Method for Unknown Vapor Characterization and Classification Using

Anal. Chem. 1999, 71, 4544-4553

Method for Unknown Vapor Characterization and Classification Using a Multivariate Sorption Detector. Initial Derivation and Modeling Based on Polymer-Coated Acoustic Wave Sensor Arrays and Linear Solvation Energy Relationships Jay W. Grate*

Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, P.O. Box 999, Richland, Washington 99352 Barry M. Wise

Eigenvector Research, Inc., 830 Wapato Lake Road, Manson, Washington 98831 Michael H. Abraham

Chemistry Department, University College London, London WCIH OAJ United Kingdom

A novel method for the characterization and classification of unknown vapors based on the response on an array of polymer-coated acoustic wave vapor sensors is presented. Unlike existing classification algorithms, the method does not require that the system be trained on all samples to be identified. Instead, the solvation parameters of the unknown vapor are estimated given the sensor responses and the linear solvation energy relationship coefficients of the sorbent polymer coatings. The vapors can then be identified from a database of candidate vapor parameters. The method is implemented in a way that is analogous to multivariate calibration with classical least squares, where the individual vapor parameters are treated as pure compounds. It is not necessary to know the vapor concentration of the vapor to perform the classification. In principle, it is possible to estimate the concentration of an unknown vapor for which the system has not been trained or calibrated. It is also possible to implement the method using inverse least-squares models, based on training samples. This new method for characterizing and classifying unknown compounds based on the responses of a multivariate sorption detector is demonstrated with synthetic data. The prevailing paradigm in the use of sensor arrays for vapor identification is that the array must be trained to recognize the vapor or vapors of interest. In the array approach, the individual sensors in the array need not be perfectly selective to particular vapors. Instead, the collective responses of the sensors in the array provide a vector of data that can be processed by statistical, neural network, or other pattern recognition methods to identify a vapor or vapors in the sample. To develop pattern recognition algorithms 4544 Analytical Chemistry, Vol. 71, No. 20, October 15, 1999

for particular vapors, the array must be tested against those vapors to generate a training set. In this essentially empirical approach, compounds not in the training set cannot be identified. This is the current state of the art in chemical vapor sensor arrays. The sensor array, however, is collecting multivariate information about how the sample interacts chemically with the sensor coatings. In principle, information should be extractable from the array response to gain knowledge of the chemical properties of the sample. By analogy, an infrared spectrum on an unknown compound may not tell you the full chemical structure of that compound, but it can be used to learn about the compound’s functional groups. In this way, the unknown can be characterized. It stands to reason that the response of a chemical sensor array to a sample can be used to characterize unknown compounds if the information encoded in the sensor array data can be transformed into well-understood chemical characteristics or parameters. Polymer-coated acoustic wave sensors represent a sensor technology that is particularly well characterized in terms of the sensors’ transduction mechanisms and the interactions of analyte species with the polymeric sensing layers.1-12 These types of sensors have been investigated as array detectors by many (1) Grate, J. W.; Frye, G. C. In Sensors Update; Baltes, H., Goepel, W., Hesse, J., Eds.; VSH: Weinheim, 1996; Vol. 2; pp 37-83. (2) Grate, J. W.; Martin, S. J.; White, R. M. Anal. Chem. 1993, 65, 940A948A. (3) Grate, J. W.; Martin, S. J.; White, R. M. Anal. Chem. 1993, 65, 987A996A. (4) Grate, J. W.; Abraham, M. H.; McGill, R. A. In Handbook of Biosensors: Medicine, Food, and the Environment; Kress-Rogers, E., Nicklin, S., Eds.; CRC Press: Boca Raton, FL, 1996; pp 593-612. (5) Grate, J. W.; Abraham, M. H. Sens. Actuators B 1991, 3, 85-111. (6) Frye, G. C.; Martin, S. J. Appl. Spectrosc. Rev. 1991, 26, 73-149. (7) Martin, S. J.; Frye, G. C.; Senturia, S. D. Anal. Chem. 1994, 66, 22012219. 10.1021/ac990336v CCC: $18.00

