Biomedical Engineering Reference
In-Depth Information
obtained through the use of theoretical models and computer-aided optimisation
methods, since computer simulations are less expensive and faster than experi-
mental studies. They can be used to explore conditions that are impossible to
achieve in the lab and they can account for the effect of more than one parameter at
time. This is why mathematical models have been extremely powerful and have
been proposed so far in the literature to analyse and interpret cryobiologically
relevant phenomena [ 26 ]. Starting from the pioneering work by Mazur [ 24 ] some
literature has been accumulated on cryopreservation modelling [ 16 ], but the main
chronological steps where significant improvements were reached are relatively
few. The bi-compartmental transport model proposed by Mazur addressed only the
osmotic behaviour of cells during the cooling stage of a cryopreservation protocol.
Thus, IIF was initially neglected whereas in the extra-cellular solution ice was
assumed to form under thermodynamic equilibrium conditions. Later, the simu-
lation of IIF dynamics was accounted for, whilst no further improvement for
describing the behaviour of the extra-cellular compartment was proposed. More
specifically, osmosis was always described on the basis of the bi-compartmental
transport model first proposed by Mazur, but ice formation inside the cells was
simulated according to the classical nucleation theory albeit considering only the
ice nucleation phenomenon [ 38 ]. Ice crystal growth inside the cells showed up
later, when Karlsson et al. [ 19 ] developed a complete physico-chemical theory
of ice formation inside biological cells by coupling the water transport model
with the theory of ice nucleation and crystal growth. More recently, Gao and
co-workers [ 44 ] proposed a modified version of Karlsson's model, where the
growing, time dependent ice volume inside a cell during cooling is accounted for
into Mazur's equation, thus coupling more tightly osmosis and IIF phenomena.
Intriguingly, a coupled description of water osmosis and CPA permeation followed
by cooling was never performed, even if it is well known that these represent two
fundamental steps taken in sequence from an experimental perspective. In this
regard, addition of a permeant CPA (i.e. CPA equilibration between the two
compartments) was always investigated separately from the cooling stage [ 1 , 3 , 7 ,
13 , 31 , 33 , 41 ].
These models and other more sophisticated ones where non-ideal, multicom-
ponent liquid aqueous solutions are taken into account, are able to predict IIF as a
function of cooling rate, ice seeding temperature, initial CPA concentration, and
CPA type. However, in all these modelling studies ice formation is either
accounted for as taking place within a single representative (i.e. average) cell, or
the PIIF is related to the nucleation rate by assuming sporadic nucleation of
identical cells [ 40 ]. In particular, the PIIF adopted in all these models is defined
and strictly valid only for the case of a crystal growth time negligible with respect
to the nucleation time [ 38 ], i.e. it is assumed that a single ice nucleus per cell is
formed and ice growth is extremely fast. All these represent relevant limitations
for interpreting the behaviour of the system under practicable operating conditions
during a standard cryopreservation process. Indeed, it is difficult to accept through
reasoning based on physical grounds that, in the relatively high number of cell
population
actually
suspended
during
a
standard
cryopreservation
protocols,
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