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LCGC NORTH AMERICA VOLUME 24 NUMBER 4 APRIL 2006
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Integration Errors in Chromatographic
Analysis, Part I: Peaks of
Approximately Equal Size
Chromatographic situations are created for varying peak resolution and
relative peak size. In this case, the peak size of the smaller peak is
always at least 5% of the larger peak, and resolution is varied from 4.0
to 1.0. All separations are then integrated, using both area and height,
by four different baseline methods: drop, valley, exponential skim, and
Gaussian skim. Integration errors are calculated using reference
calibration injections. The results demonstrate that the drop and
Gaussian skim methods produce the least error in all situations. The
valley method consistently produces negative errors for both peaks,
and the skim method generates a significant negative error for the
shoulder peak. Peak height is also shown to be more accurate than
peak area. As the relative peak size increases, resolution below 1.0 will
generate unacceptable errors, and resolution greater than 1.5 is
necessary to minimize integration errors.
I
Merlin K.L. Bicking
ACCTA, Inc., St. Paul, Minnesota
Please direct correspondence to Merlin
K.L. Bicking at [email protected]
ntegration of chromatographic peaks
(determination of height, area, and
retention time) is the first and most
important step in the data analysis part of
all chromatography-based analytical
methods. This information is used for all
subsequent calculations, including construction of calibration curves and calculation of unknown concentrations.
Clearly, any error in measurement of peak
size will produce a subsequent error in the
reported result.
Analysts have several choices for integrating peaks. Figure 1 illustrates the four
most common options for drawing the
baseline between two peaks: drop, valley,
exponential skim, and Gaussian skim.
The drop method (Figure 1a) involves
addition of a vertical line from the valley
between the peaks to the horizontal baseline, which is drawn between the start and
stop points of the peak group. The valley
method (Figure 1b) sets start and stop
points at the valley between the peaks,
thus, integrating each peak separately.
Skim procedures separate the small peak
from the larger parent with separate baselines. The parent peak is integrated from
its starting point to the apparent end of
the peak group. The small peak’s baseline
starts at the valley between the peaks, and
ends when the signal nears the baseline.
The area “under” the skimmed peak is
added to the parent peak, not the
skimmed peak. This approach has been
described also as a tangent integration
method, and the small peak variously
labeled a skim, shoulder, or rider peak.
Several variations of the skim procedure
are possible. Early integration algorithms
drew a straight line from the valley to the
end of the peak. Figure 1c shows an exponential skim baseline. An exponential
function is used to create curvature in the
skim line, in an attempt to approximate
the underlying baseline of the parent
peak. While the exact procedures used to
construct this line are a proprietary part of
modern software algorithms, they all
employ the same basic approach — use an
exponential function to draw a curved
baseline under the skimmed peak. More
recently, another skim procedure has been
developed, as shown in Figure 1d.
Although referred to as a new exponential
skim method, it will be described here as
a “Gaussian” skim, because the intent is to
more accurately reproduce the Gaussian
shape of the parent peak.
For any of these methods, analysts must
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LCGC NORTH AMERICA VOLUME 24 NUMBER 4 APRIL 2006
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(a)
Absorbance (mAU)
1000
800
600
400
200
0
0.75
0.80
0.85
0.90
0.95
1.00
0.80
0.85
0.90
0.95
1.00
0.80
0.85
0.90
0.95
1.00
0.80
0.85
0.90
0.95
1.00
(b)
Absorbance (mAU)
1000
800
600
400
200
0
0.75
(c)
Absorbance (mAU)
1000
800
600
400
200
0
0.75
(d)
Absorbance (mAU)
1000
800
600
400
200
0
0.75
Time (min)
Figure 1: Common integration baseline options: (a) drop, (b) valley, (c) exponential skim,
and (d) Gaussian skim.
also choose between measuring peak size
by either area or height. It is clear that
these integration options are likely to generate significantly different analytical
results, and analysts must decide which
approach provides better accuracy.
Despite the importance of this issue,
clear guidelines are difficult to find in the
published literature. Snyder and Kirkland
(1) discussed the relative quantitative
errors for symmetrical peaks, suggesting
that peak height with the drop method
was more accurate for poorly resolved
peaks, but area measurements were more
precise. These recommendations seem to
have been ignored largely by the chromatography community, as the drop
method with area is used almost exclusively today. Foley (2) published correction factors for pairs of tailed peaks, allowing estimation of their true areas. More
recently, Meyer published a series of
papers (3–7) dealing with several integration issues. Peaks of varying symmetries,
resolutions, and size ratios of as much as
10:1 were prepared by calculation, and
integrated using the drop method and
area (3), resulting in errors of 40% in
some cases. In another study of 10:1 peak
ratios, the height method was more accurate if the small peak was eluted first.
