published a four part series on the foundations of modeling to
address these questions, and summarize key considerations for
modeling all aspects of cells in the presence of encroaching ice
[16–19].
The scope of modeling in modern cryobiology is very broad. It
includes heat and mass transport in tissues, organs, and hybrid
systems, as well as complicated ice dynamics and formation modeling in the presence and absence of cellular systems [20–23]. It now
even includes informatics approaches to understanding cryobiological outcomes [24]. This chapter presents the now classical modeling and optimization of single-cell cryopreservation protocols and a
straightforward adaptation of these approaches to tissues. For a
complete experimental approach that includes biophysical measurement of parameters, prediction of optimal cryopreservation protocols based on those parameters and the approaches described in this
chapter, and the experimental validation and discussion of the
results of these models see the series by Kashuba et al. [25–
27]. Here we also assume no spatially dependent gradients in
temperature or concentration. These systems include the cryopreservation of most cultured cells [26, 27], gametes [28], and even
embryos and blastocysts [4], among others. In this chapter our
aims are as follows: to present the ideas needed to construct and
understand the standard single cell models and then present several
optimization schemes.
2 Model Selection
2.1 Cell Volume
Modeling cryobiological protocols depends on a thorough knowledge of the cellular state including mole fraction or concentration
of all of the intracellular components as a function of time, temperature, and protocol. Typical experiments rarely yield complete cellular state information—the available measurement is usually either
cell volume or intracellular water volume or their proxies. Moreover, cell volume limits (known as osmotic tolerance limits, see
Subheading 3.1) are usually given in terms of total cell volume. In
practice, then, cryobiological modeling takes advantage of the
relationships between the volumes of the components to describe
the total volume, and deduces the state of all intracellular constituents (see Note 2).
Modelers commonly think of a cell as a “sack of saline” surrounded by a semipermeable membrane equipped with, perhaps, a
compartment of non-transportable solids and bound water. In fact,
while models that approximate the cell in this fashion are, in general, accurate enough for cryobiological purposes, it is good to keep
in mind that the cytoplasm is full of proteins and organelles that are
connected through a complex structure of actin filaments and that
all of these components have a hydration shell of bound water. For
Mathematical Modeling and Optimization of Cryopreservation in Single Cells
131
address these questions, and summarize key considerations for
modeling all aspects of cells in the presence of encroaching ice
[16–19].
The scope of modeling in modern cryobiology is very broad. It
includes heat and mass transport in tissues, organs, and hybrid
systems, as well as complicated ice dynamics and formation modeling in the presence and absence of cellular systems [20–23]. It now
even includes informatics approaches to understanding cryobiological outcomes [24]. This chapter presents the now classical modeling and optimization of single-cell cryopreservation protocols and a
straightforward adaptation of these approaches to tissues. For a
complete experimental approach that includes biophysical measurement of parameters, prediction of optimal cryopreservation protocols based on those parameters and the approaches described in this
chapter, and the experimental validation and discussion of the
results of these models see the series by Kashuba et al. [25–
27]. Here we also assume no spatially dependent gradients in
temperature or concentration. These systems include the cryopreservation of most cultured cells [26, 27], gametes [28], and even
embryos and blastocysts [4], among others. In this chapter our
aims are as follows: to present the ideas needed to construct and
understand the standard single cell models and then present several
optimization schemes.
2 Model Selection
2.1 Cell Volume
Modeling cryobiological protocols depends on a thorough knowledge of the cellular state including mole fraction or concentration
of all of the intracellular components as a function of time, temperature, and protocol. Typical experiments rarely yield complete cellular state information—the available measurement is usually either
cell volume or intracellular water volume or their proxies. Moreover, cell volume limits (known as osmotic tolerance limits, see
Subheading 3.1) are usually given in terms of total cell volume. In
practice, then, cryobiological modeling takes advantage of the
relationships between the volumes of the components to describe
the total volume, and deduces the state of all intracellular constituents (see Note 2).
Modelers commonly think of a cell as a “sack of saline” surrounded by a semipermeable membrane equipped with, perhaps, a
compartment of non-transportable solids and bound water. In fact,
while models that approximate the cell in this fashion are, in general, accurate enough for cryobiological purposes, it is good to keep
in mind that the cytoplasm is full of proteins and organelles that are
connected through a complex structure of actin filaments and that
all of these components have a hydration shell of bound water. For
Mathematical Modeling and Optimization of Cryopreservation in Single Cells
131