2.1.

Equipment

A 10 mW helium–neon (HeNe) laser (Spectraphysics, Eugene, OR) was used as a source of visible light for aligning single capsules and optics. The experiments were conducted outdoors to provide the unobstructed 500 m path, and at night to simplify gross alignment of the optics using the visible laser. A 4 mW external cavity tunable diode laser (EOSI, Boulder, CO) was used as a source of near-IR light. Silicon charge coupled device (CCD) (Nikon) and indium antimonide (InSb) digital cameras (Cincinnati Electronics, Mason, OH) were used during the alignment process. A Stanford Research Systems programmable wave form generator was used to modulate the near-IR light at 1 kHz. A small reflecting telescope with a 9 cm aperture was used to collect light for detection. A PbS telescopic detection unit⁷ was temporarily borrowed from the 1 m near-IR telescope at the university to detect the modulated light from the capsules at 500 m. A Dell PC and 16 bit analog-to-digital converter (A/D) were employed for data collection. Digitized signals from the telescope preamplifier were cross correlated with the signal from the programmable function generator, and Fourier transformed using GLIB software in Mathematica (Wolfram Research, Champaign, IL). To reduce the total data-collection time, signals were recorded at six different wave numbers, 5797, 5764, 5731, 5682, 5634, and 5618 cm⁻¹.

The laser samples less than 1 mg of a 60 mg gelatin capsule without moving the position of the beam. Precisely aligning the invisible near-IR laser following initial alignment with a visible laser would have been very difficult without interchangeable precision micrometer mounts. These mounts were constructed for this study in the machine shop of the Department of Physics and Astronomy. The physical length of the optical path was 250 m. A 500 m total optical path was achieved by reflecting the signal from the capsules at 250 m backward using a 20 cm×25 cm first-surface aluminized mirror (see Figure 1). The distance from the laser to the capsule was 2 m, although the beam spread was low enough to permit active excitation of the capsules with the laser at distances of up to approximately 200 m.

Figure 1

A schematic diagram showing the relationship of the instruments and optical path during collection of spectra at a distance of 0.5 km from a single gelatin capsule.

An IRC-160 InSb focal plane array video camera (Cincinnati Electronics, Mason, OH) with near-IR bandpass cold filter was used for 0.5 m hyperspectral imaging of the samples. The camera frame rate was 51.44 frames/s and the photon energy response was 1800–10 000 cm⁻¹. A polarizing filter for the camera reduced specular reflectance (Polaroid, Cambridge, MA). A tunable interference filter unit⁸ with photon energy response from 4325 to 5960 cm⁻¹ was mounted on the camera. A Mitsubishi IR-M300 PtSi 256×256 CCD array camera (Cypress, CA, USA) was used at 2 m on the samples. The light source was two 250W PC37771 near-IR/IR lamps (General Electric, Cleveland, OH). A monochromator system employing a concave holographic grating (American Holographic, Fitchburg, MA) and lead sulfide (PbS) detector was used for reference spectra.⁸

Whenever a near-IR camera was used to collect tablet spectra in blister packages, two spherical silicon dioxide reflectance standards (one high reflectance, and one low reflectance) were placed in each image to control for variations in light intensity and direction. Images collected on different days and with the light sources in different locations were made comparable by adjusting the gain and offset by multiplicative scatter correction⁹ on the images so the intensities on the standards were identical. The specular reflectance on the standards was used to pin down the locations of the light sources and to supply a method to calibrate reflected specular light intensity. Diffuse reflectance from the rounded surfaces of the two standards was used to calibrate shaded areas and oblique surfaces in the images.¹⁰ Two light sources were employed for spectrometric imaging to diminish shadows and attain the greatest signal-to-noise ratio (S/N) possible. The light sources were positioned at 90° relative to one another, with the tablets at the vertex. The camera was placed between the light sources, at an angle of 45° to each. Spectral images were obtained at each wave number with the near-IR light sources turned off, and again with them turned on, to correct for the existence of other lights in the room and for blackbody photon emission from the sample.

