John P. Jasper is the chief scientific officer at Molecular Isotope Technologies, LLC, 8 Old Oak Lane, Niantic, CT 06357, jpjasper@ molecularisotopes.com

John P. Jasper

Submitted: August 10, 2015
	Accepted: October 2, 2015

Molecular isotopic engineering (MIE) is the directed stable-isotopic synthesis of chemical products for product authentication and security, as well as intellectual property protection. In tests involving naproxen manufacturing, results showed a generally excellent correspondence between observed and predicted stable-isotopic results (δ

O, and δD) for directed synthesis of a racemic mixture from its immediate precursors. The observed carbon-isotopic results can be readily explained by the laws of mass balance and isotope mass balance. The oxygen and hydrogen isotopic results, however, require additional assessment of the effects of oxygen and hydrogen exchange.

A previous study with FDA’s Department of Pharmaceutical Analysis showed that individual manufacturers of naproxen could be readily differentiated by their stable-isotopic provenance (δ

O, and δD). Results from two out of three of the samples in the latest study corresponded well to previous results, suggesting that MIE can be readily used without altering manufacturing processes other than isotopically selecting and/or monitoring reactants and products.

	Product authentication, product security, and intellectual property (IP) protection remain major concerns in the biopharmaceutical industry (1-6). The directed stable-isotopic synthesis of chemical products allows the stable-isotopic composition of materials to be predetermined to address these challenges (5). Product authentication is typically performed at three levels: overt, covert, and forensic (7). Molecular isotopic engineering (MIE) takes a forensic or analytical approach, analyzing stable isotopes in these products. Naturally abundant stable isotopes are natural tracers that occur in all matter (8). 

	Early work with FDA on the product characterization of naproxen revealed the manufacturer-level isotopic provenance of this small analgesic molecule (1,9), which was referred to as “the manufacturer’s fingerprint.” This isotopic provenance represented the convergence of the effects of the stable-isotopic compositions of starting materials and isotopic effects of the synthetic process. Rather than merely accepting the random effects of variable sourcing and synthetic process on the stable-isotopic compositions of products, MIE takes a proactive approach, purposefully directing the stable-isotopic composition of biopharmaceutical products. 

	The main rationale for MIE is to design the isotopic ranges of products for product identification and security, and also for IP considerations. As an example of MIE, the isotopic products of a later step of naproxen synthesis (

 2-Bromo-6-Methoxynaphthalene +
	Bromopropionate → ±Naproxen

	Pre-selection of three different stable-isotopic compositions of the starting material, 2-Bromo-6-methoxynaphthalene, yielded racemic naproxen products of three discrete stable-isotopic ranges. The resulting MIE naproxen is different from the substitution that would take place in deuterium labeling, in which a different isotope is substituted in a single position (10). The authors’ directed isotopic synthesis is just one example of how MIE can be used to predetermine the discrete isotopic ranges of biopharmaceutical products. 

	In principle, the MIE approach should be readily adapted to existing biopharmaceutical manufacturing operations. The only adjustment to an existing manufacturing process would be the use of starting materials or synthetic intermediates of premeasured stable-isotopic compositions. The manufacturing apparatus would remain unchanged. This approach could have broad application in securing drug identity and provenance from manufacturing plant to consumer. Molecular Isotope Technologies has developed four patented or patent-pending generations of stable-isotopic methods and technologies (5):

		Product  characterization (for both small molecules and biologics) (6, 9, 10)

		Process characterization (notably, process patent protection) (12)

		Molecular isotopic engineering or MIE (5).


	Three groups of samples were analyzed to examine the natural-abundance stable-isotopic compositions for naproxen synthesis, including the two reactants (2-bromo-6-methoxynaphalene and bromopropionic acid) and the end product (racemic naproxen).

