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1 March 2007 Mid-infrared laser measurements of aqueous glucose
Author Affiliations +
Abstract
Mid-IR semiconductor lasers of two wavelength bands, 5.4 and 9.6 μm, are applied to measure aqueous glucose concentration ranging from 0 to 500 mg/dL with Intralipid® emulsion (0 to 8%) added as a fat simulator. The absorption coefficient µa is found linear with respect to glucose and Intralipid® concentrations, and their specific absorption coefficients are obtained via linear regression. These coefficients are subsequently used to infer the concentrations and compare with known values. The objective is to evaluate the method accuracy. Glucose concentration is determined within ±21 mg/dL with 90% confidence and ±32 mg/dL with 99% confidence, using <1-mJ laser energy. It is limited by the apparatus mechanical error and not the photometric system noise. The expected uncertainties due to photometric noise are ±6 and ±9 mg/dL with 90 and 99% confidence, respectively. The uncertainty is fully accounted for by the system intrinsic errors, allowing rigorous inference of the confidence level. Intralipid® is found to have no measurable effect on glucose determination. Further analysis suggests that a few mid-IR wavelengths may be sufficient, and that the laser technique offers advantages with regard to accuracy, speed, and sample volume, which can be small, ~0.4×10-7 mL for applications such as microfluidic or microbioarray monitoring.

1.

Introduction

There have been extensive efforts to develop optical techniques to measure blood glucose concentration for diabetes care.1, 2, 3 There is also the application of in situ real-time monitoring of glucose in cell culture, microassay arrays, and bioreactors. In both applications, accuracy, reliability, and speed are important. This work is about a laser-based photometric technique that can address these issues. This work aims primarily for the latter application, but the technique is potentially applicable for blood glucose.

A broad review of minimally or noninvasive blood glucose monitoring has been given by Klonoff.4 More recently, Khalil5 and Cote6 specifically reviewed optical spectroscopy, including absorption, polarimetry, Raman, and fluorescence. Absorption spectroscopy in particular, for both the near-IR (NIR) and mid-IR (MIR) spectral region, has been explored extensively by many research groups and commercial concerns over many years. 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 For the MIR “fingerprint” region, Heise 7 in early work7 analyzed blood glucose in vitro using Fourier transform infrared (FTIR) spectroscopy. Bhandare 9 also used FTIR to investigate the spectral interference of other blood constituents on the glucose signature. In more recent works, 10, 11, 12, 13 various groups focused on using only a few key MIR wavelengths rather than the entire spectrum to determine blood glucose. Specifically, Shen 11 found that spectroscopy at just two wavenumbers, 1082 and 1093cm1 , was sufficient. Rhiel 14 employed FTIR to monitor not only glucose but also lactose and ammonia in cell culture in real time. Opposite to absorption, MIR radiation was also used to measure glucose based on Kirchoff’s law of thermal emission.15 Some other works are based on laser photoacoustic effects.25, 26

There are generally three aspects in these studies: the spectroscopy aspect is concerned with spectral features and processing algorithms; the photometry aspect is concerned with optical system engineering and measurements; and the deployment aspect is concerned with application. Photometry is often undervalued and underdeveloped compared to spectroscopy but essential for real-life applications. The MIR works, 7, 8, 9, 10, 11, 12, 13, 14, 15 for example, demonstrated the spectroscopic merit, but the photometry was simply lacking. The thermal-source FTIR instrument employed in these works is useful as a research instrument but not necessarily practical or cost effective. Thermal sources have low power spectral density, low radiance, high relative intensity noise (RIN), and large 1f thermal noise. These impose limitations on dynamic range and speed for accuracy and real-time monitoring. In comparison, the NIR approach does not have any intrinsic spectroscopic advantage, but benefits greatly from more advanced photometric technologies.

Recent progress can help exploit the MIR region. This work describes the use of advanced MIR semiconductor lasers for in vitro aqueous glucose measurements. Lasers have well-established photometric advantages including high spectral intensity, brightness, high modulation bandwidth, low RIN, and 1f noise. They are also well suited for applications such as microfluidic or microarray bioassay, as the beam can be focused to a small volume. For practicality, they are robust, compact, and potentially low cost. Advanced MIR semiconductor lasers have been demonstrated for tissue imaging,27 and application for conjunctiva glucose measurement is being developed.24 A potential issue with MIR lasers is the lack of large spectral coverage, but some studies have shown that only a few wavelengths may be needed.11, 13, 24 Also, recent progress in tunable MIR lasers 27, 28, 29, 30, 31 promises broad wavelength coverage.

Photometry is crucial to the issue of accuracy. There is often some ambiguity on what contributes to glucose measurement errors: how much it is due to spectroscopic problems (interference, distorted spectra, etc.), and how much is due to photometric errors. Glucose errors were usually determined by comparing with wet chemistry results, but not analytically linked to instrument performance. As a basic principle of system engineering, a system should fully account for all errors, and establish an intrinsic measurement confidence level based on the causal and deterministic relation with the instrument noises and errors. This is a focus of this work.

The experiments were performed for glucose in water and Intralipid® (IL) solution. It is chemically much simpler than blood, since the objective is to establish the baseline photometric merit without the issue of chemical complexity. It is hoped that it may eventually motivate blood applications with many-wavelength systems to handle the chemical complexity. The rest of the work is organized as follows. Section 2 describes the experimental approach. Results and discussion are presented in Sec. 3; and Sec. 4 presents a summary and conclusion.

2.

Experimental approach

2.1.

Overall Concept

2.1.1.

