An optical-based method is proposed for measuring the glucose concentration of samples containing scattering particles. In the proposed approach, a Stokes–Mueller reflection-based polarimetry technique is used to solve the Mueller matrices of a turbid glucose sample with circular birefringence and depolarization properties given six incident lights with different polarization states. Using an error function defined as the difference between the simulated output Stokes vectors and the experimental ones, a genetic algorithm is used to inversely derive the optical rotation and depolarization parameters of the experimental sample corresponding to the glucose concentration and scattering depolarization effect, respectively. The validity of the proposed method is demonstrated using glucose samples containing 0.02 ml and 0.04 ml lipofundin, respectively.
© 2015 Optical Society of America
Glucose levels outside of the normal range are often an indicator of a more serious medical condition. As a result, effective methods for measuring glucose concentration are urgently required. Many noninvasive measurement techniques have been proposed based on such methods and phenomena as reverse iontophoresis, ultrasound, thermal emission, light absorption spectroscopy, photo acoustic spectroscopy, fluorescence, Raman spectroscopy, and polarimetry. Park et al.  proposed a portable system for measuring the glucose concentration in urine comprising a biochemical sensor, a peripheral interface controller microcontroller, and a signal processing circuit. Tierney et al.  designed a biosensor for blood glucose concentration in which an electro-osmotic flux of glucose was induced through intact skin by reverse iontophoresis and the glucose level was then extracted from the measured flux value. However, the proposed method has several disadvantages, including a long warm-up time, a time lag compared to direct body tissue measurements, and poor accuracy. Maruo et al.  presented an in vivo near-infrared (NIR) technique for measuring blood glucose levels based on dermis spectra obtained using a fiber-optic probe at wavelengths in the range of 1430–1850 nm. Ellis and Goodacre  reviewed various proposals for the use of Raman spectroscopy to perform metabolic fingerprinting for disease diagnosis purposes. Liu et al.  developed a rapid and nondestructive method for measuring the sugar concentration in apples using Fourier transform near-infrared (FT-NIR) spectroscopy. Fuchs and Kaatze  performed molecular dynamics (MD) simulations to examine the relationship between the dielectric relaxation of carbohydrate aqueous solutions and the concentrations of D-glucose and D-fructose, respectively. Zhou et al.  presented a method for measuring the optical rotation induced by chiral molecules such as glucose and fructose in solution using a reflection polarimetry technique. In theory, such a method can also be applied to quantify the glucose concentration in the aqueous humor since the scattering coefficient of the human eye is low. However, a time lag exists between the glucose concentration in human blood and that in the aqueous humor. Furthermore, coupling the light required for polarimetric measurement purposes into the human eye is highly challenging. Jang and Fox  proposed a closed-loop optical sensor incorporating a single Faraday rotator for determining the glucose concentration in the human aqueous humor from the polarization information contained in the light reflected from the retina. Cameron et al.  used a novel dual wavelength (532 nm and 635 nm) polarimetry to perform the in vivo quantification of the glucose concentration in the aqueous humor of New Zealand white rabbits. However, as with the methods proposed in  and , the proposed method has only limited application for real-world disease diagnosis purposes in humans due to the time lag between the glucose concentration in the blood and in the aqueous humor, respectively.
In many applications, the samples used for glucose concentration measurement are not perfectly pure, but contain small quantities of scattering elements. In practice, these elements result in a depolarization effect, which must be taken into account if the glucose concentration is to be reliably estimated using polarimetry methods. Accordingly, in the present study, a new method for determining the glucose concentration of samples containing scattering particles is proposed based on a reflection polarimetry technique. In the proposed approach, experimental Mueller matrices are constructed for six incident lights with different states of polarization. Equivalent Mueller matrices are then simulated based on randomly selected values of the circular birefringence (CB) and depolarization parameters. A genetic algorithm (GA) is then applied to inversely derive the CB and depolarization parameters of the experimental sample such that the glucose concentration and scattering effects can be separately quantified.
2. MUELLER MATRICES OF OPTICAL SAMPLES WITH CIRCULAR BIREFRINGENCE WITH SCATTERING PARTICLES
The present study proposes a method for detecting the concentration of glucose in samples with scattering effects by means of a reflection polarimetry technique. The validity of the proposed method is demonstrated by comparing the results obtained for the optical rotation angle of the sample with those obtained using the transmission polarimetry method proposed by the current group in . Figure 1 presents a schematic illustration of the experimental setup considered in the present study.
