Simulation of cell shrinkage caused by osmotic cellular dehydration during freezing * N.D. Botkin,...
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Transcript of Simulation of cell shrinkage caused by osmotic cellular dehydration during freezing * N.D. Botkin,...
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:
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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
+
+
+ +
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]
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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
•
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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
Propagation under an anisotropic osmotic flux
without the curvature with the curvature
18
1
Computation time 6 sec
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.