Complete Equation for the Measurement of Organic Molecules Using

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Anal. Chem. 2008, 80, 5071–5078

Complete Equation for the Measurement of Organic Molecules Using Stable Isotope Labeled Internal Standards, Exact Matching, and Mass Spectrometry Daniel G. Burke* and Lindsey G. Mackay National Measurement Institute, Australia, 1 Suakin Street, Pymble, New South Wales 2073 Australia Highly accurate measurements of the amount of substance of organic molecules in a test material can be obtained using exactly matched calibration solutions and internal standards that are labeled with stable isotope atoms by measuring the amount ratio of analyte to internal standard using mass spectrometry. Estimating the uncertainty of quantitative measurements of organic molecules is a means of evaluating accuracy and of establishing traceability to the International System of Units (SI) and requires a measurement function that fully describes the measuring system. This paper presents the derivation of the equation (measurement function) that describes this complete measurement after the internal standard has equilibrated with the test material matrix. It is similar to the equation for inorganic measurements using isotope dilution techniques, but potential biases during chemical processing arising from whole organic molecule analysis compared to inorganic atomic analysis required greater investigation of the yield factors that occur during organic molecule measurements. In the new equation, a series of ratios of proportionality factors are used to relate the amount of substance in a test material to chromatographic peak area ratios corresponding to mass spectrometer ion current ratios. All the proportionality factors are grouped together to define a measuring system factor FX, the value of which is determined by the fundamental chemical processes affecting the yields of analyte, internal standard, and reference standard of the analyte in the measurement process. Any factors in the measurement process that affect the mole ratio of analyte to internal standard in the calibration solution differently from the test solution will result in a nonunity value for FX and a proportional bias to the measurement, and in this way FX represents the concept of recovery of the amount ratio of analyte to internal standard. Thus highly accurate measurements require FX or its constituent factors to be evaluated. In addition, the uncertainty in the evaluation of FX or of its constituent factors must be included in a complete uncertainty estimation of the analytical procedure. The many different permutations of proportionality factor ratios that may result in a unity value of FX are discussed resulting in a case for evaluating FX rather than the more common practice of evaluating individual factors for each

major stage of the measurement procedure. Since the new measurement function describes the complete chemical process that constitutes the measurement, traceability to the SI is assured when all factors in the function are measured traceably and have their associated uncertainty estimated correctly. Ignoring FX would invalidate traceability to the SI and would prevent a complete estimation of measurement uncertainty. Estimating the uncertainty of quantitative measurements of organic molecules is a means of evaluating accuracy and of establishing traceability to the International System of units SI and so is needed for the certification of reference material properties.1 Testing and calibration laboratories are now also encouraged to estimate their measurement uncertainty.2 There are two types of effects considered to contribute to total uncertainty: precision and bias. In most instances estimation of measurement precision may be readily obtained by replicate measurements, but measurement bias is more difficult to evaluate because of the multitude of factors that may contribute to bias at each step in the analysis of organic compounds. Fortunately, the effects of many of the factors that contribute to bias may be reduced through the use of internal standard methodologies and in particular by using stable isotope labeled analogues of the target molecule;3–5 however, even when internal standards are used, the reduction in bias depends on how well the specific measuring system is controlled and complete elimination of bias cannot be assumed. One of the simplest methods for evaluating bias in a measuring system is by measurement of a reference material with a property value that has been certified by a competent laboratory that has estimated the uncertainty from first principles;1 measurement bias can then easily be evaluated by subtracting the measured value from the certified value. There are many tens of thousands of compounds to be measured in an almost unlimited number of matrix materials, so in most cases a commutable * Corresponding author. Fax +61 2 9449 1653, E-mail daniel.burke@ measurement.gov.au. (1) Guide to the Expression of Uncertainty in Measurement. ISO, 1995. (2) General Requirements for the Competence of Testing and Calibration Laboratories. ISO/IEC 17025-2005. (3) Pickup, J. F.; McPherson, K. Anal. Chem. 1976, 48, 1885–1890. (4) Schoeller, D. A. Biomed. Mass Spectrom. 1980, 7, 457–463. (5) Mackay, L. G.; Taylor, C. P.; Myors, R. B.; Hearn, R.; King, B. Accredit. Qual. Assur. 2003, 8, 191–194.

