153
Anal. Chem. 1983, 55, 153-155
without having to freeze the matrix. (The first three of the four advantages of carbon given above are also given by salt matrices in SIMS (7,8, 24). A direct comparison between the carbon and salt matrices has not been made at this time.) These results show that (1) carbon can be used as a sample substrate for the SIM,S analysis of organic compounds and (2) SIMS has potential as a rapid, simple, and direct analytical technique for the characterization of carbons with adsorbed compounds. These preliminary results suggest, in addition, future SIMS analyses of real-world c,arbonsfor major adsorbed components. Already an industrial carbon has been successfully analyzed in our laboratory with the detection of the protonated molecular ion, [M HI+, arising from the impregnated amine. Other fundamental studies will include the analysis of different adsorbed compound types, such as PA(% with heteroatoms and polar side groups,
+
ACKNOWLEDGMENT The authors thank Jleff Wyatt and Joe Campana for many helpful discussions. Registry No. C, 7440-44-0;benzene, 71-43-2;toluene, 108-88-3; rn-xylene, 108-38-3;mesitylene, 108-67-8;naphthalene, 91-20-3; acenaphthylene, 208-96-8;acenaphthene, 83-32-9;9-fluorenone, 486-25-9; phenanthrene, 85-01-8; anthracene, 120-12-7;fluoran-
thene, 206-44-0; pyrene, 129-00-0.
LITERATURE CITED (1) Deltz, V. R., Ed. "Removal of Trace Contamlnants from the Alr"; Amerlcan Chemlcal Society: Washlngton, DC, 1975; ACS Symp. Ser. No. 17. (2) Rand, M. C., Greenberg, A. E., Taras, M. J., Eds. "Standard Methods for the Examination of Water and Wastewater", 14th ed.; American Public Health Association: Washington, DC, 1976. (3) Committee on the Biological Effects of Atmospheric Pollutants "Particulate Polycyclic Organic Matter", N.A.S.: Washington, DC 1972. (4) Alben, K. Anal. Chem. 1980, 52, 1821-1824. (5) Todd, R. G., Diss. Abstr. Int. 6 1971, 37, 6671. (6) . . Unoer. S. E.: Day. R. J.; Cooks, R. G.. Int. J. Mass SDectrorn. Ion Phis. 1981, 39,-231-255. (7) bay, R. J.; Unger, S.E.; Cooks, R. G. Anal. Chem. 1980, 52, 557A573A
(8)c o k , R. J. J. Vac. Scl. Techno/. 1981, 78, 737-747. (9) Barber, M.; Bordoli, R. S.;Elllott, G. J.; Sedgwick, R. D.; Tyler, A. N. Anal. Chem. 1982, 5 4 , 645A-657A. (10) Lln, L. K.; Busch, K. L.; Cooks, R. G. Anal. Chem. 1981, 53, 109-1 13. (11) Coiton, R. J.; Campana, J. E.; Barlak, T. M.; DeCorpo, J. J.; Wyatt, J. R., Rev. Sci. Instrum. 1980, 51 (12), 1685-1689. (12) Klselev, A. V.; Yashin, Y. I . "Gas-Adsorption Chromatography"; Plenum Press: New York, 1969. (13) Lewln, G. "Fundamentals of Vacuum Sclence and Technology"; McGraw-Hill: New York, 1965; p 43. (14) Grade, H.; Cooks, R. G. J. Am. Chem. SOC.1978, 700, 5615-5621.
RECEIVED for review August 3, 1982. Accepted September 21, 1982.
