drug metabolites were not observed in any of the 118 urine drug screen specimens or the 15 patients receiving cocaine anesthesia for rhinoplastic or septoplastic surgery. Applications t o Biological Samples. Analysis of urine specimens obtained from adult patients receiving cocaine anesthesia (250 mg cocaine hydrochloride topically applied to nasal mucosa) for rhinoplastic or septoplastic surgery revealed results that were not significantly different from those reported in an earlier report ( 4 ) . The observed concentration range for the initial 8-h pooled specimens for cocaine and BE were 0 to 0.3 pg/mL and 8 to 136 kg/mL, respectively. Of the 118 urine drug screen specimens, screened positive for BE by radioimmunoassay (RIA), 105 were confirmed positive by the GLC procedure, with concentrations ranging from 0.2 to 18 wg/mL for BE while cocaine concentrations did not exceed 0.1 pg/mL.
ACKNOWLEDGMENT The authors express thanks to Ralph A. Nelson, Naval Regional Medical Center, for providing urine specimens from
post-surgery patients who had received cocaine anesthesia. The authors are extremely grateful to James A. McCloskey and Pam F. Crain, Department of Biopharmaceutical Sciences, University of Utah, for performing the mass spectral analyses.
LITERATURE CITED (1) F. Fish and W. D. C. Wilson, J . Pharm. Pharmacol., Suppl., 21, 1355, (1969). (2) L. A. Woods, F. G. McMahon. and M. H. Seevers, J. Pharmacol. f x p . Ther., 101, 200 (1951). (3) A. R. McIntyre, J. Pharmacol. f x p . Ther., 57, 133 (1936). (4) J. E. Wallace, H. E. Hamilton, D. E. King, D. J. Bason. H. A. Schwertner, and S. C. Harris, Anal. Chem., 48, 34 (1976). (5) M. L. Bastos, D. Jukofsky, and S. J. Mug, J. Chromatogr.,89,335 (1974). (6) N. N. Vahnju, M. M. Baden. S. N. Valanju. D. Mulligan, and S. K. Verma. J . Chromatogr., 81, 170 (1973). (7) M. L. Bastos and D. 6. Hoffman, J. Chromatogr. Sci., 12,269 (1974). (8) S.Koontz, D. Besemer, N. Mackey, and R. Phillips, J. Chromatogr.,85, 75 (1973). (9) P. I. Jatlow in “Cocaine: Chemical, Biological, Social and Treatment Aspects”, S. J. Mule, Ed.. CRC Press, Cleveland, Ohio, 1977,pp 59-70. (IO) R. H. Greeley. Clin. Chem. (Winston-Sabm, N . C.),20, 192 (1974).
RECEIVED for review May 10, 1977. .Accepted August 1, 1977.
Reconstruction of Gas Chromatograms from Interferometric Gas ChromatographyAnfrared Spectrometry Data J. A. d e Haseth and T. L. Isenhour* Department of Chemistry 045A, University of North Carolina, Chapel Hill, North Carolina 27514
A method has been devised by which gas chromatograms are directly reconstructed from single scan GC/IR interferograms. The Gram-Schmidt vector orthogonalization process is used: first, to define from background interferograms a basis which adequately describes the inherent instrumental and collection characteristics; then second, to orthogonalize ail nonbasis single scan interferograms to that basis and measure each orthogonal component. The orthogonal component is a direct function of the total infrared absorbance of the entire spectral range with respect to the basis. The reconstructed gas chromatograms, belng derived directly from the single scan interferometric data, precisely indicate which interferograms best represent sample spectra. I t has been found that the reconstruction of G W I R chromatograms is computationally more economical than transforming the interferograms individually and searching for nonbackground spectra.