Regression Approach

Given a glucose solution with concentration Cglu , which also contains an interferent substance such as IL of concentration CIL , a system can determine these concentrations via the equation:

Eq. 1

(ĈgluĈIL)=(qglu,1qglu,2qglu,nqIL,1qIL,2qIL,n)[Δμa(λ1)Δμa(λ2)Δμa(λn)]+higherorderterms,
where Δμa(λn) is the difference between the absorption coefficient of the solution and that of pure water Δμa(λ)μa,solution(λ)μa,water(λ) , and coefficients q are quantities to be obtained from calibration. For glucose and IL solution, it is expected that μaμs in the MIR owing to Rayleigh’s λ4 scattering law and the latter can be neglected. In Eq. 1, it is seen that higher-order terms are negligible; and it is convenient to use the matrix format:

Eq. 2

Ĉ=QΔμa,
where the notation Ĉ indicates that it is only an inferred estimate of the true value C . In more complex solutions with k substances, Ĉ would represent k values of concentration.

The calibration for Q is obtained from Δμa(λ) measured for solutions of known Cglucal and CILcal based on the relation:

Eq. 3

Δμa(λ)μa(λ;Cglucal,CILcal)μa(λ;0,0) =σa,glu(λ)Cglucal+σa,IL(λ)CILcal +higher-orderterms(Cglucal2CILcal2,CglucalCILcal,),
where superscript “cal” indicates that these are calibrated concentrations, and σa,gluIL(λ) is the specific absorption coefficient. As it turns out, all higher-order nonlinear terms were found insignificant and the relation becomes:

Eq. 4

Δμacal=ΣCcal,
where:

Eq. 5

Δμacal=(Δμa,1(λ1)Δμa,2(λ1)Δμa,p(λ1)Δμa,1(λ2)Δμa,2(λ2)Δμa,p(λ2)Δμa,1(λn)Δμa,2(λn)Δμa,p(λn)),and

Eq. 6

Σ=(σa,glu(λ1)σa,IL(λ1)σa,glu(λ2)σa,IL(λ2)σa,glu(λn)σa,IL(λn)),
Ccal=(C1,glucalC2,glucalCp,glucalC1,ILcalC2,ILcalCp,ILcal),
where p is the number of calibrated samples. With linear regression, the estimate for Σ is:

Eq. 7

Σ̂=ΔμacalCcalT(CcalCcalT)1.
The relation between Q and Σ is simply: Q=(ΣTΣ)1ΣT and Eq. 2 becomes:

Eq. 8

Ĉ=(ΣTΣ)1ΣTΔμa.

2.1.2.

Uncertainty and confidence level analysis

The critical issue and an objective of this work is the confidence level of Ĉ . Mathematically, it is a function of two independent random variables Δμa and Σ . The error of Δμa is based on the photometric system: laser, detector, electronic noises, and other measurement apparatus errors. Σ is a random variable not in the sense that it varies from one measurement to the next, but that it has an intrinsic uncertainty associated from imperfect calibration.

Ideally, if Σ is very accurately calibrated, the uncertainty of Ĉ was caused only by photometric measurement error. In this case, Δμa error can be characterized by a normal distribution N(0,δ) . Then, C is also a normal distribution function with an estimated variance:

Eq. 9

variance(C)=(ΣTΣ)1δ2;orvariance(Ci)=(ΣTΣ)ii1δ2.
Realistically however, there is no perfect calibration and this is a key issue for optical glucose technologies. Calibration may have to be done frequently, as it can be unique to an individual or apparatus and can change over time. The expected variance of Σ from Eq. 7 is:

Eq. 10

variance(Σ)=(CcalCcalT)1variance(Δμacal),
which also depends on Δμacal photometric error, though it is not necessarily the same as that in Eq. 9. Photometry thus has a two-fold impact on error, and the quality of calibration is critical.

With Σ uncertainty taken into account, the uncertainty of C can then be complex, and variance can be a useless concept. For example, consider the simplest case when Σ is a normal-distribution scalar with mean σ̂ and standard of deviation u ; the probability density function (PDF) of C of Eq. 8 is then:

Eq. 11

PDF(C)=uδπ(δ2+C2u2)exp[12(Δμ2δ2+σ̂2u2)]+σ̂(δ2+CĈu2)2π(δ2+C2u2)32exp[(CĈ)22(δ2+C2u2)σ̂2] erf{σ̂(δ2+CĈu2)uδ[2π(δ2+C2u2)]12},
which has an infinite variance owing to the well-known Cauchy tail 1C2 at large C . Instead of variance, a rigorous measure is the confidence interval, e.g., the interval [Ĉa2,Ĉ+a2] given by equation
Ĉa2Ĉ+a2PDF(C)dC=0.99,
which has a 99% probability of containing the true value of glucose concentration.

If Σ uncertainties uij are small: max(uij)Σ1 , a cut-off can be imposed to keep only the Gaussian component and an approximated variance can be used:

Eq. 12

variance(C)(ΣTΣ)1δ2[I+Δμa¯2δ2u2(ΣTΣ)1],
where u=max(uij) . The uncertainty of C is degraded from Eq. 9 by an extra term (Δμa¯2δ2)u2(ΣTΣ)1 , where Δμa¯2 is Δμa mean square. The simple and intuitively obvious implication of Eq. 12 is that for any system, there is no need to measure Δμa with much less error than that of Σ and vice versa. More importantly, Σ is essentially a spectrum, and arbitrary applications of techniques such as spectral smoothing and other digital filtering, while producing appealing-looking calibration spectra, violates its integrity and may induce spectral distortion that affects the confidence of Ĉ . A merit and objective of laser-based photometry is to obtain Σ̂ and Ĉ with high confidence that is causally and rigorously linked to the system performance.

2.2.

Experimental Setup

2.2.1.

