Direct Modeling of Pore Size Evolution and Microcollapse in Lyophilization by Population Balance
In the primary drying step of pharmaceutical freeze drying, it is well known that amorphous excipients are prone to "microcollapse" as they approach their glass transition temperature. Existing modeling approaches for this phenomenon generally entail either avoidance of this regime or an experimentally burdensome empirical assessment of the resulting drop in resistance to mass transfer. In this work, we approach the phenomenon head-on by developing a population balance model for the pore size distribution of a freeze-dried cake during microcollapse and relating the pore size to mass transfer resistance. By treating the population balance with the method of moments and an interpolative closure, we keep the model's computational burden on par with the literature-standard model for primary drying. For a literature example when mass transfer resistance clearly varies as a function of product temperature during drying, we fit the population balance model to cases with high and low product temperature and show that it can then simulate intermediate cases with equal accuracy. The validity of the expression for mass transfer resistance is demonstrated with another literature data set, where pore size distributions were measured after the fact for a formulation without microcollapse; only a single parameter from this model needs to be fit to those cases to yield predictive accuracy in product temperature on par with post-hoc empirical assessment. Finally, a sophisticated multi-experiment fit for one formulation across a range of conditions yields quantitative agreement in product temperatures, suggesting that the rate constant for microcollapse behavior may be a material parameter and independent of the precise porous geometry.