T h e utilization of gas chromatography/infrared spectroscopy (GC/IR) has given the analytical chemist a powerful technique for the qualitative analysis of mixture components. Various methods ( 1 , 2) have been employed for the construction of GC/IR spectrometers. In continuous flow gas chromatography systems where GC peaks are eluted within a few seconds, conventional dispersion spectrometers are unable to complete a scan within the time constraints of the experiment. Rapid scan Fourier Transform spectrometers can collect single scan interferograms fast enough to record the spectra of GC effluents; however, the resulting data are a collection of single scan interferograms that gives no apparent indication which interferograms are to be transformed to yield the effluent IR spectra.
T o conclusively locate the interferograms of spectra of GC effluents it is necessary to transform all interferograms in the collection, plot their respective spectra, then visually discriminate between background and effluent. Clearly, this is a tedious and time consuming process. Attempts have been made to more directly locate those interferograms which correspond to spectra of GC effluents and hence save time and labor. One method is to calculate the overall absorption power of each interferogram and determine which interferograms have been attenuated by infrared active chemical components in the light path. This method, which is analogous to measuring the total ion current (TIC) in gas chromatography/mass spectrometry (GC/MS), does not produce reliable results unless the interferogram intensity is attenuated by at least 10% (3). Unfortunately, most GC/IR absorbances are well below this threshold. Another solution is to run the GC experiment twice, once in the GC/IR mode, the other time eliminating the GC/IR cell and measuring the response with a standard GC detector. All that remains is to map the GC trace onto the series of single scan interferograms and transform only those that correspond to sample effluents in the light pipe. In practice, this is not always easily achieved. Operating conditions often cannot be exactly duplicated and volume changes between the IR cell and GC detector systems offset the two data sets. Stream splitters may be useful for simultaneously recording the interferograms and the gas chromatogram. Although this method is more reliable than separately recording the two data sets, there is still a lack of precision in identifying specific sample interferograms. This paper presents a method by which the gas chromatogram may be directly reconstructed from the interferometric data, thereby identifying those interferograms to be transformed to yield mixture component spectra. The method uses ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
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the Gram-Schmidt vector orthogonalization process to establish a basis (or basis set) that represents background interferograms (Le., carrier gas only). The interferograms used to form the basis are selected from scans collected before the elution of the first sample component. The orthogonalization process removes all background information from the subsequent interferograms and measures differences between the background and any infrared active species that have been eluted. The measure of eluted species is a direct function of the total infrared absorbance (TIRA) over the entire IR spectral range. This method is quicker than the transform method and results in a gas chromatogram which pinpoints the specific interferograms to be transformed to yield spectra of the GC effluents.
THEORY T o reconstruct gas chromatograms of multicomponent samples from series of single scan interferograms it is necessary to form an appropriate discriminant. The discriminant must fulfill four basic criteria: (1) Those features common to all interferograms in the series should be ignored. This includes the light burst at zero mirror retardation and various anomalies produced by inherent characteristics of the recording instrument and the collection conditions. (2) The discriminant must be able to tolerate noise. As only single scan interferograms may be collected, the signal-to-noise (S/N) ratio is expected to be low. This ratio may be further reduced as the interferograms are often collected using a light pipe cell in which the signal intensity may be significantly attenuated. (3) The only factor distinguishable by the discriminant must be the total infrared absorbance across the spectral range of the components in the mixture. (4) The discriminant must operate on single scan interferograms without