A Comparison of Calculation Procedures for Isotope Dilution Determinations Using Gas Chromatography Mass Spectrometry? B. N. ColbyS Systems, Science and Software, Box 1620, La Jolla, California 92037, USA

M. W. McCaman Division of Neurosciences, City of Hope National Medical Center, Duarte, California 91010, USA ~~

Several data reduction procedures have been used for converting isotope ratios, determined using gas chromatography mass spectrometry, into mole ratios for the purpose of quantitating organic compounds. In this report, these procedures are shown to be related under certain limiting approximations of original sample and internal standard isotope ratios. With the understanding that some of the procedures involve approximations, the incorporationof related systematic errors into calculated mole ratio values was investigated.The results indicate that certain data reduction procedures allow quantitation over wider mole ratio ranges and with internal standards which contain a significant quantity of unlabeled impurity.

INTRODUCTION ~~

~~

During the past three decades, stable isotopically labeled materials have been used with increasing frequency as internal standards for quantitative mass spectrometry. The technique involves perturbing the normal isotopic composition of the sample by adding a known quantity of an isotopically enriched analog of the unknown. The isotope ratio of the resulting mixture falls between that of the original sample and that of the isotopically enriched material and is a measure of the mole ratio of the two substances. Although the basic principles of this isotope dilution technique are generally agreed significant differences exist in the data handling procedures used for converting the experimentally measured isotope ratios into mole ratios for sample and internal standard. The first procedure was introduced by Schoenheimer and Kittenberg and their colleagues at Columbia U n i v e r ~ i t y ~ in- the ~ early 1940s, an$ is discussed in detail by Gest et aL8 and by Campbell. It involves a calculation based on the expression

-X= c " - 1

(1) Y C where x / y is the mole ratio of sample to labeled internal standard in the mixture, and C and Co are the atom percent excesses of the mixture and pure labeled material, respectively. A second equation was given b Y Inghram" and is nicely explained by Hintenberger . Here the inole ratio is given as

t Abbreviations: GABA = y-arninobutyric acid; DNFR = dinitrofluorobenzene. $ Author to whom all correspondence should be addressed.

where R,, R, and R, are the isotope ratios of 'pure' x , 'pure' y , and a mixture of the two, respectively. An apparent third approach to the calculations is given by Jenden" and is based on a solution to the simultaneous equations

a , = xux + y a y

(3)

6 , = xb,+ y b

(4)

where a , and 6, are the abundances of isotopes a and b in a mixture of x and y . A fourth method, apparently developed intuitively and verified empirically, is given by Gordon.'* It incorporates a correction to the isotope ratio of a mixture to account for the presence of unlabeled material in the labeled standard and may be written as

L --i + U - Peak height at unlabeled m / e L Peak height at labeled m / e

(5)

where Li is the quantity of unlabeled impurity added to the mixture with an L quantity of labeled standard. The quantity of unlabeled material in the mixture from the original sample is U. A final way to treat isotope ratio data is simply to use the isotope ratio of the mixture as a direct measure of the mole ratio in the mixture or X

-=R, Y

(6)

It is not clear who developed this approach but many examples of its use appear in the l i t e r a t ~ r e . ' ~All " ~ of the equations described above for treating isotope ratio data have been used successfully. This report will discuss some of the similarities and differences between these equations and describe the effects of each mathematical treatment on the same set of experimental data.

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BIOMEDICAL MASS SPECTROMETRY, VOL. 6, NO. 6, 1979 225

B. N. COLBY A N D M. W. McCAMAN

EXPERIMENTAL Equipment The combined gas chromatography mass spectrometry was carried out using a duPont Model 321 gas chromatograph mass spectrometer coupled to a Model 320 data system for data acquisition and processing. A 5 ft long by 2 m m i.d. glass column packed with 3% OV-101 on SO/lOO mesh Supelcoport (Supelco, Inc., Bellefonte, Pennsylvania) was employed under isothermal temperature conditions at 250 "C. Electron emission was 1 ma and ionizing energy was 73 eV. A signal generator was employed as described previously. l5 For isotope ratio measurement, data was collected at m / z 252 and 254 by alternately integrating the ion current at each mass for 250ms periods. Six sample injections were made for each mole ratio value investigated and the mean and standard deviation of the 252/254 isotope ratio recorded from selected ion monitoring (SIM) area integrations. All calculations were performed using a Texas Instruments SR-52 calculator.

