a concentration greater than 8 times the maximum glucose level in normal blood, efforts were next made to detect glucose absorptions at lower concentrations. A t concentrations less than 100 mM, no glucose signals were detectable at 4400 cm-' when the single-beam spectra were referenced to the single-beam spectrum of phosphate buffer. For this reason, steps were taken to improve the detection limit by enhancing the sensitivity of the detector. A band-pass optical filter has been used to enhance the sensitivity of the detector at frequencies around 4400 cm-'. This approach enhances the sensitivity by eliminating radiation outside of the spectral region of interest, thereby utilizing the dynamic range of the detector more efficiently. In the work described here, a multilayer optical interference filter with a nominal band-pass of 5000-4000 cm-' (2000-2500nm) has been used. The percent transmittance of this fiter at 4400 cm-' is 83%. In addition, the strong absorbance of water in this region contributes to the overall band-pass of the system. The effect of the filter has been measured by comparing the noise levels of spectral 100% lines collected with and without the filter. These 100% lines were computed by ratioing two replicate spectra of water collected at 1-mm path length. Each spectrum was based on 256 coadded interfero-
Figwe 3. Spectral 100% lines wlthout (A) and with (B) the interference filter in the optical path.
grams. Figure 3 displays these 100% lines from 4800 to 4200 cm-' (2083-2381 nm). Root-mean-square (rrns) noise levels were calculated over the spectral region 4439-4351 cm-' (2253-2298 nm), corresponding to the base width of the targeted glucose band centered at 4400 cm-'. Without the filter, the rms noise level is 83.4 microabsorbanceunits. With the filter present, however, the noise level is reduced to 29.1 microabsorbance units. The filter provides a 2.9-fold reduction in the rms noise level over this spectral range. Without the filter, the glucose band at 4400 cm-' was not observed with an 88 mM glucose solution. With the filter, this band was apparent at concentrations down to 4.8 mM. Preliminary work revealed a strong dependence between the temperature of the solution and the signal-to-noise ratio of the glucose band. Initially, solution temperatures were not controlled, and considerable variation was observed in the quality of the glucose spectra. The primary effect of temperature was found to be a dramatic shift in the water absorbance. As the temperature increases, the two water bands centered at 5200 and 3800 cm-' (see Figure 2) shift to higher frequencies, thereby lowering the amount of light available in the region of the targeted glucose band. The change in the amount of light transmitted at different temperatures was measured by collecting single-beam spectra of a buffer solution at 8.8,19.9,29.0,38.5and 52.3 "C. Each of the single-beam spectra was normalized and then ratioed to the spectrum collected at 8.8 "C. The resulting transmittance spectra are plotted in Figure 4 from 4800 to 4100 cm-l(2083-2439 nm). These spectra clearly show that as the valley between the two water absorption bands shiftsto higher frequencies, more light is transmitted at higher frequencies and less light is transmitted at lower frequencies. As a result, the amount of light transmitted in the region 4439-4351 cm-' decreases significantly as the temperature increases. A decrease of nearly 20% is observed at 52.3 "C. Considering that the measurements made here are detector-noise limited, the decrease in light at higher temperatures should lead to lower precision in the measurement. Indeed, spectra collected at lower temperatures clearly possessed higher signal-to-noise ratios in the region of the glucose band. On the basis of these results, subsequent experiments were conducted with the sample temperature maintained at 20 OC. This temperature was selected as a compromise in terms of spectral quality vs experimental practicality. Design of Spectral Processing Strategy. Collection of Test Data. A series of 17 standard solutions of glucose in pH 7.2 phosphate buffer were prepared, spanning the
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Figure 5. Raw absorbance spectra for glucose standards with concentrations of 3.3 (A), 6.6 (B), 9.9 (C),and 13.2 mM (D).