However, when the small peak is eluted
second, the drop method produced a large
positive error and the skim method produced a large negative error (4). These
general recommendations were extended
to even smaller peaks (that is, 1000:1
ratios), although the better integration
method for a small second peak varied
with the relative peak widths (5). The
effects on enantiomeric analysis and calibration curves were reported (6), and a
mathematical procedure for quantification of shoulder peaks was developed (7).
Again, the recommendation that peak
height is more accurate than area has not
been put into general practice.
Although these studies provided some
excellent general guidance, they only considered the drop method when the first
peak was small, and only added straight
skimming as an option for a second small
peak. The valley and skim methods were
not evaluated. When significant errors
were noted, explanations for the errors
were incomplete or missing, especially for
the skim method.
This study seeks to provide some guidelines for analysts by creating several peak
resolution situations. The resolution
between two peaks is varied from 4.0 to
1.0 through changes in operating conditions. The relative size of the two peaks
also is changed from nearly equal size to
approximately 5% of the other peak. Each
peak pair is then integrated using each of
the four baseline options, using both area
and height, and integration errors are cal-
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LCGC NORTH AMERICA VOLUME 24 NUMBER 4 APRIL 2006
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Table I: Relative response for test
solutions*
Peak 1:
120
Absorbance (mAU)
Nitrobenzene
Second Peak Small
Mixture
100
100
100
100
100
80
90
48
18
4.6
First Peak Small
60
100
25
100
6.1
100
*Values obtained at resolution 4.0 (45%
60
40
Peak 2:
Dimethyl Phthalate
Individual components
20
acetonitrile). Largest peak is always given a
value of 100.
0
0.800
0.825
0.850
0.875
0.900
0.925
0.950
0.975
Time (min)
Figure 2: Illustration of drop method integration errors when the second peak is smaller.
The overlay shows the chromatogram for the mixture and for injections of individual components at approximately the same relative concentrations.
culated through the use of reference calibration injections. This study is concerned with situations in which the size of
one peak is at least 5% of the other peak.
A separate study will report results when
one peak is significantly smaller than the
other (8).
Finally, all resolution situations
described here were created using a liquid
chromatography (LC) system. However,
the peak shape observed is typical for any
well-behaved chromatographic system, so
application to gas chromatography (GC)
separations should be valid. Although
only one data system was used for this
study, the author believes that all modern
chromatography data systems process data
using similar procedures, and only minor
differences would be produced by other
software packages. The general conclusions still should be valid.
Experimental
Equipment: All chromatographic experi-
ments were performed using a model
1100 HPLC, equipped with a binary
pump and degasser, autosampler, column
heating compartment, and diode array
detector (Agilent Technologies, Wilmington, Delaware). The system was controlled by ChemStation software (Agilent
Technologies, Palo Alto, California, Version B.01.03). A 100 mm 4.6 mm
Hypersil C18 column, packed with 5-m
particles, was used for this study.
Reagents: Solutions of nitrobenzene
(Sigma Aldrich [St. Louis, Missouri],
99%, ACS grade) and dimethyl phthalate (Acros [Geel, Belgium], 99%) were
diluted in high performance liquid chromatography (HPLC) grade acetonitrile.
HPLC grade acetonitrile and water were
used for mobile-phase components.
Preparation of test solutions: Individual stock solutions of dimethyl phthalate
and nitrobenzene were prepared in acetonitrile, so that when diluted, the maximum peak height observed was near 1
absorbance unit (AU). This situation
minimized errors due to detector and
solution absorbance nonlinearity.
A solution of both components at
approximately equal peak area responses
was prepared by appropriate dilution.
This mixture was then diluted, first, with
a solution of dimethyl phthalate only in
acetonitrile at the same concentration as
the mixture. A combination of serial and
parallel dilutions with these two solutions
resulted in a series of test samples with a
constant concentration of dimethyl
phthalate and varying concentrations of
nitrobenzene. These solutions are referred
to in the text as “Second Peak Small,”
because nitrobenzene is eluted after
dimethyl phthalate. Similarly, the initial
mixture was diluted with a solution of
nitrobenzene only in acetonitrile, producing another series of test samples with a
constant concentration of nitrobenzene
and varying concentrations of dimethyl
phthalate. These solutions are labeled
“First Peak Small.”
Operating conditions: All experiments
were conducted at 1.50 mL/min. with an
injection volume of 5 L and a column
temperature of 40 °C. Absorbance was
monitored at 250 nm, with a bandwidth
of 100 nm, using a reference of 360 nm,
with a reference bandwidth of 100 nm.
