Modelling the Cryopreservation Process of a Suspension of Cells: The Effect of a Size-Distributed Cell Population - Computational Modeling in Tissue Engineering - page 155

þ1

R

nV; ð j g ice 50%;
g water 10%

dV

0

PIIF det ect ðÞ¼ ; ð 17 Þ

N tot

where g water ðÞ¼ V ice ðÞþ V water ðÞ

V ðÞ is the fraction of water (solid plus liquid) inside any
single size class of the cell population. This way, iced-up cells are assumed to be
experimentally detectable by human eye looking at a microscope if the ice volume
fraction reaches the value of 0.5 (i.e. g ice C 0.5), whilst a relatively small content
of intra-cellular liquid water equal to 10% vol. is maintained (i.e. g water C 0.1).
Accordingly, the PIIF detect is evaluated at each temperature level as the fraction of
frozen cells at that temperature containing a minimum amount of water (thresh-
old). The choice of a specific threshold value for g water in order to detect ice
formation is quite reasonable. Indeed, the volumetric content of total water
(V water ? V ice ) inside the cells may be relatively small with respect to the inactive
volume V b and/or CPA volume V CPA in Eq. 1 . Therefore, ice formation may not be
experimentally detectable through the darkening/flashing of the cell commonly
used in cryomicroscopy, even if the liquid water contained in the cell is completely
turned into ice (i.e. g ice ? 1 and V water ? 0). On the basis of the chosen
thresholds for g ice and g water , in general the size distribution of the cell population
may be divided in three adjacent portions, for which different fates are assumed:
unfrozen cells characterised by g ice \ 0.5; iced-up cells with a small, innocuous
amount of water when g ice C 0.5 and g water \ 0.1; iced-up cells with a large, lethal
amount of water when g ice C 0.5 and g water C 0.1.
Alternatively, as very recently proposed in the literature [ 37 ], IIF may
be measured by means of a high-speed (8000 fps) video cryomicroscopy system.
This way, cells at the initial stage of ice formation (before flashing takes place)
may be numbered and a cumulative incidence of initial IIF (PIIF init ) may be
experimentally determined. This new cumulative fraction of cells with an initial
ice formation (i.e. when g ice becomes greater than zero, g ice [ 0) is defined as:

þ1

R

nV; ð j g ice [ 0 dV
N tot

0

PIIF init ðÞ¼ ð 18 Þ

3 Results and Discussion

3.1 Experimental Validation of the Model

The most relevant aspect of the mathematical modelling of cryopreservation based
on PBM is that the different IIF temperatures experimentally evidenced for a pool
of a given cell lineage is not ascribed to statistical variations but is explained
through a deterministic criterion: the size distribution of the cell population.