added and the tubes tightly capped, then heated to 60-65 "C for 60 min in a water bath. After cooling to room temperature, 200 111H 2 0and 200 ~1 CH2C12were added. After mixing, the organic phase containing the DNP-amino acid ethyl esters was removed, transferred to 300 ~1 vials (Reactivials, Pierce Chemical Co., Rockford, Illinois) and evaporated to dryness. This extraction was carried out twice. Just prior to analysis, the residue was dissolved in 100 ~1 of ethyl acetate and 1 cl.1 was injected into the gas chromatograph mass spectrometer. The mass spectra of both the labeled and unlabeled derivatives have been reported p r e v i o ~ s l y . ' ~

RESULTS AND DISCUSSION Before it is possible to see the relationship between the various equations used to reduce mass spectral isotope dilution data, it is necessary to rewrite the equations in a common form. Since the basic intent is to determine the quantity of unknown relative to the quantity of isotopically labeled internal standard as a function of isotope ratios, the general expression selected for the equations is

Materials and reagents y- Aminobutyric acid (GABA) was obtained from Sigma Chemical Co., St. Louis, Missouri and [2,2'H2]GABA from Merck & Co., Inc., Rahway, New Jersey. Dinitrofluorobenzene (DNFB) and dichloromethane (Sequanal Grade) were obtained from Pierce Chemical Co., Rockford, Illinois; ethyl acetate was purchased from Burdick-Jackson Laboratories, Muskegon, Michigan. These solvents were used without further purification. Ethanolic HCl (1.25 N) was prepared by bubbling HC1 gas into USP-grade EtOH previously dried by distillation over calcium oxide. The HCl was dried by passing it through a scrubbing tower containing concentrated H2S04. The solution was stored at 4°C and replaced at two month intervals. DNFB was diluted 1-10 (v/v) in dry EtOH. All glassware was cleaned by boiling in 50% H N 0 3 and was rinsed extensively in glass-distilled H 2 0 prior to use.

where x/y is the mole ratio of sample t o internal standard in the mixture. The subscripts x, y and nz are used to indicate values for the original sample, pure internal standard and mixture of sample plus internal standard. Isotope ratios, R's, are defined as the abundance of isotope a divided by the abundance of isotope b.

R = -a b

To rewrite Eqn ( l ) , the appropriate abundance values are substituted for C and Co to give

x - by-b, Y

bm-bx

1

(7)

If the materials are assumed to be composed of only isotopes a and 6, then

a+b=l and

Preparation of samples Freshly prepared standard solutions of GABA and [2,2'H2]GABA were used to generate a series of seven samples ranging in mole ratio from 0.07 to 70.0. Each sample tube contained 4 nmole of [2,2-*H2]GABA. These samples were evaporated to dryness with a stream of dry Nz,and 300 pl of 0.1 M sodium carbonate buffer (pH 10) and 20 r*.l of DNFB reagent were added. After mixing, the tubes were capped and heated for 30 min in a 60-65 "C water bath, cooled to room temperature, and acidified with 300p1 of 4 N HCl. The solutions were extracted twice with 200 ~1 ethyl acetate, and the organic phases were combined and evaporated to dryness. These residues were extracted twice with 200 ~1 CH2CIZ, and the extracts were combined and evaporated to dryness. Ethanolic HCl (200 pI) was 226

BIOMEDICAL MASS SPECTROMETRY, VOL. 6, NO. 6, 1979

a=- R R+1 and

1 R+1

b=-

The effects of making this two-isotope assumption will be discusseed in detail in a subsequent section. The b abundances in Eqn (7) can now he replaced according to Eqn (9), and following a condensation of terms, x - (Ry-Rm)(Rx+ 1)

Y

(Rm-Rx)(Ry+1)

(10)