operates by extracting certain underlying frequencies from the overall spectral data. In effect, the spectra in Figure 5 can be decomposed into a series of additive harmonic signals. The base-line variation is defined by a series of low-frequency signals, while the noise superimposed on the spectral band consists of a series of high-frequency signals. Analogously, the analyte spectral band is composed largely of midrange frequencies. Thus, the digital filtering approach to the simultaneous removal of both noise and base-line artifacts is the use of a band-pass filter to extract the midrange frequencies corresponding to the analyte information. The most flexible and easily applied technique for extracting specific frequencies from data is Fourier filtering (26-32). This operation is based on a Fourier analysis of the data and is depicted pictorially in Figure 6. The upper left plot in the figure is the absorbance spectrum produced by the 6.6 mM glucose solution. While the glucose band at 4400 cm-' is apparent, a significant amount of noise is superimposed on the band, and the band itself is superimposed on a sloping base line. By applying the Fourier transform to this spectrum, the intensities of the various frequency components of the data can be obtained. In the work performed here, this operation was accomplished by zero-filling the data such that the number of points equaled a power of 2, followed by application of the fast Fourier transform (FFT) algorithm. The FFT decomposes the data into real and imaginary frequency components. These frequency components are most easily displayed by computing an amplitude spectrum. Each point in the amplitude spectrum is simply the square root of the sum of squares of the corresponding real and imaginary points. The amplitude spectrum corresponding to the 6.6 mM solution is indicated by the solid line in the upper right plot in Figure 6. The lower axis in the figure is a linear frequency scale. In keeping with common practice in the signal-processing literature, this frequency scale is given in digital frequency units, where the maximum observable frequency is defined as 0.5. An inspection of the amplitude spectrum reveals several sharp, intense features a t low frequency, along with an undulating series of signals out to the maximum frequency. The action of a Fourier filter is to remove from the amplitude spectrum the information pertaining to unwanted frequencies. This is accomplished through the definition of a frequency response function. This function defines the frequencies that the filter will pass. A typical frequency response function is indicated by the dashed line superimposed on the amplitude spectrum in the upper right of Figure 6. The
ANALYTICAL CHEMISTRY, VOL. 62, NO. 14, JULY 15, 1990
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Figure 6. Operation of Fourier filter on raw absorbance spectrum of 6.6 mM glucose. The upper left shows the raw spectrum. The upper right shows the amplitude spectrum (solid line) after Fourier transformation of the raw spectrum and the frequency response function (dashed line) for the d g b l filter. The lower left shows the resulting amplitude spectrum after multiplying the initial amplitude spectrum by the frequency response function. The lower right shows the final filtered spectrum after inverse transformation back to the original data domain.
function used here is a simple Gaussian curve, although other function shapes can certainly be used. The advantage of a smoothly varying function such as the Gaussian is that a smooth truncation of the individual frequencies is achieved. This minimizes the introduction of artifacts into the filtered data (26). The frequency response function is applied to the data by performing a complex multiplication, point by point, of the real frequency response function with the real and imaginary spectral components obtained through the FFT. The result of this operation is seen best visually, however, by multiplying the frequency response by the amplitude spectrum. The lower left plot in Figure 6 displays the product of this multiplication. The filtered result can be returned to the original data domain by applying the inverse FFT to the complex product. The result of this operation is depicted in the lower right plot in Figure 6. The absorption band at 4400 cm-' is now clearly apparent. The noise in the spectrum has been suppressed, and much of the base-line variation has been removed. It should be noted that the inverse FFT is necessary only to return the data to the form of an absorption spectrum. As the FFT is a linear transformation, the information necessary for quantitating the analyte is also present in the original filtered result. The principal advantage gained by returning the data to the form of an absorption spectrum is that familiar procedures can then be used to process the filtered absorption band. The key to the success of the Fourier filtering operation displayed in Figure 6 was the definition of the frequency response function. If an optimum frequency response were