© 1999 American Chemical Society Published on Web 09/18/1999

groups.5,13-39 Examples of acoustic wave devices used in chemical sensing applications include the thickness shear mode (TSM), the surface acoustic wave (SAW) device, and the flexural plate wave (FPW) device. Acoustic wave vapor sensors respond to any vapor that is sorbed at the sensing surface with a response that is proportional to the amount of vapor sorbed. The transduction mechanism of these sensors, which always involves a mass-loading contribution and often involves a polymer modulus change contribution, does not discriminate among sorbed species. Discrimination is dependent largely on the extent to which the applied polymer layer interacts with and sorbs particular chemical species. The interactions between vapor molecules and polymeric sorbent phases are solubility interactions, which have been (8) Martin, S. J.; Frye, G. C. Proc. of the 1992 Solid State Sensor and Actuator Workshop 1992, 27-31. (9) Grate, J. W.; Klusty, M.; McGill, R. A.; Abraham, M. H.; Whiting, G.; Andonian-Haftvan, J. Anal. Chem. 1992, 64, 610-624. (10) Grate, J. W.; Kaganove, S. N.; Bhethanabotla, V. R. Anal. Chem. 1998, 70, 199-203. (11) Grate, J. W.; Kaganove, S. N.; Bhethanabotla, V. R. Faraday Discuss. 1997, 107, 259-283. (12) Grate, J. W.; Patrash, S. J.; Abraham, M. H. Anal. Chem. 1995, 67, 21622169. (13) Carey, W. P.; Beebe, K. R.; Kowalski, B. R. Anal. Chem. 1987, 59, 15291534. (14) Carey, W. P.; Kowalski, B. R. Anal. Chem. 1986, 58, 3077-3084. (15) Carey, W. P.; Beebe, K. R.; Kowalski, B. R.; Illman, D. L.; Hirschfeld, T. Anal. Chem. 1986, 58, 149-153. (16) Ballantine, D. S.; Rose, S. L.; Grate, J. W.; Wohltjen, H. Anal. Chem. 1986, 58, 3058-3066. (17) Grate, J. W.; Rose-Pehrsson, S. L.; Venezky, D. L.; Klusty, M.; Wohltjen, H. Anal. Chem. 1993, 65, 1868-1881. (18) Rose-Pehrsson, S. L.; Grate, J. W.; Ballantine, D. S.; Jurs, P. C. Anal. Chem. 1988, 60, 2801-2811. (19) Ema, K.; Yokoyama, M.; Nakamoto, T.; Moriizumi, T. Sens. Actuators 1989, 18, 291-296. (20) Zellers, E. T.; Pan, T.-S.; Patrash, S. J.; Han, M.; Batterman, S. A. Sens. Actuators B 1993, 12, 123-133. (21) Patrash, S. J. Ph.D. Thesis, University of Michigan, 1994. (22) Patrash, S. J.; Zellers, E. T. Anal. Chim. Acta 1994, 288, 167-177. (23) Zellers, E. T.; Park, J.; Tsu, T.; Groves, W. J. Anal. Chem. 1998, 70, 41914201. (24) Zellers, E. T.; Han, M. Anal. Chem. 1996, 68, 2409-2418. (25) Zellers, E. T.; Batterman, S. A.; Han, M.; Patrash, S. J. Anal. Chem. 1995, 67, 1092-1106. (26) Schweizer-Berberich, M.; Boeppert, J.; Hierlemann, A.; Mitrovics, J.; Wiemar, U.; Rosenstiel, W.; Goepel, W. Sens. Actuators B 1995, 27, 232-236. (27) Auge, J.; Hauptmann, P.; Hartmann, J.; Roesler, S.; Lucklum, R. Sens. Actuators B 1995, 26, 181-186. (28) Rapp, M.; Boss, B.; Voigt, A.; Bemmeke, H.; Ache, H. J. Fresenius J. Anal. Chem. 1995, 352, 699-704. (29) Nakamoto, T.; Fukuda, A.; Moriisumi, T. Sens. Actuators B 1993, 10, 8590. (30) Nakamoto, T.; Fukuda, A.; Moriisumi, T.; Asakura, Y. Sens. Actuators B 1991, 3, 221-226. (31) Amati, D.; Arn, D.; Blom, N.; Ehrat, M.; Saunois, J.; Widmer, H. M. Sens. Actuators B 1992, 7, 587-591. (32) Hierlemann, A.; Wiemar, U.; Kraus, G.; Gauglitz, G.; Goepel, W. Sens. Mater. 1995, 7, 179-189. (33) Hierlemann, A.; Wiemar, U.; Kraus, G.; Schweizer-Berberich, M.; Goepel, W. Sens. Actuators B 1995, 26-27, 126-134. (34) Dickert, F. L.; Hayden, O.; Zenkel, M. E. Anal. Chem. 1999, 71, 13381341. (35) Brunink, J. A. J.; Di Natale, C.; Bungaro, F.; Davide, F. A. M.; D’Amico, A.; Paolesse, R.; Boschi, T.; Faccio, M.; Ferri, G. Anal. Chim. Acta 1996, 325, 53-64. (36) Deng, Z.; Stone, D. C.; Thompson, M. Analyst 1996, 121, 671-679. (37) Cao, Z.; Xu, D.; Jiang, J.-H.; Wang, J.-H.; Lin, H.-G.; Xu, C.-J.; Zhang, X.-B.; Yu, R.-Q. Anal. Chim. Acta 1996, 335, 117-125. (38) Ricco, A. J.; Crooks, R. M.; Osbourn, G. C. Acc. Chem. Res. 1998, 31, 289296. (39) Shaffer, R. E.; Rose-Pehrsson, S. L.; McGill, R. A. Field Anal. Chem. Technol. 1998, 2, 179-192.

modeled and systematically investigated using linear solvation energy relationships (LSERs).4,5,40,41 In this approach, vapor solubility properties are characterized and quantified by solvation parameters related to polarizability, dipolarity, hydrogen bond acidity, hydrogen bond basicity, and dispersion interactions. LSER equations correlate the log of the partition coefficient of a vapor in a polymer with the vapor solvation parameters using a series of LSER coefficients related to the polymer solubility properties. For the present purposes, the key aspect of this approach is that polymer-coated acoustic wave sensor responses are related to the solubility interactions between the polymer and the vapor, and the vapors’ solubility properties are quantified using solvation parameters. Therefore, the response vector from a polymer-coated sensor array must encode information about vapor solubility properties, and it should, in principle, be possible to transform the array data into vapor solvation parameters. These parameters would thus characterize and possibly identify vapors. Linear multivariate correlations with solvation parameters have been applied to many systems, including water/air partition coefficients,42 sorption of vapors by blood and tissue,43 toxicity of gases and vapors,40 adsorption on solid sorbents,44 adsorption on fullerene,45,46 and partitioning into gas/liquid chromatographic stationary phases.47-50 In addition, LSERs have been used to correlate various sensory measures with solvation parameters, including retention across frog olfactory mucosa, respiratory tract irritation, potency, nasal pungency thresholds, eye irritation thresholds, and odor thresholds.51-61 The partitioning of vapors (40) Abraham, M. H. Chem. Soc. Rev. 1993, 22, 73-83. (41) Abraham, M. H.; Andonian-Haftvan, J.; Du, C. M.; Diart, V.; Whiting, G.; Grate, J. W.; McGill, R. A. J. Chem. Soc., Perkin Trans. 2 1995, 369-378. (42) Abraham, M. H.; Andonian-Haftvan, J.; Whiting, G.; Leo, A.; Taft, R. W. J. Chem. Soc., Perkin Trans. 2 1994, 1777-1791. (43) Abraham, M. H.; Weathersby, P. K. J. Pharm. Sci. 1994, 83, 1450-1456. (44) Abraham, M. H.; Walsh, D. P. J. Chromatogr. 1992, 627, 294-299. (45) Abraham, M. H.; Du, C. M.; Grate, J. W.; McGill, R. A.; Shuely, W. J. J. Chem. Soc., Chem. Commun. 1993, 1863-1864. (46) Grate, J. W.; Abraham, M. H.; Du, C. M.; McGill, R. A.; Shuely, W. J. Langmuir 1995, 11, 2125-2130. (47) Abraham, M. H.; Whiting, G. S.; Doherty, R. M.; Shuely, W. J. J. Chromatogr. 1991, 587, 229-236. (48) Abraham, M. H.; Whiting, G. S.; Andonian-Haftvan, J.; Steed, J. W.; Grate, J. W. J. Chromatogr. 1991, 588, 361-364. (49) Abraham, M. H.; Whiting, G. S.; Doherty, R. M.; Shuely, W. J. J. Chromatogr. 1990, 518, 329-348. (50) Abraham, M. H.; Whiting, G. S.; Doherty, R. M.; Shuely, W. J. J. Chem. Soc., Perkin Trans. 2 1990, 1451-1460. (51) Abraham, M. H.; Andonian-Haftvan, J.; Cometto-Muniz, J. E.; Cain, W. S. Fundam. Appl. Toxicol. 1996, 31, 71-76. (52) Nielsen, G. D.; Abraham, M. H.; Hansen, L. F.; Hammer, M.; Cooksey, C. J.; Andonian-Haftvan, J.; Alarie, Y. Arch. Toxicol. 1996, 70, 319-328. (53) Abraham, M. H. In Indoor Air and Human Health; Gammage, R. B., Berven, B. A., Eds.; Lewis Publishers: New York, 1996; pp 67-91. (54) Alarie, Y.; Nielsen, G. D.; Abraham, M. H. Pharmacol. Toxicol. (Copenhagen) 1998, 83, 270-279. (55) Alarie, Y.; Schaper, M.; Nielsen, G. D.; Abraham, M. H. Arch. Toxicol. 1998, 72, 125-140. (56) Alarie, Y.; Schaper, M.; Nielsen, G. D.; Abraham, M. H. SAR QSAR Environ. Res. 1996, 5, 151-165. (57) Alarie, Y.; Andonian-Haftvan, J.; Nielsen, G. D.; Abraham, M. H. SAR QSAR Environ. Res. 1995, 134, 92-99. (58) Abraham, M. H.; Kumarsingh, R.; Cometto-Muniz, J. E.; Cain, W. S. Ann. N. Y. Acad. Sci. 1998, 855, 652-656. (59) Abraham, M. H.; Kumarsingh, R.; Cometto-Muniz, J. E.; Cain, W. S. Toxicol. Vitro 1998, 12, 403-408. (60) Abraham, M. H.; Kumarsingh, R.; Cometto-Muniz, J. E.; Cain, W. S. Toxicol. Vitro 1998, 12, 201-207. (61) Abraham, M. H.; Kumarsingh, R.; Cometto-Muniz, J. E.; Cain, W. S. Arch. Toxicol. 1998, 72, 227-232.

