517
Anal. Chem. 1900, 60,517-521
A Fuzzy Approach to Peak Tracking in Chromatographic Separations Matthias Otto* Department of Chemistry, Bergakademie Freiberg, Leipziger Strasse, 9200 Freiberg, German Democratic Republic
Wolfhard Wegscheider and Ernst P. Lankmayr Institute for Analytical Chemistry, Micro- and Radiochemistry, Technica University Graz, Technikerstrasse 4, A-8010 Graz, Austria
Recognttkn of peaks in chromatographk separations Is based on comparison of peak areas and elutlon order of peaks b e tween the trial and a reference run. To account for the Imprecision of peak areas, for the change of peak elution order, and for the uncertalnty d overlapped peak areas, the method is based on fuzzy theory as a method that can cope with uncertain and imprecise data. The appllcaMllty of the method is demonstrated for recognition of chromatograms with several overlapped peaks as well as for identlficatlon of highly imprecise peaks from HPLC separatlons of vitamlns by uslng binary and ternary mobile phases. I n the latter case correct peak identlflcatlon needs two-channel monltorlng by use of a UV absorbance and a fluorescence detector. The method is suitable for on-line peak tracking even If unknown samples are to be analyzed.
Efficient unattended optimization of high-performance liquid chromatographic (HPLC) separations is based on mathematically modeling the retention behavior of solutes in dependence on the composition of the mobile phase and requires peak identification (1-3). Commonly, the solutes are injected separately in order to evaluate their retention time pattern under the actual experimental conditions. This strategy not only is slow but is useless if sample components of unknown identity are to be separated. Much success has been achieved for the special case of multiwavelength detection with use of the photodiode array detector (4). The massive amount of data has, however, not led to an automated procedure and heavy operator interaction is required for overlapping solutes. For that reason recording of spectrochromatograms with the diode array detector will be reserved mainly to cases where overlapping peaks are to be deconvoluted (5,6) rather than to monitor the full spectrochromatogram at every optimization step. What is needed more generally, however, is a peak tracking strategy that (i) can work for single channel detectors, as well as for multichannel detectors, such as, photodiode array detection or combination of several single channel detectors, (ii) can be applied to the unknown sample itself, and (iii) can cope with unresolved peaks. This could be a major step forward in the field of unattended on-line optimization of HPLC separations. In the present work these goals are pursued by a peak tracking routine that takes into consideration the imprecision of peak area determinations, the nonadditivity of individual signals from unresolved peaks due to solute-solute interactions, and information on the solute elution order in the chromatographic experiments. The mathematical method is based on fuzzy theory (7)and has been originally developed for component identification in the UV spectral range (8). With this theory the variables 0003-2700/8S/0360-0517$01.50/0
(peak area, ilution order) are assigned a membership function ranging between 0 and 1 that characterizes the imprecision of peak areas and the variability of the solute elution order. The computer routine developed is tested for the identification of compounds in a mixture of vitamins during optimization of their separation in binary and ternary mobile phase.
THEORY The simplest procedure for peak identification can be based on comparing peak areas of a t.rial run to those of a reference run. This has been carried out by Issaq and McNitt (9), who identified the peaks by the peak area percent of each peak compared with other eluted peaks in that experiment. The procedure works well within the assumptions stated by the authors (9): “It is assumed that the peaks are Gaussian and symmetrical, with no peak frontage and tailing.” Furthermore, changes in the absorption coefficient of solutes due to change of mobile phase composition need to be negligible. The use of peak area percents also implies that the areas of all detected peaks sum up to exactly loo%, an assumption that is hardly met in complex problems, such as, in HPLC of vitamins or of tissue samples. The peak tracking method developed in this work is based on the fact that the change of chromatographic and spectroscopic conditions results potentially in asymmetrical peaks, in peaks with different absorbing properties, and in a varying number of peaks caused by component overlap (reduced peak number) or by additional peaks due to possible component decomposition. To tackle this problem a fuzzy method is used, the basic ideas of which can be found in ref 8. In the following sections the adaptation of that method to the peak tracking problem is outlined. Comparison of Fuzzy Peak Areas. With fuzzy set theory the imprecision and uncertainty of peak areas are defined as fuzzy sets over the peak areas (universe of discourse z ) by defining a so-called membership function (mf‘)m(z)renormed to the interval [0,1], e.g.