not used, unwanted information could be retained in the spectrum or needed information could be removed. In terms of the Gaussian function used here, definition of the optimum frequency response requires the definition of the mean and standard deviation of the Gaussian. These parameters determine the location and width of the filter band-pass. Several workers have studied methods to use in defining the filter frequency response. If the pure (Le., noise-free) analyte signal can be characterized, its Fourier transform can be used as a template for the filter frequency response. This is termed matched filtering (26). For spectral or chromatographic peaks, Lam and Isenhour (31) have developed a direct method for estimating an optimal frequency cutoff for the filter based on an assumed peak shape and an estimated width for the peak. Rossi and Warner (32) have used the difference between a filtered noisy signal and its noise-free counterpart as a guide to the definition of the frequency response. By computing the mean-squared error between the two signals, an objective function was obtained that served as a numerical guide to the filter development. When a filter was found that produced a minimum mean squared error, the filter development step was judged complete. The work described above focused solely on the selection of one filter parameter-the width of the frequency response. In these studies, the filters used were centered at zero frequency. On the basis of the discussion of Figure 6, however, both base-line variation and noise can be suppressed simultaneously through the use of a band-pass filter with a center displaced from zero frequency. The use of such a filter requires the selection of both a center position and width,
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however. The methods discussed above possess no capability for defining the optimum settings of two filter parameters in an automated manner. In devising a two-parameter filter design strategy, it was judged advantageous to make use of numerical optimization techniques. These procedures are computer-basedalgorithms that seek the optimum values of a set of parameters, given a response function that allows a numerical value to be obtained that characterizes the performance of any given parameter set. The key to the success of any numerical optimization study is the definition of a response function that clearly defines the optimum parameter set. In the glucose application, a response function of the type used by Rossi and Warner is not useful, as a noise-free, background-free analytical signal cannot easily be obtained. Given that the glucose band at 4400 cm-’ is superimposed on the background water absorbance, the band shape changes with concentration. In effect, the filter must be designed such that it is maximally effective at suppressing the background across the concentration range studied. Analogously, the matched filter technique discussed above cannot be used here as it cannot train the filter to work optimally across the concentration range. These concepts motivate the selection of a response function that is keyed directly to the overall goal of the workquantitation of glucose across the 1-20 mM concentration range. If the filter is successful, an absorption band of the type displayed in the lower right plot in Figure 6 will be obtained. This band can be integrated, producing a value for the absorbance area that can be related to concentration through a simple univariate calibration model based on the Beer-Lambert law. Across the range of concentrations, the use of the optimum filter will lead to a calibration model with maximum linearity. A minimum of variation will exist about the model. Thus, the best evaluationof the filter is the quality of the calibration model that ultimately results. A variety of statistics are available to assess the success of the calibration model. Utilizing simple linear regression techniques, a calibration model is formed as
A = bo + b1C
(1)
where A is a vector of integrated absorbance values, C is a vector of corresponding concentrations, and bo and bl are coefficients determined through the application of regression analysis to the A and C vectors corresponding to the set of standard glucose solutions. A smooth, sensitive function that describes the goodness-of-fitof the data to eq 1 is the t value for the statistical significance of b, (33). This value is defied as
where n
n
4 b J = [(C(ai- @ / ( n - 2))/(C(ci- V)I1” (3) i=l
i=l
