Biomedical Engineering Reference
In-Depth Information
Fig. 4 Initial condition for
Eq. 15 : cell number density
distributions of fictitious
populations of small and
large cells (adapted from [ 9 ])
2.5x10 17
2.0x10 17
n 1 0 (V)
n 0 (V)
1.5x10 17
1.0x10 17
5.0x10 16
0.0
0.0
5.0x10 -15
1.0x10 -14
1.5x10 -14
Cell volume, V [m 3 ]
largest size class. In order to better highlight the effect of a cell size distribution,
two additional fictitious, narrow, initial cell number density distributions, n 1 ð V)
and n 2 ð V), are considered. These distributions, reported in Fig. 4 , are characterised
by mean values which correspond to the smallest and the largest size class cells of
the experimental distribution given in Fig. 3 , respectively.
The complete model describing trans-membrane transport phenomena along
with IIF in a cell population involves a large number of parameters. Many of them
may be taken from the literature, whilst only five of them related to osmosis and
ice nucleation have to be determined through a fitting procedure. Ideally, they
should be fitted independently, using specifically designed experimental runs, to
reduce ambiguity whilst preserving model consistency: first the osmotic behaviour
should be determined whilst neglecting ice nucleation, then IIF may be safely
analysed. Unfortunately, the set of experimental runs necessary for this ideal fitting
procedure is not available. Therefore, all the five parameters have been determined
simultaneously by direct comparison between model results and experimental data
in terms of PIIF of 60 isolated rat hepatocytes cooled at -400C/min in a cryo-
microscopy system [ 40 ] as reported in Fig. 5 . As it can be seen, the PBM approach
is able to simulate the completion of IIF at -400C/min in the temperature range
[-15 Cto-5C] when the initial cell size distribution reported in Fig. 3 is taken
into account.
On the contrary, by adopting the two fictitious initial distributions reported in
Fig. 4 , whilst keeping constant all the other parameters and operative conditions of
the experimental run, the PIIF vs system temperature simulated by the proposed
model is completely different. Indeed, the model predicts no IIF down to -16C
for the case of a population of large cells, n 1 ð V), whilst an abrupt increase of PIIF
takes place around -5.5C, for the case of a population of small cells, n 2 ð V).
The different behaviour of cells according to their size is shown from a different
perspective in Fig. 6 . Here the path of the intra-cellular sodium chloride con-
centrations for the smallest and the largest size class cells of the initial population
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