A hydrator was assembled to control the conditions under which the tablets decomposed.¹ Tablets in blister packages were exposed to water vapor or a pH 9.0 ammonium hydroxide solution by punching a hole using a center punch through the foil support. Water absorption was determined by weighing and by near-IR spectrometry. The high-performance liquid chromatography (HPLC) for analysis of salicylic acid used a C-18 column (Analytical Sciences, Santa Clara, CA), a Prostar 320 absorbance detector (Varian, Palo Alto, CA), a model 215 pump (Varian). The software for the near-IR equipment as well as the chromatography interface was written in C++ (Microsoft, Redmond, WA) and Speakeasy (Speakeasy Computing Corp, Chicago, IL). The mobile phase was 27:73 methanol:water, 0.15 M NaH ₂ PO ₄, 5 mM tetrabutylammonium phosphate, at pH 6.0. This pH was selected to make the results comparable with our earlier study.¹ The half-life of aspirin at pH 6.0 is more than five days. Retention times for aspirin were approximately 5 min.

The tablet masses were measured with an electronic balance. Tablets were dissolved for HPLC analysis. A total of 43 tablets were selected from 172 hydrator tablets by principal component analysis (PCA) selection algorithm.¹¹ The benefit of this methodology is that it selects the best tablets to use in the calibration based on their near-IR spectra before reference measurements are obtained. The near-IR technique is fast and easy compared to the HPLC assay, making the selection of the optimum subset of tablets for the reference assay based on near-IR spectra valuable. The 172 tablets in turn came from four batches of 43 tablets, each sequence run for up to 24 h in the hydrator.¹ By staggering the starting times of each 24 h cycle it was possible to get at least six tablets in each hour of exposure from 0 to 24 h. The four runs were made over an interval of two weeks.

3.1.

Chemometrics

Principal component regression is usually accomplished with a transformation of spectra to principal axes, followed by ordinary least-squares regression. Median-based regression models such as least median of squares are a convincing alternative to mean-based regression models when the distribution of errors is unknown and likely to be non-Gaussian. Remote sensing measurements with an invisible near-IR laser are especially prone to experimental errors that have heavier tail density than the Gaussian distribution. Sample heterogeneity and alignment problems can manifest themselves as clusters of larger residuals in a robust principal component regression.

Most approaches to median regression are algorithmic and the advantageous properties arise only asymptotically. Nonparametric error depictions are often the foremost models to apply in the case of smaller sample sizes (i.e., fewer calibration spectra) where parametric specifications are hard to assert. A Bayesian method that implements accurate inference when supplied with the observed spectra is especially attractive in such cases.¹² Median-based regression is optimal just for describing skewed and unusually shaped error distributions, and such distributions can be obtained using mixtures of families of error distributions that have the accessible prior probability functions needed for Bayesian inference.

The variety of mixture densities defined in Ref. 12 allows skewness and greater variability to be properly accounted for in regression models. Furthermore, the form of the kernel of the mixture permits unambiguous quantification of skewness via a parameter denoted γ. However, the class embodying the densities f(⋅;G,γ) is merely a subset of the class F={f:f unimodal density on R with η(f )=M(f )=0}, where η(f ) and M(f ) signify the median and mode of f, respectively. The model of Kottas and Gelfand depicts all possible densities in F,¹² accomplished by designating nonincreasing densities on [0, ∞) as mixtures of uniform densities. To be precise, any nonincreasing density g on [0, ∞) can be expressed as

Kottas and Gelfand¹² employed two nonincreasing densities of the form in Eq. (1) to obtain a general depiction for a density f in F, one utilized for [0, ∞) and the other for (−∞, 0), producing

The free type of the density f(.;G₁,G₂) obviates many analytic results regarding properties of the allied distribution.⁷ Fortunately, the analytic properties can be connected to the corresponding properties of the mixing distributions G₁ and G₂. For example, the expected value and the skewness functional γ_η are given by 1/4[∫θG₂(dθ)−∫θG₁(dθ)] and (∫θG₂(dθ)−∫θG₁(dθ)]/[∫θG₂(dθ)+∫θG₁(dθ)], respectively, as long as G₁ and G₂ have a finite first moment. If η(G) denotes the median of Eq. (1), then γ_q(G₁,G₂)=[η(G₂)−η(G₁)]/[η(G₂)+η(G₁)]. When applying Eq. (2), independent Dirichlet process priors are utilized for G₁ and G₂. The posterior of γ_q(G₁,G₂) can be perfected through the posteriors of the quantiles that determine it.¹²

Separate mixing distributions can be used to control the behavior of each of the tails. Skewness can be described by more than a single parameter, enabling description of more sophisticated forms of asymmetry. However, in certain applications, such added flexibility can exaggerate the data, leading to unstable inference. Models must be selected according to context.¹² Nevertheless, the application of robust median-based regression to spectra prevents leverage effects in ordinary least-squares regression from masking experimental variables, which can lead to an increase in overall prediction error.