In eq 3, s(bl) is the estimated standard error of b,, the aiare the computed absorbance areas, the cii are the corresponding absorbance areas predicted by the calibration model, the ci are the concentrations of the calibration standards, F is the mean concentration of the calibration standards, and n is the number of calibration standards. As t increases, the overall goodness-of-fit of the data to the calibration model improves. Thus, in principle, the computed t value can be used to guide the selection of the center position and width of the frequency response of the Fourier filter. Two observed problems complicate this scheme. First, as seen in Figure 6, the filter may not remove all of the base-line variation. Second, an artifact of certain values of the filter
position and width is a slight shift in the peak maximum of the glucose band. Both of these phenomena prevent a simple integration of the signal after application of the Fourier filter. These problems can be overcome, however, through the use of a dynamic area calculation and simple linear base-line correction. After application of the filter, the first derivative of the spectrum is evaluated through the use of a central difference approximation (34). The limits of the glucose band are then taken as the zero-crossings in the derivative spectrum on either side of the peak maximum. These two inflection points are used to define a linear base line. The value of the base line a t each frequency is subtracted from the points in the filtered spectrum, thereby removing the remaining base-line variation. This simple base-line correction scheme is made possible by the removal of most of the base-line variation by the Fourier filter. The area under the baseline-corrected band is next integrated by use of Simpson’s 1/3 and 3/s rules (34). To account for band shifts, the integration is performed over a fixed frequency range on each side of the peak maximum. The above response function was tested with two numerical optimization strategies. The modified simplex approach (35-37) was used, along with Powell’s direction set method (15). A Gaussian-shaped frequency response was used,thereby defining the mean and standard deviation of the Gaussian as the two filter parameters to be optimized. The set of 36 calibration spectra was used in the optimization. Both optimization methods were executed multiple times, with the calculations being initialized with a variety of starting filter parameters. Powell’s method demonstrated less susceptibility to local minima and consistently optimized to a mean of 0.023 and a standard deviation of 0.0050. This is the frequency response plotted in the top right of Figure 6. Both parameter values are expressed in digital frequency units, as in Figure 6. The t value produced by the response function was observed to vary slightly with the width of the integration range. On the basis of a series of experiments, a total integration range of 45 cm-l was selected as optimum. As noted above, this range was centered on the peak maximum in each case. The t value produced under these conditions for the 36 spectra was 72.3. This value is indicative of an excellent goodness-of-fit. A concern in all numerical optimization studies is the question as to whether the optimum parameter values found truly represent an overall global optimum. The key factor in determining the likelihood of finding the global optimum is the degree to which the response function is well-behaved. To justify the filter parameters found, it was deemed necessary to investigate the behavior of the response function used here. This investigation was performed by evaluating the response function for all combinations of mean and standard deviation over the range 0.04.05. A step size of 0.001 was used for each parameter, resulting in a total of 2601 evaluations of the response function. Figure 7 is a three-dimensional plot resulting from this experiment. The t values produced by the response function are plotted vs the corresponding values of mean and standard deviation. The sharp peak in the plot coincides with the filter parameters found through the optimization study. The overall smooth surface of the plot leading to the optimum lends confidence to the results obtained through numerical optimization and serves to justify strongly the selection of the t value of the calibration curve as the basis for the response function. Testing of Calibration Model. Figure 8 depicts the results obtained by applying the derived Fourier filter to the four spectra in Figure 5. After filtering, the simple base-line correction procedure described above was used. The glucose band has been effectively extracted from the background
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function for the Fourier filter.
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Figure 8. Final processed spectra for glucose standards with concentrations of 3.3 (A), 6.6 (B), 9.9 (C), and 13.2 mM (D).
