# Optimal Control in Cryopreservation of Cells and Tissues

### From DFG-SPP 1253

## Project leader

Lehrstuhl für angewandte Mathematik

TU München

Tel: 089 289-16802

Fax: 089 289-16809

Email: hoffmann@ma.tum.de

Homepage:

## DFG funded assistant

Dr. Nikolai Botkin (TU München)

Zentrum Mathematik M6

Tel. 089 289 16828

Email: botkin@ma.tum.de

Varvara Turova (TU München)

turova@ma.tum.de

## Description

The proposal concerns the application of the theory of partial differential equations and optimal control techniques to the minimization of damaging factors in cryopreservation of living cells and tissues in order to increase the survival rate of frozen and subsequently thawed out cells.

Optimization and control are necessary because several competitive conditions should be
satisfied. A slow cooling causes freezing of the extracellular fluid, whereas the intracellular
fluid remains unfrozen for a while. This results in an increase in the concentration of salt in
the extracellular solution, which leads to the cellular dehydration and shrinkage due to the
osmotic flow through the cell membrane. If cooling is rapid, the water inside the cells forms small,
irregularly-shaped ice crystals (dendrites) that are relatively unstable.
If the cells are subsequently thawed out too slowly, these crystals will aggregate
to form larger, more stable crystals which may cause damage. Maximum viability is
obtained by cooling at a rate in a *transition zone*
in which the combined effect of
both these mechanisms is minimized. Thus, an optimization problem can be formulated
for a mathematical model describing the processes of freezing and thawing.
Moreover some general arguments and our experiments with freezing facilities
show that better results can be obtained, if the temperature falls not monotonically in time,
especially in the range were the latent heat is released. Thus, time dependent optimal
controls (optimal cooling protocols) are to be considered. Our numerical simulations and experiments
show that positive effects can be achieved by creating temperature gradients in the freezing area
or by forcing ice nucleation through mechanical vibration or some temperature chocks localized in a
small area (seeding). Another control tool is cryoprotective agents which vary eutectic properties
of solutions.

Additional difficulties arise when preserving solid tissues.
Significant problems are associated with the difficulty of generating heat transfer within
a large thermal mass with a complex geometry. The packing density of cells within their
extracellular matrix can
approach 80%, whereas preservation of isolated cells becomes problematic at a cell concentration
above 20%. The presence of different cell types, each with its own requirements for optimal
cryopreservation, limits the recovery of each when a single thermal protocol is imposed on all
of the cells. Extracellular ice can cause mechanical damage to the structural integrity of the
tissue. There are mechanical stresses caused by the phase volume change and the osmotic
movement of intracellular water.
Each of these is an additional source of damage, over and above those that are already know from
studies of cells in suspension. Therefore, optimization and optimal control are necessary in this
case even more then when preserving isolated cells.

The objective of this proposal
is the development of a hierarchical, coupled mathematical model that describes the most inuring
effects of cryopreservation of living cells and especially tissues. The models should be
basically governed by partial differential equations and should contain feasible controls that
can essentially reduce the inuring effects of freeing and thawing, when chosen optimally.
Such a model is expected to consist of several coupled submodels each of them covers a part of
the problem.

The first submodel is assumed to utilize mean values of thermodynamical 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.
A first version of such a model and control design are developed by the applicant and tested
in experiments on freezing of dental tissues .

The boundary temperature obtained in the previous step should be then used as the
boundary condition for the second submodel formulated in the interior of the ampoule and
describing ice formation and the concentration of salt in the milieu containing the tissue.
This submodel has to deal with spatially distributed parameters and should be governed
by partial differential equations related to phase-field models or Stefan problems.
The boundary temperature obtained from the output of the first submodel, cryoprotective
agents, vibrations, and local heat chocks are control factors here. The extensive preliminary
work of the applicant in the field of control theory for phase change
problems ensures successful and quick implementation of this part.

The third submodel coupled with the previous one should describe
ice formation inside the extracellular matrix of the tissue. This includes propagation of
the milieu in the extracellular matrix, phase change in the extracellular liquid confined
inside the extracellular matrix, and mechanical stresses arising in the tissue due to the phase
change and because of the osmotic flow through the cell membranes. The control factor here
is first of all the output of the previous submodel. Another candidate for the control
factor is an artificially created temperature gradient. The implementation of this
submodel demands the development of a new class of phase-field models which account for the
interaction of the phases with the cell membranes and the fibers of the extracellular
matrix.

The fourth, last, submodel coupled immediately with the previous one
should be based on conventional descriptions of intracellular
ice formation and cellular dehydration. The dependence of the cell membrane permeability
on the temperature and the concentration of salt and cryoprotectors is an important
element of this setting.

New methods of control theory must be developed to treat the outlined hierarchical model. The objective index has to reflect the survival rate of tissue cells, which includes accounting for biological aspects of the problem. Analytical characterization, numerical implementation, and practical verification of optimal controls through experiments is the main intention of the project. Project home page: http://www-m6.ma.tum.de/~botkin/spp1253.html