Modelling the Cryopreservation Process of a Suspension of Cells: The Effect of a Size-Distributed Cell Population - Computational Modeling in Tissue Engineering - page 146

mass balance may be replaced by a volume balance. Finally, an ideal perfect
mixing is assumed for both the intra- and the extra-cellular compartment, so that a
single concentration value may be defined for any chemical species present in the
two compartments composing the system.
According to this simplifying picture, the following equation may be adopted
for the model of a single cell:

V ðÞ¼ V water ðÞþ V b þ V NaCl þ V ice ðÞþ V CPA ðÞ ð 1 Þ

where V b and V water ðÞ represent the inactive cell volume and the volume of liquid
water, respectively. The total cell volume V ðÞ may vary due to water osmosis and
CPA permeation, being unaffected by the water phase change from liquid to ice
(i.e. DV water ¼ DV water j osmosis þ DV water j phase change and DV water j phase change ¼ DV ice ).
Typically, during cryopreservation one million of cells is suspended in 1 ml of
aqueous solution inside suitable vials. Assuming an average diameter of 10 lm for
each cell, a maximum total cytoplasmic volume of 5.2 9 10 -4 ml is obtained. It is
apparent that the capacity of the suspending aqueous solution is much higher than
the cumulative cytoplasm capacity of the suspended population of cells. Thus, the
mass transfer with the cells through cellular membrane (i.e. water osmosis and
CPA permeation) may affect only negligibly the volume and the concentrations of
the extra-cellular compartment. Then, by contrast with the cell model given in
Eq. 1 , the behaviour of the extra-cellular compartment may be described as
follows:

V ext ¼ V ext

water ðÞþ V ext NaCl þ V ext ice ðÞþ V ext

ð 2 Þ

CPA

where the total extra-cellular volume does not change with time given that
DV ext

water phase change water phase change ¼ DV ext

water ¼ DV ext

and DV ext

ice .

2.2 Temperature Dynamics

Traditionally in cryopreservation, temperature is assumed homogenous throughout
the entire system, i.e. intra- and extra-cellular spatial temperature gradients are
neglected. This is a relevant simplification from modelling perspective, since it
permits to drop the simulation of the thermal behaviour of the system thus
avoiding to couple a thermal balance to the mass one represented by Eqs. 1 and 2 .
Actually, the validity of this assumption depends on the geometries, sizes and
operating conditions specifically adopted. Certainly, the standard use from
experimental perspective of thin vials in large capacity cooling chambers justifies
this simplifying assumption of the theoretical description.
According to these considerations, the modelling of cryopreservation processes
typically assigns a specific profile to the decreasing temperature of the entire
system, regardless of any thermal accumulation, latent heat and heat transfer