3.2.1 Mazur Model
Recall that the Mazur model of ice formation during cooling is the
hypothesis that cells are likely to be ice-free if there is less than 2
∘ C
of intracellular supercooling. Karlsson shows that this is likely to be
dramatically wrong for certain combinations of concentrations and
temperatures [95]. Yet the Mazur model yields success even in
recent literature [26, 27]. Perhaps this is in part due to underestimation of the intracellular melting temperature. To wit, typically the intracellular melting temperature is modeled using the
ternary phase diagram of NaCl-CPA-Water or KCl-CPA-Water
(see Subheading 2.3). This approach overlooks large quantities of
intracellular proteins and other structures that contribute
(non-ideally) to melting point depression. Whether it is as precise
as other options or not, the Mazur model is straightforward to
implement and will at least provide order of magnitude or “ballpark” approximations of optimal cooling rates. With these caveats
in place, note that there are a variety of approaches one can use to
optimize cooling rates under this assumption.
Constant Cooling Rate Approach: This first approach yields optimal constant cooling rates (defining temperature as a linear function of time). Typically this means that cells will be cooled in a
controlled rate freezer to a specific subzero temperature and then
removed and immediately plunged into liquid nitrogen. The critical
observation here is that the amount of intracellular supercooling
before reaching the plunge temperature increases monotonically
0.1
1
10
100
1000
10
4
0
10
20
30
40
50
60
Cooling Rate C min
Survival
Mouse
embryos
Human Sperm
RBC
Mouse Sperm
Fig. 11 Cooling rate vs survival in a variety of species. Note that these cooling rates are in the “slow cooling”
regime, as the amount of CPA used in these experiments was not high enough to suppress ice nucleation
and/or significant crystal growth. Redrawn and modified from [121]
Mathematical Modeling and Optimization of Cryopreservation in Single Cells
161
Recall that the Mazur model of ice formation during cooling is the
hypothesis that cells are likely to be ice-free if there is less than 2
∘ C
of intracellular supercooling. Karlsson shows that this is likely to be
dramatically wrong for certain combinations of concentrations and
temperatures [95]. Yet the Mazur model yields success even in
recent literature [26, 27]. Perhaps this is in part due to underestimation of the intracellular melting temperature. To wit, typically the intracellular melting temperature is modeled using the
ternary phase diagram of NaCl-CPA-Water or KCl-CPA-Water
(see Subheading 2.3). This approach overlooks large quantities of
intracellular proteins and other structures that contribute
(non-ideally) to melting point depression. Whether it is as precise
as other options or not, the Mazur model is straightforward to
implement and will at least provide order of magnitude or “ballpark” approximations of optimal cooling rates. With these caveats
in place, note that there are a variety of approaches one can use to
optimize cooling rates under this assumption.
Constant Cooling Rate Approach: This first approach yields optimal constant cooling rates (defining temperature as a linear function of time). Typically this means that cells will be cooled in a
controlled rate freezer to a specific subzero temperature and then
removed and immediately plunged into liquid nitrogen. The critical
observation here is that the amount of intracellular supercooling
before reaching the plunge temperature increases monotonically
0.1
1
10
100
1000
10
4
0
10
20
30
40
50
60
Cooling Rate C min
Survival
Mouse
embryos
Human Sperm
RBC
Mouse Sperm
Fig. 11 Cooling rate vs survival in a variety of species. Note that these cooling rates are in the “slow cooling”
regime, as the amount of CPA used in these experiments was not high enough to suppress ice nucleation
and/or significant crystal growth. Redrawn and modified from [121]
Mathematical Modeling and Optimization of Cryopreservation in Single Cells
161