Optical configuration and experimental condition

The experiments were set up to measure sample absorption in the transmission mode, using a tunable MIR semiconductor lasers around 9.6μm (1040cm1) and a reference laser at 5.4μm (1850cm1) . A schematic diagram is shown in Fig. 1 . The sample is a thin liquid film held between two CaF2 windows by surface tension. One window is fixed, and the other is movable with to a stepping-motor-driven micrometer. The sample thickness was varied 50 to 250μm . The laser beam was focused to a spot 100μm diameter. The sample volume was 0.42×106mL , which is in the range for microassay or microfluidic applications.

Fig. 1

Block diagram of the glucose measurement apparatus. The lasers are wavelength-division multiplexed and focused on the liquid sample held between two optical windows with adjustable gap.

024005_1_007702jbo1.jpg

The sample temperature was not measured, but the room temperature was controlled at 25±0.5°C . Owing to very low laser power (compare Sec. 2.2.3), the instantaneous temperature rise due to laser adiabatic heating is calculated to be 0.01K , and the steady-state average temperature rise is <0.005K (based on water-specific heat of 4.186JgK and conductivity of 0.58WmK ). Thus, the laser-induced temperature rise is insignificant compared to the room temperature fluctuation. There was no effort to study temperature-dependence absorption.

2.2.2.

Sample preparation

The anhydrous glucose (Avocado Research Chemicals, Heysham, England) was used to prepare the glucose deionized water (DI) water solution samples. Glucose concentrations ranged from 0 to 500mgdL . The Intralipid® solution of 10% (Baxter, Deerfield, Illinois) was diluted by DI water and mixed with stock glucose solution. The uncertainty of glucose concentration was calculated to be 1.5mgdL based on the errors of glucose weight and water volume measurement. Intralipid ® concentration estimated uncertainty was ±0.1% , based on errors of measured liquid volume for mixing.

2.2.3.

Laser and detector

The MIR tunable laser was operated in single mode, covering from 9.5 to 9.8μm (1020 to 1050cm1 ) as illustrated in Fig. 2 , with a superposition of glucose and water absorption spectrum. The other laser has a fixed wavelength at 5.4μm , and serves as a reference. The lasers were operated in pulse mode with repetition rate from 25 to 50kHz ; the pulse width ranged from 200 to 400ns . The pulse peak power was from 5 to 10mW . The duty cycle was 0.5 to 1%, and the average power was only 25 to 100μW . A MCT photodiode (Kolmar Technologies) detector with a transimpedance amplifier (TIA) was used to measure the transmitted power.

Fig. 2

Top: Glucose absorption spectra in the 9-μm region. Bottom: Spectral output of the tunable laser used in the experiments.

024005_1_007702jbo2.jpg

2.2.4.

Data acquisition, signal processing, and system noise

An automated data acquisition system was used. The computer generated a sweep of sample thickness and the transmission was measured. The signal was digitized at 40-MHz sampling rate. The overall photometric system noise was determined from three sources: laser noise, detector noise, and electronic noise. The laser RIN power spectral density was measured to be 95dBHz , which appear to be white-noise like without 1f component down to 1Hz . The combined detector and electronic noise equivalent power spectral density was measured to be 1023W2Hz .

3.

Experimental Results

The experimental results include: 1. demonstration of large dynamic range and sample thickness for accurate measurement of absorption coefficients μa , 2. calibration for specific absorption coefficient σa,glu and its accuracy, 3. the effect of IL as an interferent substance, and 4. application to the case of fixed and unknown sample optical thickness.

3.1.

System Dynamic Range and Measurements of Absorption Coefficient

Figure 3 shows a typical transmittance that varies over 4 orders of magnitude as a function of sample thickness, which is essential for the accuracy and precision of μa obtained as the slope of the curve. The lower curve was for a glucose solution, and the upper curves were for pure water as a reference. The full signal range was actually over 6 orders of magnitude,32 but only results for thickness 120μm were shown. This signal dynamic range and sample thickness is much larger than those in FTIR-based works. Speed is also a figure of merit. The time for a typical scan was a few seconds, limited by the computer control and acquisition time. The actual laser on-time was only 120ms and its total energy was <1mJ for the entire scan.

Fig. 3

Transmittance versus sample thickness, showing typical signal dynamic range of the apparatus. Upper curve actually consists of five closely spaced lines of water transmittance at 9.54-μm wavelength. The lower curve shows that of glucose at 9.63μm .

024005_1_007702jbo3.jpg

It appears in Fig. 3 that μa can be obtained via Beer’s law: lnT=μad , where T is the transmittance and d the sample optical thickness. However, Beer’s law is not necessarily correct, especially for laser measurements. The reason is that a focused laser beam contains many transverse components, and the effective path length is longer than the actual. Proper analysis is needed and the exact theory is given in the Appendix in Sec. 5. Deviation from Beer’s law was indeed observed for a tightly focused spot and small sample thickness in this work. Calculation was performed to ensure that the results here were obtained for sufficiently large spot size and sample thickness to correctly infer the absorption coefficient.

An example of μa measurement is the upper curve of Fig. 3, which actually consists of five single-scan results but are indistinguishable on this scale. The slopes yield μa,water (9.54μm) ranging from 530.17 to 532.53cm1 , with regression standard errors (SE) varying from 0.12 to 0.29cm1 , which is 2 to 5×104 in relative terms, and the regression R2 ranges from 0.999996 to 0.999999.

More important is the residual results shown in Fig. 4 . Ideally, they should be consistent with photometric error δ as discussed in Eq. 9. Large nonrandom or systematic residuals might indicate the presence of uncontrolled variables. The residuals in Fig. 4 appear random with magnitude and behavior consistent with expected photometric system noise, which is plotted as the dashed curve in Fig. 4. It is based on the transmittance signal expressed as:

13.

