Application of the osmotic virial equation in cryobiology

Effects of different cryopreservation parameters on the differences between trypan blue and fluorescent SYTO 13/GelRed assays

Post-thaw cell viability assessment is very important in cryopreservation because it is the main assessment method used to optimize cryopreservation protocols for each cell type; hence, having standardized accurate, quick, and reliable assays for post-thaw cell viability measurements is of utmost importance. The trypan blue exclusion assay and nucleic-acid-binding fluorescence-based assays are two different methods for cell viability assessment. Both assays identify cells with damaged membranes by whether they let a compound enter the cell. In this study, these two assays are compared in the context of cryopreservation and the impacts of important cryopreservation parameters on the differences in measurements are investigated. H9c2 myoblasts were cryopreserved with different freezing protocols. Cell membrane integrities were measured immediately after thaw as well as after cryoprotectant removal by a hemocytometer-based trypan blue dye exclusion assay and a dual fluorometric SYTO 13/GelRed assay; and the results were compared. This study quantifies how (i) the absence or presence of different cryoprotectants, (ii) different cell-cryoprotectant incubation conditions, and (iii) the presence or removal of cryoprotectants after thaw affect the differences between these two viability assays.

Systematic cryopreservation study of cardiac myoblasts in suspension

H9c2 myoblasts are a cell line derived from embryonic rat heart tissue and demonstrate the ability to differentiate to cardiac myotubes upon reduction of the serum concentration (from 10% to 1%) and addition of all-trans retinoic acid in the growth medium. H9c2 cells are increasingly being used as an easy-to-culture proxy for some functions of cardiomyocytes. The cryobiology of cardiac cells including H9c2 myoblasts has not been studied as extensively as that of some cell types. Consequently, it is important to characterize the cryobiological response and systematically develop well-optimized cryopreservation protocols for H9c2 cells to have optimal and consistent viability and functionality after thaw for high quality studies with this cell type. In this work, an interrupted slow cooling protocol (graded freezing) was applied to characterize H9c2 response throughout the cooling profile. Important factors that affect the cell response were examined, and final protocols that provided the highest post-thaw viability are reported. One protocol uses the common cryoprotectant dimethyl sulfoxide combined with hydroxyethyl starch, which will be suitable for applications in which the presence of dimethyl sulfoxide is not an issue; and the other protocol uses glycerol as a substitute when there is a desire to avoid dimethyl sulfoxide. Both protocols achieved comparable post-thaw viabilities (higher than 80%) based on SYTO 13/GelRed flow cytometry results. H9c2 cells cryopreserved by either protocol showed ability to differentiate to cardiac myotubes comparable to fresh (unfrozen) H9c2 cells, and their differentiation to cardiac myotubes was confirmed with i) change in cell morphology, ii) expression of cardiac marker troponin I, and iii) increase in mitochondrial mass.

Temperature Dependence of Membrane Permeability Parameters for Five Cell Types Using Nonideal Thermodynamic Assumptions to Mathematically Model Cryopreservation Protocols

Cryopreservation is the process of preserving biological matter at subzero temperatures for long-term storage. During cryopreservation, cells are susceptible to various injuries that can be mitigated by controlling the cooling and warming profiles and cryoprotective agent (CPA) addition and removal procedures. Mathematical modeling of the changing cell volume at different temperatures can greatly reduce the experiments needed to optimize cryopreservation protocols. Such mathematical modeling requires as inputs the cell membrane permeabilities to water and CPA and the osmotically inactive fraction of the cell. Since the intra- and extracellular solutions are generally thermodynamically nonideal, our group has been incorporating the osmotic virial equation to model the solution thermodynamics that underlie the cell volume change equations, adding the second and third osmotic virial coefficients of the grouped intracellular solute to the cell osmotic parameters that must be measured. In our previous work, we reported methods to obtain cell osmotic parameters at room temperature by fitting experimental cell volume kinetic data with equations that incorporated nonideal solution thermodynamics assumptions. Since the relevant cell volume excursions occur at different temperatures, the temperature dependence of the osmotic parameters plays an important role. In this work, we present a new two-part fitting method to obtain five cell-type-specific parameters (water permeability, dimethyl sulfoxide permeability, osmotically inactive fraction, and the second and third osmotic virial coefficients of the intracellular solution) from experimental measurements of equilibrium cell volume and cell volume as a function of time at room temperature and 0 °C for five cell types, namely, human umbilical vein endothelial cells (HUVECs), H9c2 rat myoblasts, porcine corneal endothelial cells (PCECs), the Jurkat T-lymphocyte cell line, and human cerebral microvascular endothelial cells (hCMECs/D3 cell line). The fitting method in this work is based on both equilibrium and kinetic cell volume data, enabling us to solve some technical challenges and expand our previously reported measurement technique to 0 °C. Finally, we use the measured parameters to model the cell volume changes for a HUVEC cryopreservation protocol to demonstrate the impact of the nonideal thermodynamic assumptions on predicting the changing cell volume during freezing and thawing.

