Anal. Chem. 1987, 59, 1620-1626
1620
response. In the case of extremely strongly adsorbed materials on GC, it should always be possible to increase laser power until substrate ablation removes all surface adsorbates. Any technique involving a research quality Nd:YAG laser will be expensive and would not be considered routine. However, the N2 laser results indicate promise for reducing cost and complexity of a laser activated pulse voltammetry method for small electrodes. Perhaps a more fundamental long-term result of this work will be a better understanding of the relationship between surface properties and electrochemical performance. The effects of laser activation on carbon surface chemistry are presently under investigation.
LITERATURE CITED (1) Gerhardt, G. A.; Oke, A. F.; Nagy. F.; Moghaddam, B.; Adams. R. N. Brain Res. 1984, 290, 390. (2) Nagy, F.; Gerhardt, G. A.; Oke, A. F.; Rice, M. E.; Adams, R. N.; Moore, R . B.; Szentirmay, N. M.; Martin, C. R. J . Electroanal. Chem. 1985, 188. 8 5 . Hutchins-Kumar, L. D. Anal. Chem. 1986, 58, 402. Hutchins-Kumar, L. D. Anal. Chem. 1985, 5 7 , 1536. (5) Rusling. J. F. Anal. Chem. 1984, 56, 575. (6) Kumau. G. N.; Willis, W. S.: Rusiing, J. F. Anal. Chem. 1985, 57,545. (7) Thornton, D. C.; Corby, K. T.; Spendel, V. A.; Jordan, J.; Robbat, A,; Rutstrom, D. J.; Gross, M.; Ritzler, G. Anal. Chem. 1985, 5 7 , 150. (8) Laser, D.; Ariel, M. J . Electroanal. Chem. Interfacial Electrochem. 1974, 52, 291. (9) Gunsingham, J.; Fleet, B. Anawst (London) 1982, 107, 896. (IO) Hu, I.F.; Karweik, D. H.; Kuwana, T. J . Electroanal. Chem. Interfacial Electrochem. 1985, 188, 59. (11) Plock, C. E. J . Nectroanal. Chem. Interfacial Electrochem. 1989, 22, 185. (12) Taylor, R. J.; Humffray, A. A . J . Electroanal. Chem. InterfacialElectrochem. 1973, 42,347. (13) Engstrom, R. C.: Strasser, V. A. Anal. Chem. 1984, 5 6 , 136. (14) Blaedel, W. J.; Jenkins. R. A. Anal. Chem. 1974, 46, 1952. (15) Moiroux, J.; Elving, P. J. Anal. Chem. 1978, 50, 1056. (16) Wightman, R. M.; Paik, E. C.: Borman. S.;Dayton, M. A. Anal. Chem. 1978, 50, 1410. (17) Cabaniss, G. E.; Diamantis, A. A,; Murphy, W. R., Jr.; Linton, R. W.; Meyer. T. J. J . Am. Chem. Soc. 1985, 107, 1845.
(18) Wang, J.; Hutchins. L. D. Anal. Chim. Acta 1985, 767, 325. (19) Gonon, F. G.; Fombarlet, C. M.; Buda, M. J.; Pujol, J. F. Anal. Chem. 1981, 5 3 , 1386. (20) Rice, M. E.; Galus, Z.; Adams, R. N. J . Nectroanal. Chern. 1983, 143, 89. (21) Faiat, L.; Cheng, H. Y. J . Electroanal. Chem. 1983, 157. 393. (22) Stutts, K. J.; Kovach. P. M.; Kuhr, W. G.; Wightman, R . M. Anal. Chem. 1983, 55, 1632. (23) Fagan, D. T.; Hu, I.F.; Kuwana, T. Anal. Chem. 1985, 57, 2759. (24) Evans, J.; Kuwana, T. Anal. Chem. 1979, 51, 358. (25) Hershenhart, E.; McCreery, R. L.; Knight, R. D. Anal. Chem. 1984, 56, 2256. (26) Poon, M.; McCreery, R. L. Anal. Chern. 1988, 58, 2745. (27) Bishop, E.; Hitchcock, P. H. Analyst (London) 1973, 98. 475. (28) Kinoshita, K. I n Modern Aspects of Electrochemistry; Bockris, J. O’M., Conway, 8. E., White, R. E., Eds.; Plenum: New York, 1982: Vol. 14, p 557, and references therein. (29) Amatore, C.; Saveant, J. M.; Tessier, D. J. J . Hectroanal. Chem. Interfacial Electrochem. 1883, 746, 37. (30) Goldstein, E. L.; Van de Mark, M. R. Electrochim. Acta 1982, 27, 1079. (31) Adams, R. N., Electrochemistry at Solid Electrodes; Marcel Dekker: New York, 1969. (32) Petrii, 0. A.; Khomchenko, I. G. J . Electroanal. Chem Interfacial Electrochem. 1980, 706, 277. (33) Austin, D. S.;Polta, J. A,; Polta, T. Z.; Tang, A. P. C.; Cabelka, T. D.; Johnson, D. C. J . Nectroanal. Chem. Interfacial Electrochem. 1984, 168, 227. (34) Neuberger, G. G.; Johnson, D. C. Anal. Chlm. Acta 1986, 779,381. (35) Gilman, S. I n Nectroanalytlcal Chemistry. A Series of Advances; Bard, A. J., Ed.; Marcel Dekker: New York, 1967, Vol. 2, p 111. (36) Conway, B. E.; et al. Anal. Chem. 1973, 45, 1331. (37) Hoare, J. P. Nectrochim. Acta 1982, 27, 1751. (38) Hedenburg, J. F.; Freiser, H. Anal. Chem. 1953, 25, 1355. (39) Koile. R. C.; Johnson, D. C. Anal. Chem. 1979, 5 1 , 741. (40) Van der Linden, W. E.; Dieker, J. W. Anal. Chlm. Acta 1980, 179,1. (41) Lane, R. F.; Hubbard, A. T. Anal. Chem. 1976, 4 8 , 1287.