transforming them to the spectral mode. One method of forming a discriminant that meets all the above-mentioned criteria is Gram-Schmidt vector orthogonalization. T o apply the orthogonalization algorithm each interferogram must be considered an n-dimensional vector, where each resolution element or sampling point represents a different dimension. The Gram-Schmidt orthogonalization process forms an orthonormal set from a series of linearly independent vectors. If there are n linearly independent n-dimensional vectors, it is possible to form a set of n orthonormal vectors. The reconstruction of gas chromatograms requires that a series of interferograms representative of the GC background signal (Le., no sample in the light path) be collected and orthogonalized. The resulting orthonormal vector set forms a basis which is the discriminant. Once the basis has been constructed, interferograms from a GC/IR run can be individually and sequentially orthogonalized to that basis. Any orthogonal resultant is the total infrared absorbance of any infrared active chemical component in the light path when the interferogram was recorded. In other words, the orthonormal basis is a hyperplane which represents the features common to all interferograms in the series, such as instrumental characteristics, inherent interferometric characteristics, collection condition characteristics, and to a certain extent the noise. If an interferogram represents only the background, the entire interferogram should lie in the hyperplane as the interferogram should not be linearly independent. Theoretically all background interferograms should be nearly identical and the basis could be formed with a single vector. Unfortunately, the noise in these systems is so high that several vectors are needed to define the basis. An orthogonal resultant from a nonbasis background interferogram may then be attributed to noise. As noise is random, not all noise can be removed without forming a basis that spans all 1978
ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
n possible dimensions. Necessarily, the infrared absorbance factors would also lie in the n-dimensional hyperplane. If the orthogonal resultant is large relative to the resultant for noise, it is due to the total infrared absorbance of any infrared active chemical components in the light path. Thus as each interferogram in the GC/IR run becomes available it can be orthogonalized. to the existing basis and the orthogonal component recorded, thereby reconstructing the gas chromatogram. T o orthogonalize a set of n-dimensional vectors by the Gram-Schmidt process, the following derivgtion is involved: For an interferogram of n dimensions, I l = (il, ,i is, i4, &,...,in), it is first necessary to form the unit vector U1:
where ilT is the transpose of vector 1,. (The transpose of a matrix is formed by exchanging the rows with the columns of the matrix. A vector is transposed by co v r ing a row vector into a columnyector, or vice versa.) M ’ i s simply the scalar length of Il as
Now, choose any other member of the data set except fl, say 12,and write:
” -+ -+ v 2 = I2 - C l U ,
(3)
wheze c1 is a constant to be determined so that v2 is orthogonal to U1. In order for this to be true:
(4) As
GlTfil = 1, then: (5)
and
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ol,but the orthonormal vector 82is
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Unit vectors 0,and are orthonormal and form a basis. More vectors can be orthonormalized to further define the basis by using the general equations: j= 1
and
(9) Thus vectors 1, through 1, may be used to form a k-dimensional hyperplane (where k < n),that is, a basis. For a data set of m entees (interferqgrams) where k < m, the additional vectors I k + l through I,,, may be individually and
sequentially orthogonalized to the hyperplane by use of the equation:
The absolute length of the orthogonal resultant of interferwhere k + 1 Ip Im. This ogram 1, is expressed length is a measure of the total infrared absorbance of any infrared active chemical compound in the light path.
m,,