Detector settings were adjusted to ensure
that an adequate number of data points
were collected for each peak, and that
detector response time did not influence
peak shape. Triplicate injections of each
test solution were performed under each
of the operating conditions described in
the following.
The system was operated under isocratic conditions, using the following concentrations of acetonitrile in water:
45.0%, 67.5%, 75.0%, and 83.0%.
These conditions produced resolution
between the two analytes (dimethyl
phthalate and nitrobenzene) of 4.0, 2.0,
1.5, and 1.0, respectively, as measured by
the data system, using the resolution tangent method for the test sample containing approximately equal concentrations.
USP tailing factors ranged between 1.05
and 1.10, so this study presents results for
typical chromatographic peaks that show
only minimal tailing.
Calculations: Each injection was separately reprocessed using the four integration methods, recording peak areas and
heights for each injection and integration
method. Consistent integration settings
were employed to minimize variations
between different analysis conditions, and
all chromatograms were examined to
APRIL 2006 LCGC NORTH AMERICA VOLUME 24 NUMBER 4
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ensure proper placement of integration
start and stop points. In a few cases, examination of the chromatograms indicated
that a small interfering peak was present.
This peak was integrated appropriately
and its value subtracted from the observed
value before any calculations were performed.
In addition to the two component mixtures, test solutions containing each analyte individually also were analyzed. These
single-component solutions were prepared to match the highest concentration
in the test solutions (that is, 100%) and a
lower level (5%). Each of these solutions
served as a calibration reference.
Analysis under conditions generating a
resolution of 4.0 (45% acetonitrile) was
used to define the “true” value for each
test solution. That is, the relative response
of the two components was determined
from these data. The values for each test
solution are listed in Table I, with the
largest peak always assigned a value of
100. The ratio between each component
in the test solutions and its corresponding
calibration reference produced a response
factor. Then, under each set of resolution
conditions, and for each integration
method, the response for the calibration
reference was used with the response factor to calculate an expected peak response
(area or height) for those conditions.
Comparison of this expected peak
response with the actual value allowed calculation of the error for the observed
chromatogram. All values reported here
represent the percent error between the
observed and expected values. A positive
error means the observed value was higher
than expected; a negative error means
some peak area (or height) was lost.
Results
and
Discussion
The calibration reference method was
used to avoid errors due to data collection.
Different resolutions between the test
peaks were generated by adjusting chromatographic conditions — a decrease in
retention produced a decrease in resolution. Under these conditions, peak width
also decreased and the peak height
increased. However, the data system collects data at a fixed rate, so the area results
for smaller peak widths would appear to
be artificially smaller and the peak heights
Circle 46
407
would be larger, even though the same
quantity was injected. So, simple comparisons of peak areas and heights between
different operating conditions would produce inaccurate conclusions. By using a
calibration reference solution, the
response under each set of operating conditions is adjusted for the change in peak
width and height, and subsequent calculation of errors produces values that can be
compared across different levels of resolution.
A summary of integration errors is provided in Table II. The table is organized
by relative peak size and resolution. For
each of the two peaks, average integration
errors are listed for both area and height
measurements, using each of the integration methods: drop, valley, exponential
skim, and Gaussian skim. The values in
parentheses represent the standard deviation for each average, based upon three
injections. Variability was generally less
than 0.5%, indicating adequate precision
in both chromatography and integration.
