2162
Anal. Chem. 1985, 57,2162-2167
Improvement of the Limit of Detection in Chromatography by an Integration Method Robert E. Synovec* and Edward S. Yeung
Ames Laboratory-USDOE
and Department of Chemistry, Iowa State University, Ames, Iowa 50011
I n chromatography, the llmlt of detectlon (LOD), as conventlonally defined, becomes worse with Increased retention due to band broadening, for constant Injected quantlty and constant analytical detector sensitlvlty. A method Is presented that overcomes this limitation, providing better LOD values Independent of chromatographic band broadening effects. This Is based on an lntegratlon procedure that Is slmple to Implement. A LOD enhancement ratlo of roughly 5 to 20 can be obtalned for a typical chromatographic system. The method has general appllcabllity for any analytical method in whice nolse Is uncorrelated and an analytical signal is correlated along an abscissa. Examples of areas of appilcatlon include chromatography, spectral scannlng measurements, flow Injection analysis, sample Injection coupled to atomlc absorption spectrometry, and nuclear magnetic resonance.
The limit of detection (LOD) in chemical analysis is an important parameter for many analytical methods. A recent Report (1) dealt with the LOD concept, by objectively considering and providing recommendations for the use of statistics in determining the correct LOD for a given analysis. Application of static signal-to-noise theory to the detection and integration of dynamic (i.e., chromatographic) signals (2) provided good insight into understanding how the chromatographic process produces poorer LOD characteristics compared to static measurements due to peak broadening. Clarification of the LOD in chromatography has been analyzed concisely (3),with similar equations developed as in ref 2 for the effect of the chromatographic process on detection limits. By considering the chromatographic peak height as the sole constituent for detectability, much of the "peak information is wasted. Use of the entire peak area relative to the surrounding base line may better provide a statistically satisfying and objectively determined LOD. According to the international Union of Pure and Applied Chemistry (IUPAC), the LOD is the lowest concentration (or mass) of a chemical species that can be determined to be statistically different from an analytical blank (4). What is truly important is the statistical information (Le., precision and accuracy) for the final quantitative result. The purpose of this work is to study and characterize an integration method in which dynamic signals can be objectively determined at a much better LOD than if determined from a static signal LOD model ( 2 , 3 ) .The basis of the improvement will depend upon the nature of the data, with the base line behaving randomly (uncorrelated) and the eluting peaks behaving with an implicit nonrandom (correlated) behavior. Computer simulation will facilitate the development of this idea, while experimental data will be discussed afterward to compare with the simulation. In its most basic form, the method requires three steps. First, chromatographic data from a detector are collected and stored for data processing. Next, the chromatogram is base line adjusted so that the noise is centered (numerically) about zero. Finally, the chromatogram is integrated from beginning to end, point by point, to produce a secondary chromatogram.
We have found that signal detectability is greatly enhanced in the integrated chromatogram relative to the original chromatogram. The method will be more rigorously developed and studied in this work.