m(z) = [l - 2 2 ] +
(1)
where + denotes truncation to 0 if negative membership values occur; i.e. only the positive part of the function is considered. In order to enable arithmetic operations to be performed easily the mf is represented most appropriately by the LR representation (left-right) of a fuzzy set (p 53 in ref 7). The LR representation for the mf of peak areas in a trial run is
where x is the variable for the channels (wavelength axis or different kind of detector, e.g., absorbance, conductivity, and fluorescence detector), a , ( x ) > 0 and &(x) > 0 are the left and right spreads of the membership function, respectively, a(x) represents the measured peak area at channel x, and y is the 0 1988 American Chemical Society
518
ANALYTICAL CHEMISTRY, VOL. 60, NO. 6, MARCH 15, 1988
a
2 4
Figure 1. General form of a membership function for peak areas in (left-right)representation. a. and Po represent the left and right spreads, a ( x )is the measured peak area (the most probable value), y is the variable for peak area and ma fy;x) is the membership function
b.
h8 10
b 8 1 0
t
i
Figure 3. Membership function for considering the elution rank of
LR
peaks in the reference run.
for the peak area of channel x .
malize the final information to the interval [0,1] we use the degree of similarity, denoted m,, between the peak areas in the trial and reference run according to (8)
m, = [l - N/N*,J+
(6)
where N*,, is the criterion computed for the peak areas of that run that gives the greater N* value, either from the areas of the reference peak b(x) or from the areas of the trial peak
4x1 Figure 2. Fuzzy subtraction of a peak area in the trial ( a ( x ) )run from that in a reference run ( b ( x ) ) .
variable of the peak area. (The expression m,(y;x) reads “the membership function mafor the peak area y given the channel x ” . ) The same can be written for the mf of areas of the reference run L ( ( b ( x )- Y ) / C y b ( X ) ) for Y 5 b ( x ) mbb;x) = R ( b - b ( x ) ) / @ b ( X ) for ) y 2 b(x) (3) with b(x) being the measured peak area at channel x and > 0, &,(X) > 0 are again the left and right spreads of the mf, respectively. Further explanation of the LR representation is given in Figure 1. As can be seen from Figure 1the mf for describing the uncertainty of peak area need not be restricted to a symmetrical function but it can have an asymmetrical shape if this is justified based on supplementary information. It also should be mentioned that the functional type of the mf is of secondary importance as long as monotonicity is guaranteed; i.e. the larger the distance of an area value from the measured a ( x ) value, the lower should the membership value for that peak area be. Comparison of peak areas in the trial run with those of the reference run is carried out by fuzzy subtraction of both area patterns according to the following scheme: The difference of the peak areas and their related mf is computed by
{
This gives a mf around the peak differences (Figure 2) that can be used to compute a criterion N , which characterizes the deviation from the (crisp) zero function in the following way:
(5) where x1 to x p are the detector channels monitored (if discrete detector channels are used the sum over all channels is formed), Y comprises all possible area values with a nonvanishing membership function ma-b,and y is the variable of peak area difference. In other words the criterion N is a measure for the difference between peak areas of trial and reference run at all monitored channels taking into account the uncertainty of peak areas. The lower N the smaller is the peak area difference and, hence, the better the coincidence between trial and reference peak areas. The criterion can be applied with one detector as with multichannel detection (integration over x , eq 5). To nor-
