2.1.

Optical Polarimetry System and In Vitro Test Materials

The optical configuration of our experiment is shown in Fig. 1 and includes a 635-nm laser diode with maximum power of 7 mW (Power Technology Inc., Little Rock, Arkansas) as well as a green laser diode with a maximum power of 22 mW of power at a wavelength of 515 nm (Oxxius S.A. Lannion, France). Both lasers set to output 3 mW and the power at the sample was even less and, for any human studies, would be less than the FDA standards for maximum power through the eye.³³

Fig. 1

Block diagram of dual-modulation, dual-wavelength, optical polarimetry system.

The lasers were modulated using synchronous sinusoidal signals and linear amplifiers (carrier frequencies: $f_{\mathrm{c}1}=45\mathrm{kHz}$, $f_{\mathrm{c}2}=82\mathrm{kHz}$). The sinusoidal signals were generated from synchronous sine wave generator programmed in LabVIEW 10.0 (National Instruments, Austin, Texas) and implemented through a PCI-6713 card (National Instruments, Austin, Texas). The modulated beams from both sources were linearly polarized in the horizontal direction employing Glan-Thompson 100,000:1 linear polarizers (P) (Newport, Irvine, California). The individual beams propagated through respective in-house-built Faraday compensators (FCs) that serve as rotation compensators in order to achieve closed-loop feedback control. These compensators were made of 1-cm-long terbium–gallium–garnet (TGG) crystals (Deltronic Crystal Inc., Dover, New Jersey), which were inside bobbins wrapped with electrical magnetically coated coils of wire. TGG crystal was employed since it provided a high Verdet-constant to get the desired optical rotation for the generated field without losing a significant amount of optical signal transmitted through it.

The two beams were made coincident by a beam splitter/combiner. The amplitude of each beam was modulated by a ferrite-based FM at frequency of $f_{\mathrm{m}}=8.75\mathrm{kHz}$. The coil around the ferrite core was powered with an audio amplifier (Radio Shack, Fort Worth, Texas) connected in series with a $0.18\text{-}\mu \mathrm{F}$ capacitor in order to achieve resonance at the modulation frequency. The modulation frequency signal was derived from the same synchronous sine wave generator which was used to power the lasers.

Following the ferrite-based FM, the light beam proceeded through a rectangular sample cell (Starna Cells Incorporated, Atascadero California) constructed of birefringent material with a path length of 1 cm. Following the sample was a second Glan-Thompson polarizer, known as the analyzer, which was oriented perpendicular to the initial polarizer in the layout. Since the combined light beams at the two wavelengths each had separate carrier modulated frequencies ($f_{\mathrm{c}1}$, $f_{\mathrm{c}2}$), detection could be performed using a single biased photo-diode light detector (Thor Labs, Inc., Newton, New Jersey), which converted the optical signals into electrical signals.

The electrical signals were amplified by a wide-bandwidth transimpedance amplifier (CVI Melles Griot, Albuquerque, New Mexico) and fed into two lock-in amplifiers (Stanford Research Systems, Sunnyvale, California). The DC output voltages from the lock-in amplifiers served as the inputs to proportional-integral-derivative (PID) controllers programmed in LabVIEW (10.0, 32 bit, National Instruments, Austin, Texas) using a field programmable gate array, which provided real-time, closed-loop, feedback. The outputs from PID controllers were used to drive the respective FCs via linear driver circuitry. This circuitry provided sufficient current to each FC to produce rotations originating from the combination of the glucose and induced birefringence changes with motion artifact.

The in vitro studies were performed using the birefringent test cell, which was stationary or moved up and down and was filled with glucose-doped water solutions in the physiological range of 0 to $600\mathrm{mg}/\mathrm{dl}$. All samples were created from a $1000\text{-}\mathrm{mg}/\mathrm{dl}$ stock glucose solution, which was made by dissolving 1.0 g of d-glucose (EM Science) into 100-ml deionized water and was mixed with a magnetic stirrer for 8 h in order to completely dissolve the solute. The stock solution was left undisturbed for at least 3 h to allow for mutarotation before being used.

2.2.