Multicomponent solutions: Combining rules for multisolute osmotic virial coefficients

This paper presents an exploration of a specific type of a generalized multicomponent solution model, which appears to be first given by Saulov in the current explicit form. The assumptions of the underlying theory and a brief derivation of the main equation have been provided preliminarily for completeness and notational consistency. The resulting formulae for the Gibbs free energy of mixing and the chemical potentials are multivariate polynomials with physically meaningful coefficients and the mole fractions of the components as variables. With one additional assumption about the relative magnitudes of the solvent-solute and solute-solute interaction exchange energies, combining rules were obtained that express the mixed coefficients of the polynomial in terms of its pure coefficients. This was done by exploiting the mathematical structure of the asymmetric form of the solvent chemical potential equation. The combining rules allow one to calculate the thermodynamic properties of the solvent with multiple solutes from binary mixture data only (i.e., each solute with the solvent), and hence, are of practical importance. Furthermore, a connection was established between the osmotic virial coefficients derived in this work and the original osmotic virial coefficients of Hill found by employing a different procedure, illustrating the equivalency of what appears to be two different theories. A validation of the combining rules derived here has been provided in a separate paper where they were successfully used to predict the freezing points of ternary salt solutions of water.

Large Decrease in the Melting Point of Benzoquinones via High-n Eutectic Mixing Predicted by a Regular Solution Model

Decreasing the melting point (T_m) of a mixture is of interest in cryopreservatives, molten salts, and battery electrolytes. One general strategy to decrease T_m, exemplified by deep eutectic solvents, is to mix components with favorable (negative) enthalpic interactions. We demonstrate a complementary strategy to decrease T_m by mixing many components with neutral or slightly positive enthalpic interactions, using the number of components (n) to increase the entropy of mixing and decrease T_m. In theory, under certain conditions this approach could achieve an arbitrarily low T_m. Furthermore, if the components are small redox-active molecules, such as the benzoquinones studied here, this approach could lead to high energy density flow battery electrolytes. Finding the eutectic composition of a high-n mixture can be challenging due to the large compositional space yet is essential for ensuring the existence of a purely liquid phase. We reformulate and apply fundamental thermodynamic equations to describe high-n eutectic mixtures of small redox-active molecules (benzoquinones and hydroquinones). We illustrate a novel application of this theory by tuning the entropy of melting, rather than the enthalpy, in systems highly relevant to energy storage. We demonstrate with differential scanning calorimetry measurements that 1,4-benzoquinone derivatives exhibit eutectic mixing that decreases their T_m, despite slightly positive enthalpies of mixing (0-5 kJ/mol). By rigorously investigating all 21 binary mixtures of a set of seven 1,4-benzoquinone derivatives with alkyl substituents (T_m's between 44 and 120 °C), we find that the eutectic melting point of a mixture of all seven achieves a large decrease in T_m to -6 °C. We further determine that the regular solution model shows improvement over an ideal solution model in predicting the eutectic properties for this newly investigated type of mixture composed of many small redox-active organic molecules.

Comparison of dilute and nondilute osmotic equilibrium models for erythrocytes

Successful cryopreservation requires the addition of cryoprotective agents (CPAs). The addition of permeating CPAs, such as glycerol, is associated with some risk to the cells and tissues. These risks are both related to the CPA themselves (CPA toxicity) and to the volume response of the cell (osmotic damage). To minimize the potential for damage during cryopreservation, mathematical models are often employed to understand the interactions between protocols and cell volume responses. In the literature, this volume response is usually captured using ideal and dilute approximations of chemical potential and osmolality, an approach that has been called into question for cells in high concentrations of CPAs. To address this, the relevance of non-ideal and non-dilute models has been explored in a number of cell types in the presence of permeating CPAs. However, it has not been explored in erythrocytes, which have a cytosolic hemoglobin content of more than 20% by volume and are cryopreserved in 40% glycerol. Because hemoglobin has been suggested to be a highly non-ideal solute, if the non-ideal and non-dilute transport model is relevant to any cells, it should be relevant to erythrocytes. Here we investigate the use, and accuracy, of both the dilute and non-dilute models in predicting cell volume changes during CPA equilibration in erythrocytes, and demonstrate that using published values for the non-ideal and non-dilute model, applied to erythrocytes, leads to model predictions inconsistent with experimental data, whereas dilute approximations align well with experimental data.