THEORY Chromatographic Peak Model and Relationships. First, it is important to relate the area of a chromatographic peak to the maximum height of the same chromatographic peak. A Gaussian peak shape model is used here, but is applicable to other peak shapes. Each peak i in the chromatogram can be described by
where S,(t) is the height of the peak i at time t , defined by the standard deviation of the peak (ub),retention time (tR) at the maximum S,(t), volume fraction injected ( V J , and anal@ detection response factor (I?,). Albeit an exponentially modified Gaussian (EMG) function would be more rigorous in modeling a chromatographic system (5),a Gaussian model will be adequate. The area of a chromatographic peak (S,-) can be calculated from eq 1 by integrating
where integration from (tR,i - 3 4 to (tR,,+ 3u,) is sufficient in recovering 99.74% of the total area. We note that
Thus, since V, and R, are independent of time
Si,AREA = V~Ri
(2c)
Now, calculation of the maximum signal (S~,MAXHT) of a chromatographic peak is facilitated by substituting t = tR,i into eq 1 (3) Conventionally, the maximum signal (SL,MAXHT) is used to determine detectability. That is, SL,MAXHTmust be greater than a confidence value determined by considering the noise of the background ( l ) ,in order for peak i to be detectable. By use of only the S,,MAXHT of a peak for the purpose of deciding detectability, much of the "implicit" correlation about a chromatographic peak is wasted. The improvement in sensitivity by using the entire peak relative to using the maximum signal will be defined as the signal increase factor (SIF) and can be expressed by dividing eq 2c by eq 3
0003-2700/85/0357-2162$01.50/0C 1985 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 57,
NO. 12, OCTOBER 1985 2163
(4) It should be noted that no consideration of background noise has been incorporated into eq 4. For a typical chromatographic system (6), and for the purposes of simulation, the standard deviation of each peak ( u i ) can be related to retention by
(5) where r is a constant for a given chromatographic system independent of analyte or retention time and ki is the capacity factor for peak i and is defined in the conventional way
with t oequal to the dead time, typically around 150 s. In eq 5, r is on the order of 1000 or 2000 for liquid chromatography and ui calculated is in units of time (6). The result is that the SIF calculated in eq 4, which is effectively the chromatographic dilution factor for a given peak, is also in units of time. Chromatographic Base Line Model and Relationships. A chromatographic base line F ( t )can be expressed as a linear combination by
F ( t ) = mt
+ b + D ( t ) + R ( t ) + Nt
(7)
where m is the slope of linear base line drift, t is the time, b is the y intercept of a base line chosen a t t = 0 (tin&, D ( t ) is any nonlinear and essentially nonrepeating base line fluctuation such as temperature effects, R(t) is any nonlinear yet repeating base line fluctuation such as electronic ringing or pump pulsation effects, and Nt is the random noise associated with a physical measurement that can be statistically treated. Sampling a large population of N, values should yield Gaussian statistics for Gaussian experimental noise (which is frequently observed for chromatography detectors). Through proper experimental consideration and procedure, both D(t) and R(t)in eq 7 can be reduced to near zero, so they can be neglected in eq 7. For any chromatographic system there is a portion of time before any material elutes (tinjectto to),and a portion of time after any material will elute (ca. k > 7 or so). By doing a linear least-squares fit to these two portions of a chromatogram, the slope (m*)and y intercept (b*) (at t = tinject)can be calculated to good precision. Thus, a base line adjusted (BLA) chromatogram can be calculated by considering eq 7 (neglectingD(t) and R ( t ) )and sequentially subtracting the slope-intercept contribution
F*(t) = F ( t ) - m*t - b* = Nt
(8)
for t = 1, 2, 3, ...,n - 1, n for a n-point chromatogram. For the purpose of the present study, n will equal 1200, and each increment o f t denotes 1s in time. Note that there is nothing magical about using 1200 as the number of data points. It is a convenient value that provides a 20-min chromatogram from a detector with a 1-s time constant. The concept of integrating base line noise has been studied to understand the effect of various types of noise on commonly used integration techniques in chromatography (7). Maximizing the precision of quantitation was the goal in that study. Intuitively, if noise is truly random, the integration (i.e,, addition) of noise should produce a value that remains close to the mean value of F*(t) before integration, which should be approximately equal to zero (7). What we are concerned with here is how a chromatographic peak behaves relative to the background noise in both the normal and integrated time domains, for the purpose of improving detectability. One can
I
I
I
0
600 TIME
1200
CBECONDSJ
Figure 1. (A) Typical base line noise, not BLA (eq 7). (B) BLA noise (eq 8). (C) Integrated BLA noise (eq 9).
