2.1.

Absorption and Optical Rotatory Dispersion

The effects of absorption due to water, plasma proteins, and glucose in the visible and NIR were modeled using experimental data from a number of reports.^{17, 18, 19, 20} Since data for individual plasma protein absorption dispersion could not be found, the representative plasma total protein (albumin, globulin, and fibrinogen) absorption was used. Figure 1 shows the absorption coefficients ${\mu }_a\left(\lambda \right)$ per concentration of analyte ( ${\mathrm{cm}}^{-1}{\mathrm{g}}^{-1}$ dl), given as a function of wavelength for water, total protein, and glucose. To calculate the absorption due to water, the mass displacement caused by the addition of the proteins and glucose was calculated and subtracted from the mass of the initial water. This corrected mass of water was then used to calculate the absorption due to water using the data in Fig. 1. The absorption due to the total protein and glucose was calculated simply by using the mass of each constituent and the appropriate data from Fig. 1. The total absorption coefficient was then calculated as the sum of the absorption coefficients due to each component.

Fig. 1

Absorption spectra of blood plasma proteins and glucose in the visible and NIR.

The effects of optical activity due to proteins and glucose in the visible and NIR were modeled using Drude’s equation,

Fig. 2

Optical rotatory dispersion of blood plasma proteins and glucose as given by Drude’s equation in the visible and NIR.

Table 2

Parameters of Drude’s equation for plasma proteins and glucose.

Given the concentrations of the plasma components, the absorption and optical rotation signals as a function of wavelength, ${\mu }_a\left(\lambda \right)$ and $\alpha \left(\lambda \right)$ , can be calculated using the data in Fig. 1 and Eq. 2, respectively. The full polarization state of the light before and after passing through the plasma sample can be described with a four-element Stokes vector $\mathbf{S}$ .²³ The first element $I$ represents the intensity of the light beam, the second element $Q$ represents the linear polarization at 0 and $90\mathrm{deg}$ , the third element $U$ represents the linear polarization at 45 and $135\mathrm{deg}$ , and the fourth element $V$ represents the circular polarization. Polarization effects are applied to the Stokes vector using a $4\times 4$ Mueller matrix $\mathbf{M}$ . Given the input state of polarization ${\mathbf{S}}_{\mathrm{i}}$ , the output state ${\mathbf{S}}_{\mathrm{o}}$ after passing through the clear sample is calculated as

Six wavelength-dependent signals were available to build the regression model (refer to Sec. 3) and predict the glucose concentrations in clear (nonscattering) media [ ${\mu }_a\left(\lambda \right)$ and $\alpha \left(\lambda \right)$ calculated based on the constituent concentrations, and the Stokes parameters $I_o\left(\lambda \right)$ , $Q_o\left(\lambda \right)$ , $U_o\left(\lambda \right)$ , and $V_o\left(\lambda \right)$ calculated using Eq. 4 based on the input polarization]. In an experimental situation, the measurements of the Stokes parameters would be made and ${\mu }_a\left(\lambda \right)$ and $\alpha \left(\lambda \right)$ would be calculated based on measured Stokes parameters. Using the described methods to calculate absorption and optical activity, intensity and polarization spectra were generated from $500\text{to}2000\mathrm{nm}$ in $2\text{-}\mathrm{nm}$ intervals with physiological levels of the plasma constituents, and with the optical pathlength through the plasma being $1\mathrm{cm}$ . The light incident on the sample was set to be linearly polarized horizontally $({\mathbf{S}}_{\mathrm{i}}={\left[1100\right]}^{\mathrm{T}})$ . To assess the predictive abilities of the models in an experimental situation, simulated Gaussian noise was added to each signal with an increasing standard deviation. This added uncertainty simulated both source and measurement noise.

2.2.

Scattering Monte Carlo Model

To investigate the ability to predict analyte concentrations using the combined method in scattering biological media such as tissue, a Monte Carlo model for polarized light propagation in multiply scattering media was used. This validated model,^{24, 25} available for download,²⁶ records individual photon packet polarizations in the form of Stokes vectors and includes the effects of multiple scattering, absorption, optical activity, and linear birefringence. The model tracks a large number of photons as they propagate and multiply scatter through the sample, then sums the photons to calculate their macroscopic properties of interest.