3.2.

Tablet Imaging with the Bootstrap Error-adjusted Single-sample Technique (BEST)

The bootstrap error-adjusted single-sample technique (BEST) calculates distances in multidimensional asymmetric nonparametric central 68 confidence intervals in spectral hyperspace (approximately equivalent to standard deviations). The BEST metric can be thought of as a “rubber yardstick” with a nail at the center (the multidimensional mean).¹⁵ Because of the nail, the stretch of the yardstick in one direction is independent of the stretch in the other direction. This independence enables the BEST metric to describe odd shapes in spectral hyperspace (spectral-point clusters that are not multivariate normal, like the calibration spectra of many biological systems). BEST distances in standard deviations (SDs) can be correlated to sample composition to produce a quantitative moisture or degradation-product calibration in tablets, or simply used to identify regions with moisture or degradation product distributions similar to the calibration data in a spectral image. The BEST automatically detects samples and situations unlike any encountered in the original calibration, making it more accurate in such applications than typical regression approaches to near-IR data analysis. The BEST produces accurate distances even when the number of calibration samples is less than the number of wavelengths used in calibration, in contrast to other metrics that require matrix factorization. The BEST retains the direction vector of a standard deviation in hyperspace throughout all calculations, an essential characteristic for multicomponent quantification of sample composition.

The BEST calculates the integral of a probability “orbital” in hyperspace by starting at the center of the orbital and working outward in all directions at a uniform rate. The distance between the center of a calibration set orbital and a sample spectrum is proportional to the concentration(s) of the calibration set constituent(s) responsible for the vector connecting the central and sample-spectral points. The direction of the vector identifies the constituent(s) of the calibration set. The BEST direction and distance are often used to create color contour plots of the spatial distribution of moisture or degradation products. In such plots, the contours are drawn at sequential distances in SDs, and red-green-blue colors are used to denote class membership based on vector direction. The intensity of the color is proportional to the amount of substance present. Shades of blue are used to represent sample spectra similar to those already in the calibration set, while shades of red are used to represent sample spectra that contain the selected analyte. Shades of green are used to represent a second analyte or possible interfering effect. The BEST performs well as an assimilation method (a method that progressively increases its analytical performance by incorporating previously unknown samples into its calibration). The calibration samples are analyzed by another reference method in the same manner that Beer’s law is used to develop a conventional spectrophotometric calibration.

In the BEST, a population P in a hyperspace R represents the universe of possible spectrometric samples (the rows of P are the individual samples, while the columns are the independent information vectors, such as wavelengths or energies). P ^* is a discrete realization of P based on a calibration set T, which has the same dimensions as P ^* and is chosen only once from P to represent as nearly as possible all the variations present in P.

P ^* is calculated using a bootstrap process by an operation κ(T), and P ^* has parameters B and C, where C=E(P) and B are the Monte Carlo approximation to the bootstrap distribution. The expectation value, E(), of P is the center of P, and C is a row vector with as many elements as there are columns in P.

Each new sample spectrum X is analyzed by an operation ψ(T,B,X,C), which projects a discrete representation of the probability density of P in hyperspace by many–one mapping onto the vector connecting C and X. X and C have identical dimensions. The directional standard deviation (SD) is found from the projected probability density in Eq. 3

The integral over the hyperspace R is calculated from the center of P outward. The calculation of a skew adjusted is based on a comparison of the expectation value C=E(P) and C _T=med(T), the median of T in hyperspace (with the same dimensions as C) projected on the

The result of the corrected projection is an asymmetric σ that provides two measures of the standard deviation along the hyperline connecting C and X

The use of these equations in both quantitative and qualitative analysis of samples provides a number of advantages over all other methods of analysis.

