Anal. Chem. 1994,66, 1294-1301
Curve Fitting for Restoration of Accuracy for Overlapping Peaks in Gas Chromatography/Combustion Isotope Ratio Mass Spectrometry Kelth J. Goodman and J. Thomas Brenna' Division of Nutritional Sciences, Cornell Universw, Ithaca, New York 14853 The effect of graded degrees of overlap on high-precision and -accuracy carbon isotope ratios determined by gas chromatography/combustion isotope ratio mass spectrometry ( W C / IRMS) is reported. Overlapping peaks of closely matched isotope ratio (difference6 1 3 c p < ~ ~1%) were analyzed by the conventional vertical drop summation algorithm and by curve fitting using the Levenberg-Marquardt algorithm. The conventional algorithmresultedin systematic bias related to degree of overlap even though precision was not noticeably affected. The exponentially modified Gaussian (EMC) and HaarhoffVanderLinde (HVL) functions were found to model CCC/ IRMS peaks satisfactorily. Useful models over a wide range of overlap were obtained by applying consecutive HVL/HVL or HVL/EMG functions to overlappingpeaks. Accuracy was improved in most cases and was never degraded. This study demonstrates the presence of subtle bias in isotope ratio determinations of overlapping peaks and the ability of automated curve fitting to compensate for these biases. High-precisioncompound-specificisotope analysis (CSIA) is rapidly becoming an important tool in a variety of fields as diverse as organic geochemistry and biomedicine. Gas chromatography/combustion isotope ratio mass spectrometry (GCC/IRMS) and the recent introduction of liquid-based CSIA' facilitate online separation, combustion, and analysis of carbon isotopes in organic molecules. CSIA has been used primarily to investigate small changes in carbon isotope ratios that occur due to natural p r o c e ~ s e s ,although ~*~ recently the technique has been used for enriched species as ell.^,^ In GCC/IRMS, C02 generated by combustion of GC effluent is analyzed by a multicollector mass spectrometer. C02 is continuously monitored a t masses 44 (l2Cl6O2), 45 (13C1602+ 12C170160), and 46 (12C180160).A peak in each detector trace appears for each eluting compound, and thus each trace resembles the output of a flame ionization detector (FID). At the conclusion of the run, software must identify peak start/stop, substract background levels, calculate peak areas, and calculate area ratios according to theoretical considerations.6 Highly precise and accurate isotope ratio determinations depend on consistent peak and background definitions for each of the three detection channels. Total (1) Caimi, R. J.; Brenna, J. T.Anal. Chem. 1993,65, 3497-3500. (2) Matthews, D. E.; Hayes, J. M. Anal. Chem. 1918.50, 1465-1473. (3) Rautenschlein, M.; Habfast, K.; Brand, W.Stable Isotopes in Paediatric Nutritional andMetabolicResearch Intercept Limit& Andover, UK,1990. (4) Goodman, K.J.; Brenna, J. T.Anal. Chem. 1992.64, 1088-1095. ( 5 ) Guo, Z. K.;Luke, A. H.; Lee, W. P.; Schoeller, D. Anal. Chem. 1993, 65, 1954-1959. (6) Santrock, J.; Studley, S.A.; Hayes, J. M. Anal. Chem. 1985,57, 1444-1448.
1294
Analytical Chemistry, Vol. 66,No. 8, April 15, 1994
error (precision and accuracy) for isotope ratios determined in this way is routinely less than 0.05% relative standard deviation (RSD) for well-resolved peaks. Analysis of complex mixtures often results in overlapping chromatographic peaks. Specific requirements for auxiliary preparation devices (e.g., furnaces, dryers) for GCC/IRMS results in a large number of connectors between the chromatography column and the detector, which influence peak shape and generally reduce chromatographic e f f i ~ i e n c y . ~ ~ ~ Resolution is also limited by the restriction of He as carrier gas, compared with Hz. As a result, mixtures that are fully resolved by conventional GC-FID are not resolved by GCC/ IRMS. For these reasons, chromatography in GCC/IRMS tends to be more subject to overlaps than conventional GC. Mathematical curve fitting for quantitative analysis of overlapping peaks has been reported extensively in the chromatography literature. Curve fitting for well-resolved peaks is thought to offer several advantages over summation methods, including straightforward definition of peak limits and less susceptibility to noisy Previous studies focused on single detector systems and have the explicit purpose of providing improved quantitative results compared with nonfitting techniques. To date, no systematic studies of the effect of overlap on high-precision isotope determinations by GCC/IRMS have appeared. Traditional chromatograms may be thought of as a collection of well-resolved singlets with a few doublets of varying degrees of overlap. The recovery of information from doublets is the subject of this work. It is assumed that triplets and higher multiplets, although rare according to theoretical considerations,12will be resolved chromatographically. Several characteristics of a curve-fitting algorithm are desirable. (1) The algorithm should be "rugged". That is, it should yield acceptable results over a wide range of operating conditions, preferably in an automated system. (2) It must run rapidly with reasonable hardware requirements. (3) It should be useful over a wide range of overlap encountered for real samples. (4) Finally, it should improve analytical figures of merit for most peaks; those that it does not improve it (7) Guiochon, G. J . Gar Chromatogr. 1964, 139-145. (8) Maynard, V.;Grushka, E. Anal. Chem. 1972, 44, 1427-1434. (9) Anderson, A. H.; Gibb, T.C.; Littlewood, A. B. Anal. Chem. 1970,42,434440. (10) Anderson, A. H.; Gibb, T. C.; Littlewood, A. B. J . Chromatogr. Sci. 1970. 8, 6 W 4 6 . (1 1) Dyson, N. Chromatographic Integration Methods; The Royal Society of Chemistry: Cambridge, UK, 1990. (12) Davis, J. M.;Giddings, J. C. Anal. Chem. 1983, 55, 418424.
