with increasing cooling rates and with decreasing temperatures. In
particular, due to the temperature dependence of membrane water
permeability (see Eq. 23) if cells are cooled at a fixed rate, their
ability to lose water to “keep up” with extracellular ice formation
decreases with temperature.
The decision about what temperature or intracellular state to
achieve before plunging must be made before optimizing cooling
rate. This temperature or state is that which causes negligible
crystallization during plunging into liquid nitrogen. The rule of
thumb adopted by Liu et al. [4], Kashuba Benson et al. [25], and
others is that plunging when the intracellular CPA is at a “critical
concentration” of 40% w/w should be sufficiently safe, a state
usually occurring between À40 and À80
∘
C. In theory this critical
concentration percentage should be dependent on CPA and the
temperature at which this occurs, but note that cooling rates
through the most dangerous temperatures with regard to ice crystal
growth with its attendant damage are improved in this fashion as
the Leidenfrost effect is minimized due to much lower pre-plunge
temperatures, and that the thermal conductivity of ice which now
makes up a large majority of the system volume, is nearly four times
that of water.
In this case, optimization of cooling rates is simply a matter of
simulating cooling at increasing rates until the maximal supercooling exceeds 2
∘ C. This is shown In Fig. 12 where the intracellular
supercooling at three possible critical concentration CPA percentages (40, 45, and 50%) are given as a function of constant cooling
rate for mouse embryonic stem cells loaded with 1 mol/kg DMSO.
This figure demonstrates the monotonicity of supercooling as a
function of goal concentration. To generate this figure, model
(21) with temperature dependent permeability given by the Arrhenius law in Eq. 23 as well as the simple ODE: dT/dt ¼ Àb with
initial condition T ð0Þ ¼ T
0
melt , the initial melting point of the
solution, are coupled with an experimental phase diagram model
(5) or synthetic phase diagram model (9) to determine the extracellular osmolality and concentration as a function of temperature
(e.g., as in Eq. 12). In practice this optimization takes very little
computational time, though one could choose a rapidly converging
numerical optimization scheme if this was an issue.
Woelders Approach: The second and very elegant approach developed by Woelders and Chaviero [3] uses a Raoult’s law approximation of freezing point depression θ as a function of osmolality:
À θ % 1.86π. This is then solved for the osmolality π such that
θ ¼ T + p where T is the intracellular temperature and p is the degree
of intracellular supercooling allowed. In particular, the difference of
osmolalities across the membrane with the intracellular space fixed
at p degrees of supercooling will be p/1.86. Using this in Eq. 14
and assuming dS/dt % 0 yields
162
James D. Benson
particular, due to the temperature dependence of membrane water
permeability (see Eq. 23) if cells are cooled at a fixed rate, their
ability to lose water to “keep up” with extracellular ice formation
decreases with temperature.
The decision about what temperature or intracellular state to
achieve before plunging must be made before optimizing cooling
rate. This temperature or state is that which causes negligible
crystallization during plunging into liquid nitrogen. The rule of
thumb adopted by Liu et al. [4], Kashuba Benson et al. [25], and
others is that plunging when the intracellular CPA is at a “critical
concentration” of 40% w/w should be sufficiently safe, a state
usually occurring between À40 and À80
∘
C. In theory this critical
concentration percentage should be dependent on CPA and the
temperature at which this occurs, but note that cooling rates
through the most dangerous temperatures with regard to ice crystal
growth with its attendant damage are improved in this fashion as
the Leidenfrost effect is minimized due to much lower pre-plunge
temperatures, and that the thermal conductivity of ice which now
makes up a large majority of the system volume, is nearly four times
that of water.
In this case, optimization of cooling rates is simply a matter of
simulating cooling at increasing rates until the maximal supercooling exceeds 2
∘ C. This is shown In Fig. 12 where the intracellular
supercooling at three possible critical concentration CPA percentages (40, 45, and 50%) are given as a function of constant cooling
rate for mouse embryonic stem cells loaded with 1 mol/kg DMSO.
This figure demonstrates the monotonicity of supercooling as a
function of goal concentration. To generate this figure, model
(21) with temperature dependent permeability given by the Arrhenius law in Eq. 23 as well as the simple ODE: dT/dt ¼ Àb with
initial condition T ð0Þ ¼ T
0
melt , the initial melting point of the
solution, are coupled with an experimental phase diagram model
(5) or synthetic phase diagram model (9) to determine the extracellular osmolality and concentration as a function of temperature
(e.g., as in Eq. 12). In practice this optimization takes very little
computational time, though one could choose a rapidly converging
numerical optimization scheme if this was an issue.
Woelders Approach: The second and very elegant approach developed by Woelders and Chaviero [3] uses a Raoult’s law approximation of freezing point depression θ as a function of osmolality:
À θ % 1.86π. This is then solved for the osmolality π such that
θ ¼ T + p where T is the intracellular temperature and p is the degree
of intracellular supercooling allowed. In particular, the difference of
osmolalities across the membrane with the intracellular space fixed
at p degrees of supercooling will be p/1.86. Using this in Eq. 14
and assuming dS/dt % 0 yields
162
James D. Benson