Anal. Chem. I 9 8 8 , 58. 2091-2092 (3) Hutson, P. H.; Curzon, 0. Blochem. J . 1983. 211, 1-12. (4) Stamford, J. A. h l n Res. Rev. 1985, IO, 119-135. (5) Ponchon, J. C.; Cespuglb, R.; Gonon, F.; Jouvet, M.; Pujol, J. F. Anal. Chem. 1979, 51, 1483-1486. (6) Kovach, P. M.; Ewlng, A. Q.; Wllson, R. L.; Wightman, R. M. J . NeurOSCi. Methods 1984, 10. 215-227. (7) Gonon, F.; Fombarlet. C. M.; Bude, M. J.; Pujol, J. F. Anal. Chem. 1981, 53, 1386-1389. (8) Dayton, M. A.; Brown, J. C.; Stutts. K. J.; Wlghtman R. M. Anal. Chem. IgSO. 52, 946-950. (9) Slhrer, I . A. Med. Electron. Biol. Eng. 1965, 3 , 377-387. (IO) Meulemans, A.; Henzel, D.; Tran-Ba-Huy, P.; Silver, I . A. Innovations Tech. Blol. Med. 1985, 6, 353-362. (11) Armstrongdames, M.; Fox, K.; Mlllar, J. J . Neuroscl. Methods 1980, 2 , 431-434. (12) Tauc, L.; Hoffmann, A.; Tsuji, S.; Hlnzen, D. H.; Faille, L. Nature (London)l974, 250, 498-498. (13) Baux, 0.; Tauc, L. R o c . Natl. Aced. Scl. U . S . A . 1983, 80, 5126-5 128. (14) Poulaln, B.; Baux, G.; Tauc, L. Proc. Natl. Aced. Sci. U . S . A . 1988, 83. 170-173. (15) Marsden, C. A. Measurement of Neurotransmiifer Re/ease In Vlvo; Wiley: 1984; Chapter 6. (16) Kreuzer, F.; Klmmlch, H. P.; Brezlna, M. Med/cal and Blobgicel Applications of Electrochembal Devlces; Wiiey: 1980 Chapter 6. (17) . . Knox. R. J.: KnMt. R. C.: Edwards, D. I. 6iochem. Pharmacal. 1983. 32, 2149-2156 (18) Yamamoto, B. K.; Lane, R. F.; Freed. C. R. Life Scl. 1982, 30, 2155-2162. (19) Kolke, H.; Negata, Y . J . Physlol. (London) 1979, 205, 397-417.
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(20) Ono, J. K.; McCaman, R. E. Nevosclenoe 1984, 1 1 , 549-560. (21) Ekbf, B.; Lessen, N. A.; N ~ w , L.; Norberg, K.; SiesJo, B. K.; Torlof, P. Acta physbl. Sand. 1974, 01, 1-10. (22) Levln, V. A. J . Med. Chem. 1980, 23, 682-884.
Alain Meulemans Laboratoire de Biophysique Faculte de Medecine Xavier Bichat 16 rue H. Huchard, Paris 75018, France Bernard Poulain Gerard Baux Ladislav Tauc* Laboratoire de Neurobiologie Cellulaire et Moleculaire CNRS Gif-Sur-Yvette 91190, France Daniel Henzel
U 13 INSERM Hopital Claude Bernard Paris 75019, France RECEIVED for review December 11,1985. Accepted April 15, 1986. This work was supported by AIP No. 06931 to L.T.
