Mathematical modeling and control in cryopreservaton of living cells and tissues

Prof. Dr. K-H. Hoffmann. Technical University of Munich, Center of Mathematical Sciences, Chair of Applied Mathematics

Dr. N. D. Botkin. Technical University of Munich, Center of Mathematical Sciences, Chair of Applied Mathematics

DFG Priority Program SPP 1253

Kick-Off-Meeting

Optimization with partial differential equations

Thurnau, November 20 and 21, 2006

Lo

uis

Leo

pold

Bo

illy

1825

Klassifizierung der Zahntypen

Zahn mit unvollständig ausgebildeter Wurzel, von Knochen vollständig umgeben (impaktiert), normale/gekippte Achsenlage

Zahnkeim mit ausgebildeter Krone, ohne Wurzelbildung, von Knochen vollständig umgeben, normale/gekippte Achsenlage

Weisheitszähne mit WeichgewebeStadium I

Weisheitszähne mit WeichgewebeStadium II

Isolierte Zellen des Zahnfollikels

48 SI / KA / pO / d14broad flattened

small spindle-shaped

small spherical

Tooth

Cell Culture

Tissue Engineering

EctomesenchymalStem Cell

Cyto-differentiation

Tooth Repair(Connective Tissue)

Postponed repair of teeth using stored stem cells

cryopreservaton

1. Cellular dehydration and shrinkage due to the osmotic outflow through the cell membrane (slow cooling)2. Formation of intra- and extracellular irregular ice crystals (rapid cooling)3. Sudden cooling down may result in a thermal shock that can occur both above and below the freezing point

% survival % damage

cooling rate

Mechanismsassociatedwith rapidcooling

Mechanismsassociatedwith slowcooling

• A high packing density (up to 80%) of cells within their extracellular matrix

• The presence of different cell types, each with its own requirements for optimal cryopreservation

• Mechanical damage to the structural integrity due to stresses caused by ice formation inside the extracellular matrix

Additional difficulties for tissues:

Damaging factors:

Some damaging factors of cryopreservation

(SY-LAB Geräte GmbH, Austria)

- cooling system is based on gas nitrogen- chamber and sample sensors measure the responses - cooling protocols are input either manually or as a file- chamber temperature tracks the cooling protocol

Freezingchamber

Sample sensor

Computer

Chambersensor

Rack

Ampoules

Experiments with a freezer IceCube 15M

Main features:

S - supercoolingL - release of the latent heat D - sharp temperature drop

Time

Tem

pera

ture

Tf

S L D

• diminishing the duration of the latent heat release

• elimination of the sharp temperature drop

Objectives of the optimal control:

Typical response to cooling with a constant rate

Ampoule with a tissue and a milieu

• reduction of supercooling applying seeding techniques

sample temperature

chamber temperatureD shock to cells,mechanical stresses

S +Ldendritic ice crystal growth

Averaging model of the cooling process

- the averaged enthalpy density

- the volume and surface area of the ampoule

- the overall heat conductivity coefficient

- the H -T material law (from experiments)

- the temperature in the chamber

,

- the density- the specific heat

Per unit volume :

For common materialswithout phase changes:

,

measured measuredknown

!

!

Identification of :

Ampoule

sample sensor

chamber sensor

λ

λ

λ

Optimal control for DMSO solute & tissue sample: comparison with experiment

computed

experiment

optimal cooling protocol computed

S - supercoolingL - release of the latent heat D - sharp temperature drop

The averaged model works good in region D and partially L.Supercooling jump S cannot be suppressed using the averaged model.

Ampoule with a tissue and a milieu

Phase field model of the cooling process

,

is the scaled temperature

the phase function; solid liquid,

the scaled latent heatthe scaled heat conductivitythe scaled overall heat conductivitythe boundary control (add-on to the nominal profile )

on

inpl

astic

plas

tic o

r met

al

the nominal cooling profile

Caginalp, Hoffmann, Niezgodka, Pawlow, Sprekels

λ

λ

Verification of the phase field model

The temperature jumps grow wit the decrease of the cooling rate, which is consistent with experiments.

Passive suppression of the jump

the bottom of the ampoule is made of a metal the bottom of the ampoule is made of a plastic

The effect is observable but not to strong.

~ 10° C

Active tracking the nominal cooling profile

- index to be minimized

on

in

,

The behavior of the control is reasonablein the sense of minimizing : thedeviation of the temperature becomes smaller. Nevertheless this method is notacceptable because of sharp changesof the temperature.

without controlwith control

λ

Active tracking the nominal slope

- index to be minimized

on,

in

!!!

!!!

J. L. Lions. Transposing techniques:

duality L2 product

very hard to implement numerically !!!