To reduce the computational time needed to generate data, simulations were run to create a lookup-table with absorption and optical activity as the independent parameters. This lookup table approach is similar to that taken in other Monte Carlo studies.²⁷ The computational time required was reduced in two ways: first, the number of simulations needed was reduced because data for each full spectrum were not required, since each spectrum could be generated from the lookup table; second, fewer photons were needed in each simulation because the lookup table could be smoothed to reduce the noise due to the statistical nature of the model. To create the lookup table, the values of absorption and optical activity were varied in the full range of blood plasma physiological levels as found from the compiled data. The changes in refractive index and the resulting effects on scattering due to the variations in plasma components were ignored in this study to reduce the parameter space of the lookup table, and thus to reduce the number of simulations required. This approximation is a reasonable first step; in fact, its inclusion, while complicating the analysis, may actually improve the ability to predict analyte concentrations due to an increased sensitivity to concentration changes in both light intensity and polarization.¹³ In addition, refractive index dispersion in all materials was ignored to reduce the required simulations. Finally, the linear birefringence was set to zero since this effect was ignored in the current study.

Simulations were run with the absorption coefficient varied between 0 and $30{\mathrm{cm}}^{-1}$ in $2\text{-}{\mathrm{cm}}^{-1}$ increments, and with the optical activity varied between 0 and $0.009\mathrm{rad}{\mathrm{cm}}^{-1}$ (0 and $0.5157\mathrm{deg}{\mathrm{cm}}^{-1}$ ) in $0.001\mathrm{rad}{\mathrm{cm}}^{-1}$ $\left(0.05730\mathrm{deg}{\mathrm{cm}}^{-1}\right)$ increments, to create a $10\times 16$ lookup table. All simulations were run with $3\times {10}^8$ photons. The sample was a $1\times 1\times 1{\mathrm{cm}}^3$ cube with a scattering coefficient of $60{\mathrm{cm}}^{-1}$ , which is somewhat lower than that of tissues; this coefficient was selected for this feasibility investigation to reduce the simulation times by reducing the number of photons required so each photon would retain a higher degree of polarization. The scattering particles in the medium were simulated to represent spherical polystyrene microspheres with a diameter of $1.4\mu \mathrm{m}$ . The refractive indices of the scattering particles and the surrounding media were 1.59 and 1.33, respectively (corresponding to polystyrene and water), giving an anisotropy $g=0.93$ for the microspheres at $633\mathrm{nm}$ . The photons exiting the sample were binned in $1{\mathrm{mm}}^2$ detection areas with an acceptance angle of $20\mathrm{deg}$ to approximate a typical experimental measurement. The forward direction (direct transmission) was chosen as the geometry of interest to compare the results with those from the clear plasma model. As in the clear media case, the light incident on the sample was set to be horizontally linearly polarized $({\mathbf{S}}_{\mathrm{i}}={\left[1100\right]}^{\mathrm{T}})$ . Spectra were then generated with the lookup table to create wavelength-dependent Stokes parameters [ $I_o\left(\lambda \right)$ , $Q_o\left(\lambda \right)$ , $U_o\left(\lambda \right)$ , and $V_o\left(\lambda \right)$ ] between 500 and $2000\mathrm{nm}$ in $2\text{-}\mathrm{nm}$ intervals using the appropriate values for optical activity and absorption at each wavelength. The orientation of polarization $\gamma \left(\lambda \right)$ as a function of wavelength was found from the output Stokes parameters as

4.1.

