(58c) Analytical Solutions for the Modeling, Optimization, and Control of Microwave-Assisted Freeze Drying

Freeze drying, aka lyophilization, is a crucial process in biopharmaceutical manufacturing by which a product is dried (or dehydrated) via sublimation under vacuum (Fissore et al., 2018; Park et al., 2021). To achieve sublimation, the process is carried out at low temperature compared to typical dehydration and drying techniques, and hence freeze drying is better at preserving the quality and structure of heat-sensitive materials, e.g., pharmaceutical products (Barresi et al., 2009). This freeze-drying technique plays an important role in biotherapeutics (Fissore et al., 2018), including applications related to pandemics such as COVID-19 (Hammerling et al., 2021).

Freeze drying consists of three stages, namely (1) freezing, (2) primary drying, and (3) secondary drying. During freezing, the product and liquid solvent (usually water) are frozen at a very low temperature (Fissore et al., 2018; Bano et al., 2020). In this stage, the free water becomes ice crystals, whereas the bound water retains its noncrystalline state while bounded to the product molecules (Fissore et al., 2018). In primary drying, the frozen product and solvent are heated under sufficiently low pressure and temperature that the ice crystals sublimate (Pisano et al., 2010). In secondary drying, the product is heated further at higher temperature to remove much of the bound water via desorption (Velardi and Barresi, 2008; Gitter et al., 2019). Primary drying is recognized as the most time-consuming, hazardous, and expensive stage, and so is the main target for improvement and optimization (Velardi and Barresi, 2008; Pisano et al., 2010). In conventional freeze drying (CFD), the product is heated via a heating shelf located under the drying chamber or vial during the drying stages (Pisano et al., 2010; Fissore et al., 2018). To accelerate the drying process, microwave-assisted freeze drying (MFD), which relies on microwave irradiation, was studied and developed (Walters et al., 2014; Gitter et al., 2019). It was shown via both simulations and experiments that MFD can significantly reduce the drying time during primary drying, by 70%–80% while still preserving the quality of the product (Gitter et al., 2018; Bhambhani et al., 2021; Park et al., 2021). Hybrid freeze drying (HFD), which combines both CFD and MFD heating techniques, can further decrease the drying time (Park et al., 2021).

Mathematical models of freeze drying are valuable for guiding the design, optimization, and control of the freeze-drying process (Pisano et al., 2010; Fissore et al., 2015; Bano et al., 2020). Over the past few decades, mechanistic modeling of CFD has become well established (see examples and discussions in Litchfield and Liapis (1979); Mascarenhas et al. (1997); Pikal et al. (2005); Hottot et al. (2006); Velardi and Barresi (2008); Pisano et al. (2010); Chen et al. (2015); Fissore et al. (2015); Scutellà et al. (2017); Bano et al. (2020)). Mechanistic modeling of MFD is not as mature as that of CFD; only a few mechanistic models are available (Witkiewicz and Nastaj, 2010; Wang et al., 2020; Park et al., 2021). A mechanistic model for HFD can be constructed by merging the mechanistic models for CFD and MFD (Park et al., 2021).

The aforementioned mechanistic models are usually written as partial differential equations (PDEs) describing heat and mass transfer in freeze drying, so most previous research heavily relied on numerical solutions, e.g., the finite element method (Mascarenhas et al., 1997; Hottot et al., 2006; Chen et al., 2015) and the finite volume method (Park et al., 2021). Numerical solutions are useful in applications where the model equations are so complicated that analytical solutions cannot be derived. A drawback is that numerical solutions do not explicitly reveal the relationships between model parameters and solutions. The high computational cost is also a limitation for some applications, such as in model-based optimal control (Klepzig et al., 2020). Analytical solutions, on the other hand, are more accurate and easier to compute. In addition, analytical solutions allow the connections between model parameters and solutions to be interpreted clearly, which is beneficial in understanding the physics of the system and in engineering design.

This work derives exact and approximate analytical solutions to the established mechanistic model of conventional, microwave-assisted, and hybrid freeze drying during primary drying. The exact analytical solution is obtained by using the superposition principle, separation of variables, and Duhamel’s theorem. This exact solution serves as a reference solution for validating numerical or approximate results due to its highest accuracy up to machine precision. Alternatively, an approximate analytical solution is derived using the heat-balance integral (integral method). The numerical solutions are also developed based on the finite difference and finite element methods for comparison with the analytical solutions.

Results show that the exact solution is most accurate but very computationally expensive. This is because the exact solution contains infinite series and requires solving transcendental equations, which results in computational complexity. The approximate solution, on the other hand, can be written as a simple closed-form expression and does not entail any transcendental equation, which is more convenient for implementation and computationally efficient. The approximate solution produces the maximum error in drying time and temperature predictions of less than 1% in the given parameter space. It can also be computed fastest among all the solution techniques, which is about 4-fold and 200-fold faster than the numerical and exact solutions, respectively. With its high accuracy and computational performance, the approximate solution offers an efficient tool for design and optimization of the lyophilizer, which is demonstrated for parameter estimation, optimal control, and parameter space analysis.