In Fig. 1, is a polarizer and is a quarter-wave plate. In performing the experiments, and are adjusted in such a way as to produce incident light with six different states of polarization (SOPs). For each input light, the corresponding input and output Stokes vectors (i.e., and , respectively) are measured using a commercial Stokes polarimeter. According to basic Mueller calculus principles, the output Stokes vector can be formulated as
The depolarization phenomenon observed in many optical systems is generally caused by the surface roughness of the target and/or the light scattering effects . In practical glucose detection applications, the samples commonly contain lipid particles which induce scattering effects. Thus, for the glucose concentration to be reliably determined via optical methods, the resulting depolarization effect must be taken into account. For convenience, in developing the glucose concentration measurement method proposed in the present study, the analysis considers a simulating phantom with CB and low depolarization. The corresponding light propagation model in a reflection-based optical system is shown in Fig. 2.
In Fig. 2, the sample is stored in a quartz cuvette, and hence a change in the SOP and intensity of the transmitted and reflected light occurs at the quartz/sample interface, as described by the Fresnel equations . To take account of this effect in the measured values of the Stokes parameters, the Fresnel equations must be expressed in matrix form and then transformed to a corresponding Mueller matrix form. The Fresnel matrices for the transmitted () and reflected () light in Fig. 2 are given, respectively, as13]:
Note that in the present study, the refractive index for fused quartz () (fused silica) is taken to be 1.46 , while that for the pure glucose aqueous solution () is taken as 1.33 to simplify the mathematical calculations in the GA since the variation of the has slight influence on signals of the Stokes polarimetry for the low glucose concentration. Lu and Chipman  showed that in analyzing the polarization characteristics of an optical medium, the Mueller matrix can be decomposed into three factors, namely, a diattenuator, followed by a retarder, followed by a purely depolarizing factor. For the sample shown in Fig. 2, with both CB and depolarization properties, the Mueller matrix can be expressed as2, and is the Mueller matrix of the reflective interface at the bottom. It should be noticed that the reflection by the solution–quartz interface is weaker than the reflection by the quartz–air interface, and hence we neglect it and consider only the reflection by the quartz–air interface as shown in Fig. 2 in the experiment. Moreover, is the Mueller matrix of the glucose sample and has the form 2), the scattering effect can be modeled using the following Mueller matrix :
3. GENETIC ALGORITHM-BASED METHOD FOR GLUCOSE CONCENTRATION DETECTION IN SAMPLES WITH SCATTERING PARTICLES
In the present study, the CB and depolarization properties of the glucose samples are extracted using a GA based on the reflection polarimetry system shown in Fig. 1. As described in Section 2, the experiments were performed using six incident lights with different SOPs, namely, four linear polarization states, i.e., , , , and , and two circular polarization states (right-handed circular: [1,0,0,1] and left-handed circular: ). As described in the following, the experimental output Stokes vectors obtained for the sample under the six incident lights were taken as the objective function for the GA in searching for the values of the CB and depolarization parameters which minimized the following error function:17,18]. Figure 3 shows the major steps in the GA-based optimization procedure performed in the present study. As shown, the process is commenced by constructing an initial population of candidate solutions, where each solution has the form of a chromosome comprising the CB and depolarization parameters. Having compared the experimental and simulated output Stokes vectors, three nature-inspired operations are performed in an attempt to improve the quality of the optimization outcome, namely, selection, crossover, and mutation. In the selection process, a certain number of the existing solutions are selected at random for further processing. A crossover process is then performed in which two selected solutions (parents) are chosen at random, and their bit information exchanged in order to generate two further solutions (offspring). By doing so, it is anticipated that the quality of the offspring generation will be improved by accumulating the exceptional bits information of the parent generation. In the present study, the crossover process is performed using the real-valued crossover operation, i.e.,
Having completed the crossover process, the mutation operation is performed to introduce an additional variability into the population, thereby preventing the GA from converging toward a local, suboptimal solution. In the mutation process, a species string is picked at random, a mutation point is randomly selected, and the bit information is then changed. In the present study, the real-valued mutation operation is used, namely, , where is the new offspring following mutation, and random-value is a small uniformly distributed random variable. Having satisfied the termination criterion, the solution which yields the minimal value of the error function given in Eq. (10) is taken as the optimal solution and is decoded to obtain the corresponding values of the CB and depolarization parameters. Note that in this study, the size of the initial population is set as 30 and the termination condition occurs after 2000 loops.
