[124]. In fact recent work has shown that samples cooled in a
suboptimal method may be rescued by sufficiently high warming
rates [15]. Maximal warming rates for straws and most sample
containers are achieved in a circulating water bath, which provides
nearly an order of magnitude faster warming rates than warming in
air [125].
4 Conclusions
This chapter demonstrates the many facets of mathematical modeling of single-cell cryopreservation. Considerations here include the
appropriate choice of transmembrane flux model, chemical potential model, ice formation model, and others. The models considered here are only a subset of a larger system; heat and mass
transport does not exist only on the cellular level, and as such one
cannot in general ignore the effects of spatial gradients of heat and
concentration. In fact, there is a vast body of literature on the
effects of “unstirred layers” and solute-polarization on membrane
mass transport (see, e.g. [126]), but these effects have largely been
ignored in the cryobiological community. This may be because
unstirred layers are often modeled in the literature as additional
permeable membranes in series, yielding a “lumped” permeability
parameter that includes the unstirred layer. The difficulty here is
that membrane permeability measurements and the cryopreservation of cells (in sample tubes) are often performed in very different
environments (e.g., turbulent versus still), that may generate very
different unstirred layer thicknesses. There is also differential heat
transport from the outside of a sample container compared to the
inside of a sample container, even in relatively slow, quasiequilibrium cooling protocols. This may generate differential survival in cells that are particularly sensitive to cooling rates. There are
challenges to modeling CPA equilibration protocols that involve
extremely high concentrations of cryoprotectants, as the viscosity
affects diffusivity, advection, and momentum equations. While the
foundations of these models are also applicable in tissues, the heat
and mass transport models must be adapted to account for the
spatial gradients and the inherent complex geometry and structure
of the tissue.
Mathematical modeling provides cryobiologists a powerful
tool to approach general cryobiological problems, facilitating the
development of cryopreservation strategies for cells and tissues with
scientific and clinical utility. Cryobiology is also an exciting area for
applied mathematicians as it provides a rich source of interesting
and challenging clinically, biologically, and financially relevant problems that are based on classical physical models, yet require a
delicate balance of specificity and utility. The challenges of cryobiological modeling encompass analytic, computational, and
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James D. Benson
suboptimal method may be rescued by sufficiently high warming
rates [15]. Maximal warming rates for straws and most sample
containers are achieved in a circulating water bath, which provides
nearly an order of magnitude faster warming rates than warming in
air [125].
4 Conclusions
This chapter demonstrates the many facets of mathematical modeling of single-cell cryopreservation. Considerations here include the
appropriate choice of transmembrane flux model, chemical potential model, ice formation model, and others. The models considered here are only a subset of a larger system; heat and mass
transport does not exist only on the cellular level, and as such one
cannot in general ignore the effects of spatial gradients of heat and
concentration. In fact, there is a vast body of literature on the
effects of “unstirred layers” and solute-polarization on membrane
mass transport (see, e.g. [126]), but these effects have largely been
ignored in the cryobiological community. This may be because
unstirred layers are often modeled in the literature as additional
permeable membranes in series, yielding a “lumped” permeability
parameter that includes the unstirred layer. The difficulty here is
that membrane permeability measurements and the cryopreservation of cells (in sample tubes) are often performed in very different
environments (e.g., turbulent versus still), that may generate very
different unstirred layer thicknesses. There is also differential heat
transport from the outside of a sample container compared to the
inside of a sample container, even in relatively slow, quasiequilibrium cooling protocols. This may generate differential survival in cells that are particularly sensitive to cooling rates. There are
challenges to modeling CPA equilibration protocols that involve
extremely high concentrations of cryoprotectants, as the viscosity
affects diffusivity, advection, and momentum equations. While the
foundations of these models are also applicable in tissues, the heat
and mass transport models must be adapted to account for the
spatial gradients and the inherent complex geometry and structure
of the tissue.
Mathematical modeling provides cryobiologists a powerful
tool to approach general cryobiological problems, facilitating the
development of cryopreservation strategies for cells and tissues with
scientific and clinical utility. Cryobiology is also an exciting area for
applied mathematicians as it provides a rich source of interesting
and challenging clinically, biologically, and financially relevant problems that are based on classical physical models, yet require a
delicate balance of specificity and utility. The challenges of cryobiological modeling encompass analytic, computational, and
166
James D. Benson