3.1.1 Classical CPA
Equilibration Optimization
Approach
Classically, optimal CPA equilibration strategies have been multistep (piecewise constant in the context of the above section) protocols where the choice of step concentrations and durations have
typically been driven by minimizing the number of steps while
keeping cells within osmotic tolerance limits (see, e.g. Fig. 9). As
an example, suppose that we wish to equilibrate cells to a desired
final intracellular CPA molality m
des
s . In order to determine the
optimal first step, Eq. 14 or its variants are solved for the CPA
concentration m
e
s 1
that causes the cell volume to meet but not
exceed its lower or upper osmotic tolerance limit. This may be
accomplished numerically, by solving the system of differential
equations (21) and using a computational package to optimize
the choice of m
e
s that yields the appropriate minimal or maximal
volume, or analytically using either the method discussed by
Katkov or Zhang and Chen [80, 81] or the method of Benson
et al. [77]. An analytical solution will certainly be faster and
more accurate, but these advantages are of little consequence in
terms of modern computing power unless this step is part of a
much larger optimization problem (see, e.g. Lusianti et al.
[75]).
To continue to implement the method for a CPA addition
protocol, for example, if m
e
s 1
> m
des
s , assign m
e
s 1
¼ m
des
s
and the
optimal protocol will have only a single step. If not, and m
e
s 1
has
been determined, the process may be repeated, assuming that the
initial condition is characterized by a cell equilibrated with CPA of
concentration m
e
s 1
, and the concentration m
e
s 2
will be determined so
that it causes the cell volume to meet but not exceed the lower
osmotic tolerance limit. As with the first step, if m
e
s 2
> m
des
s , assign
m
e
s 2
¼ m
des
s
and the optimal protocol will have two steps. This
process may be repeated as needed until m
e
s n
> m
des
s , in which
case the optimal protocol will have n-steps. For examples of
this approach, see Gilmore et al. [7], who use examine singlestep CPA addition and removal protocols for human spermatozoa to show that a one-step CPA addition protocol will not
cause excessive shrinking if the CPA ethylene glycol is used, but
will do so if glycerol is the CPA, Agca et al. [5] who examine
the relative effects of multistep addition and dilution protocols
for mouse spermatozoa, or Mullen et al. [13] who look at
multistep CPA removal protocols for human oocytes in this
fashion.
3.2 Cooling Rate
The most widely accepted theory of damage during slow cooling
protocols where ice is allowed to nucleate, sequester water, and
concentrate the remaining solutes is called the “two-factor hypothesis” proposed by Mazur, Liebo, and Chu [119]. The hypothesis
was an attempt to explain why damage occurred during sufficiently
slow cooling protocols where the cytoplasm could concentrate
Mathematical Modeling and Optimization of Cryopreservation in Single Cells
159
Equilibration Optimization
Approach
Classically, optimal CPA equilibration strategies have been multistep (piecewise constant in the context of the above section) protocols where the choice of step concentrations and durations have
typically been driven by minimizing the number of steps while
keeping cells within osmotic tolerance limits (see, e.g. Fig. 9). As
an example, suppose that we wish to equilibrate cells to a desired
final intracellular CPA molality m
des
s . In order to determine the
optimal first step, Eq. 14 or its variants are solved for the CPA
concentration m
e
s 1
that causes the cell volume to meet but not
exceed its lower or upper osmotic tolerance limit. This may be
accomplished numerically, by solving the system of differential
equations (21) and using a computational package to optimize
the choice of m
e
s that yields the appropriate minimal or maximal
volume, or analytically using either the method discussed by
Katkov or Zhang and Chen [80, 81] or the method of Benson
et al. [77]. An analytical solution will certainly be faster and
more accurate, but these advantages are of little consequence in
terms of modern computing power unless this step is part of a
much larger optimization problem (see, e.g. Lusianti et al.
[75]).
To continue to implement the method for a CPA addition
protocol, for example, if m
e
s 1
> m
des
s , assign m
e
s 1
¼ m
des
s
and the
optimal protocol will have only a single step. If not, and m
e
s 1
has
been determined, the process may be repeated, assuming that the
initial condition is characterized by a cell equilibrated with CPA of
concentration m
e
s 1
, and the concentration m
e
s 2
will be determined so
that it causes the cell volume to meet but not exceed the lower
osmotic tolerance limit. As with the first step, if m
e
s 2
> m
des
s , assign
m
e
s 2
¼ m
des
s
and the optimal protocol will have two steps. This
process may be repeated as needed until m
e
s n
> m
des
s , in which
case the optimal protocol will have n-steps. For examples of
this approach, see Gilmore et al. [7], who use examine singlestep CPA addition and removal protocols for human spermatozoa to show that a one-step CPA addition protocol will not
cause excessive shrinking if the CPA ethylene glycol is used, but
will do so if glycerol is the CPA, Agca et al. [5] who examine
the relative effects of multistep addition and dilution protocols
for mouse spermatozoa, or Mullen et al. [13] who look at
multistep CPA removal protocols for human oocytes in this
fashion.
3.2 Cooling Rate
The most widely accepted theory of damage during slow cooling
protocols where ice is allowed to nucleate, sequester water, and
concentrate the remaining solutes is called the “two-factor hypothesis” proposed by Mazur, Liebo, and Chu [119]. The hypothesis
was an attempt to explain why damage occurred during sufficiently
slow cooling protocols where the cytoplasm could concentrate
Mathematical Modeling and Optimization of Cryopreservation in Single Cells
159