Biomedical Engineering Reference
In-Depth Information
the cells are truly identical. Moreover, it is difficult to accept that different tem-
peratures of IIF for a high number of identical cells subjected to the same freezing
cycle may be ascribed to a nucleation process stochastic in nature. In our opinion a
deterministic approach should be adopted, instead, and a more general definition of
PIIF (taking into account not only ice nucleation but also ice growth and inde-
pendently from the number of nuclei formed inside a cell) needs to be provided,
given that the comparisons between theoretical results and experimental data are
classically carried out in terms of PIIF. On the other hand, as already observed in
the literature by Mazur almost 50 years ago [
24
], large cells are characterised by
smaller surface-to-volume ratio than small ones and, therefore, are expected to
loose less water during the cooling stage. As a consequence, at any given cooling
rate, larger cells retain a higher percentage of internal water than smaller ones,
so that super-cooled conditions are reached earlier, i.e. at higher temperatures.
Following these considerations, we have recently proposed a novel theoretical
interpretation of the PIIF for a suspension of cells by addressing the effect of its
size distribution during the cooling stage of a standard cryopreservation protocol
[
9
,
10
,
11
]. The present Chapter reviews this new modelling approach and its main
implications in the cryopreservation field.
First, model equations are provided in the Model Section. For better high-
lighting our contribution to the field, the model equations are reported by starting
from the traditional description of a single cell response when cryopreserved in the
presence or absence of a CPA. Then, the single cell model is embedded in the
Population Balance Modelling (PBM) approach adopted to properly take into
account the effect of a size-distributed cell population. This naturally leads to a
novel formulation of the PIIF in line with the proposed modelling framework
where statistical variations are neglected. The mathematical description of Extra-
cellular Ice Formation (EIF) dynamics through PBM in order to remove the
classical simplifying assumption of thermodynamic equilibrium conditions for the
extra-cellular compartment is also reported. Model equations are described by
highlighting the coupling between intra- and extra-cellular compartments and the
mutual effects between all the physico-chemical phenomena that might be pro-
gressively taken into account (i.e. water osmosis, CPA permeation, IIF and EIF
dynamics through ice nucleation and crystal growth).
Later, the experimental validation of the novel modelling approach is provided
in the Results and Discussion Section, by directly comparing the theoretical results
to suitable experimental data taken from the literature, where CPA was not
used and EIF dynamics may be neglected. A two-step experimental validation is
performed in order to test model reliability more extensively. First the adjustable
parameters of the proposed model are tuned through fitting procedures
(i.e. regression analysis). Then, system behaviour measured under operating
conditions different from those used during the fitting is predicted by keeping
constant the values of the model parameters adjusted before. The novel formu-
lation of PIIF appears to be far more consistent with the experimental measure-
ments than the previous theoretical interpretations. As expected, it is found that IIF
temperature depends on the cell size, i.e. it is higher for larger cells and smaller
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