If you use SNOW for work/research presented in a publication (whether
directly, or as a dependency to another package), we recommend and encourage
the following acknowledgment:

  This research made use of SNOW, a Python package
  for the modeling of Stochastic Nucleation of Water (Deck et al., 2022).

where (Deck et al., 2022) is a citation to this paper:

  https://www.sciencedirect.com/science/article/pii/S0378517321010826

If you use SNOW for the modeling of pallet freezing in particular, which was
implemented in v1.1, we further recommend you to cite the following article:

  https://doi.org/10.1016/j.ijpharm.2022.122051

Finally, in case you are refering to the spatial freezing model implemented in
v2.0 we recommend to cite the article:

  https://doi.org/10.1016/j.cej.2024.148660

We encourage you to also include citations to the papers in the main text
wherever appropriate.


Recommended BibTeX entries for the above citations are:

@article{deck2022,
title = {Stochastic shelf-scale modeling framework for the freezing stage in freeze-drying processes},
journal = {International Journal of Pharmaceutics},
volume = {613},
pages = {121276},
year = {2022},
issn = {0378-5173},
doi = {https://doi.org/10.1016/j.ijpharm.2021.121276},
url = {https://www.sciencedirect.com/science/article/pii/S0378517321010826},
author = {Leif-Thore Deck and David R. Ochsenbein and Marco Mazzotti},
keywords = {Freezing, Freeze-Drying, Lyophilization, Nucleation, Stochastic Processes, Monte Carlo, Modeling},
abstract = {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.}
}

@article{deck2022_pallet,
title = {Stochastic ice nucleation governs the freezing process of biopharmaceuticals in vials},
journal = {International Journal of Pharmaceutics},
volume = {625},
pages = {122051},
year = {2022},
issn = {0378-5173},
doi = {https://doi.org/10.1016/j.ijpharm.2022.122051},
url = {https://www.sciencedirect.com/science/article/pii/S0378517322006056},
author = {Leif-Thore Deck and David R. Ochsenbein and Marco Mazzotti},
keywords = {Freezing, Vaccines, Mechanistic modeling, Pharmaceutical manufacturing, Ice nucleation, Stochastic nucleation},
abstract = {Biopharmaceuticals commonly require freezing to ensure the stability of the active pharmaceutical ingredients (APIs). At commercial scale, freezing is typically carried out over the course of days in pallets comprising tens of thousands of vials. The selected process conditions have to ensure both complete freezing in all vials and a satisfactory manufacturing throughput. Current process design, however, is mainly experimental, since no mechanistic understanding of pallet freezing and its underlying phenomena has been achieved so far. Within this work, we derive a mechanistic modeling framework and compare the model predictions with engineering run data from the Janssen COVID-19 vaccine. The model qualitatively reproduced all observed trends and reveals that stochastic ice nucleation governs both process duration and batch heterogeneity. Knowledge on the ice nucleation kinetics of the formulation to be frozen thus is required to identify suitable freezing process conditions. The findings of this work pave the way towards a more rational design of pallet freezing, from which a plethora of frozen drug products may benefit. For this reason, we provide open source access to the model in the form of a python package (Deck et al., 2021).}
}

@article{deck2024_spatial,
title = {Modeling the freezing process of aqueous solutions considering thermal gradients and stochastic ice nucleation},
journal = {Chemical Engineering Journal},
volume = {483},
pages = {148660},
year = {2024},
issn = {1385-8947},
doi = {https://doi.org/10.1016/j.cej.2024.148660},
url = {https://www.sciencedirect.com/science/article/pii/S1385894724001451},
author = {Leif-Thore Deck and Andraž Košir and Marco Mazzotti},
keywords = {Freezing, Nucleation, Stochastic processes, Pharmaceutical manufacturing, Process design and optimization},
abstract = {Despite its importance to multiple scientific fields and industries, the freezing process of aqueous solutions is not yet completely understood. In particular, the relationship between temperature gradients within a solution and the occurrence of stochastic ice nucleation remains elusive. To address this knowledge gap, we have derived a novel stochastic spatial freezing model from first principles. The model predicts with quantitative accuracy how temperature gradients affect the stochastic ice nucleation of sucrose solutions in vials. This motivated a detailed study of the freezing-stage in freeze-drying, revealing that a broad range of temperatures are present at the time of nucleation at different positions within the vial. This must be considered when interpreting experimental studies that measure the temperature only at a single point with a thermocouple. To ensure that both researchers and practitioners benefit from this modeling work, we provide open source access to it within our python package ethz-snow.}
}