Statistical Evaluation of Calibration Curve Nonlinearity in Isotope Dilution Gas Chromatograph!y/Mass Spectrometry Jozef A. Jonckheere and Andre P. De Leenheer" Laboratorium voor Medische Biochemie en Klinische Analyse, Farmaceutisch Instituut, 9000 Gent, Belgium
Herman L. Steyaert Seminarie voor Waarschljnlijkheidsrekeningen Mathematische Statistiek, Rijksuniversiteit Gent, 9000 Gent, Belgium
During the past 2 decades isotope dilution mass spectrometry (IDMS) has gained a very particular place as an analytical tool. This stems maiinly from the fact that an isotopically enriched analogue of the analyte is used as an internal standard. Since physicochemical properties of both analyte and internal standard are virtually identical, an optimal compensation is at woirk for losses of analyte in all analytical steps. As both products are simultaneously present in the mass spectrometer, specific determination of the analyte/internal standard mole ratio is possible by ineasuring the isotope ratio. Quantitation is performed by coniparing the degree of perturbation of the isotopic composition of unknowns with the isotope ratios of mixtures with known concentrations of analyte and internal eitandard. Although these principles are well documented ( I ) , still some confusion is apparent in many IDMS calibration procedures. Specific problems arise from the presence of unlabeled material in the internal standard and the naturally occurring isotopes of the analyte. As Pickup and McPherson explained (I),the isotope ratio (R,) in a mixture of natural and labeled product can be expressed as
where x and y are the imasses of analyte and labeled material and E and F their respective mean molecular masses. The
abundance of the main isotopic form of the analyte is represented by pi whereas qj stands for the abundance of the main isotopic form of the internal standard; pj is then abundance of naturally occurring isotopes in the analyte and qi the abundance of unlabeled material present in the internal standard. Depending on the actual values of pj and qi in a given analytical situation, the mathematical relationship varies from linear (pj negligible) to a nonlinear relationship (nonzero values for pj or pj and qi). The linear situation when pj and qi are negligible is, however, unrealistic in IDMS. It is obvious that the use of linear least-squares procedures to describe the relationship between isotope ratios and mole ratios is only valid in the special case where pj is negligible (i.e., highly labeled compounds). Assuming a linear relationship in all other cases will introduce gross errors. The use of inverse ratios or weighted linear regression (2) or log-log regression (3) only masks the effect of nonlinearity of calibration curves. To circumvent these problems of nonlinearity, many authors proposed the use of the basic IDMS equation (eq l), similar equations, or approximations of this formula, to obtain corrected isotope ratios to construct linear calibration curves ( 4 , 5 )and nonlinear curves (6) or to extrapolate the unknown concentration (7, 8). Although these "theoretical" methods have been used widely, one should realize that the accuracy of the standard
0003-2'700/83/0355-0153$01.50/00 1982 Amerlcan Chemical Soclety
154
ANALYTICAL CHEMISTRY, VOL. 55, NO. 1, JANUARY 1983
Table I. Calculation of Polynomial Regression Lines and Model Testing no.
X
Rm
std dev
1 2 3 4
16.75 33.50 67.00 134.00 268.00 536.00
0.063 1 0.0961 0.1616 0.2905 0.5404 1.0110
0.0 0.0 0.0 0.0
5 6
0.0 0.0
(1.826 x 1 0 - 3 ) x r = 0.99978 2.943 x 10-4 s = 7.357 x 10-5 R , = 3.011 X 10.’ - (1.979 x 1 0 - 3 ) x- ( 2 . 5 8 0 ~1 0 - ’ ) s 2 Xr’ = 5.450 x 10.’ S = 1.817 X l o - ’ R , = 2.992 X (1.985 X 10‘3)x . (3.123 X 1 0 . ’ ) ~-~(4.332 x 1 0 - ” ) x 3 Xr2 = 3.676 x l o - ” S = 1.838 x lo-” R , = 2.991 X l o w 2- (1.986 x 1 0 - 3 ) x(3.181 X ~ O - ^ ) . Y ‘r ~ (6.476 x 1 0 - ” ) x 3 (2.290 X 1 0 ‘ ” ) ~ ~ l r 2 = 3.854 x l o - ‘ ’ S = 3.854 X 10.’’ F test of 1 and 2 order p = 1,00000 significant F test of 2 and 3 order p = 0.99966 significant F test of 3 and 4 order p = 0.79011 nonsignificant
R , = 3.928
X
10.’
2 . ~ 2=
Flgure 1. Flow scheme for the model testing by means of an for significance.
F test
curve is completely dependent on the validity of the proposed correction formula. Approximations by assuming the presence of only two isotopic forms (4, 7), or assuming equality of pi and qj (5,6,8) all increase the inaccuracy of the quantitation. Also the determination of the pj and q ivalues is of paramount importance to the accuracy of these standard curves. As these values are determined by measuring the isotope ratio or running a mass spectrum on pure analyte and “purenlabeled product and the pj and qi values are very small in comparison with pi and qj, the respective ratios pi/pj and qi/qj will be subjected to relatively large errors.