 A late-stage synthesis of naproxen was performed by IsoSciences, LLC (King of Prussia, PA) as shown in 

 Eight samples of 2-bromo-6-methoxy-naphthalene were collected from a worldwide selection of suppliers (

). A Grignard reagent (bromopropionic acid) was acquired from Sigma-Aldrich (St. Louis, MO). Three of the 2-bromo-6-methoxynaphthalene samples, from CombiBlocks, Matrix, and Aesar, were selected for this study based on their differing 

C compositions: one high, one low, and one intermediate. Samples of racemic naproxen were synthesized from the three different starting materials.

 2-Bromo-6-methoxynaphthalene was dissolved in a round bottom flask of anhydrous toluene and anhydrous tetrahydrofuran (THF), with heating and degassing. The bromonaphthalene solution was added dropwise to the magnesium via an addition funnel. The reaction was allowed to cool to room temperature under nitrogen.

Magnesium salt formation on bromopropionic acid.

 Alpha-bromopropionic acid was dissolved in anhydrous THF. The solution was cooled to -15 °C in a dry ice/acetone bath and methyl magnesium chloride was added via syringe while maintaining the temperature below 0 °C. The temperature was kept below 0 °C until the solution was used.

 The Grignard solution was transferred into a two-neck round bottom flask with a thermometer and a septum via cannula and was then degassed. The solution was cooled in an ice bath, and the mixed magnesium halide complex was added via cannula, maintaining the temperature at 15-20 °C. The reaction was stopped after two hours. The resulting solution was then cooled in an ice bath, and a solution of 10 mL of 12N HCl in 75 mL of water was added. After stirring for five minutes, the biphasic mixture was filtered, and the filter was washed with 25 mL of toluene and 25 mL of water.

	The layers were separated, and the organic phase was extracted with 2 x 75 mL of 10% NaOH solution. The basic extracts were combined, washed with toluene (~25 mL) and filtered. To the filtrate was added 7.5 mL of methanol and 6 mL of toluene. This mixture was then acidified with concentrated HCl to a pH of 5. The resulting slurry was heated to reflux for one hour and allowed to cool overnight with stirring. The product was washed with 10 mL of water, 2 x 2 mL of toluene, and 2 x 2 mL of hexane, and dried to give an off-white solid. After drying under high vacuum for 48 hours, there was 5.8828 g (53% yield).

O, and δD) were made of each of the components of this study. In the starting material survey study, three stable-isotope ratios were measured on each of the eight samples of 2-bromo-6-methoxynaphthalene in triplicate analysis (i.e., 8 batches × 3 isotope ratios × 3 replications = 72 measurements) to assess analytical precision (13).

	Nine analogous isotopic measurements were made on the bromopropionic acid reagent (3 isotope ratios × 3 replications). Triplicate analyses were also performed for each of the three isotope ratios of the five batches of naproxen synthesized here, yielding 45 isotope measurements. Thus, a total of 126 stable-isotopic measurements of the samples were performed in this study.

 As detailed elsewhere (10), carbon (δ13C) and oxygen (δ18O) isotopic analyses were performed respectively on:

		A Carlo Erba 1108 Elemental Analyzer, interfaced using a Conflo III interface to a Thermo Scientific Delta V isotope ratio mass spectrometer (EA/IRMS)

		A Finnigan Thermal Conversion/Elemental Analyzer (TCEA) interfaced to Finnigan Delta V Plus isotope-ratio mass spectrometer (thus, a TCEA/IRMS).

 Hydrogen that is not bound to carbon in a molecule may readily exchange with other hydrogen atoms present in ambient moisture (i.e., H2O). This exchange happens even at room temperature and is difficult to control. To generate precise δD values for a given compound, the exchangeable hydrogen portions must be accounted for or controlled.

	Samples were weighed into individual 3.5 mm x 5 mm silver “boats” and  equilibrated with reference waters of known δD values to calculate the amount of exchangeable hydrogen in the sample (14). The samples were allowed to equilibrate for two hours at 50° C inside a container with an aliquot of calibrated reference water.

	The equilibration process was repeated twice on separate aliquots of sample using reference water samples that have a difference of -233‰ in δD value. After equilibration, the samples were dried overnight in a vacuum oven at 50° C, then immediately transferred to the Costech Zero Blank autosampler of a Finnigan MAT Thermal Conversion Elemental Analyzer (TC/EA) and evacuated to remove ambient moisture.