This level of precision is typical for any
chromatographic analysis based upon an
external standard method. Consequently,
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Table II: Integration errors for differing relative peak areas, resolutions, integrated method*
Peak Ratio
Pk1 Pk2
100 90
Rs Drop
2.0 -0.6
(0.04)
1.5 -1.8
(0.10)
1.0 -2.9
(0.12)
100 48 2.0 -0.7
(0.19)
1.5 -1.8
(0.14)
1.0 -2.9
(0.12)
100 18 2.0 -0.9
(0.48)
1.5 -1.7
(0.14)
1.0 -2.0
(0.20)
100 4.6 2.0 -0.6
(0.15)
1.5 -1.8
(0.09)
1.0 -1.4
(0.04)
60 100 2.0 3.7
(0.13)
1.5 3.5
(0.14)
1.0 1.7
(0.15)
25 100 2.0 -0.3
(0.25)
1.5 -1.2
(0.16)
1.0 -4.8
(0.11)
6.1 100 2.0 0.2
(2.07)
1.5 -0.6
(0.11)
1.0 -8.5
(0.23)
Peak 1 Area
Valley
-1.9
(0.16)
-6.3
(0.12)
-28.7
(0.11)
-1.84
(0.12)
-5.2
(0.13)
-23.0
(0.12)
-1.92
(0.41)
-4.0
(0.15)
-16.2
(0.10)
-1.5
(0.12)
-3.0
(0.08)
-9.2
(0.02)
2.6
(0.03)
-1.8
(0.15)
-28.7
(0.18)
-2.2
(0.18)
-7.8
(0.17)
-39.4
(0.14)
-0.9
(0.50)
-11.3
(0.13)
-56.4
(0.48)
E. Skim
-1.4
(0.09)
-4.7
(0.09)
-19.1
(0.47)
-0.2
(0.26)
1.9
(0.26)
10.8
(0.18)
-0.6
(0.54)
0.8
(0.15)
6.2
(0.18)
-0.3
(0.15)
-0.4
(0.08)
2.0
(0.05)
2.9
(0.18)
0.1
(0.15)
-17.2
(0.76)
-1.2
(0.25)
-6.4
(0.31)
-26.6
(0.40)
-0.4
(0.56)
-8.8
(0.08)
-39.5
(0.48)
Peak 1 Height
G. Skim
-0.59
(0.04)
-1.72
(0.12)
-8.99
(0.31)
-0.77
(0.19)
-1.05
(0.19)
1.34
(0.29)
-1.03
(0.47)
-1.07
(0.15)
-0.22
(0.24)
-0.69
(0.14)
-1.32
(0.09)
-1.20
(0.15)
4.23
(0.15)
3.51
(0.14)
-4.71
(0.69)
0.88
(0.25)
-1.59
(0.24)
-10.64
(0.51)
5.14
(2.38)
-1.23
(0.25)
-12.44
(0.24)
Drop
-0.3
(0.11)
-0.3
(0.17)
-0.4
(0.10)
-0.7
(0.08)
-0.6
(0.04)
-0.9
(0.18)
-0.4
(0.08)
-0.5
(0.11)
-0.4
(0.12)
0.04
(0.12)
-0.5
(0.18)
-0.7
(0.15)
3.7
(0.29)
4.3
(0.15)
4.6
(0.15)
-1.8
(0.24)
-1.8
(0.33)
-2.0
(0.02)
-0.7
(0.47)
-0.9
(0.11)
0.5
(0.32)
Valley
-0.6
(0.14)
-2.4
(0.17)
-15.8
(0.09)
-1.0
(0.09)
-2.2
(0.04)
-11.9
(0.18)
-0.6
(0.10)
-1.5
(0.11)
-7.49
(0.13)
-0.1
(0.11)
-1.1
(0.19)
-3.9
(0.17)
3.5
(0.25)
2.0
(0.15)
-14.2
(0.16)
-2.1
(0.28)
-4.4
(0.33)
-25.1
(0.12)
-0.6
(0.36)
-5.1
(0.17)
-39.3
(0.63)
E. Skim
-0.5
(0.12)
-1.2
(0.12)
-8.2
(0.38)
-0.7
(0.08)
-0.6
(0.05)
-1.0
(0.18)
-0.4
(0.08)
-0.5
(0.11)
-0.4
(0.12)
0.04
(0.12)
-0.5
(0.18)
-0.6
(0.16)
3.5
(0.31)
3.3
(0.15)
-4.9
(0.57)
-2.0
(0.23)
-3.1
(0.40)
-13.9
(0.33)
-0.8
(0.10)
-3.2
(0.13)
-22.1
(0.55)
Peak 2 Area
G. Skim
-0.26
(0.11)
-0.32
(0.17)
-1.98
(0.17)
-0.70
(0.08)
-0.58
(0.05)
-0.95
(0.18)
-0.37
(0.08)
-0.52
(0.18)
-0.45
(0.12)
0.04
(0.12)
4.32
(0.15)
-0.69
(0.12)
4.01
(0.29)
-1.83
(0.34)
2.96
(0.25)
-1.17
(0.24)
-0.91
(0.05)
-3.86
(0.18)
2.02
(0.47)
-0.91
(0.05)
-2.30
(0.37)
Drop
0.1
(0.20)
-0.02
(0.14)
1.4
(0.22)
0.3
(0.31)
0.9
(0.22)
2.2
(0.14)
0.7
(0.69)
3.4
(0.79)
4.2
(0.22)
6.6
(0.46)
0.1
(0.64)
7.7
(0.26)
-0.5
(0.23)
-0.2
(0.24)
-0.1
(0.20)
-0.2
(0.13)
0.01
(0.24)
0.02
(0.05)
-0.9
(0.16)
-0.8
(0.14)
-1.4
(0.31)
Valley
-0.9
(0.07)
-6.2
(0.12)
-26.0
(0.13)
-1.2