Comments on Improvement of the Limit of Detection in Chromatography by an Integration Method Sir: Recently, Synovec and Yeung described a method of improving the limit of detection in chromatography by numeric integration of the chromatogram (1). They showed both by theoretical means and by computer simulation that the signal-to-noise ratio is improved by integration of a chrqmatogram when the base line noise has certain characteristics. The amplitudes of both signal and noise are increased by integration, but the signal increases more than the noise. Noise Sources. The authors model the chromatographic base line, F ( t ) ,by F ( t ) = mt b D ( t ) R(t) Nt (1) where mt is a linear base line drift, b is a base line offset, D(t) and R(t)are base line fluctuations,and Nt is random detector noise. The noise, N,, at each data point is sampled from a Gawian and each sample is independent of all others. This sampling procedure is equivalent to assuming that all time constants in the system are much shorter than the sampling interval. In the frequency domain, this type of noise is uniformly distributed throughout the spectrum and is commonly known as white noise. The authors refer to it as “uncorrelated”noise because there is no correlation from point to point in the base line. In contrast, the chromatographic signal is “correlated” because a peak spreads over a large number of data points. Most of the power of white noise is at frequencies substantially higher than the frequencies of chromatographic peaks. And, as the duration of the peak increases, an ever larger portion of the white noise spectrum falls beyond the frequency range of the chromatographic signal. The base line fluctuation terms, D ( t ) and R(t),are in the same frequency range as are chromatographic peaks. These fluctuations are of the same duration as typical chromatographic peaks and so could be mistaken for peaks. Base line offset and linear drift are examples of noise at extremely low frequencies, well below those of chromatographic peaks. Offset and drift in a chromatogram may be
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inconvenient, but they cannot be mistaken for peaks. Signal-to-Noise Ratio Improvement. All methods to improve the signal-to-noise ratio involve some means of distinguishing between signal and noise. In this case, the distinguishing feature is frequency. Most of the signal in a chromatogram is contained within a limited frequency range. Power outside this range is mostly noise. Accentuating signal containing frequencies while discriminating against noise containing frequencies gives an improved signal-to-noise ratio. Integration is but one of many methods of altering the power of one frequency range relative to another. It discriminates against the higher frequencies and so eliminates much of the white noise from a chromatogram. The base line offset and drift corrections applied before integration discriminate against the extremely low frequencies. Mid-range frequencies,such as those composing chromatographic peaks, pass through this integration fiiter. The method is equivalent to a Fourier transform smoothing procedure (2). Except for a normalization factor, differentiation is equivalent to multiplication by io in the frequency domain (3).Thus, an integral, I @ ) ,may be computed from a chromatogram, C ( t ) ,by I(t) = FT1[FTIC(t)]/io] where FT and FT-’ are the direct and inverse Fourier transforms, respectively. Division by the independent frequency variable, o,in eq 2 accentuates the low frequency content of the signal relative to the high frequencies. Signal-to-noise ratio and the limit of detection are improved in the integral because the signal is less affected by this division than is the noise. The Fourier transform method of computing an integral attenuates power at low frequencies less than it does power at high frequencies. The running total method increases power at low frequencies more than it does power at high frequencies. The difference between these two methods is just a normalization factor and both change the 0 1986 American Chemical Soclety
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ratio of low frequency to high frequency power identically. There are many other smoothing filters applicable to chromatography. All in some way block frequencies containing mostly noise while passing frequencies containing mostly signal. The Fourier transform is the most general way to compare filters in which the feature distinguishing between signal and noise is frequency, but computations can often be done more easily in the time domain. The integration filter is slightly less effective than most frequency-based filters because it does not attenuate the noise-containing frequencies as strongly. For example, the trapazoidal filter (2)sets the power to zero at all frequencies beyond a selected cutoff frequency. An integration has no specific cutoff frequency but gradually decreases power with increasing frequency and retains some power even at the highest and noisiest frequencies. Noise in the same frequency range as the signal cannot be discriminated against by any method based on frequency. In the model used here a small portion of the white noise power is in the same frequency range as the chromatographic peaks and so passes through the integration fiiter. Most of the base line fluctuation noise is in the signal‘s frequency range, and so also passes through the filter. The base line noise model as used in this paper is unrealistic in that it assumes that base line fluctuations can be reduced to near zero through proper experimental consideration and procedure and so can be neglected. Often noise sources responsible for base line fluctuation are initially insignificant in comparison to other sources, but once these other sources have been controlled or the noise resulting from them filtered out of the chromatogram, then the originally unimportant base line fluctuations may become the dominate noise and thus determine the limit of detection. The authors state that the detection limit improvement resulting from integration is “independent of any data averaging that might be applied to the chromatogram”. This statement is misleading since data averaging is another smoothing technique that discrimatesagainst high frequencies. The same noise can be removed by either procedure. Noise remains for integration to remove only if data averaging (or other smoothing) has not been applied with a low enough cutoff frequency. Commonly, smoothing a t the data acquisition stage by either an electronic time constant or a data averaging procedure has a relatively high cutoff frequency. Noise filters implemented within data acquisition electronics and software are conservatively designed to avoid any possibility of attenuating sharp peaks and, thus, limiting application of the instrument. An instrument user, however, may change the data averaging cutoff frequency to more closely match the requirements of a particular application. Dramatic improvements in signal-to-noise ratio can be obtained this way if the chromatographic peaks are significantly broader than those assumed by the original filter. One of the advantages of the integration method is that the degree of improvement increases with increasing retention time. More strongly retained chromatographic peaks have greater band duration and thus lower frequency. Other methods exist to accomplish the same thing. For example, data averaging over a variable time interval provides a variable degree of smoothing to match the variable chromatographic band duration ( 4 ) .
