2.1.

Experimental System

A supercontinuum source (SC46, YSL photonics, China) with a broadband from 600 to 1700 nm is used as light source in the custom-built NIR spectrometer. As shown in Fig. 1, the system is equipped with a noncollinear acousto-optic tunable filter (TEAF10-1.0-1.8-S, Brimrose Company, Maryland) as the light splitting element to provide a spectral response range from 1100 to 1400 nm (9091 to $7143{\mathrm{cm}}^{-1}$) with a resolution of 2 nm, an InGaAs infrared detector (G12181-210K, Hamamatsu Photonics K.K., Japan), and a 16-bit data acquisition board (PCI-6220, National Instrument Inc.). The fiber used in the experiment is multimode fiber, its core diameter is $200\mu \mathrm{m}$, and its numerical aperture is 0.22.

Fig. 1

Schematic illustration of the measurement system.

2.2.

Samples

NIR diffuse reflection spectra were recorded for the scattering media of 10% Intralipid solution^20,21 with different glucose concentrations, which are from 0 to $2000\mathrm{mg}/\mathrm{dL}$ with an interval of $200\mathrm{mg}/\mathrm{dL}$.

2.3.

Measurements

Twenty-two detection distances of SD separations were set from 0.47 to 3.095 mm with an interval of 0.125 mm. The surrounding temperature was controlled at 25°C. All samples were measured 5 times within 48 h, and the samples were tested in a random order every time. The samples were placed on a magnetic plate and stirred for 1 min and then allowed to settle for 3 min before recording. The test time was arranged to start at 9:16 am and 9:16 pm of the first day, 9:15 am and 9:15 pm of the second day, and 9:16 am of the third day. We call the five times tests first, second, third, fourth, fifth tests, respectively. Measuring all SD separations and wavelengths for one sample took $\sim 50\mathrm{s}$ and all 11 samples took 10 min.

Every separate test from the five tests can be considered as a short-term test, where the $R^{\prime }$ can demonstrate $R_{\mathrm{G}}^{\prime }$ caused by glucose. But between the five tests, which lasted for 48 h, the $R^{\prime }$ for a same sample can demonstrate $R_{\mathrm{D}}^{\prime }$ caused by light source drift.

2.4.

Defining Relative Changes in Diffuse Reflection Light Intensity

Figure 2 shows the diffused light in a scattering medium, where $I_0$ is the incident light intensity, and $I$ is the diffuse reflection light intensity.

Fig. 2

The schematic of the diffused light in scattering medium.

The relative variation of diffuse reflection light is defined as $R^{\prime }$ in Eq. (1), where $I$ is the diffuse reflection light intensity and $\mathrm{\Delta }I$ is the variation of $I$ as referenced to an initial state of the scattering medium. $I_r$ denotes the diffuse light intensity for reference. $\mathrm{\Delta }I$ can be induced by glucose or disturbances

2.5.

Linearity of Relative Changes in Diffuse Reflection Light Intensity

Using diffuse equation (DE), i.e., the first-order approximate radiative transfer equation (P1-RTE),²² and Monte-Carlo (MC) simulation,²³ the linearity of $R^{\prime }$ can be demonstrated using their results. The diffused light intensity can be calculated using Eq. (2), which is solved for a semi-infinite medium by setting an extrapolative boundary

In the glucose measurement, $R_{\mathrm{G}}^{\prime }$ can be calculated after the diffuse light intensity values with different glucose concentrations are estimated by using Eq. (2) or an MC stimulation program.