Modeling the Simultaneous Transport of Multiple Cryoprotectants into Articular Cartilage Using a Triphasic Model

Cryopreserving articular cartilage by vitrification can increase the availability of tissue for osteochondral allograft transplantation to treat cartilage defects. Developing well-optimized vitrification protocols can be supported by mathematical modeling to reduce the amount of trial-and-error experimentation needed. Fick's law has been used to model cryoprotectant diffusion, but it assumes ideal, dilute solution behavior, neglects water movement, and assumes diffusion of each cryoprotectant is independent of the presence of other cryoprotectants. The modified triphasic model addresses some of these shortcomings by accounting for water movement and the nonideal, nondilute nature of cryoprotectant vitrification solutions. However, it currently only exists for solutions containing a single cryoprotectant. As such, we extend the modified triphasic model to include two permeating cryoprotectants so that simultaneous diffusion occurring in vitrification protocols can be more accurately modeled. Using previously published experimental data, we determine suitable values for the fitting parameters of the new model. We then model a successful vitrification protocol for particulated cartilage cubes by calculating concentration, freezing point, vitrifiability, and strain profiles at the end of each loading step. We observe that Fick's law consistently underestimates cryoprotectant concentration throughout the cartilage compared to the modified triphasic model, leading to an underestimation of tissue vitrifiability. We additionally observe that simultaneous diffusion of cryoprotectants increases the permeation rate of each individual cryoprotectant, which Fick's law fails to consider. This suggests that using the two-cryoprotectant modified triphasic model to develop vitrification protocols could reduce excess exposure to cryoprotectants and improve preserved tissue outcomes.

Infiltration from Suspension Systems Enables Effective Modulation of 3D Scaffold Properties in Suspension Bioprinting

Bioprinting is a biofabrication technology which allows efficient and large-scale manufacture of 3D cell culture systems. However, the available biomaterials for bioinks used in bioprinting are limited by their printability and biological functionality. Fabricated constructs are often homogeneous and have limited complexity in terms of current 3D cell culture systems comprising multiple cell types. Inspired by the phenomenon that hydrogels can exchange liquids under the infiltration action, infiltration-induced suspension bioprinting (IISBP), a novel printing technique based on a hyaluronic acid (HA) suspension system to modulate the properties of the printed scaffolds by infiltration action, was described in this study. HA served as a suspension system due to its shear-thinning and self-healing rheological properties, simplicity of preparation, reusability, and ease of adjustment to osmotic pressure. Changes in osmotic pressure were able to direct the swelling or shrinkage of 3D printed gelatin methacryloyl (GelMA)-based bioinks, enabling the regulation of physical properties such as fiber diameter, micromorphology, mechanical strength, and water absorption of 3D printed scaffolds. Human umbilical vein endothelial cells (HUVEC) were applied as a cell culture model and printed within cell-laden scaffolds at high resolution and cell viability with the IISBP technique. Herein, the IISBP technique had been realized as a reliable hydrogel-based bioprinting technique, which enabled facile modulation of 3D printed hydrogel scaffolds properties, being expected to meet the scaffolds requirements of a wide range of cell culture conditions to be utilized in bioprinting applications.

Permeability and Osmotic Parameters of Human Umbilical Vein Endothelial Cells and H9C2 Cells under Non-ideal Thermodynamic Assumptions: A Novel Iterative Fitting Method

Cryopreservation is the use of very low subzero temperatures to preserve cells and tissues for later use. This is achieved by controlled cooling in the presence of cryoprotectants that moderate the amount of ice formed. Mathematical modeling of the cryopreservation process is a useful tool to investigate the different variables that affect the results of this process. The changing cell volume during cryopreservation can be modeled using cell membrane water and cryoprotectant permeabilities and the osmotically inactive fraction of the intracellular contents. These three cell-specific parameters have been found previously for different cell types under ideal and dilute assumptions, but biological solutions at subzero temperatures are far from ideal and dilute, especially when cryoprotectants are included. In this work, the osmotic virial equation is used to model the changing cell volume under non-ideal assumptions, and the intracellular environment is described using the grouped solute, which consists of all impermeant intracellular solutes grouped together, leading to two additional cell-specific parameters, the second and third osmotic virial coefficients of the grouped solute. Herein, we present a novel fitting method to efficiently determine these five cell-specific parameters by fitting kinetic cell volume data under non-ideal assumptions and report the results of applying this method to obtain the parameters for two cell types: human umbilical vein endothelial cells and H9C2 rat myoblasts.

