Chemical Kinetics of Radiolabelling Reactions.

The application of chemical kinetics is one of the most powerful and versatile tools for investigating reaction mechanisms in complex mixtures. Kinetic studies are commonplace in traditional synthetic chemistry but are seldom used in radiopharmaceutical sciences. When deriving standard reaction rate laws, the focus is normally placed on calculating the chemical concentration of different species over time. In radiopharmaceutical synthesis, the desired product is one of the radioactive components of the mixture. Reaction conditions are optimised to obtain the radioactive product in the highest activity yield. When short-lived radionuclides are used, radioactive decay during the reaction window means that the maximum activity yield does not necessarily coincide with the chemical or decay-corrected radiochemical yields. To account for this difference in the kinetic models, it is shown how standard integrated rate laws can be modified to incorporate the contribution from radioactive decay. An example is then presented to show how radiochemical kinetics can be used to model complex systems, like [18 F]FDG radiosynthesis, that involve parallel or competing reactions at the different chemical scales of the radionuclide and substrate. Increased knowledge of reaction rates, and a more wide-spread application of radiochemical kinetics, can facilitate the development of new radiolabelling reactions. Accurate identification of maximum activity yields using kinetic models also has the potential to improve the optimisation and radiochemical efficiency of all current and future radiopharmaceutical syntheses.


Introduction
Chemical kinetics is the study of the rate (or velocity) of chemical reactions during which reactants are converted to products along a reaction coordinate. (1,2) Kinetic experiments usually involve measuring the concentration of a species versus time. Other macroscopic or 'classical' variables that influence reaction rates include changes in temperature, pressure, solvent composition, ionic strength, heat and entropy etc. When empirical data on reaction kinetics are combined with thermodynamics, molecular spectroscopy and transition state theory, it allows complex chemical processes to be deconvoluted into the underlying elementary steps that define our concept of reaction mechanisms. Methods derived from statistical thermodynamics can also be used to bridge the gap between the macroscopic and microscopic (or quantum chemical) world. Quantum descriptions allow reaction mechanisms to be explained at the atomic or molecular level. Kinetic investigations that integrate classical and quantum theories are referred to as the study of molecular reaction dynamics.
Kinetic experiments have access to a rich pool of analytical techniques. Traditional methods for monitoring the concentration of a species over time include time-resolved electronic absorption, infrared, NMR or fluorescence-emission spectroscopy, electrochemistry, potentiometry, and radioactive probes coupled with chromatography or chemical separation. The first use of radioactivity to measure kinetics can be traced back to the work of George de Hevesy and the invention of the radiotracer principle in 1923. Later, he received the Nobel Prize in Chemistry in 1943 for using radioactivity to study various chemical processes including transport and metabolism in plants and animals. (3)(4)(5) Radiotracers are ideally suited for studying kinetics for several reasons. (6) First, accurate detection and quantification of radioactivity is possible even when species are present at extremely low concentrations (sub-femtomolar). Second, this high sensitivity means that radiotracers can be used at concentrations that do not perturb the original kinetic (or thermodynamic) process under investigation. Finally, many different radionuclides are available which means that it is possible to select or synthesise radiotracers that are chemically identical to the reacting species. In this respect, organic compounds labelled with 3 H(t1/2 = 12.32 y), 14 C(t1/2 = 5700 y), 32 P(t1/2 = 14.268 d), and 35 S(t1/2 = 87.37 d) are frequently used in drug development to measure target binding properties in vitro, as well as absorption, distribution metabolism and excretion in vivo. Biological assays that use radiotracers include, for example, the measurement of enzyme-catalysed reactions using 32 P-or 35 S-labelled adenosine triphosphate. (7) The use of radiopharmaceuticals for diagnostic imaging or molecularly target radionuclide therapy is a core discipline in modern Nuclear Medicine. In contrast to the use of radiotracers in chemical or enzymatic assays, radiopharmaceutical synthesis usually involves working with radionuclides that have comparatively short half-lives. (8,9) For instance, imaging agents for positron emission tomography (PET) often utilise 11 C(t1/2 = 20.364 min.), 13 N(t1/2 = 9.965 min.), 18 F(t1/2 = 109.7 min.), 64 Cu(t1/2 = 12.701 h), 68 Ga(t1/2 = 67.71 min.), and 89 Zr(t1/2 = 78.41 h). Here, the radioactive product is the desired species. The challenge is that the entire synthetic process must be completed on a time scale that gives sufficient activity for further use in imaging or therapy. Radiopharmaceutical production can be divided into two time-critical steps: i) radiochemical synthesis, and ii) quality control and release of the isolated product. Optimisation of quality control procedures is beyond the scope of this article. Due to the time constraints, efficient radiopharmaceutical synthesis is strongly dependent on reaction rates. Radiolabelling reactions are frequently 'optimised' empirically to give the highest decay-corrected radiochemical yield (RCY) in the shortest time, but actual reaction rates (and rate constants) are rarely determined.
The application of advanced chemical kinetics in radiopharmaceutical science has great potential to improve and streamline the process of optimising new (and existing) radiolabelling reactions. The problem is that kinetics models that focus on the activity of a radioactive component of the mixture as the primary variable are absent from standard texts. (1,2,(10)(11)(12)(13)(14) In this perspective, the theory of chemical kinetics is extended to account for radioactive decay during the reaction/process. Analytical solutions and graphical plots are used to illustrate that maximum chemical and radiochemical yields do not coincide with maximum activity yield in first-and second-order radiochemical reactions with short-lived radionuclides. By extension, it is shown that the decay-corrected RCY is not the most appropriate variable to use when optimising these radiolabelling reactions. Finally, examples using data from the clinical production of four PET radiopharmaceuticals: [ 11 C]raclopride,(15-17) 2-deoxy-2-[ 18 F]fluoro-D-glucose ([ 18 F]FDG), (18) [ 68 Ga]GaPSMA-11, (19) and the antibody-based agent [ 89 Zr]Zr-DFO-J591 are presented ( Figure   1). (20)(21)(22) The primary motivation of this work is to show how radiochemistry can be integrated more closely with traditional curricula of the chemical sciences for mutual benefit. Ultimately, a more wide-spread use of chemical kinetics would help radiochemists to identify new reactions that have potential in radiochemistry, and reduce the time spent optimising reactions when new precursors/substrates or reactions become available.

