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

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 usesthereof. European Patent. Date f publication and mention of the grant of the patent 04.06.2008.

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

lipid double layer

protein molecules

hydro-carbon chains

Very low heat conductivity but very good semi-permeability

pore

wit

h a

cell

insid

e

A cell located inside of extracellular matrix3

- 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

cout

frozen extracellular liquid

cell with intracellular liquid

unfrozen extracellular liquidwith the increased salt con-centration

osmotic outflow tends to balanceintra- and extracellular salt con-centration

cin

Osmotic dehydration of cells during freezing 4

Frémond, M. Non-smooth thermomechanics. Springer-Verlag: Berlin, 2002.

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.

Hamilton-Jacobi equation

Difference scheme for viscosity solutions of H-J equations:

Hamiltonian:

H-J equation:

Greed function:

6

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

+

+

+ +

Simulations: regularization techniques

2D 3D

8

Simulations: exact H-J equations

9

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

g

Accounting for curvature-dependent normal velocity

Analog of the Gibbs-Thomson conditionin 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

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

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

Removing swallow tails: complicated cases

14

Propagation under isotropic osmotic flux

without accounting for the curvature with accounting for the curvature

15

, 1

Computation time 6 sec

Comparison of reachable fronts

without the curvature with the curvature

16

17

without the curvature with the curvature

Reachable fronts for isotropic osmotic flux (movie)

Propagation under an anisotropic osmotic flux

without the curvature with the curvature

18

1

Computation time 6 sec

without the curvature with the curvature

19

Comparison of reachable fronts

20

without the curvature with the curvature

Reachable fronts for anisotropic osmotic flux (movie)

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.