Modelling the Cryopreservation Process of a Suspension of Cells: The Effect of a Size-Distributed Cell Population - Computational Modeling in Tissue Engineering - page 148

;

L p ðÞ¼ L p ; ref exp E
< T ðÞ 1

1

ð 6 Þ

T ref

where E is the apparent activation energy, and the pre-exponential factor L p,ref
represents the value of hydraulic permeability at a given, reference temperature
T ref .
When a permeant CPA is used, cell volume changes in response not only to
water osmosis but also to CPA permeation. In this case, a rather more complex
description results, since water osmosis and CPA permeation are coupled phe-
nomena from the material transport perspective. More specifically, if water and
CPA are co-transported (i.e. water and CPA molecules are transferred through
cell membrane by travelling along the same transport channels) the Kedem and
Katchalsky formalism [ 21 ] may be adopted to simulate cell volume variation:

osmosis

dV

dt ¼ dV water

þ dV CPA
dt
¼ L p ðÞ A ðÞ< T

dt

þ r

m int

NaCl ðÞ m ext

m int
CPA

ðÞ m ext

NaCl ðÞ CPA ðÞ

ð 7 Þ

Here r (called the reflection coefficient) is the adjustable parameter that takes
into account the interactions between water and CPA transport across cellular
membrane.
Equation 5 or 7 are used to determine cell volume V ð t Þ in presence or absence
of CPA, respectively. To this aim, the concentrations of NaCl and CPA in the
intra- and the extra-cellular terms of the driving forces appearing in Eq. 5 or 7
have to be determined. These concentrations are calculated as inversely propor-
tional to the liquid water volumes, i.e. V water ð t Þ and V ext

water ð t Þ , correspondingly,
which are determined from Eqs. 1 and 2 . Of course, NaCl and CPA concentrations
are also proportional to their corresponding contents (i.e. osmoles) in the intra- and
the extra-cellular compartments. In this regard, whilst the content in both com-
partments of the not-permeant solute NaCl is constantly equal to a given initial
value (i.e. the so-called isotonic condition), the intra-cellular CPA content may
change due to permeation and, according to Kedem and Katchalsky formalism, it
is calculated as:

dV
dt

m int
CPA

ðÞþ m ext
CPA

dn CPA
dt

ðÞ

¼ 1 r ð Þ

2 ð 8 Þ

þ P CPA ðÞ A ðÞ m ext ðÞ m int

CPA

ðÞ

CPA

where n CPA represents the number of CPA moles inside the cell, whilst P CPA is the
permeability of CPA that shows an Arrhenius-like temperature dependence as the
hydraulic permeability L p reported in Eq. 6 .
At this point it is worth noting that, some assumptions detailed above cannot be
made and should be actually avoided, even though they are widely used in the
technical literature of cryopreservation modelling. Among these, the van't Hoff