Anal. Chem. 1990, 62, 717-721
717
Characterization of Overlapped Chromatographic Peaks by Their Second Derivative. The Limit of the Method Eli Grushka* and Dror Israeli
Department of Inorganic and Analytical Chemistry, The Hebrew University, Jerusalem, Israel
The second derhratlve of a chromatographic signal can aid In the recognltlon of composite peaks. I n the case of a twocomponent composlte, the second derlvative has three maxIma and two mlnlma. The tkne positlon of the second of the three maxlma Is indlcatlve of the point where the peak envelope of each component cross each other. Thls second maxknum can be used to trlgger the end of Integration of the first component and the beglnnlng of the Integration of the second component. I n theory, the second derivative approach works well. The present paper Investigates the llmlts of the method. I t was found that, depending on the separatlon between the two components In the composite, there Is a peak height ratlo below which the method faHs to recognize the existence of a composite system. The greater the separation, the smaller the helght ratlo. At 4a separatlon, the mlnlmum helght ratlo that the method can stili recognize as a composlte Is around lo4. However, the useful range of the approach may extend to a lower range of helght ratios due to the small changes In the area with relatively large changes In the peak helght ratio. Noise also affects the useful range of the method. I n general, as the noise increases, the method falls at hlgher peak helght ratlos. A surface descrlblng the useful range of the method Is given as a function of the slgnal-to-nolse ratio, the separatlon, and the helght ratio.
INTRODUCTION The differentiation of the chromatographic detector signal, either in the time domain or in the wavelength axis, can be used to gain some important information on the purity of the eluting peak. Among the first to describe the differentiation of the chromatographic signal were Ashley and Reilley ( 1 ) . They used time-based derivatives to electronically sharpen and deconvolute the peaks. Grushka (2,3) used derivatives in the time domain to characterize severely overlapped chromatographic peaks. Zelt and his group ( 4 ) and Traveset and his co-workers (5) used spectrometers capable of taking derivatives of spectra as detectors in high-performanceliquid chromatography (HPLC) and in thin-layer chromatography (TLC). Berridge (6) used time-based derivatives of the chromatogram as an aid to optimize the mobile phase selection in HPLC. Fell and his co-workers (7-10) have used derivatives to obtained qualitative and quantitative information from the chromatogram. Gerow and Rutan (11)used derivatives with Kalman filters in order to smooth TLC chromatograms. Grant and Bhattacharyya (12)used derivatives to determine the purity of chromatographic peaks. Ebel(13) discussed briefly the use of derivatives in deconvolution techniques. Recently, Grushka and Atamna (14) have shown that the time-based second derivative can be used as a criterion to terminate and initiate the area integration of strongly overlapped peaks (0.5 R, 0,75). Although overlapped peaks with resolutions between 0.5 and 0.75 can be recognized by the eye as a composite, an integrator may have difficulties in recognizing the existence
of multiple components. A composite made up of two peaks with resolution between 0.5 and 1is characterized by a second derivative having three maxima and two minima. The time position of the second maximum of the second derivative is a good indication of the crossover point of the concentration profiles of the two components. Therefore, the time position of the second maximum of the second derivative can be used as a criterion to stop the integration of the first peak in the composite and start the integration of the second peak. Grushka and Atamna have shown that for noiseless Gaussian peaks this approach works well (14). The areas thus measured were a linear function of the amounts of the components in the composite. More importantly, the areas were independent of the separation between the peaks (14). Since the preliminary results were promising (14),the limitation of the method must be examined to check the useful range of the approach. The second derivative method is valid as long as it has three maxima and two minima. The method fails once the system of five extrema collapses into three extrema as is the case with severely overlapped peaks. Failure of the method is due to four main reasons: (a) the resolution is below a critical value so that the two components severely overlap, (b) the height of one of the components is below a critical value so that it disappears under the envelope of the major component, (c) the noise in the detector signal is large enough to interfere with the differentiation process, and (d) the chromatographic peaks are not Gaussians but of other asymmetric shapes which grossly distort the second derivative curves. The present paper