(0.11)
-7.8
(0.12)
-34.0
(0.20)
-2.0
(0.36)
-10.9
(0.06)
-47.8
(0.28)
-2.4
(0.12)
-23.6
(0.31)
-76.0
(0.47)
-1.1
(0.19)
-5.1
(0.12)
-20.0
(0.22)
-0.6
(0.09)
-3.3
(0.13)
-12.9
(0.07)
-1.0
(0.12)
-2.6
(0.01)
-7.5
(0.28)
Peak 2 Height
E. Skim
0.9
(0.32)
2.8
(0.16)
16.2
(0.20)
-0.7
(0.17)
-6.3
(0.12)
-24.2
(0.67)
-1.1
(0.42)
-8.3
(0.09)
-33.6
(0.33)
1.6
(0.43)
-19.0
(0.24)
-56.1
(0.12)
0.1
(0.21)
1.4
(0.15)
9.1
(0.21)
0.2
(0.14)
0.9
(0.17)
4.8
(0.05)
-0.7
(0.25)
-0.6
(0.16)
0.3
(0.32)
G. Skim
0.06
(0.19)
0.37
(0.10)
6.57
(0.19)
0.40
(0.34)
0.08
(0.29)
-6.61
(0.70)
1.15
(0.75)
2.31
(0.13)
-5.79
(0.06)
9.05
(0.56)
2.89
(0.40)
0.83
(0.93)
-0.51
(0.22)
-0.15
(0.15)
2.74
(0.37)
-0.21
(0.13)
-0.02
(0.19)
1.21
(0.13)
-0.89
(0.14)
-1.03
(0.15)
-1.24
(0.30)
Drop
0.2
(0.07)
-0.2
(0.17)
-0.2
(0.17)
0.3
(0.06)
0.5
(0.11)
1.4
(0.28)
0.8
(0.16)
1.0
(0.08)
4.0
(0.06)
2.6
(0.27)
1.9
(0.18)
14.9
(0.15)
-0.1
(0.39)
-0.8
(0.14)
-1.1
(0.23)
0.1
(0.22)
-0.4
(0.31)
-0.9
(0.07)
-0.3
(0.09)
-1.3
(0.12)
-1.5
(0.17)
Valley
-0.19
(0.11)
-2.4
(0.17)
-15.0
(0.15)
-0.2
(0.02)
-2.9
(0.13)
-20.2
(0.33)
-0.1
(0.13)
-4.7
(0.06)
-31.8
(0.36)
-0.7
(0.14)
-11.2
(0.41)
-64.8
(0.82)
-0.3
(0.38)
-2.3
(0.13)
-11.0
(0.24)
-0.1
(0.23)
-1.4
(0.30)
-6.7
(0.09)
-0.4
(0.16)
-1.7
(0.13)
-3.8
(0.17)
E. Skim
0.2
(0.07)
-0.2
(0.17)
-0.3
(0.18)
0.05
(0.03)
-2.0
(0.13)
-12.6
(0.60)
0.3
(0.15)
-3.0
(0.09)
-19.4
(0.10)
1.1
(0.29)
-7.6
(0.36)
-41.4
(0.15)
-0.1
(0.39)
-0.8
(0.14)
-1.2
(0.23)
0.1
(0.22)
-0.4
(0.32)
-1.0
(0.08)
-0.3
(0.09)
-1.3
(0.11)
-1.5
(0.17)
G. Skim
0.21
(0.07)
-0.13
(0.14)
-0.28
(0.18)
0.30
(0.06)
0.48
(0.13)
-1.92
(0.54)
0.82
(0.16)
1.06
(0.06)
-1.04
(0.19)
2.56
(0.27)
1.80
(0.22)
-0.64
(0.22)
-0.06
(0.39)
-0.74
(0.17)
-1.19
(0.23)
0.06
(0.22)
-0.39
(0.33)
-0.96
(0.08)
-0.31
(0.09)
-1.20
(0.06)
-1.49
(0.17)
All numbers are percentages.
*Values represent the difference between the measured response and the expected value, as a percent of the expected value. Numbers in parentheses are
the standard deviation for three injections.
any errors less than about 2% should be
considered negligible, and for the purposes of this discussion, such results represent no error. Also, recognize that by
reporting relative errors rather than
absolute errors, constant errors will be
magnified for the smaller peak.
Resolution was evaluated over the range
from 2.0 to 1.0, as this range represented
separation conditions for a majority of
chromatographic systems. While “baseline” resolution (Rs 1.5) is considered
an ideal compromise between separation
quality and analysis time, many methods
will require larger resolution values. In
other cases, chromatographic limitations
force the chromatographer to accept resolution values of 1.0. Lower resolution is
generally not desirable.
Integration Errors for the Drop
Method
The drop method is probably the most
common integration procedure in use.