The 10% Intralipid solution’s parameters²⁴ ${\mu }_{\mathrm{a}}$, ${\mu }_{\mathrm{s}}$, $g$, and $n$ used in the equation and MC simulation are shown in Table 1. The variations of the parameters caused by glucose can be determined according to the equations in Kohl’s papers.^25,26 The variation of the absorption coefficient induced by the glucose concentration can be expressed as²⁵ $\mathrm{\Delta }{\mu }_{\mathrm{a}}=({ϵ}_{\mathrm{g}}-f_{\mathrm{w}}^{\mathrm{g}}·{ϵ}_{\mathrm{w}})\mathrm{\Delta }{c}_{\mathrm{g}}$, where ${ϵ}_{\mathrm{g}}$ and ${ϵ}_{\mathrm{w}}$ are the molar extinction coefficients of glucose and water, respectively, and $f_{\mathrm{w}}^{\mathrm{g}}$ is the displacement factor between glucose and water. The variations of the refractive index $n$ and anisotropic factor $g$ induced by the glucose concentration are expressed as²⁵ $\mathrm{\Delta }n=2.5\times {10}^{-5}\mathrm{\Delta }{c}_{\mathrm{g}}$, $\mathrm{\Delta }g=8.45\times {10}^{-6}\mathrm{\Delta }{c}_{\mathrm{g}}$. For Intralipid solutions, the variation of the scattering coefficient ${\mu }_s$ induced by the glucose concentration is²⁶ $\mathrm{\Delta }{\mu }_{\mathrm{s}}=-2m{\mu }_{\mathrm{s}}\mathrm{\Delta }{c}_{\mathrm{g}}$, where $m=(1.569\times 10-5\lambda +0.001)/100$, and $\lambda$ is the wavelength (nm).

Table 1

The parameters of 10% Intralipid used in P1-RTE and MC at 1100 to 1400 nm.

Figures 3 and 4 show the results of $R_{\mathrm{G}}^{\prime }$ for the 1100-nm diffuse light from the SD separations of 0 to 3.5 mm.

Fig. 3

The result of $R_{\mathrm{G}}^{\prime }$ calculated using P1-RTE for the medium of 10% Intralipid solution at 1100 nm when glucose concentration varies between in 200 and $2000\mathrm{mg}/\mathrm{dL}$ and the solution without glucose is the reference. The FRP locates at 1.6 mm.

Fig. 4

The result of $R_{\mathrm{G}}^{\prime }$ calculated using MC simulation for the medium of 10% Intralipid solution at 1100 nm when glucose concentration varies between in 200 and $2000\mathrm{mg}/\mathrm{dL}$ and the solution without glucose is the reference. The FRP locates at 0.9 mm.

The $R_{\mathrm{G}}^{\prime }$ in Figs. 3 and 4 can show an approximate linearity to SD separation. Glucose would induce a simultaneous change in ${\mu }_{\mathrm{a}}$ and ${\mu }_{\mathrm{s}}^{\prime }$ of the tested medium, and the two factors would induce an approximately linear change in $R^{\prime }$ to SD separation. The linearity caused by glucose could be a comprehensive result.

It also can be found that the diffuse light intensity around the FRP would be relative stable and insensitive to the change of glucose concentration. The values of FRP appear different in Figs. 3 and 4 because of the differences of the two theoretical practices using DE and MC simulations.

3.1.

Result of $R_G^{\prime }$ in One Short-Term Test

The FRP is found out by the phantom experiment. The relative variations $R_{\mathrm{G}}^{\prime }$ were calculated for the samples with the glucose concentration of 200 to $2000\mathrm{mg}/\mathrm{dL}$ using Eq. (1) when the $I_{\mathrm{r}}$ is from the sample without glucose. Figure 5 shows some typical results of $R_{\mathrm{G}}^{\prime }$.

Fig. 5

The results of $R_{\mathrm{G}}^{\prime }$ for the first test in the medium of 10% Intralipid solution when glucose concentration varies between in 400 and $2000\mathrm{mg}/\mathrm{dL}$ with the interval of $400\mathrm{mg}/\mathrm{dL}$ and the solution without glucose is the reference at 1100 nm. The FRP locates at $\sim 1.9\mathrm{mm}$.