Vitrification of particulated articular cartilage via calculated protocols

Preserving viable articular cartilage is a promising approach to address the shortage of graft tissue and enable the clinical repair of articular cartilage defects in articulating joints, such as the knee, ankle, and hip. In this study, we developed two 2-step, dual-temperature, multicryoprotectant loading protocols to cryopreserve particulated articular cartilage (cubes ~1 mm3 in size) using a mathematical approach, and we experimentally measured chondrocyte viability, metabolic activity, cell migration, and matrix productivity after implementing the designed loading protocols, vitrification, and warming. We demonstrated that porcine and human articular cartilage cubes can be successfully vitrified and rewarmed, maintaining high cell viability and excellent cellular function. The vitrified particulated articular cartilage was stored for a period of 6 months with no significant deterioration in chondrocyte viability and functionality. Our approach enables high-quality long-term storage of viable articular cartilage that can alleviate the shortage of grafts for use in clinically repairing articular cartilage defects.

Mathematical Modeling and Optimization of Cryopreservation in Single Cells

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.

Foundations of modeling in cryobiology-III: Inward solidification of a ternary solution towards a permeable spherical cell in the dilute limit

In the previous two manuscripts we outlined the general theory of heat and mass transport in a cell-liquid-ice system with general boundaries and nonideal and nondilute assumptions. Here we simplify the models considerably by presenting a reduction to a spherically symmetric system-a spherical cell with an encroaching spherical ice front. We also reduce to linear approximations of the nonideal nondilute models, essentially assuming dilute and ideal conditions. We derive the resulting nondimensional combined heat and mass transport model for a ternary solution and present numerical solutions. We include an analysis of the effects of varying some nondimensional parameters on rates of ice growth with comments on the necessity of models that account for spatially varying quantities in cryobiology.

Using engineering models to shorten cryoprotectant loading time for the vitrification of articular cartilage

Osteochondral allograft transplantation can treat full thickness cartilage and bone lesions in the knee and other joints, but the lack of widespread articular cartilage banking limits the quantity of cartilage available for size and contour matching. To address the limited availability of cartilage, vitrification can be used to store harvested joint tissues indefinitely. Our group's reported vitrification protocol [Biomaterials 33 (2012) 6061-6068] takes 9.5 h to load cryoprotectants into intact articular cartilage on bone and achieves high cell viability, but further optimization is needed to shorten this protocol for clinical use. Herein, we use engineering models to calculate the spatial and temporal distributions of cryoprotectant concentration, solution vitrifiability, and freezing point for each step of the 9.5-h protocol. We then incorporate the following major design choices for developing a new shorter protocol: (i) all cryoprotectant loading solution concentrations are reduced, (ii) glycerol is removed as a cryoprotectant, and (iii) an equilibration step is introduced to flatten the final cryoprotectant concentration profiles. We also use a new criterion-the spatially and temporally resolved prediction of solution vitrifiability-to assess whether a protocol will be successful instead of requiring that each cryoprotectant individually reaches a certain concentration. A total cryoprotectant loading time of 7 h is targeted, and our new 7-h protocol is predicted to achieve a level of vitrifiability comparable to the proven 9.5-h protocol throughout the cartilage thickness.

Aqueous Rechargeable Metal-Ion Batteries Working at Subzero Temperatures

Aqueous rechargeable metal-ion batteries (ARMBs) represent one of the current research frontiers due to their low cost, high safety, and other unique features. Evolving to a practically useful device, the ARMBs must be adaptable to various ambient, especially the cold weather. While much effort has been made on organic electrolyte batteries operating at low temperatures, the study on low-temperature ARMBs is still in its infancy. The challenge mainly comes from water freezing at subzero temperatures, resulting in dramatically retarded kinetics. Here, the freezing behavior of water and its effects on subzero performances of ARMBs are first discussed. Then all strategies used to enhance subzero temperature performances of ARMBs by associating them with battery kinetics are summarized. The subzero temperature performances of ARMBs and organic electrolyte batteries are compared. The final section presents potential directions for further improvements and future perspectives of this thriving field.

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.