will examine theoretically the first three limitations, and it will establish some guidelines showing the useful range of the method. THEORETICAL CONSIDERATIONS Before going into the theoretical analysis, several parameters need to be defined. In this study, composite peaks are assumed to be made up of two Gaussians. Since the two components elute very closely, we assume that the peak widths are the same. In the following treatment we will assume that the concentration of the second component in the composite is equal to, or smaller than, the first component. Since Gaussian peaks are symmetrical, the conclusions reached by using the above assumption can be extended to cases where the fiist component is the smaller of the two. The height ratio is defined here as the height of the second peak divided by the height of the first peak, H 2 / H l . Thus, very low height ratios mean very small amounts of the second component in the composite. The separation is given in terms of the standard deviation of the peaks. Separation and Height Limitation. The limits placed on the method by the separation between the componentsand by the peak height are related. The height of one peak at which it will disappear under the second peak in the composite is a function of the separation between the peaks. To find the limit imposed by the height ratio of the peaks and the separation between them, we need to calculate the point at which the five extrema in the second derivative collapse into three: two maxima and one minimum. Due to the transcendental nature of the equation describing the second de-
0003-2700/90/0362-0717$02.50/0@ 1990 American Chemical Society
718
ANALYTICAL CHEMISTRY, VOL. 62, NO. 7, APRIL 1, 1990 A
1.00
t
I
Height ratio = 0.8
7
200
1 0.25 W
z a
6
5 .o
s
Lo W
- 0 150
75
0
225
300
Time
0.00 1.50
2.00
2.50
3.00
3.50
4.00
Separation (In units of 17)
B
Height ratio = 0.5
r------
Figure 1. Peak height ratio at which the second derivative fails. The dashed lines enclose a region of ambiguity (see text). C
rivative of a composite made up from two Gaussians, numerical techniques must be employed to find this point of collapse. Figure 1 shows the height ratio, at which the second derivative method fails as a function of the separation between the components in the case of noiseless Gaussian peaks. For cases where the height ratio of the components is above the solid line, the method is successful in recognizing and quantifying composites. Of course, the greater the separation, the smaller the height ratio at which the method fails. At a 4a separation the theoretical height ratio limit is about 1 X 10”. Under normal conditions, 40 separations do not present difficulties for integrators. However, when one of the peaks in the composite is very small, the perturbation that it causes on the composite envelope is very minute and conventional integrators will frequently miss it. Unexpectedly, Figure 1 indicates that, for noiseless Gaussians, the second derivative method can characterize composites even at the 10000:l height ratio. In the separation range of about 4-1.75u, the height ratio at which the method fails increases monotonically although not linearly. At 1 . 7 5 separation ~ there is a sudden and marked increase in the height ratio at which the method fails. The fact that the method works at separations less than 2u is quite surprising. A t such separations, the two components in the composite are completely merged and conventional integrators loose their effectiveness. Yet, the second derivative can discern the two components provided that the height ratio of the peaks is above the critical value. At a separation of 1.75a, that critical value is about 0.24. Below the separation of 1 . 7 4 the ~ second derivative can still be used, but its behavior is more complicated. Above the solid line of the curve in Figure 1, the second derivative behaves in the “normal” manner in so far as the time position of its second maximum occurs before the actual maximum of the second component. However, below the solid line, there is a range of height ratios and separations where the second derivative shows again three maxima and two minima. That range is bounded by the broken lines in Figure 1. In this region, the second maximum in the second derivative occurs after the actual maximum position of the second component. Figure 2 compares four composites with four height ratios at a separation of 1 . 7 ~ The . first component in the composite was assumed to have a retention time of 120 (in arbitrary units) and a standard deviation of 20. Therefore,the retention time of the second component is 154. The height ratio of 0.8 (Figure 2A) represents a case above the solid line of Figure 1. The second maximum in the second derivative curve occurs at time 143. At height ratio of 0.5 (Figure ZB), which is below the solid line in Figure 1 but above the broken line, the second derivative has only two maxima. While the derivative indi-
2
c
0’25
W
?
