Chapter 4
Mathematical Modeling and Optimization
of Cryopreservation in Single Cells
James D. Benson
Abstract
Cryobiology is a multiscale and interdisciplinary field. The scope and scale of interactions limit the gains
that can be made by one theory or experiment alone. Because of this, modeling has played a critical role in
both explaining cryobiological phenomena and predicting improved protocols. Modeling facilitates understanding of the biophysical and some of the biochemical mechanisms of damage during all phases of
cryopreservation including CPA equilibration and cooling and warming. Moreover, as a tool for optimization of cryopreservation protocols, modeling has yielded many successes. Modern cryobiological modeling
includes very detailed descriptions of the physical phenomena that occur during freezing, including ice
growth kinetics and spatial gradients that define heat and mass transport models. Here we reduce the
complexity and approach only a small but classic subset of these problems. Namely, here we describe the
process of building and using a mathematical model of a cell in suspension where spatial homogeneity is
assumed for all quantities. We define the models that describe the critical cell quantities used to describe
optimal and suboptimal protocols and then give an overview of classical methods of how to determine
optimal protocols using these models. We include practical considerations of modeling in cryobiology,
including fitting transport models to cell volume data, performing optimization with cell volume constraints, and a look at expanding cost functions to cooling regimes.
Key words Mass transport, Boyle van ’t Hoff, Chemical potential, Freezing point depression, Phase
diagram, Virial equation, Optimization
1 Introduction
Theoretical and practical inroads from mathematical modeling in
cryobiology began over 50 years ago. At the intersection of biology,
engineering, physics, chemistry, and mathematics, and like other
similar fields, the relationship between biology and biophysics
forged in cryobiology has been fruitful for both fields.
The first foundational cryobiological model was Mazur’s biophysical model of intracellular state as a function of cooling rate
proposed in 1963 [1]. This model lent support to an experimentally verifiable theory of cell death as a function of too-high cooling
rates and set the stage for modelers to use similar approaches to
Willem F. Wolkers and Harrie ¨ tte Oldenhof (eds.), Cryopreservation and Freeze-Drying Protocols, Methods in Molecular Biology,
vol. 2180, https://doi.org/10.1007/978-1-0716-0783-1_4, © Springer Science+Business Media, LLC, part of Springer Nature 2021
129
Mathematical Modeling and Optimization
of Cryopreservation in Single Cells
James D. Benson
Abstract
Cryobiology is a multiscale and interdisciplinary field. The scope and scale of interactions limit the gains
that can be made by one theory or experiment alone. Because of this, modeling has played a critical role in
both explaining cryobiological phenomena and predicting improved protocols. Modeling facilitates understanding of the biophysical and some of the biochemical mechanisms of damage during all phases of
cryopreservation including CPA equilibration and cooling and warming. Moreover, as a tool for optimization of cryopreservation protocols, modeling has yielded many successes. Modern cryobiological modeling
includes very detailed descriptions of the physical phenomena that occur during freezing, including ice
growth kinetics and spatial gradients that define heat and mass transport models. Here we reduce the
complexity and approach only a small but classic subset of these problems. Namely, here we describe the
process of building and using a mathematical model of a cell in suspension where spatial homogeneity is
assumed for all quantities. We define the models that describe the critical cell quantities used to describe
optimal and suboptimal protocols and then give an overview of classical methods of how to determine
optimal protocols using these models. We include practical considerations of modeling in cryobiology,
including fitting transport models to cell volume data, performing optimization with cell volume constraints, and a look at expanding cost functions to cooling regimes.
Key words Mass transport, Boyle van ’t Hoff, Chemical potential, Freezing point depression, Phase
diagram, Virial equation, Optimization
1 Introduction
Theoretical and practical inroads from mathematical modeling in
cryobiology began over 50 years ago. At the intersection of biology,
engineering, physics, chemistry, and mathematics, and like other
similar fields, the relationship between biology and biophysics
forged in cryobiology has been fruitful for both fields.
The first foundational cryobiological model was Mazur’s biophysical model of intracellular state as a function of cooling rate
proposed in 1963 [1]. This model lent support to an experimentally verifiable theory of cell death as a function of too-high cooling
rates and set the stage for modelers to use similar approaches to
Willem F. Wolkers and Harrie ¨ tte Oldenhof (eds.), Cryopreservation and Freeze-Drying Protocols, Methods in Molecular Biology,
vol. 2180, https://doi.org/10.1007/978-1-0716-0783-1_4, © Springer Science+Business Media, LLC, part of Springer Nature 2021
129