## Abstract

Polarimetric determination of glucose is known to be strongly affected by scattering in turbid media. Other effects like fluctuations of light source emission and sample absorption also deteriorate glucose predictability. This work presents a measurement setup using a real-time data processing method to address these problems. The approach uses the frequency-dependent intensity components created when the polarization of the incident light is periodically modulated by a Faraday rotator. The efficacy of the proposed method was verified experimentally for a glucose range of 0 − 500 mg/dl. It was shown that the approach reduces the prediction errors in slightly turbid media from 35.7 mg/dl down to 1.17 mg/dl. In a similar way, the glucose predictability for fluctuating light source emission was improved from ±16.16 mg/dl to ±1 mg/dl and for varying sample absorbance from ±15.69 mg/dl to ±1.23 mg/dl, respectively. Therefore, considerable improvement of robustness and reproducibility of glucose determination was demonstrated.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Over the past years diabetes continues to be a growing problem. According to the World Health Organisation’s (WHO) annual report on diabetes, the proportion of people afflicted with diabetes nearly doubled from 1980 until 2014 [1]. Diabetes encompasses a group of metabolic diseases arising from low insulin production or a faulty glucose response [2]. A resulting insufficient insulin level can lead to high glucose levels (hyperglycemia) and cause damage to kidneys, eyes, nerves or heart [3]. To minimize long-term health complications it is necessary to diagnose diabetes as early as possible [1]. In this case, continuous glucose monitoring (CGM) can help diabetic patients to better control their blood glucose levels [4].

Polarimetry uses the ability of glucose to rotate polarized light depending on wavelength, pH-value and temperature and has been explored for many years [5]. One of the most challenging tasks in noninvasive glucometry is to acquire an appropriate signal in a turbid sample. Substantial progress in this field has been made by different groups, particularly by Nirmalaya Ghosh, Alex Vitkin and Steven Jacques [6–9] who developed several scattering models using Monte Carlo calculations [6–9] to study the behavior of polarized light in turbid media. Experimental investigations with Intralipid [10] and polystyrene microspheres [11] were carried out using a photoelastic-modulated (PEM) polarimeter and lead to close agreement between theoretical and measured values [7]. The working group of Alex Vitkin showed that a certain polarization remains in forward direction even for highly scattering media [6, 9, 12]. This remaining polarization was proven to be linearly proportional to the glucose concentration with a factor of proportionality depending on the concentration of scattering particles [11].

Additionally, the low optical rotation of physiological glucose levels constitutes another obstacle, even for measurements in clear media. This requires very robust measurement setups for accurate determination. Since rotation-dependent transmitted light intensity is also influenced by the absolute amount of emitted light, as shown more detailed in subsection 2.2, we present an approach to increase the reproducibility and robustness of a Faraday-modulated polarimeter setup. For this purpose we evaluate specific frequency components to allow real-time compensation of fluctuations in absolute light intensity. Furthermore, we demonstrate that this method can be applied to reduce the disturbance impact of scattering by turbid media on glucose predictability. This is an alternative method to other approaches using closed loop systems [3, 13–16] and PEM-modulated polarimeters [7,9,11,17,18] as described in subsection 2.2.

## 2. Theoretical background

#### 2.1. Optical activity

The optical activity of molecules like glucose or proteins is caused by their chirality. The specific optical rotation depending on the wavelength is called optical rotatory dispersion (ORD) and is unique for every type of optically active molecule. The specific rotation angle *α* can be calculated for a certain wavelength *λ* by using Drude’s equation as follows [5,19]:

Strength and direction of rotation are defined by the constant *A*, while the absorption center wavelength *λ _{c}* determines the unique spectral distribution and can be found as constants in the literature [20].

*pH*-value and temperature

*T*have also been reported to influence the rotation [5]. The resulting rotation Φ for a concentration

*c*and a given path length

*l*can be expressed as [21]:

As a result, the absolute rotation caused by the sample is proportional to the sample concentration and optical path length.

