Many reported noninvasive glucose experiments follow a procedure where spectra are collected periodically during an oral glucose tolerance test. The corresponding glucose transient is quantified using reference glucose concentrations measured in a series of blood samples. Glucose concentrations are assigned to each spectrum through interpolation. Generally, a time lag is added to account for dynamics between the concentration of glucose in blood and interstitial fluid. This lag time is a consequence of the fact that the noninvasive spectrum is obtained primarily through interstitial fluid rather than blood, and some time is required for glucose transport from the blood to the tissue. In many cases, different lag times are tried until the best results are obtained.
The resulting noninvasive spectra are combined with the assigned glucose concentrations to prepare a calibration model that predicts in vivo glucose concentrations. Procedurally, the full set of spectral data is randomly divided into two groups, where one group is used to generate the calibration model and the second group is used to validate model accuracy. This first group is called calibration data and is used to establish a calibration vector that represents the best correlation between spectral variance and glucose concentration predictions. The predicted concentration of glucose is obtained as the inner product between the sample spectrum and this calibration vector. The primary objective is to generate a calibration vector that provides accurate and robust predictions of glucose concentration from noninvasive spectra collected subsequently to the calibration data.
Statistical methods, such as partial least-squares (PLS) regression or artificial neural networks, are typically used to generate the calibration vector for a targeted analyte. In these methods, variations within spectral data are statistically correlated to changes in glucose concentration. The working assumption is that the spectral variations that correlate most strongly with changes in concentration can be used to predict glucose concentrations in spectra generated from subsequent samples. It is important to understand, however, that there is no conditional requirement that the principal spectral variance originates from the molecular structure of glucose. Indeed, these algorithms do not require molecule specificity, only statistical significance in the correlations. By design, these algorithms can discern tiny correlations within these data sets, even if the correlation originates from ambiguous sources, such as noise or spectral drift.
The efficiency of the PLS algorithm to establish correlations between spectral variance and analyte concentration makes this a popular method for generating calibration vectors for noninvasive measurements. Unfortunately, the blind acceptance of results of this algorithm can result in calibration models that are based on drifts, noise, and spurious correlations. Powerful insight can be gained by understanding the origin of the information embedded within the calibration vector.
PLS Calibration Model from Noninvasive Rat Spectra
A calibration model based on the PLS algorithm was developed from noninvasive spectra collected from the animal model. In this analysis, 360 noninvasive spectra were randomly split with 306 spectra (85%) used as calibration data and the remaining 54 spectra (15%) used for model validation (prediction data). The PLS calibration vector was determined over a spectral range of 4900–4200 cm-1
with nine PLS factors. This number of factors was deemed optimal by comparing the standard error of cross validation (SECV) as a function of number of factors for 50 repeated leave one-third-out cross validation calculations.8
SECV was calculated according to the following equation for each repeat of the one-third left-out splitting of calibration data and for each number of factors:
is the assigned glucose concentration for spectrum i
in the one-third left-out spectra, i
is the corresponding concentration of glucose predicted from the calibration model for spectrum i
, and nCV
is the number of spectra in the one-third left-out cross validation, which is 102 spectra in this case. A lower SECV indicates greater model accuracy as the difference between assigned and model-predicted glucose concentrations is smaller.
shows the impact of number of factors on model performance. As is typical of PLS calibration models, the standard error decreases to a minimum as more factors are used in the analysis. For these data, the minimum is reached at 12 factors. An F
-test statistical analysis is used to determine the fewest number of factors for which the SECV is not significantly different than the minimum at the 90% confidence level. For these data, the optimum number of factors is nine. The calibration data set has a sufficient number of independent spectra to support this number of factors. In general, six independent spectra are needed to justify each factor in the calibration on the basis of degrees of freedom.10
In this experiment, calibration data consist of 204 spectra (306 – 102) and, therefore, can accommodate a maximum of 34 factors.
Effect of number of factors used in the PLS calibration model on model performance, showing minimum standard errors with 12 factors.
The final parameter to be established for the calibration model is the lag time between arterial blood glucose measurements and assigned tissue levels of glucose for each spectrum. In this work, a 15-minute lag time was applied to the arterial glucose readings, which is consistent with values measured with implantable biosensors.11
Results of the PLS calibration model are presented in
in the form of a glucose–time profile. In this plot, the predicted concentrations of glucose are plotted as a function of time where circles represent glucose concentrations predicted from spectra used in calibration data and triangles represent concentrations obtained from prediction spectra. The solid black line represents the estimated tissue glucose levels, which correspond to the measured blood glucose concentrations shifted 15 minutes to the right. The standard error of prediction (SEP) for prediction data points is 1.98 mM
across all predictions, where SEP is calculated according to the following equation:
is the assigned glucose concentration for spectrum i
in the prediction data set, i
is the corresponding concentration of glucose predicted from the calibration model for spectrum i
, and np
is the number of spectra in the prediction data set (54 in this experiment).