EXPERIMENTAL SECTION Instead of artificially transforming the data to a linear model, we tried to describe the relation between isotope ratios and mole ratios by means of polynominal regression, since the basic IDMS equation (eq l),a rational function, can be seen as R, = ao + ai(x/y) + ~ z ( + / Y ) ’+ ... ~ , ( + / Y Y (2) the degree (n)depending on the actual values of pj and qi. For a given data set, we therefore calculate different polynomial functions starting from a first degree (linear equation) up to a fourth degree. The use of higher order polynomials is not advisable, to avoid oscillating of the curve through the measuring points. The residuals around the different models, i.e., the absolute differences between given mole ratios and the values calculated by the polynomial, are then used for a statistical evaluation for goodness of fit. This is done by testing the difference between the residuals of two consecutive models against the residuals of the highest degree model by means of an F test for significance (p = 0.95). The flow scheme of the model testing is given in Figure 1. Each model is tested against a higher order model and the best one is then tested again with the following higher order model. This test indicates that, if the lower order model is true, there is only 5% chance to choose the wrong model. Also the use of weighted regression has been included to compensate for the nonconstant variance of the analytical data points. The calculation procedure, as described, was tested by applying it to synthetic data generated with eq 1,for the analytical situation of a previous IDMS study carried out in our laboratory (9). In this assay Bromhexine, a mucolytic drug (C14HzoNzBrz)was quantitated with its trideuterated analogue of the following isotopic purity: do,0.80%; dl, 3.69%; dz, 3.84%; d3,91.67%. The test has not been developed for the fragment ions used in this study since differences in fragmentation efficiency due to isotope effecta make the theoretical developmentmore difficult. By means of a computer-programbased on the probability theory of Pickup and McPherson ( I ) , the different parameters of eq 1were calculated for the molecular ions. These values were entered in eq 1to obtain the isotope ratios covering the range 1.67-53.6 ng of Bromhexine/53.6 ng of labeled analogue. The data points obtained were then used in the regression analysis program, to give the four polynomials and the subsequent model test. The function thus produced was further used to recalculate the mole ratios which are listed against the theoretical data points (Table I). It is seen that the model appropriately described the curvature of the calibration curve.
R , given 0.063 08 0.096 08 0.161 55 0.290 45 0.540 40 1.011 00
Third Order .Y calcd 16.748 29 33.501 48 67.001 4 1 133.998 45 268.000 4 1 535.988 12
std dev 0.0028 0.0025 0.0026 0.0029 0.0033 0.0036
Given the evidence on a theoretical example, the program has been applied to the experimental data of another published IDMS study ( 4 ) . It should be mentioned that in this study no replicate measurements were given, so that a nonweighted regression analysis had to be performed. As is evident however from statistical literature (IO),the accuracy of the calibration curve is almost invariably increased when weighting factors are incorporated, takiig into account the experimentallydetermined variances at each measurement point. The given isotope ratios were reentered in the chosen model and, for the purpose of comparison, also in the linear equation. The difference between the calculated mole ratios and the given mole ratios is graphically represented for both the higher order equation and the linear equation (Figure 2). Clearly the higher order equation was statistically more appropriate t o describe the IDMS calibration curve. Also the correlation coefficient has been calculated and indicated. As has been stressed recently ( I I ) , the use of this correlation coefficient, as a means of evaluating goodness of fit of linear regression, should clearly be discouraged. Its statistical significance indicates only that there exists “anrelationship between x and y values, without evaluating linearity. This is clearly seen in Figure 2 where despite a very high correlation coefficient the higher order model is more appropriate as proved by the F test for significance.
RESULTS AND DISCUSSION This study proves that the eventual curvature of IDMS calibration curves can be described very accurately by means of higher order polynomials. The ability to check different models allows one to adapt the same calculation procedure regardless the actual analytical situation, i.e., regardless the degree of labeling of the internal standard of the interference from naturally occurring isotopes of the analyte. This is especially important in those cases were, due to low efficiency of the synthesis, a large amount of unlabeled, or uncompletely
Anal. Chem. 1983, 5 5 , 155-157
the data is not always possible due to transformation of the experimental data. Our proposed method is even more valuable in cases where some chromatographic separation of analyte and internal standard (using highly labeled compounds or high resolution capillary columns) occurs. This effect destroys the validity of the basic IDMS equation and subsequent calculation procedures based on this formula. Indeed the ion overlap is only partial, and no estimation of the interferences can be obtained by measuring pure product and/or internal standard separately. I t is obvious that the presented polynomial regression analysis with model-testing requires a reasonable computational facility (12). Modern mass spectrometers, however, all incorporate powerful computer systems which can handle this problem very easily. The above summarized regression analysis by multiple polynomials is worked out in a computer program, RAMP, which is available on request in FORTRAN IV or HPLBASIC.