0003-2700/94/036&1294$04.50/0
0 1994 American Chemical Society
should not degrade. We report here an experimental study of the effects of overlapping peaks on accuracy and precision of GCC/IRMS-determined isotope ratios and a curve-fitting strategy applicable to recovery of isotope information from overlapping peaks.
EXPERIMENTAL SECTION Instrumentation. A Finnigan MAT 252 high-precision GIRMS instrument interfaced to a Varian 3400 GC via a ceramic combustion furnace was used for the analysis. A detailed description of this instrumentation can be found el~ewhere.~ Briefly, the effluent from the capillary GC column enters a furnace and is combusted quantitatively to C02. After drying, analyte-derived C02 is admitted into the electron impact ion source of the high-precision GIRMS system. The mass spectrometer is operated at its full accelerating potential of 10 kV and a source chamber pressure of 4 X 10“ Torr. Faraday cup detectors with dedicated amplifiers and counters continuously monitor each of the three major ion beams. For this work, the integration time was 0.25 s, as is routine in our laboratory. The GC was operated with the injector ( T = 250 “C) in split mode with a split ratio of 20: 1 and He carrier flow of 45 cm/s through the column. A J&W DB-WAX 30 m X 0.32 mm X 0.5 pm fused silica capillary column was used. The oven temperature program was isothermal. Test Chromatograms. Methyl tridecanoate (Me1 3:O) and butylated hydroxytoluene (BHT; Sigma Chemical Co, St. Louis, MO) were used as test compounds. Stock solutions of each pure component were prepared in bulk and used throughout the experiment to ensure the isotopic integrity of the samples. Mixtures were prepared from the stock solutions in themassratios l:l,lO:l,and 1:lO. Theseratioswerechosen as representative of the extremes of mass ratios to which this work applies. Also, for integration routines that employ both perpendicular drop and tangent-skim algorithms, an absolute abundance ratio of 10 commonly represents the decision threshold between the two methods.I3 Concentrations were adjusted to 5 pg/pL for each component of the 1:l mixture and the more abundant components of the unequal mixtures (1:lO and 1O:l). The concentrations of the less abundant components of the unequal mixtures were adjusted to 0.5 pg/ pL. An injection volume of 1 pL yielded a peak signal of 3.5 V in the mass 44 channel for the 1:l components and the major components of the unequal mixtures. Graded degrees of overlap may be produced by double injection or by manipulation of chromatographic parameters. The chief advantage of double injection, used primarily for liquid chromatography (LC), is optimal preservation of peak shape.14 This approach is not practical for GC since peaks are sharp and maintaining reproducibility of overlap is difficult. For this reason, graded overlap was produced by increasing the column temperature from 220 to 250 OC for isothermal runs. This method generated highly reproducible overlap for the eluting compounds by degrading resolution. Peak shapes were not noticeably different for individual compounds run at each temperature, although least-squares fits did show some small distortion, as reflected by r2. This distortion was well within that observed for injection of widely differing quantities (13) Papas, A. N.; Tougas, T. P. Anal. Chem. 1990, 62. 234239. (14) Jeansonne, M. S.; Foley, J. P. J. Chromafogr. 1989, 461, 149-163.