.
,
~
100
~ 0.00
W
.Eb
W
L
I I
0 W
,,
I
,
v) 0
9I ,
-0.25
0 0
75
150
225
300
Time
C
Height ratio = 0.3
..E W
b
n
p
loo--
Lo W
I
0
75
150
225
300
Time
D
Height ratio = 0.2
l-----7
0’25
W
E
a
E b
E 0
Lo
0
75
150
225
300
Time
Figure 2. Four composites and their second derivatives. I n all cases the separation is 1.70. The detector signal scale Is in arbitrary units. The solid curve is the composite; the dashed curve is the second deriiative. Four different Mght ratios are given: (A) height ratio above the solid line in Figure 1, (B)height ratio below the s o l i line but above the broken lines, (C) height ratio within the broken lines, and (D) height ratio below the broken lines.
cates, in this case, that the peak is a composite, it does not allow the recognition of peak-start and peak-end. The height ratio of 0.3 (Figure 2C) is inside the area bounded by the broken lines. Here, the second derivative curve has again three maxima. However, the second maximum occurs at time 157, which is past the peak maximum. Thus, while the method can be used to initiate and/or terminate the integrator, the areas recovered will be erroneous. The height ratio of 0.2 (Figure 2D) represents a case below the broken lines. The
ANALYTICAL CHEMISTRY, VOL. 62, NO. 7, APRIL 1, 1990
second derivative still indicates the existence of a composite peak, but it cannot be used for quantitative purposes since it has only two maxima. The region bounded by the broken lines in Figure 1 is small. On the separation axis it extends from 1 . 7 4 ~to 1 . 6 ~ .The region is widest at 1 . 7 4 ~separation; it covers height ratios from about 0.41 to 0.24. The existence of this region is troublesome since it may lead to erroneous qualitative results. However, the region may be recognized by the fact that the second maximum occurs on the bottom part of the tail of the composite and by the positive second minimum of the second derivative curve. To be useful, the method should be able not only to indicate the end and start of peaks but also to give areas which are proportional to the concentration injected, even at very low height ratios. Figure 3 shows graphs of second peak area as a function of height ratio for four separations: 4a, 3a, 2a, and 1 . 7 ~ At . each separation, the figure gives three lines: (a) the solid line is the area measured under the composite starting at the time position of the second maximum of the second derivative, (b) the short-dashed line represents the theoretical area of the complete second peak, and (c) the long-dashed line is the area of the second component only, from the time position of the second maximum in the second derivative. The lines were least-squares fitted to areas generated with the computer. In all cases the fits were very good with correlation coefficients better then 0.997. Table I gives the data obtained from the least-squares procedure for the composite and for the second component when the first peak height is 10 (arbitrary units) and a is 20. The line obtained from the area of the complete (-a to +m) second peak is identical a t all separations: the slope is 501.198, the intercept is around lo4, and the correlation coefficient is unity. Figure 3 and Table I show some important points. A t 4a separation the method is linear down to height ratios of 1 x 10”’. However, because of the nonzero intercept, and the relatively gentle slope, changes in the areas with changes in the height ratios are small at height ratios below 0.01. In the height ratio range of 0.01-0.001 the method should be used with care; the area changes between 27.237 and 22.773. Below a height ratio of 0.001, the method is not usable at all: the area changes between 22.773 at H z / H l of 0.001 and 22.351 at H2/H1 of 0.0001. The situation is much the same for 3a separation, although the method fails at a higher height ratio due to the smaller separation. The linear range extends to a height ratio of 0.008. However, below H 2 / H l of 0.01 the qualitative information obtained from the method is not reliable. At 2a separation the method is, again, linear all the way to the failure point at H 2f H l of 0.15. At this separation, the whole linear range yields valid qualitative information. A t 1 . 