Anal. Chem. 1994, 66, 2829-2834
Error Propagation in Isotope Dilution Analysis As Determined by Monte Carlo Simulation K. Y. Patterson,?**C. Veillon,* and T. C. O'Haver'ft Department of Chemistry and Biochemistry, University of Maryland, College Park, Maryland 20742, and Beltsville Human Nutrition Research Center, ARS, USDA, Beltsville, Maryland 20705
Error propagation in isotope dilution analysis (IDA) can be studied with Monte Carlo (MC) simulation. In inductively coupled plasma mass spectrometry with ion-countingdetection, the limiting error in the measured isotope ratio can be described with Poisson statistics. Taking into account this error in the measured isotope ratio, parameters for IDA can be optimized. The utility of MC simulation for IDA is illustrated in the optimization of conditions for zinc analysis. The minimum imprecision in the determinationof zinc by IDA was found for the internal standard enriched with the isotope having the lowest natural abundance. In addition to optimizing analytical conditions, MC simulation can provide information on the theoretical detection limits in double and triple IDA for stable isotope tracers. Isotope dilution analysis (IDA) is a powerful tool for the analyst when accuracy and precision are important for either inorganic or organic analysis. Unlike most analytical methods that make relative measurements, e.g., calibrating instrument response against standards, IDA is an absolute method requiring only the measurement of isotope ratios for calibration of the internal standard and analyte determination. To make isotope ratio measurements, isotope or mass-specific analysis is needed-either neutron activation analysis or mass spectrometry. Thermal ionization mass spectrometry has been used for many years in IDA, but extensive sample preparation is often needed and sample throughput is limited. The development of the inductively coupled plasma mass spectrometer (ICPMS) has made it possible to more conveniently use IDA for inorganic analysis. The basic principles and problems of IDA mass spectrometry have been described in a review article by Fassett and Pau1sen.I Briefly, it is possible to obtain a form of an element that has one stable isotope enriched relative to its natural abundance. For example, as it occurs in nature, zinc is composed of only 4.1%zinc with mass 67, yet zinc enriched in the 67Zn isotope to as much as 94% can be obtained commercially. A determination of the amount of an element in a sample can be made by IDA when a known amount of the isotopically enriched element is added as an internal standard. The isotope in which the internal standard is enriched is referred to as the spike isotope. In IDA, the internal standard is equilibrated with the element present in the sample and a measurement is made of the ratio between a reference (unenriched) isotope of the University of Maryland. Beltsville Human Nutrition Research Center. (1) Fassett, J. D.; Paulsen, P. J. Anal. Chem. 1989, 61, 643A-649A +
0003-2700/94/0366-2829$04.50/0 0 1994 American Chemical Society
element and the spike isotope. The measured ratio, RM,can be used to calculate the amount of the element in the sample as follows:
where& and As are the atomic abundances for the reference isotope in the natural element and in the internal standard, respectively; BN and Bs are the atomic abundances for the spike isotope in the natural element and internal standard, respectively; and M N and MS are the moles of the natural element in the sample and the moles of the internal standard added to the sample, respectively. Rearranging, the moles of the element present in the sample can then be calculated since RM is measured and the other terms are known.
With a few exceptions,' the atomic abundances of the isotopes in the natural, terrestrial elements are essentially constant and have been published.2 The atomic abundances of the isotopes in the internal standard are constant, the values being supplied by the producer or determined experimentally by the user. The error in M N ,the amount of element determined to be present in a sample, is the result of determinate (systematic) errors in all the terms and indeterminate (random) errors in the measured ratio, RM,and in the amount (mass) of internal standard added. Neglecting systematic errors and the random error contribution from the amount of internal standard added, the indeterminate error in MN is a consequence of the imprecision in the measurement of the isotope ratio, RM.The imprecision in M N cannot be less than the imprecision in RM. Calculations have been made3" showing that the imprecision in RM is magnified in the IDA calculation of M N with the amount of magnification depending on the value of the isotope ratio, RM. Even though the imprecision in M N will always (2) de Bi&vre,P.; Barnes, I. L. Int. J . Mass Spectrom. Ion Processes 1985, 65, 21 1-230. (3) Heumann, K. G. In Inorganic Mass Spectromefry; Adams, F., Gijbels, R., VanGrieken, R., Ed.; John Wiley & Sons: New York, 1988; pp 301-375. (4) Colby, B. N.; Rosecrance, A. E.; Colby, M. E. Anal. Chem. 1981, 53, 19081911. (5) van Heuzen, A. A.; Hoekstra, T.; van Wingerden, B. J . Anal. At. Specfrom. 1989, 4, 483-489. (6) Jamieson, R. T.; Schreiner, G. D. L. In ElectromagneticallyEnrichedIsofopes and Mass Spectrometry; Smith, M. L., Ed.; Butterworths Scientific Publications: London, 1956; pp 169-176.