	Several reference standards accompanied each batch of samples, including a polyethylene standard that has no exchangeable hydrogen and is therefore unaffected by ambient moisture. In the TC/EA, the samples were reduced at 1400 °C in the presence of glassy carbon. The resulting hydrogen was then separated from other gases via a gas chromatograph and transmitted into an isotope ratio mass spectrometer (IRMS) for isotopic analysis to obtain the δD values. Using post-analysis calculations (15), the δD value of the non-exchangeable hydrogen could be quantified from the equilibrated sample data sets.

 Carbon (and all other) isotopic results are expressed in δ values (‰ = parts per thousand differences from international standards), as expressed in 

C ratio of an International Atomic Energy Authority standard  (IAEA, known as “VPDB” [Vienna Pee Dee Belemnite], whose 13C/12C ratio has been defined as the official zero point of the carbon-isotopic scale).

O and D/H values are given relative to IAEA Vienna Standard Mean Ocean Water (VSMOW) standard, which gives the zero points of the oxygen and hydrogen-isotopic scales.

 Because all measurements in this study were made in triplicate, the averages and 1σ-standard deviations are reported here for the observed isotopic data in 

 Two sigma standard deviations are shown for the deviations from mass balance and isotope mass balance.

	Characteristic one sigma (1σ) standard deviations for the isotope  measurements reported in this study were: δ13C (±0.03‰), δ18O (±0.09‰), and δD (±1.0‰) as shown in 

Stable isotopic composition of reactants.

O, and δD compositions of eight samples of the reactant 2-bromo-6-methoxy-naphthalene measured in triplicate are shown in 

, and those of bromopropionic acid are shown in 

The stable isotopic records of naproxen synthesis.

 The directed stable-isotopic synthesis of naproxen is discussed in two parts:  the mass-balance/isotope-mass balance (MB/IMB) component, and then the deviations (if any significant) from MB/IMB. Because these results are compared to the MB/IMB frame of reference, that topic is briefly described.

Comparing observed versus predicted isotopic values.

 The laws of mass balance and isotope mass balance (14) provide a primary frame of reference for assessing the results of the naproxen isotopic synthesis. The basic mathematics of MB/IMB (15) is summarized in 

	Isotopic Fractionation (one component in excess):

 = number of moles of compounds A, B, and C;

 = isotopic  compositions of compounds A, B, and C; and

 The observed carbon isotopic results and the predicted MB/IMB results for the naproxen synthesis are shown in 

 Observed and predicted values align well and will be further examined below.

 The observed oxygen isotopic results and the predicted MB/IMB results for the naproxen synthesis are shown in 

 Observed and predicted values deviate slightly from each other and will be further examined below.

 Observed isotopic results and predicted MB/IMB results for the naproxen synthesis are shown in 

 In one case, observed and predicted values deviate notably from each other and will be further examined below.

Mass balance and isotope mass balance: correspondence and deviations.

 The correspondence to and deviations from MB/IMB 

 are examined here to account for the three isotope ratios examined.

C values versus the predicted values on the basis of MB/IMB is shown in 

 The excellent correspondence indicates that the carbon-isotopic synthesis is consistent with the MB/IMB model.

Oxygen and hydrogen: fractionation due to equilibration with water.

 The O and H data, however, both show significant differences between observed and predicted values. In both cases, the observed values are isotopically enriched relative to the predictions. The oxygen data argue against direct incorporation of water into the samples, because local water should have a δ

	However, if the water is incorporated through an equilibrium isotope effect, then one might suggest that 18O would favor being bound to the carboxyl position (the more stable bonding environment), whereas 16O would favor remaining in the water phase.