7 separation, ~ the method is linear and valid down to the height ratio of 0.6. While this range of linearity is rather small, it permits the analyst to recognize and to quantitate composites that conventional integrators could not handle. A t 4a and 3a separations the composite area, as measured from the time position of the second maximum in the second derivative, is larger than that of the area of either the complete Gaussian or the area, from the point of integration initiation, of the second component only. The reason that the area of the composite is greater than that of the complete Gaussian is as follows. The peak recognition scheme by the second derivative causes a truncation of some of the area from the second component and the inclusion of some area due to the first component. The contribution of the area of the first solute in the composite is greater than the area loss due to the truncation of the second solute. The only case where the area lost equals the area gained should occur when the two components are of equal height. Indeed, Figure 3 shows that
A
719
4a SEPARATION 500
400
300
E 200
100 0 0.00
0.20
0.40
0.60
0.80
1.00
0.80
1.00
Height ratio
B
3 4 SEPARATION 500 400
S
e
300 200
100 0
0.00
0.20
0.40
0.60
Height ratio
C
2 0 SEPARATION
I
0.00
0.20
0.40
0.60
0.80
1.00
080
1.00
Height ratio
D
1 . 7 0 SEPARATION
loot
I
O L
0.00
0.20
0.40
0.60
Height ratio
Figure 3. Linearity of the measured areas as a function of the height ratio. The solid line describes the area of the composite from the time position of the second maximum of the second derivative, the longdashed line describes the area only of the second component from the above time and the shortdashed line describes the area of the complete Gaussian. Four separations are shown: (A) 4a,(B) 3a,(C) 2a, and (D) 1.70.
at the height ratio of 1the area of the composite (from the time position of the second maximum in the second derivative) equals the area of the complete Gaussian. As the separation decreases, see Figure 3 at 2a and 1.70 separations, the composite area becomes smaller than that of the complete Gaussian. As the separation decreases, the second maximum in the second derivative tends to shift closer to the position of the peak maximum. Therefore, the area of the second component that is lost in the recognition process
720
ANALYTICAL CHEMISTRY, VOL. 62, NO. 7, APRIL 1,
1990
Table I. Slopes, Intercepts, and Correlation Cofficients Obtained from Least-SquareFits of the Integrated Area to the Height Ratio of the Peaks" 4a
slope int coir coef
composite 2nd comp composite 2nd comp composite 2nd comp
479.876 491.379 22.199 0.164
0.999985 0.999996
30
481.344 468.124
21.551 -0.940 0.999972 0.999985
2a
515.186 449.084 -16.266 -31.260 0.99931
0.99942
1.70
745.206
575.151 -239.008 -170.184
0.9985 0.9992
The entries for "composite" refer to area under the composite envelope; "2nd comp" refers to areas belonging only to the second component. All areas are calculated from the time position of the second maximum in the second derivative curve. far exceeds the area of first component that is included in the area of the composite. The result is that the area of the composite is smaller than that of the complete Gaussian. Only when the height ratio is unity does the area lost equal that gained and the composite area will, as before, equal that of the complete Gaussian. That behavior is observed in Figure 3. The slope of the line due to the composite area becomes larger as the separation decreases. The intercept becomes smaller and, then, more negative. Between 4a and about 2.5a, the changes in the slope and the intercept are not too great. Therefore, in that separation range, calibration curves are independent to a large extent from the separation. More on this point was discussed by us previously (14). As the separation decreases below 2.5a, the changes in the slope and intercept become larger and larger. From a practical point of view the increasing dependence of the slope and intercept on the separation means that the chromatography has to be controlled tightly if the quantitative results are to be meaningful. The short linear range of the method at small separations and the need for tight chromatographic control stress again the fact that there are no substitutes for good chromatography: if the resolution is too small, one is better off trying to improve the chromatographic system rather than manipulating, numerically or electronically,the detector signal. Limitation due to Noise. The preceding discussion