EXPERIMENTAL Several single scan interferogram data sets were supplied by L. V. Azarraga of the Environmental Protection Agency, Athens, Ga. ( 3 ) . These data are of GC separable mixtures, run on two different GC/IR light pipes. One light pipe measures 0.7 mm in diameter by 52 cm in length, the second is 2 mm in diameter with a length of 30 cm. The data were collected on a Digilab FTS-14 spectrophotometer using a mercury cadmium telluride detector. The interferograms were written directly onto magnetic tape during collection. All programming was written in Fortran and the computations were performed at the University of North Carolina Computation Center on an IBM Series 360 Model 075 and an IBM Series 370 Model 155 computers. RESULTS AND DISCUSSION T o reconstruct gas chromatograms from GC/IR interferometric data using the Gram-Schmidt orthogonalization process, it is preferable to define the basis with as few vectors as possible. The fewer the vectors to adequately define the basis, or the lower the dimensionality of the hyperplane, the greater the discriminating ability of the basis. Also, the fewer the vectors used in constructing the basis, the lower the number of subsequent computer operations. Other variables besides the number of basis vectors may be optimized for the reconstruction procedure. The number of data points for each interferogram in the data sets is 2048, which defines the dimensionality. The nature of the interferogram, being collected in the displacement rather than the spectral mode, makes it possible to reduce the number of dimensions or data points used. Directly related to the number of dimensions is the displacement of these data points within the interferogram. That is, if the number of dimensions is fewer than that available in the interferogram, the data points may be selected at a considerable displacement from the first collected datum of the interferogram. Consequently. there are three variables to optimize: the number of basis vectors, the dimensionality, and the displacement. The data set used to optimize the reconstruction algorithm is a sample mixture of six components run on the 0.7-mm diameter light pipe. It was determined empirically that all six peaks can be resolved using 50 basis vectors, a dimensionality of 100 (Le., 100 consecutive data points from the available 2048) and a displacement of 35 data points from the zero mirror retardation point on the trailing edge of the interferogram. The maximum valued data point was considered to be close to zero mirror retardation and was used as its equivalent. Displacement from the first datum was not used as a reference point as the zero mirror retardation can vary with respect to this datum from instrument to instrument. This point also can be easily adjusted or altered on a single instrument. The first variable to be optimized was the dimensionality; the number of basis vectors was held constant a t 50 as was the displacement at 35. The dimensionality was varied from 25 through 500 with data collected every 25 dimensions up to 100, then every 50 dimensions through 500. A measure of the success of each variation was needed to quantify the results. The measure that was chosen was the signal-to-noise ratio for the GC peak with the least intensity, but still greater
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Flgure 1. Plot of the resolution vs. change in the number of dimensions. The number of basis vectors was held constant at 50 and the displacement was fixed at 35 data points from the zero retardation point. The sampie used was a six-component mixture run on a 0.7-mm diameter by 52-cm light pipe
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than the noise, multiplied by the total number of peaks resolved above the noise. This measure is defined as the Resolution. A plot of the dimensionality vs. resolution is shown in Figure 1. As can be seen, the resolution rises sharply between 25 and 50 dimensions and levels off beyond 300 dimensions. As the resolution at dimensionality 100 is within a few percent of that at 300, the number of dimensions was optimized a t 100. An integral aspect of the optimization procedure is to minimize the number of computational operations. Holding the dimensionality down to 180 far outweighs the extra resolution gained at 300 if the number of extra computations is considered. The next variable optimized was the displacement. A positive displacement WBS considered to be on the trailing edge of the interferogram with respect to the zero mirror retardation point. A negative displacement is one on the leading edge of the zero mirror retardation point with the possibility of the sample points encompassing the light burst. The dimensionality was Tied at 100 and the number of basis vectors held at 50. The data resulting from this optimization are plotted in Figure 2. Several observations can be made from the results. When the points used span the light burst, the resolution is very low. The data in the region of displacement 10 to 100 appears to be scattered. As the values found in this region were reproducible over a series of measurements, it was ascertained that these data represent a series of maxima and minima with the overall maximum a t a displacement of 60. Beyond a displacement of 100, the curve appears to be a ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