Figure 6 shows the FRPs of 1100 to 1400 nm by MC when ${\mu }_{\mathrm{s}}$ changes $\pm 5\%$. The location of FRP is relatively stable when the scattering properties of the medium change a little. The FRPs of MC are different with the experiments because MC is based on the assumption of a semi-infinite medium, while it is actually closer to an infinite medium in the experiments.

Fig.6

FRP of 1100 to 1400 nm by MC when ${\mu }_{\mathrm{s}}$ changes $\pm 5\%$.

3.2.

Result of $R_D^{\prime }$ with Instrument Drift During 48 h

Using the diffuse reflection light intensity results from the first test as the $I_{\mathrm{r}}$ value, the $R_{\mathrm{D}}^{\prime }$ results can be acquired for all samples to demonstrate their drifts in the second to fifth tests. For every sample, the $R_{\mathrm{D}}^{\prime }$ primarily shows the influence from the instrument drift.

Some typical $R_{\mathrm{D}}^{\prime }$ results are shown in Fig. 7 and the $R_{\mathrm{D}}^{\prime }$ from different SD separations appear approximately similar. It can, therefore, be assumed that there is an equivalent $R_{\mathrm{D}}^{\prime }$ for an arbitrary pair of SD separations of 0.47 to 3.095 mm as shown in

Fig. 7

The result of $R_{\mathrm{D}}^{\prime }$ calculated at 1100, 1180, 1260, and 1340 nm for the second to fifth test in the medium of 10% Intralipid solution as the first test is for reference: (a) 1100 nm, (b) 1180 nm, (c) 1260 nm, and (d) 1340 nm.

Based on Eq. (3), the light source intensity drift can be reduced by differential calculation between the $R_{\rho }^{\prime }$ of two SD separations. As shown in

The glucose-related information in $\mathrm{d}{R}_{\mathrm{G}}^{\prime }$ after differential calculation is a combination of the glucose information from ${\rho }_1$ and ${\rho }_2$. If one of the SD separations ${\rho }_2$ is the FRP, $R_{\mathrm{G}\&D,{\rho }_2}^{\prime }$ does not contain any glucose information, i.e., $R_{\mathrm{G}\&D,{\rho }_2}^{\prime }\approx R_{\mathrm{G},{\rho }_2}^{\prime }$. It can be derived that $\mathrm{d}{R}_{\mathrm{G},{\rho }_1-{\rho }_2}^{\prime }$ would be the corrected $R_{\mathrm{G}}^{\prime }$ at ${\rho }_1$, i.e., $\mathrm{d}{R}_{\mathrm{G},{\rho }_1-{\rho }_2}^{\prime }\approx R_{\mathrm{G},{\rho }_1}^{\prime }$. So, the correction can be described as

$\rho$ is any SD separation where the spectral data need to be corrected.

Equation (5) can also be described as

Therefore, using Eq. (6), the change of light intensity at any position ($\mathrm{\Delta }{I}_{\rho }$) can be estimated by the drift at the FRP ($\mathrm{\Delta }{I}_{\mathrm{FRP}}$) when it takes the FRP as the reference position.

It should be noted that the characteristics of the synchronic drift might not occur with weaker diffuse light signals, which are reflected at the distances far from the light source. For the wavelengths where the medium has stronger absorption, the available SD separations would be very small.

3.3.

Correcting based on the FRP

From Eq. (3), the FRP can directly indicate the drift $R_{\mathrm{D}}^{\prime }$ even for all available positions. When the measurement is only disturbed by the drift of incident light intensity or glucose concentration variation, the drifted $R_{\mathrm{G}}^{\prime }$ can be corrected for all SD separations when the $R_{\mathrm{G}\&D}^{\prime }$ deducts the drift $R_{\mathrm{D}}^{\prime }$ at ${\rho }_{\mathrm{FRP}}$, as shown in Eq. (5).