RECEIVED for review January 2, 1987. Accepted March 23, 1987. This work was supported by the donors of the Petroleum Research Fund, administered by the American Chemical Society, and by the OSU Materials Research Laboratory. Partial support from The Chemical Analysis Division of the National Science Foundation is also acknowledged.
Maximum Likelihood Quantitative Estimates for Peaks: Application to Photoacoustic Spectroscopy P. E. Poston and J. M. Harris*
Department of Chemistry, University of Utah, Salt Lake City, Utah 84112
The method of maximum Ilkellhood provldes an optlmum technique for combining measurements of dlff erlng uncertalnty. The method is approprlate for the quantitative lnterpretatlon of peaks from analytical measurements where the peak shape Is known or can be measured. Maxlmum Ilkellhood estlmates of sample concentration are derived in this work for several classes of noise havlng both homogeneous and Inhomogeneous variance. The theory Is tested by uslng numerically simulated data and then applled to detection of weak absorbances in ilqulds using pulsed laser-excited photoacoustlc spectroscopy. Maxlmum llkeiihood estimates produce detection limits whlch are a factor 5 lower than a gated measurement at the peak maximum, in agreement wlth theoretical predictions.
A large fraction of the data produced by analytical chemistry methods appears in the form of peaks rising from a base line as a function of time, wavelength, or applied voltage. 0003-2700/87/0359-1620$0 1.50/0
Numerous examples could be listed from spectroscopy, chromatography, voltammetry, and flow injection analysis. To determine the amount of sample or analyte responsible for the observed signal is a standard goal of a quantitative analytical procedure. For data that appears in the form of peaks, a number of strategies may be implemented to obtain an estimate of the sample concentration. The most straightforward approach is a measurement at a single point, usually a t the signal maximum, corresponding to the peak height ( I ) ; this strategy may be adequate when the signalto-noise ratio is large. Recently, the theory and practice of peak integration or peak area measurement were investigated for improving quantitation in chromatographic applications (2-4). Since integration has the property of averaging noise while summing up the available signal, it is a simple but potentially powerful method for reducing detection limits when flicker (l/f)noise is not predominating. The effectiveness of a data processing strategy generally improves as more “a priori” knowledge is utilized in the analysis. The concept has been implemented in quantitation 0 1987 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 59, NO. 13, JULY 1, 1987
of signal peaks by use of a “matched fiiter” where the expected shape of a peak is cross-correlated with the raw data (5-8). The zero-displacement value of this correlation function yields an optimal estimate of signal magnitude or sample concentration when the noise in the data is constant, that is, when the variance of the error distribution is independent of signal amplitude. The fact that cross-correlation of the data with the model peak shape is an optimal quantitative estimate can be shown either by minimizing the squared deviations in the time domain (6,8) or by maximizing the signal-to-noise ratio in the frequency domain (7-9). In this paper, we derive maximum likelihood or weighted least-squares estimates for the sample quantity of concentration responsible for an observed peak given an expected peak shape. The result of this time domain strategy to fitting signal peaks is equivalent to the frequency domain, “matched filter“ approach when the noise power or signal variance is constant; however, the maximum likelihood approach can provide optimal quantitation estimates even with noise sources which depend on signal amplitude such as shot noise or proportional noise. Maximum likelihood estimates of sample concentration are derived for each of the above classes of noise, as well as mixed noise sources. The theory is first tested by using numerically simulated data with pseudorandom Gaussian noise. The technique is then applied to improving detection of weak absorbances in liquids using pulsed laserexcited photoacoustic spectroscopy, the signals from which are characterized by a mix of constant and proportional noise. Maximum likelihood estimates of sample concentration produce detection limits which are a factor 5 lower than gated measurement at the peak maximum, in agreement with theoretical predictions.