under identical conditions. The degree of overlap is expressed as”%valley”, calculatedas 100(valleyheight)/(peakheight).’d Conditions were established to produce 10,40,and 70%valley for the 1:l mixture. A 70% valley is observed in dense chromatograms of complex mixtures and should be particularly useful for GCC/IRMS applications to enriched tracer molecules requiring lower precision. The 1O:l and 1:lO mixtures were analyzed under each set of conditions used for the 1:l case, and in subsequent discussion are designated 10, 40, or 70%valley with respect to the 1:1 case. Pure compounds were run separately under conditions used for 10%valley and plotted as “0% valley”. Me13:O eluted first under these conditions. Data Analysis: ConventionalMethod. Data was analyzed by the conventional method using vendor-provided software (Finnigan ISODAT). The peak detect algorithm processes only one data channel for the detection and definition of peak start and stop. Peaks are detected by calculating the slope of a rolling regression line drawn through five consecutive data points and comparing the result to a user-defined threshold slope (”slope sensitivity”). Peak stops are set when the slope falls below threshold, after a peak start has been detected. The slope sensitivity was set to a single intermediate value that produces satisfactory results in our hands for well-resolved peaks at signal-to-noise ratios above 50. Backgrounds are determined as the average of four adjacent points beginning five points before the start of the peak. Once the peak start and stop have been defined for a single channel, the values are extrapolated to the other two channels. Peak maxima are determined for all channels and a time shift is used to correct for chromatographic separation of i s ~ t o p e s . ~Peak J ~ areas are calculated by summing the difference between baseline and signal for the region defined by the peak start and stop. Ratios and associated calculations are made using areas produced for each peak.16 For overlapping peaks, the valley minimum is assigned as the leading peak stop and the trailing peak start. When applied to overlapping peaks, the routine resembles the “perpendicular drop” method.lIJ3 Summing of consecutive readings between peak limits after subtraction of baseline is referred to as the “summation” method.17 The ratios of areas corresponding to individual components are taken for each trace (44,45, and 46) to produce R45 (area 45/area 44) and R46 (area 46/area 45). R45 and R46 are both required for calculating 6I3C, as R46 is used for an 170 correction that subtracts the contribution of [12C170160] to the m / z 45 signals6 For compounds of low oxygen content, such as those considered here, most of the 0 of COZis derived from the combustion furnace and is therefore invariant. The small contribution from the test compounds, Me1 3:O and BHT, is also of identical isotopic composition within experimental error. Calculations show that relatively large changes in the mass 46 channel do not appreciably alter the 613Ccalculations. For these reasons, and to simplify the calculations for the curve fitting, traces of mass 44 and 45 only were fitted and the data will be presented without correction for I7O. The standard notation for expression of high-precision gas isotope ratio mass spectrometry results is the 6I3C notation,
-
(15) Freeman, K. H.; Hayes, J. M.; Trendel, J. M.; Albrecht, P. Nature 1990,343, 254-256.
(16) Ricci, M., private communication, 1993. (17) Anderson, D.J.; Walters, R. R. J. Chromatogr. Sci. 1984, 22, 353-359.
Analytical Chemistry, Vol. 66,No. 8, April 15, 1994
1295
which requires the 170correction. In this work, we use 645 defined as
proven to be an acceptable model for real (asymmetric) chromatographic peaks, especially when extra column effects are p r e ~ e n t . ’ ~The , ~ ~form of this function is EMG
where R s p and ~ RPDBare the m / z 44 and 45 signal ratios corresponding to the sample and standard, Pee Dee Belemnite, respectively, with RPDB= 0.01 1 237 2*0.000 009 0.l8 Data Analysis: CurveFitting. Raw data files were converted to ASCII files, containing four columns of data corresponding to time, and counts (arbitrary units) for each of three detectors m / z 44,45, and 46. A BASIC program wasthen used to split the data files into three separate files, each containing time and one of the signals. Data fitting was performed using a Compaq 386/20e desktop computer equipped with a math coprocessor, 4 Mbytes of RAM, and a 210-Mbyte hard drive. The DOS computer program Peakfit 3.0 (Jandel Scientific), an implementation of the Levenberg-Marquardt algorithm, was used for all curve fitting.19 Given a set of initial values, this