introduce the idea of an integrated base line (IBL) at each time interval t (i.e., data point) in a chromatogram t
t
I B L ( t ) = CF*(t)= E N , t=l
t=l
(9)
where IBL(t) is the running-total integration of F*(t). Figure 1displays an example of eq 7,8, and 9. It is important to note that a base line adjusted to be centered about zero is necessary to have the IBL stay close to zero. Although the noise appears larger in the IBL compared to the normal base line in Figure 1, it will be shown later that the signal in an integrated chromatogram relative to a normal chromatogram more than compensates for this increase. For the IBL, the units for the vertical scale are actually “relative signal X time” and not just the “relative signal”, as is the case for the normal base line. Notice that eq 9 (and some others to follow) is a running-total expression and not a running average. Using a running total will not decrease the resolution (Le., separation information) of closely eluting chromatographic peaks, since no averaging of information is done, whatsoever. Objective determination of confidence limits for the IBL chromatogram must be considered. Calculation of the standard deviation of the base line fluctuations for any base line (IBL or otherwise) will be made by considering the time span of an “event” of chromatographic elution (1). The chosen event time span will be 60 data points (i-e.,seconds) for this simulation. Going back to eq 5 it can be seen that 60 points should be adequate for most k values in a realistic separation. Thus, the average value (u)of a section j of a base line can be calculated by P t
uj
=
t” = t“D
(10)
P
with j = 1, 2 , 3 , 4 , ..., m - 1, m, and p = 60 such that m X p = 1200, and with t b = 1 + (0’- 1)p)and tf = p + (0‘- 1)p). Then, the standard deviation ( u ) for each base line section can be calculated by
uj
=
\
“P-1
)
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ANALYTICAL CHEMISTRY, VOL. 57, NO. 12, OCTOBER 198‘5
for j = 1,2, 3, ..., m - 1, m, and with tb and t f as in eq 10, and
(E
“i)
for m sections of length p , where t is the index synonymous with time. From eq 12, 8, is the average standard deviation of a “series” of blank or background measurements in general form. For a BLA chromatogram, ii, = uN is calculated by using eq 10-12 with X ( t ) = F*(t). For the integrated BLA chromatogram, ax = uI is calculated similarly with X ( t ) = IBL(t). Thus, in the absence of any peaks, a comparison of the base line noise for a normal chromatographic base line and an integrated chromatographic base line can be made by relating UN and up Assuming the same number of standard deviations is used in defining detectability, a noise increase factor (NIF) can be defined as
NIF =
(13)
cI/QN
eq 13 should adequately describe the theoretical relationship between the noise in a normal chromatogram and the noise in the integrated chromatogram. It is anticipated that the NIF of eq 13 will be greater than one. Combining the Chromatograph Peak and Base Line Models. For the simulation of a “real” chromatogram containing peaks and base line noise, a convolution of both of these features must be made. A convolution of chromatographic peaks with base line noise to form a chromatogram is made by combining eq 1 and eq 7. A simulated normal chromatogram C ( t ) is produced by
C(t) = S(t) + F(t) (14) C(t) is a sequential array of elements with several peaks. Each peak (indexed by i) is defined by a retention time ( t ~ , and i) a peak width (ui, eq 5 and eq 6). For a BLA chromatogram, eq 14 will be C(t) =
[ V,Ri
LTi(2T)1/2 exp{
-(t
- tR,J2
2a;2
}]+
Nt (15)
The exponential part essentially vanishes for It - tR,il> 3ui. Also, Nt represents the random base line noise. Also an integrated chromatogram I ( t ) can be a calculated by using eq 15 t
I(t) = C a t )
(16)
t=l
where I ( t ) is the running-total integration of C ( t ) . The comparison in detectability between C ( t ) and I ( t ) is our main concern. It is observed from eq 4 alone that one has a better detectability in the integrated chromatogram, while eq 13 alone suggests a better detectability in the normal chromatogram. An improvement factor (IMP) for the integrated chromatogram relative to the normal chromatogram can be calculated using eq 4,5 , 6, and 13 as follows:
The value of IMP increases as the retention time increases, showing greater improvement for later eluting peaks compared to earlier eluting peaks. The constant r decreases as the efficiency of a column decreases, suggesting that the poorer the column, the better the improvement (larger IMP). The value of IMP is independent of any data averaging that might be applied to a chromatogram. However, it is important to have a chromatogram that has enough points to maintain the chromatographic information for the resolution of closely eluting species and to achieve good precision in the final
quantitation. This is realized experimentally by detecting the event with an instrument that has an appropriate time constant. As a suggested minimum, the width (u) of the earliest chromatographic peak should be at least 10 time constant intervals. Then a data collection rate the same as the detector time constant will produce the optimum IMP possible for a given detector time constant. The ratio uN/uI is determined empirically by characterizing the detected background noise (without injection).