THEORY For a series of n independent measurements of a sample concentration, ci, each having an uncertainty given by a standard deviation, uc,,the estimate of the mean concentration, E, which maximizes the likelihood of having made the n observations (IO) is
This least-squares expression weights each measurement by the inverse of the measurement variance. The uncertainty of this weighted average is characterized by a variance u z , which can be found by using propagation of errors (IO)
The maximum likelihood formalism can be used to obtain optimal concentration estimates from signal peaks when some known signal shape is expected from the measurement. The signal shape may be obtained by modeling the physical process responsible for the measurement ( I I ) , obtained by linear regression from a series of standard samples (IZ),or by averaging the signal from a single pure standard (8). The use of weighted linear least squares to obtain a concentration estimate requires that the signal, zit from a sample having a true concentration, E , be proportional to a shape function, gi, which is independent of concentration, such that 2 , = ?gi+ ei, where e; is the error in the signal. Under this condition, each data point in time provides an independent measure of the sample concentration ci = zJg,
(3)
the uncertainty of which depends on the nature of the errors, ei, in the measured signal. Constant Noise. When the amplitude of the noise or error is independent of the signal amplitude, then the signal
1621
standard deviation is constant, u,, = 6,. The uncertainty of the concentration as determined a t each point, i , using eq 3 can be found by a simple propagation of errors, uc, = u,/gi. Substituting these values of the standard deviation into the weighted least-squares formula of eq 1,provides a maximum likelihood estimate of the sample concentration
(4) 1
1
This result corresponds to calculating the zero-displacement value of the cross-correlation between the signal and the shape function, divided by a constant which is proportional to the mean square value of the shape function. This result is identical with the “matched filter” estimate of signal magnitude, previously described (5-8). The power of the maximum likelihood strategy becomes more evident when dealing with noise sources having inhomogeneous variance, such as shot or proportional noise described in the following sections. Shot Noise. If a signal peak arises from a very weak current or a faint optical signal, then the signal uncertainty may be dominated by the small number of electrons or photons being detected. Under these circumstances, the standard deviation of the signal is predicted by Poisson statistics (IO) to be proportional to the square root of the mean signal amplitude, uz, = K’(z~)’/~ = k’(cgL)1/2.The uncertainty in concentration at each point, i, is found by propagation of errors through eq 3, where u,, = k’(c/g,)’/2.The concentration uncertainty for shot noise limited signals depends on the square root of the inverse peak shape amplitude rather than simply the inverse of the amplitude as in the constant noise case. Substituting this concentration uncertainty relationship into eq 1yields the following optimal estimate of sample concentration:
(5) When the signal uncertainty is dominated by shot noise, therefore, the maximum likelihood estimate for the concentration is the area divided by a constant corresponding to the area of the peak shape function. Proportional Noise. When the predominant noise present in a signal arises from fluctuations in the measurement sensitivity, such as amplifier gain noise or excitation source noise, the rms noise amplitude or standard deviation of the signal . increases in proportion to signal size, u,, = ks, = k ( ~ g , )The corresponding uncertainty in the concentration obtained at each point using eq 3 can again be found from propagation of errors. This yields a simple result, uc, = u,,/g, = kE, indicating that the error in concentration is constant, independent of i. Substituting this result into eq 1 shows that the concentration measurements, c,, are unweighted in the maximum likelihood estimate, which corresponds to weighting the observed signal values, z l , by the inverse of the peak shape function
Mixed Noise Sources. While proportional noise may dominate the uncertainty of a signal in regions where the signal amplitude is large, it is unusual to find the entire waveform governed by proportional noise particularly in regions where the amplitude is small. In the vicinity of the base line with either a small value of gi or E , it is probable that signal uncertainty will not continue to decrease in proportion to signal amplitude but will rather reach a constant noise limit. Since the proportional noise and constant noise would, in general, be derived from different, independent sources, their contributions tothe overall signal variance would equal the sum of the two noise variances
1622
ANALYTICAL CHEMISTRY, VOL. 59, NO. 13, JULY 1, 1987
(7) This somewhat more complex form of the signal variance can be propagated through eq 3 to determine the concentration uncertainty which is substituted into eq 1. The resulting weighted least-squares estimate of the sample concentration is given by