Analytical Chemistry, Vol. 71, No. 20, October 15, 1999

4545

into sorbent polymers at 298 K has been investigated with LSERs (correlation coefficients were typically 0.99),41 and these LSER equations have been used to estimate the responses of polymercoated SAW vapor sensors.12 In addition, LSERs have been developed that correlate the responses of polymer-coated SAW devices to vapor solvation parameters.62 These yield LSER coefficients related to partitioning and detection of vapors with polymer films on SAW device surfaces. In this paper, we will describe a data-processing method for characterizing analytes for which the “spectrum” or pattern from a multivariate instrument has not been determined in advance from experimental calibrations. Our method is based on a multivariate detection method related to thermodynamic partitioning, and we will derive the approach for polymer-coated acoustic wave vapor sensors. More generally, however, the method relates to the interpretation of data from a multivariate detector where the response of each channel of the detector can be modeled by a linear relationship correlating response with sample descriptors. The descriptors of unknown samples are then extracted from the instrument response. The ability to extract these descriptors in order to chemically classify compounds that were not in the training of the instrument represents a significant advance beyond current empirical classification methods. THEORY LSER Models. The equilibrium distribution of a vapor between the gas phase and a polymeric sorbent phase on the sensor surface is given by the partition coefficient, K, which is the ratio of the concentration of the vapor in the sorbent polymer, Cs, to the concentration of the vapor in the gas phase, Cv.

K ) Cs/Cv

(1)

The response of a mass-sensitive acoustic wave sensor to absorption of a vapor into the polymeric sensing layer is related to the partition coefficient as shown in eq 2.63-67

∆fv ) n∆fsCvK/F

(2)

The sensor’s response to the mass of vapor absorbed, a frequency shift denoted by ∆fv, is dependent on the frequency shift due to the deposition of the film material onto the bare sensor (a measure of the amount of polymer on the sensor surface), ∆fs, the vapor concentration, the partition coefficient, and the density of the sorbent phase, F. If the observed response is entirely due to massloading, n ) 1. This result is observed if the acoustic wave sensor is a TSM device with an acoustically thin film.10,11,68 This result can also be observed if a SAW device has an acoustically thin film of a polymer with low initial modulus.7,10,11 If a modulus decrease of the polymer due to vapor sorption also contributes (62) Patrash, S. J.; Zellers, E. T. Anal. Chem. 1993, 65, 2055-2066. (63) Janghorbani, M.; Freund, H. Anal. Chem. 1973, 45, 325-332. (64) Edmunds, T. E.; S.West, T. Anal. Chim. Acta 1980, 117, 147-157. (65) McCallum, J. J.; Fielden, P. R.; Volkan, M.; Alder, J. F. Anal. Chim. Acta 1984, 162, 75-83. (66) Wohltjen, H. Sens. Actuators 1984, 5, 307-325. (67) Grate, J. W.; Wenzel, S. W.; White, R. M. Anal. Chem. 1991, 63, 15521561. (68) Martin, S. J.; Frye, G. C. Proc. IEEE Ultrason. Symp. 1991, 393-398.

4546


to the frequency shift, n can be some number greater than 1, with values from 2 to 4 suggested for certain polymers.9-11 Whatever the value of n, the observed response is proportional to the amount of vapor sorbed as expressed by the partition coefficient. The LSER method for understanding and predicting polymer/ gas partition coefficients is based on eq 3, which expresses log K as a linear combination of terms that represent particular interactions.4,5,40 H H 16 log K ) c + rR2 + sπH 2 + a∑R2 + b∑β2 + l log L

(3)