assumed that the peaks were noiseless. The presence of noise, however, should have deleterious effects on the second derivative method. To check the effect of noise, we generated composite peaks with noise. The noise levels given are in terms of peak-to-peak levels and not in terms of root mean square. In the preceding treatment, it will be assumed that a time region in the chromatogram was isolated to contain only a suspected composite peak. Thus, we assume that there is a prior recognition of the peak beginning and end. The following procedure checks to see if the peak is a true single peak or a composite. Derivatives magnify the noise and the second derivative trace is noisier than the original detector trace. Therefore, the second derivative approach may either converge to the wrong system of maxima and minima thus identifying wrongly a composite peak or miss entirely the presence of a composite. To utilize the derivative method, a smoothing process must be employed. There are several ways to smooth data. We chose the following procedure to smooth the derivative data and to identify the correct combination of maxima and minima: (a) Noisy chromatograms were generated and stored in a digitized manner in the computer. (b) By use of the Savitzky-Golay method (15),the second derivative was computed by using a filter window whose width is a function of an estimated peak width. The filter used a second order polynomial. The examples that will be shown shortly use a 17point window. (c) Next, the highest point in the second derivative trace is located and height thresholds are established at f0.5 of the highest point. (d) All maxima and minima that exceed the thresholds are found. (e) The extrema
Figure 4. A noisy composite and its smoothed second derivative. The dashed lines give the threshold limits for searching minim and maxima. The separation between the two components is 3a;the height ratio is 0.33; SIN ratio is 10. The existence of three maxima alternating with two minima ensures the identification of the composite.
are searched for a pattern of successive three maxima alternating with two minima. This last step leads to one of several possible courses of action. ( f l ) If alternating three maxima and two minima are found, then a composite is declared and the time position of the second maximum in the second derivative curve is used to stop the integration of the first peak and start the integration of the second peak. (E)If more than five extrema are located, then the method fails due to noise. Further attempts to locate the composite are pursued, if desired, outside the procedure described here. (f3) If five extrema are found but the two minima do not alternate with the three maxima, the method fails due to noise. (9) If there are less than five extrema, the treshold is halved and steps d-f are repeated. This iteration process continues until one of the conditions in steps fl-f3 is met. Figure 4 gives an example of the procedure. Figure 4a shows the composite that is made up of two Gaussian peaks a t a separation level of 3a,height ratio of 0.33, and signal-to-noise (S/N) ratio of 10. Figure 4b gives the smoothed second derivative of the composite. The dashed lines indicate the height thresholds. The figure gives the results of the second iteration. In the first iteration, the thresholds were higher and the algorithm found only two maxima and two minima. In the second iteration, alternating three maxima and two minima were found outside the height thresholds. At this stage the algorithm recognized the composite, the iteration process was stopped, and the integration can proceed. The small maxima and minima inside the height thresholds are due to noise and they were rejected from the count of maxima and minima.
ANALYTICAL CHEMISTRY, VOL. 62, NO. 7, APRIL 1, 1990
721
A
\
/ \
4 15
Nolre
Ratio
Sspatatlon ~
Figure 6. A surface describing the noise-related limit of the second derivative method. The surface is plotted as a function of S/N ratio, separation, and height ratio.
limiting S/N ratio is about 20.
Flgure 5. A ndsy composite and its S
r " e d
second derivative. The
dashed lines give the threshold limits for searching minima and maxima. The separation between the two components is 3a;the height ratio is 0.33; S/N ratio is 5. The existence of four maxima above the threshold causes the method to fail.
