Mathematical modeling of the stage of freezing in technology of lyophilized drugs

The purpose of this work is to demonstrate the existing possibilities and methodology of mathematical modeling of the freezing phase in the technology of lyophilization of vials filled with a solution containing a pharmaceutical substance and auxiliary substances. The freezing process is crucial for the subsequent stages of primary sublimation and the drying stage. So it is at the stage of freezing that ice crystals are formed and a certain microstructure forming the pores for subsequent stages and influencing the speed of all stages in the future. The developed methods allow estimating the kinetics of cooling, freezing, calculating the sizes of ice crystals and determining the permeability of the sublimation layer. Of course, for these calculations it is necessary to know the conditions and values of the variables, calculated and measured during the pilot freezing. However, modeling allows to reduce the time of development and optimization of the process, increases the speed of transfer of the technological process to other equipment. In this review, formulas are analyzed, the thermal conductivity equations for each cooling zone, equations for estimating the crystal size and pore size in the dried layer, and calculating the permeability of the sublimated layer. The article draws conclusions about the perspective of modeling methods for the freezing phase, examines the equations used in modeling and demonstrates the model of supercooling, crystal formation and other mass and heat exchange processes during the cooling and freezing stage. The calculations presented in this paper are confirmed by references to experimental data and have great practical value.

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