AnalyticalChemistry, Vol. 66, No. 18,September 15, 1994 2829
be greater than that of R M the , imprecision in M Nmost closely approaches the imprecision in RM when the value of R M is equal to the geometric mean of the reference isotope to spike isotope ratio in the internal standard and in the naturally occurring element. The geometric mean can be calculated from the atomic abundances of the isotopes in the internal standard and the natural element. This suggests that if the imprecision in RM is constant regardless of its value, the geometric mean as described above should be the optimum value for RMin order to minimize the error in M N ,the amount of analyte. Knowing the optimal value for RM and the approximate amount of analyte in a sample, the amount of internal standard to be used in an analysis can be estimated. The situation becomes more complex in IDA by ICPMS, where the imprecision of R Mis not a constant. Adriaens et al.’ pointed out that when signal detection is based on pulse or ion counting, the standard deviation of the ratio measurement, R M ,is value dependent; e.g., its value is the result of the standard deviation in the count rate measurements of each of the isotopes. These can be estimated with Poisson statistics. For example, the standard deviation for the count rate, acA, of the reference isotope is as follows: (3)
where t A is the signal collection time and CAis the count rate for the reference isotope, A. There are two factors to be considered in optimizing the IDA-minimizing the imprecision in R Mand minimizing the magnification of that imprecision in the calculated value for M N . Determination of the optimum value for RMunder these circumstances can be approached numerically,’ but a useful alternative, which avoids complex mathematical computations, is Monte Carlo simulation. Monte Carlo (MC) techniques have been used in many situations where mathematical methods are inconvenient or i m p o s ~ i b l e , including ~,~ error propagation in electrochemistrylo and in IDA for uranium and plutonium reference materials.’ An important benefit of using MC simulation is that it is possible to investigate realistic situations and arrive at answers without having to generate closed-form mathematical solutions. The results, obtained by multiple recalculations, in the past have required the power of mainframe computers and custom computer programs.8 Recently, significant improvements in the speed and memory of personal computers, and in the sophistication of available programs, have made techniques such as MC simulation more convenient. The optimization of analytical parameters for IDA by MC simulation presented here uses a commercial spreadsheet program and a personal computer. Monte Carlo Simulation. The approach of the MC simulation for optimizing IDA, referring to eq 2, is todetermine at which value of RM(the measured ratio of the reference to spike isotopes) the imprecision in M N is at a minimum. This is done by recomputing M N many times while R Mvaries in value as defined by counting statistics around a predetermined (7) Adriaens, A. G.; Kelly, W. R.; Adams, F. C. Anal. Chem. 1993,65,66(M63. (8) Schwartz, L. M . Anal. Chem. 1975, 47, 963-964. (9) Giiell, 0. A.; Holcombe, J. A. Anal. Chem. 1990, 62, 529A-542A. (IO) Efstathiou, C. E.; Hadjiioannou, T. P. Anal. Chem. 1982, 54, 1525-1528. (11) Rizhinskij, M. V.; Vitinskij, M . Y. Kernenergie 1990, 33, 84-88.