Mathematical Description of the System

Without laser modulation, the intensity detected by the detector for each wavelength in the closed-loop system is derived by Jones vectors and matrices and can be described as^27–29

For the dual-modulation, dual-wavelength system, the lasers were modulated by sinusoidal signals at frequencies of $f_{\mathrm{c}1}=45\mathrm{kHz}$, $f_{\mathrm{c}2}=82\mathrm{kHz}$, respectively, and were also synchronously modulated by the Faraday rotator at a frequency of $f_{\mathrm{m}}=8.75\mathrm{kHz}$. Due to the crossed-polarized nature of the alignment and according matrices for the optical components presented in the block diagram in Fig. 1, the detected signal of each wavelength can be represented as

In the presence of an optical active element such as glucose, each wavelength of light experiences rotation, ${\varphi }_i$, which is a constant and depending on the concentration of sample and the wavelength of light used, a different sideband frequency component $f_{\mathrm{c}1}\pm f_{\mathrm{m}}$, $f_{\mathrm{c}2}\pm f_{\mathrm{m}}$ was observed, which had a proportional increase with the increase in the rotation, ${\varphi }_i$. Two lock-in amplifiers were used to lock into the detected signals at the frequency of interest, namely, $f_{\mathrm{c}1}+f_{\mathrm{m}}$ and $f_{\mathrm{c}2}-f_{\mathrm{m}}$ to extract the information carried by each wavelength, which in turn was processed to obtain the concentration of the glucose in the sample. Since the lock-in amplifiers locked in to the sum of the modulation frequency and one of the carrier frequencies, phase-sensitive detection was used. In order to measure the amplitude of the signals at a specified reference frequency, the frequency and phase stability had to be maintained to prevent signal drift, and hence a synchronous sinusoidal signal generator programmed in LabVIEW 10.0 and implemented through PCI-6713 was used to generate the carrier signals, modulation signal, and reference signals.

2.3.

Theoretical Description for Overcoming Motion Artifact

Mathematically, as noted above, the operation of each of the single wavelengths is based on Eq. (3). From Eq. (3), it is evident that with any optically active sample the signal has DC terms and, at each wavelength, has major frequency components at $f_{\mathrm{c}1},f_{\mathrm{c}2},2f_{\mathrm{m}},f_{\mathrm{c}1}\pm 2f_{\mathrm{m}},f_{\mathrm{c}2}\pm 2f_{\mathrm{m}}$ and without an optically active sample there are no components at $f_{\mathrm{c}1}\pm f_{\mathrm{m}},f_{\mathrm{c}2}\pm f_{\mathrm{m}}$, because the rotation, ${\varphi }_i$, due to glucose is zero. So when an optically active element such as glucose is present the rotation, ${\varphi }_i$, will cause the intensity of detected signal to vary at the frequencies of $f_{\mathrm{c}1}\pm f_{\mathrm{m}},f_{\mathrm{c}2}\pm f_{\mathrm{m}}$ and the signals at these frequencies proportionally increase with the increase in the rotation, ${\varphi }_i$, depending on the concentration of sample and the wavelength ($i=1$ or $i=2$) of light used. So locking into one of sideband frequency signals provides glucose concentration information carried by each wavelength.

In a closed-loop system, if there is minimal birefringence (i.e., not enough to make the polarization become circular) and the birefringence of the sensing site did not change, the glucose concentration could be extracted from the feedback voltage to the FCs from either wavelength signal, since each compensates for the rotation in polarization due to glucose. In this case, ${\varphi }_{\mathrm{g}i}$, (rotation at wavelength $i$ due to glucose) is equal to ${\varphi }_{\mathrm{f}i}$ (rotation at wavelength $i$ fed back to the FC), although the rotation is different for each wavelength and follows Drudes law. However, when motion artifact is introduced into the system, ${\varphi }_i$ becomes a function of birefringence, glucose, and the feedback of the Faraday for a given wavelength: ${\varphi }_i={\varphi }_{\mathrm{g}i}+{\varphi }_{\mathrm{b}i}-{\varphi }_{\mathrm{f}i}$, where, ${\varphi }_{\mathrm{b}i}$ is the rotation induced by the birefringence from each wavelength, but the relationship between ${\varphi }_{\mathrm{b}1}$ and ${\varphi }_{\mathrm{b}2}$ are fixed as long as the wavelengths are known because birefringence is a function of both the wavelength and birefringence $|n_e-n_o|$ for a fixed path length through sample. Birefringence for an individual wavelength can be expressed as ${\varphi }_{\mathrm{b},i}(t)=\frac{2\pi }{{\lambda }_i}|n_e-n_o|L(t)$, which is constant with wavelength at a given position, a behavior attributed to the form birefringence of the cornea.^34–36