As noted earlier, this technique is conceptually similar to the historic fraction
collection procedure, and most analysts
would view it intuitively as the best
option. However, examination of Table II
reveals that, while the drop method often
produces smaller errors than, say, the valley method, there are several important
situations where the drop method produces significant errors.
In particular, when the difference in
peak size becomes larger, the errors for the
410
LCGC NORTH AMERICA VOLUME 24 NUMBER 4 APRIL 2006
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(a)
400
Absorbance (mAU)
350
Mixture
300
250
Shoulder response
added to parent peak
200
150
100
Parent peak
Observed end
50
0
0.775
Actual end
0.800
0.825
0.850
0.875
0.900
0.925
0.950
0.975
Time (min)
(b)
400
Absorbance (mAU)
350
Mixture
300
250
200
150
100
50
0
Parent peak
0.850
0.900
0.950
Time (min)
Figure 3: Integration errors for the skim methods: (a) exponential skim, (b) Gaussian skim.
A chromatogram of the parent peak only is overlaid on the chromatogram of the mixture.
smaller peak increase. When the second
peak is small, its integrated area is higher
than expected, while the larger first peak
shows a small decrease. These errors generally get larger as the resolution
decreases.
There are two important effects that
contribute to these integration errors.
First, as the relative peak size increases, the
apparent valley between the peaks shifts
toward the smaller peak. This shift can be
shown readily by analyzing perfectly symmetrical (that is, Gaussian) peaks (see reference 4 for a good review of these concepts). The change tends to create a
negative error for the smaller second peak
and a positive error for the larger peak.
Interestingly, the apparent peak width for
the smaller peak will also change, which
suggests that any data system calculations
for the small peak (width, efficiency, and
so forth) are likely to be in error. When
the peaks are tailed, however, the tail of
the first peak extends further into the elution region of the second peak, resulting
in an added positive error for the smaller
second peak, with a corresponding loss of
area from the first peak. This effect is
shown in Figure 2, in which single component injections have been overlaid on
the mixture chromatogram. Note that
tailing from the larger peak adds additional area to the smaller peak. You can see
the change in the location of the valley
toward the smaller peak. The location
(retention time) of the smaller peak also
changes.
Thus, these two effects will tend to balance each other, with the eventual error
being a complex function of relative peak
size and tailing. In most cases, the tailing
effects will dominate, resulting in a positive bias for the small peak. However, situations can occur where the two effects
exactly balance, resulting in no error.
Unfortunately, it is difficult to predict
such situations. When the first peak is
small, the effects of tailing are generally
absent, although the influence of the
larger peak produces a loss in the first
peak’s area at high area ratio, probably due
to the change in the valley position. These
results are qualitatively consistent with
earlier work (3). Slight differences with
other studies probably reflect the variabilities from working with real chromatographic data rather than artificial peaks.
Finally, it is important to note that integration errors for height measurements
are smaller than the corresponding area
measurements in almost every situation.
The use of height for quantitative measurements has been abandoned by many
analysts, in the belief that area measurements are either more precise or more
accurate. The data presented here do not
support such an argument. While area
measurements for severely tailed peaks
can be more reproducible than height, for
most chromatographic systems with only
moderate tailing, height measurements
offer a clear accuracy advantage for poorly
resolved peaks (resolution less than 1.5),
as the overlap contribution to peak height
from the tailing produces less error than
the area contribution. This conclusion is
general for all integration methods and
relative peak sizes, and is consistent with
previous studies (3–5).
Integration Errors for the Valley
Method
The data in Table II indicate that the valley method produces negative errors in
every resolution situation. The errors are
particularly large (greater than 25%) for
many combinations. For example, as the
second peak gets smaller, the area errors
for the large peak are reduced from
28% to 16%, while the area errors
for the small peak increase from 26%
to 76% at resolution equal to 1.0.
Similar trends are observed at resolution
equal to 1.5, but the errors are smaller.
Comparable results are found when the
small peak is eluted first.
It is relatively easy understand these
errors by looking at Figure 2, and imagining the baselines being drawn to the valley
from the start of peak 1 and from the end
of peak 2. The loss of area is particularly
important for the smaller peak. The
height of the valley above the baseline
(due to decreasing resolution) is responsi-
LCGC NORTH AMERICA VOLUME 24 NUMBER 4 APRIL 2006
Integration Errors for the Skim
Method
The skim integration method normally is
applied in situations in which a small
peak is resolved partially from a larger
peak. Often called a “shoulder” peak, the
smaller peak is integrated so that the
integrator attempts to follow the baseline
produced by the larger peak. Any area
“under” the baseline for the skimmed
peak is added to the parent peak, while
the area above becomes the skimmed
peak.
