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Abstract
A differential Mueller matrix polarimetry technique is proposed for obtaining non-invasive (NI) measurements of the glucose concentration on the human fingertip. The feasibility of the proposed method is demonstrated by detecting the optical rotation angle and depolarization index of tissue phantom samples containing de-ionized water (DI), glucose solutions with concentrations ranging from 0~500 mg/dL and 2% lipofundin. The results show that the extracted optical rotation angle increases linearly with an increasing glucose concentration, while the depolarization index decreases. The practical applicability of the proposed method is demonstrated by measuring the optical rotation angle and depolarization index properties of the human fingertips of healthy volunteers.
© 2017 Optical Society of America
1. Introduction
With rising obesity levels around the world, diabetes has emerged as a major concern with serious health and economic implications. Consequently, reliable methods for its testing and diagnosis are urgently required. Of the various methods available, non-invasive (NI) techniques based on measuring the glucose concentration in human blood are particularly attractive due to their accuracy and painless aspects. Existing methods for the NI determination of glucose can be broadly classified into two groups, namely those which track a particular molecular property of glucose and those which track the effects of glucose on the tissue and blood properties [1]. Methods of the former type typically track such intrinsic molecular properties as the optical rotation angle, the near-infrared (NIR)/mid-infrared (MIR) absorption coefficient, the Raman shift, the NIR photoacoustic absorption, and so forth. Such methods assume the ability to detect glucose in tissue or blood independent of other body components or the body’s physiological state. Song et al. [2] proposed a method for estimating the concentration of blood glucose based on multi-wavelength NIR spectra using a NIR diffuse reflectance spectroscopy technique based on three wavelengths (850 nm, 950 nm and 300 nm). The results showed that the proposed method was capable of determining the blood glucose concentration level with a mean absolute deviation of 80 mg/dL. Coté and Vitkin [3] calculated the optical rotation angle of turbid scattering media such as glucose solutions using a transmitted polarimetry technique. Timmerman et al. [4] developed a point-of-care NI system for glucose monitoring based on Raman spectroscopy. The experimental results showed that the measured glucose values had an average error of as little as 0.9 nM.
The second group of methods for diabetes monitoring is based on the effects of glucose in changing the optical properties of human tissue (e.g., the light scattering coefficient, the refractive index of the interstitial fluid, and the rate of acoustic propagation). Bruulsema et al. [5] proposed a NI glucose measurement method based on diffuse reflectance measurements of the skin obtained at a distance of 1-10 mm from a point source. Larin et al. [6] presented a NI blood glucose monitoring technique based on optical coherence tomography (OCT). It was shown that a sharp linear decrease in the OCT signal slope occurred as the blood glucose concentration increased. Maier et al. [7] investigated the effect of the glucose concentration level on the refractive index of extracellular tissue fluid and showed that the scattering coefficient μs reduced as the glucose concentration increased. Kohl et al. [8] also employed diffusion theory to examine the physical background of the effect of glucose on the transport of light in tissue-simulating phantoms. The results showed that the absorption coefficient μa was insensitive to the glucose concentration. However, the scattering coefficient μs decreased linearly as the glucose concentration increased due to a corresponding increase in the refractive index, which affected the scattering properties of the particles suspended in solution.
In general, the results presented in the studies above confirm the basic feasibility of NI techniques for glucose monitoring and diagnosis purposes. However, NI devices are presently not widely used in clinical diabetes applications due to their poor precision, robustness, stability and analytical performance compared to that of conventional blood glucose meters. Consequently, much work remains to be done in improving the performance of NI glucose monitoring systems such that they provide a more viable approach for clinical diagnosis. In a previous study [9], the present group proposed a transmitted Stokes-Mueller matrix polarimetry technique for glucose sensing in low-scattering media based on a differential Mueller matrix formalism. In the current study, an analytical model is further derived to extract the optical rotation angle (γ) and depolarization index (Δ) of glucose samples with a high scattering effect using a reflective optical system. The feasibility of the proposed approach is demonstrated using aqueous phantom tissue samples comprising de-ionized (DI) water, glucose solutions with concentrations ranging from 0~500 mg/dL and 2% lipofundin. In addition, the practical applicability of the proposed method is verified by performing NI glucose concentration measurements on the human fingertip.