For the case of minimal, nonvarying birefringence (without motion), the relation between the feedback voltage and glucose concentration is linear, a linear regression is then used to predicted glucose concentrations, and a predicted concentration model for individual wavelength takes the form: ${\text{Glucose}}_{\mathrm{pred}}=\text{slope}*[V(t)]+b$, where $V(t)$ is the compensation voltage signal applied to the FC for each wavelength used to null the detected signal via a closed-loop PID controller, ${\text{Glucose}}_{\mathrm{pred}}$ is the predicted concentration of the sample, $\mathrm{slope}$ and $b$ are the slope and intercept of calibration line determined by the regression model for each wavelength. As stated, since the rotation due to glucose varies with wavelength following Drudes equation, the values for each are not identical. For the dual-wavelength system, a multiple regression model was used to determine the glucose concentration and the model had the form: ${\text{Glucose}}_{\mathrm{pred}}={\mathrm{slope}}_1*[V_1(t)]+{\mathrm{slope}}_2*[V_2(t)]+b$, where $V_1(t)$ and $V_2(t)$ are the feedback voltage obtained to nullify the system for the corresponding dual-wavelength system and ${\mathrm{slope}}_1$, ${\mathrm{slope}}_2$, and $b$ are the calibration coefficients calculated by multiple linear regression (MLR) model.^30,31 This MLR model has been shown previously to minimize the effect of time-varying birefringence caused by motion using the dual-wavelength polarimetric system due to the difference in the signal from glucose rotation with wavelength (Drudes equation) and the relative common mode noise in each wavelength from time-varying birefringence.

3.1.

Speed Determination of the System

As mentioned earlier, a previous attempt to minimize the effect of time-varying corneal birefringence in the eye that used a dual-wavelength system was only modulated at 1.09 kHz and required 100 ms time constant setting on the lock-in amplifiers, resulting in around 300 ms response time to stabilize the system.²⁹ With our new system the time response was measured by visualizing the response speed of the feedback voltage produced by the control system. One widely used measure of the speed of response is the 10% to 90% rise time or the amount of time the system takes to go from 10% to 90% of the steady-state. For the response speed studies, $300\mu \mathrm{s}$ was the lock-in amplifier time constant that was used. In Fig. 2, the feedback voltages as a function of time for the single wavelength systems (515 and 635 nm, respectively) are depicted. The plot indicates that the PID control system can reach stability in less than 10 ms, which is $\sim 30$ times faster compared with the real-time, dual-wavelength approach presented by Malik and Coté.²⁹ This shows that the laser intensity modulation combined with Faraday modulation and an adaptive PID controller can improve the response speed of the system.

Fig. 2

Stabilization of 515 and 635 nm PID control system without motion.

3.2.

System Performance without Motion

Although it has been shown in the previous section that the new system can provide enhancement in speed, the performance in terms of sensitivity to glucose measurements over the same range as shown previously but requiring less time needs to be determined. The performance of the system was first tested and evaluated for glucose-doped water solutions in the motionless system. In particular, a 1-cm rectangular quartz cuvette that allowed for sample to be easily pipetted into was mounted in a fixed holder. The experiments were performed for individual glucose samples between the concentration range of 0 to $600\mathrm{mg}/\mathrm{dl}$. The sample cell was in a fixed position for each sample in the motionless system.