Clear Media

The percent error values for the clear plasma model are shown in Figs. 4, 5, 6 . They demonstrate the improved predictive abilities of the combined intensity and polarization approach. The descriptor block contained 400 spectra between 500 and $2000\mathrm{nm}$ in $2\text{-}\mathrm{nm}$ intervals, with each spectrum having a corresponding glucose concentration in the response block. For reference, a typical human blood glucose level is $0.18\mathrm{g}∕100\mathrm{ml}$ (from Table 1); to be within the generally acceptable level of error for glucose monitoring of 15%³¹ (some other reports indicate 20%³²), the predictive error should be less than $0.027\mathrm{g}∕100\mathrm{ml}$ . Figure 4 plots the results from regressing $\alpha \left(\lambda \right)$ and ${\mu }_a\left(\lambda \right)$ individually, with increasing noise added to the signals using conventional PLS, and the combined regression results using U-PLS and MB-PLS. In this case, the standard deviation of the added noise was specified as a percentage of the mean value of each intensity and polarization signal. This was done to corrupt the signals at approximately the same rate, because the signals differed considerably in magnitude. Figure 4 shows that regressing the absorption spectra ${\mu }_a\left(\lambda \right)$ produced much better predictive results (i.e., less corrupted by noise) than regressing the optical activity spectra $\alpha \left(\lambda \right)$ . This is due to the similarity in the ORD of the plasma components, as shown in Fig. 1, making them difficult for the PLS regression to separate. However, when the absorption and optical activity spectra were regressed with the combined methods (either with U-PLS or MB-PLS), there was considerable improvement over regressing the absorption spectra (or ORD) alone. Both combined methods produced an approximately 25% reduction in percent error with the maximum added noise over ${\mu }_a\left(\lambda \right)$ alone. As hypothesized, the combination of the information in both these spectra decreased the predictive error. Given the poor results of the optical activity spectra regression, the large improvement when it was combined with the absorption spectra was somewhat surprising. Evidently, enough information was contained in the optical activity spectra to complement the absorption spectra and improve the combined regression results.

Fig. 4

Percent error as a function of simulated noise for ${\mu }_a\left(\lambda \right)$ and $\alpha \left(\lambda \right)$ in clear media regressed individually and in combination using PLS, U-PLS, and MB-PLS. Points are prediction results and lines are a guide for the eye.

Fig. 5

Percent error as a function of simulated noise for $I\left(\lambda \right)$ , $Q\left(\lambda \right)$ , and $U\left(\lambda \right)$ in clear media regressed individually using PLS. Points are prediction results and lines are a guide for the eye.

Fig. 6

Percent error as a function of simulated noise for $I\left(\lambda \right)$ , $Q\left(\lambda \right)$ , and $U\left(\lambda \right)$ in clear media combined and regressed using PLS, U-PLS, and MB-PLS. Points are prediction results and lines are a guide for the eye.

To test the regression methods on more experimentally realistic signals (i.e., on results from a typical polarimeter¹³), the Stokes vectors of the output light from the clear plasma model were calculated using Eq. 4 with input horizontally and linearly polarized light $({\mathbf{S}}_{\mathrm{i}}={\left[1100\right]}^{\mathrm{T}})$ . The results from regressing the Stokes parameters individually [including the intensity $I\left(\lambda \right)$ ] using conventional PLS with increasing noise added to each parameter are shown in Fig. 5. Stokes parameter $V\left(\lambda \right)$ (circular polarization) was omitted because it contained no information in the present model. Predictions with the parameter $U\left(\lambda \right)$ produced poor results because the input light was horizontally polarized, yielding small values of $U\left(\lambda \right)$ that were rapidly corrupted by the added noise. A small change due to the rotation of the polarization from optical activity was present; however, this small change was quickly swamped by the added noise. Predictions using either $I\left(\lambda \right)$ or $Q\left(\lambda \right)$ parameters produced similar and far superior results. The predictive errors using $I\left(\lambda \right)$ and $Q\left(\lambda \right)$ were very similar since they contained nearly identical information due to the fully horizontal polarization of the input light $[I_i\left(\lambda \right)=Q_i\left(\lambda \right)]$ . Stokes parameter $Q\left(\lambda \right)$ was attenuated by the absorption and slightly reduced while the optical activity while the polarization vector was rotated. In this case, even though $Q\left(\lambda \right)$ was affected by both absorption and optical activity, there was no improvement gained over $I\left(\lambda \right)$ , which contained only absorption information. This suggests that even though $Q\left(\lambda \right)$ was affected by both absorption and optical activity, it was difficult for PLS to decouple the effects.