2. Differential Mueller matrix formalism for high-scattering turbid media
An optical sample can be described by the matrix formulation S = MS′, where S is the Stokes vector of the output light, M is the 4x4 Mueller matrix of the sample, and S′ is the Stokes vector of the input light. The general form of this relation is given as
The use of four input lights, namely three linear polarization lights (0°, 45° and 90°) and one right-hand circular polarization light, is sufficient to yield all the equations required to solve the sample matrix M in Eq. (1). The corresponding input Stokes vectors are given as S′0° = [1,1,0,0]T, S′45° = [1,0,1,0]T, S′90° = [1,-1,0,0]T and S′R = [1,0,0,1]T, respectively. The output Stokes vectors are then obtained as
Assuming that the illuminating beam propagates along the z-axis of a right-handed Cartesian coordinate system, the differential Mueller matrix of the sample given in Eq. (1) can be obtained from an eigenvalue analysis of M as follows [10]:
where v and λ are the eigenvectors and eigenvalues of matrix M, respectively. The Mueller matrix of a turbid medium with circular birefringence (CB) properties has the form [11]where γ is the optical rotation angle. Furthermore, the differential Mueller matrix of the CB sample can be expressed asSimilarly, the differential Mueller matrix of a sample with depolarization effects can be formulated aswhere the anomalous depolarization is characterized by the differential parameter η′v. By equating the differential Mueller matrix in Eq. (6) with that given in Eq. (9), the optical rotation angle can be obtained asSimilarly, the differential Mueller matrix describing the depolarization effect can be written asPerforming inverse differential calculation [10], the macroscopic Mueller matrix describing the Dep effect is obtained aswhere e22 and e33 are the degrees of linear depolarization and e44 is the degree of circular depolarization. Finally, the depolarization index can be expressed asIn order to derive the differential Mueller matrix required to extract the optical rotation angle (γ) and depolarization index (Δ) of any unknown or complex turbid medium, the physical properties of the medium are assumed to be completely homogeneous. In other words, the sample is treated as a black box and its properties are represented using the Mueller matrix only. However, in applying the Mueller matrix to extract the sample properties, the optical path length is directly related to the sample thickness, and hence the extracted values of the optical rotation angle and depolarization index are also affected. Consequently, in a future study, Monte Carlo theory will be incorporated into the present analytical model in order to obtain more quantitative predictions of the glucose concentration from the experimentally derived values of the optical rotation angle and depolarization index.3. Experimental setup and results
In a previous study [9], a Stokes-Mueller matrix polarimetry system was employed in a transmission mode to evaluate the glucose concentrations of low-scattering samples consisting of 5 mg of microspheres dispersed in 200 ml glucose aqueous solutions. In the present study, a similar system was used in a reflection mode to quantify the glucose concentration of both laboratory and biological samples with high scattering effects. As shown in Fig. 1, the system consisted of two main components, namely a polarization state generator (PSG) and a polarization state analyzer (PSA). The PSG comprised a stabilized He-Ne laser source (SL 02/2, SIOS Co., wavelength 632.99 nm), a polarizer (GTH5M, Thorlabs Co.) to extract only linear polarized light from the laser source, an EO modulator (ONESET Co., Model 350-50) driven by a saw-tooth waveform voltage, a quarter-wave plate (QWP0-633-04-4-R10, CVI Co.) to perform linear polarization scanning, and a half-wave plate (AHWP10M-600, Thorlabs Co.) to generate circular polarization light. The PSA consisted of two EO modulators (ONESET Co., Model 350-50) driven by a saw-tooth waveform voltage, a polarizer (GTH5M, Thorlabs Co.). Finally, a photo-detector (New Focus, Model 2001) interfaced with a data acquisition (DAQ) card (NI USB-6366) was used to measure the Stokes vectors of the light emerging from the sample. The sampling rate and resolution of the DAQ card were 2 MS/s and 16 bits, respectively. Furthermore, the sampling frequency was set as 80 kHz and the number of measurement samples was set to 100,000 in the continuous sample acquisition mode.
In performing the experiments, the amplitude of the saw-tooth voltage used to drive the EO modulators was set to 0.635 V in every case. In addition, the frequency of the saw-tooth voltage for the EO modulator in the PSG was set as 40 Hz, i.e., twenty times higher than that used in the previous study [9]. The scanning time was thus less than 0.1 s. The frequency of the saw-tooth driving voltage for the EO modulators in the PSA was set as 4 kHz (i.e., 100 times higher than that used for the EO in the PSG) in order to obtain 100 Stokes vector values per scanning period. Notably, the high scanning speed of the PSG also minimizes the effects of environmental noise and instabilities of the input voltage, respectively, and therefore improves the accuracy of the measurement results. The frequencies of the EO modulators in the PSA were synchronized using an arbitrary waveform generator (TGA1244, TTid). Finally, the incident angle was set as 55° in every case.