H H 16 are In this relationship, R2, πH 2 , ∑R2 , ∑β2 , and log L solvation parameters that characterize the solubility properties of the vapor,40 where R2 is a calculated excess molar refraction parameter that provides a quantitative indication of polarizable n and p electrons; πH 2 measures the ability of a molecule to H stabilize a neighboring charge or dipole; ∑RH 2 and ∑β2 measure effective hydrogen bond acidity and basicity, respectively; and log L16 is the liquid/gas partition coefficient of the solute on hexadecane at 298 K (determined by gas/liquid chromatography). The log L16 parameter is a combined measure of exoergic dispersion interactions that increase log L16 and the endoergic cost of creating a cavity in hexadecane leading to a decrease in log L16. The LSER equation for a particular polymer is determined by regressing measured partition coefficients for a diverse set of vapors on that polymer against the solvation parameters of the test vapors. The regression method yields the coefficients (s, r, a, b, and l) and the constant (c) in eq 3. These coefficients are related to the properties of the sorbent polymer that are complementary to the vapor properties. The polymer LSER coefficients will henceforth be referred to as the polymer parameters. LSER coefficients for the polymers are generally obtained by regression of partition coefficients determined by gas chromatographic measurements, but they may also be determined from the responses of a masssensitive acoustic wave device with a thin film of the polymer.41,62 Using acoustic wave sensor responses to determine the LSER coefficients would represent an initial training on known compounds. Solvation parameters have been tabulated for some 2000 compounds, and LSER equations derived from chromatographic measurements at 298 K have been reported for 14 sorbent polymers suitable for use on acoustic wave devices.41 These data can be used to set up a matrix of calculated log K values for hundreds of vapors on those polymers. These log K values can be converted to K values and then used to estimate sensor responses according to eq 2.12 Such estimates can be used to select polymers offering sensitivity to particular vapors and to estimate limits of detection. Classical Least-Squares Approach. Classification of vapors using sensor array responses to obtain vapor solvation parameters can be formulated in a manner analogous to classical least-squares (CLS) formulations used in absorbance spectroscopy.69 Matrix R (samples by channels), containing the responses of a spectrometer, is modeled as

(69) Beebe, K. R.; Pell, R. J.; Seasholtz, M. B. Chemometrics: A Practical Guide; John Wiley and Sons: New York, 1998.

R ) CS

(4)

where C is a matrix of concentrations (samples by analytes) and S is a matrix of pure component spectra (analytes by channels). If S is known, the concentrations C can be obtained given R.

C ) R ST(SST)-1

(5)

Calculation of log K values and sensor responses from LSERs can be reformulated in matrix algebra notation as follows. Matrix L, containing log K values, can be calculated according to eq 6.

L ) VP + 1c

(6)

Matrix V (number of vapors by 5) contains the vapor solvation parameters, and matrix P (5 by number of polymers) contains the polymer parameters. The constants of the LSER equations are given by the vector c (1 by number of polymers), and 1 is a vector of ones (number of vapors by 1). Conversion of the predicted log K values according to eq 6 to estimated sensor responses, assuming mass-loading responses, can then be represented by eq 7.

R ) C 10(VP + 1c)D-1F

(7)

Matrix R (vapors by polymers) contains the estimated response values as frequency shifts for particular vapor/polymer combinations. Equation 7 is similar to eq 2 (n ) 1), where C (number of vapors by number of vapors) is a diagonal matrix of the concentrations of the vapors and F (number of sensors by number of sensors, or number of polymers by number of polymers) is a diagonal matrix of the ∆fs values of the sensors. Similarly, D (number of polymers by number of polymers) is a diagonal matrix of the polymer densities. The superscript of -1 denotes the inverse of the matrix. As just set out, matrix R is a set of predicted sensor responses that can be used for evaluation and modeling. Alternatively, eq 7 can be rearranged to solve for V using a matrix R containing the observed responses from an array of sensors to various single vapors. A single vector within R represents the pattern vector for a vapor. As we shall set out below, the pattern vector can be used to determine the solvation parameters of the test vapor provided that the required properties of the sensors are known. The properties of the polymer films that are required are the polymer densities, the thicknesses of the films on the sensors in terms of ∆fs, and the polymer parameters, represented in D, F, P, and c above. Sensors for which these properties are known shall be defined as “characterized” sensors. Rearranging, taking the log of both sides and then subtracting 1c from both sides of eq 7, one obtains

log(C-1RDF-1) - 1c ) VP

(8)

To solve for vapor solvation parameters in V, it is necessary to remove the P matrix from the right side of eq 8. Since P is not a square matrix, and inverses are only defined for square matrices, one cannot simply multiply by the inverse of P. However, both

sides can be multiplied by PT(PPT)-1, the pseudoinverse of P, yielding

{log(C-1RDF-1) - 1c}PT(PPT)-1 ) V

(9)

The superscript T denotes the transpose of a matrix. It is important to note that the PPT term represents a 5 by 5 square matrix of the same rank as P. It should be easily invertible provided that the P matrix is of full rank; i.e., the set of polymers exhibits independent variations in all five polymer parameters. The PPT term must be well conditioned, and the stability of the approach requires that a diverse set of polymers representing all the solubility properties of the LSER model is included in the array. According to eq 9, the responses of an array of characterized sensors to a vapor of known concentration could be used to determine the solvation parameters of the test vapor. However, in the identification of an unknown vapor, the concentration would not be known. Therefore, the real question is whether the solvation parameters for a vapor can be determined without the concentration; i.e., can one solve for the parameters in V without C? To show how this can be done, two new matrices must be defined. The matrix Va is the matrix V augmented by the log of the vapor concentrations. Thus, this matrix has a column containing log of vapor concentrations in addition to the five columns containing vapor solvation parameters. In matrix algebra,

Va ) [V log(diag(C))]

(10)

Similarly, a matrix Pa is defined as the matrix P augmented by a vector of 1’s of appropriate dimension (1 by number of polymers). Thus, this matrix contains a row of 1’s at the bottom in addition to the five rows of polymer parameters. In matrix algebra,

Pa ) P 1

[ ]

(11)

Using these new matrices, eqs 12-14 can be derived following the approach in eqs 7-9.

