Geodesic
Modeling osmotic de- and rehydration of living cells using Hamilton--Jacobi eqytions and reachable set techniques
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 5, pp. 308-315

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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.
Keywords: сryopreservation of cells, osmotic effect, mathematical model, Hamilton–Jacobi equations, finite-difference scheme, reachable set. 
@article{TIMM_2010_16_5_a34,
     author = {V. L. Turova},
     title = {Modeling osmotic de- and rehydration of living cells using {Hamilton--Jacobi} eqytions and reachable set techniques},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {308--315},
     publisher = {mathdoc},
     volume = {16},
     number = {5},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2010_16_5_a34/}
}
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V. L. Turova. Modeling osmotic de- and rehydration of living cells using Hamilton--Jacobi eqytions and reachable set techniques. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 5, pp. 308-315. http://geodesic.mathdoc.fr/item/TIMM_2010_16_5_a34/