Figure 8(a) shows the $R_{\mathrm{G}\&D}^{\prime }$ results, including $R_{\mathrm{G}}^{\prime }$ and $R_{\mathrm{D}}^{\prime }$, for $1000\mathrm{mg}/\mathrm{dL}$ glucose solution samples referencing to the solution without glucose. The corrected $R_{\mathrm{G}}^{\prime }$ result is shown in Fig. 8(b).

Fig. 8

$R^{\prime }$ results of the all tests at 1100 nm for the 10% Intralipid solution with $1000\mathrm{mg}/\mathrm{dL}$ glucose concentration: (a) $R_{\mathrm{G}\&D}^{\prime }$ before correction and (b) $R_{\mathrm{G}}^{\prime }$ after correction.

Using the correction method based on FRP, the $R^{\prime }$ results from all samples and all tests are corrected. Figure 9 shows the correction results for some typical samples in the last test, which started 48 h after the first test. And all the samples’ $R^{\prime }$ are calculated using Eq. (1), in that the $I_r$ is the reference light intensity measured in the solution without glucose.

Fig. 9

The $R^{\prime }$ results of the fifth test at 1100 nm for the 10% Intralipid solution with glucose concentration of 400 to $2000\mathrm{mg}/\mathrm{dL}$ with the interval of $400\mathrm{mg}/\mathrm{dL}$: (a) $R_{\mathrm{G}\&D}^{\prime }$ before correction and (b) $R_{\mathrm{G}}^{\prime }$ after correction.

According to Eq. (4), the light source drift can be reduced by differential calculation, $\mathrm{d}{R}_{\mathrm{G},{\rho }_1-{\rho }_2}^{\prime }=R_{\mathrm{G}\&D,{\rho }_1}^{\prime }-R_{\mathrm{G}\&D,{\rho }_2}^{\prime }$. But the result related to the two positions $\mathrm{d}{R}_{\mathrm{G},{\rho }_1-{\rho }_2}^{\prime }$, will be different from any result of a single position. Figure 10 shows two results after correction. As can be seen, they are different from the short-term test shown in Fig. 5.

Fig. 10

The $\mathrm{d}{R}_{\mathrm{G},{\rho }_1-{\rho }_2}^{\prime }$ results of the fifth test at 1100 nm for the 10% Intralipid solution with glucose concentration of 400 to $2000\mathrm{mg}/\mathrm{dL}$ with the interval of $400\mathrm{mg}/\mathrm{dL}$: (a) $\mathrm{d}{R}_{\mathrm{G},{\rho }_1-{\rho }_2}^{\prime }$ of ${\rho }_2=1.0\mathrm{mm}$ and (b) $\mathrm{d}{R}_{\mathrm{G},{\rho }_1-{\rho }_2}^{\prime }$ of ${\rho }_2=2.5\mathrm{mm}$.

In Fig. 10, the horizontal axis shows that ${\rho }_1$, and ${\rho }_2$ are the reference positions. In Fig. 10(a), ${\rho }_2$ is the near light source, ${\rho }_2=1.0\mathrm{mm}$, and FRP is at 1.9 mm. In Fig. 10(b), ${\rho }_2$ is far from the light source, ${\rho }_2=2.5\mathrm{mm}$. As can be seen, choosing ${\rho }_2$ far from the light source does not contribute to a better result due to the increase of noise after differential calculation, although the glucose information increases at the same time. The selection of ${\rho }_1$ and ${\rho }_2$ could deeply influence the result of absolute change of light intensity $\mathrm{d}{I}_{\mathrm{G},{\rho }_1-{\rho }_2}^{\prime }$. The detection limit can be evaluated by the comparison of $\mathrm{d}{I}_{\mathrm{G},{\rho }_1-{\rho }_2}^{\prime }$ and the level of noise.

3.4.