10.1021/ac800270u CCC: $40.75 Published 2008 by the American Chemical Society Published on Web 06/03/2008

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certified matrix reference material is not available. The thorough examination of critical factors involved in quantitative organic measurements has led to the development of a measurement function that more fully describes the measuring system and may form a theoretical basis for the improved preparation of reference materials. Isotope dilution mass spectrometry (IDMS) using one reverse step (double IDMS) has been used for high-accuracy measurements of the amount of substance of the elements and has been recognized as a primary ratio technique with the potential for producing measurements that are traceable to the SI.6,7 Measurement equations for the application of double IDMS to inorganic and gas analyses have been derived using various nomenclatures.8–10 Equation 1 below is a succinct summary of those equations reproduced with variable notations changed to those used in this paper. It describes the measurement of the amount of substance of an element by calibration of mass spectrometer ion current ratios using a blend prepared by mixing a reference standard of the element and an internal standard

nX ) nZ

nY RY - RB RB,c - RZ nY,c RB - RX RY - RB,c

∑R ∑R

iX

(1)

iZ

where, nX ) amount of substance of analyte (in moles) in a sample of test material; nZ ) amount of substance of reference standard of analyte (in moles) in calibration blend; nY ) amount of substance of internal standard (in moles) added to a sample of test material; nY,c ) amount of substance of internal standard (in moles) added to calibration blend; RB ) amount ratio of selected isotopes in sample blend solution; RB,c ) amount ratio of selected isotopes in calibration blend solution; RX ) amount ratio of selected isotopes of element in test material; RY ) amount ratio of selected isotopes in internal standard; RZ ) amount ratio of selected isotope in reference standard; ∑RiX ) sum of all the isotope amount ratios of the analyte in the test material; ∑RiZ ) sum of all the isotope amount ratios of the reference standard. In this equation, n may be replaced by N, where N is the symbol for number of entities, e.g., atoms or molecules, since N/NA = n (where NA is Avogadro’s number). Though not stated explicitly in the literature, we believe the intent has been to define nX as the amount of substance of an atomic element in a test material (e.g., soil, food, body fluid) as opposed to the solution analyzed by mass spectrometry and as such it has been assumed that there is no change in isotope ratios of the element when it is transferred from the sample to the analytical solution. Ion current ratios produced from the analytical solution by the mass spectrometer are related to isotope amount ratios in the test material by a K factor that includes isotope fractionation and that is measured using certified reference (6) Milton, M. J.; Quinn, T. J. Metrologia 2001, 384, 289–296. (7) Richter, W. Accredit. Qual. Assur. 1997, 2, 354–359. (8) De Bie`vre, P.; De Laeter, J. R.; Peiser, H. S.; Reed, W. P. Mass Spectrom. Rev. 1993, 12, 143–172. (9) Milton, M. J. T.; Wielgosz, R. I. Metrologia 2000, 37, 199–206. (10) De Bie`vre, P. In Trace Elemental Analysis in Biological Specimens; Herber, R. F. M., Stoeppler, M., Eds.; Elsevier: Amsterdam, The Netherlands, 1994; pp 169-183.

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materials.8,9,11,12 The correction for K may be avoided in most cases when RB is matched with RB,c and RX ) RZ and so K is often omitted from the equation.11 However, early work in this field by De Bie`vre clearly explains the requirement for evaluating K when highly accurate measurements are needed.13 In addition, since the test material matrix is completely destroyed during chemical processing, K has not been defined as incorporating matrix effects. Methods for calculating the uncertainties associated with this equation and for selecting the optimum conditions for minimum uncertainty have been well documented.9,13–19 Equation 1 then has been derived for polyisotopic elements and requires internal standards of the same element with one or more of the isotopes enriched and has fundamental assumptions about isotopic homogeneity and complete digestion of test material matrix. Measurements of organic molecules using stable isotope labeled molecules as internal standards differ in the application of isotope dilution theory. Instead of measuring steady state signals at the m/z values of the isotopes of atomic elements, area ratios of chromatographic peaks obtained by monitoring the m/z values corresponding to the analyte and internal standard molecules are measured. In addition, eq 1 may describe only the last step in the analytical scheme (instrumental analysis) and assumes that isotope ratios of the elements do not change during sample workup. In contrast, the measurement of small quantities of organic molecules in a test material commonly requires several steps before instrumental analysis of the solution containing the organic molecule. As a result the amount ratio of analyte to isotope labeled internal standard has a higher probability of change since the isotope is part of a different molecule and thus the differences between the calibration and test solution may affect this ratio to a higher degree than for isotopes of the same element. The nature of the calibration equation for organic measurements has been discussed for over 30 years. Pickup and McPherson developed equations based on binomial probability theory that were similar to historical work by Hintenberger and Webster on isotope dilution of elements.3,20,21 They found that the calibration graph is a curve with some special cases which are straight lines. Two calibration equations (including that of Pickup and McPherson) were compared by Sabot et al.15 They found that there was a difference between experimental and calculated results which could not be elucidated. Thus, application of classical isotope dilution theory to this type of analysis may hinder the development of a complete equation to describe the analysis of complex organic (11) Henrion, A. Fresenius J. Anal. Chem. 1994, 350, 657–658. (12) Milton, M. J. T.; Wielgosz, R. I. Rapid Commun. Mass Spectrom. 2002, 16, 2201–2204. (13) De Bie`vre, P. In Advances in Mass Spectrometry; Daly, N. R., Ed.; Heydon and Son: London, U.K., 1997; Vol. 7A, pp 395-447. (14) Colby, B. N.; Rosecrance, A. E.; Colby, M. E. Anal. Chem. 1981, 53, 1907– 1911. (15) Sabot, J.-F.; Ribon, B.; Kouadio-Kouakou, L.-P.; Pinatel, H.; Mallein, R. Analyst 1988, 113, 1843–1847. (16) Thienpont, L. M.; Van Nieuwenhove, B.; Stockl, D.; De Leenheer, A. P. J. Mass Spectrom. 1996, 31, 1119–1125. (17) De Bie`vre, P.; Peiser, H. S. Fresenius J. Anal. Chem. 1997, 359, 523–525. (18) Kipphardt, H.; De Bie`vre, P.; Taylor, P. D. P. Anal. Bioanal. Chem. 2004, 378, 330–341. (19) Catterick, T.; Fairman, B.; Harrington, C. J. Anal. At. Spectrom. 1998, 13, 1009–1013. (20) Hintenberger, H. In Electromagnetically Enriched Isotopes and Mass Spectrometry; Smith, M. L., Ed.; Butterworth: London, U.K., 1956; p 177. (21) Webster, R. K. In Advances in Mass Spectrometry; Waldron, I. D., Ed.; Pergamon: London, U.K., 1959.