Activity yield as the key variable
Before examining the chemical kinetics of radiolabelling reactions, it is important to define the variables. Current nomenclature rules and parameter definitions given in the consensus report by Coenen et al. are used throughout.(23) By definition, the RCY is the amount of activity in the product expressed as a percentage (%) of the starting activity used in the considered process (for example, synthesis, separation, etc.), where both quantities refer to the same radionuclide, and are decay corrected to the same point in time before the calculation is performed. The RCY is an important value in radiochemical synthesis, but it is not the most useful variable when studying or optimising radiochemical kinetics where the goal is normally to obtain the maximum activity of the desired radioactive product.
Consider a quantitative reaction between a radioactive species (A), and a radiolabelling precursor/substrate (B), that gives a single isolated product (P) with no material losses (Equation 1). The second-order rate constant is given by k2 (M -1 s -1 ).
At the end of the reaction, the decay-corrected RCY is equal to the chemical yield (calculated using the radioactive species as the limiting reagent), and both equal 100%. In reactions where no appreciable radioactive decay occurs during the experiment, such as those involving long-lived radionuclides and/or very short reaction times, optimisation based on the chemical yield or decaycorrected RCYs is sufficient. However, in processes where radioactive decay in experimental window in non-negligible (as is the case for the synthesis of many PET radiotracers), decay-corrected RCYs are insufficient to describe the evolution of the activity associated with different radiochemical species over time.
The absolute and relative activity of a radioactive product at any time point is given by the non-decay-corrected activity yield. Activity yield is defined as, '…the overall activity of a radioactive product isolated from the production of a radiolabelled compound or radiopharmaceutical.'(23) Activity yields are normally expressed in units of activity (Bq) but values can also be normalised to the initial activity and reported as a percentage (%). This step facilitates comparison with chemical yields (%) and is convenient for kinetic analysis.
It is important to note that the chemical yield and the activity yield differ when radioactive decay is non-negligible during the process. Although high chemical yields are desirable, the maximum activity yield does not necessarily coincide with peak chemical conversion. For instance, once the reaction given by Equation 1 is complete, the chemical yield (and RCY) are 100% at all time points, but the activity yield is time-dependent and decreases by the factor, ) * + ,-. . Here, A0 is the initial activity in Bq, l / s -1 is the decay constant, and t is the time difference between the initial and final activity measurements. A chemically efficient, quantitative reaction is of no use in radiochemical synthesis if it proceeds too slowly and the activity has decayed before the product can be isolated/used. Therefore, optimisation of radiolabelling reactions requires identifying conditions that give the highest activity yield. Methods for calculating activity yields based on standard reaction kinetic schemes are presented in the following sections.