Correction Results of all Spectra at 1100 to 1400 nm

The diffuse reflection spectra in the wavelength ranging from 1100 to 1400 nm are corrected using the reference position of FRP. Similarly, the absolute light intensity can be corrected using the corrected $R^{\prime }$ and its initial value. Figure 11 shows the absolute light intensity results for $1000\mathrm{mg}/\mathrm{dL}$ from 1100 to 1400 nm before and after correction for the SD separation of 0.595 mm. The coefficients of variation (CVs) of the spectra from the original and corrected results are compared in Fig. 12. It is clear that the CVs improved from 0.01 to 0.025 before correction to less than 0.004 after correction.

Fig. 11

Diffuse reflection spectra of the 10% Intralipid solution with $1000\mathrm{mg}/\mathrm{dL}$ glucose concentration before and after correction at SD separation of 0.595 mm: (a) before correction and (b) after correction.

Fig. 12

Comparison of CVs of diffuse reflection spectra of the 10% Intralipid solution with $1000\mathrm{mg}/\mathrm{dL}$ glucose concentration before and after correction at SD separation of 0.595 mm.

3.5.

Modeling and Predicting Glucose Concentrations Using Corrected Spectra

Partial least square and leave-one-out cross validation are used for modeling and predicting the glucose concentrations in 10% Intralipid. It is necessary to use the spectra from those positions, which are far away from the reference FRP, because their signals should be sensitive to glucose. For the wavelengths of 1100 to 1400 nm, the FRPs distribute between 1.2 and 2.0 mm. Therefore, the SD separations of 1.2 to 2.0 mm are not available. The data from greater than 3.2 mm could not be used due to the weak signals at some wavelengths. By randomly choosing spectra from the five tests for every sample, we can obtain a set of data to test the prediction with the disturbance of instrument drift. Using these data, the prediction has an unsatisfactory root mean square error of cross validation (RMSECV) result in the cross validation, as shown in the first RMSECV column result of Table 2. When all of the spectra are corrected using our approach, the RMSECV values are closer to the best values obtained from the spectra recorded in one test, which takes $\sim 10\min$ and can be called “short-term monitoring.” The average RMSECV values of the five short-term tests are listed in the third RMSECV column of Table 2. Moreover, the SD separations, which are near the FRP, should not be used for the glucose measurement, since there the signals got a worse prediction result due to being insensitive to the glucose concentration’s variation.

Table 2

RMSECV results of the prediction using the reference position of FRP.

Table 3 shows the RMSECVs of prediction after differential measurement between two SD separations. Choose ${\rho }_1=0.595$ as the detection position, and the referenced positions are the SD separations that are less than FRP, near FRP, and greater than FRP, respectively. For the wavelengths of 1100 to 1400 nm, FRP is between 1.2 and 2.0 mm (based on short-term experiment). The result shows that if ${\rho }_2$ is less than FRP, the differential result could decrease the glucose-related information because the two positions are too close to each other, and the prediction error is bigger than the result only using one position of ${\rho }_1$. If ${\rho }_2$ is near to FRP, glucose information would be lost less and the RMSECVs using two positions are similar to the result using only one position of ${\rho }_1$. For the ${\rho }_2$ far away from the FRP, the noise from ${\rho }_2$ greatly increases and the RMSECVs decrease after differential of the two positions.

Table 3

RMSECV results of the prediction using other positions as reference.

In Tables 2 and 3, the prediction of glucose after using the reference position of the accurate FRPs or of the positions that are near to FRP show good results, which is closer to the result from the short-term test. This is because the spectral data collected near the FRP carry little glucose-related information, so the corrected $R^{\prime }$ may reserve most original glucose information at ${\rho }_1$. The prediction accuracy of selecting accurate FRPs or the positions near FRP as reference are similar. In real use, we suggest the FRPs of the subject should be measured first and then select the position that is easy to measure near the FRP. But the correction results using an accurate FRP have advantages of building a model because the corrected spectra and original spectra can be used for mutual prediction.