Analytical optimal controls for the state constrained addition and removal of cryoprotective agents

Cryobiology is a field with enormous scientific, financial, and even cultural impact. Successful cryopreservation of cells and tissues depends on the equilibration of these materials with high concentrations of permeating chemicals (CPAs) such as glycerol or 1,2 propylene glycol. Because cells and tissues are exposed to highly anisosmotic conditions, the resulting gradients cause large volume fluctuations that have been shown to damage cells and tissues. On the other hand, there is evidence that toxicity to these high levels of chemicals is time dependent, and therefore it is ideal to minimize exposure time as well. Because solute and solvent flux is governed by a system of ordinary differential equations, CPA addition and removal from cells is an ideal context for the application of optimal control theory. Recently, we presented a mathematical synthesis of the optimal controls for the ODE system commonly used in cryobiology in the absence of state constraints and showed that controls defined by this synthesis were optimal. Here we define the appropriate model, analytically extend the previous theory to one encompassing state constraints, and as an example apply this to the critical and clinically important cell type of human oocytes, where current methodologies are either difficult to implement or have very limited success rates. We show that an enormous increase in equilibration efficiency can be achieved under the new protocols when compared to classic protocols, potentially allowing a greatly increased survival rate for human oocytes and pointing to a direction for the cryopreservation of many other cell types.