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Simulation of cell shrinkage caused by osmotic cellular dehydration during freezing * N.D. Botkin, V. L. Turova Technische Universität München, Center of Mathematics, Chair of Mathematical Modelling Garching, Germany * Supported by the DFG, SPP 1256 4th INTERNATIONAL CONFERENCE ON HIGH PERFORMANCE SCIENTIFIC COMPUTING March 2-6, 2009, Hanoi, Vietnam

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Isolated stem cells derived from dental follicle. Stem cells derived from tooth follicle A wisdom tooth with soft tissue. 1 J.Schierholz, oN.Brenner, H.-L.Zeilhofer, K.-H.Hoffmann, C.Morsczeck. Pluripotent embryonic-like stem cells derived from teeth and uses thereof. European Patent. Date f publication and mention of the grant of the patent 04.06.2008.

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Cellular dehydration and shrinkage due to the osmotic outflow through the cell membrane (slow cooling) One of damaging factors of cryopreservation Reduction of the effect: cryoprotective agents such as dimethyl sulfoxide (DMSO) or ethylene glycol (EG) lowering the freezing point. The reason: an increase in the con- centration of salt in the extracellular solution because of freezing of the extracellular water. 2

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lipid double layer protein molecules hydro-carbon chains Very low heat conductivity but very good semi-permeability pore with a cell inside A cell located inside of extracellular matrix 3

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- the Stefan condition (mass conservation law) - the osmotic outflow - the fluid density - the normal velocity of the cell boundary - the unfrozen part volume of the pore - the intracellular salt concentration (constant) - the extracellular salt concentration (variable), - the unfrozen water content W c out frozen extracellular liquid cell with intracellular liquid unfrozen extracellular liquid with the increased salt con- centration osmotic outflow tends to balance intra- and extracellular salt con- centration c in Osmotic dehydration of cells during freezing 4 Frémond, M. Non-smooth thermomechanics. Springer- Verlag: Berlin, 2002.

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Implementation The cell region is searched as the level set of a function: 1. Hamilton-Jacobi equation with a regularization term: - Minkowski function of. 2. Exact Hamilton-Jacobi equation: Botkin, N.D. Approximation schemes for finding the value functions. Int. J. of Analysis and its Applications. 1994, Vol. 14, p. 203-220. Numerical techniques from: 5 Souganidis, P. E. Approximation schemes for viscosity solutions of Hamilton – Jacobi equations. Journal of Differential Equations. 1985, Vol. 59, p. 1-43.

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Hamilton-Jacobi equation Difference scheme for viscosity solutions of H-J equations: Hamiltonian: H-J equation: Greed function: 6

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decreases in If monotonically, then The desired monotonicity can be achieved through the following transformation of variables: where is a sufficiently large constant.,. Conditions of the convergence use the right approximations to preserve the Friedrichs property: 7 + + + +

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Simulations: regularization techniques 2D 3D 8

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Simulations: exact H-J equations 9

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Drawbacks Time- and space-consuming numerical procedures. The region to discretize should be much larger than the object to simulate. Accounting for the tension effects (proportional to the curvature) is hardly possible. 10 Development of more advantageous numerical techniques

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g Accounting for curvature-dependent normal velocity Analog of the Gibbs-Thomson condition in the Stefan problem Accounting for the curvature can alter the concavity/convexity structure of the Hamiltonian Hamiltonian Conflict control setting 11 N.N.Krasovskii, A.I.Subbotin

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Conflict control setting The objective of the control is to extend the initial set as much as possible. Reachable set from on time interval [0,t] 12

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Construction of reachable fronts Local convexity Local concavity V.S. Patsko, V.L.Turova. Level sets of the value function in differential games with the homicidal chauffeur dynamics. Int. Game Theory Review. 2001, Vol.3 (1), p.67-112. Reachable fronts 13

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Removing swallow tails: complicated cases 14

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Propagation under isotropic osmotic flux without accounting for the curvature with accounting for the curvature 15, 1 Computation time 6 sec

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Comparison of reachable fronts without the curvature with the curvature 16

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17 without the curvature with the curvature Reachable fronts for isotropic osmotic flux (movie)

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Propagation under an anisotropic osmotic flux without the curvaturewith the curvature 18 1 Computation time 6 sec

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without the curvature with the curvature 19 Comparison of reachable fronts

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20 without the curvature with the curvature Reachable fronts for anisotropic osmotic flux (movie)

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21 Conclusions Very fast numerical method for the simulation of the cell shrinkage under anisotropic osmotic flows is developed. The method allows us to compute the evolution of the cell boundary with accounting for the tension effects. These techniques are suitable for computing of front propagation in problems where the shape evolution is described by a Hamilton-Jacobi equation with the Hamiltonian function being neither convex nor concave in impulse variables.

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