#### 2.2. Polarimeter principle

The polarimeter used in this work is shown schematically in Fig. 1. It consists of a light source, two crossed polarizers, a cuvette, a light detector and utilizes a Faraday rotator for modulation. The emitted light from the light source (a) is linearly polarized by the first polarizer (b). The polarization state is then sinusoidally modulated by the Faraday rotator (c) and additionally influenced by the sample (d). A second polarizer (e) is arranged perpendicular to the first one (b). All the transmitted light through the second filter is detected by a detector (f) and the signal subsequently processed with a PC (g). The rotation Φ caused by the sample overlaps with the periodically modulated rotation with the frequency *ω* and modulation depth Θ* _{m}* at the modulation frequency

*ω*. The transmitted electrical field amplitude

_{m}*E*at the time

_{t}*t*was already described in the literature normalized to a constant unit incident electrical field with an amplitude of 1 [22]:

In a more general way, the incident light intensity is represented by the emitted electrical field amplitude from the light source *E*_{0} and the sample transmission *T*. This generalizes Eq. 3 to:

Due to the small physiological rotation of a few millidegrees [4] and modulation depths of approximately 1 deg, the approximation sin (*x* ) ≈ *x* simplifies Eq. 4 to [22]:

As ${I}_{t}\propto {E}_{t}^{2}$, the detected light intensity *I* can be expressed as:

With 2 sin^{2}(*x* ) = [1 − cos (2*x*)] Eq. 6 delivers the final Eq. 7 describing the detected intensity *I _{t}* at a certain time

*t*depending on sample rotation Φ and maximum Faraday modulation depth Θ

*[4,5,23]:*

_{m}There are three contributions to the total intensity, all of which are proportional to the squared reference electric field *E*_{0} and the sample transmission *T*, which is influenced by light source emission, detector sensitivity and sample absorption. The first intensity component is constant in time, the second oscillates with the frequency of the Faraday rotator *ω* and the third shows the double frequency 2*ω*. The amplitudes of the three frequency components can be extracted and expressed as

In contrast to *I* (*ω*), *I* (2*ω*) is not influenced by the sample rotation Φ [5] and has already been proposed to monitor the overall transmitted intensity, e.g. sample absorption [24]. Building the ratio of *I _{t}*(

*ω*) and

*I*(2

_{t}*ω*) leads to:

Like *I*_{t}(*ω*) the ratio $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ is linearly dependent on the sample rotation Φ. Light intensity fluctuations on numerator and denominator of the ratio $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ are the same, because the intensity of each frequency component shows the same dependence on absolute amount of emitted light. Hence, the ratio is independent on fluctuations of absolute light intensity and is only dependent on the rotation of the sample and the Faraday modulation depth Θ* _{m}*.

Since the first polarizer in Figure 1 generates linear polarized light, a perpendicular amplitude emitted by the light source is diminished by the factor of the polarizer’s extinction ratio. For Glan-Thompson polarizers as used in this setup (see subsection 3.1), the extinction ratio is 100,000 : 1. This leads to a negligible influence of the perpendicular amplitude on the signal. Consequently, only the parallel amplitude provides a contribution to the absolute light intensity *E*_{0}. A variation of light source polarization state, which is a common problem for lasers, can therefore be treated similar to a fluctuation of absolute light source intensity. Hence, the ratio $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ should also be able to compensate for drifts of light source polarization state.

Influences on the measured sample rotation Φ like mechanical torsion of the polarization filter mounts can affect the measurement, leading to a non-perpendicular orientation of the two polarizers. While the temperature influence on the Verdet constant of various optical glasses is reported to be approximately 10^{−4} K^{−1} [25] and small in comparison to temperature-induced specific sample rotation of approximately 10^{−1} K^{−1} [5], the current of the Faraday rotator has to be kept as stable as possible, because it has a direct influence on the modulation depth Θ* _{m}*.

As the overall light intensity is present in all frequency components, other ratios involving the DC-part e.g. $\frac{{I}_{t}(DC)}{{I}_{t}(\omega )}$ as already suggested by Jie et. al. [26] or $\frac{{I}_{t}(DC)}{{I}_{t}(2\omega )}$ might be conceivable to compensate for fluctuations of light intensity. However, these approaches have significant disadvantages in comparison to our method like introducing low-frequency noise and a nonlinear relation to sample concentration as can be seen from the extracted frequency components of Eq. 7.