Inherent in this expression are the original definitions of Co and C which stipulate isotope b as the more @ Heyden & Son Ltd, 1979

CALCULATION PROCEDURES FOR ISOTOPE DETERMINATIONS

abundant isotope of the pure internal standard, i.e. the ‘label’ isotope. Although this seems to indicate that the isotope ratio definition, a / b , is essential, the inverse definition, R = b / a , produces to the exact same form of Eqn (10). This means that n o physical distinction need be made between isotopes a and b concerning which is the ‘label’ isotope. Equation (10)is also produced when Eqn ( 2 ) is rearranged. In the development of Eqn ( 2 ) according to Hintenberger,* no physical significance was assigned to isotopes a and 6, so again the isotope ratio definition can be made arbitrarily. This is also true for the simultaneous Eqns ( 3 ) and ( 4 ) which, when solved, produce Eqn (10).An excellent review of Eqn (10)has been given by Pickup and McPherson.’6 To rewrite Eqn ( 5 ) in the desired form, it is necessary to assume that the labeled and unlabeled materials have equal molecular weights. This assumption is not absolutely correct, but is a good approximation in many cases, and was implied when the equation was originally presented. Using this assumption, Eqn (6) can be rewritten as

where the appropriate abundance values have been substituted for ‘peak heights’. On rearrangement and with substitution, x -R,-R,

Y

RY+1

If the inverse isotope ratio definition ( R = b / a ) were used, Eqn (5) would become

rather than Eqn ( 1 1 ) . This means that the physical significance of the isotopes cannot be ignored in defining the isotope ratio; b must be the ‘label’ isotope with this equation. Determining the relationship between Eqns ( 1 1 ) and ( 1 2 ) and Eqn (10)is possible when it is recalled that zero contribution to the ‘peak height at labeled m / e ’ due to ‘labeled’ material in the sample was originally implied. This zero contribution cannot be correct in an absolute sense because both sample and internal standard are composed of naturally occurring isotopes. It may be a useful approximation, however, the validity of which depends on the extent to which R , approaches either zero or infinity (depending on the ratio definition). Taking the limits of Eqn (10) as R , + m and as R,+O clearly shows that Eqns ( 1 1 ) and ( 1 2 ) are, in fact, approximations of Eqn (10)in those limits of R x .

With Eqn (lo), the only restriction on the relative magnitudes of R,, R , and R , was that R , must not equal R yand that R , must occur between R , and R,. For Eqns (11 ) and ( 1 2 ) , however, the relationships R , > R, > R , and R , < R , < R , must be satisfied, respectively. This limitation is a purely mathematical concern, however, since conversion from one form of the equation to the other can be achieved simply by substituting the inverse isotope ratio definition. Two limiting approximations similar to those above are possible, but apparently have not been reported previously. They are

Although these may seem to be of purely academic interest, it will be shown that they can be useful in practical situations. Here restrictions on R , and R , are R , > R , > R , for Eqn ( 1 5 ) and R , < R , < R , for Eqn (16). A next step in simplifying Eqn (10) is to take its combined limits as R , + 00 and R , + 0 or as R , + 0 and R y+ co.When this is done

R,-m

and the most simplified means of treating isotope ratio data, Eqn (6), is generated. Here the restrictions on R,, R , and R , are R , > R , > R , and R , < R , < R yfor Eqns (17)and (18), respectively. The reason for these restrictions is perhaps clearer than it was for the single limit approximations of Eqn (10) because of the relative directions in which the limits of R , and R , are taken. Also, the reciprocal relation between the isotope ratio definitions is now obvious. All the previous discussion has been based on the assumption that the compound under investigation had only two isotopes. This is clearly not the true situation for organic compounds. If Eqn (10) is rewritten to include other isotopes and if those isotopes are included as the sum of the abundances of other isotopes, 0, it becomes

For spectra which contain fragment ions this other isotope concept can be expanded further to also include those ions. If the sums of the abundances of all other peaks in the pure sample and in the internal standard are equal (0,= O,), however, the (1 - 0 , ) / ( 1 - 0,) term becomes unity and Eqn (19) reduces to Eqn (10). In @ Heyden & Son Ltd, 1979