algorithm systematically searches parameter space to find parameters yielding the minimum error sum of squares. A single chromatogram was fitted manually using each of several test chromatographic functions to establish initial parameter values. These values were then used as initial values for all subsequent fits. Chromatograms for each detection channel were fitted in a batch-processing mode using several functions. The batch mode permits a user-selectable prefit routine that fits every other point as a rapid “coarse” fit, followed by fine adjustments using all the data. This procedure results in more rapid fitting than use of all the data from the outset. Each function is expressed using peak area as an independent, adjustable parameter. Isotope ratios were calculated from the best-fit peak area parameter. A variety of functions have been proposed for modeling of chromatographic peaks; therefore it was necessary to determine which functions most satisfactorily modeled GCC/ IRMS data. Functions were evaluated based on the fit parameters r2 and mean square ratio ( F statistic), and comparing the calculated mass and area ratios to the known ratios of the mixtures. Time for fitting was also taken into account. Chromatographic elution profiles previously have been modeled as a Gaussian distribution. Under real chromatographic conditions, peaks are often skewed and not modeled well by symmetric functions such as a Gaussian.2b24 Peak models that take asymmetry into account improve accuracy and precision relative to symmetric model^.^^-^^ The exponentially modified Gaussian (EMG) function, a mathematical convolution of a Gaussian with an exponential decay, has (18) Craig, H. Geochem. Cosmochim. Acta 1957, 12, 133-149. (19) Press, W. H.; Flannery, B. P.; Teukolsky, S.A.; Vetterling, W. T. Numerical Recipes: The Art ofScientific Computing, Cambridge University Press: New York, 1986. (20) Schmauch, L. J. Anal. Chem. 1959, 31, 225-230. (21) Johnson, H. W., Jr.; Stross, F. H. Anal. Chem. 1959, 31, 357-365. (22) Giddings, J. C.Anal. Chem. 1963, 35, 1999-2002. (23) Gladney,H.M.;Dowden,B.F,Swalen, J. D.Anal.Chem. 1969,41,883-888. (24) Pauls, R. E.; Rogers, L. B. Anal. Chem. 1977, 49, 625-628. (25) Haarhoff, P. C.; Van Der Linde, H . J. Anal. Chem. 1966, 38, 573-582. (26) Cheder, S. N.; Cram, S.P. Anal. Chem. 1973, 45, 1354-1359. (27) Pauls, R. E.; Rogers, L. B. Sep. Sci. 1977, 12, 395-413. (28) Barber, W. E.; Carr, P. W. Anal. Chem. 1981, 53, 1939-1942. (29) Foley, J. P.; Dorsey, J. G. Anal. Chem. 1983, 55, 73C737. (30) Foley, J. P.; Dorsey, J. G. J . Chromatogr. Sci. 1984, 22, 40-46. (31) Foley, J. P. Anal. Chem. 1987, 59, 1984-1987. (32) Jeansonne, M. S.;Foley, J. P. J . Chromatogr. Sci. 1991, 29, 258-266.
1286 Analytical Chemistry, Vol. 66, No. 8, April 15, 1994
f(x) =
2
a,-x %+ T][er(---)
exp[ 2a3
x-a,
a,
2lI2a2 2lI2a3
+ 11
(2) where a0 is area, a1 is elution time, a is width of the Gaussian, a3 is the exponential constant which determines distortion in this model, and erf is the error function. Estimates of peak area, chromatographic efficiency, and maximum attainable efficiency, can be obtained from these function parame t e r ~ . The ~ ~ EMG , ~ ~ model, ~ ~ ~ one of the most commonly employed chromatographic functions for modeling GC peaks, has also been successfully applied to situations where baseline resolution has not been a ~ h i e v e d . ~ ~ - ~ ~ Other functions considered here are the (a) HaarhoffVanderLinde (HVL), which was developed for nonideal gas chromatography and is capable of modeling peaks that exhibit “fronting” or tailing,25(b) Giddings (GID), a function designed for conditions in which diffusion and extracolumn effects are absent and different kinetic rates of adsorption and desorption are the primary sources of band b r ~ a d e n i n gand , ~ ~(c) nonlinear chromatographic (NLC), which was recently considered for modeling ideal 6 function analyte loading.37 The forms of these equations are as follows: HVL
GID
NLC
f(x)=
-[ 00
1-
2‘3’
with (33) Wu, N . S.;Cai, C. P.; Yang, Y. Chromotographia 1990, 30, 220-222. (34) Binsheng, L.; Pcichang, L. HRC CC, J . High Resolur. Chromotogr. Chromotogr. Commun. 1987, 10,449454. (35) Foley, J. P. J . Chromotography 1987, 384, 301-313. (36) Giddings, J. C. Dynamics of Chromatography; Marcel Decker;New York, 1965. (37) Wade, J. L.; Bergold, A. F.; Carr, P. W. Anal. Chem. 1987.59, 1286-1295.
=
1.0 .
C
.P 0.8
v1
u
1O:l
1:1
-
0.8
-E-
0.4
+ CO2Std P0.k
iB
QCPoak
0
E 0.2 0
z
0.0 121
123
125
Time
127
120
(8)
Flgure 1. Signal from mass 44 channel Illustratingsignal decay of GC peak and COPstandard after closing of a valve. The GC signal decays -3-fold slower than that resulting from the valve closing.
