The goal of this work is to estimate the approximate time, $t$, that a reactive API will produce a concentration level of an impurity, $\alpha$, at a temperature, $T$, and humidity, $H$. \begin{equation} t=\hat{\alpha}(\alpha; T, H) \end{equation} A possible method to obtain this relationship is to integrate the reaction rate, $r_{\alpha}$, for the impurity production over time. \begin{equation} \alpha = \int_0^t \frac{d \alpha}{dt}dt \equiv \int_0^t r_{\alpha}(\{p_i\}_{i=1}^N; T, H)dt \end{equation} Here, $\{p\}_{i=1}^N$ denotes a set of N other possible parameters that the reaction depends upon (which will be denoted as $\{p_i\}$ for short). Given a model for the reaction rate, $r_{\alpha}$, the above equation can be integrated to give the impurity as a function of time, which can be used to determine an approximate shelf life. The difficulty is thus estimating an accurate kinetic model. Fortunately, observations of many reactions have lead to several assumptions help simplify the estimation of a kinetic model.
One assumption is that the temperature dependence of the reaction kinetics can be separated from the concentration dependence, and takes the form of an exponential function, which was first proposed by Arrhenius. The ASAP literature further suggests thatthe humidity dependence can also be separated as an exponential function. Thus, the temperature and humidity are often separted out into a rate constant, $k$. \begin{equation} r_{\alpha}=k(T,H;A,E)f(\{p_i\}) = A \exp^{-\frac{E}{RT}+BH}f(\{p_i\}) \end{equation}
The parameters, $p_i$, are often other reactants, which can be similarly expressed using reaction kinetic models. Therefore, the solution is obtained by the integration of the coupled reactions for all the species in the reaction. \begin{equation} \frac{d p_i}{dt} = k_i \hat{p_i}(\{p_j\}) \end{equation} Since the tolerable level of impurity is usually low, it is reasonable to assume that the concentration of the components involved in the reaction do not deviate substantially from the initial values. Thus, the above reactions can be approximated by a truncated Taylor series. \begin{equation} \hat{p_i} \approx \sum_j \frac{\partial p_i}{\partial p_j}\bigg|_{j \ne i}(p_j-p_{j,o}) = \sum_j \nu_{ij}(p_j-p_{j,o}) \end{equation} Here, $\nu_{ij}$ is the stoichiometric coefficient ratio between reactants $i$ and $j$. Substiting this into the the coupled equations. \begin{eqnarray} \frac{d \alpha}{dt} = k_\alpha \nu_{\alpha j} \sum_j (p_j-p_{j,o}) \\ \frac{d p_i}{dt} = k_i \nu_{ij} \sum_j (p_j-p_{j,o}) \end{eqnarray}
Integration of the above equations appxroximates the changes of impurity concentration for small step changes, and is exact if all the kinetics are first order. Moreover, the equations can be easily solved numerically or can be analytically solved using an orthogonal tranformation. Thus, this is a reasonable starting point for reactions with sufficient measurement data, such that the reaction mechanism (i.e. the $\nu_ij$) can be determined, as well as the temperature and humidity dependence parameters ($\{E_i\}$, and $\{B_i\}$) be estimated. However, this may not be suitable for the impurity analysis, which has very limited data.
The idea of isoconversion is to esitmate the temperature and humidity parameters without developing a mechanistic model. First, the imprutiy reaction rate is written by separating out the temperature and humidity from the other reactant dependencies. \begin{equation} \frac{d \alpha}{dt} = A \exp^{-\frac{E}{RT}+BH}f(\{p_i\}) \end{equation} Next assume that each reactant species can be parametized by the impurity concentration. \begin{equation} p_j = \hat{p_j}(\alpha) \end{equation} An example of a parameterization often used in kinetics courses is that the concentration of the reactant decreases proportionally (stoichiometrically) to the product (i.e. $p_j = p_{j,o} - \nu_{j\alpha} \alpha$). Substituting the genral parametized relationships into the reaction eqation gives. \begin{equation} \frac{d \alpha}{dt} = A \exp^{-\frac{E}{RT}+BH}f(\{\hat{p_j(\alpha)}\}) \equiv A \exp^{-\frac{E}{RT}+BH} f_\alpha(\alpha) \end{equation} Integrating the above gives. \begin{equation} \int_0^\alpha \frac{d \alpha}{f_\alpha(\alpha)} \equiv g(\alpha) = \int_0^t A \exp^{-\frac{E}{RT}+BH} dt = A \exp^{-\frac{E}{RT}+BH}t \end{equation} Finally, rearranging renders the desired result. \begin{equation} \ln t = \frac{E}{RT} + B H + \ln \left[\frac{g(\alpha)}{A}\right] \end{equation} The above gives the expression for time, $t$, to reach an impurity concentration, $\alpha$ for a specified temperature, $T$, and humidity, $H$. The idea is to make a series of measurements at different conditions ($T$, $H$) holding the samples until they achieve equal degredation, such that $g(\alpha)$ remains constant, and thus leave only $E$, $B$, and $g(\alpha)/A$ as unknowns. Once the three parameters are regressed from experimental data, the time shelf life can be estimated.
The assumptions of the isoconversion method are:
Lastly, if the the time dependence of the individual $p_i$ are coupled as a series of ordinary differential equations(as was demonstrated in the mechanicstic section). \begin{equation} \frac{d p_i}{dt} = k_i \hat{p_i}(\{p_j\}) \end{equation} If one the equations is rate limiting and the dependence of the reactants in this equation can be parametrized by the degredation extent, then the result reduces to the same conclusion as seen in the previous two assumptions.
This added analysis shows that the previous assumption 2 can be augmented to include temperature dependence for degredation reactions that have a multiplicative dependence on other reactants, for small degredation extents with a dominant pathway, and for a series reaction where one intermediate is rate limiting. One of these assumptions would likely be satistifed degredation reactions dominated by a particular pathway over all temperature and humidity enviroments measured. Contrarily, the assumptions would be invalid for competing reactions that have strong temperature dependence over the range of experiments.
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