Modeling osmotic de- and rehydration of living cells using Hamilton–Jacobi eqytions and reachable set techniques

The paper describes mathematical models of the osmotic shrinkage and swelling of living cells during freezing and thawing. The cell shape is searched as the 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. The Hamilton–Jacobi equation is then solved numerically in two and three dimensions using a monotony preserving finite-difference scheme. A generalized variant of the Stefan condition accounting for tension effects in the cell membrane is also considered, and the corresponding cell shape evolution is computed in two dimensions using a reachable set technique arising from conflict control approach.