Modelling the Cryopreservation Process of a Suspension of Cells: The Effect of a Size-Distributed Cell Population - Computational Modeling in Tissue Engineering - page 149

equation that describe the osmotic pressure in analogy with the ideal-gas equation
(see Eq. 4 ) is not valid when cooling at low temperatures, where very concentrated
solutions may be obtained due to liquid water sequestering for ice formation.
In this latter case, the concentrations should be replaced by activities to express
the osmotic driving forces of Eqs. 5 - 7 , thus accounting for non-ideal and/or
concentrated liquid mixtures through the Debye-Huckel theory [ 36 ] or the osmotic
virial equation [ 8 ].
An additional assumption that may not be justified is represented by water and
CPA co-transport across cellular membrane, i.e. the Kedem and Katchalsky for-
malism given in Eqs. 7 and 8 . Indeed, in natural biological membranes co-transport
is often unlikely and/or negligible, since transport channels are highly selective and
stereo-specific [ 20 , 22 , 26 ]. In such a case, the two-parameter formalism reported by
Kleinhans [ 22 ] may be adopted thus making the experimental validation of the model
easier by reducing the number of adjustable parameters (the reflection coefficient
r is abandoned).

2.4 Intra-Cellular Ice Formation

In order to evaluate the volume of liquid water in a cell V water (t) through Eq. 1 ,an
independent equation is required to determine the intra-cellular volume of ice
V ice (t). This task is accomplished by modelling IIF dynamics. To this aim,
a mechanistic scenario of nucleation and diffusion-limited growth of ice crystals is
typically adopted.
In this context, if cryopreservation is carried out following standard protocols
(i.e. slow cooling and low CPA concentrations), ice nucleation may take place
only at relatively high temperatures, when crystal growth is relatively fast.
Therefore, the rapid ice crystal growth in association with a small control volume
(i.e. cell volume) generate only a very limited number of ice crystals inside a cell
(typically no more than one). On the other hand, when cooling down at higher
rates IIF is hindered kinetically, so that the intra-cellular liquid solution eventually
reaches vitrification at low temperatures. This aspect, clearly prevents the appli-
cation of a continuum modelling approach (i.e. PBM) to simulate ice formation
inside a cell.
As a consequence, the intra-cellular ice volume V ice (t) appearing in Eq. 1 has to
be determined necessarily through a discrete modelling approach, by following the
fate of any single ice nucleus that is first formed, and then grows as time increases
[ 19 ]:

8
<
:

0 if N ice ðÞ¼ 0

N ice ðÞ

P

V ice ðÞ¼ ; ð 9 Þ

3

4p
3

½ r i ðÞ if N ice ðÞ 1

i ¼ 1