As an alternative to Faraday rotators, other modulators like Pockels cells [27] or photoelastic modulators (PEMs) [6, 7, 9, 11, 17] have been used in many other setups. A PEM is a piezoelectrically-driven crystal which modulates the phase of incident light along the modulation axis [17]. The PEM is modulated sinusoidally, commonly with a frequency *ω*_{PEM} = 50 kHz [6, 7, 9, 11]. Similar to the Faraday-modulated polarimeter presented here, harmonics of *ω*_{PEM} arise from the signal leading to frequency components of *I*(DC), *I*(*ω*_{PEM}) and *I*(2*ω*_{PEM}) [6, 11, 28]. Since the *I* (DC) part in the PEM approaches is a measure for absolute light intensity [18] while *I*(*ω*_{PEM}) and *I*(2*ω*_{PEM}) comprise only polarization information [11], the ratios $\frac{I({\omega}_{\text{PEM}})}{I(\text{DC})}$ [17], $\frac{I(2{\omega}_{\text{PEM}})}{I(\text{DC})}$ [11] and $\frac{I({\omega}_{\text{PEM}})}{I(2{\omega}_{\text{PEM}})}$ [17] have been successfully used to increase stability and to improve their signal-to-noise ratio (SNR). We show that with $\frac{I(\omega )}{I(2\omega )}$ a similar method can be applied for Faraday-modulated polarimeters to increase the robustness in the presence of slightly turbid media and varying absolute light intensity.

## 3. Material and methods

#### 3.1. Polarimeter setup

The experimental setup used in this study is shown schematically in Figure 1. To demonstrate the compensation ability of fluctuating light source emission and sample absorbance we used a green LED (LT W5SM, Osram Light GmbH, Germany) with a center wavelength of 528 nm and a full width at half maximum (FWHM) of 33 nm connected to an LED driver. The reference signal for the driver was generated by a DC power supply (E3631A, Hewlett Packard, USA). The light was linearly polarized by a Glan-Thompson polarizer (GTH10, Thorlabs GmbH, Germany), with an extinction ratio of 100,000 : 1. The light passing the first polarizer was then modulated by a Faraday rotator. This rotator utilizes a 65 mm SF-59 rod, chosen due to its small temperature dependence compared to commonly used Terbium gallium garnet (TGG) [25]. The rotator’s coil has 475 turns and was driven at 2 A_{pp} by a current driver. The current fluctuations during the measurements were ±100 *µ*A (±0.005 %). The sinusoidal reference signal for the coil driver was generated by a function generator (33120A, Hewlett Packard, USA) with a frequency stability of 2 ppm/deg and 10 ppm/90 days. According to other scientific contributions [5, 21, 29], a frequency of 1.09 kHz was chosen to avoid any overlap of harmonics of 50 Hz.

A flow-through cuvette which can be flushed externally was located behind the Faraday rotator. Since a larger path through the sample leads to an enhanced rotation, we chose a path length of 50 mm to obtain a high sensitivity [30]. A second Glan-Thompson polarizer (GTH10, Thorlabs GmbH, Germany) was placed behind the cuvette. The transmitted light was measured by a photodiode detector (DA36A, Thorlabs GmbH, Germany) and processed by an A/D converter (NI9205, National Instruments GmbH, Germany) while the signals were evaluated by LabVIEW 2016. The whole setup was placed inside a temperature-controlled chamber at 37 ± 0.1 °C to reduce temperature induced drift effects like mechanical torsion.

In order to detect the different frequency components *I _{t}*(

*DC*),

*I*(

_{t}*ω*) and

*I*(2

_{t}*ω*), we used a Fourier transform which was performed by the FFT express VI in LabVIEW 2016.) After a recording time of 1 s corresponding to 50,000 signal samples, the FFT was calculated using RMS values and a Hann-Filter. This procedure was repeated 20 times and the resulting FFT values were then averaged. The intensity components were then extracted as the amplitude of the Fourier spectra at the modulation frequency

*ω*(1.09 kHz) and at 2

*ω*(2.18 kHz) with a bandwidth of 1 Hz. Commonly the Faraday reference signal is used as input signal for a lock-in amplifier to detect

*I*(

_{t}*ω*) [4,5,23]. In order to extract both frequency components

*I*(

_{t}*ω*) and

*I*(2

_{t}*ω*), this would require an additional reference signal with the frequency 2

*ω*and the correct phase, and a separate lock-in amplifier. This would increase the complexity of the setup and introduce further sources of error affecting the precision of the system.

#### 3.2. Sample preparation

All samples were created by mixing glucose with red dye and fat emulsion stock solutions and distilled water. Because of its major influence on ORD [5], the *pH*-value was stabilized for all stock solutions at a value of 7.4 using phosphate buffer (Morphisto GmbH, Germany).

Five liters of glucose stock solution with a concentration of 1000 mg/dl were prepared by weighing D-glucose powder (Sigma Aldrich, Germany) and dissolving it in distilled water. 50 g of glucose were measured with a high precision balance (Kern EW120-4NM, Kern & Sohn GmbH, Germany). Five liters of distilled water were added under weight control (PCE-BSH 10000, PCE Instruments GmbH, Germany) with an accuracy of ±0.6 g. The resulting uncertainty in stock solution concentration was ±0.012 %. To fully achieve mutarotation equilibrium between *α*- and *β*-glucose we waited 24 h at a constant temperature of 37 ± 0.1 °C before performing the measurement which has been reported to be sufficient [3].

In order to show the influence of sample absorption we chose Allura Red AC (Sigma Aldrich, Germany) as a red dye that displays a broad absorption peak at 504 nm, close to the green LED’s center wavelength of 528 nm. Due to its high absorption the dye concentration needed for the stock solution was 4 mg/dl. To ensure a precise concentration, the preparation of the solutions was divided into two steps. First, five liters of 200 mg/dl stock solution similar to glucose solutions were created, dissolving 1000 ± 1 mg in 5000 ± 0.6 ml distilled water. Then 10 ± 0.0001 ml were diluted with 4900 ± 0.6 ml distilled water. The resulting mixing error of Allura Red AC stock solution is ±0.1 %.

The effects of turbid media were evaluated by using ClinOleic 20 % (Baxter GmbH, Germany) which is similar to Intralipid and offers a scattering coefficient of *µ _{s}* ≈ 1,120 cm

^{−1}at the LED wavelength of 532 nm [31]. In other publications measurements with

*µ*between 20 cm

_{s}^{−1}and 100 cm

^{−1}[9], 20 cm

^{−1}[6] and 60 cm

^{−1}[7] have been reported. However, the glucose concentrations in those investigations were approximately 0−180,000 mg/dl [7], 21,600 mg/dl [6] and 360−16,200 mg/dl [9] and thus orders of magnitude above the physiological level. The glucose concentrations of 0 − 500 mg/dl used for the measurements presented here are approximately 100 times lower. In this first attempt we therefore created a comparable scenario by using a 200 times lower

*µ*of 0.25 cm

_{s}^{−1}. Consequently, a stock solution of 55.8 mg/dl ClinOleic corresponding to

*µ*= 0.625 cm

_{s}^{−1}was created by diluting 1.116 ± 0.003 ml with an Acura 826 XS pipette into 2 ± 0.0006 l of distilled water, which equals to an error of ±0.4%.

Different concentrations were automatically created by mixing stock solutions of glucose with Allura Red AC, ClinOleic and distilled water using three high precision syringe pumps (neMESYS 290N, Cetoni GmbH, Germany), equipped with 25 ml glass syringes (ILS GmbH, Germany). After dosing their complete volume into the cuvette, the syringes were automatically refilled with stock solutions. For all measurements samples of 50 ml were injected into the cuvette to ensure minimal contamination with prior samples. After each measurement the cuvette was additionally flushed with 50 ml distilled water. The measurement was performed 30 s after the flow was stopped to ensure stationary conditions. As for the measurement setup, the syringe pumps and the stock solutions were kept inside the temperature controlled box at 37 ± 0.1 °C during the entire measurement.

The accuracy of the dosing system was determined gravimetrically by using distilled water at 25 °C and sample volumes of 1 ml, which represents the smallest delivered volume during a typical measurement. The resulting absolute combined uncertainty was 2.1 *µ*l of delivered volume (bias) + 0.14 % (precision).

#### 3.3. Measurement protocol

The theoretical considerations, described in subsection 2.2, were experimentally investigated by different measurements. First we evaluated the effect of glucose concentration variations, LED current fluctuations and sample absorbance on *I _{t}*(

*ω*),

*I*(2

_{t}*ω*) and the ratio $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ separately. In a second step, measurements in slightly turbid samples were carried out in order to show that the method is also applicable for this scenario. Although absorbance plays a minor role in comparison to turbidity with respect to a future application (e.g. noninvasive glucometry in tissue), the effect was investigated to show the robustness of the setup.

The glucose prediction accuracy was then examined under both stable and fluctuating light source emission as well as for changing sample absorbance and varying turbidity to compare the results for the use of the conventional *I _{t}*(

*ω*) with the ratio It $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$. For this purpose, two sets of identical measurements were collected for each experiment and split into calibration and validation data. A linear regression between signal intensity and concentration was performed on the calibration data to predict concentrations of the validation data while their deviation from the regression line is considered as the standard error of prediction (SEP).

All samples were measured in a random order to avoid any temporal cross correlation influence, like for example those caused by temperatures fluctuations, as described by other authors [3,4,32]. According to Eq. 7, both amplitudes *I _{t}*(

*ω*) and

*I*(2

_{t}*ω*) were extracted by Fourier transform. An overview of the performed measurements is given in Table 1. In experiment #1 we measured glucose concentrations in the range of 0 − 500 mg/dl in 12.5 mg/dl increments at a constant LED current of 400 mA. The signals

*I*(

_{t}*ω*) and

*I*(2

_{t}*ω*) were monitored to prove that

*I*(2

_{t}*ω*) is independent of the sample rotation Φ according to Eq. 7.

Measurement #2 was performed to show that fluctuations of absolute light intensity have a similar influence on *I _{t}*(

*ω*) and

*I*(2

_{t}*ω*), while the ratio of both intensity components remains constant. Under real operating conditions (e.g. clinical setting), we expect variations of the LED intensity of at least ±10 % due to junction temperature fluctuations, aging and binning. The input current was thus varied between 360 mA and 440 mA to simulate the corresponding intensity changes, while keeping the glucose concentration constant at 500 mg/dl.

Similar to #2, in measurement #3 the influence of absorption by Allura Red AC on the frequency components was investigated. The Allura Red AC concentration was varied in the range of 0 − 0.2 mg/dl in 0.01 mg/dl increments while the glucose concentration remained at 500 mg/dl. This corresponds to the expected absorbance variability due to remaining blood cells, blood proteins, etc. when measuring glucose concentration in filtrated blood plasma. Comparable concentrations of proteins have already been used in other publications [30].

The influence of turbid media on the *I*(*ω*) and *I*(2*ω*) frequency components was investigated in measurement #4 by using ClinOleic solution. Since glucose measurements in turbid media have been performed for much higher concentrations [6,7,9], we used lower scattering coefficients of *µ _{s}* = 0.25 cm

^{−1}corresponding to 22.32 mg/dl ClinOleic suspension with respect to the physiological glucose range used in this investigation. Similar to the LED influence in measurement #1, the ClinOleic concentration was varied by ±10 % leading to

*µ*= 0.225 − 0.275 cm

_{s}^{−1}, while the glucose level was kept constant at 500 mg/dl.

Measurement #5 continues with the analysis of glucose predictability at a stable LED current of 400 mA. Therefore two similar sets of 30 concentrations each in the range 0 − 500 mg/dl in 100 mg/dl increments were created. Each concentration was measured 5 times for calibration and validation. The first set was used as calibration data to create a linear fit for *I _{t}*(

*ω*) and

*I*(2

_{t}*ω*) as a function of glucose concentration, respectively. These regression models were used to predict the concentrations of the remaining validation data and to compare the results of

*I*and

_{t}ω*I*(2

_{t}*ω*). In order to evaluate the impact of fluctuations in absolute light intensity we measured the same concentrations as for #5, but with varying LED currents of ±10 % by measurement #6 in Table 1. Each of the 5 repetitive concentrations of calibration and validation data was measured at a different LED current in the range of 360 − 440 mA in 20 mA steps. As a result, 30 unique combinations of glucose concentrations and LED currents were generated for calibration and validation, respectively. The SEP with conventional

*I*(

_{t}*ω*) was compared with that of measurement #5 without extra disturbances.

Measurement #7 was performed to investigate the influence of sample absorbance on glucose SEP. The samples were spiked with 0 − 0.2 mg/dl Allura Red AC while the LED current was kept constant at 400 mA. The prediction by linear regression was performed as for #5.