BIOMEDICAL MASS SPECTROMETRY, VOL. 6, NO. 6, 1979 227

B. N. COLBY AND M. W. McCAMAN

reality, 0, and 0, are very nearly equal, a fact that can be verified by inspecting at the spectra of unlabeled compounds and their labeled analogs. Consequently, it seems reasonable to ignore the other isotopes and ion fragments when performing the calculations. Since Eqns (6),(11)and (12)are all approximations of Eqn (10) in certain limits of R,and R,, it is important to realize that data processed using them will result in the incorporation of a certain degree of systematic error into the calculated mole ratios. The extent of this error incorporation should be expected to vary depending on the range of mole ratios involved and the extent to which the isotope ratio values of sample and internal standard deviate from 0 and CO. In order to evaluate the level of impact these assumptions might have on final values, each of the equations was applied to the experimentally determined isotope ratios given in Table 1. For the purpose of simplicity, only data calculated for isotope ratios where R,>R,>R, will be discussed from this point on [Eqns (lo), (13), (15) and (17)]. Calculations have been done using the equations for the inverse isotope ratio definition but they do not alter, in any way, any of the conclusions which are drawn from the data presented. When the data reduction equations were applied to the isotope ratios in Table 1 and the results plotted on linear scales, the curves shown in Fig. 1 were produced. Since the data cover such a wide range of values, it is necessary to use two sets of scales to do this. Note that both sets of x/y values calculated using the approximation that R , + 0, are highly nonlinear at high mole ratio values. This should be expected because, as an increasing quantity of sample is present in the mixture, it becomes less appropriate to ignore labeled material inherently present in the sample. The two curves which make no assumption about R,, however, are linear in as much as can be seen from the graph. Interestingly, the assumption the R y + 0 has little effecton the shape of the curve. It is apparently much less important to correct for the quantity of unlabeled material in the internal standard than it is t o correct for the quantity of labeled material in the original sample. In order to evaluate the linearity of the curves shown in Fig. 1 more carefully, linear regression lines were determined. The deviations in the data from those regression lines are plotted in Fig. 2. Three significant factors should be noted here. First is the major difference in nonlinearity between the curves derived using the assumption that R,+ 00 and those which were not. This is a reconfirmation of the observation made

i

loo(

i

Mole Ratio Figure 1. Linear plots of data calculated using Eqns l O ( x ) , 13 (O), 15 (0) and 17 (@).

using Fig. 1. Second, note that the curves overlap in two cases. The correlation coefficients for these overlapping lines (Table 2) are, in fact, essentially identical. This means that the overlapping cases produce equally ‘linear’ straight lines, i.e. correcting for the presence of

Table 1. Experimental isotope ratio data for mle’s 2521254 GABA derivative: R, = 26.5* 1.42% SD [2,2-’HZIGABA derivative: R, = 0.0490* 3.84% SD GABA/[2,2-’H~1GAEA Mole ratio

Isotope ratio of of mixture (R,)

% SD

0.0700 0.233 0.700 2.33 7.00 23.3 70.0

0.1388 0.3294 0.8462 2.568 6.403 13.73 20.38

6.76 3.75 1.57 0.85 1.27 0.70 1.25

228 BIOMEDICAL MASS SPECTROMETRY, VOL. 6,

NO. 6, 1979

-101

,

I

,

,

0.I

I

10

100

Mole ratio

Figure 2. Absolute deviation of linear regression line for data calculated using Eqns 10 and 15 ( x ) and Eqns 13 and 17 (0).