124
T(u, u ) = e-'~e-tZo((2ut)'/2) dt
(5b)
where a0 is peak area, a1 is elution time, a2 is peak width, a3 is distortion (for HVL and NLC), and Zn are modified Bessel functions of the first kind. Peak Shapes. The degradation of chromatographic efficiency by dead volumes has been considered previously.7~~ Most chromatographic detectors are placed at the exit of the chromatography column to minimize dead volumes and unnecessary broadening due to diffusion after the separation. In GCC/IRMS instrumentation, additional connections are required to interface the capillary column with the mass spectrometer. In the present instrument, this corresponds to eight connections between the GC column and the mass spectrometer and includes two changes in tubing diameter to accommodate a furnace and water trap. An open split, a device used to moderate pressure, is placed in-line prior to the mass spectrometer and peak shape is sensitive to the flow at this split. Finally, the tight ion source of GIRMS instruments is designed to maximize residence time, in order to maximize sensitivity. The net effect of this hardware is to distort peak shapes appreciably from that emerging from the GC column. The trailing section of a sharp, fast-eluting, well-behaved GC peak is plotted in Figure 1, along with the falloff in signal from closing of the valve that admits calibrated C02 directly to the ion source. The half-life for the calibrated C02 signal is tl12 = 170 ms compared with t l p = 530 ms for the peak and demonstrates that the ion source responds more rapidly to changes in signal than required for most capillary GC peaks. The source of peak broadening resides in the chromatographic separation and interface, rather than the mass spectrometer ion source.
RESULTS AND DISCUSSION The chromatograms for the various test mixtures and degrees of overlap are presented in Figure 2. The absolute retention times from chromatogram to chromatogram have been adjusted for comparison purposes but the peak shapes and widths with respect to time are unaltered. Single Peaks. The four separate functions, EMG, GID, HVL, and NLC were screened for goodness of fit and fitting time. Both the r2and F statistics are commonly employed to evaluate goodness of fit. R2 alone is not sufficient to define the quality of fit, as it is an index of noise as well as degrees of freedom in the model relative to the number of data
1:lO
128
124
128
Time (s)
Flgure 2. Chromatograms for the three mixtures and three overlap condkions Investigated in this work. Me13:O elutes prior to BHT in these chromatograms. Table 1
EMG HVL NLC
GID
rk
F
0.999 48 f 0.OOO 26 0.997 85 f O.OO0 77 0.997 88 f O.OO0 76 0.99120 0.003 74
41864 f 14615 9627 f 2779 9762 f 2837 4167 i 3820
*
"Mean f SD ( n = 6).
point^.^^,^^ The F statistic, however, is adversely affected by irrelevant degrees of freedom in the Therefore, both the r2 and F statistics were calculated to evaluate goodness of fit, and residuals were plotted for inspection. Six typical chromatograms were fitted with each function to evaluate the appropriate function to model GCC/IRMS peaks. No less than 40 data points were used for each fit. The data are presented in Table 1 as means and standard deviations. The EMG fits resulted in significantly greater mean r2 and F statistics (pairwise t, 95% confidence). The residual plots for the EMG fits were random. The HVL and NLC resulted in comparable fits, which were poorer than the EMG. Inspection of the residuals indicated that the peak tail was not modeled well for either the HVL or NLC. In addition, the NLC function required 66 s per mass channel compared with 25 s for the H function, a substantial 2.6-fold difference. The GID function gave the poorest fit based on both r2 and F, as might have been anticipated based on the assumptions of that model and the present application. The isotope ratios calculated from fits with each function did not reveal any clear trend in absolute value or precision. Based on these results, the GID and NLC functions were eliminated from further consideration and the EMG and HVL functions were applied to unresolved peaks. For convenience, we refer to these functions by their first letters in the following. Fitting Characteristics of Overlapping Peaks. The E and H functions were screened for goodness of fit for overlapping peaks. In addition to r2 and F, the accuracy of the peak area ratios was used as an index of goodness of fit. At higher degrees of overlap the EE combination broke down. (XU refers to the application of function X to the first eluting peak (38) Gans, P. Data Firring in the Chemical Sciences; John Wiley & Sons: New York, 1992. (39) Lancaster, P.; Salkauskas, K. C u m and Surface Fitting, Academic Prcss: New York, 1986.