S=S¯+noise=S¯(1+noiseS¯)
ln(S)ln(S¯)=ln(1+noiseS¯)noiseS¯=[ndet2+(S¯nRIN)2]12S¯,
where ndet and nRIN are the detector noise and laser RIN, respectively. At small sample thickness, the transmitted signal is strong and the noise is dominated by laser RIN, since S¯nRINndet and the noise is
nRIN[NRIN(f)×Δf]1223×104,
where NRIN(f) is the laser RIN power spectral density mentioned in Sec. 2.2.4, and Δf=250Hz is the bandwidth. At large thickness, the reverse is true: the signal is weak and dominated by detector noise. The dashed curve and data are in excellent agreement, verifying the causal relation between μa measurement accuracy and the basic photometry performance.

Fig. 4

Residuals of regression fit of the water transmittances shown in Fig. 3. The dashed curve shows the calculated system noise, which agrees with the observed residuals.

024005_1_007702jbo4.jpg

However, more extensive measurements with repeated scanning indicated another source of error unrelated to photometry. It was the mechanical variation of the sample thickness control assembly. This mechanical error was intermittent but sufficiently frequent to degrade μa precision to 1 to 2×103 . It could be traced to the 90-nm scan-to-scan variation of the micrometer, independently observed via interferometry. This variation was actually within the manufacturer specification of 100-nm repeatability. Another potential source of errors is the thermal expansion of the assembly, which was built on a 5-cm -long steel frame with a thermal expansion of 60nmK . If repeated scanning heated the frame, or room temperature varied by 1°C , this could also contribute to errors, although evidence suggested it was less likely than the former.

Observation of this mechanical error actually underscores the importance of establishing a baseline photometric accuracy. This type of micrometer is usually adequate for many applications, but the reason for its error to be detected was because of the high sensitivity and accuracy of photometric measurements. An implication is that the full advantage of laser accuracy can be exploited for cases without moving parts or with high mechanical precision, such as microfiuidic channels or optical waveguide probes.

3.2.

Specific Absorption Coefficient of Glucose (without Intralipid® )

Calibration for specific glucose absorption coefficient σa,glu was obtained for solutions without IL to establish the baseline glucose accuracy without interferent. It was derived from the differential absorption Δμa,gluμa(Cglucal)μa(0) as described by Eqs. 3, 4, 5, 6, 7. A challenge is that Δμa,glu is a just small difference (6%) between two large absorption slopes, as seen in Fig. 3. Figure 5a shows Δμa,glu(λ) as a function of Cglu from 0 to 500mgdL at λ=5.4 , 9.63, and 9.54μm . The resulting regression statistics based on Eq. 3 are listed in Table 1 . There are two main results: the validity of the linear model, and the accuracy of σa,glu calibration.

Fig. 5

(a) Absorption difference between glucose solution and DI water as a function of glucose concentration for different laser wavelengths. The regression-fit slopes are indicated in the figure. (b) The residuals of the fit along with their RMS.

024005_1_007702jbo5.jpg

Table 1

Regression results of differential absorption coefficients of glucose in water.

Part A - With nonlinear coefficients
Wavelength (μm) CoefficientStandard error (SE)Student t -test P(H0:coef=0) R2 Adjusted R2
9.63Linear0.058340.0014739.7 1.8×1010 0.99954
Quadratic 3.6×106 3.7×106 0.98 0.3560.99942
9.54Linear0.052880.0014436.8 3.2×106 0.99983
Quadratic 2.4×106 3.4×106 0.720.5120.99975
Part B - With linear coefficient only R2 only
9.63Linear0.056970.000433131.5 4.4×1016 0.99948
9.54Linear0.053880.000333161.5 1.7×1010 0.99981
5.4Linear0.000170.000180.9450.3690.09031

The linear model discussed in Sec. 2.1 is clearly valid, as part A of Table 1 shows that the nonlinear terms ( Ca,glu2 -dependency) for both the 9.63- and 9.54-μm results are insignificant, as indicated in column “ P(H0:coef=0) ” (which indicates the probability for the null H0 hypothesis that the nonlinear coefficient= 0 ).

The linear regression results are shown in part B of Table 1, with σ̂a,glu(λ)=5.697×102 and 5.388×102cm1mgdL for 9.63 and 9.54μm , respectively. Their uncertainties are given in the SE column, which are 4.3×104 and 3.3×104cm1mgdL , and in terms of relative unit, 0.76 and 0.62%. Ideally, σ̂a,glu SE should be determined only by Δμa photometric error. Unfortunately, the mechanical error dominated these results. As evidenced in Fig. 5b, the root-mean-squared (RMS) residual error of Δμa,glu(λ) is 0.29cm1 for λ=9.59.6μm , which is consistent with mechanical errors for ten scan averages. The photometric error contribution should be 0.05cm1 .Without the mechanical error, the expected relative SE of σ̂a,glu(λ) should have been 1.4×103 and 1.1×103 .

If this calibration is used to infer glucose concentration in unknown samples based on the simplified Eq. 8: Ĉ=[σ̂a,glu]1Δμa , the confidence level can be rigorously established based on Eq. 11, and a calculation result is shown in Fig. 6 . The upper solid curves represent the confidence level for an actual worst-case mechanical error, and the lower dashed curves are for the hypothetical photometric-only error. For both, the calculation was for a single-scan error of Δμa . The results show that given the mechanical error, glucose could be determined within ±21mgdL with 90% confidence and ±62mgdL with 99% confidence. For hypothetical photometry-only errors, the results would be ±6 and ±9mgdL for 90 and 99% confidence, respectively.

Fig. 6

Calculated confidence interval for inferred (predicted) glucose concentration. The upper solid curves are for actual errors; the lower curves are hypothetical confidence intervals if there were only photometric errors.

