Kinetics of coupling water and cryoprotectant transport across cell membranes and applications to cryopreservation

Foundations of modeling in cryobiology-II: Heat and mass transport in bulk and at cell membrane and ice-liquid interfaces

Modeling coupled heat and mass transport in biological systems is critical to the understanding of cryobiology. In Part I of this series we derived the transport equation and presented a general thermodynamic derivation of the critical components needed to use the transport equation in cryobiology. Here we refine to more cryobiologically relevant instances of a double free-boundary problem with multiple species. In particular, we present the derivation of appropriate mass and heat transport constitutive equations for a system consisting of a cell or tissue with a free external boundary, surrounded by liquid media with an encroaching free solidification front. This model consists of two parts-namely, transport in the "bulk phases" away from boundaries, and interfacial transport. Here we derive the bulk and interfacial mass, energy, and momentum balance equations and present a simplification of transport within membranes to jump conditions across them. We establish the governing equations for this cell/liquid/solid system whose solution in the case of a ternary mixture is explored in Part III of this series.

Numerical solution of inward solidification of a dilute ternary solution towards a semi-permeable spherical cell

Modeling a cell's response to encroaching ice has informed the development of cryopreservation protocols for four decades. It has been well documented that knowledge of the cellular state as a function of media and cooling rate faciliate informed cryopreservation protocol design and explain mechanisms of damage. However, previous efforts have neglected the interaction between solutes and the encroaching ice front and their effects on the cell state. To address this, here we examine the cryobiologically relevant setting of a spherically-symmetric model of a biological cell separated by a ternary fluid mixture from an encroaching solid-liquid interface. The cell and liquid regions contain cell membrane impermeable intracellular and extracellular salts, respectively, a cell membrane permeable solute commonly used in cryopreservation protocols known as a cryoprotective agent (CPA), and water as a membrane permeable solvent. As cooling and solidification proceed the extracellular chemical environment evolves and leads to mass transport across the cell membrane. Consequently, both the solidification front and the cell membrane are free boundaries whose dynamics are coupled through transport processes in the solid, liquid and cell regions. We describe a numerical procedure to solve this coupled free-boundary problem based on a domain transformation and method of lines approach. We also investigate how the thermal and chemical states inside the cell are influenced by different cooling protocols at the external boundary. Finally, we observe that the previously unaccounted-for partial solute rejection at the advancing solid-liquid interface increases the CPA and salt concentrations in the extracellular liquid as a function of the interface speed and segregation coefficients, suggesting that previous model predictions of the cell state during cryopreservation were inaccurate.

A highly-occupied, single-cell trapping microarray for determination of cell membrane permeability

Semi- and selective permeability is a fundamentally important characteristic of the cell membrane. Membrane permeability can be determined by monitoring the volumetric change of cells following exposure to a non-isotonic environment. For this purpose, several microfluidic perfusion chambers have been developed recently. However, these devices only allow the observation of one single cell or a group of cells that may interact with one another in an uncontrolled way. Some of these devices have integrated on-chip temperature control to investigate the temperature-dependence of membrane permeability, but they inevitably require sophisticated fabrication and assembly, and delicate temperature and pressure calibration. Therefore, it is highly desirable to design a simple single-cell trapping device that allows parallel monitoring of multiple separate, individual cells subjected to non-isotonic exposure at various temperatures. In this study, we developed a pumpless, single-layer microarray with high trap occupancy of single cells. The benchmark performance of the device was conducted by targeting spherical particles of 18.8 μm in diameter as a model, yielding trap occupancy of up to 86.8% with a row-to-row shift of 10-30 μm. It was also revealed that in each array the particles larger than a corresponding critical size would be excluded by the traps in a deterministic lateral displacement mode. Demonstrating the utility of this approach, we used the single-cell trapping device to determine the membrane permeability of rat hepatocytes and patient-derived circulating tumor cells (Brx-142) at 4, 22 and 37 °C. The membrane of rat hepatocytes was found to be highly permeable to water and small molecules such as DMSO and glycerol, via both lipid- and aquaporin-mediated pathways. Brx-142 cells, however, displayed lower membrane permeability than rat hepatocytes, which was associated with strong coupling of water and DMSO transport but less interaction between water and glycerol. The membrane permeability data reported here provide new insights into the biophysics of membrane transport such as aquaporin expression and coupling transport of water and solutes, as well as providing essential data for the ultimate goal of biobanking rare cells and precious tissues.