The results from combined regression methods for the Stokes parameters are shown in Fig. 6, along with the result for the regression of $I\left(\lambda \right)$ alone (from Fig. 5) for comparison. Both U-PLS and MB-PLS produced lower predictive errors than regressing $I\left(\lambda \right)$ alone. In both cases, combining either $I\left(\lambda \right)$ and $Q\left(\lambda \right)$ or $I\left(\lambda \right)$ , $Q\left(\lambda \right)$ , and $U\left(\lambda \right)$ resulted in similar predictive errors, yielding little additional information in the $U\left(\lambda \right)$ signal as discussed above. However, MB-PLS produced significantly lower predictive errors than U-PLS. The reductions in percent error for U-PLS and MB-PLS were approximately 8% and 15%, respectively, at the maximum noise level. These results differed from the results of predicting using $\alpha \left(\lambda \right)$ and ${\mu }_a\left(\lambda \right)$ , where U-PLS and MB-PLS prediction accuracy were nearly the same. Evidently, the creation of the consensus super-block in MB-PLS by combining the common information in each descriptor block leads to better predictive power in the case of the Stokes parameters. This is likely due to the mix of absorption and optical activity information contained in Stokes parameters $Q\left(\lambda \right)$ and $U\left(\lambda \right)$ as opposed to the pure optical activity and absorption channels $\alpha \left(\lambda \right)$ and ${\mu }_a\left(\lambda \right)$ . This mixing may make it difficult for the regression methods to decouple the effects when building the predictive models. These results suggest that regressing absorption combined with optical rotation, rather than the measured Stokes parameters, provides better predictive abilities [an improvement of approximately 25% for combined $\alpha \left(\lambda \right)$ and ${\mu }_a\left(\lambda \right)$ versus an improvement of approximately 15% for the combined Stokes parameters]. However, when optical rotation is calculated based on measured Stokes parameters, the magnitude of noise will increase as the parameters are divided in Eq. 5. Therefore, as shown later, the potential improvement is diminished. To summarize, it is apparent from these results that combining the information contained in the intensity and polarization spectra improves the predictive abilities for glucose in simulated clear media.

4.2.

Scattering Media

The results of predictive calculations using the data generated by the Monte Carlo model for scattering media are shown in Figs. 7, 8, 9 . Similar to the clear media, the descriptor block contained 400 spectra between 500 and $2000\mathrm{nm}$ in $2\text{-}\mathrm{nm}$ intervals with each spectrum having a corresponding glucose concentration in the response block. As also was the case for the clear media, a reduction in predictive error was achieved by combining the intensity and polarization signals. Figure 7 plots the percent error for the Stokes parameters $I\left(\lambda \right)$ , $Q\left(\lambda \right)$ , $U\left(\lambda \right)$ , and $V\left(\lambda \right)$ , plus the orientation of the polarization $\gamma \left(\lambda \right)$ calculated using Eq. 5, with each regressed individually. In contrast to Fig. 5, the results for the parameter $V\left(\lambda \right)$ are now shown, since scattering produced some transfer of linear-to-circular polarization [giving a nonzero value of $V\left(\lambda \right)$ ] and therefore some information is now contained in $V\left(\lambda \right)$ . As expected, when using single information channels, regression using $I\left(\lambda \right)$ produced the lowest predictive error. The other Stokes parameters and $\gamma \left(\lambda \right)$ suffered (to various extents) in predictive ability as a result of the depolarizing effects of multiple scattering. This depolarizing reduced the values of the Stokes parameters, leading to large corruption, relative to $I\left(\lambda \right)$ , by the added simulated noise. The fraction of retained polarization for the incident fully linear (horizontally) polarized light was approximately 50% after propagating through the media. As a result, the values of the Stokes parameters $Q\left(\lambda \right)$ , $U\left(\lambda \right)$ , and $V\left(\lambda \right)$ were significantly reduced. Because the incident light was horizontally polarized, $Q\left(\lambda \right)$ yielded much lower predictive errors than either $U\left(\lambda \right)$ , $V\left(\lambda \right)$ , or $\gamma \left(\lambda \right)$ , although the percent error was still higher than that for $I\left(\lambda \right)$ due to depolarization.

Fig. 7

Percent error as a function of simulated noise for $I\left(\lambda \right)$ , $Q\left(\lambda \right)$ , $U\left(\lambda \right)$ , $V\left(\lambda \right)$ , and $\gamma \left(\lambda \right)$ in scattering media regressed individually using PLS. Points are prediction results and lines are a guide for the eye.