	With that, the following equilibrium is plausible, and the forward direction is favored, as shown in 

	Thus, the isotope mass balance equation is solved by considering the O from the original methoxynapthalene, plus the Grignard reagent contribution of one unaltered O and one carboxylic acid O that has been equilibrated with water. For accounting purposes, the three oxygens are A, B, and C - the methoxynapthalene (A), the carbonyl (B), and the exchangeable OH (C), as shown in 

 + 1000) specifically for the carboxyl-OH group.  The best fit solution indicates that α = 1.018.

Comparison of results with literature data


	From the outset, the authors are comparing stable-isotopic data for racemic naproxen with the only other naproxen-isotope data (viz., S-naproxen) (10) that, to the authors’ knowledge, exists. Assuming that the isotopic fractionation of naproxen is small between the racemic mixture and the purified enantiomer (S), the authors make the present comparison but acknowledge that this assumption must be tested in ongoing research.

	The carbon- and oxygen-isotopic results of the present syntheses of naproxen are superimposed on pre-existing data (6) in 

 Although there was no intent to reproduce the pre-existing naproxen-isotope data, the present naproxen data fall within the range of the pre-existing data.

	In addition, two of the present naproxen values (Matrix Scientific and Alfa Aesar) lie within approximately 2σ of pre-existing results (namely, “India Manufacturer B” and “India Manufacturer A,” respectively). By contrast, the Combi-Blocks-sourced naproxen does not lie near any of the pre-existing clusters of naproxen data, plausibly because no such naproxen was obtained for the earlier study.

Product identification, security, and IP protection


	MIE allows unprecedented stable-isotopic definition of chemical products from isotopically-known starting materials. In fact, its use in naproxen synthesis permits the precision of compound production to within a few tenths of a permil for carbon and oxygen and approximately one permil for hydrogen when the ranges of starting materials may span tens of a permil. Such narrow delimitation of products’ isotopic fingerprint decreases their vulnerability to various IP infringements. MIE thus allows for the design and synthesis of drug molecules with discrete stable isotopic composition for a wide range of stable isotopes.

	Starting with a small survey suite of readily-available reactants, various chemical products can be produced via existing chemical processes. The only difference from pre-existing processes is that the stable-isotopic compositions of the reactants and products are now measured either offline or online.

	The major result of MIE is to generate chemical products of narrowly-delimited isotopic ranges as compared to the seemingly random distribution of typically-produced products in which no explicit effort is made to delimit their compositions. In other words, MIE allows for the design of a unique and characteristic isotopic array or internal “bar code” or “fingerprint” for a drug molecule. The potential implications for product authentication, supply chain custody, security, and anticounterfeiting are enormous.

	Furthermore, because MIE-designed drug molecules are essentially new chemical entities, MIE has some potentially interesting IP implications. Consider for example, a conventionally-synthesized, but isotopically-labeled drug molecule, where the resulting product is a new entity that was not previously found in nature. By using MIE, one can go beyond merely positionally-labeling a drug molecule with an isotope to design new molecules (9), rationally and selectively, with far more complex (i.e., multipositional), and, thus, highly specific isotopic fingerprints.


	Consistent  with principles of mass balance and isotope mass balance, directed stable-isotopic synthesis (or, Molecular Isotopic Engineering) permitted the production of racemic naproxen of pre-determined isotopic compositions (C,O,H) for reasons of product authentication, security, and, perhaps, IP protection. A small, worldwide survey of a key naproxen intermediate (2-bromo-6-methoxynaphthalene) gave a wide range of C, O, and H isotopic values for the present starting material. 

	Mass balance and isotope mass balance (MB/IMB) account for the carbon-isotopic relationship between the reactants and product naproxen very well. In addition to MB/IMB considerations, the equilibration between O and H and naproxen is readily accounted for by equilibrium isotopic exchange with reaction water.

	In general, the use of existing synthetic manufacturing methods indicate that MIE should generate products in predetermined isotopic ranges for product authentication and security, and may present a new mode of pharmaceutical patenting: “isotopic composition of matter.”

AUTHORS’ NOTE: Pending patent applications are on file with respect to the technology, which has already been granted US Patents No. 7.323,341 and 8,367,414 B2.