R ) 10(VaPa+1c)D-1F

(12)

log(RDF-1) - 1c ) VaPa

(13)

{log(RDF-1) - 1c}PaT(PaPaT)-1 ) Va

(14)

Equation 12 is essentially the same as eq 7, except that the log of the vapor concentrations has been placed in the exponential term. This is equivalent to placing the concentration in front of the exponential term as in eq 7, since multiplying by a constant is the same as adding to a log term. It is assumed in these equations that all the sensors in the array give responses that are linear with concentration within the concentration range being considered. Then the difference in pattern from one concentration to another is simply a common multiplicative factor across all sensors. Also, in eq 14, PaPaT must be invertible. The PaPaT term will be 6 by 6 and should be easily invertible provided that the matrix of polymer parameters P is of full rank (5) and that none Analytical Chemistry, Vol. 71, No. 20, October 15, 1999

4547

of the five polymer parameters are constant over all the polymers. This means the set of polymers in the array must be diverse, as previously noted in connection with eq 9. The power of eq 14 is that it is not necessary to know the concentration of the vapor independently in order to solve for the vapor solvation parameters. Instead, the solvation parameters and log of the concentration of an unknown vapor can be solved for simultaneously using the responses of an array of characterized sensors. The vapor parameters can then be compared with a tabulation of vapor parameters to classify the vapor. This represents a fundamentally different way to classify vapors from array data. Provided that the array has “characterized sensors”, vapors can be classified that the instrument has never been trained on. In addition, the concentration of a vapor can be estimated even if its identity is unknown and no experimental calibrations on that vapor have been performed. In this approach, all the vapor parameters are solved for simultaneously. It is similar to a classical least-squares solution in absorbance spectroscopy, where the observed response, R, is used to obtain the concentrations C given the analyte pure component responses S. Inverse Least-Squares Approach. An alternative approach is to solve for each vapor parameter individually. This is the inverse least-squares (ILS) approach, where an individual parameter, y, is modeled as a weighted sum of the responses

y ) Xb

(15)

where X is the measured response and b is a vector of weights, generally determined by regression:

b ) X+y

(16)

where X+ is the pseudoinverse of X. This pseudoinverse is defined differently depending upon the type of regression to be used.69,70 In multiple linear regression (MLR, i.e., ordinary least squares)

X+ ) (XTX)-1XT

(17)

In systems where the variables in X are expected to collinear, other pseudoinverses are used such as those defined by principal components regression (PCR) or partial least-squares (PLS) regression. In the case under consideration, y would correspond to one of the five vapor solvation parameters or concentration and X would be the (log) array response. In this system, collinearity is expected any time there are more than five sensors in the array and MLR would not be an appropriate technique for developing a model of the form in eq 15. In this work, PLS will be used. EXPERIMENTAL SECTION Modeling Data. A matrix of predicted log K values was calculated beginning with a table of solvation parameters for 280 vapors. The parameters were taken from published tabulations.40,42 Vapors included alkanes (24), cycloalkanes (11), alkenes (including dienes and cycloalkenes) (18), terminal linear alkynes (7), (70) Wise, B. M.; Gallagher, N. B. J. Process Control 1996, 6, 329-348.

4548 Analytical Chemistry, Vol. 71, No. 20, October 15, 1999

Table 1. Polymers in the Arraya abbrev

description

PIB poly(isobutylene) PECH poly(epichlorohydrin) OV25 OV202 PVPR PVTD PEM SXCN PEI SXPYR FPOL SXFA

an OV stationary phase an OV stationary phase poly(vinyl proprionate) poly(vinyl tetradecanal) poly(ethylene maleate) an OV stationary phase poly(ethylenimine) a polysiloxane fluoropolyol a polysiloxane

characteristics nonpolar aliphatic hydrocarbon material slightly basic ether linkages and slightly dipolar chloromethyl groups polarizable phenyl groups dipolar nonbasic trifluoropropyl groups moderately basic esters acetal and residual alcohol groups dipolar basic ester linkages dipolar basic cyanopropyl groups basic amine linkages basic dipolar aminopyridyl groups strong hydrogen bond acid strong hydrogen bond acid

a The LSER ceofficients for these polymers, indicating chemical diversity, have the following ranges: parameter, range; r, -1.0 to 0.67; s, 0.37-2.7; a, 0.18-6.78; b, 0.0-4.25; l, 0.72-1.0.

fluoroalkanes (2), chloroalkanes (21), chloroalkenes (5), bromoalkanes (10), iodoalkanes (7), ethers (8), aldehydes (11), ketones (12), esters (15), nitriles (8), amines (12), nitroalkanes (7), dimethylamides (2), alkanoic acids (6), alcohols (14), fluoro alcohols (3), thiols (7), sulfides (3), organophosphorus compounds (2), aromatic hydrocarbons (11), chlorobenzenes (4), bromo- and iodobenzenes (6), various aromatic compounds with oxygencontaining functional groups (7), various aromatic compounds with N-containing functional groups (4), phenols (22), and pyridines (11). The solvation parameter ranges represented by these vapors were the following: parameter, range; R2, -0.64 to 1.453; πH 2, H 16 0-1.33; ∑RH 2 , 0-0.77; ∑β2 , 0-1.06; and log L , 1.2-5.5. LSERs and densities for a diverse set of 12 polymers were taken from previous papers.12,41 These polymers are listed in Table 1. The matrix of log K values was converted to a matrix of estimated sensor responses, assuming mass-loading responses, 250 kHz of material on each sensor, and a concentration of 5000 mg/m3 for each vapor. This produced a matrix of estimated responses, R, for use in modeling studies. For some purposes, this matrix was divided into a training set with 195 vapors and a prediction set containing 85 vapors. Vapors from each of the various compound classes were distributed proportionately between the training and prediction sets. In addition, each vapor was labeled with a compound class chosen from the list above. Calculations. After setting up the original matrices for V, P, L, and R in an Excel spreadsheet, all further calculations were performed in MATLAB version 5.2 (The MathWorks, Natick MA) with PLS_Toolbox 2.0 (Eigenvector Research, Manson, WA). RESULTS AND DISCUSSION Modeling Studies by Classical Least-Squares-Characterized Sensors. A matrix R (12 by 280) containing vapor sensor responses ∆fv was calculated as described above, where V (5 by 280) contained solvation parameters for 280 diverse vapors, P (12 by 5) contained polymer parameters for 12 diverse polymers, and vector c (1 by 12) contained the constants for those polymers. Additional details are given in the Experimental Section. This matrix was used as the basis for modeling studies to investigate approaches for determining vapor parameters from sensor array responses. Two subsets of the 12-polymer set were also examined

Figure 1. RMSEP for the five vapor LSER parameters and concentration as a function of fraction proportional noise in the response for the CLS model.

