Biomedical Engineering Reference
In-Depth Information
2.6 The Dynamics of a Size-Distributed Population of Cells
The equations reported above for cell volume variation, IIF dynamics and EIF at
thermodynamic equilibrium conditions have been proposed in the literature during
the past decades. This set of equations allowed researchers to predict the ice
formation inside the intra-cellular compartment as a function of different operating
conditions as cooling rate, seeding temperature, initial CPA concentration and
CPA type, but always taking into account a single, average cell size, i.e. addressing
only a mono-sized population of cells. In this context, the striking experimental
evidence that inside a system at homogeneous temperature identical cells show
different temperatures of IIF even if subjected to the same freezing cycle, was
explained by assuming that the nucleation process is stochastic in nature.
Accordingly, the PIIF was related to the nucleation rate by assuming a sporadic
nucleation of identical cells [
38
,
40
].
In his pioneering work on osmotic behaviour of cells during cryopreservation,
Mazur observed that, large cells are characterised by a smaller surface-to-volume
ratio than small ones, and, therefore, are expected to loose less water during the
cooling stage [
24
]. Consequently, at any given cooling rate, larger cells retain an
higher percentage of internal water than smaller ones, so that super-cooled con-
ditions are reached earlier, i.e. at higher temperatures. Even though this aspect was
already highlighted almost 50 years ago, it has never been accounted for when
modelling cryopreservation until the authors recognised that a population of cells
of the same lineage is always distributed in size [
9
]. This is mainly due to the
mitotic cycle progression rather than the biological diversity. According to this
picture, a population of differently sized cells subjected to the same freezing
protocol may exhibit different IIF temperatures. Specifically, one would expect
that the IIF temperature of large cells is higher with respect to that one of small
cells. In other words, PIIF should be related to the initial size distribution of a cell
population, and the sporadic nucleation of identical cells should be discarded in
favour of a deterministic criterion.
To this aim, a population of cells characterised by a size distribution needs to be
considered and the dynamics of this distribution during the cryopreservation
protocol needs to be quantitatively described. In the framework of a continuum
modelling approach, this task may be accomplished by means of the following
PBM:
o
nV;
ðÞ
ot
þ
o
G
v
ðÞ
nV;
ðÞ
½
¼
0
oV
;
ð
15
Þ
nV;
ðÞ¼
n
0
ðÞ
at t
¼
0;
8
V
2
0
; þ1
½
nV;
ðÞ¼
n
1
; t
ð
Þ¼
0
8
t
where n(V; t) represents the cell number density distribution (i.e. n(V; t) dV is the
number of cells that at time t possess a volume between V and V ? dV), whilst
n
0
(V) refers to the initial size distribution of the cell population at isotonic con-
ditions. The PBM formulation inherently ensures a constant number of cells N
tot
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