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	When referring to this article, please cite it as J. Jasper et al., “Evaluating Practical Uses of Molecular Isotopic Engineering," 

Articles

The authors' directed isotopic synthesis is one example of how molecular isotopic engineering can be used to predetermine the discrete isotopic ranges of biopharmaceutical products.

Evaluating Practical Uses of Molecular Isotopic Engineering

Measurements of the abundances of naturally occurring stable isotopes in pharmaceutical materials can be used to quantitatively characterize the sources of the products and the synthetic processes used to produce them (1, 2). Since 2001, industry has had increased interest in the use of stable-isotopic analyses in industrial pharmacology. The results of reconnaissance studies have led to detailed examinations of the mechanisms that determine the observed isotopic compositions. The initial, observational work started with a field study to differentiate individual batches of drug products (3). This work led to an investigation of stable isotopic analyses as monitors of process consistency during drug manufacturing. In that study, various batches of two over-the-counter analgesic drugs were analyzed for isotopic consistency. Batch-to-batch variations in stable-isotopic compositions indicated that either the sources of the starting materials or aspects of the manufacturing process had changed.  

To follow up, blind isotopic studies were initiated by the Division of Pharmaceutical Analysis of the US Food and Drug Administration (St. Louis, MO) to determine whether isotopic analyses of drugs could be used to identify the manufacturing source. In the first of two studies, researchers demonstrated that stable-isotopic analyses could distinguish among individual manufacturers and batches of four active pharmaceutical ingredients (APIs) produced by nine manufacturers (4). A second study examined the ability of stable-isotopic analyses to differentiate manufacturer sources of an API as it was transported across national borders. All six manufacturers of the API (naproxen) were recognized successfully (5).  

The results demonstrate the utility of stable-isotopic analysis for monitoring and identifying pharmaceutical products and manufacturers, thus providing a new tool for product security and drug authentication. It also can be used to monitor process consistency (

the changes or modifications that occur during the various synthetic steps in a pharmaceutical drug-manufacturing process).  

Another area that is of interest for stable-isotopic analysis is its use as a forensic tool in cases of process patent protection. Two sets of variables are pertinent: the stable-isotopic compositions of the starting materials and synthetic intermediates, and the isotopic fractionations that occur during production. Understanding the extent of isotopic fractionation that occurs at each step in a synthetic route can provide valuable insight into which process was used.  

In this article, the authors focus on means of constraining the major variables (

 the precursors and synthetic pathways) that determine the isotopic compositions of pharmaceutical materials. Ideally, isotopic compositions of pharmaceutical products will be predictable, and the synthetic processes used will be recognizable. In a broader sense, these studies of process authentication bear on process-patent protection and on process analytical chemistry, including process consistency. 

To fully elucidate the isotopic fractionations that occur between reaction steps, it is ideal to have at least 0.1 mg of all intermediates and products for each stable-isotope ratio being measured.  

N ratios, ~0.1-mg solid samples were weighed and placed into tin cups that were crimped tightly closed. The analytical system used was an elemental analyzer–isotope-ratio mass spectrometer (EA-IRMS), and consisted of an elemental analyzer (EA, Carlo Erba 1108, Carlo Erba, Italy), a continuous-flow interface (Conflo II, Thermo Fisher Scientific, Bremen, Germany) and an isotope-ratio mass spectrometer (Finnigan MAT Delta Plus XL, Thermo Electron, Waltham, MA). The oxidation furnace of the EA was operated at 1020 °C, the reduction furnace temperature was at 650 °C, and the chromatographic column was heated at 60 °C. Isotope ratios are reported in terms of δ

C values relative to the international Vienna Pee Dee Belemnite (VPDB) standard and δ

N values relative to the international Air, both of which are IAEA standards. The δ notation is explained later in this article. 