1979
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Flgure 4. Reconstructed gas chromatogram of the six-component sample using the optimum values from Figures 1, 2, and 3. The dimensionality is 100, the number of basis vectors and the displacement are both 60 decaying wave that eventually dies out at about a displacement of 500. The displacement was thus optimized a t 60. So far two variables have been optimized which have the following implications: only 100 consecutive data points need be selected for reconstruction; and, these 100 data points are selected from the region of the interferogram close to zero retardation where the major modulations have died down but modulations due to discrete absorption bands are still strong. With the displacement set a t 60 and the dimensionality fixed a t 100 the number of basis vectors was varied. Figure 3 shows that a maximum is reached a t 60 basis vectors. The resolution declines after 60 basis vectors because the number of dimensions spanned by the basis becomes sufficiently high that much of the discriminating ability is lost. The result of the gas chromatogram reconstruction using the values of the three variables optimized is shown in Figure 4. This sample contains the following six components; bis-2-chloroethyl ether, acetophenone, methylsalicylate, 2,3,5-trimethylphenol, acenaphthene, and 2,4,6-trimethylphenol, eluted in this order. The concentration of the sample is 0.5 bg/pL with a total sample size of 0.8 bL. The solvent peak was not recorded. The orthogonal component values of the first 60 interferograms are omitted from the chromatogram as they were used to form the basis. All six chromatographic peaks are clearly visible in the plot even though the overall signal-to-noise ratio is low at approximately five-to-one. Therefore, there may be a direct correlation between spectral signal-to-noise and chromatogram signal-to-noise in which, the S / N of the 1980
ANALYTICAL CHEMISTRY, VOL. 49,NO. 13, NOVEMBER 1977
z w 0
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Flgure 5. Reconstructed gas chromatogram of a multicomponent sample using a 2-mm diameter by 3 0 c m light pipe. The dimensionality is 100, the displacement is 60, and the number of basis vectors Is 5 chromatogram can equal the S / N of the spectrum with the highest S/N ratio, but not exceed that value. The spectra that correspond to the GC peak maxima in Figure 4 indicate that the spectral S/N ratio is equivalent to the corresponding GC peak S/N ratios. The spectral S/N ratio is calculated on the signal-to-noise ratio of the largest infrared peak to the noise. Because the spectra exhibited single strong bands, it is not known if the signal-to-noise ratio of the reconstructed gas chromatogram is always equal to the signal-to-noise ratio of the spectrum in, for example, a spectrum with several nearly equal intensity strong bands. By way of comparison, a GC/IR chromatogram was reconstructed from a multicomponent sample recorded using the 2-mm diameter light pipe. The resulting chromatogram shown in Figure 5 indicates that the overall signal-to-noise ratio is approximately ten-to-one. The reconstruction was optimized for this sample and the best values for the dimensionality and displacement were again found to be 100 and 60, respectively. The number of basis vectors was optimized at 5 . This sample of methylated phenolic derivatives of paper mill waste has a total sample size of 1.0 1L;again, the solvent peak was not recorded. The large bore of this light pipe (2-mm diameter) does not significantly attenuate the infrared radiation compared to the 0.7-mm diameter light pipe and, hence, has a higher signal-to-noise ratio. Even though the larger bore light pipe has the advantages of higher signal-to-noise and fewer computations per interferogram for the reconstruction, it has a distinct disadvantage. When a gas chromatograph separation was performed on the same sample as in Figure 5 using a flame ionization detector and omitting the light pipe, approximately 30 peaks were resolved. The FID chromatogram was compared with the transform of all interferograms which led to the conclusion that the flow characteristics of the light pipe were very poor. It was later determined that there was a large dead volume a t the light pipe/gas chromatograph junction which was the cause of the poor flow (3). If the flow were to be corrected, i t is expected that all chromatographic peaks would be adequately resolved.