molecules due to the increased number of atoms and possible combinations of their isotopes that may occur at each mass value compared to what occurs with the elements. This paper details the derivation of an equation that describes the measurement of organic molecules and that includes by definition all chemical workup steps prior to instrument analysis. A series of ratios of proportionality factors relate the amount of substance of analyte in a test material matrix to chromatographic peak area ratios of mass spectrometer ion currents in a similar way to original work by De Bie`vre and Debus on isotope dilution analysis of the elements.22 In the present work, all the proportionality factors are grouped together to define a measuring system factor FX, the value of which is the result of the chemical interactions affecting the yield of analyte, internal standard, and reference standard of the analyte in the measurement process. The equation does not cover the equilibration of internal standard molecule with sample material which is a fundamental requirement of this measuring system and must be assessed separately. There has been some discussion of the concept of recovery and its role in assessing the accuracy of analytical procedures,23–26 but since it does not yet have a single well defined meaning, use of the term recovery may lead to ambiguity. Instead, a measuring system factor FX is defined for the case of measurements of organic molecules using the internal standard method with adequately matched calibration solutions. The many different permutations that may result in a unity value of FX, and thus zero bias in the measuring system, are discussed, and a standard addition experiment is presented as the most effective method for evaluating FX. This method is analogous to the surrogate recovery method described in the harmonized guidelines.23 The measuring system factor does not equate to the accuracy or recovery of the method since not all possible biases are included (e.g., equilibration of internal standard with sample and purity of reference material), but it must be evaluated and its uncertainty must be included in a complete uncertainty estimation of the measurements from the analytical procedure since it is potentially a directly proportional contributor to measurement bias. Highly accurate results can only be obtained when all biases and precision effects are quantified, and the results are traceable to the SI only when the values of all terms in the measurement equation can be traced through an unbroken chain of references to an SI standard. It may seem reasonable to omit evaluation of FX in cases where a blank material is available for use as a calibration matrix and where carbon-13 labeled analogues of the target molecules are available for use as internal standards (e.g., many of the dioxins); however, if FX is not evaluated, no claims about accuracy and traceability can be made since this factor is a crucial part of the measurement function. Also, the cost of evaluating FX using a standard addition experiment is relatively low, further supporting the case for proper estimation of measurement uncertainty and establishment of traceability to the SI. (22) De Bie`vre, P. J.; Debus, G. H. Nucl. Instrum. Methods 1965, 32, 224–228. (23) Thompson, M.; Ellison, S. L. R.; Fajgelj, A.; Willetts, P.; Wood, R. Pure Appl. Chem. 1999, 71, 337–348. (24) Koscielniak, P. Anal. Lett. 2004, 37, 2625–2640. (25) Dybkaer, R. Accredit. Qual. Assur. 2005, 10, 302–303. (26) Thompson, M.; Wood, R. Accredit. Qual. Assur. 2006, 10, 471–478.