For mixed proportional and constant noise, therefore, the signal contribution to maximum likelihood estimate varies between being weighted inversely with the peak shape amplitude and being weighted in proportion to the peak shape amplitude depending on the relative size of the two noise sources. This model for mixed proportional and constant noise was found to accurately describe the uncertainty in pulsed laser-excited photoacoustic spectroscopy, where proportional noise is related to shot-to-shot variation in the laser pulse energy and the acoustic sensitivity while a constant amplitude noise from the detector electronics dominates when the sample absorbance is small. E r r o r Estimates. For any of the above methods of concentration or amplitude estimation, the uncertainty of the estimate can be determined by using eq 2, where the concentration variance associated with each data point in the ~, on the uncertainty of the signal, uCc2= peak, u ~ , depends az,2/g,2. For mixed constant and proportional noise, the concentration estimate determined using Equation 8 should exhibit a variance given by at2
= 1/C[gi2/(k2z,*
+ az2)]
(9)
i
This formula is appropriate for predicting the concentration variance with any relative fraction of constant and proportional noise. It provides a theoretical basis to test the improvement in precision which a maximum likelihood approach affords for peak quantitation in general and for photoacoustic detection in particular. EXPERIMENTAL SECTION Numerical Simulations. Numerical studies were carried out on a Leading Edge Model D personal computer running Fortran 77. A Gaussian distributed pseudo-random-number generator from the NAG subroutine library was used to produce model noise having a mean of zero and a standard deviation of unity. Different sequences were added to each data set and scaled to the desired signal-to-noise ratio. Constant noise, the standard deviation or root mean square (rms) amplitude of which is independent of signal intensity, was modeled by multiplying the output from the random number generator by a single scaling factor before adding it to model signals. Proportional noise was scaled in proportion to the signal magnitude. Photoacoustic Measurements. The diagram of the photoacoustic spectroscopy experiment, used to test the above concepts in maximum likelihood estimation, is shown in Figure 1. A Nd:Yag laser pumped dye laser system (QuantaRay Model DCR-2) was used for sample excitation. The 1.06-pm, 20-Hz output of the Nd:YAG laser was frequency doubled to 532 nm and used to pump the dye laser which was operated at 560 nm and a pulse energy of 3 mJ. The dye laser beam was focused with a 45-cm focal length lens. The sample cell was positioned approximately 5 cm before the beam waist to avoid nonlinear optical effects, dielectric breakdown, and bubble formation. The cylindrical sample cell was constructed of 15 mm i.d. Pyrex tubing, 5.0 cm in length, with two glass tee sections on top t o accommodate the piezoelectric transducer assembly and filling port, respectively. To a third tee section below the cell was attached a standard Teflon stopcock. An aspirator was connected to the stopcock to facilitate easy changing of samples during data collection. This also ensured that the cell remained in exactly
I
i
I
L
C
&I trigger
Figure 1. Block diagram of pulsed-laser excited photoacoustic spectrometer: L is a 451m focal length lens, C is the sample cell, and PZT is the piezoelectric transducer. Details of the instrumentation are described in the text.
the same position relative to the laser beam throughout the experiment. The piezoelectric transducer assembly was based on the design of Tam and Pate1 (13). The output from the piezoelectric transducer was amplified with a 3-MHz bandwidth preamplifier having a voltage gain of approximately 200, as previously described (14). Background signals arising from ambient acoustic noise and vibrations were removed with a four-pole active filter having a low frequency cutoff at approximately 10 kHz, constructed from Analog Devices Model 509 operational amplifiers. The signal from the filter was fed to a 60-MHz oscilloscope (Tektronix Model 2213) for monitoring and a signal averager (Nicolet Model 1170) for data acquisition. The signal was digitized at 5 MHz for 256 channels using a flash converter plug-in unit (Nicolet Model 1134); 256 photoacoustic transients were collected and averaged for each experiment. The averaged transient signal was transferred via a serial interface (DEC Model DlV11-J) to a DEC 11/23 microcomputer for analysis. All data analyses were performed on the 11/23 using Fortran IV routines. Azulene (Aldrich) was chosen as a sample analyte for this study due to its high photostability in the visible region. The solvent, carbon tetrachloride, glass distilled, was vacuum degassed in an ultrasonic cleaner prior to mixing solutions. Degassing was virtually complete after about 10 min. Solutions were made by dilution of a single stock solution, the absorbance of which was verified with a Cary 17D spectrophotometer. Sample solutions were filtered with PTFE 0.5-pm-pore filters made by Millipore, which resulted in greatly reduced stray light from particulates. A typical data gathering and manipulation sequence is as follows. First, six replicates of the solvent blank were acquired. In general, six replicates were taken since F values and Student’s t values do not change significantly beyond this point. This was followed by the lowest absorbing sample (six replicates),and then another set of solvent blank measurements was made. The two blank data sets were averaged together, and this background signal was subtracted from the sample signal. This procedure was repeated for successively higher absorbance samples. Continuous blank measurements were made to avoid uncertainties from any slowly changing background signal level. All data were corrected for dc offset by determining the average of the transient and subtracting that value from the entire transient. The data were also converted to the piezoelectric signal in volts by dividing by the preamplifier gain, the sensitivity of the flash converter plug-in unit, and by the number of averages. The first 10 p s of the fitting function, before the arrival of the first acoustic pulse, was arbitrarily set to zero to avoid having rf noise from the Q switch influence the quantitative results. RESULTS A N D DISCUSSION Numerical Studies. The use of maximum likelihood estimates for quantitation of analytical peaks was f i s t evaluated by using numerically generated data so that the noise characteristics could be controlled. Peaks were simulated as Gaussian functions having an arbitrary maximum of unity and differing amounts of praportional and constant noise. Two examples are shown in Figure 2, parts a and b, where the ratios of the rms proportional noise amplitude and the rms constant