Active averaged tracking the nominal slope

Averaging the model equations yields:

This regularizes the adjoint equations.

without controlwith controloptimal temperature profile: opt

Temperature (without control)

Temperature (with control)

Ampoule with a tissue and a milieu

plas

ticpl

astic

or m

etal

Phase function (without control)

Phase function (with control)

Towards a full coupled controlled model

Optimal coolingprotocol

Control of the temperature response using averaged models (reported techniques).

Ice formation in theextracellular matrixof tissue. Phase-fieldmodels. (Garcke, Knoll, van Duijn)

Formation of intracellular ice. Thermodynamicmodels of ice nucleationand cellular dehydration(Karlsson, Mazur, Toner )

1

2

3

4

ampoule

ice

solute

tissue

Ice formation in the milieu. Controlled Stefan problems andphase field models (Hoffmann, Niezgodka, Pawlow, Sprekels)

In work !

Implemented !

The second submodel formulated in the interior of the ampoule should describe ice formation and the concentration of salt in the milieu containing the tissue. The boundary temperature obtained from the output of the previous submodel, cryoprotective agents, vibrations, and local heat chocks are control factors here. (Hoffmann, Chen, Lishang, Niezgodka Pawlow, Sprekels)

The first submodel utilizes mean values of thermodynamic parameters to describe the boundary temperature of the ampoule with a milieu containing a tissue sample. The control here is the temperature regime in the freezing chamber containing the ampoule. (Hoffmann, Botkin, Pykhteev, Thie)

The third submodel coupled with the previous one has to describes ice formation inside the extracellular matrix of the tissue including mechanical stresses. Control factors here are the output of the previous submodel and artificially created temperature gradients. (Garcke, Knoll, van Duijn )

The last submodel coupled immediately with the previous one should be based on conventional descriptions of intracellular ice formation and cellular dehydration. (Karlsson, Mazur, Toner )

1

2

3

4

Short characteristics of the submodels

!

Ying Xu, J. M. McDonough, K. A. Tagavi, Dayong Gao. Two-Dimensional Phase-Field Model Applied to Freezing into Supercooled Melt. Cell Preservation Technology. Jun 2004, Vol. 2, No. 2: 113-124.

K.-H. Hoffmann, J. Sprekels, and M. Niezgodka. Feedback Control via Thermostats of Multidimensional Two-Phase Stefan Problems. Journal of Nonlinear Analysis: Theory, Methods and Applications, Vol. 15, No.10 (1990), 955-976.

K.-H. Hoffmann and Jiang Lishang. Optimal control of a phase field model for solidification. Numer. Funct. Anal. Optimiz., 13 (1&2) (1992), 11 - 27.

Some references

I. Pawlow. Optimal control of dynamical processes in two-phase systems of solid-liquid type. Numerical analysis and mathematical modelling. Banach Center Publ., 24, PWN, Warsaw, (1990) 293-319.

C. J. van Duijn and R. J. Schotting. Brine transport in porous media: On the use of Von Mises and similarity transformations. Scientific report of Delft Technical University, Centre for Mathematics and Computer Science (CWI), Modelling, Analysis and Simulation (MAS): MAS-R9724, (1997).

H. Garcke. On mathematical models for phase separation in elastically stressed solids. Habilitationsschrift. Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelm-Universität Bonn, Bonn 2000.

C. Morsczeck, T, W. Götz, J. Schierholz, F. Zeilhofer, U. Kühn, C. Möl, C. Sippel, K.-H. Hoffmann. Isolation of precursor cells (PCs) from human dental follicle of wisdom teeth. Matrix Biology 24 (2005) 155-165.

M. Thie, O. Degistirici, B. Faßbender, J. Siemonsmeier, N. Botkin. Zahn-Einfrier- Behälter. Patent submitted 7.08.2005 to Deutchen Patent- und Markenamt.

Peter Mazur. Principles of Cryobiology. In: Barry J. Fuller, Nick Lane and Erica E. Benson, editors, Life in the Frozen State, CRC Press 2004, BocaRaton, New York, Washington, D.C. pp. 3-65.

D. Knoll, D. Kothe, and B. Lally. A New Nonlinear Solution Method for Phase Change Problems. LANL Report LA-UR-98-2350 & Numerical Heat Transfer Part B, 35(4), (1999).

Jens. O. M. Karlsson, Ernest G. Cravalho, Inne H. M. Borel Rinkes, Ronald G. Tompkins, Martin L. Yarmush, and Mehmet Toner. Nucleation and growth of ice crystals inside cultured hepato-cytes during freezing in the presence of dimethyl sulfoxide. Biophys J. 1993 Dec; 65(6):2524-36.

Mehmet Toner, Ernest G Cravalho and Marcus Karel. Thermodynamics and kineticsof intracellular ice formation during freezing of biological cells. Journal of AppliedPhysics, 67, 3, 1990, pp. 1582-1593