A mathematical framework for developing freezing protocols in the cryopreservation of cells

When cooling cells to preserve them during cryopreservation, cooling too quickly results in the formation of lethal intracellular ice, while cooling too slowly amplifies the toxic effects of the cryoprotective agents (CPAs) added to slow down ice formation. We derive a mathematical model for cell cryopreservation to understand and quantify these observations. We assume that the system has a spherical geometry of three different regions: ice, extracellular liquid medium, and cell. The two interfacial boundaries separating the three regions can move and must be determined as part of the solution. The presence of CPA lowers the freezing point of the system, and the cell membrane moves due to the osmotic pressure difference across the membrane. We use a combination of numerical and asymptotic methods to determine how the temperature, the CPA concentration, and concentration of an ion species internal and external to the cell evolve during cooling for a range of cooling rates across different timescales. We introduce two metrics to characterize the cell damage caused by freezing, accounting for supercooling and CPA toxicity. Given cell properties and the operating protocol of the cryopreservation process, we show how the damage metrics can be used to predict an optimal cooling rate. Our asymptotic analysis provides a computationally efficient framework from which to determine this optimal rate.