DERIVATION OF EQUATIONS Sample Preparation. The types of measurements considered in this paper are those that require the mass fraction of a well defined molecule to be calculated in a given test material which may be either solid or liquid and in which the distribution of the target molecule in the material is not known. The measurement techniques for trace amounts of nonvolatile organic molecules have a requirement for the compound to be in solution, and to achieve this it must be extracted from the test material matrix into a solvent compatible with analytical measurement. If interferences are present in the extract, further processing of the test material extract may be necessary; then the solution may be subsampled and fractionated and the fractions possibly subsampled and derivatized to produce a solution which is ultimately used for the measurement. That is, the amount of substance of compound X in the test material extract solution after chemical workup (nX,e, where subscript “e” represents the word “extract”) is proportional to the amount in the test material (nX) through a proportionality factor (kX), the value of which is determined by losses in the workup, e.g., extraction, fractionation of the extract, and possible derivatization of the extract fraction. For this discussion, all proportionality factors shall be assumed to be independent of the amount of substance though this is not necessarily the case, and if low uncertainty measurements are needed, the sensitivity of the proportionality factors to amount of substance should be investigated. The relationship between amount of substance in the test material and in the extract from the test material after chemical processing can be expressed algebraically as nX,e ) kXnX

(2)

The yield of analyte during sample workup may be highly variable and analyte losses may be significant, but the use of an internal standard has the potential to eliminate the bias on the measurement due to losses during extraction and analysis and may also improve precision. In this type of analysis, an amount of internal standard (nY) can be equilibrated with a sample of the test material before any chemical processing occurs, and then the mole ratio of analyte to internal standard in the test material is used to measure the amount of analyte. The earlier in the analytical scheme that the internal standard is added, the more steps that are controlled and the more chemically similar the internal standard to the target molecule the closer the control will be. When the internal standard is a stable isotope labeled analogue of the target molecule, the physicochemical properties of the molecule that determine workup and analysis yield may not be significantly altered so there is a high probability that factors affecting the yield of the target molecule will also affect the labeled analogue to the same extent. However, there is no absolute requirement to use a stable isotope labeled internal standard; as long as the relevant physicochemical properties of the internal standard result in the ratio of proportionality factors being the same in test and calibration solutions then this does not contribute to bias in the measurement. The amount of internal standard added to the test material need not be accurately known if a separate calibration blend is prepared using a similar amount of the same internal standard (nY,c) and an accurately known amount of a reference standard of Analytical Chemistry, Vol. 80, No. 13, July 1, 2008

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the analyte (nZ) that is identical to the target molecule. To reduce the potential sources of bias, the amount of reference standard added to the calibration blend is matched as closely as possible to the amount of analyte in the sample of test material used for measurement so that the instrument response is identical (within the degree of matching) for test material and calibration blends. The amount of substance X in the test material can now be expressed as shown in eq 3. The complete derivation of eq 3 is given in the Supporting Information.

nX )

nX,enY,cenZnYkYkZ nY,enZ,enY,ckXkY,c

(3)

where nX,e ) amount of substance of analyte (in moles) in the extract from the sample of test material after workup; nY,e ) amount of substance of internal standard (in moles) in the extract from the sample of test material after workup; nZ,e ) amount of substance of analyte in reference standard (in moles) in calibration blend extract after workup; nY,ce ) amount of substance of internal standard (in moles) in the calibration blend extract after workup; kY ) proportionality factor for yield of internal standard from sample blend into extract solution after workup; kY,c ) proportionality factor for yield of internal standard from calibration blend into extract solution after workup; kX ) proportionality factor for yield of analyte from sample blend into extract solution after workup; and kZ,c ) proportionality factor for yield of analyte in reference standard from calibration blend into extract solution after workup. Equation 3 describes the relationship between the amount of substance in the sample material to the amount of substance extracted from the sample material into the solution including all extract fractionation and derivatization procedures that may be needed for effective analysis by gas chromatography/mass spectrometry (GC/MS) or liquid chromatography/mass spectrometry (LC/MS). Calibration of Analytical Instrument. Instrumental analysis of organic molecules by LC/MS or GC/MS has some significant differences from analysis of inorganic elements by, for example, inductively coupled plasma mass spectrometry (ICPMS). The organic molecules in solution are chromatographically separated before individual components enter the mass spectrometer; there, ionization and mass to charge separation results in ion current signals for each molecule at characteristic m/z values that are monitored over time as the LC or GC column eluant enters the mass spectrometer. The signal intensity is proportional to the amount of substance entering the mass spectrometer, and signal intensity is measured as peak areas (the sum of intensity over time) of ion currents at the selected characteristic m/z values. When there are no overlapping ions between analyte and internal standard, the chromatographic peak area of the characteristic ion signal (a) is proportional to the amount of substance of analyte in the solution analyzed (nX,e). This can be expressed algebraically as a ) KX,enX,e

(4)

where KX,e is the proportionality factor covering losses during instrument analysis and denoted in upper case to distinguish from the proportionality factors involved in sample workup given in eq 5074