EXPERIMENTAL SECTION Chromatographic simulation, quantitation, and statistical calculations were all done on a PDP 11/10 minicomputer (Digital Equipment Corp., Maynard, MA). All software was written inhouse except for the utilization of a pseudo-random-number generator designed specifically for a 16-bit computer (8). The pseudo-random-number generator produces numbers evenly distributed between 0 and 1,so the distribution was transformed into a normal distribution (Gaussian) by a suitable method (9). The noise generated was treated statistically,and the distribution was found to be Gaussian by calculating the fraction of noise for a population between one, two, and three standard deviations. All equations pertaining to the chromatographic simulation are given in the Theory section. Experimental noise was collected for two sets of conditions. To ensure that problems due to long-term fluctuationswould not affect the data, a 10 ms data collection rate (to produce a 1200 point data set) was used. A 12-s base line was produced. An Amperex (North American Philips) 56-DVP photomultiplier tube was operated at either 1500 V (room lights on) or 2100 V (room lights off) with a Hamner (Princeton, NJ), Model NV-13-P, high-voltage power supply. The photomultipler tube output signal was sent into a Princeton Applied Research (Princeton, NJ), Model HR-8 lock-in amplifier with termination at 100 kQ. A 10-ms output time constant was used with a 100 kHz measurement lock-in frequency supplied by a Wavetek (San Diego, CA) Model 162 wave generator. The signal output from the lock-in amplifier was collected by the computer with a LPS-11 laboratory interface at a 10-ms data collection rate. The resulting data files were analyzed as before and found to also show a Gaussian distribution. RESULTS AND DISCUSSION T o this point, equations have been left in general terms, except for defining the data arrays to be 1200 points. For the simulated data to be presented it is important to note that what is important is the relative values in comparing a normal chromatogram to an integrated chromatogram. Calculation of a “working” NIF (eq 13) is facilitated by statistical analysis of three base line noise chromatograms (without peaks). Table I contains the results of applying eq 10,11, and 12 to the normal and the integrated chromatograms for typical base line noise, that is Gaussian noise in the normal chromatogram. A normal base line should be BLA and appears like Figure lb, while the integrated base line appears as in Figure 1C. By analysis of two more base lines similar to the one in Table I, uN was found to be 98.67 (f0.38) and uI was 274.13 (h31.82). The important quantity is the, ratio CTI/UN,namely, NIF = 2.78. Simulation of chromatographic peaks should suggest realistic column performance. A typical value for the column performance constant in eq 5 and eq 17 is r = 1500, along with t o = 150 s. Simulated chromatographic peak data were calculated using eq l , 2, and 3 to form “noise-free” chromatograms (Table 11). Five, well-resolved peaks were generated such that each analyte has the same detector response factor ( R J ,such as with an indirect mode detector. Volume fractions are chosen for the purpose of discussion and should be used only to compare the normal and integrated chromatograms. The theoretical IMP factors are shown in Table 11, suggesting a marked improvement in detectability in the integrated chromatogram. Notice that the improvement increases with increased retention time. This is because the integration
ANALYTICAL CHEMISTRY, VOL. 57, NO. 12, OCTOBER 1985
2165
Table I. Typical Base Line Noise Data normal chrom
integrated chromatogram
interval, s
Uj"
Ujb
Uj"
bid
1-60 61-120 121-180 181-240 241-300 301-360 361-420 421-480 481-540 541-600 601-660 661-720 721-780 781-840 841-900 901-960 961-1020 1021-1080 1081-1140 1141-1200
-7.61 -21.08 1.05 -1.49 3.93 5.66 11.57 6.98 -8.37 -2.93 16.55 -7.30 -21.41 -1.52 1.05 8.01 -2.91 -4.07 2.48 0.73
73.51 96.30 98.66 95.48 108.55 105.40 100.48 105.72 100.44 101.99 99.63 104.04 96.21 107.97 85.03 102.39 113.15 106.42 92.39 90.26
-231.72 -1190.20 -2046.08 -1415.66 -1420.20 -1307.92 -729.67 -290.26 -150.69 -700.70 23.39 -20.96 -1247.43 -1556.82 -1495.66 -1004.27 -1024.23 -1069.80 -1517.20 -1052.04
199.34 448.92 248.78 299.04 295.80 230.81 396.52 280.69 394.25 210.88 239.18 189.35 392.38 253.66 102.61 207.81 302.38 196.50 195.25 150.25
AV = 8;
I
261.72
99.20
aEquation 10, X ( i ) = F*(t). bEquation 11, X(i) = F*(t). cEquation 10, X ( i ) = IBL(t). dEquation 11, X ( i ) = IBL(t). eEauation 12.
