Control design in cryopreservation of living cells

A mathematical model of ice formation in living cells during freezing is considered. Application of an appropriate averaging technique to partial differential equations describing the dynamics of water-ice phase transitions reduces spatially distributed relations to a few ordinary differential equations with control parameters and uncertainties. Such equations together with an objective functional that expresses the difference between the amount of ice inside and outside of a cell are considered as a differential game where the aim of the control is to minimize the objective functional, and the aim of the disturbance is opposite. A stable finite-difference scheme for computing the value function is presented. Based on the computed value function, optimal controls are designed to produce cooling protocols ensuring simultaneous freezing of water inside and outside of living cells. Such a regime provides balancing the pressures inside and outside of cells, which prevents cells from injuring.