Analytical Chemistry, Vol. 66,No. 8, April 15, 1994
1297
1:l 3
-8-
1 190
Y Flt
-25
1 185
-30 1180
* 102
108
114
-35
1175
lr:
\
r 7
m v
Time (s)
1190
Flgure 3. Best fit for the application of the EMG/EMG function combination to a 1:1, 70% valley chromatogram. The tail of the leadingEMG function extends into the second peak, leadingto erroneous abundance and isotope ratios.
-
j
-10 (I)
I
1185
1180 1175
Y Exp. 0
20
40
60
0
20
40
60
Percent Valley
Flgure 5. Isotope ratios obtained by summation and fitting for equal abundance mixtures. Open symbols refer to summation data and are repeated in all cells for comparison. Closed symbols refer to fitted data. CirciesrefertoMel30, ~Md-~eiutesearlierthanBKT, represented by squares. Error bars are 95% confidence limits (using t statistic).
CD
101
106
Time
111
(8)
in
101
1OB
111
Time (s) Figure (a, top) Best fit for the HVLIHVL combination to a 10: 70 % valley chromatogram. The leading edge of the trailing HVL function extends into the first peak. (b, bottom) Best fit for the HVL/EMG combination to the same data.
of a doublet, with Y applied to the trailing peak.) The EE fit of the 1:l 70% valley is presented in Figure 3. This fit resulted in a superior r2and Fstatistic compared with the H H combination. The first E function, however, distorts and tails excessively into the second peak, resulting in inaccurate peak areas. In contrast, the H H combination yielded a poorer r2 but better estimates for the peak areas. Application of the H H combination to the other overlap chromatograms revealed that the H H combination did not provide accurate estimates for the 1O:l 70% valley, shown in Figure 4a. The area for the second H peak area was overestimated because it fronted into the first peak to compensate for inability of the H function to model the tail of the first peak. Since the E was susceptible to errors due to excessive tailing and the H due to fronting, the HE combination was tested. A plot of the HE combination model for a 1O:l 70%valley is presented in Figure 4b. In this plot, all experimental points are coincident with those of the model. This combination did not show any obvious fronting or tailing distortions for any of the conditions and, therefore, was used along with H H and EE for fitting all overlaps. The fourth and remaining combination of EH was evaluated for completeness. The nine overlap conditions of Figure 2 were fitted by use of each of these four combinations of functions. Three or four replicate chromatograms were obtained and fitted for 1298
Analytical Chemistry, Voi. 86, No. 8, April 15, 1994
each overlap condition. Fits were evaluated for quality of (a) isotope ratios and (b) single-channel ( m / z44) area ratios, for quantitative analysis. To evaluate accuracy, results were compared to those obtained by summation and offline combustion. All differences in means were tested for significance by use of the pairwise t test at a 95% confidence le~el.~’J Isotope Ratios: Equal Abundance. Figure 5 presents R45 and corresponding 645 obtained by summation and fitting for the four combinations of functions. Each data point represents a mean of at least three replicates fitted independently. The summation data are repeated in each cell for convenient comparison. Error bars represent 95% confidence limits, and when absent, they are contained within the symbol. The 645s and standard deviations for these data are presented in Table 2. Isotope ratios for all degrees of overlap are significantly different when determined by summation than for 0% valley and appear to be systematically related to degree of overlap. At 70%valley, the apparent isotope ratio of the leading peak (Me13:O)isdepletedbyabout645 = -lO’%,whiletheisotope ratio of the trailing peak (BHT) is enriched by a corresponding level. This unexpected result is observed even though the difference in carbon isotope ratio between these compounds is only 645 = -0.30. It is significant to note that the precision for most of these highly inaccurate ratios is not seriously degraded, suggesting that precision is not a good indicator for accuracy for the summation method. Fitted and summation data for the leading peak at 0% valley are not significantly different for any of the functions. Significant differences were detected for 0% valley trailing peak using the EE and EH fits (645 = -26.67 f 0.189) compared with summation data (645 = -27.51 f 0.097). For all function combinations, the fitted ratios are of significantly improved accuracy relative to the summation ratios for 10 and 40% valley. All functions except the EE combination resulted in improved accuracy at the 70%level. (40) Snedecor, G. W.; Cochran, W. G. Statistical Methods, 8th ed.; Iowa State University Press: Ames, IA, 1989.
