Mathematics of Cryopreservation: from tissue preparation to freezing and ice formation

Monday, June 14 at 5:45pm (PDT)
Tuesday, June 15 at 01:45am (BST)
Tuesday, June 15 09:45am (KST)

SMB2021 SMB2021 Follow Monday (Tuesday) during the "MS03" time block.
Share this


Robyn Shuttleworth (University of Saskatchewan, Canada), James Benson (University of Saskatchewan, Canada)


Cryopreservation is the process of freezing and storing cells or tissues at very low temperatures. Successful cryopreservation requires several challenging and interdependent steps, where failure in one step often results in complete destruction of the material. Some of these challenges include osmotic effects such as dehydration, solution toxicity damage, ice crystal formation, and thermal stress fractures. Whilst there is often high success with single cell cryopreservation, larger tissue samples prove much more challenging. The cryopreservation of large tissues would enable the prolonged, stable storage of whole organs. This would be beneficial for many reasons, with one of the most compelling being thorough testing and safe delivery of donor organs to their recipients. Because cryopreservation is a series of biophysical and biochemical events, it has a long history of success driven by modeling. Therefore, the aim of this mini symposium is to bring together the mathematical approaches used to analyze the different protocols required for successful cryopreservation. These models will highlight and tackle the challenges we face at each stage of the cryopreservation process and give insight into how we are progressing towards whole organ cryopreservation.

Adam Higgins

(Oregon State University, United States)
"Rational design of less toxic cryoprotectant solutions for cryopreservation"
Cryoprotectants (CPAs) are essential components of vitrification mixtures because they promote formation of a non-crystalline glassy state. However, CPAs can be toxic, and it remains a challenge to identify minimally toxic CPA mixtures for vitrification. This difficulty stems from two main issues. First, there are many different CPA types that can be combined in an infinite number of ways to create vitrification mixtures. It is therefore impractical to empirically determine the best vitrification mixture from among this infinite set. Second, the mechanisms of CPA toxicity are not well understood, making it difficult to identify promising mixtures using conceptual reasoning. To address these issues, we have developed a mathematical model of CPA toxicity that accounts for specific and nonspecific toxicity mechanisms, as well as formation of complexes between CPA pairs. We fit this model to experimental data for cultured endothelial cells exposed to five common CPAs [i.e., glycerol (Gly), dimethyl sulfoxide (DMSO), ethylene glycol (EG), propylene glycol (PG) and formamide (FA)], as well as their binary mixtures. The resulting best-fit model parameters were examined using Sloppy Model analysis to provide clues about the toxicity mechanisms. The results suggest that FA and Gly have the highest specific toxicity, PG exerts the most nonspecific toxicity, and that complexes between Gly-FA, DMSO-FA and Gly-EG affect the toxicity of mixtures containing these CPAs. To examine the predictive ability of the model, we predicted the toxicity of ternary CPA solutions, which resulted in reasonable agreement with experimental data. We then combined the toxicity model with a previously published model of glass formation in CPA mixtures to predict promising compositions for vitrification. The combined model predicts that the least toxic CPA cocktail that will result in formation of a glass is a mixture of Gly, FA and DMSO at concentrations of 7.5, 2.1 and 1.4 molal, respectively.

Ross Warner

(Oregon State University, United States)
"A general strategy for modeling the distribution of cryoprotectants in tissues"
The ability to successfully cryopreserve any biological specimen would undoubtedly change the face of modern medicine and scientific research. Single cell cryopreservation is a difficult problem by itself, but cryopreservation of complex specimens—mainly tissues and organs—is arguably an order of magnitude more difficult and is an active area of research. Vitrification is a promising avenue for successful complex specimen cryopreservation, but toxicity remains a major hurdle to overcome, as vitrification requires a high concentration of cryoprotectants (CPAs) to completely suppress ice formation. In the past, our group has leveraged mathematical modeling to minimize CPA toxicity for single cells. To do so, we developed a toxicity cost function and used mathematical optimization to minimize its value, which resulted in the prediction of a novel vitrification protocol that was experimentally verified to be less toxic than conventional methods. To extend this promising approach to tissues, an appropriate mass transfer model is needed. Fick’s law is commonly used, but it is limited due to its dilute assumption, as well as not accounting for tissue-specific phenomena such as fixed electrical charges, tissue size changes, and the coupling between cell membrane and extracellular mass transfer. In this work, we propose a general modeling paradigm for mass transfer in tissues. To accomplish this, we augmented an acellular mixture theory model in the literature for articular cartilage by incorporating cellular effects. With this augmentation, we show that the model can not only predict changes in CPA concentration and tissue size for the low cell density, rigid tissue of articular cartilage but also for the high cell density, elastic tissue of pancreatic islets. As such, this modeling paradigm is a promising general tissue model that can be used to further our mathematical optimization approach to cryopreservation and to better understand observations during tissue cryopreservation.

Fatemeh Amiri

(University of Saskatchewan, Canada)
"Agent based tissue modeling of ice propagation"
We model intracellular ice formation (IIF) in large multicellular tissues using Monte Carlo and agent-based modeling techniques. The previous implementa- tions have not allowed for within-tissue cell phenotype (i.e. parameter) het- erogeneity, nor have they coupled the models with key substrate diffusion and reaction equations. Therefore, to account for these critical differences and to understand IIF in large tissues, we have developed and validated a Monte Carlo method. In this model the tissue is described by a regular lattice in which each lattice site represents a cell, and intercellular ice propagation is allowed only between nearest neighbors. In our approach, each cell in the tissue is considered as an agent using the open source software PhysiCell, a multicellular system simulator which is designed to model tissues involving many interacting cells in multi-substrate 3D-microenvironments. We have validated the Monte Carlo method against theoretical Markov chain model for linear two-cell, four-cell and 2 × 2-cell constructs. Unlike the Markov model that involves exponential computational complexity associated with the tissue size, the Monte Carlo model has been successfully applied for large tissues with high numbers of cells. We also investigate the effects of tissue size on IIF in large tissues constructs and model IIF in mouse embryos.

Janet A. W. Elliott

(University of Alberta, Canada)
"Thermodynamics of Cell and Tissue Cryopreservation"
Cryobiology is the study of the effects of low temperature on biological systems with a major application being cryopreservation—the use of extremely low temperatures for the effectively indefinite storage of cells and tissues for later use. Cryoprotectants are used that modify the amount and location of ice formation during cryopreservation procedures. The ice–cryoprotectant-solution phase diagram and the osmotic and cryoprotectant transport across cell membranes and across tissues during cryoprotectant addition/removal and cooling/warming play crucial roles in whether or not cells survive. Thermodynamics is a broadly applicable subject whereby equations describing relationships among properties are derived from a few core postulates using multivariable calculus. Over more than 20 years our group has been developing equilibrium thermodynamic and nonequilibrium thermodynamic (transport) equations to describe cryobiological processes and gain insight to optimize cryopreservation protocols for a variety of cells and tissues. This talk will briefly introduce various areas of our prior and current work in cryobiological thermodynamics.

Hosted by SMB2021 Follow
Virtual conference of the Society for Mathematical Biology, 2021.