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3. The subscript “e” is used to denote the extract after workup. Since this proportionality factor represents yield of instrument signal from the test material extract, it includes the phenomenon of “mass bias” as specified in previous work.5 In some instances, overlapping mass spectral signals from analyte and internal standard renders eq 4 invalid. This occurs when the analyte and its isotopically labeled analogue have very similar retention times and the stable isotopically labeled internal standard differs in molecular weight from the analyte by only one or two Daltons or when the analyte contains multiple atoms of elements that have significant proportions of naturally occurring stable isotopes (e.g., chlorine). Then, the observed signal intensity at the two characteristic m/z values, i.e., one from the analyte and one from the stable isotopically labeled internal standard, may contain contributions from both compounds, and thus the signals are the sums of the contributions from the natural and labeled molecules. The contribution from each molecule to the total signal intensity can be calculated if the ratios of the signal intensities at the two characteristic m/z values in the analyte (denoted R′X) and internal standard (denoted R′Y) molecules can be measured. Since R′X and R′Y are constant ratios, the amount of analyte can still be measured even when there is significant overlap between mass spectral ions from the analyte and internal standard. Deriving the relationship between signal intensity (as measured by peak area) and amount of analyte in this way leads to eq 5. For consistency with previous literature concerning IDMS equations, the primed notation R′ shall be used to represent the measured chromatographic peak area ratio of the selected ions from the analyte and internal standard. nX,e ) R′XnY,e

KY,e (R′Y - R′B) KX,e (R′ - R′ ) B

(5)

X

where KX,e ) proportionality factor for yield of analyte at instrument detector; KY,e ) proportionality factor for yield of internal standard at instrument detector; R′B ) peak area ratio of selected ions in test material extract solution after workup; R′X ) peak area ratio of selected ions in analyte molecule; R′Y ) peak area ratio of selected ions in internal standard molecule. An analogous equation can be derived for the calibration solution, and the two equations can be combined to give eq 6 when the same internal standard solution is used in both test material and calibration blend solutions. The full derivation of eq 6 is given in the Supporting Information. Using the notation KB,e for the ratio of proportionality factors KY,e/KX,e makes this equation consistent with previous work.9,12

nX,e ) nZ,e

nY,e R′X KB,e R′Y - R′B R′B,c - R′Z nY,ce R′ KB,ce R′ - R′ R′ - R′ Z

B

X

Y

(6)

B,c

where R′Z ) peak area ratio of selected ions in reference standard molecule; R′B,c ) peak area ratio of selected ions in calibration blend extract solution after workup; KB,e ) ratio of proportionality factors KY,e/KX,e in test material extract solution; KB,ce ) ratio of proportionality factors KY,ce/KZ,e in calibration blend extract solution. Equation 6 describes the amount of substance of analyte in the test material extract after workup in terms of measured peak

Table 1. Numerical Examples of Matrix Effects on Proportionality Factor Ratios matrix for test material and calibration blend

KY,e

KX,e

KY,ce

KZ,e

KB,e

KB,ce

KB,e/KB,ce

same different

0.99 0.99

0.90 0.90

0.99 0.99

0.90 0.99

1.10 1.10

1.10 1.00

1.00 1.10

area ratios and amounts of substance of internal standard and reference substance in blend extracts after workup. These amounts of substance in the extract after workup are not known since the proportionality factors are not known and thus this equation is not yet in a form that can be used for calculation of the amount of substance of analyte in the sample material. The proportionality factors of this instrument analysis step, KB,e and KB,ce, encompass the chemical reactions that may occur in the chromatograph and mass spectrometer and include any degradation and irreversible adsorption during chromatography and the fundamental processes effecting ionization, transmission, and ion detection efficiency in mass spectrometry. The equivalence of KB and KB,ce relies on the relevant physicochemical properties that influence the ratio of proportionality factors in the test material extract solution being the same in the calibration blend extract solution. Even though the values of the constituent proportionality factors for analyte and internal standard molecules may be different in the sample and calibration solution, KB and KB,ce can still be equivalent and thus cancel from eq 6 if the ratios of the constituent proportionality factors are equivalent. However, if a proportionality factor is different in test and calibration solutions, then a bias is present in the measurement. This is illustrated with numeric examples in Table 1 in which two hypothetical scenarios are compared; when test and calibration materials are composed of the same matrix, the ratios of proportionality factors may be unity even when the proportionality factors are not quantitative. However, if test and calibration materials are composed of different matrixes and an improvement in yield (i.e., an increase in KZ,e) is observed in the calibration solution, then the ratio of proportionality factors will not be unity and a bias will be present in the measurement. When it can be shown that the analyte and the reference standard of the analyte are identical and the test and calibration extracts after workup are also identical, then KB,e = KB,ce by definition. However, when calibration and test solutions are different and/or when analyte and reference molecules may not be identical (e.g., due to isotopic differences), the ratios of proportionality factors cannot be assumed to be identical and so must be retained in the equation. Complete Measurement Function for Organic Molecules. In order to obtain an equation linking the amount of substance of analyte in the test material to measured peak area ratios, eq 6 can be substituted into eq 3 then simplified to eq 7. The complete derivation for eq 7 is given in the Supporting Information. ′

nX )

′

′

′

B

X

Y

B,c

FX )