ANALYTICAL CHEMISTRY, VOL. 59, NO. 13, JULY 1, 1987
1623
Table I. Maximum Likelihood Concentration Estimates for Numerically Simulated Peaks d SCP
UCpe
10
2
3.8 8.5 12.7 21.2 29.6
2.2 x 1.1 x 7.4 x 4.5 x 3.2 x
10-2 10-2 10-3 10-3 10-3
3.1 X 1.6 X lo-* 1.0 x 10-2 6.2 x 10-3 4.4 x 10-3
2.1 x 10-1 1.4 X lo-' 9.6 X 5.9 x 10-2 4.2 X
2.6 X 1.2 x 7.8 X 4.7 x 3.4 x
4.2 5.3 6.2 9.3 15.6
2.1 x 10-2 1.7 X 1.6 X 1.0 x 10-2 6.3 x 10-3
3.9 x 10-2 3.1 X 2.9 X 1.9 x 10-2 1.2 x 10-2
1.7 X 1.5 X 1.6 X 1.1 x 6.5 X
lo-' lo-' 10-1 10-1
2.4 X 1.9 x 1.6 X 1.1 x 6.4 X
lo-'
9.2 12.7 12.9 13.2 13.2
10-1
10-2 10-2
lo-'
8.3 8.1 9.9 10.2 10.4
10-1
lo-' 10-1
Ratio of proportional to constant noise at peak maximum. * Signal-to-noise ratio at peak maximum. Relative standard deviation of weighted least-squares concentration estimate determined using eq 9. Sample relative standard deviation of concentration estimates, n = 10. e Relative standard deviation of peak height concentration estimate determined using eq 10. f Improvement in precision of maximum likelihood estimate over peak height measurement. noise amplitude at the peak maximum are 10.0 and 2.0, respectively. Comparing these plots shows that the signal-tonoise ratio of the data dominated by proportional noise is considerably more independent of signal amplitude. For both examples, the weighted least-squares fit is also plotted given by ii= Egi) where the concentration estimated, E, is determined from eq 8. For an ideal fit, the weighted residuals, ( i i - zi)/uE,, should form a uniform distribution about zero spanning the t limits of the Gaussian distribution. While a few outliers are found, the residuals are generally random and confined to a range of f3uz,, as shown by the example in Figure 2c. As a test of the theory developed in the section above, the uncertainty of the concentration estimate predicted by eq 9 can be compared to a sample standard deviation obtained from replicated numerical results. For two relative fractions of proportional and constant noise and five different signal to noise ratios, this comparison was made with ten replicates at each condition to determine the sample standard deviation; the results are reported in Table 1. In all cases, the predicted and observed standard deviations were indistinguishable at 95% confidence, according to an F test (IO). A second goal of these numerical simulations was to determine the improvement in precision of a maximum likelihood estimate relative to a more naive concentration determination, such as a measurement of peak height. The concentration estimate derived from the peak height, cp = z p / g p , is drawn from a population having a variance which is a single term at the maximum or peak value, i = p , from the error expression given in eq 9. This prediction of the variance in the concentration estimate for single measurements of peak height was compared to the sample variance of replicated numberical results, listed in Table I as their corresponding standard deviations. The predicted and observed variances were again indistinguishable at the 95% confidence level according to an F test. The validity of the variance predictions of eq 9 and 10, evident from the numerical studies, allows one to evaluate the improvement in precision of the maximum likelihood estimate over a simple peak height measurement. The improvements in precision are calculated from the ratios of concentration standard deviations, ucp/u:,and are also listed in Table I. The main trend found in these results is that the maximum likelihood method is more effective for signals increasingly dominated by proportional noise. This is to be expected since, for the 100%proportional noise case, the signal to noise ratio of each point in the peak is constant and contributes at equal concentration precision to the concentration estimate. Under these circumstances, one expects that the improvement in concentration standard deviation over a peak height mea-
I
b
1
0.S
0
31 I
I 1
2 1
0 -1
-2
-'I ' I -4
-s! 0
,
,
40
,
,
,
, 120
10
nm
,
, 160
,
, 200
,
,
I
240
(ARB. U N ~ S )
Figure 2. Maximum likelihood quantitative estimates for numerically simulated peaks. The ratios of the rms proportional noise amplitude and the rms constant noise amplitude at the peak maximum are 10.0 and 2.0 for parts a and b, respectively; the heavy line indicates the best fit of the data (Eg/). Part c shows the scaled residual error for part b [ ( E a - Z , ) / ~ ~ , I .
surement would reach a limiting value of n112,corresponding to 256112 = 16 for these studies; this result was indeed observed in a numerical simulation by setting the constant noise var-
1624
ANALYTICAL CHEMISTRY, VOL. 59, NO. 13, JULY 1, 1987
1so
0.4
F
$
u
1
20
lo
i o -10
-20 -SO
-0.3
f 0
40
20 TIME (CISEC)
20
Figure 3. Average photoacoustic transient for azulene in CCI,, 2 = 4.7 X cm-’. The standard deviations of each of the 256 points are plotted as symmetric error bounds about the origin using a heavy
line.