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CALCULATION PROCEDURES FOR ISOTOPE DETERMINATIONS

Table 2. Correlation coefficients of regression lines Equation number

Ea uat io n

Assumption

R,

- R, R”+ 1

13

xly=

15

xly=

17

xly=R,

R,(RX+1) Rx- Rm

Linear

Logarithmic

R,< R, R,+a

0.94105 0.99200

R,> R, R,+O

0.99994 0.99862

Rx+m R,+O

0.94104 0.99436

unlabeled material in the pure internal standard has no significant effect on the ‘linearity’ of the straight line. Thus far, only the absolute deviations in x / y have been considered. Relative or percent deviations must be considered in a slightly different way. This is because linear regressions tend to minimize absolute deviations in x / y , ” i.e. d ( x / y ) .In order to minimize relative deviations in x / y , i.e. d ( x / y ) / ( x / y )logarithmically , weighted data can be used since

This is accomplished by using log-log scales in graphical applications or by operating on the logarithms of the data for regression analysis. This approach has been suggested by S ~ h o e l l e r ’based ~ on empirical evidence demonstrating minimized percent error in values calculated using a set of hypothetical isotope ratios. When this approach is used, the curves shown in Fig. 3 are produced. Note that curves which make the assumption that R , + 0 now exhibit nonlinearity at low mole ratio values. This can be explained using the same kind of reasoning that explained nonlinearity in Fig. 1 . That is, at low mole ratios there is a greater relative quantity or percent of internal standard than there is sample and consequently, the amount of unlabeled material present in the internal standard cannot be ignored. On log-log scales at a mole ratio of 0.07, the 49% difference in the

x / y values calculated using the assumption that R y + 0 versus not using that assumption, is visually apparent. On the linear scales (Fig. 1),49% of0.70 or 0.035 is very small and is just varely visible. Curvature at the high mole ratio end of the plot in Fig. 3 is present for the same reason that it was in Fig. 1 . Clearly, if relative or percent deviations in x / y are of interest, it is less appropriate to make the assumptions that R,+ w or R , + 0 than it is not to make the assumptions. That is to say, labeled material naturally present in the original sample and unlabeled material present in the internal standard may both have a significant impact on the calculated sample to internal standard mole .ratio. When the logarithms of the data are subjected to linear regression analysis and the relative deviations plotted in a fashion similar to that of Fig. 2, it becomes clear that corrections for R , f 00 and R , # 0 are appropriate (Fig. 4). It is only for data derived using Eqn (10) that relative deviations from linearity are uniformly small. In both the absolute case (linear scale) and relative case (logarithmic scale), however, acceptable results may be produced using any or all of the approximations if the range of mole ratios involved is limited to values where the assumptions are valid. An advantage from making the appropriate corrections may be realized, however, when a wide range of mole ratio values is being investigated or when labeled internal standards which contain significant amounts of unlabeled impurity are used. On inspection of Fig. 4, the data derived using Eqn (10) appear to have a small and perhaps significantly upward curve. In order to evalute whether this could be due to statistical fluctuations in the isotope ratio measurements rather than some systematic error, a propagation of errors analysis was carried out using the standard deviation data presented in Table 1 . This was done using standard procedure^^^.'^ which show the propagated errors, d ( x / y ) ,for Eqn (10)to be

0.6

-2 52

0.4

I

0.2

-0 m

-$

0

0

n“

-0.2

-0.4 0.01

I

I

I

0.I

I

10

I 100

Mole r a t i o

Figure 3. Logarithmically scaled plots of data calculated using Eqns 10 (x), 13 ( 0 ) , 1 5 (0) and 17 (0).

@ Heyden & Son Ltd, 1979

0.I

I

10

I00

Mole r a t i o

Figure4. Relative deviation from logarithmically weighted linear regression line for data calculated using Eqns 10 (x), 13 (0). 15 (0) and 17 (0).

BIOMEDICAL MASS SPECTROMETRY, VOL. 6, NO. 6, 1979 229

B. N. COLBY AND M. W. McCAMAN

z

0.18

009

006

c

Q -0.03

I

LT

-0.061 1

1

0.I

I I

I

10

I

100

Mole ratio

Figure5. Relative error between the data calculated using Eqn 10 and the theoretical calibration curve. The brackets indicate propagated random error calculated using the *I SDvaluesfrom Table 1.