Impedance and conductivity of bovine myocardium during freezing and thawing at slow rates - implications for cardiac cryo-ablation

Increasing impedance during freezing might be a valuable marker for guiding cardiac cryo-ablation. We provide model based insights on how decreasing temperature below the freezing point of tissue relates to the percentage of frozen water. Furthermore, we provide experimental data for comparing this percentage with the increase in impedance. Measurements were performed on a bovine tissue sample at frequencies between 5 and 80 kHz. Slow cooling and heating rates were applied to minimize temperature gradients in the myocardial sample and to allow accurate assessment of the freezing point. Computer simulation was applied to link impedance with temperature dependent conductivities. The osmotic virial equation was used to estimate the percentage of frozen water. Measurements identified the freezing point at -0.6 ^∘C. At -5 ^∘C, impedance rose by more than a factor of ten compared to that at the freezing point and the percentage of frozen water was estimated as being 89%. At -49 ^∘C impedance had increased by up to three orders of magnitude and ice formation was most pronounced in the extracellular space. Progressive ice formation in tissue is reflected by a large increase in impedance, and impedance increases monotonically with the percentage of frozen water. Its analysis allows for experimental assessment of factors relevant to cell death. Solid ice contributes to the rupture of the micro-vasculature, while phase shifts reflect concentration differences between extra- and intracellular space driving osmotic water transfer across cell membranes.

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.

Bacterial Ice Nucleation in Monodisperse D2O and H2O-in-Oil Emulsions

Ice nucleation is of fundamental significance in many areas, including atmospheric science, food technology, and cryobiology. In this study, we investigated the ice-nucleation characteristics of picoliter-sized drops consisting of different D2O and H2O mixtures with and without the ice-nucleating bacteria Pseudomonas syringae. We also studied the effects of commonly used cryoprotectants such as ethylene glycol, propylene glycol, and trehalose on the nucleation characteristics of D2O and H2O mixtures. The results show that the median freezing temperature of the suspension containing 1 mg/mL of a lyophilized preparation of P. syringae is as high as -4.6 °C for 100% D2O, compared to -8.9 °C for 100% H2O. As the D2O concentration increases every 25% (v/v), the profile of the ice-nucleation kinetics of D2O + H2O mixtures containing 1 mg/mL Snomax shifts by about 1 °C, suggesting an ideal mixing behavior of D2O and H2O. Furthermore, all of the cryoprotectants investigated in this study are found to depress the freezing phenomenon. Both the homogeneous and heterogeneous freezing temperatures of these aqueous solutions depend on the water activity and are independent of the nature of the solute. These findings enrich our fundamental knowledge of D2O-related ice nucleation and suggest that the combination of D2O and ice-nucleating agents could be a potential self-ice-nucleating formulation. The implications of self-nucleation include a higher, precisely controlled ice seeding temperature for slow freezing that would significantly improve the viability of many ice-assisted cryopreservation protocols.

Cryoprotectant kinetic analysis of a human articular cartilage vitrification protocol

We recently published a protocol to vitrify human articular cartilage and a method of cryoprotectant removal in preparation for transplantation. The current study's goal was to perform a cryoprotectant kinetic analysis and theoretically shorten the procedure used to vitrify human articular cartilage. First, the loading of the cryoprotectants was modeled using Fick's law of diffusion, and this information was used to predict the kinetics of cryoprotectant efflux after the cartilage sample had been warmed. We hypothesized that diffusion coefficients obtained from the permeation of individual cryoprotectants into porcine articular cartilage could be used to provide a reasonable prediction of the cryoprotectant loading and of the combined cryoprotectant efflux from vitrified human articular cartilage. We tested this hypothesis with experimental efflux measurements. Osteochondral dowels from three patients were vitrified, and after warming, the articular cartilage was immersed in 3 mL X-VIVO at 4 °C in two consecutive solutions, each for 24 h, with the solution osmolality recorded at various times. Measured equilibrium values agreed with theoretical values within a maximum of 15% for all three samples. The results showed that diffusion coefficients for individual cryoprotectants determined from experiments with 2-mm thick porcine cartilage can be used to approximate the rate of efflux of the combined cryoprotectants from vitrified human articular cartilage of similar thickness. Finally, Fick's law of diffusion was used in a computational optimization to shorten the protocol with the constraint of maintaining the theoretical minimum cryoprotectant concentration needed to achieve vitrification. The learning provided by this study will enable future improvements in tissue vitrification.