Figure 5 gives a case where the approach fails. The composite is similar to that shown in Figure 4, except that the S/N ratio here is 5. The composite is shown in Figure 5a and the derivative in Figure 5b. Figure 5b shows the second iteration, and in the present case there are four maxima above the threshold and two minima below. The method fails since there are more than three maxima. The previous example demonstrates clearly the difficulties associated with noise. In the case of noiseless peaks, the second derivative approach will have no problems in recognizing a composite at 3a separation and height ratio of 0.33. Severe noise makes this recognition very difficult. Therefore, it would be of interest to obtain the lower limit of the second derivative method to recognize composites of noisy peaks. Using the algorithm described above to locate the extrema of the second derivative curve, we examined the limits at which the method fails. Figure 6 describes a surface in the coordinate system of S/N ratio, separation, and height ratio. On the surface, the procedure described here recognizes composites in 90% of the attempts made. Above the surface, the method is very reliable. Below the surface the method fails too frequently to be of practical value. The features of the surface are as expected. In general, as the separation decreases, the limiting S / N ratio becomes higher and the method is less sensitive to the presence of double peaks. For example, at the height ratio of 1 and a separation of 4a,the limiting S/N ratio is about 3, while at a separation of 20 the limiting S/N ratio is about 8. Similarly, as the height ratio decreases, the limiting S/N ratio increases. For example, at a separation of 2a and a height ratio of 1,the limiting S/N ratio is 8 while at the height ratio of 0.2, the
CONCLUSIONS The second derivative can be used to recognize and to quantify composite peaks that are made up of two Gaussian peaks. In cases where the S/N ratio is very high and the separation is moderate (between 30 and 4a), the method can be used to obtain quantitative information about the components in the composite even at height ratios of 100 to 1. However, noise in the signal will decrease the effective range of the method. At very noisy signals and small separations, the limit of the method seems to occur at a height ratio of 0.3. The method described here to overcome noise effects is a first attempt to deal with the problem. The approach uses a rather simplistic algorithm to locate the maxima and minima of the second derivative. More ambitious search routines might be more efficient in identifying the extrema profile of the second derivative. With such search routines, the useful range of the second derivative method can be enlarged and composites of smaller height ratios can be quantified. LITERATURE CITED Ashley, J. W., Jr.; Reilley, C. N. Anal. Chem. 1985, 35, 626, 627. Grushka, E.; Monacelli, G. C. Anal. Chem. 1972, 44, 404-489. Grushka, E. Anal. Chem. 1972, 44, 1733-1738. Zelt, D. T.; Owen, J. A.; Marks, G. S. J . Chromatogr. 1980, 189, 209-216. Traveset, J.; Such, V.; Golzalo, R.; Gelpi, E. J . Chromatogr. 1981, 203, 51-58. Berridge, J. C. Chromatographia 1982, 16, 172-174. Fell, A. F.; Scott, H. P.; Gill, R.; Moffat, A. C. Chromatographla 1982, 16, 69-78. Fell, A. F.; Scott, H. P.; Gill, R.; Moffat, A. C. J . Chromatogr. 1983, 273, 3-17. Fell, A. F.; Scott, H. P.; Gill. R.; Moffat, A. C. J . Chromatoar. - 1983. 282, 123-140. Clark, B. J.; Fell, A. F.; Scott, H. P.; Westerlund, D. J . Chromatogr. 1QBA. 286. 261-273 . ., ., -. . -. .. Gerow, D. D.; Rutan, S. C. Anal. Chim. Acta 1986, 184, 53-64. Grant, A.; Bhattacharyya, P. K. J . Chromatogr. 1985, 347, 219-235. Ebel, S. Chromatographia 1988, 22, 373-378. Grushka, E.; Atamna, I. Chromatographla 1987, 24, 226-232. Savitzky, A.; Golay, M. J. E. Anal. Chem. 1964, 36, 1627-1639.
RECEIVED for review September 5,1989. Accepted December 18, 1989.