Table 2 7% valley
SUM 0 10 40 70
I
Me13W
BH’P
-27.21 f 0.46 -29.48 f 0.17 -34.53 f 0.13 -37.99 i 0.33
-27.51 i 0.097 -26.98 f 0.26 -23.00 i 0.54 -20.12 f 1.1
-27.28 f 0.18 -27.84 f 0.16 -27.72 f 0.26 -22.24 f 5.5
-26.67 f 0.19 -27.49 f 0.34 -28.31 f 0.49 -34.16 f 6.1
-27.26 i 0.70 -28.03 f 0.16 -28.99 f 0.29 -26.56 f 0.56
-26.67 f 0.19 -27.36 f 0.38 -26.80 f 0.21 -28.59 f 0.76
f l.1801 l1.185 lgot
tu
EE 0 10 40 70
HE 0 10 40 70
I
I
I
I
Ti
A i t - I t i l
1q?1IL 1
1%: 1175
-30 -35
0
20
40
HH 0 10 40 70
x
II
60
0
20
40
60
Percent Valley
-27.26 i 0.70 -28.17 f 0.13 -28.83 f 0.48 -27.27 f 0.69
-26.83 i 0.12 -27.39 f 0.19 -26.93 f 0.24 -28.26 f 0.88
-27.28 f 0.18 -27.95 f 0.14 -25.87 f 0.70 -24.14 f 3.66
-26.83 f 0.12 -27.91 f 0.24 -30.34 f 1.2 -32.07 i 4.0
7 7
it
4
it
{
-25
EH 0 10 40 70 a
645 f SD. l.lgo 1.185
The EE combination gives very poor precision at 70% valley. The E function is known to break down a t high degrees of overlap (>45% valley), and so this result is consistent with previous reports. This appears to be due to the tendency of this function to tail excessively from the leading peak into the trailing peak. The E H combination is consistent with this interpretation as it results in the poorest precision and accuracy at the 40% valley. Isotope ratios determined using the EE and EH functions cross over between 10 and 40% valley. In contrast, the H E and H H functions recover isotope ratios similarly to one another and more accurately than the other combinations. The isotope ratios for the H E and H H cross over between 40 and 70% valley. For this data set these functions appear to provide superior recovery of isotope ratios. We have previously considered the accuracy of GCC/ IRMS carbon isotope determinations referenced to conventional combustion and dual-inlet GIRMS a n a l y ~ i s .An ~ offset of up to 613C = 1% is observed for GCC/IRMS analyses referenced to calibrated COz. For this reason, the use of isotopically calibrated standards within the chromatography mixture has been ~uggested.~ In any case, the offset observed for fitted data up to 40% valley is small and within this accuracy limit. Isotope Ratios: Unequal Abundance. Fit and summation results for the 10:1 mixture is shown in Figure 6a. The scaling is identical to that used in Figure 5 for comparison, and again the summation results are plotted for all fitting combinations. The summation peak detection algorithm detected two peaks for the 40% valley in only a single case and failed to detect two peaks for any of the 70% valley replicates. However, the presence of the smaller peak significantly lowered the calculated isotope ratio for the large peak in the 10% case. In the single 40% valley case the presence of the smaller peak altered the major peak by 645 = -2%. The summation results for the smaller peak result in substantially greater degradation of both precision and accuracy than for the larger peak.
t
-25
-30
1.180
-35
1175 0
20
40
60
0
20
40
60
Percent Valley
Flgure 8. Isotope ratios obtained by summation and fitting for the (a, top) 1 0 1 and (b, bottom) 1:lO abundance mixture. The symbol key Is given in the legend to Figure 5.
No significant differences are detected between fitted and summation ratios for either 0 or 10% valley for the leading peak, while the EE and E H combinationsprovide similar ratios, which significantly improvethe accuracy of the smaller trailing peak. The H E and H H combinations give similar results for which isotope ratios are reversed relative to the other combinations. Not for any fitted case is the accuracy degraded relative to summation. Data for the 1:lO mixture is presented in Figure 6b. In this case, the summation peak detect algorithm successfully identified all peaks. Summation and fitted ratios for the larger trailing peak are significantly different at the 0% valley only. In contrast to the 1O:l case, the precision and accuracy of the isotope ratio of the larger peak is affected by less than 1% for all degrees of overlap using the summation method. For the smaller leading peak, summation ratios for all degrees of overlap are significantly different from the 0% valley, and the trend toward lower isotope ratios is significant, though smaller than the trend of the smaller peak in the 10:1 data. At the 10 and 40% valleys, the improvement in ratio for the fitted data is statistically significant. Quantitative Analysis. Ratios of fitted peak areas for the mass 44 channel were constructed to evaluate the preservation of quantitative information. Area ratios were constructed for summation and fitted data using the leading peak in the numerator and are presented graphically in Figure 7 for the equal abundance data (1:l). The summation ratios show a Analytical Chemistry. Vol. 66, No. 8, April 15, 1994
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Figure 7. Area ratios for the equal abundance data calculated using the mass 44 signal: Mel3:O/BHT, 1:1. Error bars are 95 % confidence limb. Closed symbols refer to fltted results: EE (trlangle), HE (circle), HH (square), and EH (diamond); Open squares refer to Summation results.