Non-ideal solution thermodynamics of cytoplasm

Quantitative description of the non-ideal solution thermodynamics of the cytoplasm of a living mammalian cell is critically necessary in mathematical modeling of cryobiology and desiccation and other fields where the passive osmotic response of a cell plays a role. In the solution thermodynamics osmotic virial equation, the quadratic correction to the linear ideal, dilute solution theory is described by the second osmotic virial coefficient. Herein we report, for the first time, intracellular solution second osmotic virial coefficients for four cell types [TF-1 hematopoietic stem cells, human umbilical vein endothelial cells (HUVEC), porcine hepatocytes, and porcine chondrocytes] and further report second osmotic virial coefficients indistinguishable from zero (for the concentration range studied) for human hepatocytes and mouse oocytes.

Modeling and optimization of cryopreservation

Modeling plays a critical role in understanding the biophysical processes behind cryopreservation. It facilitates understanding of the biophysical and some of the biochemical mechanisms of damage during all phases of cryopreservation including CPA equilibration, cooling, and warming. Modeling also provides a tool for optimization of cryopreservation protocols and has yielded a number of successes in this regard. While 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.

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.

Comparison of non-ideal solution theories for multi-solute solutions in cryobiology and tabulation of required coefficients

Thermodynamic solution theories allow the prediction of chemical potentials in solutions of known composition. In cryobiology, such models are a critical component of many mathematical models that are used to simulate the biophysical processes occurring in cells and tissues during cryopreservation. A number of solution theories, both thermodynamically ideal and non-ideal, have been proposed for use with cryobiological solutions. In this work, we have evaluated two non-ideal solution theories for predicting water chemical potential (i.e. osmolality) in multi-solute solutions relevant to cryobiology: the Elliott et al. form of the multi-solute osmotic virial equation, and the Kleinhans and Mazur freezing point summation model. These two solution theories require fitting to only single-solute data, although they can make predictions in multi-solute solutions. The predictions of these non-ideal solution theories were compared to predictions made using ideal dilute assumptions and to available literature multi-solute experimental osmometric data. A single, consistent set of literature single-solute solution data was used to fit for the required solute-specific coefficients for each of the non-ideal models. Our results indicate that the two non-ideal solution theories have similar overall performance, and both give more accurate predictions than ideal models. These results can be used to select between the non-ideal models for a specific multi-solute solution, and the updated coefficients provided in this work can be used to make the desired predictions.

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.

Rationally optimized cryopreservation of multiple mouse embryonic stem cell lines: II--Mathematical prediction and experimental validation of optimal cryopreservation protocols

In Part I, we documented differences in cryopreservation success measured by membrane integrity in four mouse embryonic stem cell (mESC) lines from different genetic backgrounds (BALB/c, CBA, FVB, and 129R1), and we demonstrated a potential biophysical basis for these differences through a comparative study characterizing the membrane permeability characteristics and osmotic tolerance limits of each cell line. Here we use these values to predict optimal cryoprotectants, cooling rates, warming rates, and plunge temperatures. We subsequently verified these predictions experimentally for their effects on post-thaw recovery. From this study, we determined that a cryopreservation protocol utilizing 1M propylene glycol, a cooling rate of 1°C/minute, and plunging into liquid nitrogen at -41°C, combined with subsequent warming in a 22°C water bath with agitation, significantly improved post-thaw recovery for three of the four mESC lines, and did not diminish post-thaw recovery for our single exception. It is proposed that this protocol can be successfully applied to most mESC lines beyond those included within this study once the effect of propylene glycol on mESC gene expression, growth characteristics, and germ-line transmission has been determined. Mouse ESC lines with poor survival using current standard cryopreservation protocols or our proposed protocol can be optimized on a case-by-case basis using the method we have outlined over two papers. For our single exception, the CBA cell line, a cooling rate of 5°C/minute in the presence of 1.0M dimethyl sulfoxide or 1.0M propylene glycol, combined with plunge temperature of -80°C was optimal.

Cryopreservation of articular cartilage

Cryopreservation has numerous practical applications in medicine, biotechnology, agriculture, forestry, aquaculture and biodiversity conservation, with huge potentials for biological cell and tissue banking. A specific tissue of interest for cryopreservation is the articular cartilage of the human knee joint for two major reasons: (1) clinically, there exists an untapped potential for cryopreserved cartilage to be used in surgical repair/reconstruction/replacement of injured joints because of the limited availability of fresh donor tissue and, (2) scientifically, successful cryopreservation of cartilage, an avascular tissue with only one cell type, is considered a stepping stone for transition from biobanking cell suspensions and small tissue slices to larger and more complicated tissues. For more than 50years, a great deal of effort has been directed toward understanding and overcoming the challenges of cartilage preservation. In this article, we focus mainly on studies that led to the finding that vitrification is an appropriate approach toward successful preservation of cartilage. This is followed by a review of the studies on the main challenges of vitrification, i.e. toxicity and diffusion, and the novel approaches to overcome these challenges such as liquidus tracking, diffusion modeling, and cryoprotective agent cocktails, which have resulted in the recent advancements in the field.