Thus, the degree of similarity m, approaches 1 if the peak areas of the trial and reference run are identical. Considering Peak Elution Order. One of the main difficulties of chromatographic optimizations arises from the change in the elution order of solutes in different mobile phases (1-3). Although this is one of the most challenging reasons to develop automatic peak tracking routines the peak elution order can on the other hand be used as a source of information for identification of peaks. This is due to the fact that in a chromatographic system the change of peak elution order is normally limited with respect to a definite solute to, say, A3 retention positions, and only rarely is it observed that the first eluting solute will elute at the end of the chromatogram if the mobile phase composition is changed. Thus, the elution order in the reference run can be taken as a “typical” elution order, and peak tracking can be improved by comparing the elution order in the trial run with that in the reference run. This comparison is based on fuzzy sets over the elution order as the universe. The fuzzy sets are described by a mf of the form
m(t) = [l - c l t - til]’
(8)
where t is the elution rank of the peak tested for membership to the ith peak in the reference run and c is a constant normalizing m(t) to the interval [0,1]. Figure 3 gives this mf in a chromatographic run with 10 peaks for a normalization factor c = 1/9 considering the 10th (Figure 3a) or the 4th (Figure 3b) peak as the reference peak under study. In the trial run the membership value with respect to elution rank is assigned by intersecting the mf for the elution order of peaks in the reference run mb(t)with the mf for the elution rank of the trial peak m,(t) giving as the result the following membership value me(t) = min (mb(t),m,(t)) (9) Note that the mf for the elution rank in the trial run is taken as crisp-the opposite to “fuzzy”-with m,(t) = 1 for the elution rank t and zero for all other ranks. For aggregating the results from comparing the peak areas (eq 6) and the elution order of peaks (eq 9) the arithmetic mean has been calculated as
m, = (m, + me)/2
(10)
Thus, the final results are expressed as mean membership values that range between 0 and 1, reflecting the quality of coincidence between the reference and the trial peak area patterns.
ANALYTICAL CHEMISTRY, VOL. 60, NO. 6, MARCH 15, 1988
Establishing the Reference Run. The run with the highest peak count is normally chosen as the reference run. The reference run is selected among a set of predefined chromatographic experiments or-if a sequential strategy is applied-the run with the currently highest peakcount is taken as the reference run. Problems may arise if relatively unstable components, such as vitamins, are to be investigated. Then a threshold for the expected number of peaks should be fixed or other means should be considered to avoid optimizing against an unrealistic peak number. Handling Overlapped Peaks. Summing of individual reference peaks for representing unresolved trial peaks has already been described by Issaq and McNitt (9) for the case of comparing peak area percents. All trial peaks that cannot be ascribed to a certain reference peak are compared to additive combinations of reference peaks in order to evaluate the best peak area percent fit. This principle has been transferred in the present work to summing uncertain peak areas as outlined in the following. Combinations of uncertain reference peak areas are produced by adding up fuzzy numbers (peak areas) having the mf mb(u;x)as stated in eq 3. Addition of two fuzzy peak areas results in a peak area sum that is more uncertain than are the individual peak areas. In analogy to fuzzy subtraction as used in eq 4,fuzzy addition of peak area b l ( x ) and bz(x) of the reference run gives the mf (7, 10)
U ( b , ( x ) + b,(x) + 4) {R((y ( b , ( x ) + b&))/(P, + @*)I) - Y)/(Cu,
mbl+bz
( y ; x )=
-
(11)
where a l p z and &,& represent the left and right spreads of peak areas b,(x) and b,(x), respectively. The sum of peak areas is then compared against the candidate trial peak areas as described in eq 5 and 6. In the present computer algorithm up to three peaks can be combined to match the unused sample peaks, the extension to a greater number being straightforward.