absorbance in each case, and the noise superimposed on the spectra has been suppressed. Figure 9 is a plot of the calibration model obtained by performing the dynamic area calculation on the 36 calibration spectra. The point corresponding to each calibration spectrum is indicated by an open circle in the figure. The calibration model has a slope of (6.19 f 0.08) X and a y intercept of (4.46 f 0.95) X lo4. The plot is highly linear throughout the 1-20 mM concentration range, producing an R2for the regression of 99.4% and, as noted above, a t value of 72.3 for the significance of the slope. The F value for the significance of the regression is 5231. The variation about the regression line is random with concentrationand with respect to the order of the data collection. The derived calibration model was tested by employing it to predict the concentrations of the 15 spectra withheld from the filter development and calibration calculations. The integrated areas corresponding to these points are plotted as the solid circles in Figure 9. Without exception, these points lie on the regression line within the variation seen in the calibration spectra. Table I compares the predicted and known concentrations of the solutions. The predicted value for each solution is taken as the mean of the predicted values
Table I. Accuracy of Glucose Determinations
actual, mM
predicted, m M
std dev, m M
4.3 6.0 7.9 11.2 14.4
4.1
6.3 7.8 11.2 14.3
0.07 0.3 0.5 0.9 0.6
% error
5.3 4.4
1.7 0.047 0.98
for the three replicate measurements. The standard deviation of the predicted values is listed, along with the percent error of the prediction. The predictions are highly accurate, with the maximum prediction error equal to 0.3 mM. The mean percent error is 2.5%. Finally, the reproducibility of the methodology was examined by comparing processed spectra for three sets of three replicates of an 8.5 mM glucose solution. In this experiment, a reference spectrum of the pH 7.2 phosphate buffer was collected initially in the conventional manner. Next, the sample cuvette was filled with the glucose solution (sample A), and the single-beam spectrum was collected as normal with 256 scans (trial 1). Two more spectra were then collected (trials 2 and 3) before this sample was removed and discarded. The sample cuvette was flushed and refilled with a fresh aliquot from the same stock 8.5 mM glucose solution (sample B), and the same procedure was used to collect three spectra for this sample (trials 1-3). The entire procedure was repeated a third time for sample C. Each glucose single-beam spectrum was ratioed to the same reference spectrum, and the resulting transmittance spectra were converted to absorbance units. The Fourier filtering, base-line correction, and dynamic area calculations were performed on these spectra. The resulting nine processed spectra are plotted in Figure 10. Considerable variation is observed between the samples and between the trials within the samples. Variations are evident in the position, width, and height of the resulting absorption band. No trends could be identified in these variations. The pooled relative standard deviation for the entire set of computed areas is 6.9%. Propagation of the errors corresponding to the uncertainties in the parameters of the calibration model and the area measurement reveals a relative standard deviation of 7.8% for the predicted glucose concentration at 8.5 mM. This high degree of variation in the processed area is likely caused by the small absorbance values. Such low absorbances are sensitive to slight variations in the data collection environment (i.e., interferometer alignment). Collection of a reference spectrum immediately prior to the sample would be expected to improve the measurement
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ANALYTICAL CHEMISTRY, VOL. 62,NO. 14, JULY 15, 1990
sponse is a powerful technique for obtaining an optimally filtered spectrum.
LITERATURE CITED
-2.04 4410
4440
4420
4400
4580
i 4360
4340
FREQUENCY (cm-')
Flgure 10. Processed spectra for 8.5 mM glucose solutions showing three trials for each of three samples: sample A (solid lines), sample B (dashed lines), and sample C (dotted lines).
precision. In addition, a higher intensity light source would be expected to improve the measurement precision by allowing longer optical path lengths, thereby providing higher absorbance values.
CONCLUSIONS The results described above clearly indicate that NIR spectroscopy can be used to quantitate physiological levels of glucose in an aqueous matrix. The successful quantitation of glucose in this concentration range represents a 50-fold reduction in the glucose detection limit when compared to previously published work. Given that the background absorbance of water is the principal NIR spectral interference in whole blood, this work embodies a first key step in the development of a noninvasive glucose sensor. The next phase of the work is to examine the impact of other blood constituents on the analysis. By increasing the complexity of the sample matrix, the selectivity of the current glucose analysis scheme can be firmly established. A central question to be addressed is whether or not the current univariate calibration scheme provides sufficient selectivity for clinical measurements or if a multivariate approach is required. In addition, the influence of experimental parameters such as spectral resolution, interferometer stability, and solution pH must be evaluated. Last, it should be noted that the Fourier fiitering procedures used here are general and can be applied in other spectral processing applications. The use of numerical optimization techniques to guide the definition of the filter frequency re-
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Received for review December 15, 1989. Accepted April 16, 1990. This work was supported by the National Institutes of Health under Grant 1-R03-RR04583-01. Portions of this work were presented at the 1989 Meeting of the Federation of Analytical Chemistry and Spectroscopy Societies, Chicago, IL, October, 1989.