Normally, skimming is performed only
on peaks that are poorly resolved.
Although there appears to be no formal
definition of when a peak becomes a
“shoulder” peak, it is proposed here that
this definition be reserved for situations
where the resolution is less than or equal
to one, and the smaller peak area is less
than about 50% of the larger peak. In
reviewing the data in Table II, clearly,
skimming need not be applied when the
resolution is 2.0, and it would be unlikely
even when the resolution was 1.5. So, the
Error (%)
(a)
20
10
0
-10
-20
-30
-40
(b)
0
Error (%)
data at resolution equal to 1.0 are of most
interest.
The results indicate that exponential
skimming of the small peak produces a
substantial negative bias, with errors of
more than 25% in most cases. For the
largest peak ratio (smallest second peak),
the area error is greater than 50% when
the small peak is second, and almost 40%
when the small peak is eluted first. Height
errors for these situations are more than
40% and 20%, respectively. Clearly, the
exponential skim integration method is
not valid for shoulder peaks with a resolution of 1.0, when the shoulder peak size is
greater than about 5% of the parent peak.
Skim integration errors have been
reported (4,5), but no explanations have
been offered. The data obtained in this
study provide an explanation, as shown in
Figure 3a. The data system sees the end of
the peak group as being after the shoulder
peak, where the baseline returns to prepeak 1 levels. All of the area under the
shoulder peak is added to the parent peak.
However, when the parent peak is overlaid
with the mixture peak, it is evident that
the parent peak actually ends much
sooner than what is observed, and the
actual response due to the parent peak is
considerably lower than what is projected
from the skim baseline. This difference
results in a loss of area (or height) from
the shoulder peak (negative error), and a
gain in area for the parent peak (positive
error). The error becomes even larger as
the shoulder peak becomes smaller or the
resolution decreases further. Under these
conditions, shoulder peaks could be integrated using the drop method. Although
the drop method generates a slight positive error, the magnitude of the errors is
considerably smaller than for the skim
method.
As noted earlier, the Gaussian skim
method is a newer approach to dealing
with shoulder peaks. In general, the software drops the Gaussian skim line to baseline at a much earlier time than the exponential skim method, whereas in the
exponential skim method, the skim line
merges with the existing (observed) baseline. The Table II errors suggest that this
approach results in much lower integration errors, especially when the second
peak is small.
The overlay in Figure 3b reveals that
the Gaussian skim line does indeed do a
-5
-10
-15
-20
(c)
Error (%)
ble for the error. Any errors from tailing
effects are considerably less important.
Again, peak height measurements produce significantly less error than area
measurements, especially for resolution
better than 1.5, and when the peaks are of
similar size. However, even it these situations the errors are often greater than
10%.
It is important to note that the valley
method does have some validity in chromatography, although not in the situations described here. In chromatograms
with multiple peaks and complex baselines, the valley method might indeed
produce less error than the drop method.
For example, unresolved matrix components or wandering baselines can create
artificially high valleys between peaks,
resulting in a significant positive error if
the drop method is used. In such situations, the real baseline more closely follows a valley-to-valley path. However,
identification of this phenomenon
requires careful review of the baseline, and
is often missed by a less experienced analyst. When peak widths are significantly
different, the more narrow peak can be
more accurately integrated using the valley method, although skimming would
probably be a better choice (5).
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40
20
0
-20
-40
-60
-80
-100
(d)
20
Error (%)
412
0
-20
-40
-60
-80
Small peak relative area
Figure 4: Summary of integration errors
when the second peak is small: Errors for (a)
peak 1 (larger peak) using area, (b) peak 1
(larger peak) using height, (c) peak 2
(smaller peak) using area, and (d) peak 2
(smaller peak) using height. Rs 1.
better job of following the actual parent
peak. A slight negative area error is produced when the second peak is small, but
the magnitude of the error is much more
acceptable. The area errors are near -10%
when the first peak is small, because the
skim line rises too quickly. However, the
overall correction is much better than the
exponential skim method.
As with the other methods, the heightbased errors are smaller than the areabased measurements. No significant error
is observed when the second peak is small,
and a small negative error may be present
when the first peak is small. This fortuitous situation arises because the skim line
is at or near the baseline level in the region
near the small peak’s maximum, and the
integration errors disappear. The Gaussian
skim–height method thus becomes equivalent to the drop–height procedure for the
small peak.