First-order and pseudo-first-order radiochemical reactions
The majority of radiolabelling reactions involve a bimolecular process (Equation 1). Exceptions include Szilard-Chalmers-type processes where a radioactive product is isolated after unimolecular degradation of an irradiated target material or radioactive parent species. Szilard-Chalmers reactions are not relevant for PET radiochemistry and are not considered further. In addition, the kinetic analysis of first-order reactions follows the same derivation as pseudo-first-order reactions, the latter of which is given below.   Table 1 [vide infra], and Supporting Information Table S1). In this example, the theoretical maximum activity yield, AP(max.), is ~82% and occurs at ~24 min., but the maximum chemical conversion (here, where both the chemical yield [P], and RCY, equal 100%) does not occur until ~45 min. Interestingly, when AP reaches a peak, the total activity in the sample is ~86% which means that the radiochemical purity (RCP) is only ~96%. If the reaction is allowed to proceed until complete chemical conversion, the radiochemical purity (RCP) approaches 100% asymptotically, but at the expense of decreasing the theoretical AP(max.) to ~75% at 45 min. These data illustrate the compromise that must be made between activity yields and RCP when optimising radiolabelling reactions.
When new reactions are developed, their potential use in radiochemistry is governed by many factors. These factors include (among others): the chemically efficiency (yield) of the process; accessibility, stability, solubility and reactivity of the radiolabelling precursors; regio-and stereo-selectivity of the products; and functional group tolerance / substrate scope etc. Reaction rates are also crucial in deciding if a new reaction will work in a radiochemical setting. Figure 2B shows the variation in AP versus time, as the overall reaction rate changes by variation in kobs from 0.02 s -1 to 0.0002 s -1 . The rates of inherently slow reactions can be increased by using more precursor, but solubility and competing side reactions often place limits on the maximum value of  Figure 2B suggest that for successful application in 18 Fradiochemistry, new reactions should aim have observed rate constants for nucleophilic SN2 substitution by fluoride anions at least ~0.001 M -1 s -1 (pink line). This lower boundary to the value of kobs ensures that the reaction reaches maximum AP within a reasonable reaction time of <60 min.
To maintain this limit on kobs when using lower amounts of radiolabelling precursor, the inherent reaction rate (given by the value of k2) in general must be >0.1 M -1 s -1 .
Similar analyses can be performed for different radionuclides. Pseudo-first-order kinetics calculated using the same starting conditions as employed in Figure 2A, but switching the radionuclide to 15 O, 13 N, 11  Values of the corresponding AP(max.) and RCP are given in Supporting Information Table S1. For long-lived radionuclides like 89 Zr, and to a lesser extent 64 Cu, activity yields mirror closely the chemical yield or RCY in the region before AP reaches a maximum. For example, when AP reaches a theoretical peak, corrections for activity yield constitute a difference of <1% for 89 Zr, and <4% for 64 Cu, versus the chemical yield or decay-corrected RCY. Therefore, for most radiolabelling processes that use radionuclides where the physical half-life is >10 times the reaction window, effective reaction optimisation can be accomplished on the basis of standard chemical yields or RCYs, without introducing significant loss of activity in the final product.
For short-lived radionuclides like 15 O, 13 N, 11 C, 18 F and 68 Ga, a pronounced difference occurs between AP(max.) and the chemical yield or RCY. Analysis indicates that under otherwise equivalent conditions, for 11 C-radiolabelling reaction to achieve a theoretical AP(max.) of ~90%, the observed reaction rate must increase by a factor of 10 compared with processes that involves 18 F. Hence, for applications in radiopharmaceutical synthesis, new 11 C-radiolabelling reactions should aim to have higher inherent reactions rates with an approximate value of k2 >1.0 M -1 s -1 . A 10-fold increase in rate is readily attainable in 11 C-and 13 N-radiochemistry, but the situation for 15 O is more challenging. The very short half-life of 15 O requires that (ideally) the reaction goes to completion in ~10 to ~15 s. This means that only very fast and efficient reactions, with k2 > ~10 2 M -1 s -1 can be considered for radiosynthesis with 15 O. To place these rate constants in context, strain-promoted, copper-free 'click' reactions are some of the fastest bioorthogonal reactions known, and are commonly used for imaging applications in vitro and in vivo using pretargeted strategies.(24-31) Reported second-order rate constants between azides or tetrazine derivatives, and various strain-promoted alkene or alkyne reagents range from ca. 0.1 to >10 3 M -1 s -1 ( Figure   4).