024005_1_007702jbo6.jpg

It is also clear in part B in Table 1 that the linear coefficient for 5.4μm is within the SE, suggesting that glucose has no significant absorption at this wavelength. Thus, it can be used as a reference for samples of fixed but unknown thickness. This is further discussed in Sec. 3.4. Also evident in Fig. 5b is that the RMS error for λ=5.4μm is smaller than that of 9.5 to 9.6-μm by a factor proportional to μa at these wavelengths, which are 233cm1 at 5.4μm and 541cm1 at 9.6μm . This behavior indicates that Δμa error is not a constant-amplitude additive term, but is consistent with the mechanical error, which is proportional to μa .

3.3.

Effect of Intralipid® Interference

A matrix of samples with glucose concentration ranging from 0 to 500mgdL and IL concentrations from 0 to 8% were measured at λ=9.63 and 5.4μm . The differential absorption coefficients Δμa are listed in Table 2 and plotted in Fig. 7 . Each measurement resulted from a ten-scan average. The symbols are experimental data, and the planes are the linear regression fits. It is clear that two substances have nearly orthogonal absorption behaviors at these wavelengths. Glucose shows strong absorption at 9.63μm , while IL has little absorption; the reverse is true at 5.4μm . Interestingly, IL has a negative absorption coefficient at both wavelengths. This can be interpreted that the IL volume that displaces water has a lower absorption coefficient than water. Similar negative effects were reported for blood at 4.6 and 5.4μm in a previous work.32

Fig. 7

Water-subtracted absorption coefficient as a function of glucose and Intralipid® concentration for two wavelengths. The symbols represent measurements, and the planes are linear regression fits.

024005_1_007702jbo7.jpg

Table 2

Water-substracted absorption coefficients (cm−1) for various glucose and Intralipid® concentrations (absorption of pure water in parentheses).

Intralipid® concentration
λ Glucose concentration (mg/dL)0%2%5%8%
9.63μm (water μa:541.1cm1 ) 00.00 0.96 2.92 4.01
1006.084.82.671.65
20011.6710.338.587.04
30017.4815.7314.0012.12
40022.7421.3319.4517.44
50028.2626.5725.4324.34
5.4μm (water μa:233.91cm1 ) 00.00 7.98 17.58 25.88
1000.15 8.30 17.45 26.15
2000.26 8.14 17.55 26.24
3000.11 8.18 17.48 26.46
4000.01 8.21 17.69 26.35
500 0.03 8.53 17.29 26.45

The analysis according to Eq. 7 was applied to obtain matrix Σ of Eq. 6. The main results include: 1. the validity of the linear model in Eq. 4; 2. calibration matrix Σ ; and 3. the effect of IL on the precision and accuracy of glucose determination.

3.3.1.

Probability of nonlinear terms

The regression results for nonlinear terms are shown in Table 3 , part A. For 9.6μm , a t -test clearly shows that various nonlinear terms can be neglected. However, the results for 5.4μm indicates a probable quadratic term for CIL2 with P(H0:coef=0)=1.2×108 . This quadratic term persisted even when glucose is eliminated from the regression model, as shown in the last two rows of Table 3, part A. Further analysis, however, suggests that this nonlinear term may not be as probable as suggested by the statistics. The regression residuals for data of different glucose concentrations are not random, but systematically form clusters with respect to each value of IL concentration. A possibility for this peculiar result is that each set of samples with the same IL but different glucose concentrations were measured under similar room temperature and mechanical conditions. But sets with different IL concentrations were measured at different times and conditions, and thus had different errors. This systematic error would generate false quadratic behavior only for IL but none for glucose. Also, for both wavelengths, the cross-correlation terms CgluCIL are insignificant, indicating that their absorptions are simply additive to each other.

Table 3

Regression results for glucose and Intralipid® absorptions.

Part A - With nonlinear coefficients
Wavelength (μm) CoefficientStandard error (SE)Student t -test P(H0:coef=0) R2 Adjusted R2
9.63 σglu 0.057720.0011351.3 7.7×1022 0.99963
σIL 0.639 0.0737 8.68 4.9×108 0.99953
σglu(2) 2.1×106 2.5×106 0.821 0.422
σIL(2) 0.01470.009521.540.14
ρ 1.5×104 1.1×104 1.39 0.18
5.4 σglu 0.00047 0.00122 0.38 0.7060.99960
σIL 4.035 0.0802 50.3 1.1×1021 0.99950
σglu(2) 2.8×107 2.7×106 0.10.919
σIL(2) 0.09860.01049.52 1.2×108
ρ 4×105 1.2×104 0.335 0.741
5.4 σIL 4.087 0.06354 64.3 1.53×1026 0.99957
σIL(2) 0.10240.0089 11.5 8.9×1011 0.99953
Part B - With linear coefficient only
Wavelength (μm) CoefficientStandard error (SE)Student t -test P(H0:coef=0) R2 Adjusted R2
9.63 σglu 0.056320.000305184.5 1.4×1036 0.99954
σIL 0.5500 0.0192 28.7 6.3×1019 0.99950
ANOVA
Degree of freedomSum of squareMean of square F ratio P(H0:coef=0)
Model257532876.4723871 2×1037
Error222.6510.1205
Total245756
Covariance matrix 104×[9.3×1040.0380.0383.68]
5.4 σglu 0.00169 0.000742 2.28 0.0330.99755
σIL 3.3084 0.0466 71 1.7×1027 0.99732
ANOVA
Degree of freedomSum of squareMean of square F ratio P(H0:coef=0)
Model2636631834472 1.9×1029
Error2215.70.712
Total246381
Covariance matrix 103[5.5×1040.0220.0222.17]

3.3.2.