Individual solid samples of 0.08–0.10 mg were mixed with 0.5 mg V

, weighed into tin cups, and sealed. The analysis was accomplished using a second EA–IRMS, that was composed of an elemental analyzer (Eurovector, model 3000r), an interface (Dilutor, Micromass Ltd.), and an isotope-ratio mass spectrometer (Isoprime, Micromass Ltd.). The oxidation furnace of the EA was operated at 1030 °C. The gas flows were 10 mL O

/min and 150 mL He/min. Isotope ratios are reported in terms of δ

S values relative to the international Canyon Diablo troilite (CDT) standard. 

Before analysis, solid samples were equilibrated for several days at ambient temperature with water vapor by exposure to the laboratory atmosphere to fully exchange labile H/D sites and any water present in a hydrated form (6–9). Following equilibration, ~0.2 mg-samples were weighed into silver boats which were then crimped tightly. For solid samples, researchers used a thermal conversion–elemental analyzer (TCEA, Finnigan, Thermo) interfaced to an isotope-ratio mass spectrometer (TCEA–IRMS, Finnigan Delta Plus XL, Thermo). The same system was used for liquid samples, but a direct liquid-injection port was fitted in place of the autosampler. The TCEA pyrolyzes the samples at 1350 °C to quantitatively generate H

 and CO, which are separated chromatographically at 85 °C. Isotope ratios are reported in terms of δD values and δ

O values relative to the international Vienna Standard Mean Ocean Water (VSMOW) standard. 

Terminology for isotopic relationships between precursors and products. 

 Isotopic calculations are based on only two systems of equations. The first system uses mass balances and the second involves integrated forms of rate equations that pertain only to kinetically controlled isotopic fractionations. Equations describing mass balances are exact when cast in terms of fractional abundances [

C)]. In contrast, the assessments of differential rates are based on isotope ratios (

C). When these systems must be blended, either approximations or equations with multiple terms are used. Additional information is available in reference 10. 

The relevant isotopic parameters are stoichiometry (

), isotopic abundance (δ), the magnitude of the isotopic effect (ε), and a variable related to the conversion of reactants to products (

). The following is an explanation of these terms. 

, represents the stoichiometry of the reaction, specifically the number of atoms of a given element (

carbon) in a molecule involved in the reaction			

 A measure of isotopic abundance, δ, is usually reported as the difference in parts per thousand, or permil (‰), from an international standard. The δ can be negative or positive, depending on whether the sample is enriched or depleted in the heavy isotope relative to the standard. In the case of carbon, for example, the difference is calculated as: .

C ratio in the standard. Thus, δ is linearly proportional to the isotopic ratio in the sample. Standards are available from the International Atomic Energy Authority and a standard for each isotope is used to determine the zero point of an abundance scale for that isotope. Standards include average seawater for H and O, calcium carbonate for C, air for N, and a meteorite for S. When the sample is depleted in the heavy isotope relative to the standard, δ is negative. When the sample is enriched, it has a positive value. If it has the same isotopic abundance, then δ = 0 (11). 

  The ε value is a measure of the magnitude of an isotope effect. Its value depends on details of the reaction and on the relative mass difference between isotopes. Effects are largest for D 

 H and smaller for heavier elements. In general, the values of ε are specific to individual positions within the molecules involved. They are largest at the reaction site, much smaller at neighboring positions, and usually not measurable elsewhere. Like δ, ε relates to the isotopic difference between two materials (

reactant and product) and is usually expressed in permil. For kinetic isotope effects in the system used in this article, ε = –10‰, which means that a reaction site bearing the heavy isotope reacts 10 parts per thousand (or 1%) more slowly than a site bearing a light isotope. For equilibrium isotope effects, ε

 = 15‰ would mean that, at equilibrium, A was enriched in the heavy isotope by 15 parts per thousand relative to B. In this instance, A and B refer to specific atomic positions that can be related by a chemical equilibrium. 

Variable related to the fractional conversion of reactants to products (f).

 is a measure of the reaction's progress. It is the most important variable governing fractionations caused by isotope effects. Its value ranges from 1 to 0 and depends on factors such as temperature, pressure, or the availability of reactants. In equilibria (A ⇔ B), 

 indicates the position of the equilibrium, with 

 = 1 indicating a complete conversion to B and, at any position, 

 indicates the portion of reactant X which remains unconsumed, with 

 → 0 as the reaction proceeds to completion.  