KB,e kYkZ KB,ce kXkY,c

(8)

Defining the measuring system in this way may broaden the scope of this measurement technique since eq 8 clearly shows that the combined effect of the proportionality factors determines measuring system bias and that this may be evaluated as a single factor in addition to the product of its constituents. When the relevant physicochemical properties of the internal standard and analyte molecules are similar enough to give a consistent measurement system factor, it is possible that molecules other than stable isotope labeled analogues of the analyte may be effective as internal standards. This can be of practical significance when the stable isotope labeled analogue is not available or may be relatively expensive. Applying the Complete Measurement Function. In practice, most organic substance measurements are made in mass fractions, and then mole quantities are obtained by dividing the mass fraction value by compound molecular mass calculated from published atomic weights. In order to make eq 7 directly applicable to laboratory usage, it may be converted to mass fraction format to give eq 9; the symbol w is used for mass fraction (mass per mass) as recommended by IUPAC.

wX ) wZ

mYmZ (R′Y - R′B)(R′B,c - R′Z)R′X F mXmY,c (R′ - R′ )(R′ - R′ )R′ X B

X

Y

B,c

(9)

Z

where wX ) mass fraction of target molecule in test material; wZ ) mass fraction of reference substance in solution added to calibration matrix; mY ) mass of internal standard solution added to test material; mZ ) mass of reference standard solution added to calibration matrix; mX ) mass of test material extracted; and mY,c ) mass of internal standard solution added to calibration matrix. When the overlap between analyte and internal standard ion signals is insignificant, the complete equation simplifies to eq 10 as derived in the Supporting Information. The critical value of R′X at which its effect on wX becomes insignificant, and thus at which eq 10 can be used, is dependent on the difference between R′X and R′B and the precision with which the difference can be measured.5

wX ) wZ

mYmZ R′B FX mXmY,c R′

(10)

B,c

′

(R Y - R B)(R B,c - R Z) R X KB,e nZnYkYkZ (R′ - R′ )(R′ - R′ ) R′ KB,ce nY,ckXkY,c

ratios since all the terms on the right-hand side of the equation can be measured. This occurs because the terms involving the measurement of amounts of substance in the extracts from both the test material and the calibration blend cancel, i.e., nX,e, nY,e, nZ,e, nY,ce cancel as can be seen in the Supporting Information. For simplicity, the proportionality factors may be combined into a single measurement system factor FX,

(7)

Z

Equation 7 shows that the amount of substance in the test material can be measured using mass spectrometer ion current

Thus eqs 9 and 10 are complete descriptions of the measurement system after equilibration of internal standard with test material, and the measurement uncertainty can be obtained from the combination of uncertainties of each factor in this equation. Analytical Chemistry, Vol. 80, No. 13, July 1, 2008

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In addition, the definition of this complete equation enables measurements made in this way to be traceable to the SI when the measurement system factor can be evaluated traceably. Fundamental Assumptions. Equations 9 and 10 hold within certain boundary conditions or fundamental assumptions that were mentioned in the derivations above and are summarized here for convenience. Compliance with these boundary conditions must be established for correct application of these equations. The assumptions are 1. All proportionality factors are independent of the relevant amount of substance present in the test material and the calibration blend, and their extracts, within the achieved level of matching. In most measuring systems for organic molecules, the proportionality factors will be concentration dependent and the degree of matching required to minimize this effect must be critically examined. 2. The test material is homogeneous to the level of subsampling used for the measurement, and the internal standard molecule equilibrates uniformly with the target analyte molecule in both sample and calibration blend matrixes. 3. The ratios R′X,R′Y, and R′Z are constant and characteristic of the measuring system. Measuring System Factor. The factor FX is a measure of how well the mole ratio of analyte to internal standard in the test material and calibration blend is conserved through the entire measurement process. All of the surface and solution chemistry encountered by the analyte, internal standard, and reference substance molecules during the sample workup and instrument analysis steps is encompassed by this factor. While accurate measurement of sample and calibration solution masses and of peak area ratios may be rudimentary, all of the factors that may influence the values of the proportionality factors constituting FX are seldom known and thus FX often cannot be explicitly measured. However, differences in sample and calibration solution matrixes are common, and thus differences in proportionality factors between sample and calibration solution may be a major source of bias in the measuring system and so FX must be evaluated. Mathematically, there are three different ways in which the proportionality factors may cancel to give unity for FX. First the trivial case when the values of all proportionality factors are equal is found when test and calibration blend materials are identical, test and calibration extract processing is identical, analyte and reference substance are identical, and the amount of substance of analyte and reference standard are identical. Then KB,e ) KB,ce and kY ) kY,c and kX ) kZ, thus FX ) 1. In this case, the measurement system factor must be unity by definition. Second, FX can be unity when the products of all the proportionality factors in the numerator and denominator of eq 8 are equivalent, i.e., KB,ekYkZ ) KB,cekXkY,c. In this case, it is possible that the values of corresponding proportionality factors may be different but their products may be equal. This may occur when losses in analyte or internal standard at one step in the measurement process are compensated by different yields of reference standard or internal standard at a different step in the process. For example, if the calibration blend matrix was different from the sample blend matrix and this resulted in a lower yield of 5076