The motionless system was calibrated, and then the glucose concentrations were calculated using a simple linear regression model of the data acquired without motion.³⁷ In each experiment, three runs of data were recorded for the two wavelengths, 635 and 515 nm, respectively. As it was shown in Sec. 2, the relation between the measured signal in volts and sample concentration is linear, so least-squares linear regression was used for the validation of the model. The predicted versus actual glucose concentrations were plotted for the single wavelength system as shown in Figs. 3 and 4, respectively. It can be observed that the plots indicate the dependency of glucose on one wavelength only; the slopes of this fitted line are slope1 = 2591.12, slope2 = 2605.89, slope3 = 2574.22 for the green (515 nm) light and for the red (635 nm) light are slope1 = 2609.30, slope2 = 2570.36, slope3 = 2598.10, which remained relatively constant. One can see that a high degree of linearity exists given the correlation coefficients for the green (515 nm) light are $r_1=0.9990$, $r_2=0.9993$, $r_3=0.9993$ and for the red (635 nm) light are $r_1=0.9985$, $r_2=0.9993$, $r_3=0.9990$, respectively. Further, the average standard error of prediction (SEP) for the validation of the glucose-doped water was $6.48\mathrm{mg}/\mathrm{dl}$ for the green light and $7.18\mathrm{mg}/\mathrm{dl}$ for the red light. The feedback voltages for both wavelengths were applied in an MLR model, which was used to predict the glucose concentration, and Fig. 5 shows the estimated glucose concentration as a function of the actual glucose concentration when both wavelengths are taken into account. The slopes of MLR model are $\mathrm{slope}11=-2091.5$, slope12 = 748.47, $\mathrm{slope}21=-1463.52$, slope22 = 1664.27, $\mathrm{slope}31=-1703.72$, and slope32 = 1362.41 for the different runs. Using the MLR model, the information from both wavelengths was combined to improve the results without motion and indeed one can see that a high degree of linearity still exists for all three cases since each correlation coefficient exceeds 0.9996 and the SEP is slightly less than using each wavelength alone ranging from 4.23 to $5.63\mathrm{mg}/\mathrm{dl}$, with no outliers present in any of three data runs.

Fig. 3

Predicted glucose concentration as a function of actual glucose concentration for the 515-nm laser without motion using a single linear regression model.

Fig. 4

Predicted glucose concentration as a function of actual glucose concentration for the 635-nm laser wavelength without motion using a single linear regression model.

Fig. 5

Predicted glucose concentration as a function of actual glucose concentration for the combined 635 and 515 nm laser wavelengths without motion using an MLR model.

3.3.

System Performance with Motion from Time-Varying Birefringence

The experiments were conducted to test the performance of the system for glucose solution in the sample cell while introducing motion into the system. For these experiments, the test cell was mounted on a programmable translation stage (Thorlabs, Newtown, New Jersey) and the stage was controlled by a T-Cube DC Servo Motor Controller (Thorlabs, Newtown, New Jersey) and allowed to move up and down to simulate the birefringence changes of the cornea due to motion artifact. The speed and distance of the translation of the motor stage were controlled by a visual basic program.

The system with motion was calibrated and the glucose concentrations were extracted using a linear regression model. This was done for glucose concentrations from 0 to $600\mathrm{mg}/\mathrm{dl}$.

In order to compare the results of the single wavelength closed-loop and dual-wavelength closed-loop system, actual versus predicted glucose concentration for glucose-doped water coupled with a varying birefringence signal using the single wavelength closed-loop system for each wavelength (515 and 635 nm) is plotted in Figs. 6 and 7. As shown in Figs. 6 and 7, the glucose concentrations cannot be predicted very well using a single closed-loop system, since the change in the birefringence of the sample masks the optical rotation due to glucose in the sample. Thus, the single wavelength model has an SEP over $100\mathrm{mg}/\mathrm{dl}$ in the presence of birefringence changes due to motion.

Fig. 6

Actual versus predicted glucose concentration for glucose-doped water in the presence of time varying birefringence due to motion using a single wavelength closed-loop system for 515 nm.

Fig. 7

Actual versus predicted glucose concentration for glucose-doped water in the presence of time varying birefringence due to motion using a single wavelength closed-loop system for 635 nm.

The predicted concentrations results for the dual-wavelength experiments using MLR calibration are shown in Fig. 8. The predicted calibration results when both wavelengths are used to compensate for the correlated noise due to motion showed a high degree of linearity with all correlation coefficients exceeding 0.9972 and the standard errors of prediction of 13.0, 13.7, and $13.9\mathrm{mg}/\mathrm{dl}$ as shown in Fig. 8. No outliers were present in any of the data sets. Although the SEP was greater than the motionless system, it was still as good as the previous slower system, and a very simple sample was still within the range needed for noninvasive glucose monitoring.

Fig. 8

Predicted glucose concentration as a function of actual glucose concentration with motion due to time-varying birefringence calculated using an MLR regression model for 635 and 515 nm wavelengths.