Modelling the Cryopreservation Process of a Suspension of Cells: The Effect of a Size-Distributed Cell Population - Computational Modeling in Tissue Engineering - page 152

the ice was seeded [ 32 ]. As explained by Petersen et al. [ 32 ], if the temperature at
which EIF sets in is reduced, cell dehydration decreases and IIF increases similar
to the case of using higher cooling rates. Needless to say that, generally speaking,
EIF does not take place at a single temperature but instead it involves a range of
temperatures, starting from an initial value. The latter one along with the size of
the temperature range involved actually depend on EIF dynamics and the adopted
cooling rate, unless EIF is triggered by seeding ice on purpose at a specific tem-
perature as commonly done in the cryo-microscopic analysis addressed in the
papers cited just above. Thus, not only the prediction of a single temperature, but
the more general description of the EIF dynamics is necessary when aiming to
develop a reliable and comprehensive modelling of the cryopreservation process.
When removing the restrictive assumption of thermodynamic equilibrium thus
taking into account EIF dynamics, the classic nucleation theory and diffusion-
controlled growth may be adopted, thus following the same constitutive laws used
above to simulate IIF. The only difference between the two compartments is that
the extra-cellular control volume is large enough to reach a sufficient number of
ice crystals, i.e. a population of ice crystals, thus justifying the continuum-
modelling approach through a PBM that we adopted [ 11 ]. On the basis of these
considerations, V ext

ice ðÞ is

calculated as being proportional to the third-order

moment of the extra-cellular ice crystal size distribution n ext

ice r; ð :

ice ðÞ¼ Z

þ1

4
3 p r 3

V ext n ext

ice r; ð dr :

ð 13 Þ

0

The distribution n ext
ice

r; ðÞ represents a number density distribution (i.e. n ext

ice r; ðÞ
dr is the number of extra-cellular ice crystals that at time t have a radius between
r and r ? dr). The dynamics of n ext

ice r; ðÞ is described by the following 1-D PB
equation and the corresponding initial and boundary conditions:

on ext

¼ 0

ice r; ðÞ
ot

þ o
or

G ext ðÞ n ext

ice

r; ðÞ

8
<

n ext
ice

r; ðÞ¼ 0 tt ¼ 0 8 r 2 r ; ext ; þ1 ½ Þ:

ð 14 Þ

Þ¼ B ext

ðÞ

:

r ; ext ; t at r ¼ r ; ext n ext

ice

0

ð 8 t [ 0

G ext ð r ; ext Þ

The quantities B ext 0 ðÞ and G ext ðÞ are the ice crystal's nucleation and growth
rates, respectively. In the initial condition reported above it is assumed that ice
crystals are not present in the extra-cellular compartment when cooling starts at
t ¼ 0. Later, they are formed and increase their size starting from the critical radius
r ; ext . The value of the boundary condition at the lower end of the domain
n ext
ice

r ; ext ; t

ð Þ depends on both the nucleation and growth rates [ 35 ].