I
10'00
u l l
l
r-
I
0
600 TIME
I
1200
CSECONo8)
and R,= Flgure 2. (A) Simulated chromatogram, with Vi= 9 X 5 X 10' for all five peaks (eq 15). (B) Simulated chromatogram, with VI = 3 X and R, = 5 X 10' for all five peaks (eq 15). (C) and R, = 5 X 10' for Simulated chromatogram, with V, = 1 X all five peaks (eq 15).
Table 11. True Data and Theoretical Improvement Factor (IMP) for Simulated Chromatograms IMP (theoretical): peak no. 1 2
3 4 5
peak
tR, s
300 450 600 750 900
max height"
areab
S
364.2 210.3 148.7 115.2 94.0
5000.0 5000.0 5000.0 5000.0 5000.0
4.94 8.55 12.10 15.62 19.13
"Equation 3 and eq 5 with to = 150 s, Ri = 5 X lo', Vi = 1.0 X bEquation 2C with Ri= 5 X lo', Vi = 1.0 X W3. CEquation 17 using uI/uN = 2.78, to = 150 s and r = 1500. method compensates for the band broadening effect due to chromatographic retention. Now it is possible to combine both base line noise and chromatographic peak data to produce a simulated "real-life" chromatogram, via eq 15. This is shown for successive dilutions in Figure 2. The peak data given in Table I1 correspond to the BLA chromatogram shown in Figure 2C. Note that the detectability does not appear very favorable, especially for the later eluting peaks in Figure 2C. By performing the running-total integration of the data in Figure 2A-C, using eq 16, we obtain integrated chromatograms. The results are shown in Figure 3. Note the vertical scale used for each integrated chromatogram relative to the scale used in Figure 2. The improvement in detectability in the integrated chromatograms relative to the normal chromatogram is quite obvious. It is necessary to study the precision and accuracy that this integration method provides, as compared to conventional approaches. Statistics concerning precision and accuracy can be addressed by simulating multiple injections via a fixed S ( t ) in eq 14 but varying the base line noise function F ( t ) . Thus, it is assumed that no uncertainty in V iexists, so the uncertainties in quantitation due only to the data handling can be studied. Three integrated chromatograms using eq 16 on three different C(t) arrays for the same "sample" are shown in Figure 4. Note that Figure 4A is for the same sample as Figure 3C.
O
I
W
I k
I
I -
r - 4
a 4 W
I
1
I
0
600
TIME
[SECONDS)
1200
Flgure 3. (A) Integrated chromatogram, integration of Figure 2A (eq 16). (B) Integrated chromatogram, integration of Figure 28 (eq 16). (C) Integrated chromatogram, integration of Figure 2C (eq 16).
Determining the statistics of precision and accuracy in the quantitation of each signal will provide insight into reproducibility, but first, an explanation on show the signal is determined in this integration method is required. Earlier, a uI value was stated. From this value a LOD for the integrated domain can be calculated as some number (typically 5 ) multiplied by uI. Deflections in an integrated chromatogram exceeding this LOD constitute an analytical signal. For a given analytical signal, a region on each side of the signal relatively close to having a slope of zero will be observed as can be seen in Figure 5 . The quantity corresponding to the LOD, ~ U I ,is shown for convenience. Since the slope may not be exactly zero in the two regions on either side of the signal, a least-squares linear regression is calculated for each section. The extrapolated linear regression lines are
2166
ANALYTICAL CHEMISTRY, VOL. 57, NO. 12, OCTOBER 1985
Table 111. Precision (Percent Relative Standard Deviation) for Average of Three Trials
peak no.
max htb ( t R not assumed)
height at tRC
peak aread this worke
Sample 1” 1 2 3 4 5
7.4 6.4 11.7
14.0 59.5 46.6
3.4 19.1 39.2
f f
f f
f f
8.3 6.8 8.6 18.8 20.8
1 2 3 4 5
3.9 7.5 6.0 5.4 8.7
5.5 4.4 4.3 6.4 32.5
2.8 2.4 3.0 6.4 6.2
Sample 2a
a I
J
0
-L-#
4.5 18.2 12.0 45.1 16.3
Sample 3“
0
600
TIME
[SECONDS)
1200
Figure 4. Integrated chromatograms (eq 16) of the same peak data but different noise data in eq 15, for V, = 1 X R, = 5 X lo6.
