(84d) Modelling the Effect of Ice Nucleation Stochasticity on Batch Heterogeneity of Freezing Processes of Vials on a Shelf

Freezing and freeze-drying are widely used processes to improve the stability and thus shelf life of pharmaceutical formulations. Ice nucleation is the first step of the freezing process; it is an activated process that requires cooling of the product beyond its equilibrium freezing temperature. It occurs at different times and temperatures for individual vials, even in case they experience identical cooling profiles. This variance was identified as a main source of batch heterogeneity across frozen vials on a shelf [1]. Such heterogeneity is a major risk for industrial processes involving freezing steps since several product attributes depend strongly on the freezing history of each vial: The ice crystal morphology in a vial depends both on the temperature at which ice nucleation occurs and on the time required for complete solidification after nucleation, i.e. the solidification time [2]. Finally, stochasticity of ice nucleation depends, to some extent, on the process conditions: Generally, one observes lower average nucleation temperatures and a broader distribution in dust-free GxP settings compared to the laboratory, thus rendering the transfer of a process design from laboratory to production scale challenging. To mitigate the effects of this stochasticity, process technologies have been developed that artificially induce ice nucleation at the same time in all vials, so called controlled nucleation techniques [1].

While there is experimental evidence that these techniques indeed reduce heterogeneity across the shelf, experimental studies to quantify heterogeneity are challenging due to the requirement of analyzing several product attributes in a statistically significant number of vials. Therefore, a shelf-scale model of the freezing process is required to deepen the understanding of batch heterogeneity and to optimize the process design. To the best of our knowledge, such model is not available in literature and the theoretical understanding of the relevant mechanisms leading to heterogeneity is limited. Besides the stochasticity of ice nucleation, variability in heat transfer was identified as major source of heterogeneity, in the form of spatial differences across the shelf and due to geometric imperfections of vials [4].

We thus propose a modelling framework for the freezing process of an arbitrary number of vials on a shelf to understand and identify all potential mechanisms of batch heterogeneity with a focus on the stochasticity of ice nucleation. With the help of this model, we investigate the effect of various process parameters on heterogeneity to aid process design and upscaling.

Methods

The model describes the freezing process of vials on a shelf in a mechanistic way taking into account the geometry of the system, heat transfer, and ice nucleation. Each vial is considered as thermally homogeneous while it evolves from liquid to a partially frozen, and to a completely frozen state. The governing equations account for the vial’s physical state and ice nucleation is described as an inhomogeneous Poisson process [5].

While experimental studies on batch heterogeneity mostly focus on measuring product attributes, we characterize heterogeneity based on three characteristic quantities of the freezing process. These are the nucleation time, nucleation temperature and solidification time for each vial of the simulated system.

We have simulated sets of freezing experiments and generated the corresponding histograms of these quantities. To identify potential mechanisms for heterogeneity, we investigated the effect of ice nucleation parameters and their variability, heat transfer parameters and their variability and process control strategies on these distributions.

Results

We have performed simulation studies of a system comprising 7x7 vials on a shelf, a typical batch size for experimental studies in laboratory scale freeze-dryers. In addition to the direct effect of the stochasticity of ice nucleation on heterogeneity, we account also for an indirect role played by ice nucleation, namely heat transfer between vials triggered by the heat released upon nucleation. Since nucleation is stochastic, the occurrence of such thermal effects adds to the heterogeneity across the shelf. Figure 1 illustrates both these effects on the quantities characterizing the freezing process, expressed in terms of the intensity of the interaction. We use the heat transfer coefficient between vials, as parameter to quantify this interaction.

In case of thermally independent vials, i.e. k_int = 0, the distributions of the characteristic quantities correspond to the freezing process of single vials: The only source of variability between vials is the stochasticity of ice nucleation. For sufficiently strong interaction, i.e. k_int > 10 Wm^-2K^-1, we observed a bimodal nucleation time distribution as shown in Fig. 1a. In this case, a first group of vials nucleated randomly across the shelf and delayed the nucleation events of the remaining vials. These early nucleating vials then act as “local hotspots” on the shelf that temporarily slow down the cooling process of neighboring vials. The formation of distinct populations corresponding to early and late nucleating vials can be seen in the bivariate distribution of nucleation times and solidification times in Fig. 1d. These populations differ both in nucleation and solidification times, thus creating a significant heterogeneity across the batch.

The effect of thermal interaction on the nucleation temperature distribution was smaller than for the other two quantities; we only identified a weak trend towards higher temperatures for increasing interaction as shown in Fig 1a-c. This observation, namely that in some situations the distributions of nucleation time and of temperature behave differently, is of importance for process monitoring. Commonly, the two distributions are used interchangeably; nucleation times in literature are detected with a camera and converted into temperatures based on reference temperature measurements [3]. Based on our model results, we hypothesize that such approach may be misleading and suggest the direct measurement of both quantities, e.g. via infrared thermography.

We can use these outcomes to improve the freezing process design. In case of freeze-drying processes, a homogeneous crystal morphology across the shelf is required. The crystal morphology correlates directly with the solidification time, so that a narrow distribution of the solidification times across the shelf is desired. Consequently, one should aim to operate the process in a regime with low thermal interaction. In practice, this may be achieved by arranging vials in a geometry with less direct contact and by arranging them directly on the shelf without tray.

In general, we can apply this modelling framework to study not only the impact of the stochasticity of ice nucleation, but all potential sources of heterogeneity. Our model can be extended with ease to simulate the additional heat transfer to edge vials, as well as a random variability of the vial-to-shelf heat transfer coefficient.

Conclusion

We present for the first time a mechanistic model for the freezing of an arbitrary number of vials on a shelf that is capable of capturing the stochasticity of ice nucleation and its effects on the freezing process. It proves to be a useful tool in improving the understanding of batch heterogeneity in freezing processes. The model enables us to assess the impact of process parameters, such as varying cooling profiles or techniques like controlled nucleation, on the freezing outcome. It also allows the simulation of the freezing process for both GxP and non-GxP settings based on the specific ice nucleation kinetics of each setting; thus aiding upscaling and technology transfer between freezing devices. The developed model thus serves as a powerful simulation tool in the design and optimization of freezing and freeze-drying.

Acknowledgement

The authors thank The Janssen Pharmaceutical Companies of Johnson & Johnson for the support in the course of this project.

References

[1] A. Arsiccio et al.: Vacuum Induced Surface Freezing as an effective method for improved inter- and intra-vial product homogeneity, European Journal of Pharmaceutics and Biopharmaceutics 128 (2018) 210–219.

[2] A. Arsiccio, A. A. Barresi, and R. Pisano: Prediction of Ice Crystal Size Distribution after Freezing of Pharmaceutical Solutions, Crystal Growth & Design (2017), 17, 4573−4581.

[3] L. C. Capozzi, R. Pisano: Looking inside the ‘black box’: Freezing engineering to ensure the quality of freeze-dried biopharmaceuticals, European Journal of Pharmaceutics and Biopharmaceutics 129 (2018) 58–65.

[4] B. Scutella et al.: How Vial Geometry Variability Influences Heat Transfer and Product Temperature During Freeze-Drying, Journal of Pharmaceutical Sciences 106 (2017) 770-778.

[5] G. M. Maggioni and M. Mazzotti: Modelling the stochastic behaviour of primary nucleation, Faraday Discussions (2015), 179, 359.