The precision of isotopic analyses is typically calculated by two methods. Pooled standard deviations of raw data are typically computed from sets of duplicate or triplicate measurements (12). From those pooled standard deviations, standard deviations of mean values pertaining to specific substances are calculated. More specifically, the standard deviation of a mean value is the pooled standard deviation divided by 

 is the number of measurements performed on a given sample (13). For carbon, nitrogen, oxygen, and sulfur, the resulting 95% confidence intervals for a result are typically in the range of ±0.1 to ±0.4‰. For hydrogen, the 95% confidence interval is typically ±3‰. 

The precise quantitation of stable-isotopic compositions in pharmaceutical intermediates and products requires both mass balance and isotopic fractionation equations that are applicable to both single and multistep reaction sequences. We start from the most basic requirement of mass balance and then consider isotopic fractionations in a single reaction. Finally, we discuss applications to the protection of process patents. 			

 For A + B → C, in which reactants A and B are quantitatively converted to product C, two mass balances can be written: 

 are molar amounts of carbon (or any other element) in A, B, and C and the isotopic compositions of that carbon (or any other element) in A, B, and C are given by δ

 carbon in = carbon out) whereas Equation 2 is an isotopic mass balance (

C out). Under the conditions postulated (quantitative conversion), the isotopic composition of C can be computed from those of A and B (10, 11). 

 For isotopic fractionations, calculations must take into account factors such as reaction completeness and isotope effects. These effects will cause the isotopic composition of C to differ from that computed using the mass balance equation and assuming quantitative conversion of reactants to products. 			

To provide a concrete example, assume that A is present in excess whereas B, the limiting reactant, is quantitatively converted to product. In that case: 

 represent the numbers of carbon atoms (or any other element of interest) in A, B, and C. Because A is not quantitatively converted to product, the isotopic compositions of the A-derived positions in C can differ from those in the initial reactant. In this case, the isotopic offset is expressed as ΔA. As Figure 1 shows, its value depends on the isotope effects and on the fraction of A that remains unconsumed. If the reaction conditions, particularly the magnitude of the excess of A, are consistent, ΔA will be constant. Because the 

 values are known exactly, ΔA can be determined from Equation 3 after isotopic analysis of the reactants and product (

 do not affect the values of ΔA. Accordingly, once ΔA is known for a given reaction and a set of conditions, it is necessary only to know two of the δ values to compute the third. Thus, for example, when ΔA, δ

, are known, the isotopic value of the product (δ

				If neither A nor B is consumed completely during the course of the reaction, and if the rate of the chemical reaction (or position of the chemical equilibrium) is sensitive to isotopic substitution on both reactants, it is necessary to consider values of ΔA and ΔB: 

  If reaction conditions cannot be manipulated so that 

) can be independently driven to completion (

 zero), it will be possible to determine only the sum, 

. From theoretical considerations (14), Δ

 can be evaluated separately for all values of 

Of course, isotopic fractionations such as those previously described accumulate during the different steps of a multi-step synthesis scheme. They can, however, be individually and systematically differentiated, not only for multiple reactants but also for multiple isotopes. To provide an example, consider carbon-isotopic fractionations in a hypothetical four-step sequence: 

The authors describe the methods by which precise analyses of stable-isotopic abundances can be used in security and forensic applications for pharmaceutical materials. These methods include product and process authentication of raw materials, pharmaceutical intermediates, drug substances, formulated drug products, and synthetic pathways. Collectively, these methods can be used to investigate and mitigate patent infringement. In the future, more complete examples will be presented containing full isotopic results and the application of the methods described in this article.

Process Patent Protection: Characterizing Synthetic Pathways by Stable-Isotopic Measurements

The use of stable isotopic analyses to characterize drug products before they leave the manufacturer is explored as a means to identify counterfeit products.