few percent bias (low) relative to the fully resolved case. The four fitted combinations result in a systematic trend for which the 10% valley ratios are slightly underestimated, with overestimation for the higher degrees of overlap. The order of overestimation is EH > EE > H H > HE, and the best accuracy is obtained with the H E combination. Precision degrades with increasing overlap in all cases. The confidence limits plotted in Figure 7 are constructed from the standard deviations of the pooled replicates for each fit and, when calculated in this way, are not significantly different from the 0% valley. However, close inspection of the data reveals that most of the variability could be attributed to random variation from run to run, which could not be attributed tovariation in signal or injection volume. Therefore, each of six differences in means among the four combinations was calculated for each replicate, and the resulting mean differences were tested as significantly different from zero. In every case, except one with particularly high variability, differences were found to be significant between combinations (95% confidence limits). This indicates that quantitative analysis using curve fitting is sensitive to choice of function. The four differences were calcula ed between the summation results and each function. Differences for all combinations except H E were statistically significant. The HEcombination data at the 10 and 70%valleys were not statistically different from the summation results and were marginally significant (P < 0.03) at the 40% valley. These results indicate that quantitative information is best preserved by the H E combination under these conditions. Mass 44 area ratios for fitting of the unequal abundances are presented in Figure 8. Figure 8a shows the area ratios resulting from 1O:l data set. As expected, the precision is substantially poorer than in the 1:l case. However, mean differences in area ratio are more dramatic than in the 1:l case, as many of the mean ratios are statistically different. As expected, most area differences (14 of 18) were statistically significant. The summation ratio is severely underestimated for both degrees of overlap for which there are data (10 and 40%). The 10% valley is the only overlap in which there are replicate summation data, and they are not statistically different from any of the fits. Figure 8b presents the 1 : l O data. Here all ratios are systematically shifted to lower values as a function of overlap and there is much better agreement among fits and summation. None of the areas resulting from these fits are significantly
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Percent Valley Figure 8. Area ratios for unequal abundance data calculated using the mass 44 signal: Mel3:0/BHT of (a,top) 1 0 1 and (b, bottom) 1:lO. The symbol key Is given In the legend to figure 7.
different from the summation data. However, all but one fit is statistically different from one another at 10% valley. At the higher degrees of overlap only 1 of 12 comparisons was statistically significant. These results indicate that choice of functions is important for trailing smaller peaks at all degrees of overlap; for leading small peaks the choice of functions is important only for low degrees of overlap. This difference in behavior in both isotope ratio and abundance calculations for the unequal abundance mixtures probably arises because of the asymmetric shape of the GCC/IRMS peaks considered here. In nearly all regards, the 1:lO mixture is better behaved than the 1O:l mixture, suggesting that the large tail in the latter case is not optimally modeled by these functions. Functions show a tendency to compensate for deficiencies in the modeling of the remaining peak. This was shown dramatically earlier and probably operates here on a subtle level. The overall greater effectiveness of the HE combination over the wide range of overlaps and abundance ratios may be due to the resistance of the H function to tail and the E function to front. The H H combination was also highly effective but was less effective than HE in some cases.
CONCLUSIONS Overlapping peaks detected by conventional algorithms are systematically distorted in isotope ratio even for closely matched compounds, though high precision is maintained. Further, small trailing peaks can significantly affect the apparent isotope ratio of the major peak. Curve fitting is effective in restoring isotope ratios for overlap as high as 70% valley, and its effectivenessdepends on the relative abundances and elution order as overall peak areas become more unequal. In none of the cases investigated did the fitted data result in poorer accuracy than the conventional summation method; the fitted data were either improved or not degraded. Our data indicate that the HVL/EMG or HVL/HVL combina-
tions provide the best fits of the functions considered in detail and are most robust and rapid for automated fitting of GCC/ IRMS chromatograms. Small trailing peaks are the most difficult to model, apparently because they are obscured by the tail of the preceding peak. Small leading peaks are more easily modeled, and their minor effects on the major trailing peakcan be effectively removed within the experimental error of these measurements. Finally, chromatographic peak shapes depend on specific chromatographic configuration and conditions, for which other functions may yield improved performance.
ACKNOWLEDGMENT This work was supported by NIH Grant GM49209 and by thesupport Of NIH the USDA-cSRS* K*J.G* Training Grant DK07158.
Received for review October 6, 1993. Accepted January 27,
l994.@
Abstract published in Aduance ACS Absfracrs, March 1, 1994.
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