in some experiments. Removal of FPOL and SXFA from the 12polymer set yielded a 10-polymer set lacking a strong hydrogen bond acid polymer. Thus, this represents a less diverse polymer set.71 Removal of PVPR and PVTD from the 12-polymer set yielded a 10-polymer set that preserved chemical diversity in the array. Initial calculations were carried out with all vapors at 5000 mg/ m3 concentration. Given characterized sensors (i.e., D, F, P, and c known), the vapor parameters, V, can be calculated from R to machine accuracy. This is simply a rearrangement of the original calculations to obtain R. Then matrix R was modified so that the vapors were at random concentrations between 0 and 5000 mg/ m3. Given characterized sensors, it was verified that Va could be calculated from R, obtaining the vapor parameters and the vapor concentrations correctly to machine accuracy. Plots of predicted parameters and concentrations against the actual parameters and concentrations are perfectly linear with slopes of 1. These calculations began with essentially perfect noiseless data. The effect of measurement noise on the determination of vapor parameters and concentration was investigated by adding noise to the sensor responses in R. The added measurement noise was proportional to the response and was normally distributed. The noise was added independently across the polymers. Carey and Kowalski reported measurement noise of 2-5% relative to the magnitude of the response for TSM vapor sensors and used a value of 3% in simulations of TSM sensor arrays.14 Frye and co-workers have shown data for repeated exposure of a SAW vapor sensor that indicate measurement noise on the order of 1% or less.72 Vapor parameters and concentrations were calculated by solving for Va, and the errors in these results were determined (71) Grate, J. W.; Kaganove, S. N.; Patrash, S. J. Anal. Chem. 1999, 71, 10331040.

as a function of the added measurement noise. The root-meansquare errors of prediction (RMSEP) for each of the parameters and the concentration are plotted versus fraction noise in the data (e.g., 0.1 indicates that the standard deviation of the noise was 10% of the sensor signal) in Figure 1. Each line on the plot corresponds to a different set of polymers. The solid line includes the 12 polymers in Table 1. With the exception of concentration, the errors grow approximately linearly with noise, as would be expected. Concentration errors grow approximately exponentially with noise. This is a result of the fact that the log of the concentration is predicted, and it must be transformed. The results for a set of 10 diverse polymers are similar to those for the 12polymer set, but a 10-polymer arrray lacking hydrogen bond acid polymers gives poorer results, especially for the ∑βH 2 parameter (as might be expected). The errors in the original solvation parameter scales can be H H taken as about 0.03 unit for the π∑H 2 , ∑R2 , and ∑β2 parameters. The error for the log L16 parameter can be taken as 0.1 unit or less. These parameters are all related to free energies and were determined from experimental data on partitioning or complexation equilibria.73-77 The R2 parameter is different, since it is calculated from molar refraction values for liquids and extended (72) Frye, G. C.; Gilbert, D. W.; Colburn, C.; Cernosek, R. W.; Steinfort, T. D. Field Screening Methods Hazard. Wastes Toxic Chem., Proc. Int. Symp. 1995, 2, 715-726. (73) Abraham, M. H.; Grellier, P. L.; Prior, D. V.; Duce, P. P.; Morris, J. J.; Taylor, P. J. J. Chem. Soc., Perkin Trans. 2 1989, 699-711. (74) Abraham, M. H.; Grellier, P. L.; Prior, D. V.; Morris, J. J.; Taylor, P. J. J. Chem. Soc., Perkin Trans. 2 1990, 521-529. (75) Abraham, M. H.; Whiting, G. S.; Doherty, R. M.; Shuely, W. J. J. Chromatogr. 1991, 587, 213-228. (76) Abraham, M. H.; Fuchs, R. J. Chem. Soc., Perkin Trans. 2 1988, 523-527. (77) Abraham, M. H.; Grellier, P. L.; McGill, R. A. J. Chem. Soc., Perkin Trans. 2 1987, 797-803.


4549

Figure 2. Average number of vapors within solvation parameter error bound of 2 times the standard error as a function of the noise in the frequency shift response of the array. The lower trace (solid line) represents the analysis using all 12 polymers, a diverse set. The upper trace (short dashes) represents the results using a 10-polymer set lacking strongly hydrogen bond acidic polymers fluoropolyol and SXFA. The middle trace (long dashes) was created using a diverse set of 10 polymers (PVPR and PVTD left out).

by a group contribution scheme.50 The parameter errors in Figure H H 1 for π∑H 2 , ∑R2 , and ∑β2 are approximately 0.06, 0.02, and 0.03, respectively, for 20% noise in the sensor responses. This is comparable to the error in the original parameters. The log L16 error at 20% sensor noise is somewhat larger at 0.3-0.4 log unit. Once sensor responses in R have been used to solve for Va, the found solvation parameters can be matched to tabulated solvation parameters for known vapors. The effect of measurement noise on this matching process for vapor identification was examined. Given the prediction error information just described, it is reasonable to construct error bounds of 2 times the RMSEP around each of the vapor parameters for each vapor in Va. This is equivalent to a 2 standard deviation bound around the predictions. For each vapor, it is possible to determine how many other vapors in Va fit within this bound. The optimal answer is one, where the only vapor that fits within the error bound is the correct one. As the noise increases and the error bounds increase, more vapors will fit. The results of this analysis are shown in the lowest trace in Figure 2, plotting the average number of matches for each vapor as a function of the added measurement noise. Here we are considering the lower (solid) line on the plot for all 12 polymers. For noise levels up to about 10%, typically two or fewer vapors are within the error bound, suggesting the ability to identify the correct vapor will be pretty good up to this noise level. Above this, the number of vapors within the error bounds tends to grow more rapidly. Nevertheless, even at 20% noise, the number of vapors fitting the solvation parameters within error bounds is still limited (about 5 or 6). It is worth noting that this is a conservative evaluation of identification “precision”, since independently derived limits define a larger space than a group determination of the error bounds. Vapors within some compound classes tend to have larger numbers of vapors fitting within the error bounds for each vapor than those in other compound classes. For example, there are an average of 15 vapors, all alkanes, fitting within the error bounds for each alkane at the 20% noise level. This result is due to the 4550


