# Freezing of Living Cells: Mathematical Models and Design of Optimal Cooling Protocols

Chapter
Part of the International Series of Numerical Mathematics book series (ISNM, volume 160)

## Abstract

Two injuring effects of cryopreservation of living cells are under study. First, stresses arising due to non-simultaneous freezing of water inside and outside of cells are modeled and controlled. Second, dehydration of cells caused by earlier ice building in the extracellular liquid compared to the intracellular one is simulated. A low-dimensional mathematical model of competitive ice formation inside and outside of living cells during freezing is derived by applying an appropriate averaging technique to partial differential equations describing the dynamics of water-to-ice phase change. This 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. 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 applied to the problem. On the base of the computed value function, optimal cooling protocols ensuring simultaneous freezing of water inside and outside of living cells are designed. Thus, balancing the inner and outer pressures prevents cells from injuring. Another mathematical model describes shrinkage and swelling of cells caused by their osmotic dehydration and rehydration during freezing and thawing. The model is based on the theory of ice formation in porous media and Stefan-type conditions describing the osmotic inflow/outflow related to the change of the salt concentration in the extracellular liquid. The cell shape is searched as a level set of a function which satisfies a Hamilton-Jacobi equation resulting from a Stefan-type condition for the normal velocity of the cell boundary. Hamilton-Jacobi equations are numerically solved using finite-difference schemes for finding viscosity solutions as well as by computing reachable sets of an associated conflict control problem. Examples of the shape evolution computed in two and three dimensions are presented

## Keywords

Optimal cooling rate differential game Hamilton-Jacobi equations finite-difference scheme reachable sets

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## Authors and Affiliations

• Nikolai D. Botkin
• 1
Email author
• Karl-Heinz Hoffmann
• 1
• Varvara L. Turova
• 1
1. 1.Department of MathematicsTechnische Universität MünchenGarchingGermany

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