F u
ia f
10
o
-10
-20
v
w o
3
z
0
0.02
0.04
0.06
0.08
0.1
0.12
PHOTOACOUSTIC SIGNAL AMPLITUDE (mV3
Figure 4. Photoacoustic signal variance vs. the square of signal magnitude. Heavy line is the linear least squares best fit to eq 7.
iance in the simulation equal to zero. Application to Photoacoustic Measurements. In order to apply weighted least squares analysis to photoacoustic data, the linearity of the measurement must be established. Photoacoustic data were gathered on six azulene samples prepared in carbon tetrachloride uniformly spanning the absorbance to 4.7 X IO4 cm-I. Measuring the amplitude range of 1.9 x a t the maximum of the largest photoacoustic peak yielded a linear calibration plot having a correlation coefficient for an unweighted linear least squares fit, r = 0.987, and no systematic deviation from linearity. The slope of the calibration line was 0.70 i 0.05 V/AU and the intercept was 5 i 11 yV, indicating that the line passes through the origin within experimental error. T o properly compute a maximum likelihood estimate of sample concentration from an experimental data set, unlike the numerical simulation studies where the magnitudes of the proportional and constant noise were known in advance, one must determine the magnitudes of these errors from replicate measurements. This task was accomplished by averaging six replicates of the photoacoustic measurement on the highest absorbing sample, A = 4.7 X low4cm-l, and determining the reproducibility of each point along the photoacoustic transient. A plot of the average signal and the standard deviation shown as error bounds in Figure 3 reveals significant proportional character in the noise along with a small constant noise contribution which keeps the error bounds from vanishing when the signal size is small. If the proportional and constant noise sources are independent, then one would expect the signal variance to follow the relationship of eq 7. This was assessed by rearranging the data from Figure 3 as a histogram of signal
1
0
20
40
TIME (CISEC)
Flgwe 5. Maximum I i k e l i i absorbance estimates for photoacoustic signals. Photoacoustic transients from azulene in CCI, in parts a and b have absorbances A = 4.7 X cm-’ and 1.9 X cm-’, respectively; the best fits to the data (&,) are shown as heavy lines. Part c shows the scaled residual error for part b [(Eg,- z,)/cT,,].
variance plotted vs. the square of the signal magnitude as shown in Figure 4. The slope of this plot is the proportionality factor, k2,for the proportional noise variance, the square root of which is k = 0.21 f 0.01. The intercept is the constant noise variance, the square root of which is vz = 7.8 MV. The primary source of the constant noise was traced to the piezoelectric detector and its amplification electronics since nearly the same level of constant noise was observed with the laser beam blocked. Some of the proportional noise appears to be related to pulse-to-pulse fluctuations in the laser energy; if the Q switch hold-off voltage is adjusted to be less than optimal, leading to erratic pulse energy fluctuations, the proportional noise factor increases significantly. Proportional noise can also arise from convection, driven by laser heating of the illuminated region, giving rise to gradients in the solvent density which would affect the photoacoustic response (13). The average photoacoustic transient for the highest concentration sample plotted in Figure 3 was also used as the model for the shape function, gi,in order to obtain maximum likelihood quantitative estimates for lower concentration samples including several sets of replicates of the solvent blank. The results of these studies are summarized in Figure
ANALYTICAL CHEMISTRY, VOL. 59, NO. 13, JULY 1, 1987 ~~
1625
~
Table 11. Maximum Likelihood Absorbance Estimates for Photoacoustic Signals"
A, cm-'
A, cm-' 4.7 (f0.1) x 10-5 1.9 (f0.2) X solvent blank 1 solvent blank 2 solvent blank 3 solvent blank 4
4.7 x 1.5 X 6.0 X 5.4 X 4.0 X 1.8 X
10-5c
10" 10" 10" 10"
q,cm-'
si,
cm-'
2.2 x 10" 2.0 X 10" 2.2 X lo4 2.1 X 10" 2.1 X 10" 2.0 X 10"
2.7 x 2.3 X 2.3 X 2.3 X 1.6 X 1.7 X
uAp,cm-'
io"
1.4 x 10-5 1.2 X 10" 1.2 X 1.2 X 1.2 X 1.2 X
10" 10" 10" 10"
10"
sAp,cm-'
8.3 x
io"
7.8 X 10" 6.1 X lo4 1.6 X 6.0 X 10" 1.1 X
LODA~ 1.1 x 9.2 X 9.4 X 9.2 X 6.7 X 7.2 X
10-5 10" 10"
lo4 10" 10"
LoDApb
3.4 x 10-5 3.2 X 2.4 X 6.2 X loT5 2.4 X 4.4 X loT5
aAplud
6.2 6.0 5.4 5.7 5.7 5.8
sAp/sA
3.0 3.4 2.6 6.9 3.7 6.7
'The results for each sample represent six replicate trials. For definitions of terms, see footnotes in Table I. bLimits of absorbance detection defined as (14), LODA = 2t(one-sided)Sk cResultsrepresent the net sample absorbance since the blank was subtracted from these photoacoustic signals.
20
b
s -4
-2
0
2
5414
4
WeiQhted Residual (t-"due)
Figure 6. Histogram of the scaled deviations from photoacoustic transient of Figure 5c. Solid line is the model Gaussian errw distribution with no adjustment of parameters (zero mean, unit standard deviation, area equals 256 observations).