Second-order radiochemical reactions
It is important to note that for standard biomolecular reactions, the use of a second-order kinetic scheme is almost always correct, even when the reactions are performed under pseudo-first order  Table   1 [vide infra] and Supporting Information Table S2).
Most radiolabelling reactions with 68 Ga and 89 Zr should be modelled as a second-order process. In fact, it is likely that many of optimised radiolabelling reactions that use metal-based radionuclides follow second-order kinetics. The reason for this is simple -radiometal chemists usually perform test reactions to evaluate the molar activity of the radionuclide source, and the final product. On scaling up the reactions for production, starting conditions are usually chosen to ensure that the final RCYs and RCPs are close to 100%. This approach has the potential advantage of eliminating the need to perform lengthy purification steps to increase RCP before the product can be used. Although, even in processes that avoid purification, the presence of other chemical impurities must be tested before a product can be released.
In both preclinical and clinical work, 68 Ga-radiolabelling reactions are usually complete The striking feature is that radiometallation with 68 Ga 3+ and 89 Zr 4+ ions using these acyclic chelates occurs spontaneously at reaction rates that are ~10 2 to ~10 4 times faster than most reactions used in 11 C-and 18 F-radiochemistry. This is exemplified by the fact that radiometal ion chelation occurs under mild conditions (room temperature) whilst many 11  The reader should note that metal ion complexation reactions are not always rapid.
Thermodynamic barriers mean that efficient radiometallation of cyclic chelates like DOTA, and the cross-bridged variant CB-TE2A, with 64 Cu 2+ requires heating to around 60 -95 o C. (32) Nevertheless, the high thermodynamic driving forces for complexation, and the relative kinetic ease of displacing mono-or didentate ligands with multidentate chelates, mean that radiometal ions are often the first choice for accessing radiotracers (like many radiolabelled peptides) that cannot be radiolabelled efficiently with existing 11 C-or 18 F-chemistry.