Linear coefficients

With the nonlinear terms excluded, the linear regression results are shown in Table 3, part B. For both, the F -ratio statistics indicated that the linear model alone was quite sufficient to account for the data. The coefficients σ̂a,glu(λ) and σ̂a,IL(λ) along with the contour of 95% joint-confidence ellipse (based on F -ratio distribution) and individual 84% confidential intervals (based on t -distribution) are plotted in Fig. 8 . For 9.63μm , the σ̂a,glu value of 0.0563cm1mgdL is comparable to the value 0.05697 for no-IL obtained in Sec. 3.2. For 5.4μm , σ̂a,glu is clearly not significant.

Fig. 8

Specific absorption coefficients of glucose and Intralipid® for 9.63μm (left) and 5.4μm (right) from regressions of measurements in Fig. 7. The ellipses represent the 95%-joint-confidence region; the rectangles represent the 84%-confidence interval of each coefficient.

024005_1_007702jbo8.jpg

3.3.3.

Inferred glucose and intralipid concentration

With calibrated coefficients Σ , the reverse inference of glucose and IL concentration can be obtained via Eq. 8. There was no need to prepare unknown samples given the available extensive number of samples. The results are shown in Fig. 9 . The filled circles represent known values, the open squares with center dots represent inferred values. The residuals of all 24 concentrations are shown in the scatter plot of Fig. 10 .

Fig. 9

Inferred concentrations of glucose and Intralipid® (open square symbols with a dot in the center) versus their known values (filled circles) for the 24 samples.

024005_1_007702jbo9.jpg

Fig. 10

Discrepancies of inferred concentrations versus known values for glucose and Intralipid® shown in Fig. 9. The vertical dashed lines represent the root mean square of glucose errors.

024005_1_007702jbo10.jpg

It appears from Figs. 9 and 10 that the residuals of IL concentration do not seem to be random around the expected values. All inferred IL concentrations in Fig. 9 are either higher or lower than expected values, forming two distinctive clusters in Fig. 10. This reflects the systematic error discussed before for the 5.4-μm IL data.

More important are the glucose results. The RMS of all residual glucose errors shown in Fig. 10 is 5.7mgdL , indicated by the two thick dashed lines. Figure 11 plots the glucose RMS error for each IL. Here, the result in Sec. 3.2 is just a subset with CIL=0% . These RMS errors are consistent with the mechanical errors, and there is no obvious correlation between glucose RMS and CIL , indicating that IL has no effect on glucose error. The result is hardly surprising, considering that the two substances have orthogonal absorption behaviors at the two wavelengths.

Fig. 11

Glucose concentration errors versus Intralipid® concentration show no discernible trend. In other words, IL does not affect glucose measurement accuracy.

024005_1_007702jbo11.jpg

Ellipsoid confidence contours can be calculated for inferred glucose and IL concentrations; however, they are not relevant with regard to IL, owing to the IL systematic errors, and IL is also not of interest. Therefore, the bivariate PDF P(C{Cglu,CIL}) can be integrated over CIL to yield PDF for Cglu only, and numerical approximation can be used to obtain the confidence interval. The result is shown in Fig. 12 . It is remarkably similar to the result without IL in Fig. 6, confirming that IL does not interfere with glucose accuracy. Comparison of Figs. 12 and 10 shows that all 24 glucose data points fall within the calculated 99% confidence level, and 23 out of 24 (96%) fall within the calculated 90% confidence level.

Fig. 12

Calculated confidence interval for inferred glucose concentration with an arbitrary IL concentration between 0 to 8%. The upper solid curves are for a single scan; the lower curves are for a ten-scan average.

024005_1_007702jbo12.jpg

3.4.

Approach for Fixed and Unknown Thickness

In applications such as monitoring a microfluidic channel, microassay array, or diffuse scattering from tissues, the sample effective optical thickness d is usually fixed or unknown. The system must infer the concentrations without varying the thickness. For these cases, the algorithm is slightly modified from that in Sec. 2.1 by treating d as an additional unknown variable. The details are described in Appendix B in Sec. 6. The essential result is similar to Eq. 8:

Eq. 14

ĈX=(ΣXTΣX)1ΣXTlnS,
where ĈX is modified from Ĉ by including d , ΣX is modified by including water absorption, and a key difference is to use lnS instead of Δμa , where S is the optical signal that can be transmission or backscattering.

It is sufficient to consider an example to illustrate the principle, and this is summarized in Table 4 . The first two rows show the numerical values of ΣX , (ΣXTΣX)1 and lnS(λ) , which are obtained from the transmittance data of an aqueous glucose sample set at a fixed assembly thickness. The last two rows show inferred value d̂=150.6μm and Ĉglu=570mgdL , to be compared with known values 150μm and 500mgdL , respectively. There is good agreement for d̂ , but Ĉglu is a bit off, which is likely due to some unaccounted optical loss (Appendix B in Sec. 6.).

A crucial aspect of this result is the choice of wavelengths. A requirement of Eq. 14 is that (ΣXTΣX)1 is not badly conditioned, which in principle is no different from Eq. 8. In practice, however, it is a more critical requirement, since ΣX contains μa,water(λ) , which provides spectral contrast to infer d . Since μa,water(λ) is large, the λ ’s must be chosen such that μa,water(λ) has variation much larger than their uncertainty. In the second row of Table 4, the small-font numbers for (ΣXTΣX)1 indicate its uncertainty. Their small uncertainty values ensure that (ΣXTΣX)1 is meaningful, owing to the large water absorption difference between 5.4 and 9.6μm . In comparison, any wavelength set in the laser tuning range 9.5 to 9.8μm was not as useful, since water absorption varies too slowly over this range. Both this and the IL results suggest that large wavelength diversity can be advantageous for accuracy and confidence, as discussed in Appendix B Sec. 6.