5 and Table 11. In order to assess the impact of the data analysis on the detection of weak absorbing solutions, the quantitative results are presented not in terms of concentration of the particular chromophore detected but rather in terms of absorbance per unit path length, which is equal to the product of molar absorptivity and concentration. Path length is factored out of the sample absorbance since the magnitude of the acoustic signal does not depend on path length as long as i t exceeds the diameter of the piezoelectric transducer, which is 0.4 cm in this work. The capability of a maximum likelihood estimate to extract quantitative information from noisy data is illustrated by the cm-l sample in Figure results in Figure 5. For the 1.9 X 5b; for example, the maximum likelihood estimate is in error by only 20% despite the peak signal to noise ratio being only 1.6. The scaled residuals, shown in Figure 5c, are random and of a magnitude expected for the experimental error, indicating that mast of the quantitative information has been successfully extracted. As a test of the Gaussian model for the error distribution used as the underlying basis to obtain a maximum likelihood estimate of eq 1,a histogram of the scaled deviations of Figure 5c is plotted in Figure 6 along with the model error distribution with no adjustment of parameters (zero mean, unit standard deviation, area equals number of observations). The noise of the photoacoustic signals is remarkably Gaussian, which accounts in part for the agreement between the experimental results and theoretical predictions described below. The experimental results are quantitatively summarized in Table I1 where standard deviations of peak height and maximum likelihood estimates, predicted by eq 9 and 10, respectively, are compared with sample standard deviations obtained from replicate measurements. As with the simulation studies, the expected errors could not be distinguished, using an F test at 95% confidence, from those predicted by theory. The sample standard deviations of peak height measurements, s+ appear to be more erratic than those derived from fitting the complete photoacoustic transient, SA,probably due to the
dependence of the former on the small number of particular data points drawn into the sample. The limits of absorbance detection (14) for maximum likelihood and peak height determinations are also reported in Table I1 based on the observed sample standard deviations of the replicated results. There does not appear to be a significant increase in the limits of detection for absorbance measurements made with samples exhibiting absorbance background as high as 5 X 10" cm-' which is consistent with the predictions of eq 9 and 10. The improvement in quantitative precision a t the blank level over a peak height measurement given by a maximum likelihood estimate of the complete transient record was predicted by eq 9 and 10 to be a factor of 5.6 (f0.2) on average while the observed improvement based on replicate measurements was found to average about 5.0 (f2). The uncertainty of this latter result is related to the rather erratic behavior of the peak height standard deviations, discussed above. The photoacoutstic transients making up the four sets of solvent blank replicates were processed without any background or blank subtraction step. The four determinations of the blank absorbance, A , each of which is the average of six measurements, exceed the decision limit (14) for detecting in all four trials. nonzero absorbance, [ t(one-sided)s~/6~/*], If the four results are averaged together, the absorbance of carbon tetrachloride at 560 nm may be estimated, A = 4.3 (f0.8) X lo4 cm-'. This is similar in magnitude to results obtained at 514.5 nm using thermal lens methods which produced an average value of 5.6 (f0.6) X lo4 cm-' (17, 18). As a final test of the stability of the signal background and its influence on the results, the target fitting function, g,, was first orthogonalized against the blank signal using a GramSchmidt orthonormalization procedure (15, 16) before application of the weighted least-squares fit. Orthonormalization has been shown to be a powerful method for discriminating against large fluctuating backgrounds from signals having characteristic time behavior different than an absorption signal, such as stray light on the transducer or rf noise (15). Such a fitting function can be adapted to a maximum likelihood estimate if the weighting factors from the noise distribution are derived independently of the orthonormalized model to which the data will be fit. The photoacoustic data were carried through this procedure and compared with the results in Table 11;the precision and accuracy of the estimated sample absorbances, however, were indistinguishable from those obtained without prior orthogonalization of g,. One can conclude that fluctuations in the background signal in this particular case are not dominating the uncertainty of the results.
CONCLUSIONS The method of maximum likelihood provides an optimum technique for combining measurements of differing uncertainty which is appropriate for the quantitative interpretation of peaks from analytical measurements where the peak shape is known or can be measured. A propagation of errors cal-
Anal. Chern. 1987, 59, 1626-1632
1626
culation allows one to predict both the uncertainty in the quantitative results produced by this method and the improvement in precision relative to other methods. The importance of maintaining a reproducible peak shape for this approach to data analysis should not be overlooked. In our early attempts to apply the method to photoacoustic detection, we used a sample cell which was removed between measurements to change samples. This procedure did not maintain a reproducible spatial relationship between the excitation beam and the piezoelectric transducer and produced random phase shifts in the acoustic waveform between measurements. In retrospect, it is not surprising that these earliest attempts to apply the method of maximum likelihood met with little success since the peak shape was not stable.
ACKNOWLEDGMENT The authors wish to express their appreciation to D. Heisler, D. Plumb, and M. Scott for their help with the detector electronics and data acquisition. H. Morrow is acknowledged for his fabrication of the photoacoustic cell. LITERATURE CITED (1) Sweileh, J. A.; Cantwell, F. F. Can. J . Chem. 1985, 63, 2559-2563. (2) Taraszewski, W. J.; Haworth, D. T.; Pollard. 6. D. Anal. Chim. Acta 1984, 157. 73-82.