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Table 2. Numerical Examples of Possible Scenarios Giving FX ) 1 scenario

KB,e

KB,ce

kY

kX

kY,c

kZ

FX

factors equivalent product of factors equivalent quotient of factors equivalent (eq 11) quotient of factors equivalent (eq 12)

0.99 0.99

0.99 0.50

0.90 0.50

0.95 0.99

0.90 0.90

0.95 0.90

1.0 1.0

0.80

0.80

0.90

0.60

0.45

0.30

1.0

0.90

0.90

0.90

0.90

0.50

0.50

1.0

internal standard molecule from the sample blend but there was lower ionization suppression of the reference standard in the calibration blend, then these effects would result in a lower kY and KB,ce that may at least partially compensate for each other. Although it is common for sample and calibration blend matrixes to be different, it may seem implausible for the resulting differences in proportionality factors to exactly cancel. A similar situation may occur if an isobaric interference were to mask a low extraction yield. Third, FX can also be unity when the quotients of proportionality factors are equal. The two ways in which the quotients can be equivalent are illustrated by eqs 11 and 12. KY,e KX,e ) KY,ce KZ,e

and

kY kX ) kY,c kZ,c

(11)

KY,e KY,ce ) KX,e KZ,e

and

kY kY,c ) kX kZ

(12)

The relationships in eq 11 may occur when the test material and calibration matrixes are equivalent so the same factors operate in sample and calibration blends for the same compounds and is similar to the first case described above except that here the proportionality factors may be different but their ratios may be equivalent. An example of this situation is when the internal standard molecule has a different yield or ionization efficiency than the analyte or because of a difference in retention time, the internal standard may be subject to a different level of ionization suppression in LC/MS. The relationships in eq 12 may occur when the sample and calibration matrixes are different, but the ratio of their proportionality factors in either matrix is equivalent because the relevant physicochemical properties of the internal standard are similar to the analyte’s for the purposes of the measurement technique. An example of this situation is found when the test material matrix imparts a carrier effect on the analyte molecule that is absent when the calibration blend is prepared from a blank solvent. Numerical examples to illustrate the different possibilities outlined above are given in Table 2. Only those relevant chemical factors (e.g., solubility in extraction solvent, losses during volatilization in GC, ionization efficiency and suppression in LC/MS) that may lead to a difference in proportionality factors between sample and calibration blends are relevant in this context, so there may be many different matrixes that do not affect these properties. Since there are many different permutations of proportionality factor ratios that may result in FX = 1, it may be more effective to evaluate FX using the procedures discussed below than the more

common practice of evaluating individual proportionality factors for each major sep of the measurement procedure. FX and Recovery: Measuring Proportionality Factors. In the absence of other biases, the measurement system factor FX defined in this paper is the inverse of the recovery defined in the IUPAC guidelines. Since the measured value of wX includes the contribution from FX, the value of FX may be obtained by using a matrix reference material.23 In the ideal case, a matrix reference material may be available with the same matrix as the test material and in which the mass fraction of target analyte,wX,ref, is traceable to the SI with low uncertainty. This implies that FX for the reference material has been effectively evaluated. In addition, the relevant chemical properties of the calibration blend must match those of the reference material and the test material in order to ensure that the same proportionality factors apply to each. The mass fraction of analyte in the matrix reference material can be measured experimentally with the same system as that used to measure the test material. All the terms on the righthand side of eq 9 or 10 can then be assigned a measured value except for FX, so FX can then be obtained by dividing wX,ref by the experimentally measured value, wX,obs, as shown in eq 13. A complete derivation of eq 13 is given in the Supporting Information. FX )

wX,ref wX,obs

(13)

In reality, appropriate commutable matrix reference materials are rare and it is difficult to find materials where it can be assumed that the FX implicit in the certified reference value of the measured quantity will be the same as the FX for the measuring system that is used for the test material. Where such a commutable matrix reference material is not available, a standard addition experiment may enable measurement of FX. In this experiment, nx is measured in the sample blend in the normal way by comparison to a calibration solution and, separately, the reference standard is added to the sample blend to give a standard addition blend; the amount of substance of analyte in the standard addition blend nX,tot is measured and should be the sum of analyte and added reference material. The measurement of the amount of substance of analyte in the standard addition blend (the measurement contains the effects of FX) should be equal to the sum of the amount of substance of analyte measured in the sample blend (that measurement also contains the effects of FX) and the amount of substance of analyte added as reference standard. Note that both the measurements of the amount of substance incorporate the measurement system factor FX as shown in eq 14. nX,totFSAB ) nX,sabFX + nQ,sab