Selected data from Table II are shown
graphically in Figures 4 and 5. In Figure
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LCGC NORTH AMERICA VOLUME 24 NUMBER 4 APRIL 2006
Error (%)
(a)
60
40
20
0
-20
-40
-60
Error (%)
(b)
60
40
20
0
-20
-40
-60
(c)
Error (%)
20
10
0
-10
-20
-30
Error (%)
(d)
0
-2
-4
-6
-8
-10
-12
Small peak relative area
Figure 5: Summary of integration errors
when the first peak is small: errors for (a)
peak 1 (smaller peak) using area, (b) peak 1
(smaller peak) using height, (c) peak 2
(larger peak) using area, and (d) peak 2
(larger peak) using height. Rs 1.
4, integration errors for each method are
shown for the situation in which the size
of the second peak is varied, relative to the
first peak. Error results are organized by
peak and integration type (area or height).
Although only values for resolution equal
to 1.0 are shown, the shape of the curves
is similar for resolutions of 1.5 and 2.0.
The only significant difference is the magnitude of the error scale. Figure 5 illustrates similar data when the size of the first
peak is changed.
In all curves, it is evident that the drop
and Gaussian skim methods provide less
error in most situations, and height errors
are almost always smaller than area errors.
While the drop method produces essentially no error for many of the combinations, there are some combinations where
none of the traditional methods produces
an accurate result. For example, when the
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second peak is about 5% of the first peak,
and the resolution is 1.0, the smallest
error (drop method) is still greater than
5%. A negative error of similar value is
observed when the small peak is eluted
first. While such errors would be considered insignificant in a trace analysis
method, if this were a high accuracy
analysis, such as a pharmaceutical formulation, this level of error would likely be
unacceptable. The Gaussian skim
method, however, provides a much better
estimate of peak size, when height is
measured. Also, it should be noted that
the curves suggest a trend in the data at
smaller relative peak areas than are discussed here. The integration situation
becomes more complicated when the
small peak is less than 5% of the large
peak. These results are discussed in a separate study (8).
For the large peak (Figures 4a, 4b, 5c,
and 5d), integration errors are expected to
be smaller or insignificant. Indeed, this is
observed in most cases. However, there
are some situations where the integration
of the large peak produces significant
errors, and analysts must be aware of these
issues if integration of the large peak is
important.
Finally, it should be mentioned that all
of the integration errors discussed here
arise from the poor resolution between
the two peaks. As noted earlier (6,7),
errors would occur if the analysis involved
a calibration standard with only one peak,
but the actual sample included the second
peak. If the calibration standards have
both peaks present, then errors in the
samples are likely to be somewhat less.
However, samples rarely will have the
same relative size as the standards, so some
integration errors are likely to result,
although their magnitude would be difficult to predict. Greater errors are likely to
occur as one peak gets significantly
smaller than the other, as resolution
decreases, or as tailing increases.
Conclusions
As expected, integration errors are largely
absent when the resolution is 2.0. While
none of the integration methods produces a significant error, the drop
method is most convenient. For resolutions between 1.5 and 1.0, either the
drop or Gaussian skim methods also produces the smallest error when the peaks
are of similar size (the smaller peak is at
least 5% of the larger peak). The valley
method consistently results in large negative errors, particularly for the small
peak. The exponential skim procedure
generally is invalid under these conditions, as it produces a significant negative
error for the shoulder peak. In general,
height measurements generate smaller
errors than area measurements. So, for
the situations described here, the drop
method using height would be the best
choice. A Gaussian skim procedure
would also be an acceptable choice for
these situations. However, there is less
experience with this new integration
method, and its consistent implementation across a variety of software packages
may be problematic until more information is obtained.
Finally, it is important to recognize that
some errors are beginning to increase even
at resolution equal to 1.5, especially when
one of the peaks is significantly smaller
than the other. For this reason, if minimization of integration errors is important, then minimum resolution should be
2.0, rather than 1.5. When the resolution
is 1.0, the magnitude of some integration
errors will be unacceptable. Resolution
less than 1.0 should be avoided, as the
magnitude of the integration errors significantly increases. References
(1) L.R. Snyder and J.J. Kirkland, in Introduction
to Modern Liquid Chromatography (WileyInterscience, Hoboken, New Jersey, 1979),
pp. 45–48, pp. 543–545.
(2) J.P. Foley, J. Chromatogr. 384, 301–313
(1987).
(3) V.R. Meyer, J. Chromatogr. Sci. 33, 26–33
(1995).
(4) V.R. Meyer, LCGC 13(4), 252–260 (1995).
(5) V.R. Meyer, Chromatographia 40, 15–22
(1995).
(6) V.R. Meyer, Chirality 7, 567–571 (1995).
(7) S. Jurt, M. Schär, and V.R. Meyer, J. Chromatogr., A 929, 165–168 (2001).
(8) M.K.L. Bicking, in press, LCGC.