I
1
U
i I
1 2 3 4 5
1.4 2.8 2.4 0.47 8.5
1.5 5.9 3.7 10.6 6.2
0.68 1.3 2.4 2.5 1.9
0.94 0.83 1.0 2.1 2.0
“Peak number as in Table 11: sample 1, Vi = 1 X sample 2, sample 3, Vi = 9 x low3;samples 1, 2, and 3, Ri = 5 X lo6. bMethod I. “Method 11. dMethod 111. eMethod IV. fundefined, since U Q > Q.
Vi= 3 x
I
m
I/
/ i i
LOO
600 700 [SECONDS]
600 TIME
Figure 5. Demonstration of the quantitation of a signal in an integrated chromatogram: V, = 1 X R, = 5 X loe, and t , = 600 s.
shown for the signal in Figure 5. The distance between the two lines a t the inflection point of the analytical signal curve is the area of the original chromatographic peak and, thus, the signal of the analyte in the integrated chromatogram. Precision of an analytical method can be discussed by using the relative standard deviation. The standard deviation for the quantitation of three trials (ua) relative to the average for the three trials (Q), multiplied by 100, will be used
RSD% =
)(:
100%
Accuracy, likewise, is discussed by comparing the true value ( T ) , given in Table 11, relative to the average of three trials (Q). A relative difference of these two quantities can be calculated by
RD% =
(7)
100%
Note that the sign is important for this quantity in comparing analytical methods. Three other analytical methods used to quantitate chromatographic data can be compared to our integration method. Method I employs defining the peak height as the measurement of the largest signal on an interval in a normal chro-
matogram known to contain a peak, but exact knowledge of the retention time is not known. Method I1 also determines the peak height in a normal chromatogram, but the retention time must be known to the same precision as the data acquisition rate. In method 11 the peak height is measured at precisely the retention time, thus not necessarily providing the largest signal for a given peak. Method I11 provides the peak area by summing the signals in a normal chromatogram, but only those signals greater than or equal to some confidence level (2). This method, in essence, treats each data point as a separate event and suggests no implicit correlation of the data for an eluting chromatographic peak. For method 111, only those data points greater than 3aN are added to the total for a given peak. The integration method we have described here will be labeled method IV. Results were obtained for the application of the four quantitation methods for the three samples shown in Figure 2, using three arbitrarily chosen base line noise arrays. The standard deviation ( g a ) of the three signals obtained for a given peak was calculated, and the “true” values were known (Table 11) so eq 18 and eq 19 could be applied. The results for studying precision are given in Table 111,while the accuracy data are given in Table IV. The following comparisons relative to method IV can be made from the data of these two tables. Method I clearly produces a signal that is very inaccurate, biased to higher values compared to the true values, although the precision is reasonably good. Method I1 has no expected bias, but the precision and the accuracy are both relatively poor, Because of the confidence level criterion imposed upon method 111, the peak area is substantially less than the true values, as previously reported (2). The precision of method I11 is comparable to method IV, except at low volume fractions. In general, the predicted improvement in detectability (IMP,eq 17) is substantiated by the observations made from Tables I11 and IV. For sample 1,the peaks at 750 and 900 s cannot be quantitated by methods 1-111, while the new integration method (method IV) works quite well. It is interesting to note that the signals shown in Figure 4, together with statistics given in Tables I11 and IV (sample 1, method
ANALYTICAL CHEMISTRY, VOL. 57, NO. 12, OCTOBER 1985
Table IV. Relative Percent Deviation (RD%)from True Value for Average of Three Trials
peak no.