The direct influence of turbid media on glucose predictablity was investigated in measurement #8. Each of the 6 different glucose values in the range of 0 − 500 mg/dl was combined with 5 different scattering coefficients of *µ _{s}* = 0.225 − 0.275 cm

^{−1}. The resulting 30 unique combinations were measured two times to obtain calibration and validation data.

## 4. Measurements and results

#### 4.1. Proof of theoretical considerations

In order to illustrate the relative changes in the certain frequency components, intensities in the following graphs are expressed in percent of variation relative to their maximum values within the measurement range. The results for glucose level variation of measurement #1 in Figure 2 reveal a linear dependency of *I _{t}*(

*ω*) on glucose concentration which corresponds to Eq. 7. Furthermore,

*I*(2

_{t}*ω*) remains constant for glucose variations in the range of 0 − 500 mg/dl. As a result, the ratio $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ shows the same relative linear behavior as with

*I*(

_{t}*ω*). This indicates that $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ delivers the same results as

*I*(

_{t}*ω*) under perfectly stable conditions and can be used for linear regression models.

A variable absolute light intensity simulated by variations of LED currents in the range of 360 − 440 mA shown in Figure 3 (left) reveals a strong influence on *I _{t}*(

*ω*) and

*I*(2

_{t}*ω*).

The ratio It $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ stays constant during this range and the effect of fluctuating light source emission could be reduced by more than 99 %. This corresponds to Eq. 8, where the ratio It $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ was shown to be independent of absolute light intensity.

An increasing Allura Red AC concentration causes a similar decline in *I _{t}*(

*ω*) and

*I*(2

_{t}*ω*) shown in Figure 3 (right). The application of It $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ reduces this effect significantly, although a slight slope in the ratio remains for high absorbance. The reason for this effect could not be fully explored yet and needs further investigation. A possible reason could be optical activity of the Allura Red dye. After consulting the manufacturer, they could not provide any information about this. Nonetheless, this variation is negligible compared to the effect observed on the single frequency components.

The measured intensities of *I _{t}*(

*ω*) and

*I*(2

_{t}*ω*) in Figure 4 rapidly decline with increasing scattering coefficient, while $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ stays constant and compensates for more than 99 % of the turbidity-induced variation, similar to the measurements #2 and #3.

#### 4.2. Concentration determination

The results for measurement #5 with stable LED current are shown in Figure 5.

The goodness of glucose prediction is commonly expressed by the SEP and the regression coefficient *R*^{2} as performed by Cameron et. al, 1997 [4] or Clarke et. al, 2013 [3]. The SEP for a number of *n* concentrations is calculated by the standard deviation of the predicted concentration *C _{p}* against their actual concentrations

*C*.

_{a}*R*^{2} is the coefficient of determination and is a statistical measure of how close the data are to the regression line. In the case of a linear least square regression, it is equivalent to the square of the correlation coefficient *r* [33]. In glucose prediction this is the correlation between predicted glucose values and the regression curve. A value of 1 means perfect correlation while values with 0 are not correlated. An SEP of ±1.05 mg/dl and a high linearity of *R*^{2} = 0.99996 were achieved by using *I _{t}*(

*ω*). The prediction via $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ delivers slightly improved results with an SEP of 0.92 mg/dl and

*R*

^{2}= 0.99998. According to the results with fluctuations of absolute light intensity in Figure 3 and Eq. 8, the ratio brings greater advantages for higher light fluctuations. Due to high efforts in setup stabilization by current driver and high precision Hewlett Packard E3631A power supply for the LED reference signal, the LED current was as stable as 400±0.3 mA during the entire 12 h experiment.

According to Eq. 7 and subsection 4.1, the greatest advantage of the compensation method is expected for a highly fluctuating LED current. The prediction accuracy for the glucose determination measured at LED currents of 360 − 440 mA can be seen in Figure 6. Compared to the results obtained using a stable LED current in Figure 5, the SEP using only *I _{t}*(

*ω*) deteriorates significantly from ±1.05 mg/dl to ±16.16 mg/dl and the regression coefficient decreases from 0.99996 to 0.99148.

The application of $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ for the regression model shows a highly improved predictability. An SEP of ±1.00 mg/dl and *R*^{2} = 0.99998 were achieved. These are very similar results compared to those obtained with undisturbed light intensity, although in this case the LED current was fluctuating between 360 mA and 440 mA. This measurement shows that a glucose prediction with the use of It $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ is highly independent of the light source emission. It was also performed with an LED current variation of ±50 % equivalent to 200 − 600 mA to show the benefit of the ratio also for high fluctuations. As a result, the SEP could be reduced from ±82.28 mg/dl to ±1.03 mg/dl.

