• cell number density distribution;
  • population balance;
  • intracellular ice formation;
  • membrane permeability


A novel model able to quantitatively describe intracellular ice formation (IIF) as a function of temperature in a cell population characterized by a size distribution is proposed for the case when a permeant cryoprotectant agent (CPA) is present. As such, the model represents the ideal extension of the one recently proposed by the authors for interpreting the behavior of a size-distributed population of cells during cryopreservation without CPA. To resemble the experimental procedure actually adopted, the former model is coherently modified for the simulation of the initial equilibration stage (i.e., a one-step CPA permeation kinetics into cells initially under isotonic conditions to be suspended in a water/salt/CPA solution at ambient temperature) and of the subsequent cooling stage at constant rate down to −60°C. Specifically, the size distribution dynamics of a cell population in response to water osmosis, CPA permeation, and IIF is simulated by means of a suitable population balance approach. Water and CPA intracellular transport equations based on the classical Kedem and Katchalsky formalism are coupled to the quantitative description of nucleation and diffusion-limited growth of ice crystals in the framework of a one-dimensional population balance equation. The driving forces of all the physico-chemical phenomena involved are evaluated by taking into account the relevant ternary phase diagram, whereas the viscosity of the cytoplasm solution is properly accounted as a function of the salt and CPA concentrations. After a reasoned choice of the values of model parameters, the behavior of isolated rat hepatocytes suspended in a water/sodium-chloride/glycerol solution is simulated and analyzed at different, practicable (i.e., low) constant cooling rates and CPA equilibration concentrations. It is confirmed that differently sized cells in a single population exhibit different IIF temperatures under the same operative conditions even in the presence of CPA. Correspondingly, the probability of IIF results to be a function of the initial size distribution of the cell population. Depending on the specific operative conditions adopted, the size distribution and the osmotic properties of the cell lineage at hand, IIF at −60°C may be lethal for a fraction of the cell population (i.e., larger size classes), or it may not reach a dangerous level for the intermediate size class cells, while it will not even take place for the smaller ones. Finally, even if no comparison with experimental data is provided, an original and physically sound explanation for several, well-known experimental evidences, which appeared in a number of articles available in the literature of cell cryopreservation with CPA, is comprehensively given. © 2010 American Institute of Chemical Engineers AIChE J, 2011