CONCLUSIONS Reconstructing gas chromatograms by the Gram-Schmidt orthogonalization process is a lengthy computation; however, it i s computationally more economical than the fast Fourier transform (FFT). Let us assume that the FFT obeys the ideal situation where 2N logz 2N computer operations (Le., multiplications, divisions, additions, subtractions, etc.) are required for a transform, where N equals the number of data points in the interferogram. Transforming each interferogram of 2048 points, 49 152 computer operations are required. The
spectrum must still be plotted, viewed, and evaluated. If lo00 scans have been collected for a GC/IR run, transforming involves over 49 million computer operations. T o calculate the number of computer operations for the Gram-Schmidt method there are two parts: first, the number of operations to form the basis; second, the number of operations to orthogonalize each remaining interferogram to the basis. The number of operations to form the basis is 2D(2n 1) 1, where D is the number of dimensions and n is the number of vectors. The number of operations for each subsequent orthogonalization is D(4n - 1) + 1. For 100 dimensions and 60 basis vectors, the number of computer operations to form the basis is 24201. The number of operations for each subsequent orthogonalization is 23 901. A collection of 1000 single scan interferograms would require a total of 22.5 million computer operations, or about 46% of the total FFT operations. By comparison, if only five basis vectors are used with a hundred dimensions as in the case of the 2-mm diameter light pipe, only 1.9 million operations need be performed for 1000 scans. This is 3.8% of the corresponding number of FFT computer operations. Consequently the orthogonalization reconstruction method requires much less computer time. An alternate gas chromatogram reconstruction method has recently been presented ( 4 ) . In this method small interferograms, 512 points, are transformed to give 32 wavenumber resolution spectra. The total absorbance is calculated for various bands or windows in the spectrum and plotted vs. interferogram number to give chemical functional group chromatograms. These fast Fourier transforms (FFT) are computed during collection and require over 10.2 million computer operations per 1000 spectra which makes the technique comparable in number of computations to the Gram-Schmidt algorithm. However, Coffey and Mattson did not indicate the detection limits of the small FFT technique and it is not known if the method can be applied to high noise single scan interferograms as presented in this paper; the topic is under investigation in this laboratory. Another unknown quantity in the FFT technique is that it might be possible to miss entire absorption peaks with a resolution of 32 wavenumbers. Many of the spectra transformed in the GramSchmidt study had baseline peak widths of only 20 to 25 wavenumbers. Until these unknowns are answered, it is not possible to definitively compare the two techniques. It has been found that once a set of optimum conditions for a light pipe has been determined, they remain constant and appear to be a function of the physical cell and not the operating conditions. The dimensionality and displacement
+ +
also appear to be constant from GC/TR cell to GC/TR cell leaving only the number of basis vectors to be optimized. If the operating conditions remain unchanged, a basis may be retained and used on other sample runs. When gas chromatograms are reconstructed using the Gram-Schmidt method, the infrared spectrometer is a true GC detector measuring the total infrared absorbance of the effluent in the same manner a flame ionization detector measures total oxidizable carbon. The chromatograms are then plotted as a function of the integrated total infrared absorbance (TIRA) over the entire spectral range of the spectrometer. This method of gas chromatogram reconstruction should provide very accurate quantitative infrared analysis as Beer’s law measurements need not be made on a single absorption peak; rather, the total infrared absorbance of all absorption peaks can be simultaneously measured directly from the reconstructed gas chromatogram. The Gram-Schmidt algorithm may also be used to resolve overlapped GC/IR peaks. One or more components may be mathematically removed from the unresolved peak by adding interferograms of those components to the basis and performing a new reconstruction of the chromatogram. Research is currently being carried out on these problems in this laboratory.
ACKNOWLEDGMENT The authors acknowledge Jonathan Brezin of the Department of Mathematics, University of North Carolina a t Chapel Hill, for his helpful discussions. The authors also thank Leo V. Azarraga, Environmental Protection Agency, Athens, Ga. for the GC/IR data.
LITERATURE CITED (1) P. R. Griffiths, “Chemical Infrared Fourier Transform Spectroscopy”, Why-Interscience, New York, N.Y.. 1975. (2) L. V. Azamaga and A. C. McCall, “Infrared Fowier Transform SpcVomeby of Gas Chromatograph Effluents”, Environmental Protection Technology Series, EPA 660/2-73-034 (1974). (3) L. V. Azarraga, Environmental Protection Agency, Athens, Ga., personal communication. (4) P. Coffey and D. R. Mattson, “A Programmable Specific GC Detector-A GC-FTIR System Capable of On-the-Fiy Functional Group Differentiation”, presented at the International Conference on Fourier Transform Infrared Spec!~oscopy, University of South Carolina, Columbia, S.C., June 20-24, 1977.
RECEIVED for review June 21,1977. Accepted August 19,1977. This paper was presented in part a t the International Conference on Fourier Transform Infrared Spectroscopy a t Columbia, S.C., June 1977. The financial support of the Environmental Protection Agency is greatly appreciated.
ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
1981
