Modelling the Cryopreservation Process of a Suspension of Cells: The Effect of a Size-Distributed Cell Population - Computational Modeling in Tissue Engineering - page 154

(i.e. R þ1
0

ðÞ dV ¼ R þ1 n 0 throughout

0 nV; ðÞ dV ¼ N tot ). G v (V)
appearing in Eq. 15 is the rate of change of the cell volume due to water osmosis
and CPA permeation. A negative growth rate G v (V) shifts the cell number density
distribution n(V;t) towards smaller volumes, thus simulating cells shrinkage in a
hypertonic environment. In reverse, cell volume expansion in a hypotonic extra-
cellular solution is reflected by a positive G v (V).
The rate of cell volume variation G v (V) may be evaluated by means of Eq. 5 for
the case of CPA absence or Eq. 7 when a CPA is used. Actually, all the equations
reported above for the case of a single, average cell may be adopted when
addressing a population of cells characterised by a size distribution, paying
attention to make proper distinctions. Basically, any single sized class of cells
composing the distribution of the cell population behaves independently by fol-
lowing the same equations given above for the simulation of water osmosis, CPA
permeation and the discrete IIF approach. As such, any single sized class of cells is
characterised at any time during cryopreservation by its own intra-cellular water
content, NaCl and CPA concentrations. Even the IIF and the extent of water phase
change may differ from class to class of cells, whilst all of them communicate
through water osmosis and CPA permeation with the same extra-cellular
compartment which is characterised by a single, homogenous value of chemical
species concentrations and EIF extent.

the process

2.7 Probability of Intra-Cellular Ice Formation

Experimentally, the temperature at which IIF occurs at given cooling conditions is
commonly determined looking at cells under a microscope from the darkening
of the cell due to light scattering of the small ice crystals that form inside.
The number of iced-up cells is then evaluated using cryomicroscopy, and the
corresponding PIIF is computed at each temperature level as the fraction of cells
(with respect to the total initial ones) cumulatively frozen at that temperature.
When accounting for a size-distributed population of cells, the model of spo-
radic nucleation of identical cells used so far in the literature to define the PIIF has
to be discarded in favour of a deterministic criterion that takes into account the
different extent of IIF from class to class of cells, i.e. the distribution of IIF [ 9 , 10 ].
To this aim, it is assumed that cells are iced-up when their corresponding internal
ice volume fraction g ice ðÞ¼ V ice ðÞ

V ðÞ V b reaches the value of 0.5. Accordingly, the PIIF

may be defined as:

R

þ1

nV; ð j g ice 0 : 5 dV

0

PIIF ðÞ¼

: ð 16 Þ

N tot

Besides, a detectable value for the probability of internal ice formation may be
defined as: