# Rate Equations and Kinetic Models

Drug substances undergo chemical degradation by various pathways and mechanisms, depending on their chemical structures. The rate of chemical degradation is determined by various factors contributing to the rate equations. Drug substances can be stabilized by inhibiting the degradation through control of these factors, as will be described in Section 2.3. Here, methods for describing the chemical degradation rate of drug substances on the basis of their kinetics are presented. Included are methods for kinetically analyzing an observed degradation curve under specific experimental conditions, obtaining rate constants, and predicting degradation rates under alternative conditions on the basis of this information. This section deals only with prediction of the chemical degradation rate of drug substances themselves. The prediction of the stability of dosage forms (for the purpose of estimating the shelf lives of drug products), in which physical degradation may also play a role, will be described in Chapter 4.

Obtaining reliable drug degradation data requires the development and validation of a stability-indicating assay. Once a stability study is initiated, one attempts to use a set of conditions that allows one to obtain a summary parameter, such as a rate constant, by kinetic analysis of a degradation versus time curve under these specific and controlled conditions. A kinetic model is selected to describe the degradation curve, and arate constant is calculated by fitting the observed degradation curve to a suitable rate equation according to the assumed model. This section describes the selection of the kinetic model(s) and the calculation of a rate constant. Figure 6. Eyring plots for the hydrolysis of CI-988 and analogue (compound I). (Reproduced from Ref. 248 with permission.)

2.2.3.1. Kinetic Models to Describe Drug Degradation in Solution

The generalized rate expression for drug degradation is represented by the rate equation that was given earlier (Eq. 2.4). When a drug substance, D, degrades via a certain mechanism in which reactants A, B, . . . participate, the degradation rate generally depends on the concentrations of the various reactants A, B, . . . and D according to Eq. (2.10), assuming that all the reactants are involved directly or indirectly in the rate-controlling step.

When the concentrations of A, B, . . . are maintained constant, that is, when the change in their concentrations during the reaction is negligible owing to their being present at much higher concentrations than drug D, or when these species are components that are maintained constant through the use of buffers, such as hydronium ion, the degradation rate is often described by

When n equals 0, 1, or 2, the reaction is said to be a pseudo-zero-, pseudo-first-, or pseudo-second-order reaction (pseudo-nth-order for higher order reactions), respectively. If the concentration of an additional reactant other than drug D is not constant during the reaction, the reaction order becomes n + 1.

Kinetic models generally used for drug stability prediction usually follow pseudo-zero-, pseudo-first-, or pseudo-second-order kinetics. Drug degradation higher than second order is rarely seen. Even complex degradation pathways involving multiple consecutive or parallel reactions can be represented by the combinations of zero-, first-, and second-order reactions. General kinetic models describing drug degradation are elaborated below.

2.2.3.1.a. Simple Pseudo-First-Order Reaction and Zero-Order Reaction k

The differential rate equation for a pseudo-first-order reaction is

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