fact that alkanes are distinguished from one another only by their log L16 values (i.e., they are very similar to one another), the data set contains many alkanes, and many isomers are included. For all other vapor classes, the results are much better, and the results averaged over all vapors, shown in Figure 2, are skewed to higher values by the poorer results for alkanes. The plots in Figure 3 show the average number of vapors fitting within the error bounds with the correct vapor for each compound class, based on modeling with all 12 polymers. We shall define a compound fitting within the error bounds for another compound as an error. In-class errors and out-of-class errors are indicated. The results are quite good at 10% noise and a diverse set of polymers. Except for ethers and ketones, most errors are within class. At 20% noise, out-of-class errors increase somewhat, especially for ethers, ketones, and aldehydes, all vapors with basic oxygen-containing functional groups. Because the derivation for this analysis approach indicates that a diverse set of polymers is required, the accuracy of vapor identification was also examined using a less diverse polymer set. The two hydrogen bond acidic polymers were removed and the results with this 10-polymer set were determined, as shown in Figure 2. Because these hydrogen bond acidic polymers are not commercially available, this set represents the type of less diverse array that will most likely occur. As seen in the graph, vapor classification is not too bad at measurement noise of 5% or less, but it becomes significantly degraded relative to a diverse array at measurement noise above 10%. To demonstrate that this effect is related to diversity rather than polymer number, the same analysis was done with a 10-sensor array that included the hydrogen bond acidic polymers. This array gives results similar to those of the diverse 12-sensor array (see Figure 2). The array lacking hydrogen bond acids yielded more out-of-class errors than the diverse arrays, as found by examining plots (not shown) similar to those in Figure 3. At 10% noise, overall results are not bad, but out-of-class errors are notable for esters, ethers, ketones, and aldehydes. At 20% noise, there are large numbers of out-ofclass errors in most compound classes. The reason for the effect of polymer diversity on the prediction error is suggested by the form of eq 12. Note the (PaPaT)-1 in the equation. If the matrix PaPaT is ill-conditioned, the problem will be subject to considerable numerical instability. Small changes in the response due to noise will result in large changes in the predictions, an undesirable effect. The amount of ill-conditioning present can be assessed by calculating the condition number of the matrix. The condition number is the ratio of the largest to smallest singular value of the matrix. When all 12 polymers are considered, the condition of PaPaT is 5947. When the hydrogen bond acid polymers are removed, the condition number jumps to 9562. This increase in the condition number is, in part, responsible for the increase in prediction errors. The condition number of the PaPaT matrix was calculated when PVPR was left and PVTD was 5998. Thus, leaving out these polymers had little effect on the condition of the matrix. Overall, these results demonstrate the concept that a sensor array consisting of characterized sensors should be able to classify an unknown vapor in terms of its solvation parameters and match it to a limited number of vapor candidates. The technique can also provide an estimate of the unknown vapor concentration. The

Figure 3. Average number of errors per vapor according to compound classes (number of errors in class/number of members in class), showing in-class and out-of-class errors. Errors are defined as vapors whose parameters fit within solvation parameter error bound of another vapor, where the error bounds are taken as 2 times the standard error. Number of compounds in each class is indicated. Results are shown for 10 and 20% noise levels in the frequency shift response of the array.

concentration estimation, however, is much more sensitive to the measurement noise. The derivation for this approach assumes that patterns are constant regardless of vapor concentration; i.e., sensor calibration curves are linear. The tolerance for noise in solving for vapor parameters and matching to known vapors suggests that the method may also tolerate moderate nonlinearity in sensor calibration curves. Modeling Studies by Inverse Least Squares. Modeling was also carried out using ILS methods to determine models for each individual vapor parameter from sensor responses as given in eq 15. In this approach, the sensor response data can be empirically used without knowing the polymer parameters. In other words, one need not have characterized sensors as described above. The matrix of sensor responses to particular vapors in V was divided into training and prediction sets. Models were developed using PLS with six latent variables, training on R and C to get V. PLS models developed for each vapor solvation parameter with the sensor responses in the training set were able to predict the parameters for the vapors in the prediction set to machine accuracy. However, this test was based on perfect data. The effect of measurement noise was investigated by adding noise to both the training set and the prediction set. PLS models were developed using the training set data with noise added. Then the ability to predict the vapor parameters of the vapors in the prediction set using the “noise-added” response data was tested. The results are shown in Figure 4. These results are very similar to those for the CLS models shown in Figure 2. In fact, the ILS models perform modestly better than the CLS models. Thus, it appears reasonable that one could train on sensor responses to develop models to predict vapor solvation parameters even if the polymer parameters are not known. These models could then be used to classify unknowns that were not in the training.

Application Considerations. A polymer-coated acoustic wave device responds to any molecules captured at the surface, including those that adsorb as well as those that absorb. The selectivity for a vapor adsorbing at a polymer/surface interface or a bare surface (if the film is not continuous) will be different from that of vapor absorbing into the polymer.78,79 Therefore, the LSER for the polymer may not effectively model the adsorption of vapor at interfaces, nor will it necessarily model the response of a sensor for which significant adsorption effects occur. The method described will be most effective if the ratio of polymer volume to sensor surface area is maximized and the surface is minimally adsorptive. This suggests the use of acoustic wave devices such as the TSM or FPW sensor that employ thicker polymer films (thickness in absolute terms, not in terms of frequency shift on application). A SAW device tends to use thinner films, and practical film thicknesses decrease with increasing frequency at the same time the sensitivity to adsorbed mass in increasing. Higher frequency SAW devices are not preferred for this method. Polymer-coated acoustic wave devices may or may not have a modulus contribution to their vapor responses. Modulus contributions may be avoided (n ) 1 in eq 2) by choosing TSM devices with acoustically thin films. Because the acoustic wave device has some sensitivity to adsorption, and may include modulus contributions that are specific to each polymer, it may be advantageous to obtain the polymer parameters from LSERs derived from sensor response (78) Bodenhofer, K.; Hierlemann, A.; Noetzel, G.; Weimar, U.; Goepel, W. Anal. Chem. 1996, 68, 2210-2218. (79) McGill, R. A.; Grate, J. W.; Anderson, M. R. In Interfacial Design and Chemical Sensing; Mallouk, T. E., Harrison, D. J., Eds.; ACS Symposium Series 561; American Chemical Society: Washington, DC, 1994; pp 280294.