3.1 Glucose concentration detection in phantom solutions
Tissue phantom samples were prepared by mixing de-ionized (DI) water with 10 ml glucose solutions (100 mg/ml-Merck Ltd) with concentrations ranging from 0 ~500 mg/dL in 100 mg/dL increments, and 2% lipofundin (lipofundin MCT/LC1 20%, B|Braun). Additional samples were also prepared consisting of DI water, glucose solutions with concentrations ranging from 0~100 mg/dL in 20 mg/dL increments and 2% lipofundin to determine the minimum detection limit of the proposed method. Note that previous studies have reported that 2% of lipofundin is sufficient to accurately reproduce the scattering effect of human skin and tissue [12-13]. Figure 2 shows the experimental results obtained for the optical rotation and depolarization properties of the samples. As shown in Fig. 2(a), the optical rotation angle γ increases linearly with an increasing glucose concentration. The standard deviation of the experimental γ values obtained over four repeated tests for each glucose sample was found to be ± 8.4 × 10−3°. In addition, the sensitivity of the measured γ values was determined to be 6 × 10−5°/(mg/dL). This value compares to a sensitivity of 1 × 10−5°/(mg/dL) obtained in a previous study using a transmission-mode optical system [9] (inside a 1-mm thickness quartz container with a 40-mm optical path length) and can be attributed to the shorter optical path length in the present reflective system, which results in an improved measurement accuracy. It is noted that the sensitivity of the measured value of γ is defined here as the rate of change in the value of γ with an increasing glucose concentration. Figure 2(b) shows the experimental results for the depolarization index Δ of the various samples. It is seen that Δ decreases linearly with an increasing glucose concentration. The standard deviation of the experimental Δ values obtained over four repeated tests for each sample was found to be ± 8.0 × 10−2. In addition, the sensitivity of the depolarization index measurements was determined to be 6 × 10−5/ (mg/dL). Note that the sensitivity of the measured depolarization index is defined here as the rate of change of the measured value of Δ with an increasing glucose concentration. The reduction in the depolarization index with an increasing glucose concentration is consistent with the results obtained using a conventional diffusion technique [14] and is reasonable since, as the glucose concentration increases, the refractive index of the aqueous solution also increases. As a consequence, the velocity of light through the sample reduces and the scattering properties of the particles suspended in the solution are thus also reduced [7-8].
Fig. 2 Experimental results for extracted values of: (a) γ with standard deviation of ± 8.4 × 10−3° over four repeated tests; and (b) Δ with standard deviation of ± 8.0 × 10−2 over four repeated tests for aqueous phantom samples with glucose concentrations ranging from 0 ~500 mg/dL and 2% lipofundin.
In general the results presented in Fig. 2 confirm the feasibility of the proposed technique for performing glucose sensing using just two optical parameters, namely the optical rotation angle and the depolarization index. The optical rotation angle represents the circular birefringence property of the glucose sample, while the depolarization index describes the scattering effects caused by the scattering particles within the sample. In general, the depolarization index of turbid media varies as a function of the optical path length traveled by the light through the sample. Thus, both parameters (i.e., the optical rotation angle and the depolarization index) are important in obtaining reliable estimates of the glucose concentration [15]. Overall, the present results show that the proposed technique has the ability to detect glucose concentrations as low as 20 mg/dL. Notably, this value compares with a minimum detection limit of 100 mg/dL for eye glucose concentration detection [16] and 50 mg/dL for blood glucose and water glucose concentration detection [15,17].
3.2 NI measurement of glucose concentration on human fingertip
The practical feasibility of the proposed technique was evaluated by measuring the optical rotation angle and depolarization index of the extracellular tissue on the human fingertip of four healthy volunteers. The tests were performed in both a normal glucose condition and an enhanced glucose condition. In the latter case, the volunteers were asked to swallow 550 mL of sugared water, and glucose detection was then performed 15 minutes later. In both cases, a glass plate with a thickness of 0.5 mm was pressed onto the fingertip such that the reflected light fell on the detector after passing through the PSA. Tables 1 and 2 show the results obtained for γ and Δ, respectively, over four repeated tests for each volunteer. (Note that the X symbols in the two tables indicate that no result was obtained due to the effects of noise and / or errors.) As shown, the average optical rotation angle γ increases after the ingestion of sugared water for all four volunteers other than volunteer 3. Moreover, the average depolarization index Δ reduces in every case. The results are therefore in good qualitative agreement with the extracted values of γ and Δ reported in Section 3.1 for the tissue phantom samples. However, while the results presented in Section 3.1 indicate that the values of γ and Δ have a similar sensitivity to the glucose concentration, those presented in Tables 1 and 2 show that the depolarization index is more sensitive to the blood glucose concentration than the optical rotation angle. The difference between the two sets of results may stem from the calibration procedure used in the experiments, the moisture content of the skin, the partial contact between the glass plate and the skin, and the change in glucose concentration within the volunteer’s body over time.