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Analytical Chemistty, Vol. 66, No. 18, September 15, 1994
Table 1. Atomlc Abundances Expressed as Percentages for Natural and Isotoplcally Enrlched Zlnc, and Atomlc Welghts
natural Zna
67Zn enriched zincb
@Zn enriched zind
70Zn enriched zincb
64 66 67 68 70
48.6 27.9 4.1 18.75 0.62
1.113 1.95 94.6 2.28 0.054
0.32 0.21 0.10 99.36 0.01
4.44 2.75 0.51 3.69 88.61
atomic wt
65.40
66.90
67.90
69.46
isotope mass
de Bibvre, P.; Barnes, I. L. Int. J . Mass Spectrom. Ion Processes Atomic abundances are given for specific lots.
1985, 65, 211-230.
mean value. The value for RMused in each recompution is determined by the following equation: (4)
where Ni and Nj are numbers from normally distributed sets of random numbers having a mean value of 0 and a standard deviation of 1. Each recomputation of MN uses new values for N i and Nj. The standard deviation and relative standard deviation (5% rsd) for M N ,at a given mean value for R M can , then be determined based on the many values for M N that are obtained in the simulation. A series of simulations, each with another mean value of RMacross the range of possible values (the reference to spike isotope ratios in the internal standard and in the natural element define the range), makes it possible todetermine which value of R M results in a minimum imprecision in the mean value of M N . The mean value of M N is constant throughout the series of simulations and M s changes for each tested mean value of RM to maintain the equality in eq 2.
EXPERIMENTAL SECTION To illustrate how MC simulations can be used, the optimization of zinc analysis by IDA is presented. Zinc has several stable isotopes of widely varying abundances (Table l ) , and zinc enriched in the each of the isotopes is available commercially (Oak Ridge National Laboratory, Oak Ridge, TN). Three series of simulations were performed, each using one of the three lowest abundance zinc stable isotopes as the internal standard. The reference isotope was 66Zn, as recommended by H e ~ m a n n .The ~ optimum R Mwas determined for each internal standard, and the results were compared to ascertain whether there was a difference in the minimum imprecision for M N under identical conditions. Table 2 gives thevalues to be used in eq 2 for the simulations and the range of values of RMthat could be tested for each internal standard. The atomic abundances for the internal standards were from the values for the isotopically enriched zinc listed in Table 1. The value chosen for RMdetermined CAand CB.Since at high count rates the ICPMS detectors may exhibit gain suppression,l* the maximum count rate for the simulations was 300 000 counts/s. When R Mwas > 1, CA (1 2 ) CHANNELTROM Electron Multiplier Handbook for Mass Spectrometry Applications; Galileo Electro-Optics Corp.: Sturbridge, MA, 1991,
Monte Carlo Simulation for Isotope Dilution Analysis Using 6'Zn Enriched Internal Standard I
e.
I
Results of the Mdnte Carlo Simulation for ialue of R,entered meanvalue sd % rsd M, (pnolNatZn) 0.030583 0.0017 0.0836
RM (66167)
1.199980
I
0.0815
1
0.0679
I
I
c. Monte Carlo simulation calculations (first 6 of 2000) MN I , R I C,' j C,' (uno1 Nat Zn) 0.030564
I
I
I
Ni
1.1994
1300102.86
I 250215.34
0.030563
1.1994
299963.62
25010 1.10
I
0.030592
1.2003
300010.65
I
0.030583
1.2000
299910.68
0.030620
1.2012
300227.96
249943.95
1.61
0.030623
1.2013
299955.83
249700.27
I
AN As
BN Bs RMrangec
1.67 0.78
249947.65
0.08
-0.41
249927.94
-0.63
spike isotope 68Zn
0.73
1 -0.26
Table 2. Values Used for the Monte Carlo Slmulallona 67Zn
Ni
I
70Zn
0.279b 0.0195 0.041 0.946
0.279 0.0021 0.1875 0.9936
0.279 0.0275 0.0062 0.8861
0.0206-6.805
0.0021-1.488
0.031-45.0
0 MN, (mean) 2 bg (30.6 nmol) of zinc. IUS,adjusted to maintain equality in eq 2. b Fractionalatomic abundance. 66Znreference isotope.
= 300 000 counts/s and CBwas adjusted to give the chosen ratio, and when R Mwas
