Stochastic shelf-scale modeling framework for the freezing stage in freeze-drying processes

Freezing and freeze-drying processes are commonly used to improve the stability and thus shelf life of pharmaceutical formulations. Despite strict product quality requirements, batch heterogeneity is widely observed in frozen products, thus potentially causing process failure. Such heterogeneity is the result of the stochasticity of ice nucleation and the variability in heat transfer among vials, which lead to unique freezing histories of individual vials. We present for the first time a modeling framework for large-scale freezing processes of vials on a shelf and publish an open source implementation in the form of a python package on pypi. The model is based on first principles and couples heat transfer with ice nucleation kinetics, thus enabling studies on batch heterogeneity. Ice nucleation is assumed to be an inhomogeneous Poisson process and it is simulated using a Monte Carlo approach. We applied the model to understand the individual pathways leading to batch heterogeneity. Our simulations revealed a novel mechanism how ice nucleation leads to heterogeneity based on thermal interaction among vials. We investigated the effect of various cooling protocols, namely shelf-ramped cooling, holding steps and controlled nucleation, on the nucleation and solidification behavior across the shelf. We found that under rather general conditions holding schemes lead to similar solidification times, as in the case of controlled nucleation, thus identifying a potential pathway for freezing process optimization.