EXPERIMENTAL SECTION Apparatus and Computations. HPLC experiments were performed with Waters 6000M pumps, an automatic injection system Waters WISP 710A, and a Perkin-Elmer TriDet trifunctional detector. The detector enables UV, fluorescence, and conductivity data to be acquired. Only UV and fluorescence data have been used for further data treatment. For system automatization a Waters 840 chromatography data station with a Digital Equipment 360 personal computer was used. The raw chromatographic data were stored on hard disk by the computer. Waters BRIDGE software was used to convert files that contained peak areas and these data were imput to the computer program written in compiled BASICPLUS-2. Chromatographic separations were carried out on a 150 X 4.6 mm i.d. HPLC column filled with 5-pm Spherisorb ODS2. Flow rates were maintained a t 1.0 mL/min throughout all experiments. Samples. The following chemicals were used for analysis: riboflavin, procain hydrochloride, folic acid, nicotinamid, pyridoxine hydrochloride, rutin, riboflavin-5-phosphate, ascorbic acid, niacin, thiamin hydrochloride. They were analytical reagent grade and had been obtained from various sources. Chromatography. The experimental design for adjusting mobile phase compositions was based on a triangle of equal elution strength. The necessary elution strength was evaluated in methanol/water mixtures. T o adjust the k’to values between l and 10, an elution strength corresponding to 60% methanol was required. According to Glajch and Kirkland (IO)seven ios-eluotropic mobile phases were applied having
519
the following contents of organic modifiers: O(reference run)-35% tetrahydrofuran (THF); 1-50% acetonitrile (ACN); 2-6070 methanol (MeOH); 3-22% MeOH, 22% THF; 4-27% MeOH, 27% ACN; 5-2170 ACN, 21% THF; 6-15% MeOH, ACN, and T H F each. Computing Algorithm. The general scheme of the peak tracking routine is based on the following steps: (1)Data input: the user decides on the number of detector channels and the type of membership functions to be used for peak elution order and for peak area comparison. To simplify the procedure in the present application, the spreads for characterizing the uncertainty of the peak areas are taken as relative percent of the actual peak area (normally 10%). So only the percent uncertainty must be specified by the user. Then, the peak area data for the initial two runs are read into the computer memory. The run with the highest peak count is used as the reference run. (2) Discrete membership values for the defined membership function of peak comparison (eq 1) are computed and tabulated a t 13 values of y for later use. (The increments for computing the membership values can be chosen by the user. Thirteen values have been found to provide a good compromise between close approximation to the model of the membership function and a yet handleable number of discrete values to be used for the computer time intensive numerical integration over the membership functions. The interval is set to 1 and then transformed to the spreads of the actual membership function.) (3) Every trial peak is tested against all reference peaks by computing the criterions N and N*,, according to eq 5 and 7 by using the tabulated membership values and numerical integration over the whole universe (peak area domain) by Simpson’s method (11). (4)N is checked to see if it is less than N*,, and the mf m, is calculated from N and N*,,, (eq 6). If N > N*,, the procedure is restarted at step 3 with exchanged peak areas of the trial and reference run. (5) The membership value for the peak elution ranks is computed by use of eq 9 and the membership values of m, and me are combined to give the final membership value m, according to eq 10. (6) Every trial peak is assigned to the reference peak with the highest membership value. If this membership value is higher than a given threshold, e.g. 0.90, the peak is assumed to match definitely the reference peak. ( 7 ) If all peaks of the trial run have been assigned to peaks of the reference run, the algorithm reads the next trial run and continues at step 3. Otherwise all unused trial peaks are tested versus combinations of all unused reference peaks by adding up combinations of two or, in a second cycle, three peaks and restarting at step 3. (8) Again peak assignment is performed and those that reach the predefined value of the membership value are fixed to the overlapping candidate reference peaks. Otherwise peak combination is restarted a t step 7 . (9) Finally, the peak assignment is reported together with the membership values as a measure for the quality of peak identification. (10) The next run is read in and the procedure continues at step 3.