Table 1. Optical rotation angle γ of extracellular tissue on human fingertip of four healthy volunteers
Table 2. Depolarization index Δ of extracellular tissue on human fingertip of four healthy volunteers
A further series of experiments was performed to measure the optical rotation angle and depolarization index of the extracellular tissue on the human fingertip of three healthy volunteers at three different periods of the day defined relative to the standard meal times, i.e., breakfast (around 9.00), lunch (around 12.00) and dinner (around 17.00). It has been reported that the human glucose level drops around 2 hours after a meal [18]. Consequently, the experimental measurements were performed in two glucose level conditions: (1) a low glucose level condition (i.e., the measurements were obtained at lunch time (12:00) and dinner time (17:00)); and (2) a high glucose level condition (i.e., the measurements were obtained within two hours after breakfast (i.e., at 10:00) and lunch (i.e., at 14:00). The measurements for the three volunteers were obtained sequentially 15 minutes one after another. In all cases, a glass plate with a thickness of 0.5 mm was pressed onto the fingertip such that the reflected light fell on the detector after passing through the PSA.
Figures 3(a) and 3(b) show the results obtained for γ and Δ, respectively, over four repeated tests for each volunteer. As shown in Fig. 3(a), the optical rotation angle values increase after meal times (i.e., 10:00 and 14:00), but decrease before meal times (i.e., 12:00 and 17:00). Meanwhile, Fig. 3(b) shows that the depolarization index increases before meal times (i.e., 12:00 and 17:00), but decreases after meal times (i.e., 10:00 and 14:00) [18]. The results are therefore in good qualitative agreement with the extracted values of γ and Δ reported in Section 3.1 for the tissue phantom samples. Furthermore, Fig. 3 shows that the values of γ and Δ vary in the ranges of 0.025° ~0.25° and 0.2 ~0.8, respectively. The standard deviations of the measured values of γ and Δ obtained over four repeated tests for each volunteer are shown in Table 3 and are seen to be approximately ± 0.045° and ± 0.13, respectively. The deviation of the extracted results stems most likely from the calibration procedure used in the experiments, the partial contact between the glass plate and the skin, the moisture content of the skin, the different properties of the skin of each volunteer, and other unknown environmental noise factors.
Fig. 3 Experimental results for extracted values of: (a) γ-optical rotation angle and (b) Δ-depolarization index of extracellular tissue on human fingertip of three healthy volunteers at four different times of the day.
Table 3. Standard deviations of four measured values of γ and Δ
4. Conclusions
This study has proposed a differential Mueller matrix polarimetry technique for estimating the glucose concentration of aqueous samples based on the extracted values of the optical rotation angle (γ) and depolarization index (Δ). The validity of the proposed method has been demonstrated by measuring the glucose concentration of tissue phantom samples comprising DI water, glucose solutions with concentrations ranging from 0~500 mg/dL and 2% lipofundin. The results have shown that the proposed method enables the γ and Δ values to be measured with standard deviations of 8.4 × 10−3° and 8 × 10−2, respectively, over the considered glucose concentration range. Moreover, the proposed technique has a minimum glucose detection limit of 20 mg/dL. The practical applicability of the proposed method has been demonstrated by obtaining NI measurements of the γ and Δ properties of the extracellular tissue on the human fingertips of healthy volunteers. The results have shown that the measured values of γ and Δ vary in the ranges of 0.025° ~0.25° and 0.2 ~0.8, respectively. Moreover, the extracted values of γ and Δ have standard deviations of approximately ± 0.045° and ± 0.13, respectively. In general, the extracted values of γ and Δ are in good qualitative agreement with those obtained for the tissue phantom samples. To the best of the authors’ knowledge, the method presented in this study represents the first reported attempt to perform NI glucose monitoring using just the optical rotation angle and depolarization index parameters. In a future study, the quantitative results obtained using the proposed method for the γ and Δ values of samples with various glucose concentration levels will be verified by means of Monte Carlo simulations.
Funding
Ministry of Science and Technology of Taiwan (MOST) 104-2221-E-006-125-MY2, 104-3113-E-006-002, and 105-3113-E-006-002.
Acknowledgments
The authors gratefully acknowledge the financial support provided to this study by the Ministry of Science and Technology of Taiwan (MOST) under Grant Nos.104-2221-E-006-125-MY2, 104-3113-E-006-002 and 105-3113-E-006-002. The research was also supported in part by the Ministry of Education, Taiwan, under the “Aim for Top University Project” of National Cheng Kung University (NCKU), Taiwan.
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