Laeven, J. M.; Smit, H. C. Anal. Chim. Acta 1985, 176, 77-104. Synovec. R. E.; Yeung, E. S. Anal. Chem. 1985, 5 7 , 2162-2167. Mann, C. K.; Goieniewski, J. R.; Sismandis, C. A. Appl. Spectrosc. 1982, 36, 223. Lam, R. B. Appl. Spectrosc. 1983, 3 7 , 567-569. Dyer, S.A.; Hardin, D. S. Appl. Spectrosc. 1985, 3 9 , 655-662. Nickolaisen, S. L.; Bialkowski, S. E. J . Chern. I n f . Cornput. Sci. 1986, 2 6 , 57-59. Papoulis. A. Probability, Random Variables, and Stochastic Process e s ; McGraw-Hill: New York, 1965. Bevington, P. R. Data Redoction and Error Analysis for the Physical Sciences; Mc(3raw-Hill: New York, 1969. Carreira, L. A.; Antcliff, R. R. In Advances in Laser Spectroscopy; Garetz, B. A,, Lombordi, J. R., Eds.; Heyden: London, 1982; Chapter 6. Honigs, D. E.; Hieftje, G. M.; Hirschfeld, T. Appl. Spectrosc. 1984, 38, 317-322. Tam, A. C.; Patel, C. K. N. Appl. Opt. 1979, f8. 3348-3357. Currie, L. A. Anal. Chem. 1988, 4 0 , 586-593. Bialkowski, S.E. Anal. Chem. 1986, 58, 1706-1710. Brown, C. W.; Obremske, R. J.; Anderson, P. Appl. Spectrosc. 1986, 4 0 , 734-742. Dovichi, N. J.; Harris, J. M. Anal. Chem. 1981, 5 3 , 106-109. Carter, C. A.; Harris, J. M. Appl. Opt. 1984, 2 3 , 476-481.
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RECEIVED for review November 17, 1986. Accepted March 5,1987. This work was funded in part by the National Science Foundation under Grant CHE85-06667. Fellowship support (to J.M.H.) from the Alfred P. Sloan Foundation is also acknowledged.
Trace Determination of Nitrogen-Containing Drugs by Surface Enhanced Raman Scattering Spectrometry on Silver Colloids Edith L. Torres' and J. D. Winefordner*
Department of Chemistry, University of Florida, Gainesville, Florida 3261 1
The evaiuatbn of swface enhanced Raman scattering (SERS) spectroscopy on silver coliokls as an analytical technique Is presented. The use and performance of a slmple and inexpensive devke for Raman studles are evaluated by taking advantage of the enhanced signals observed In thls new technique. The results demonstrate that SERS on sliver coliolds can be successfully used for the quaiitatlve determination d N-contakrlng drugs such as p-aminobenrolc acid (PABA), phenytoln, uracll, and uracil derivatives as well as N-containing polynuclear aromatic hydrocarbons, such as 2-amkrofiuorene. Analytical cailbratlon curves of PABA and 2-amlnofluorene yield straight lines with the slopes of thelr log-log plots close to unity. The absolute lbnlts of detection for these compounds were 35 and 7 ng, respectively. Reasons for the rather poor correlation between signal and concentration for the other adsorbates studied are discussed. For some compounds, SERS has the hlgh Identitication power Inherent in all vibratlonai techniques, wlth the added advantage of being nearly as sensitive as many iumlnescence techniques presently used for trace analyticai purposes.
Since the discovery of the technique now known as surface enhanced Raman scattering (SERS)spectroscopy in 1974 most of the research reported has been oriented towards the unPresent address: Smith Kline & French Lab. Co., Call Box SKF, Cidra, PR 00639.
derstanding of the phenomenon itself (1-7). Little attention has been given to the development of the technique as an analytical method for the identification and quantitation of molecular species present at trace concentration levels. The technique of SERS consists of observing the regular Raman process at or near metallic surfaces, which are properly roughened, detecting in this way Raman bands, which can be amplified by factors ranging from 3 to 6 orders of magnitude. Its sensitivity, therefore, is similar to that of luminescence techniques with the added advantage of spectral selectivity. The simplicity of instrumentation needed to perform Raman studies, i.e., UV-visible sources, UV-visible monochromators, UV-visible detectors, glass and quartz optics, and cells, and a wide range of solvents including aqueous solutions make Raman spectrometry and especially SERS a valuable analytical method. Recently, Pemberton and Buck (8)reported the detecton of the anion diphenylthiocarbazone adsorbed at the surface of a silver electrode. By use of the multiplicative effects of surface enhancement and resonance enhancement, concentration levels as low as M were reported to give spectra with acceptable signal-to-noise (S/N) ratios. Following this line, Tran (9) reported subnanogram detection of dyes spotted on filter paper that was previously treated with colloidal silver hydrosols. Detection limits were in the picogram range level and the calibration plots exhibited linear responses. Vo-Dinh et al. (1&12), however, have directed their research toward the development of practical substrate materials that can be easily prepared and that can provide data with sufficient reproducibility and accuracy for analytical purposes.
0003-2700/87/0359-1626$01.50/00 1987 American Chemical Society
