however, one uses either Eq. 21 or, after dividing the data by
isosmotic volume, Eq. 25 with a numerical solver such as Matlab
or Mathematica to define estimated time-volume pairs as a function
of water and solute permeabilities. The sum of squared errors
between model prediction and data are calculated and used in a
numerical minimization routine such as “fmincon” in Matlab or
“FindMinimum” in Mathematica.
It is a good practice to perform trial fits to get an estimate of
good fit parameters, and a good starting point is parameter values
from the literature. This serves a number of purposes. First, it
ensures that model implementation is correct and experimental
data have been treated appropriately. Second, it gives the modeler
a better feel for the data: it is always advisable to hand fit a selection
of a data set to ensure that there are no problems with the data at
large, and to serve as a comparison of parameters to algorithmically
fit datasets. Finally, this initial estimate of good fit parameters and
the requisite numerical “exploration” of the relation between parameters, fits, and data give the modeler insight into reasonable upper
and lower bounds for parameters. These bounds can be implemented in the code and reduce fitting time and importantly, erroneous
fits. This happens most frequently when insufficient data at the
volume extrema are provided. The sum of squares function frequently has two local minima, one in an “expected” region and one
with an artificially high value for L p (see Fig. 7).
2.6 Ice Formation
Models
While the modeling of extracellular ice has a long and active history
in and beyond the cryobiological literature, there are essentially two
models of intracellular ice formation used to predict the likelihood
that a particular modeled cryopreservation protocol will cause the
formation of potentially lethal intracellular ice. The first model was
defined in Mazur’s seminal paper [1]. Mazur’s method for prediction of intracellular ice is based on an experimentally observed and
modeling-justified statement that the likelihood of intracellular ice
increases dramatically when the cellular solution is supercooled to
2
∘ C below its melting temperature.
More modern approaches to formulate models to predict intracellular ice formation have been proposed by Toner et al. [93] and
Karlsson et al. [94, 95]. In particular they use the hypothesis that
the growth rate of ice crystals is limited by the diffusivity of water
during cooling. Combined with temperature and viscosity dependent stochastic models of ice nucleation that include the likelihood
of nucleation within the cell, on the cell membrane, and outside of
the cell, the model provides repeatable predictions of the likelihood
that a cell will undergo intracellular ice formation given a particular
cooling protocol. Because of the complexity this model is not
provided here.
Mathematical Modeling and Optimization of Cryopreservation in Single Cells
153
isosmotic volume, Eq. 25 with a numerical solver such as Matlab
or Mathematica to define estimated time-volume pairs as a function
of water and solute permeabilities. The sum of squared errors
between model prediction and data are calculated and used in a
numerical minimization routine such as “fmincon” in Matlab or
“FindMinimum” in Mathematica.
It is a good practice to perform trial fits to get an estimate of
good fit parameters, and a good starting point is parameter values
from the literature. This serves a number of purposes. First, it
ensures that model implementation is correct and experimental
data have been treated appropriately. Second, it gives the modeler
a better feel for the data: it is always advisable to hand fit a selection
of a data set to ensure that there are no problems with the data at
large, and to serve as a comparison of parameters to algorithmically
fit datasets. Finally, this initial estimate of good fit parameters and
the requisite numerical “exploration” of the relation between parameters, fits, and data give the modeler insight into reasonable upper
and lower bounds for parameters. These bounds can be implemented in the code and reduce fitting time and importantly, erroneous
fits. This happens most frequently when insufficient data at the
volume extrema are provided. The sum of squares function frequently has two local minima, one in an “expected” region and one
with an artificially high value for L p (see Fig. 7).
2.6 Ice Formation
Models
While the modeling of extracellular ice has a long and active history
in and beyond the cryobiological literature, there are essentially two
models of intracellular ice formation used to predict the likelihood
that a particular modeled cryopreservation protocol will cause the
formation of potentially lethal intracellular ice. The first model was
defined in Mazur’s seminal paper [1]. Mazur’s method for prediction of intracellular ice is based on an experimentally observed and
modeling-justified statement that the likelihood of intracellular ice
increases dramatically when the cellular solution is supercooled to
2
∘ C below its melting temperature.
More modern approaches to formulate models to predict intracellular ice formation have been proposed by Toner et al. [93] and
Karlsson et al. [94, 95]. In particular they use the hypothesis that
the growth rate of ice crystals is limited by the diffusivity of water
during cooling. Combined with temperature and viscosity dependent stochastic models of ice nucleation that include the likelihood
of nucleation within the cell, on the cell membrane, and outside of
the cell, the model provides repeatable predictions of the likelihood
that a cell will undergo intracellular ice formation given a particular
cooling protocol. Because of the complexity this model is not
provided here.
Mathematical Modeling and Optimization of Cryopreservation in Single Cells
153