4551

Figure 4. RMSEP for the five vapor LSER parameters and concentration as a function of fraction proportional noise in the response for the ILS models.

data. This approach has been described by Zellers,62 and LSER equations with small standard deviations and high correlation coefficients were obtained from SAW sensor data. These good results suggest that any modulus effects were effectively subsumed into the LSER equation constant or coefficients. Using sensor data to obtain polymer parameters involves a calibration against many vapors of known solvation parameters. Once this training is complete, the array can be used to obtain characterization information about vapors that were not in this training set. As noted in the derivation and the experimental results, the set of polymers should be diverse.5,71 In the present study, we have illustrated this principle in terms of accoustic wave vapor sensors as the sorption detectors in an array. Other multivariate sorption detectors, such as multicolumn gas chromatographs, arrays of carbon-loaded polymer composite chemiresistors with current readout,80,81 optical vapor sensor arrays82-84 with responses related to vapor absorption that can be modeled by LSERs, or multivariate sorption detectors yet to be invented, might be treated with this new classification method. Improvement in descriptor scales, linear models, and multivariate sorption detectors that decrease errors and noise will increase the power of this data analysis technique. Discussion. This work describes a new classification method for a multivariate detector where the response of each channel (80) Doleman, B. J.; Lonergan, M. c.; Severin, E. J.; Vaid, T. P.; Lewis, N. S. Anal. Chem. 1998, 70, 4177-4190. (81) Lonergan, M. C.; Severin, E. J.; Doleman, B. J.; Beaber, S. A.; Grubbs, R. H.; Lewis, N. S. Chem. Mater. 1996, 8, 2298-2312. (82) Michael, K. L.; Taylor, L. C.; Schultz, S. L.; Walt, D. R. Anal. Chem. 1998, 70, 1242-1248. (83) Johnson, S. R.; Sutter, J. M.; Engelhardt, H. L.; Jurs, P. C.; White, J.; Kauer, J. S.; Dickinson, T. A.; Walt, D. R. Anal. Chem. 1997, 69, 4641-4648. (84) White, J.; Kauer, J. S.; Dickinson, T. A.; Walt, D. R. Anal. Chem. 1996, 68, 2191-2202.

4552


can be modeled with a linear relationship based on a set of sample descriptors. Unknown samples can then be characterized and classified in terms of those descriptors. The method is derived and modeled for a multivariate detector whose responses are related to the amount of vapor sorbed by a sensor coating, using solvation parameters as the descriptors and LSERs as the linear relationships. Both CLS and ILS methods have been demonstrated. It is not necessary that the unknown sample has been in a training set or that its concentration is known. The ability to chemically classify a compound not in a training set represents a fundamental breakthrough for chemometric analysis of multivariate detector data. A fielded instrument such as a sensor array, for example, could characterize and possibly identify an unexpected compound. In the past, sorption data for a vapor on multiple gas chromatographic stationary phases has been used in combination with “polymer parameters” describing the stationary phases to obtain values for vapor solubility parameters to be assigned to known vapors.75,85 The method was not used to characterize or identify unknowns, nor was the concentration issue addressed. The use of CLS to obtain multiple descriptors for purposes of classification is rather unusual. CLS is almost always used to obtain concentrations of multiple analytes in a mixture.69 Typically, concentration is the only parameter determined per analyte. In our case, we effectively model a single vapor as a mixture of five “pure components” related to each solvation parameter, yielding the five parameters for the test vapor. Normally, CLS requires that all analytes are known and pure “spectra” can be determined for each one experimentally (directly or indirectly). In the case of a sensor array, the “spectrum” is a pattern. In our approach, it (85) Patte, F.; Etcheto, M.; Laffort, P. Anal. Chem. 1982, 54, 2239-2247.

is not necessary to know the “spectrum” or pattern of the analyte from experimental instrument determinations. It is sufficient that the analyte can be modeled as a combination of the five “pure components” related to each solvation parameter. In this way, CLS is extended to unknown analytes to obtain characterizing information. If these characteristics are to be matched to particular compounds, it is necessary to know the solvation parameters of candidate compounds, but not the “spectrum” or pattern from the measurement instrument. This approach has a number of interesting features relevant to multivariate analysis. The training requirements are changed. Given characterized sensors as defined above, compounds can be characterized and classified even if they have not been in a training set. Given low measurement noise, one can potentially even determine the concentration of a vapor even if you do not know what it is and have never run an experimental calibration for it. The CLS approach is well suited to including redundancy and responding to faults. Redundant sensors in the array have the same polymer parameters and polymer densities, but the amount on each sensor may vary. If a redundant sensor fails in a recognizable way, the sensor can simply be dropped from the array data analysis. The remaining sensors are all characterized and the matrix algebra proceeds. Even if the faulty sensor is not directly redundant in the sense that it has the same polymer as another sensor, it may still be possible to drop it and proceed with the matrix algebra provided that the remaining sensors still represent a sufficiently diverse set. In conventional classification approaches, loss of a sensor may require development of a new pattern recognition algorithm. In principle, the CLS approach may also aid the standardization problem. If the sorption detectors respond as expected on the basis of absorption of vapor within the bulk of the polymer, then a polymer batch can be characterized for its LSER coefficients. This batch can then be used in the production of many sensor arrays. Given a measure of the amount

of polymer on each sensor, the “calibration” is effectively transferred. The polymer amount could be determined during coating by the acoustic wave device frequency shift or, alternatively, by calibration against one or a few test vapors to get the effective coating thickness. Once the solvation parameters of the detected vapor are determined, they can be used to calculate other properties for that compound, such as air/water partition coefficient, sorption in blood and biological tissues, respiratory tract irritation in mice, eye irritation and nasal pungency thresholds in man, and possibly also odor thresholds in man.40,42,43,51-61 These could be useful in environmental characterization, toxicology, and environmental health. For example, a headspace measurement in the vadose zone giving the solvation parameters and concentration of the vapor could then be used to determine the groundwater concentration using the LSER equation for the air/water partition coefficient. In general, because array responses are related to solvation parameters, and solvation parameters can be related to other properties and parameters, it also follows that array responses could be correlated directly with those other properties and parameters. ACKNOWLEDGMENT This authors are grateful for funding from the United States Department of Energy Office of Nonproliferation and National Security, NN-20, and from the Office of Environmental Science and Technology within the Department of Energy Office of Environmental Management. The Pacific Northwest National Laboratory is a multiprogram national laboratory operated for the Department of Energy by Battelle Memorial Institute. Received for review March 30, 1999. Accepted July 15, 1999. AC990336V


4553

Method for Unknown Vapor Characterization and Classification Using

Recommend Documents