Resolution. The authors state that “using a running total will not decrease the resolution”. This is incorrect because resolution is dependent upon the highest frequency components of the chromatographic signal. The integration method accentuates the lowest frequency components of the signal at the expense of higher frequencies. The simulated chromatograms all have well-separated peaks, so this loss of resolution is not obvious and does not interfer with quantitation in the simulations. Quantitation of an integral requires the presence of a horizontal region on both sides of a step signal. base line resolution is required in the chromatogram to obtain horizontal regions in the integral. a pair of peaks with a valley, but not base line, between them are readily distinguishable in the unintegrated chromatogram but give a pair of steps with only an inflection point, no horizontal region, between them in the integral. As the depth of the valley is reduced, the inflection point becomes less distinct and eventually disappears leaving only a single step in the integral visually indistinguishable from that due to a single component. Chromatographic software packages generally find peaks in the unintegrated chromatogram where they are more distinct. Only after their starting and ending points have been determined are they integrated to improve the limit of detection. Integration does not throw away any information during smoothing. No frequencies are set to zero power and it is always possible to get back the original chromatogram with its original resolution by differentiation. But, in the integral form the higher frequency components of the signal are less distinct and resolution is degraded. This trade-off between resolution and signal-to-noiseratio during integration or differentiation of signals is well-known in other signal processing applications. For example, UVvisible spectra of organic substances generally consist of broad featureless absorption bands. First and second derivatives of these spectra are useful in analysis because they show more structure as subtle inflection points in the spectra become peaks and valleys in the derivatives (5). Higher order derivative spectra are impractical,however, because of their poor signal-to-noiseratios. Derivatives of polarographic waves (6) and potentiometric titration curves (7)have been used similarly.
LITERATURE CITED (1) Synovec, R. E.; Yeung, E. S. Anal. Chem. 1885, 57, 2162-2167. (2) Lephardt, J. 0. In Transform Techniques h Chemistry; Griffiths, Peter R., Ed.; Plenum Press: New York, 1978; p 294. (3) Champeny, D. C. Fourier Transforms and Their Physical Appiications : Academic Press: New York. 1973; p 17. (4) Saadat, S.; Terry, S. C. Am. Lab. (FairfieM, Coflfl.) 1984, 16(5), 90-101. (5) O’kver, T. C. Anal. Chem. 1979, 5 1 , 91A-100A. (6) Kelley, M. T.; Jones H. C.; Fisher D. J. Anal. Chem. 1959, 3 1 , 1475. (7) Willard, H. H.; Merrltt, L. L., Jr.; Dean, J. A.; Settle, F. A., Jr. Instrumental M e w s of Analysis, 6th ed.: D. Van Nostrand: New York, 1981; pp 667-670.
John B. Phillips Department of Chemistry & Biochemistry Southern Illinois University Carbondale, Illinois 62901
RECEIVEDfor review November 14, 1985. Accepted May 5, 1986.