When this relation is used to determine propagated random error, all the data fall within *1 SD of the straight line. Consequently, the curvature of Eqn (10) data in Fig. 4 could be the result of random error propagation alone. If comparison is carried one step further and the data plotted against a theoretical calibration curve having a slope of 1.0 and an intercept at the origin (Fig. 9, it is interesting to find that the data fit the theory within 1 SD. This is considerably better a reement with theory than has been used previously 1 8 to verify the isotope dilution concept in applications using Gas Chromatography Mas Spectrometry for the determination of organic compounds. The important thing to note, however, is that propagated errors tend to increase as x / y deviates from a value of about one. This is to be expected for two reasons: first, measurement precision is

*

less at both high and low R values14 and second, the magnitude of correction required to account for isotopic impurities present in the original sample and the internal standard increases at both high and low mole ratios values. These propagated errors must not be ignored but neither should they be overestimated. Systematic errors generated by making inappropriate assumptions about R, and R, can introduce much more error into calculated values than propagated errors. Propagated errors are at a minimum over portions of the calibration curve which are inherently linear. When an extended linear range is required and abundance corrections made using an appropriate equation, it is important to monitor propagated errors. If they become unacceptably large, the range of applicability for the corrections has been reached and other approaches for solving the problem should be investigated. If a body of statistical information is not available for a given instrument and application, substituting an estimated precision value based on previous experience with the instrumentation, may be a useful approximation. In conclusion, an expanded useful analytical range for quantitative isotope dilution Gas Chromatography Mass Spectrometry is available when appropriate corrections are made for isotopic impurities in the original sample and in the internal standard. This is significant when isotopically impure internal standards are used or when a wide range of concentration values are anticipated. The range over which these corrections are effective is governed by propagated errors which should be monitored as an indication that the effective range has not been exceeded.

Acknowledgement This work was supported in part by Grant BMS 75-06762 awarded by the National Science Foundation and Grants NS 09339 and NS 12116 awarded by the National Institutes of Health.

REFERENCES 1. J. H. Beynon, Mass Spectrometry and its Application to Organic Chemistry, pp. 98-1 02. Elsevier, Amsterdam (1960). 2. H. Hintenberger, Electrornagnetically Enriched Isotopes and Mass Spectrometry, ed. by M . L. Smith, pp. 177-189. Butterworths, London (1956). 3. R. Schoenheimer, S. Ratner and D. Rittenberg, J. Biol. Chem. 130,703 (1939). 4. D. Rittenberg and G. L. Foster, J. Biol. Chem. 133,737 (1940). 5. S. Graff, D. Rittenberg and G. L. Foster, J. Biol. Chem. 131,745 (1940). 6. G. L. Foster, J. Biol. Chem. 159, 431 (1945). 7. D. Shemin, J. Biol. Chem. 159,439 (1945). 8. H. Gest, H. D. Kamen and J. M . Reiner, Arch. Biochern. 12,273 (1947). 9. 1. M . Campbell, Bioorg. Chern. 3,386 (1974). 10. M . G. Inghram, Annu. Rev. Nucl. Sci. 4, 81 (1954). 11. D. J. Jenden, M . Roch and R. A. Booth, Anal. Biochem. 55,438 (1973). 12. E. K. Gordon, J. Oliver, K. Black and I. J. Koplin, Biochem. Med. 11,32 (1974).

230 BIOMEDICAL MASS SPECTROMETRY, VOL. 6, NO. 6, 1979

13. H. R. Sullivan, P. G. Wood and R. E. McMahon, Biomed. Mass Spectrom. 3 , 212 (1976). 14. D. A. Schoeller, Biomed. Mass Spectrom 3, 265 (1976). 15. B. N. Colbyand M. W . McCaman, Biomed. Mass Spectrom. 5, 215 (1978). 16. J. F. Pickup and K. McPherson, Anal. Chem. 48, 1885 (1976). 17. D. G. Peters, J. M. Hayes and G. M. Hieftje, ChemicalSeparations and Measurements, p. 32. W. B. Saunders Co., Philadelphia (1974). 18. D. P. Shoemaker and C. W. Garland, Experiments in Physical Chemistry, p. 34. McGraw-Hill, New York (1962). 19. P. J. DeBievre and G . H. Debus, Nucl. Instrum. Methods 32, 224 (1965).

Received 29 August 1978 @ Heyden & Son Ltd, 1979

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