Parallel or competing radiochemical reactions
The methods used to derive activity yields in pseudo-first-order and second-order radiolabelling reactions can be readily adapted to any kinetic scheme. As illustrated, some radiolabelling reactions, including 68 Ga-radiolabelling of peptides and 89 Zr-radiolabelling of antibodies can be considered as 'ideal' in that they go to completion rapidly with essentially no side-product formation. This is not the case for the synthesis of most 11   dihydro-4H-pyran-4-one from elimination. The remaining ~24% of the mass balance was associated with unidentified side-products. From these data, it can be deduced that the SN2substitution rate is ~8.4 times slower than the main chemical degradation pathway (elimination) and ~5.8 times slower than hydrolysis pathway that yields glucose. Using these data, combined with the reaction scheme given by Equations 12 and 13, and starting conditions derived from data on the clinical production (Table 1) Figure 7B. Species P1 is assigned to the desired product

Initial conditions in radiolabelling reactions: defining [A]0
The molar activity problem Previous sections introduced the theory of radiochemical kinetics but the accuracy of a kinetic model in describing a reacting system relies on knowledge of the starting conditions. In standard, non-radioactive chemical kinetics, the initial concentrations can be controlled easily, but in radiochemistry the situation is more complex. Table 1  The starting concentration of precursor [B]0 is usually well-known. The initial activity of species A is also measured routinely. Conversion of the activity of species A into the number of moles (or concentration) requires accurate determination of the molar activity, As / MBq nmol -1 .
For preclinical work with many radioactive metal ions, the apparent molar activity can be measured by using isotopic dilution assays. (34) In contrast, in most clinical processes that use 11 C or 18 F, the molar activity of the radionuclide source is not normally measured. Representative molar activities for 11 C and 18 F (    (34) j Note that the mole ratio and theoretical maximum yields for work with antibodies are calculated based on the number of moles of accessible chelates. This value is equal to the number of moles of antibody, n(mAb), multiplied by the experimentally measured number of accessible chelates per mAb.

Radiolabelling reaction rates
How fast are current radiopharmaceutical reactions? By using radiochemical kinetic models coupled with the starting conditions given in Table 1, it is possible to estimate how fast a reaction must be for it to be potentially useful in radiopharmaceutical synthesis. The estimated ranges of observed second-order rate constants (k2-values) derived from radiolabelling reactions using 11 C, 18  Radiometallation reactions depend on the nature of the chelate/complex but are usually fast with estimated rate constants that can exceed k2 >10 3 M -1 s -1 . These values mean that radiolabelling with metal radionuclides can often be performed in stoichiometric ratios leading to second-order kinetics.
When developing new reactions for potential use in radiochemistry, reaction rate is one of the critical factors that will determine if the process can deliver sufficient activity yield. If measured reaction rates fall within the range of existing radiolabelling technologies, it is likely that the new process can be adapted for use in radiopharmaceutical synthesis. Fast reaction rates are usually desirable because they facilitate higher radiochemical conversion and higher isolated activity yields. However, slower or more complex reactions in which the desired radiochemical species is not the main radioactive component of the mixture can also be used when low activity yields are acceptable. For instance, when a single imaging dose is required (ca. ~185 to ~740 MBq), activity yields of <1% may still provide enough product, so long as the reaction can be scaled for use with high initial activities.

Conclusions
Chemical kinetic theory has been extended to describe reactions in which different reactants and products undergo radioactive decay during the observation window. Corrections for radioactive decay are straight-forward, and the methodology can be applied to kinetic schemes of any order or complexity. Evaluation of clinical data on current radiopharmaceutical products reveals that, contrary to common perceptions, not all radiolabelling reactions obey pseudo-first-order kinetics.
Second-order kinetic schemes are appropriate for many radiolabelling reactions that involve complexation of 68 Ga 3+ , 89 Zr 4+ and other radiometal ions. Fluorination reactions using [ 18 F]fluoride, and some 11 C-radiolabelling reactions are likely to follow pseudo-first-order kinetics but the variability in the molar activity of the radionuclide source, and potential contaminants introduced in the reaction, may switch the kinetics toward second-order. Radiochemical kinetics can be used to model complex reactions, like [ 18 F]FDG radiosynthesis, that involve parallel or      (Table S2).