Figure 4 presents the details of the error evaluation process performed in the GA optimization procedure. Initially, random values of the CB and depolarization parameters of the glucose sample are assigned within the following ranges: , 0–180°; , 0–1; , ; , ; , ; and , . The corresponding Mueller sample matrix is then constructed. Having obtained this matrix, the output Stokes vectors corresponding to the six incident lights are computed by multiplying the sample matrix by the corresponding input Stokes vectors. The simulated output Stokes vectors are then compared with the experimental output Stokes vectors in order to determine the quality of the simulated optical sample properties.
Figure 5 presents the simulation results obtained for a pure glucose solution without scattering effects. Note that the incident angle of the illuminating light is assumed to be 65° and 70°. Also, for the case of no scattering events, the best-fit should be a unit matrix. The simulation of the experimental sample matrix is constructed using Eq. (6), and then the related parameters are extracted by GAs based upon the similar flowchart in Fig. 4. It is seen that a good agreement exists between the extracted values of the optical rotation angle and the input values. Note that the standard deviation of the optical rotation angle is estimated as 0.0021°.
Figure 6 presents the simulation results obtained for a glucose sample with scattering effects. Note that two incident angles are used in the GA solution procedure, namely, 65° and 70° in order to obtain more objective functions for exact solutions when measuring the complicated sample. Note also that the sample Mueller matrix is constructed using Eqs. (6)–(8). A good agreement is once again observed between the input values of the optical rotation angle and scattering effects and the extracted values. It is noted that the average minimum error in the GA solution procedure is around . Thus, the validity of the Mueller matrix model containing both CB and depolarization effects is confirmed. Note that the standard deviation of the optical rotation angle is estimated as 0.0023°.
4. EXPERIMENTAL SETUP AND RESULTS
The present experiments were performed using a He–Ne laser with a center wavelength of 632.8 nm (JDS Uniphase, Model 1144P) as the illumination source. The optical system comprised a polarizer (PLP-12.7B-VIS, LAMBDA, extinction at 420–680 nm), a quarter-wave plate (WPQ10M-633, Thorlabs, at 633 nm), and a commercial Stokes polarimeter (PAN5710-VIS, Thorlabs). It is noted that orientations of all the polarized components need to be calibrated carefully according to the axis of the Stokes polarimeter, and the calibration is investigated by the Stokes polarimeter. Furthermore, the quartz container can be replaced by a mirror to confirm the exact incident angle of the input light by checking the experimental and simulation Stokes vectors. The experiments commenced by determining the optical rotation angles of pure glucose aqueous solutions comprising various amounts of glucose powder (Reag. Ph Eur, D(+)-Glucose, Merck KGaA) dissolved in deionized (DI) water. The experiments were then repeated using simulating phantom comprising glucose powder, DI water, and lipid particles (Lipofundin MCT/LCT 20%, BRAUN). In performing the initial experiments, the incident angle of the illuminating light was set as 65° and 70° and the sample was stored in a quartz cuvette with a thickness of 1 cm. Moreover, aqueous solutions with glucose concentrations with a range of 0–0.8 M were prepared. Figure 7 presents the results obtained for the variation of the optical rotation angle with the glucose concentration. Note that results are presented for both the reflection-based optical system proposed in this study and the transmission-based system proposed by the current group in . For both systems, a high degree of linearity exists between the optical rotation angle and the glucose concentration in the absence of scattering effects. According to , the optical rotation angle is proportional to , where is the specific rotation of glucose at a particular wavelength, is the glucose concentration, and is the distance traveled by the light in the optical system. The results presented in Fig. 7 indicate that the specific rotation of glucose at a wavelength of 633 nm is equal to in the transmission-based optical system and in the reflection-based system. It is noted that these values are in good qualitative agreement with the value of reported in previous studies .