1. No analytical assumptions are required to make the problem solvable. Other chemometric methods typically assume that no discriminating variable (wavelength) is a linear combination of other discriminating variables, that the covariance matrices for all spectral groups are approximately equal, and that each group is drawn from a population that is normally distributed on the discriminating variables. None of these assumptions is usually true and violations of these assumptions increase the likelihood of producing incorrect quantitative and qualitative analytical results.

2. This nonparametric assimilation method can be used with full spectra of samples, which often include thousands of variables that describe each sample. The large global memory on the newest supercomputers has made possible the manipulation of images involving millions of spectra at thousands of wavelengths. No wavelength selection procedures or data compression techniques, such as principal-axis transformation or Fourier transformation, are required by this assimilation method in order to analyze complete spectra. Collinearity in the discriminating variables (wavelengths) does not degrade the results. (Collinearity disrupts conventional matrix techniques like the Mahalanobis method, which relies on matrix factorization to produce a distance in SDs.)

3. The vector CX in the assimilation method identifies the sample components. The metric is calculated by a highly parallel code that can be distributed across as many processors as are available, and can be used in computerized searches of spectral libraries for qualitative analysis of real samples. The length of the vector is proportional to the concentrations of the constituents. Thus, quantitative analytical capabilities are also provided by the same assimilation method, which can still be distributed across as many processors as are available.

4. The BEST metric is not only more accurate and precise than the Mahalanobis metric (the metric commonly used in near-IR spectrometry), the BEST metric is often calculated more rapidly as well. The matrix inversion required by the Mahalanobis metric is usually accomplished by algorithms whose complexity (in terms of number of operations required) increases as the number of wavelengths cubed. In contrast, the complexity of the BEST metric increases linearly with the number of wavelengths.

The assimilation model developed by the BEST can be compressed if desired with singular value decomposition procedures to conserve memory. The model can also be converted into a hash table and hash function by calculating the distance and direction from the center of the calibration set to each of the replicates, and then used in ASIC microchips or on laboratory PCs.

4.1.

Distance Study

Figure 2 shows the bright visible and near-IR images of the capsules on the 500 m path. The intensity of the signals received suggests that spectrometry over even greater distances is possible. Spectra were collected individually from 24 hard gelatin capsules. Six of these capsules were not exposed to formaldehyde, while six were exposed to 150 ppb formaldehyde for 2 h, another six for 4 h, and the last six for 8 h. Principal component regression of the spectra versus exposure time to formaldehyde produced r²=0.89, with the standard error of estimate (SEE)=1.17 h and SEP=1.17 h. The calibration line is shown in Figure 3. The process of cross-linking in gelatin capsules is not homogeneous, and cross-linking tends to occur most extensively in the portions of the capsule closest to the source of moisture and aldehyde, where the concentrations are highest. The laser beam is relatively narrow in comparison to the capsule. The r² would probably have been higher if the entire capsule had been illuminated by the laser source and analyzed spectrometrically instead of less than 1 mg of the sample (previous full-capsule experiments in the laboratory yielded r²=0.963 ).⁴ In addition, a smaller wave number range was scanned than is usually employed for capsules to save the time required to change laser heads for extended wave number range. Finally, because the experiment is conducted outdoors, some low frequency noise is introduced in the system by wind. Most of the wind-induced motion occurs in the distant mirror, which is supported by a tripod. A 20 lb sandbag was hung from the tripod to reduce motion and the chance that the tripod would blow over in an elevated location. This solution was not 100 effective in preserving alignment of the invisible near-IR laser beam over extended periods, however. For this reason, robust (median-based) regression was employed to enable detection of alignment problems in the samples.

Figure 2

The visible light image (top) and near-IR light image (bottom) shows the intensity of the signal received over the 0.5 km path from a single gelatin capsule.

Figure 3

The actual vs predicted exposure time to formaldehyde calculated using the spectra acquired at a distance of 0.5 km.