MR spectroscopy measurement of the diffusion of dimethyl sulfoxide in articular cartilage and comparison to theoretical predictions

Cartilage cryopreservation requires optimal loading of protective solutes, most commonly dimethyl sulfoxide (DMSO), to maximize chondrocyte survival. Previously, diffusion models have been used to predict the distribution of solutes in tissue samples, but the accuracy of spatiotemporal predictions of these models have not been validated with empirical studies and remains unknown.

OBJECTIVE

In this study, magnetic resonance spectroscopic imaging was used to measure the spatial and temporal changes in DMSO and water concentrations in porcine articular cartilage plugs, throughout 1 h of solute loading.

DESIGN

A custom NMR spectroscopic imaging pulse sequence provided water and DMSO concentration images with an in-plane spatial resolution of 135 μm and a temporal resolution of 150 s, repeated for 60 min throughout DMSO loading. Delayed gadolinium-enhanced magnetic resonance of cartilage (d-GEMRIC) imaging provided fixed charge density and spin-density imaging provided water density images prior to DMSO loading.

RESULTS

The measured spatial and temporal distribution of DMSO in three different samples was compared to independent predictions of Fick's law and the modified triphasic biomechanical model by Abazari et al. (2011) with the empirical data more closely agreeing with the triphasic model.

CONCLUSION

Dynamic NMR spectroscopic imaging can measure spatial and temporal changes in water and cryoprotectant concentrations in articular cartilage. The modified triphasic model predictions for the interstitial distribution of DMSO were confirmed and its advantage over the predictions by Fick's law model, which is commonly used in the literature of cryobiology, was demonstrated.

Vitrification of intact human articular cartilage

Articular cartilage injuries do not heal and large defects result in osteoarthritis with major personal and socioeconomic costs. Osteochondral transplantation is an effective treatment for large joint defects but its use is limited by the inability to store cartilage for long periods of time. Cryopreservation/vitrification is one method to enable banking of this tissue but decades of research have been unable to successfully preserve the tissue while maintaining cartilage on its bone base - a requirement for transplantation. To address this limitation, human knee articular cartilage from total knee arthroplasty patients and deceased donors was exposed to specified concentrations of 4 different cryoprotective agents for mathematically determined periods of time at lowering temperatures. After complete exposure, the cartilage was immersed in liquid nitrogen for up to 3 months. Cell viability was 75.4 ± 12.1% determined by membrane integrity stains and confirmed with a mitochondrial assay and pellet culture documented production of sulfated glycosaminoglycans and collagen II similar to controls. This report documents successful vitrification of intact human articular cartilage on its bone base making it possible to bank this tissue indefinitely.

Mathematical modeling of cryoprotectant addition and removal for the cryopreservation of engineered or natural tissues

Long-term storage of natural tissues or tissue-engineered constructs is critical to allow off-the-shelf availability. Vitrification is a method of cryopreservation that eliminates ice formation, as ice may be detrimental to the function of natural or bioartificial tissues. In order to achieve the vitreous state, high concentrations of CPAs must be added and later removed. The high concentrations may be deleterious to cells as the CPAs are cytotoxic and single-step addition or removal will result in excessive osmotic excursions and cell death. A previously described mathematical model accounting for the mass transfer of CPAs through the sample matrix and cell membrane was expanded to incorporate heat transfer and CPA cytotoxicity. Simulations were performed for two systems, an encapsulated system of insulin-secreting cells and articular cartilage, each with different transport properties, geometry and size. Cytotoxicity and mass transfer are dependent on temperature, with a higher temperature allowing more rapid mass transfer but also causing increased cytotoxicity. The effects of temperature are exacerbated for articular cartilage, which has larger dimensions and slower mass transport through the matrix. Simulations indicate that addition and removal at 4°C is preferable to 25°C, as cell death is higher at 25°C due to increased cytotoxicity in spite of the faster mass transport. Additionally, the model indicates that less cytotoxic CPAs, especially at high temperature, would significantly improve the cryopreservation outcome. Overall, the mathematical model allows the design of addition and removal protocols that insure CPA equilibration throughout the sample while still minimizing CPA exposure and maximizing cell survival.