4.

Summary and Conclusion

This work describes the use of mid-IR semiconductor lasers to measure aqueous glucose concentration and examines the issue of accuracy. Compared with many previous works 7, 8, 9, 10, 11, 12, 13, 14 that focused mainly on spectroscopy, this work aims at photometric engineering to demonstrate the merit of the laser-based technique as opposed to others such as FTIR. Specifically, the work focuses on the laser properties of high signal-to-noise ratio, large dynamic range, and speed. The laser capability to focus onto a small sample volume (4×107mL) is also suitable for microfluidic and microarray bioassay applications.

The approach measures the absorption coefficients of aqueous glucose and Intralipid® solutions of various concentrations for different wavelengths (5.4, 9.54, and 9.63μm ). Linear regression is applied to obtain the specific absorption coefficients for each solute, which are then applied to infer their concentrations and compare with known values.

The important issue is accuracy, and the errors are shown to be causally and deterministically related to the basic instrument noises and system errors. There are no unaccounted sources of error. This causal relation allows inferring measurement confidence level based on rigorous system understanding instead of phenomenological assessment. The results show that glucose concentration could be determined ±21 and ±32mgdL with 90 and 99% confidence, respectively, in a single measurement. These values improve to ±9.5 and ±15mgdL with ten measurements. These uncertainties were not limited by photometry, but by a larger mechanical error of the sample thickness control assembly. This mechanical error is not fundamental, and in applications without this problem, the expected uncertainty due to photometry-only error would be ±6 and ±9mgdL for 90 and 99% confidence level, respectively, for a single measurement. Intralipid® up to 8% concentration had no effect on the glucose measurement.

The choice of wavelengths is also evidently important. The large difference between 5.4 and 9.63μm offers large spectral contrast between glucose, IL, and water that contributes to the measurement confidence. In general, for the laser method, fewer λ ’s entails a simpler system, whereas FTIR offers the entire spectrum. However, an insight from these results is that with regard to accuracy, measurements with many wavelengths but noisy is not necessarily better than those with fewer wavelengths but low noise. Other recent MIR works 10, 11, 12, 13, 24 even with FTIR also suggest that a few wavelengths might be sufficient. The laser approach can thus be applicable and practical.

5.

Appendix A: Deviation from Beer’s Law for Focused Laser Beam

In transmission or diffuse backscattering, the laser light coherence property may require the full optical solution for accurate interpretation of the result. For an absorptive thin film with parallel interfaces, i.e., a lossy etalon, the transmittance is:

15.

tTE={1+[1exp(2idk2)](k1k2)24k1k2}1exp(idk2)
tTM={1+[1exp(2idk2)](ε2k1ε1k2)24ε1ε2k1k2}1exp(idk2)
where TE and TM denote the transverse electric and magnetic polarization, respectively, and d is the sample thickness. For all other quantities, the subscripts 1 and 2 correspond to the window and sample, respectively. ε(n+iκ)2 is the complex dielectric constant; k=εk02β2 with k0=2πλ , and β is the transverse wave number of the beam parallel to the interface.

For small β , the paraxial approximation can be used:

Eq. 16

k=εk02β2εk0(1β22εk02)(n+iκ)k0β22nk0(1iκn),

Eq. 17

knk0(1β22n2k02)+iκk0(1+β22n2k02)nk0cosθ+iκk01cosθ,
where θ is the beam propagation angle in the medium. Substituting the imaginary component in exp(idk2) of Eq. 15 results in the absorption term:

Eq. 18

exp(dκk0)1cosθ=exp(12μad1cosθ);(μa2κk0),
which indicates that the beam experiences a larger loss factor 1cosθ associated with a longer path length. A focused laser beam is a superposition of many plane waves with nonvanishing transverse β , therefore it experiences a higher loss than a normal-incident plane wave.

In this work, the index of refraction of the CaF2 window is very close to that of water, and there is little difference between TE and TM transmissions. Thus the net beam transmittance is:

Eq. 19

T=dβxdβyA(βx)A(βy)tTE2,
where A(βx) , A(βy) are the power spectral density functions of transverse spatial frequencies βx , βy . For an x - y Gaussian beam:

Eq. 20

A(β)=w2πexp(w2β22).
With paraxial approximation, applying Eq. 19 with Eq. 20 yields:

Eq. 21

T=exp(μad)wx(wx2+μadk02n22)12wy(wy2+μadk02n22)12,and

Eq. 22

Texp(μad)k02n22wxwyμadforsmallwx,wy.
From Eq. 21, Beer’s law would be obtained for large beam waists: wx2 , wy2μadk02n22(=μadλ24π2n22) . But for small beam waists, Eq. 22 shows a deviation by the term k02n22wxwyμad , which would lead to substantial error of μa if not included.

In this work, the semiconductor laser beam is not Gaussian and has x - y asymmetry. A numerical calculation for a modeled laser beam produces results shown in Figs. 13, 14, 15 . Figure 13 shows that the transmittance is superficially similar to Beer’s law on a large scale. But as evidenced in Fig. 14, the residuals from the exponential fit clearly deviate from Beer’s law, even if the interference fringes are ignored. The deviation is worse for smaller spot size as expected. The “effective” absorption coefficient deviates from the actual coefficient as shown in Fig. 15. Considering that the largest differential glucose absorption is only 28.5cm1 in this work, a deviation of 4cm1 would result in an error of 14%.

Fig. 13

Numerical calculation of transmission loss (transmittance) of a semiconductor laser beam with different spot sizes (ranging from 10 to 30μm ) as a function of sample thickness. The small differences between them are more visible in Fig. 14.