RESULTS AND DISCUSSION A Simple Case Study. As a first application, the experimental data used by Issaq and McNitt (9) with their computer program were evaluated. The data are reviewed in Table I. The first run is selected as the reference run. Since the reproducibility of peak area measurements was within lo%, the fuzzy peak tracking routine works accurately. The peak
ANALYTICAL CHEMISTRY, VOL. 60, NO. 6, MARCH 15, 1988
520
assigned to trial peaks. The low membership value (0.721), however, signals a lack of correspondence between trial and reference peak area. A Worst Case Study. The performance of the fuzzy peak tracking routine was further evaluated for data from optimization experiments with 10 different vitamins. The problem with this class of substances arises from their instability leading to poor peak area reproducibility as well as to additional peaks from solute decomposition. Peak area data were collected from seven mobile phases as described in the Experimental Section. The chromatograms were recorded at two channels (UV absorbance at 254 nm and fluorescence based on excitation at 254 nm and emission monitored at >280 nm). These data are given in Table 111. Owing to detector cell geometry, stray light from the light source adds up to a fluorescence base-line signal, thus resulting in negative peak areas when a UV absorbing solute lacks fluorescence emission. At first, peak identification was tried by peak areas (without referring to elution order) monitored by UV absorbance. The results are given in Table IV (column a) along with the true peak elution order in the first column for each run. The peak tracking method does not work properly and wrong peak assignments (labeled by brackets) are observed for 4-6 peaks out of 10 peaks present. Adding the fluorescence detector as a second channel improves the correct number of peak assignments just marginally (Table IV, column b). In the case of peak tracking by considering peak elution order, identification of peaks work better with respect to the first six peaks even by monitoring UV absorbance solely. (Table IV, columns c). However, wrong identification occurs with peaks 7-9, which change elution order from one mobile phase to the other. Referring to peak elution order, however, guarantees that the assignment of a peak eluting a t the end of the chromatogram, e.g., peak no. 7, is unlikely to be associated with the peak no. 1as happened in the first run for cases a and b. From Table IV it can be seen that some of the peaks are recognized twice or several times. This is caused by membership values less than the predefined threshold (here mthreshold = 0.85) for fixing recognition of a peak. The most appropriate threshold can be found empirically from a number of test runs (as done in the present case), it can be fixed by
Table I. Peak Area Data for Seven Chromatographic Runs According to Reference 9
peakno.
1
2
1 2 3 4 5 6 7
6 4 20 15 14 21 19
20 30 10 40
oeak area Dercent 3 4 5 10 19.5 14.9 14.1 40
5.9 4.1 30.1 14.8 14.1 20.9 19.1
5.8 4.2 20.1 14.9 14.1 21.9 18.6
6
7
48 6 15 14 17.4
io 20 30 40
assignment result is given in Table I1 along with the computed final measure-the aggregated membership values according to eq 10. Note that only the single peaks have been verified with values greater than 0.9 in a single program loop and overlapping peaks were found in the second or third (run number 6) loop. The membership function for accounting for elution order was of the type in eq 8 with c = 30. The spreads were taken as 10% of the actual peak area and the threshold for deciding on successful peak assignment was set to a membership value of 0.90. The results are in agreement with those in ref 9; i.e. accurate peak assignment has been achieved for all peaks that coincide with the reference peaks within the experimental error of about 10%. Difficulties are encountered, as demonstrated also in ref 9 with run number 6 since neither the first eluting peak (48 area percent) nor the last eluting peak (17.4 area percent) match one of the reference peaks or reference peak combinations within the actual precision. The fuzzy method reflects this bad match automatically by extraordinary low values for the membership value, Le., 0.831 and 0.854, respectively. Such a situation cannot be ruled out with experimental optimizations and the algorithm could substitute peak identities whose membership values are less than 0.9 with, e.g., the mean elution rank calculated from the remaining six different runs. The same is true with peak 3 in run 4 where the peak area of 30.1% cannot be correlated with the remaining reference peak 3 (20%) and therefore the assignment results only from the fact that all other reference peaks have already been
Table 11. Peak Assignment to the Chromatographic Runs 2-7 with Run No. 1 as the Reference Run (cf. ref 9)" peak no.