max htb (tR n o t assumed)
height a t tRe
peak aread
this worke
Sample '1
1 2 3 4 5
+38.4 +82.1 +134.7
f f
-4.2 -12.0 -30.7
53.0 -84.1 -86.6
f f
f f
+1.3 +11.8 +9.8 +2.8 -14.2
Sample 2O
1 2 3 4 5
+10.3 +22.7 +43.0 +50.8 +62.7
-1.3 -3.9 -10.3 -38.4 +24.1
-6.8 -15.9 -30.8 -50.0 -71.1
+0.43 +3.9 +3.3 +0.92 -4.7
-0.91 -3.9 -5.8 -8.4 -16.3
+0.14 +1.3 +1.1 +0.31 +1.6
Sample 3"
1 2 3 4 5
+2.3 +6.7 +12.3 +14.0 +16.7
-0.45 -1.3 -3.4 -12.8 +8.0
O-fSarne as Table 111.
IV), are at an average signal-to-noiseratio (SIN) of 7.30. For this system at the LOD (SIN = 2), a volume fraction of 2.74 X lo4 is calculated. Comparatively, the other three methods are essentially useless even at a volume fraction of 1 X Because the success of this integration method is strongly dependent upon the behavior of the original base line, it is worthwhile to study a few experimentally obtained "real" noise arrays. A few typical detection systems were studied and the results supported the data previously provided by the simulation. One of the detection systems studied is described in the Experimental Section. The importance of minimizing the effects of D ( t ) and R(t) in eq 7 through proper experimental considerations cannot be emphasized enough. In applying this integration method, it is essential to use the base line adjustment procedure as discussed in the Theory section. Once
'
2167
base line adjusted, it is a simple task to generate the running-total integrated chromatogram from a normal chromatogram. Similar to any other quantitative method, rejection of spurious results can be made by proper statistical tests. That is, anomalous features in an integrated chromatogram can be dealt with just as is conventionally done in a normal chromatogram for such things as glitches, pseudopeaks, impurity peaks, etc. This integration method should give better LOD values, that are essentially independent of chromatographic dilution effects. Furthermore, this method can be readily incorporated into typical data handling systems currently in use for chromatography. Finally, the method should have general applicability for any data set in which the experimental noise, be it Gaussian or otherwise, is uncorrelated along a given abscissa and an analyical signal is correlated along the same abscissa. The abscissa may be time, wavelength, frequency, etc.
ACKNOWLEDGMENT R.E.S. wishes to thank Joel D. Kress for helpful discussion of the reserch idea and W. J. Kennedy, Jr., for assistance in generating Gaussian noise. LITERATURE CITED (1) Long, G. L.; Winefordner, J. D. Anal. Chem. 1983, 5 5 , 712A-724A. (2) Taraszewski, W. J.; Haworth, D. T.; Pollard, B. D. Anal. Chlm. Acta 1984, 157,73-81. (3) Foley, J. P.; Dorsey, J. G. Chromatographia 1884, 18, 503-511. (4) "Nomenclature, Symbols, Unlts and Their Usage in Spectrochemical Analysis-11" Spectrochlm. Acta, Part 6 1978, 336, 242-245. (5) Foley, J. P.; Dorsey, J. G. J . Chromatogr. Scl. 1984, 22, 40-48. (6) Fritz, J. S.; Scott, D. M. J . Chromafogr. 1983, 271, 193-212. (7) Smlt, H. C.; Walg, H. L. Chromatographia 1975, 8 , 311-323. (8) Wichmann, 6. A.; Hill, I.D. Appl. Stat. 1982, 31, 188-190. (9) Muller, M. E. "Generation of Normal Deviates"; Technical Report No. 13; Statlstical Techniques Research Group, Department of Mathematics, Princeton University.
RECEIVED for review April 8, 1985. Accepted June 13, 1985. The Ames Laboratory is operated for the U S . Department of Energy by Iowa State University under Contract No. W-7405-eng-82. This work was supported by the Office of Basic Energy Sciences. R.E.S. thanks the American Chemical Society Analytical Division and specifically the Society for Analytical Chemistry of Pittsburgh for supporting this work, in part, through a Fellowship.
