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Simulation of cell shrinkage caused by osmotic

4th INTERNATIONAL CONFERENCE ON HIGH PERFORMANCE SCIENTIFIC COMPUTING March 2-6, 2009, Hanoi, Vietnam. Simulation of cell shrinkage caused by osmotic cellular dehydration during freezing *. N.D. Botkin , V. L. Turova. Technische Universität München,

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Simulation of cell shrinkage caused by osmotic

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  1. 4th INTERNATIONAL CONFERENCE ON HIGH PERFORMANCE SCIENTIFIC COMPUTING March 2-6, 2009, Hanoi, Vietnam 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

  2. 1 Stem cells derived from tooth follicle Isolated stem cells derived from dental follicle. A wisdom tooth with soft tissue. 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.

  3. 2 One of damaging factors of cryopreservation Cellular dehydration and shrinkage due to the osmotic outflow through the cell membrane (slow cooling) The reason:an increase in the con-centration of salt in the extracellular solution because of freezing of the extracellular water. Reduction of the effect:cryoprotective agents such as dimethyl sulfoxide (DMSO) or ethylene glycol (EG) lowering the freezing point.

  4. 3 A cell located inside of extracellular matrix pore with a cell inside Very low heat conductivity but very good semi-permeability hydro-carbon chains lipid double layer protein molecules

  5. 4 Osmotic dehydration of cells during freezing - the osmotic outflow frozen extracellular liquid - the fluid density W unfrozen extracellular liquid with the increased salt con- centration - the normal velocity of the cell boundary cin cell with intracellular liquid cout osmotic outflow tends to balance intra- and extracellular salt con- centration • the unfrozen part • volume of the pore - the intracellular salt concentration (constant) - the Stefan condition (mass conservation law) - the extracellular salt concentration (variable) , - the unfrozen water content Frémond, M.Non-smooth thermomechanics. Springer-Verlag: Berlin, 2002.

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

  7. 6 Hamilton-Jacobi equation Hamiltonian: H-J equation: Greed function: Difference scheme for viscosity solutions of H-J equations:

  8. 7 Conditions of the convergence monotonically, then If decreases in use the right approximations to preserve the Friedrichs property: + , + . The desired monotonicity can be achieved through the following transformation of variables: + + is a sufficiently large constant. where

  9. 8 Simulations: regularization techniques 2D 3D

  10. 9 Simulations: exact H-J equations

  11. 10 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. Development of more advantageous numerical techniques

  12. 11 Accounting for curvature-dependent normal velocity Analog of the Gibbs-Thomson condition in the Stefan problem g g z s = - g Hamiltonian z g , s = g Accounting for the curvature can alter the concavity/convexity structure of the Hamiltonian Conflict control setting N.N.Krasovskii, A.I.Subbotin

  13. 12 Conflict control setting z The objective of the control is to extend the initial set as much as possible. Reachable set from on time interval [0,t]

  14. 13 Construction of reachable fronts Reachable fronts z • • Local concavity Local convexity V.S. Patsko, V.L.Turova. Level setsof the valuefunction in differential gameswith the homicidalchauffeurdynamics. Int. Game Theory Review. 2001, Vol.3 (1), p.67-112.

  15. 14 Removing swallow tails: complicated cases

  16. 15 Propagation under isotropic osmotic flux 1 z , with accounting for the curvature without accounting for the curvature d =0.005 Computation time 6 sec

  17. 16 Comparison of reachable fronts without the curvature with the curvature

  18. 17 Reachable fronts for isotropic osmotic flux (movie) without the curvature with the curvature

  19. 18 Propagation under an anisotropic osmotic flux 1 z without the curvature with the curvature d =0.005 Computation time 6 sec

  20. 19 Comparison of reachable fronts without the curvature with the curvature

  21. 20 Reachable fronts for anisotropic osmotic flux (movie) without the curvature with the curvature

  22. 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 concavein impulse variables.

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