run 2
1 2 3 4 5 6
3 (0.892) 4 + 5 (0.892) 1 + 2 (0.892) 6 + 7 (0.892)
run 3 1
+ 2 (0.925)
2 (0.904)
4 (0.907) 5 (0.908) 6 7 (0.908)
+
I
I
run 4
run 5
run 6
run 7
1 (0.922) 2 (0.925) 3 (0.721) 4 (0.923) 5 (0.925) 6 (0.924) 7 (0.925)
1 (0.919) 2 (0.919) 3 (0.925) 4 (0.924) 5 (0.925) 6 (0.922) 7 (0.922)
2 + 3 + 6 (0.831) 1 (0.908) 4 (0.908) 5 (0.908) 7 (0.854)
1 + 2 (0.925) 3 (0.908) 4 + 5 (0.908) 6 + 70 (0.892)
The membershb values according to ea 10 are given in Darentheses. Table 111. Chromatographic Experiments for Separation of 10 Vitamins by Variation of Mobile-Phase Composition ref peak no. 1 2
3 4
5 6 n
8 9 10
A,,
2.002 3.408 2.271 0.342
AF
-0.0383 -0.0541 -0.0299 0.0347 1.824 -0.0471 2.174 2.542 1.734 -0.0226 0.0827 0.738 0.0743 -0.0020 1.133 -0.0264
Alrv
run 1 AF
1.499 3.050 2.532 0.306 1.866 2.275 0.0656 0.0631 2.00 1.009
-0.0273 -0.0457 -0.0323 0.0356 -0.0453 2.832 -0.001 59 0.0965 -0.0207 -0.0204
run 2 A,,
AF
2.189 3.246 2.685 1.81 0.318 2.087 0.077 0.0802 1.802 1.136
-0.368 -0.477 -0.04 1 4 -0.0736 0.0515 2.629 -0.00227 0.0919 -0.0198 -0.0238
run 3 AUv AF 1.882 3.132 2.673 0.249 1.756 2.214 0.087 1.848 1.848 0.994
-0.0309 -0.0505 -0.032 0.0422 0.0415 2.782 -0.0019 -0.0200 -0.0200 -0.0242
1.717 3.068 2.286 0.326 4.801 2.399 0.0701 0.0769 1.787 1.224
-0.0269 -0.0472 -0.0429 0.0465 -0.0627 3.237 -0.0020 0.0763 -0.0124 -0.0247
1.532 2.936 2.57 0.283 1.618 2.341 0.0744 0.0630 1.87 0.919
-0.0233 -0.0467 -0.0384 0.0513 -0.0613 2.932 -0.0019 0.0691 -0.0275 -0.022
1.366 2.712 2.45 0.283 1.544 2.216 1.805 0.0601 0.0667 0.922
-0.0249 -0.0418 -0.0363 0.0499 -0.0219 2.865 -0.0200 -0.00154 0.0738 -0.0233
521
ANALYTICAL CHEMISTRY, VOL. 60, NO. 6, MARCH 15, 1988 Table
IV. P e a k Assignment of Vitamin Solute P e a k s of Six Trial R u n s t o a Reference Run C h r o m a t o g r a p h e d in T H F / W a t e r ,
35/65 (v/v)
run 2
run 1 peak no.
theor
a
1
(7) 2 3 4 5 6 9
2 3 4 5 6
7
6 9 10
peak no.
theor
3 4 5
3 4 5 6 9
6 7 8 9
10
8 (1) 10
8
8 9 10
b
7
a
run 4 b
c
d
theor
d
theor
a
b
c
d
1
1
1
1
1
1
1
2 3 4 5 6 9
2 3 5 4
2 3 5 4 6
2 3 4 5
7
2 3 4 5 6 9 7 8
(9)
10
2 3 4 5 6 9 7 8 10
c
d
3 4 5 9 8
8
9 8
(6) 10
7
7
2 3 5 4 6 (8) 8 7
10
10
10
10
10
c
d
theor
a
c
d
theor
3 4 5 6
3 4 5
3 4 5
3 4 5
3 4 5
9 8
3 4 (7) (3) 9 8
(5) 10
(8) 8
6
8
8
7 10
(5)
7
7
10
10
10
10
6 9
run 3
c
1
(8) 8
(2)
b
2 3 4 5 6
3 4 5 6 9 8 7
3 4
a
6
run 5 b
9 8
6 (8)
7
a
run 6 b
6
6
6
9 8
7
8
(8) 8
3 4 5 6 7
(5)
7
7
10
10
10
9 8 10
(8) (9) 10
3 4 (7) 6 9
6 7 10
"UV channel, without mf, for elution order. *UV and fluorescence channel, without elution order mf. 'UV channel, with use of mf for elution order. dUV and fluorescence channel, with use of mf for elution order.