024005_1_007702jbo13.jpg

Fig. 14

Residual of fits of the transmittance curves in Fig. 13 with Beer’s law, showing clear deviation from the law for small laser spot sizes. The oscillations are simply the interference fringes.

024005_1_007702jbo14.jpg

Fig. 15

Errors of absorption coefficient if Beer’s law is applied to the results in Fig. 13. Smaller laser spot size results in larger errors, which can be as much as 0.8% in this case.

024005_1_007702jbo15.jpg

Table 4

Example of inferring unknown optical thickness.

λ1=5.4μm , λ2=9.63μm
ΣX=(μa,w(λ1)σa,glu(λ1)μa,w(λ2)σa,glu(λ2)) (ΣXTΣX)1 lnS(λ)
(233.910.17541.156.97) 103×(0.01850.176±3.9×104±5×1020.1761.98±5×102±0.07) (3.5238.635)
(d̂Ĉglud̂)=(ΣXTΣX)1ΣXTlnS=(0.015060.008578)
With proper unit conversion
(d̂Ĉglu)=(150.6μm570mgdL);(dCglu)=(150μm500mgdL)

This error does not just occur on the slope of the log transmittance, but also on its amplitude. Thus, for cases with fixed-thickness samples, the error on transmittance amplitude is similarly as severe and correction is necessary.

6.

Appendix B: Approach for Fixed or Unknown Sample Effective Optical Thickness

In general, the transmission or diffuse backscattered return signal can be expressed as:

Eq. 23

S(λ)=Q(λ)exp[μa(λ)d],
where Q(λ) is a loss factor that may or may not be known, and μa(λ) is a sum of absorptions:

Eq. 24

lnS(λ)=μa,w(λ)di=1kσa,i(λ)Cid+η(λ),
where η(λ)lnQ(λ) , μa,w(λ) is the baseline absorption coefficient of the medium (which can be a solvent other than water), d can represent single or round-trip, and Ci for i=1 to k denotes concentration of substance of interest.

In controlled applications with fixed, known d and η , Eq. 24 can be written:

Eq. 25

1d{η(λ1)ln[S(λ1)]η(λ2)ln[S(λ2)]η(λn)ln[S(λn)]}[μa,w(λ1)μa,w(λ2)μa,w(λm)]=[σa,1(λ1)σa,2(λ1)σa,k(λ1)σa,1(λ2)σa,2(λ2)σa,k(λ2)σa,1(λn)σa,2(λn)σa,k(λn)](C1C2Ck),
and a solution similar to Eq. 8 is obtained:

Eq. 26

Ĉ=(ΣTΣ)1ΣT[1d(ηlnS)μa,w].
If d is unknown, Eq. 24 can be written:

Eq. 27

{η(λ1)ln[S(λ1)]η(λ2)ln[S(λ2)]η(λn)ln[S(λn)]}=[μa,w(λ1)σa,w(λ1)σa,1(λ1)σa,k(λ1)μa,w(λ2)σa,w(λ2)σa,2(λ2)σa,k(λ2)μa,w(λn)σa,w(λn)σa,1(λn)σa,k(λn)](dC1dCkd).
If η is negligible, Eq. 27 leads to Eq. 14. If η is unknown, the problem cannot be solved. However, η(λ) usually represents optics effects such as light collection loss, Fresnel loss, scattering loss, or chromatic dispersion, and therefore can be a slowly varying function of wavelength. For the sake of discussion, suppose that it can be parametrized to the first order:

Eq. 28

η(λ)=η0+ρ1λ,
then Eq. 27 can be modified to yield:

Eq. 29

{ln[S(λ1)]ln[S(λ2)]ln[S(λn)]} =[μa,w(λ1)σa,1(λ1)σa,k(λ1)λ11μa,w(λ2)σa,1(λ2)σa,k(λ2)λ21μa,w(λn)σa,1(λn)σa,k(λn)λn1](dC1dCkdρ1η0),
which allows inferring all three unknown parameters d , η0 , and ρ1 .

A crucial requirement relevant to the wavelengths used in this work is that (ΣXTΣX)1 should have low uncertainty. Let each row or column of ΣX be denoted as a vector s . If the wavelength range is too narrow, various vector si can become close to each other, (ΣXTΣX)1 can be nearly degenerate with large uncertainty, and the measurements would have poor confidence. A nominal criterion for the vectors to be “far” from each other is to use the quasi-Mahalanobis distance:

Eq. 30

min[siγsju2,γ]min({p=1r[sipγsjp]2u2(sip)+γ2u2(sjp)}12,γ)1,
where mxp are the vector elements, u2(sxp) are their uncertainties, and γ is a proportional factor.

Applying this criterion in this case explains the reason why the 5.4- and 9.6-μm wavelength combination produced the best results in Sec. 3.4. The quasi-Mahalanobis distance for ΣX columns with 5.4 and 9.63μm is 93, compared to only 3.5 to 6 for any set of wavelengths in the 9.5 to 9.8-μm range.

Acknowledgment

This work was supported in part by the DARPA/ARO under contract DAAD-190010361, the Air Force Research Laboratory under contract F29601-00-2-0058, and the State of Texas through the THECB and the Texas Center for Advanced Materials and Center for Superconductivity. We thank the late Pradeep Johnson, C. Peng, G. P. Luo, and A. Seetheraman for valuable assistance.

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©(2007) Society of Photo-Optical Instrumentation Engineers (SPIE)
Bujin Guo, Yi Wang, Yang Wang, and Han Q. Le "Mid-infrared laser measurements of aqueous glucose," Journal of Biomedical Optics 12(2), 024005 (1 March 2007). https://doi.org/10.1117/1.2714283
Published: 1 March 2007
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