Glucose–time plot for PLS calibration model showing calibration points (blue circles), prediction points (red triangles), and arterial blood glucose concentrations with a 15-minute lag time (black line).
A noteworthy aspect of the time profile in
is the large scattering of points just before the glucose transient. This scattering of points corresponds to the period between 1300 and 1345 hours when the fiber-optic interface was purposely removed and resecured before collecting each spectrum. Resetting the interface before each spectrum creates a significant amount of variance, which is likely caused by slight differences in various components of the tissue matrix within the optical region. Changes in the amount of water, protein, fat, and scattering properties of skin are evident in spectra collected during this period as characterized elsewhere.9
The large variations associated with resetting the fiber-optic interface contributes to the large SEP (1.98 mM
) noted earlier for this calibration model.
PLS Calibration Model with Incorrect Glucose Concentration Assignments
An interesting exercise is to repeat the PLS analysis of noninvasive spectral data after purposely assigning incorrect glucose concentrations to each spectrum. Incorrect glucose concentrations should result in poor models and an inability to predict glucose concentrations accurately. Two different types of incorrect glucose assignments are used to illustrate the effectiveness of this approach.
First, glucose concentrations are assigned randomly to spectra over a concentration range of 5 to 35 mM and the PLS calibration model is regenerated by the same procedures described earlier. Results are presented in for the resulting PLS model. These results are presented as a concentration correlation plot where the concentration of glucose predicted from the PLS model is plotted as a function of the randomly assigned glucose concentration. No systematic correlation is apparent in this plot and the SEP is 13.3 mM, which is essentially the same as the standard deviation of the randomly assigned glucose concentrations. These results indicate that this PLS model has no ability to predict glucose, as expected for incorrect glucose assignments.
Concentration correlation plot for a PLS calibration model generated from noninvasive spectra with randomly assigned incorrect glucose concentrations showing calibration points (blue) and prediction points (red).
The second type of incorrect glucose assignment studied here is to shift the glucose transient forward by 2 hours. Instead of the initial transient starting at 1400 hours, glucose concentrations are assigned as if the transient had started at 1200 hours. Basically, the entire time profile is shifted forward by 2 hours and reference concentration values obtained before 1300 are wrapped around to the end of the transient. In this experiment, the blood glucose concentration actually measured for the arterial blood sample collected at 1300 hours is assigned incorrectly to the noninvasive spectrum collected at 1100 hours.
Results from the corresponding PLS calibration model are presented in
. This incorrect PLS model uses 13 factors and appears to provide excellent predictions of glucose concentrations for both calibration and prediction data points. The computed SEP is only 3.2 mM
. Furthermore, a Clarke error grid analysis12
of these model predictions (plot not shown) reveals that 99% of the model results (calibration and prediction data combined) fall within the acceptable A–B regions, whereas only 1% of these predictions falls in the C region and none of the observations falls in the D or E regions.
Figure 5. Glucose–time plot for a PLS calibration model generated from nonrandom but incorrectly assigned glucose concentrations showing calibration points (blue circles), prediction points (red triangles), and arterial blood glucose concentrations incorrectly (more ...)
Results from the Clarke error grid analysis and the ability to closely track the glucose transient rival or surpass published accounts of alleged noninvasive glucose measurements. The problem is, of course, the values reported by this model are invalid. Because the glucose assignments are incorrect, the resulting calibration model cannot possibly be predicting glucose concentrations on the basis of glucose-specific information.
It is noteworthy that random glucose assignments did not produce an apparently functional calibration. The time dependency of this second type of incorrect assignment produces temporal correlations within spectral data that correlate to the time-dependent variations of the glucose concentrations, thereby providing a basis for the PLS regression model and an apparently functional calibration.
Most importantly, the false calibration model represented in calls into question the validity of the calibration model with the correct glucose assignments (). Without further analysis, it is impossible to conclude that glucose is being measured in owing to the known incorrectness of the model represented in . Most literature reports that claim successful noninvasive glucose measurements are subject to this criticism because no additional proof is provided to corroborate findings from the PLS model alone.