referring to subjective experience, or it can be deduced from the uncertainty of the measurements characterized by the actual spreads. In the latter case an upper threshold follows from the definition of the criterion N (eq 5 ) and the degree of similarity m, (eq 6). If all peaks can be matched to reference peaks with high membership values, no peak would occur repeatedly in the peak list. For further explanation the actual membership values obtained for trial run no. 1 in Table IV are given for case a (only UV channel, without considering elution order) and for d (all information used), i.e., the mfs for the assigned peaks 1-10 are for case a, 0.744,0.739,0.786,0.843,0.849,0.779, 0.728, 0.850, 0.781, and for case d, 0.780,0.894, 0.869,0.893, 0.921, 0.888, 0.774, 0.839, 0.721, 0.890. As can be followed, in case d only 4 out of 10 peak assignments do not reach the predefined threshold membership value of 0.85 whereas in case a 9 out of 10 peak assignments have lower final membership values. Thus, in case a, many more possibilities exist for ascribing reference peaks to the trial peaks leading to several wrong peak assignments. Correct peak recognition can be achieved for the poorly reproducible vitamin data only if all available information is simultaneously applied to the identification problem, i.e., two-channel detection, accounting for fuzzy peak areas and peak elution order (Table IV, column d). These experiments show that comparison of peak areas at one channel should be considered as a first approximation for correct peak recognition only and any supplementary information from other channels or from a photodiode array detector should be included.
CONCLUSIONS Application of the fuzzy theory approach to the peak
tracking problem enables identification of peaks to be carried out even in cases of imprecise peak area measurements, of changing elution order of peaks, and of nonadditive peak overlap due to solute-solute interaction. Compared to single-channel detection two-channel or multichannel detection can much improve the rate of correct peak identification and may be obligatory if peak area measurements are of poor reproducibility as shown for peak tracking of vitamins or if peaks have similar areas. In the presented examples the exploitation of information on peak elution order has increased the accurate peak assignment. Further chromatographic systems need be studied, however, to allow unattended evaluation of the membership function for characterizing the peak elution sequence.
LITERATURE CITED (1) Schoenmakers, P. Optimization of Chromatographic Selectivity;Elsevier: Amsterdam, 1986. (2) Berridge, J. C. Techniques for the Automated Optimization of HPLC Separations; Wiley: Chichester, 1986. (3) Otto,M.; Wegscheider, W. J . Chromatogr. 1983, 258, 11-12. (4) Drouen, A. C. J. A.; Billiet, H. A. H.; De Galan, L. Anal. Chem. 1985. 57, 962-968. (5) Vandeginste, B.; Essers, R.; Bosman, T.; Reijnen, J.; Kateman, G. Anal. Chem. 1985, 5 7 , 971-985. (6) Otto,M.; Wegscheider, W.; Lankmayr, E. P. Anal. Chim. Acta 1985, 171. 13-31. (7) Dubois, D.; Prade, H. Fuzzy Sets and Systems: Theory and Applica tion; Academic: New York, 1980. (8) Otto. M.: Bandemer, H. Anal. Chim. Acta 1986. 191. 193. (9) Issaq, H. J.; McNitt, K. L. J . Liq. Chromatogr. 1982, 5 , 1771-1785 (IO) Glajch, J. L.; Kirkland, J. J. J . Chromatogr. 1982, 238,269-280 (11) Selby, S. M StandardMathematical Tables; CRC Press: New York, 1974.
RECEIVED for review May 8, 1987. Accepted November 1987.
3,