The variation of Allura Red AC concentration in the range of 0 − 0.2 mg/l led to an accuracy of ±15.69 mg/dl utilizing *I _{t}*(

*ω*), as seen in Figure 7. The SEP could be reduced to ±1.23 mg/dl for application of $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$. To show the influence of highly absorbing media, this measurement was also performed for a tenfold increase of Allura Red AC concentration with 0 − 2 mg/l where an improvement of ±1.37 mg/dl instead of ±110.68 mg/dl could be achieved.

The glucose predictability in presence of slightly varying turbidity from 0.225 − 0.275 cm^{−1} shown in Figure 8 deteriorates to ±35.7 mg/dl when only *I _{t}*(

*ω*) is used. An improved SEP of ±1.17 mg/dl similar to the results of clear media was achieved by the application of $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$. The regression coefficient increases from 0.95889 to 0.99996, respectively. However, the detected intensity rapidly decreased for higher scattering coefficients. In the current setup the light intensity provided by the LED is not sufficient for higher turbidity. Further investigations are planned in this context.

## 5. Discussion and conclusion

We presented a method utilizing the ratio of the frequency components *I _{t}*(

*ω*) and

*I*(2

_{t}*ω*) to create a robust polarimeter which is insensitive to fluctuations of absolute light intensity. With this setup we demonstrated, that reliable glucose determination is possible in slightly scattering media. A variation of LED current from 360 mA to 440 mA at a constant glucose level of 500 mg/dl reveals a similar influence on

*I*(

_{t}*ω*) and

*I*(2

_{t}*ω*). The ratio $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ stays constant for a constant glucose concentration at any LED current.

The glucose predictability was first evaluated for a constant LED current of 400 mA. This delivered an SEP of ±1.05 mg/dl and *R*^{2} = 0.99996 using uncompensated intensity *I _{t}*(

*ω*) and ±0.92 mg/dl and

*R*

^{2}= 0.99998 for compensated $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ ratio. Under stable LED current of 400 ± 0.3 mA and no sample absorption, similar results were achieved for

*I*(

_{t}*ω*) and $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$. Under highly fluctuating LED currents from 360 to 440 mA the glucose predictability worsens to ±16.16 mg/dl and

*R*

^{2}= 0.99148 for conventional

*I*(

_{t}*ω*). The application of $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ on the same dataset leads to an improved SEP of ±1.00 mg/dl and

*R*

^{2}= 0.99998 which is similar to measurements with stable LED current. A varying Allura Red AC concentration delivers a glucose SEP of ±15.69 mg/dl and

*R*

^{2}= 0.99235 for uncompensated

*I*(

_{t}*ω*). This accuracy is improved to ±1.23 mg/dl and

*R*

_{2}= 0.99998 by using $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$.

The use of $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ instead of *I _{t}*(

*ω*) brings a great advantage with respect to varying light intensity without additional hardware expense or further measurement time in order to take reference values. The prediction error for highly fluctuating LED currents of 360 − 440 mA and sample absorption up to 16 % could be reduced by more than 99 %. Under these conditions the SEP was even better than previously reported in the literature for stationary cuvettes of ±4.34 mg/dl [4] and ±5.4 mg/dl [3], while our pathlength was five times larger which brings more sensitivity according to Eq. 2.

In this paper we demonstrated that the application of $\frac{{I}_{t}(\omega )}{{I}_{t}(2\omega )}$ introduces a significant robustness improvement with respect to fluctuations of absolute light intensity. Furthermore, we showed also the glucose predictability could be improved from ±35.7 mg/dl to ±1.17 mg/dl in the presence of varying slight turbidity. This method is especially suitable for future low cost applications to overcome the need of highly stabilized and thus expensive light sources.

However, the scattering coefficients of 0.225 − 0.275 cm^{−1} used in this initial investigation were low in comparison to other publications [6,7,9]. Further measurements and most likely additional hardware modifications have to be carried out in order to improve the system performance in this matter.

## Acknowledgments

The authors would like to thank Mrs. Johanna Schmelter and Dr. Christian Damiani for manuscript editing.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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