4 h
7
m U .r
42
3
W
L 0
c W
. +
2
> x I
D
1
W 42
m 7
2 U
0
7
m
-. v
>
-1
x
E 0 .r
+
-2
m
.r
> W n
-3
155
OY 0
= 1 . 3 9 t 5.34
Y = 2.38 10-1
-1.89
-4
2.33
lo-'
r
10-1 X t 1.02
= 0.99,
X
X2
23.3
7.0
X/Y theoretical
Flgure 2. Difference between calculated and given mole ratios for a
linear and polynomial regression line, constructed from the data in ref describing an IDMS assay for y-aminobutyric acid (GABA) with [2,2-*H,]GABA as internal standard.
4,
labeled, product is present. Also the setup of analyses with internal standards with low mass increment is facilitated. In contrast with the calibration methods based on expressions or approximations of the basic IDMS equation, no initial estimates of the amount of unlabeled product and/or influence of naturally occurring isotopes are necessary. Regardless the validity of the "theoretical" imodel used, this step greatly reduces the accuracy of such calibration procedures, because of the experimental error involved in determining these small abundances. Also, proper statistical handling of
LITERATURE CITED (1) Pickup, J. F.; McPherson, K. Anal. Chem. 1976, 4 8 , 1885. (2) Schoeller, D. A. Blomed. Mass Spectrom. 1976, 3 , 265. (3) Van Langenhove, A.; Costello, C. E.; Biller, J. E.; Biemann, K.; Browne, T. R. Clin. Chim. Acta 1981, 175,263. (4) Colby, B. N.; McCaman, M. W. Biomed. Mass Spectrom. 1979, 6. 225. (5) Bush, E. D.; Trager, W. F. Blomed. Mass Spectrom. 1981, 8 , 211. (8) Min, 8.H.; Garland, W. A.; Khoo, K. C.; Torres, G. S., Biomed. Mass Spectrom. 1978, 5 , 692. (7) Gambert, P.; Lallemant, C.; Archambault, A,; Maume, B. F.; Padieu, P. J. Chromatogr. 1979, 1 , 162. (8) Siekmann, L. "Quantltatlve Mass Spectrometry in Life Sciences 11"; De Leenheer, A., Roncuccl, R., Van Peteghem, C., Eds.; Elsevier: Amsterdam, 1978; p 3. (9) Jonckheere, J. A.; Thlenpont, L. M. R.; De Leenheer, A. P. Biomed. M s s Spectrom, 1980, 7 , 582. (IO) Schwarz, L. M. Anal. Chem. 1979, 51, 6. (11) VanArendonk, M. D.; Skoaerboe. R. K.: Grant, C. C. Anal. Chem. 1981, 53, 2349. (12) Mendenhall, W. "The Deslgn and Analysis of Experiments"; Wadsworth: Belmont, 1968; Chapter 7.
RECEIVED for review April 12,1982. Resubmitted August 16, 1982. September 24,1982. This work was supported through a N.F.S.R. grant to J.J. and a research contract from F.G.W.O. (No. 3.0011.81).
Desorption of Riadon from Activated Carbon into a Liquid Scintillator Howard M. Prichard" and Koenraad Marien The University of Texas School of Public Health, P.O. Box 20186, Houston, Texas 77025
Activated carbon has long been noted for its ability to adsorb radon from the surrounding air and has often been exploited in measurement techniques requiring radon concentration. Radon entrained on the carbon can be measured directly by y spectrometry or can be removed from heated carbon for counting in an a scintillation cell. The latter approach ( I ) is far more sensitive than y counting but does involve a considerable amount of sampke manipulation. As part of an effort to develop a passive integrating radon sampler based on activated carbon adsorption, we sought a counting method that would have a level of sensitivity approaching that of the a scintillation cell while involving minimal sample treatment. If radon could be reproducibly removed from activated carbon by simple desorption in a solvent such as toluene, then liquid scintillation counting 'would be a promising 0003-2700/83/0355-0155$0 1.50/0
analytical method for large-scale applications such as personal dosimetry in uranium mines. Other potential applications include the analysis of activated carbon radon flux detectors or radon entrained on low temperature activated carbon traps. From theoretical considerations, it is reasonable to expect that the general approach might be applicable to the analysis of other radioactive noble gases, such as 133Xeand 86Kr.
EXPERIMENTAL SECTION Initial desorption experiments were performed with sufficiently high radon concentrations to permit analysis by conventional y spectrometry. Ten-gram lots of a large grained, low density activated carbon (Nuchar WVL 8x30 Mesh) were rinsed in methyl alcohol to remove fines and then dried at 110 O C for 24 h. The prepared carbon was exposed to radon gas, sealed in 22-mL glass liquid scintillation vials, and held for at least 3 h to allow for 0 1982 American Chemical Soclety
