Mathematical Modeling

Abstract

Solution crystallization is becoming increasingly important as separation, purification, and controlled particle formation of biomolecules in pharmaceutical applications, with high potential as a green and scalable method at an industrial level. As this book's previous chapters discussed already, a large amount of attention is paid to the fundamental aspects of nucleation and growth, as triggering the crystallization of macromolecules is often challenging. Once the suitable ranges of the key thermodynamic conditions are identified, such as the solvent system, precipitating agent, concentration, pH, and temperature, the toolsets of the crystallization process engineering, relying on mathematical modeling, can support the process understanding, design, optimization, and control/operation.As a primer to this chapter, one must keep in mind that the results drawn from the relatively small number of proteins and other biomolecular systems studied in detail in the past indicate that a strong correspondence exists between the growth behavior and mechanisms of the biomolecule crystals with theories accumulated into the crystallization of inorganic small-molecule crystals. Although there is no guarantee that all protein and other biomolecular systems analyzed in the future will obey the same or similar fundamental rules, given the available evidence, a generic crystallization modeling and control overview will be given here. This chapter is divided into two main sections. Firstly, the mathematical modeling and simulation aspects will be described, which can help to understand, analyze, and optimize the processes on a manufacturing level. Secondly, control solutions will be presented for crystallizers that enable the realization of the desired products.

Keywords: Biomolecular crystallization; Crystallization process design; Metastable intermediate phase; Nucleation and growth kinetics; Particle size distribution; Population balance modeling; Process analytical technology; Solubity modeling.