References

Bano, G., De-Luca, R., Tomba, E., Marcelli, A., Bezzo, F., Barolo, M., 2020. Primary drying optimization in pharmaceutical freeze-drying: A multivial stochastic modeling framework. Ind. Eng. Chem. Res. 59, 5056–5071.

Barresi, A.A., Pisano, R., Fissore, D., Rasetto, V., Velardi, S.A., Vallan, A., Parvis, M., Galan, M., 2009. Monitoring of the primary drying of a lyophilization process in vials. Chem. Eng. Process. 48, 408–423.

Bhambhani, A., Stanbro, J., Roth, D., Sullivan, E., Jones, M., Evans, R., Blue, J., 2021. Evaluation of microwave vacuum drying as an alternative to freeze-drying of biologics and vaccines: The power of simple modeling to identify a mechanism for faster drying times achieved with microwave. AAPS PharmSciTech 22, 52.

Chen, X., Sadineni, V., Maity, M., Quan, Y., Enterline, M., Mantri, R.V., 2015. Finite element method (FEM) modeling of freeze-drying: Monitoring pharmaceutical product robustness during lyophilization. AAPS PharmSciTech 16, 1317–1326.

Fissore, D., Pisano, R., Barresi, A.A., 2015. Using mathematical modeling and prior knowledge for QbD in freeze-drying processes, in: Jameel, F., Hershenson, S., Khan, M.A., Martin-Moe, S. (Eds.), Quality by Design for Biopharmaceutical Drug Product Development. Springer, New York, pp. 565–593.

Fissore, D., Pisano, R., Barresi, A.A., 2018. Process analytical technology for monitoring pharmaceuticals freeze-drying – a comprehensive review. Dry. Technol. 36, 1839–1865.

Gitter, J.H., Geidobler, R., Presser, I., Winter, G., 2018. Significant drying time reduction using microwave-assisted freeze-drying for a monoclonal antibody. J. Pharm. Sci. 107, 2538–2543.

Gitter, J.H., Geidobler, R., Presser, I., Winter, G., 2019. Microwave-assisted freeze-drying of monoclonal antibodies: Product quality aspects and storage stability. Pharmaceutics 11.

Hammerling, M.J., Warfel, K.F., Jewett, M.C., 2021. Lyophilization of premixed covid-19 diagnostic RT-qPCR reactions enables stable long-term storage at elevated temperature. Biotechnol. J. 16, 2000572.

Hottot, A., Peczalski, R., Vessot, S., Andrieu, J., 2006. Freeze-drying of pharmaceutical proteins in vials: Modeling of freezing and sublimation steps. Dry. Technol. 24, 561–570.

Klepzig, L.S., Juckers, A., Knerr, P., Harms, F., Strube, J., 2020. Digital twin for lyophilization by process modeling in manufacturing of biologics. Processes 8, 1325. doi:10.3390/pr8101325. Litchfield, R., Liapis, A., 1979. An adsorption-sublimation model for a freeze dryer. Chem. Eng. Sci. 34, 1085–1090.

Mascarenhas, W., Akay, H., Pikal, M., 1997. A computational model for finite element analysis of the freeze-drying process. Comput. Methods Appl. Mech. Eng. 148, 105–124.

Park, J., Cho, J.H., Braatz, R.D., 2021. Mathematical modeling and analysis of microwave-assisted freeze-drying in biopharmaceutical applications. Comput. Chem. Eng. 153, 107412.

Pikal, M.J., Mascarenhas, W.J., Akay, H.U., Cardon, S., Bhugra, C., Jameel, F., Rambhatla, S., 2005. The nonsteady state modeling of freeze drying: In-process product temperature and moisture content mapping and pharmaceutical product quality applications. Pharm. Dev. Technol. 10, 17–32.

Pisano, R., Fissore, D., Velardi, S.A., Barresi, A.A., 2010. In-line optimization and control of an industrial freeze-drying process for pharmaceuticals. J. Pharm. Sci. 99, 4691–4709.

Scutellà, B., Plana-Fattori, A., Passot, S., Bourlès, E., Fonseca, F., Flick, D., Tréléa, I., 2017. 3D mathematical modelling to understand atypical heat transfer observed in vial freeze-drying. Appl. Therm. Eng. 126, 226–236.

Velardi, S.A., Barresi, A.A., 2008. Development of simplified models for the freeze-drying process and investigation of the optimal operating conditions. Chem. Eng. Res. Des. 86, 9–22.

Walters, R.H., Bhatnagar, B., Tchessalov, S., Izutsu, K.I., Tsumoto, K., Ohtake, S., 2014. Next generation drying technologies for pharmaceutical applications. J. Pharm. Sci. 103, 2673–2695.

Wang, W., Zhang, S., Pan, Y., Yang, J., Tang, Y., Chen, G., 2020. Multiphysics modeling for microwave freeze-drying of initially porous frozen material assisted by wave-absorptive medium. Ind. Eng. Chem. Res. 59, 20903–20915.

Witkiewicz, K., Nastaj, J.F., 2010. Simulation strategies in mathematical modeling of microwave heating in freeze-drying process. Dry. Technol. 28, 1001–1012.