(14)

where nQ,sab ) amount of substance of reference standard added to the standard addition blend; nX,sab ) amount of substance of analyte in standard addition blend calculated from measured wX; nX,tot ) total amount of substance of analyte and reference standard measured in standard addition blend; and FSAB ) measurement system factor in standard addition blend. When it can be readily shown that the sample blend matrix is not significantly changed by the addition of the reference

molecule, e.g., by addition of the same amount of blank solvent or when subsequent steps in the measurement process negate the influence of the addition (e.g., solvent extraction after addition of standard), then the measurement system factor in the standard addition blend can be equated to the same factor in the sample blend, that is FSAB ) FX and thus

FX )

nQsab nX,tot - nX,sab

(15)

The standard deviation in this method of measurement of FX includes an estimate of method precision since multiple sample and calibration blends must be prepared for the standard addition experiment. Where possible, the method precision component should be removed from this standard deviation when it is used as an estimate of the uncertainty in the value of FX to avoid double counting of uncertainty components. In this laboratory, it has been regular practice to evaluate many of the proportionality factors that constitute FX and to include these factors in the measurement equation for the purposes of estimating measurement uncertainty.27 However, since the evaluation of FX can be readily accomplished by a standard addition experiment, evaluation of constituent proportionality factors may only be necessary when the uncertainty in evaluating FX leads to a total uncertainty that is not fit for purpose or when the value of FX is significantly different from unity and indicates an unreliable measuring system. In these cases, improvements in measurement accuracy may only be achieved when the constituent proportionality factors are specifically evaluated. An example of estimating FX from a standard addition experiment is provided in the Supporting Information from work reported in ref 27. The alternative approach of measuring the constituent proportionality factors can be employed when a blank material identical to the test material is available. This enables a series of experiments to be performed to measure the different proportionality factors by adding the reference standard and internal standard to the calibration blend at different stages of chemical processing, thus eliminating one or the other of the proportionality factors; the value of these factors will be obtained by any change in wX subsequently measured provided that the additions themselves do not contribute significant differences to the blend matrixes. The values for the constituent proportionality factors may then be used in eq 8 to calculate FX. In this way it may also be possible to measure the effect of equilibration of target analyte and internal standard with the test matrix when blank material is available, and this would allow estimation of that source of bias that is not covered by the equations in this paper. CONCLUSIONS Equations 9 and 10 provide a complete description of the measurement process for organic molecules using appropriate internal standards after equilibration of internal standard with the sample matrix. The derivation of these equations retained the proportionality factors that describe separately the yields of both the analyte and internal standard molecules at each stage in the (27) Mackay, L. G.; Burke, D.; Liu, F.-H.; Sousou, N.; Vamathevan, V. V.; Cuthbertson, J.; Mussell, C.; Myors, R. B. Accredit. Qual. Assur. 2007, 12, 475–482.

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measurement procedure, and thus they provide an accurate model for estimating measurement uncertainty. The ratio of all proportionality factors was combined into a single term, FX, which represents the bias in the entire measurement process. There are many scenarios that may lead to zero bias in the measurement process, i.e., FX ) 1, because only the quotients of the constituent proportionality factors of eq 8 must cancel as shown in eqs 11 and 12. One of the most common scenarios leading to bias is when the difference between test and calibration solutions results in different proportionality factors for either analyte or internal standard molecules. A theoretical argument has been presented that a standard addition procedure would enable ready evaluation of FX and could be used in a simple top down approach to evaluating the uncertainty of many organic measuring systems. When the value of FX is considerably different from 1 or when the uncertainty in evaluating FX is not fit for purpose, the proportionality factors for intermediate analytical steps should be evaluated and may then be used to identify which constituent analytical process may need improvement. The measurement uncertainty obtained by applying the procedures defined in the Guide to the Expression of Uncertainty in Measurement (GUM) to eq 9 or eq 10 is by definition the uncertainty of the entire measurement process after equilibration

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of internal standard with sample.2 When each term in the equation including the measurement system factor FX is measured traceably and their uncertainties are properly estimated, the wX value calculated from the equation is traceable to the SI. Thus this derivation of the measurement equation for organic molecules provides a means for establishing traceability to the SI and shows that it may be possible to make highly accurate measurements using an internal standard that does not contain stable isotopes provided FX is unity. ACKNOWLEDGMENT The authors thank Professors Paul De Bie`vre and Brynn Hibbert for their encouragement and helpful reviews of the draft manuscript. SUPPORTING INFORMATION AVAILABLE Complete derivations for eqs 3, 6, 9, 10, and 13 and an example of the estimation of FX from a standard addition experiment. This materialisavailablefreeofchargeviatheInternetathttp://pubs.acs.org.

Received for review February 7, 2008. Accepted April 16, 2008. AC800270U