These results suggest that active-excitation spectrometry is a technique that is ripe for further research in pharmaceutical analysis. The active-excitation technique is particularly applicable to closed vessels where passive excitation does not reach. Indeed, NASA’s Jet Propulsion Laboratory is working on a multifunctional active-excitation spectral analyzer (MAESA) for missions to planets like Pluto.¹⁶ Pluto is so far from the sun that passive excitation would not be effective. The MAESA uses a laser to illuminate either a point or a line on a target. Raster scanning permits hyperspectral images to be collected. With the addition of a diffraction grating on the detection end of the system, active excitation can also be used to obtain Raman or fluorescence spectra. The NASA device features a wavelength response from 500 to 2500 nm, low power consumption, and a portable package design that operates near room temperature.

Figure 4 shows illustrative spectra of an aspirin tablet (dotted line), the blister package (solid line), an aspirin tablet in a blister package (dot-dashed line), and an aspirin tablet in a blister package with a leak exposing the tablet to moisture (dashed line). These spectra were obtained from single samples with the nonimaging spectrometer. The major peaks of the packaging are at 5700–5800 and 4000–4350 cm⁻¹. The major spectral peaks of acetylsalicylic acid, at about 6050 and 4200–4800 cm⁻¹, are apparent through the packaging. The most distinguishing spectral feature of salicylic acid arising through aspirin decomposition appears at 6370 cm⁻¹.¹ The major water-absorbance peak appears at approximately 5200 cm⁻¹. This section of the spectrum contains only modest interference from the packaging, which simplifies the imaging of tablets. Calibration lines were computed for water and salicylic acid. The calibration lines were created using data obtained at 0.5 m from 43 tablets selected from 172 by the PCA method of Svensson, Josefson, and Langkilde.¹¹ The standard error of estimate (SEE) for water was 0.05 of tablet mass and the SEP was 0.06 of tablet mass (r²=0.96). For salicylic acid, the SEE=0.06 of tablet mass and SEP=0.06 of tablet mass (r²=0.98). The correlation is calculated between the concentrations predicted from near-IR spectra and the concentrations determined by reference methods, i.e., weighing (for water) and HPLC (for salicylic acid).

Figure 4

Representative spectra of aspirin tablet (dotted line), the blister package (solid line), an aspirin tablet in a blister package (dot-dashed line), and an aspirin tablet in a blister package with a leak exposing the tablet to moisture (dashed line).

The PC (principal component) plot in Figure 5 portrays water uptake by ten packaged tablets in the field of packaged tablets shown in the contour plot in the inset. The tablets were imaged at 8, 16, and 24 h after a hole was punched in the foil backing. The times are marked on each tablet in the plot. The end point was at the same time for all tablets, and the starting time (i.e., when the hole was punched) was varied. The tablet spectra were collected through the blister packaging, and the spectra are shown after multiplicative scatter correction. The spectra were used to calculate the contour plot in the inset of Figure 5. The contours connect pixels with the same distance in multidimensional standard deviations (SDs) from the calibration tablets centered near the origin in the PC plot. The PC scores of the spectra show that the changing signal from water over time is readily detected through the blister packaging using a near-IR camera. In Figure 5 a distance less than 3.8 SDs from the calibration set of normal tablets is displayed as light gray contour. A distance greater than 3.8 SDs in the direction of the spectrum of water (shown by the arrow) is colored progressively darker. The intensity of the black color is correlated to the magnitude of the distance in SDs between each pixel spectrum and the center of the spectra of control (dry) tablets. The maximum distance is 8.6 BEST SDs (corresponding to the tablet with the blackest density in the contour plot, and highest spectral peak at 5200 cm⁻¹). Previous studies in nonpharmaceutical applications have suggested that there might be a minimum number of pixels required on a sample (e.g., 16) to achieve an acceptable S/N.¹

Figure 5

Contour plot with lines drawn in bootstrap error-adjusted single-sample technique standard deviations (SDs) for ten packaged tablets in a field of 1286 packaged control tablets. Data are not interpolated from the array. Gray contour lines are <3.8 SDs from control tablets. Darker coloration begins at 3.8 SDs and peaks at its blackest at 8.6 SDs.

Figure 5 suggests that the upper limit on the number of tablets that can be imaged concurrently on a 256×256 array is approximately 1300. With 1296 tablets in the camera field of view, at least 16 pixels can be applied in simultaneous sampling of each tablet, which is enough spectra on each tablet to get an effective S/N and form contours for imaging. Principal component analysis of the tablet spectra showed loadings with spectral features that correspond to salicylic acid and acetylsalicylic acid.¹