MATHEMATICAL TREATMENTS OF ICE FORMATION AND VITRIFICATION DURING CRYOPRESERVATION

Ice crystal nucleation and growth during cryopreservation is problematic, as ice drives cell damage and death. Cooling of biological systems to the glass transition temperature of water is commonly performed in an effort to vitrify the system in question, mitigating ice crystal formation. This is aided by the use of cryoprotective agents (CPAs), which are small molecule solutes that hinder ice formation. Toward improved design of cryopreservation protocols, the mathematical underpinnings of ice formation and vitrification are of interest. In this chapter, we provide an overview of mathematical models used to describe the nucleation of ice crystals and their subsequent growth, as relevant to cryopreservation. We derive classical relationships commonly used to describe liquid-to-solid phase transformation rates (the Avrami equation). By coupling with population balance models used to describe the size distribution evolution of growing ice nuclei, we overview extensions of classical approaches to the analysis of non-isothermal systems, such as those with rapid cooling rates commonly encountered in cryopreservation, and to the analysis of growth of nonspherical, fractal-like nuclei. We also discuss critical cooling rates and rewarming rates in the presence of CPAs needed to avoid ice formation, as well as outstanding issues in the mathematical modeling of phase change during cryopreservation. An emphasis throughout the chapter is on first-principles derivations of relevant equations and model frameworks geared toward physical scientists and engineers interested in understanding phase change kinetics as they pertain to cryopreservation.