A model of diffusion-limited ice growth inside biological cells during freezing

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Abstract
A theoretical model for predicting the kinetics of ice crystallization inside cells during cryopreservation was developed, and applied to mouse oocytes, by coupling separate models of (1) water transport across the cell membrane, (2) ice nucleation, and (3) crystal growth. The instantaneous cell volume and cytosol composition during continuous cooling in the presence of glycerol were predicted using the water transport model. Classical nucleation theory was used to predict ice nucleation rates, and a nonisothermal diffusion‐limited crystal‐growth model was used to compute the resulting crystallization kinetics. The model requires knowledge of the nucleation rate parameters Ω and κ, as well as the viscosity η of a water‐NaCl‐glycerol solution as a function of both the composition and temperature of the solution. These dependences were estimated from data available in the literature. Cell‐specific biophysical parameters were obtained from previous studies on mouse oocytes. A sensitivity analysis showed that the model was most sensitive to the values of κ and η. The coupled model was used to study the effect of cooling rate and initial glycerol concentration on intracellular crystal growth. The extent of crystallization, as well as the crystal size distribution, were predicted as functions of time. For rapid cooling at low to intermediate glycerol concentrations, the cells crystallized completely, while at high concentrations of glycerol, partial or total vitrification was observed. As expected, the cooling rate necessary for vitrification dropped with increasing glycerol concentration; when cells initially contained ∼7.5 M glycerol, vitrification was achieved independent of cooling rate. For slow cooling protocols, water transport significantly affected the results. At glycerol concentrations greater than ∼3 M, the final intracellular ice content decreased with increasing glycerol concentration at a fixed cooling rate. In this regime, the cooling rate at which a critical amount of ice was formed increased as more glycerol was used. When less than ∼3 M glycerol was initially present in the cell, an increase in glycerol concentration was predicted to cause an increase in the final intracellular ice content at a given cooling rate. In this regime, the critical cooling rate decreased with increasing glycerol concentration. These predictions clarify previous empirical observations of slow freezing phenomena.