Global stability and exact solution of an arbitrary-solute nonlinear cellular mass transport system

The prediction of the cellular state as a function of extracellular concentrations and temperatures has been of interest to physiologists for nearly a century. One of the most widely used models in the field is one where mass flux is linearly proportional to the concentration difference across the membrane. These fluxes define a nonlinear differential equation system for the intracellular state, which when coupled with appropriate initial conditions, define the intracellular state as a function of the extracellular concentrations of both permeating and nonpermeating solutes. Here we take advantage of a reparametrization scheme to extend existing stability results to a more general setting and to a develop analytical solutions to this model for an arbitrary number of extracellular solutes.

Foundations of modeling in cryobiology-I: concentration, Gibbs energy, and chemical potential relationships

Mathematical modeling plays an enormously important role in understanding the behavior of cells, tissues, and organs undergoing cryopreservation. Uses of these models range from explanation of phenomena, exploration of potential theories of damage or success, development of equipment, and refinement of optimal cryopreservation/cryoablation strategies. Over the last half century there has been a considerable amount of work in bio-heat and mass-transport, and these models and theories have been readily and repeatedly applied to cryobiology with much success. However, there are significant gaps between experimental and theoretical results that suggest missing links in models. One source for these potential gaps is that cryobiology is at the intersection of several very challenging aspects of transport theory: it couples multi-component, moving boundary, multiphase solutions that interact through a semipermeable elastic membrane with multicomponent solutions in a second time-varying domain, during a two-hundred Kelvin temperature change with multi-molar concentration gradients and multi-atmosphere pressure changes. In order to better identify potential sources of error, and to point to future directions in modeling and experimental research, we present a three part series to build from first principles a theory of coupled heat and mass transport in cryobiological systems accounting for all of these effects. The hope of this series is that by presenting and justifying all steps, conclusions may be made about the importance of key assumptions, perhaps pointing to areas of future research or model development, but importantly, lending weight to standard simplification arguments that are often made in heat and mass transport. In this first part, we review concentration variable relationships, their impact on choices for Gibbs energy models, and their impact on chemical potentials.

An improved model for nucleation-limited ice formation in living cells during freezing

Ice formation in living cells is a lethal event during freezing and its characterization is important to the development of optimal protocols for not only cryopreservation but also cryotherapy applications. Although the model for probability of ice formation (PIF) in cells developed by Toner et al. has been widely used to predict nucleation-limited intracellular ice formation (IIF), our data of freezing Hela cells suggest that this model could give misleading prediction of PIF when the maximum PIF in cells during freezing is less than 1 (PIF ranges from 0 to 1). We introduce a new model to overcome this problem by incorporating a critical cell volume to modify the Toner's original model. We further reveal that this critical cell volume is dependent on the mechanisms of ice nucleation in cells during freezing, i.e., surface-catalyzed nucleation (SCN) and volume-catalyzed nucleation (VCN). Taken together, the improved PIF model may be valuable for better understanding of the mechanisms of ice nucleation in cells during freezing and more accurate prediction of PIF for cryopreservation and cryotherapy applications.

The effect of solution nonideality on modeling transmembrane water transport and diffusion-limited intracellular ice formation during cryopreservation

A new model was developed to predict transmembrane water transport and diffusion-limited ice formation in cells during freezing without the ideal-solution assumption that has been used in previous models. The model was applied to predict cell dehydration and intracellular ice formation (IIF) during cryopreservation of mouse oocytes and bovine carotid artery endothelial cells in aqueous sodium chloride (NaCl) solution with glycerol as the cryoprotectant or cryoprotective agent. A comparison of the predictions between the present model and the previously reported models indicated that the ideal-solution assumption results in under-prediction of the amount of intracellular ice at slow cooling rates (<50 K/min). In addition, the lower critical cooling rates for IIF that is lethal to cells predicted by the present model were much lower than those estimated with the ideal-solution assumption. This study represents the first investigation on how accounting for solution nonideality in modeling water transport across the cell membrane could affect the prediction of diffusion-limited ice formation in biological cells during freezing. Future studies are warranted to look at other assumptions alongside nonideality to further develop the model as a useful tool for optimizing the protocol of cell cryopreservation for practical applications.