The Nonsteady State Modeling of Freeze Drying: In-Process Product Temperature and Moisture Content Mapping and Pharmaceutical Product Quality Applications

Introduction. Theoretical models of the freeze-drying process are potentially useful to guide the design of a freeze-drying process as well as to obtain information not readily accessible by direct experimentation, such as moisture distribution and glass transition temperature, Tg, within a vial during processing. Previous models were either restricted to the steady state and/or to one-dimensional problems. While such models are useful, the restrictions seriously limit applications of the theory. An earlier work from these laboratories presented a nonsteady state, two-dimensional model (which becomes a three-dimensional model with an axis of symmetry) of sublimation and desorption that is quite versatile and allows the user to investigate a wide variety of heat and mass transfer problems in both primary and secondary drying. The earlier treatment focused on the mathematical details of the finite element formulation of the problem and on validation of the calculations. The objective of the current study is to provide the physical rational for the choice of boundary conditions, to validate the model by comparison of calculated results with experimental data, and to discuss several representative pharmaceutical applications. To validate the model and evaluate its utility in studying distribution of moisture and glass transition temperature in a representative product, calculations for a sucrose-based formulation were performed, and selected results were compared with experimental data. Theoretical Model. The model is based on a set of coupled differential equations resulting from constraints imposed by conservation of energy and mass, where numerical results are obtained using finite element analysis. Use of the model proceeds via a “modular software package” supported by Technalysis Inc. (Passage™/Freeze Drying). This package allows the user to define the problem by inputing shelf temperature, chamber pressure, container properties, product properties, and numerical analysis parameters required for the finite element analysis. Most input data are either available in the literature or may be easily estimated. Product resistance to water vapor flow, mass transfer coefficients describing secondary drying, and container heat transfer coefficients must normally be measured. Each element (i.e., each small subsystem of the product) may be assigned different values of product resistance to accurately describe the nonlinear resistance behavior often shown by real products. During primary drying, the chamber pressure and shelf temperature may be varied in steps. During secondary drying, the change in gas composition from pure water to mostly inert gas is calculated by the model from the instantaneous water vapor flux and the input pumping capacity of the freeze dryer. Results. Comparison of the theoretical results with the experiment data for a 3% sucrose formulation is generally satisfactory. Primary drying times agree within two hours, and the product temperature vs. time curves in primary drying agree within about ± 1°C. The residual moisture vs. time curve is predicted by the theory within the likely experimental error, and the lack of large variation in moisture within the vial (i.e., top vs. side vs. bottom) is also correctly predicted by theory. The theoretical calculations also provide the time variation of “Tg–T” during both primary and secondary drying, where T is product temperature and Tg is the glass transition temperature of the product phase. The calculations demonstrate that with a secondary drying protocol using a rapid ramp of shelf temperature, the product temperature does rise above Tg during early secondary drying, perhaps being a factor in the phenomenon known as “cake shrinkage”. Conclusion. The theoretical results of in-process product temperature, primary drying time, and moisture content mapping and history are consistent with the experimental results, suggesting the theoretical model should be useful in process development and “trouble-shooting” applications.