Fig. 8

Percent error as a function of simulated noise for $I\left(\lambda \right)$ , $Q\left(\lambda \right)$ , $U\left(\lambda \right)$ , and $\gamma \left(\lambda \right)$ in scattering media combined and regressed using PLS and U-PLS. Points are prediction results and lines are a guide for the eye. Note that the points overlap for $I\left(\lambda \right)$ regressed using PLS and $I\left(\lambda \right)$ and $\gamma \left(\lambda \right)$ combined and regressed using U-PLS.

Fig. 9

Percent error as a function of simulated noise for $I\left(\lambda \right)$ , $Q\left(\lambda \right)$ , $U\left(\lambda \right)$ , and $\gamma \left(\lambda \right)$ in scattering media combined and regressed using PLS and MB-PLS. Points are prediction results and lines are a guide for the eye.

The predictive errors for the combined regression methods are shown in Figs. 8 and 9, along with the results for the regression of $I\left(\lambda \right)$ alone for comparison (from Fig. 6). In addition to the combinations of the Stokes vectors, the orientation of the polarization vector $\gamma \left(\lambda \right)$ as calculated by Eq. 5 was also combined and regressed with $I\left(\lambda \right)$ using U-PLS and MB-PLS. All the combined methods with Stokes parameters improved the predictive power; however, the improvement was more modest than that for clear media calculations. In addition, little difference was observed between the percent errors for U-PLS and MB-PLS, as was seen for the clear media, with the overall reduction in error being approximately 8% for all combined Stokes parameter prediction methods. Also, the combination of $I\left(\lambda \right)$ and $\gamma \left(\lambda \right)$ through U-PLS or MB-PLS produced little or no improvement. In fact, the predictions when using U-PLS shown in Fig. 8 overlapped with predictions when using $I\left(\lambda \right)$ alone. This is likely due to an increase in noise when calculating $\gamma \left(\lambda \right)$ through Eq. 5, which led to a decrease in predictive power [i.e., the effective error on $\gamma \left(\lambda \right)$ was approximately two times that for the other parameters]. As previously discussed, the lower improvements are due to the depolarizing effects of multiple scattering and the subsequent reduction in the Stokes parameters $Q\left(\lambda \right)$ , $U\left(\lambda \right)$ , and $V\left(\lambda \right)$ . However, even with the lower improvement for the combined approaches in scattering media, the results still demonstrate the potential for the method. A significant improvement in prediction is still achieved with the combination of intensity and polarization information from multiply scattering media. Future work will further investigate the influence of scattering in an effort to determine at what level of scattering an improvement is still realized. Since this rather large reduction in improvement has already been observed from clear media to a scattering coefficient of $60{\mathrm{cm}}^{-1}$ , it would seem unlikely that predictions with more realistic tissue-scattering values $(\sim 100{\mathrm{cm}}^{-1})$ would exhibit significant improvement. However, this is yet to be determined, and smoothing or other preprocessing steps could be used to reduce the signal noise. The present results demonstrate the potential for the methodology in clear or moderately scattering biological fluids such as blood plasma, or extra-cellular fluid and tissues such as the aqueous humor of the eye.

Four additional areas have been identified for further investigation and will be explored in future work. First, the effects of refractive index matching due to variations in constituents¹³ will be added to the model to investigate how these effects influence the predictive abilities of the regression techniques. The changes in the refractive index and resulting changes in the scattering properties of the media will influence both the intensity and polarization of the light propagating through the media and may add some additional sensitivity to the constituent concentration changes. It is possible that this may improve the predictive power of this methodology. Second, the influence of other common biological optical effects such as linear birefringence²⁵ on the abilities of the regression methods will also be investigated. Linear birefringence has the effect of transferring the polarization state from linear to circular or vice versa, and will interfere with the effects of optical activity. The extent to which this influences the abilities of the methodology will be investigated by employing a method of polar Mueller matrix decomposition to decouple the two effects.^{33, 34} This will be done in both clear and scattering media with increasing scattering coefficients. Third, the effect of detection geometry and wavelength range will also be investigated to determine the optimal experimental parameters for the measurement of intensity and polarization signals.^{11, 13} This will involve both simulations and experimental work. Fourth, experimental testing must be completed to validate the combining methodology, because the current results are for simulated measurements only. This will involve the design, construction, and testing of a combined spectral intensity and polarimetry system suitable for experimental measurements.