Analysis of Calibration Vectors
The calibration vector is the final product of the PLS algorithm and represents the component of sample spectra that best predicts analyte concentrations. The inner product of the calibration vector and a sample spectrum produces a scalar quantity that represents the predicted concentration of the targeted analyte in the sample. Both the shape and the magnitude of the calibration vector are important for measurement accuracy. The shape defines measurement selectivity and the magnitude defines measurement sensitivity.
A calibration vector can be thought of as a filter that passes information related to the analyte while suppressing signals originating from interferences. In the simplest case where the analyte and interferences have nonoverlapping signals, the shape of the calibration vector will match the spectrum of the analyte. In near-infrared spectroscopy, however, the analyte signal generally overlaps with signals from the matrix components and the resulting shape of the calibration vector is considerably more complex. The filtering properties of a calibration vector with over-lapping spectra must maximize the amount of analyte information that passes while minimizing the passage of interfering signals. The impact of an overlapping and interfering signal can only be minimized by dividing the interfering signal in half, integrating each half separately, and then subtracting these two areas. Using this strategy, the final integrated signal from an interfering signal sums to zero regardless of the concentration of the interfering substance. In practice, this process to remove overlapping signals is accomplished with a combination of positive and negative components to the calibration vector.
Examples of calibration vectors are presented in where shows the calibration vector for the PLS calibration model based on correct glucose assignments and presents the calibration vector for the model based on the nonrandom incorrect glucose assignments described earlier. Both have positive and negative features, which are necessary to pass analyte information and suppress interfering signals. A more detailed interpretation of the shape of these calibration vectors is nearly impossible owing to the chemical complexity of the sample matrix and the degree of spectral overlap between matrix components. Nevertheless, the difference in the shapes of these two calibration vectors is highly significant.
Comparison of PLS (black) and NAS (red) calibration vectors for PLS models generated from noninvasive spectra with correct glucose assignments (A) and nonrandom incorrect glucose assignments (B).
Superimposed on each PLS calibration vector in
is the calibration vector generated by the net analyte signal (NAS) computed for glucose in this rat skin tissue matrix.9
The term NAS was first introduced by Lober to describe that component of the analyte spectrum that is orthogonal to all interfering sources of spectra variance within the data set.13,14
The NAS, therefore, represents the selective component of the analyte spectrum relative to the spectral background generated by the spectral variance created by changes in the sample matrix, instrumentation, and environment.
The concept of the NAS calibration vector presented in
and the procedures for its creation are described in detail elsewhere.9
Briefly, the NAS is determined from a collection of baseline noninvasive spectra and the pure component spectrum of glucose dissolved in water. For the NAS vector presented in
, baseline noninvasive spectra were collected during the period of the experiment when the glucose concentration is essentially constant. This constant period extends from 11 to 14 in
. Because glucose levels are constant during this period, variance in baseline noninvasive spectra is assigned to nonglucose-dependent sources and includes variations created by the instrumentation, environment, and sample. This nonglucose-dependent spectral variance is characterized by a set of principal component factors. The NAS vector is obtained as the residual vector after projecting the glucose spectrum vector onto the set of baseline principal component vectors. This process determines the portion of the glucose spectrum that is unique compared to the nonglucose-dependent spectral variance incorporated in the set of baseline spectra.
A comparison between calibration vectors generated by the PLS algorithm and the NAS calculations reveals a major difference between the PLS calibration vectors created from correct and nonrandom incorrect glucose assignments.
shows that the PLS calibration vector corresponding to the correct glucose assignments is very similar in shape and magnitude compared to the NAS calibration vector. The similarity between these vectors indicates that the PLS calculation with the correct glucose assignments correctly weighs the glucose selective portion of sample spectra. This finding is significant because the two calibration vectors were generated by using entirely different data analysis procedures. The PLS calibration vector is based on a statistical regression to establish the best correlation between spectral variance and glucose concentration. In contrast, the NAS calibration vector is based on a set of factors that describe the spectral variance and the known absorptivity spectrum of glucose without recourse to a regression step.9
The convergence of these approaches is significant and lends confidence that both models are in fact based on glucose-specific information present in in vivo
However, shows that there is absolutely no similarity between PLS and NAS calibration vectors when incorrect glucose assignments are used in the PLS calculations. The high frequency features below 4400 cm-1 in strongly suggest that only noise is being modeled in the spectral region most sensitive to glucose, as shown in . The poor overlap between these calibration vectors is consistent with a lack of glucose-specific information in the PLS calibration model.