In performing the experiments for cases with scattering effects, the samples were stored in quartz cuvettes with a thickness of 0.75 cm and the Stokes vectors were measured for the incident angle at 65° and 70°. Figure 8 presents the experimental results obtained for the variation of the optical rotation angle with the glucose concentration given glucose samples containing 0.02 ml lipofundin. A good agreement is once again observed between the results obtained using the transmission-based and reflection-based optical systems. In both optical systems, a high degree of linearity between the optical rotation and the glucose concentration is found despite the presence of scattering particles. According to the results obtained using the transmission-based and reflection-based optical systems, the specific rotation of glucose is equal to and , respectively. It is noted that these values are slightly higher than those obtained for the pure glucose samples, i.e., and , respectively. The difference between the two sets of results is attributed to the increased optical path length in the samples containing lipofundin due to scattering effects . Note that the standard deviation of the specific rotation is estimated as .
Figure 9 presents the corresponding results obtained for glucose samples containing 0.04 ml of lipofundin for the incident angle at 65° and 70°. From inspection, the specific rotation is found to be given the use of the transmission-based optical system and given the use of the reflection-based optical system. It is noted that these values are higher than those obtained for the samples with a lower lipofundin content of 0.02 ml. This is thought to be the result of a greater number of scattering events as the lipofundin content increases, which gives rise to a longer optical path length within the sample . Note that the standard deviation of the specific rotation is estimated as . It is noted that the average minimum error in the GA solution procedure is around 0.086.
Figure 10 shows the experimental results obtained for the variation of the degree of depolarization with the glucose concentration for the samples containing 0.02 ml and 0.04 ml lipofundin, respectively. It is observed that for a given glucose concentration, the degree of depolarization increases with an increasing lipofundin content due to a greater number of scattering events. For both samples, the degree of depolarization reduces with an increasing glucose concentration. An increase in the glucose concentration will increase its refractive index and also reduce the refractive-index mismatch between the glucose solution and lipofundin if the refractive index of the lipofundin is higher than the refractive index of the glucose solution and still remains the same. Since the scattering coefficient of the sample depends on the refractive-index mismatch between the solutions and scatters [20–22], the reduced refractive-index mismatch will result in a reduced scattering coefficient. It is noted that this finding is consistent with the results reported by Kohl et al.  for simulating phantoms, and Maier et al.  and Bruulsema et al.  for in vivo tests on human volunteers.
5. CONCLUSIONS AND DISCUSSION
This study has presented a Stokes polarimetry method for determining the glucose concentration of samples containing scattering elements. In the proposed approach, a GA is used to extract the circular birefringence and depolarization parameters of the glucose sample based on a comparison of the experimentally determined output Stokes vectors and the corresponding simulated output Stokes vectors. The validity of the proposed method has been demonstrated using aqueous samples with glucose concentrations ranging from 0–0.8 M and lipofundin concentrations of 0.02 ml and 0.04 ml, respectively. The results have shown that for all of the samples, the optical rotation angle increases linearly with an increasing glucose concentration. Furthermore, for a given glucose concentration, the specific rotation angle increases as the lipofundin content increases due to the greater number of scattering events, which increase the optical path length within the sample. Finally, the degree of polarization increases with an increasing lipofundin content but reduces with an increasing glucose concentration.
The present study has considered the relatively simple case of glucose samples containing only low concentrations of scattering particles. However, in many practical applications such as human tissue and fruit, the propagation of light within the sample is highly complex due to the high degree of scattering and absorption. In the biomedical field, the light propagation behavior of polarized light in bio-tissue is commonly examined by means of Monte Carlo simulations. Thus, in a future study, a new measurement system based on both Monte Carlo simulations and Stokes–Mueller matrix analysis will be proposed to measure the optical parameters of highly scattering media.
Ministry of Science and Technology, Taiwan (MOST) (104-2221-E-006-114-MY2, 104-3113-E-006-002); Ministry of Education, Taiwan, R.O.C.
The authors gratefully acknowledge the financial support provided to this study by the Ministry of Science and Technology of Taiwan. This research was, in part, supported by the Ministry of Education, Taiwan, R.O.C. The aim is for the Top University Project to the National Cheng Kung University.
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