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

Thermodynamic and kinetic models can provide a wealth of information on the physical response of living cells and tissues experiencing cryopreservation procedures. Both isothermal and nonisothermal models have been proposed so far, accompanied by experimental verification and cryoapplications. But the cryoprotective solution is usually assumed to be dilute and ideal in the models proposed in the literature. Additionally, few nonisothermal models are able to couple the transmembrane transport of water and cryoprotectant during cooling and warming of cells. To overcome these limitations, this study develops a whole new set of equations that can quantify the cotransport of water and cryoprotectant across cell membranes in the nondilute and nonideal solution during the freezing and thawing protocols. The new models proposed here can be simplified into ones consistent with the classic models if some specific assumptions are included. For cryobiological practice, they are applied to predict the volumetric change for imprinting control region (ICR) mouse spermatozoa and human corneal keratocytes in the freezing protocol. The new models can determine the intracellular concentration of cryoprotectant more precisely than others by abandoning the assumptions such as dilute and ideal solutions and nonpermeability of membranes to cryoprotectant. Further, the findings in this study will offer new insights into the physical response of cells undergoing cryopreservation.

Application of the multisolute osmotic virial equation to solutions containing electrolytes

The prediction of multisolute solution behavior of solutions containing electrolytes is important in many areas of research, including cryopreservation. In this study, the use of a novel form of the osmotic virial equation for multisolute solutions containing an electrolyte is investigated and compared to a rigorous electrolyte solution theory, the Pitzer-Debye-Huckel equation. For aqueous solutions containing a small molecule (either dimethyl sulfoxide or glycerol) and sodium chloride, the multisolute osmotic virial equation, which utilizes only two parameters to capture the electrolyte solution behavior, is shown to be as accurate as the Pitzer-Debye-Huckel equation, which utilizes six empirical parameters and multiple functions to capture the electrolyte solution behavior. In addition, an approach based on the multisolute osmotic virial equation to investigate the effect of electrolyte concentration on macromolecule solution behavior is presented and applied to aqueous solutions of hydroxyethyl starch and sodium chloride. The multisolute osmotic virial equation is shown to be an accurate, straightforward predictive solution theory for important multisolute solutions containing electrolytes.

Intracellular ice formation in confluent monolayers of human dental stem cells and membrane damage

Dental pulp stem cells (DPSCs) are of interest to researchers and clinicians due to their ability to differentiate into various tissue types and potential uses in cell-mediated therapies and tissue engineering. Currently DPSCs are cryopreserved in suspension using Me(2)SO. However, preservation as two and three dimensional constructs, along with the elimination of toxic Me(2)SO, may be required. It was shown that intracellular ice formation (IIF), lethal to cells in suspensions, may be innocuous in cell monolayers due to ice propagation between cells through gap junctions that results in improved post-thaw recovery. We hypothesized that innocuous IIF protects confluent DPSC monolayers against injury during cryopreservation. The objective was to examine the effects of IIF on post-thaw viability of both confluent monolayers and suspensions of DPSCs. Confluent DPSC monolayers were assessed for the expression of gap junction protein Connexin-43. IIF was induced on the cryostage and in the methanol bath at different subzero temperatures. Membrane integrity and colony-forming ability were assessed post-thaw. Confluent DPSC monolayers expressed Connexin-43. In cell suspensions, 85.9+/-1.7% of cells were damaged after 100% IIF. In cell monolayers, after 100% IIF, only 25.5+/-5.5% and 14.8+/-3.3% of cells were damaged on the cryostage and in the methanol bath respectively. However, DPSC monolayers exposed to 100% IIF showed no colony-forming ability. We conclude that confluent monolayers of DPSCs express the gap junction-forming protein Connexin-43 and upon IIF retain membrane integrity, however lose the ability to proliferate.

A biomechanical triphasic approach to the transport of nondilute solutions in articular cartilage

Biomechanical models for biological tissues such as articular cartilage generally contain an ideal, dilute solution assumption. In this article, a biomechanical triphasic model of cartilage is described that includes nondilute treatment of concentrated solutions such as those applied in vitrification of biological tissues. The chemical potential equations of the triphasic model are modified and the transport equations are adjusted for the volume fraction and frictional coefficients of the solutes that are not negligible in such solutions. Four transport parameters, i.e., water permeability, solute permeability, diffusion coefficient of solute in solvent within the cartilage, and the cartilage stiffness modulus, are defined as four degrees of freedom for the model. Water and solute transport in cartilage were simulated using the model and predictions of average concentration increase and cartilage weight were fit to experimental data to obtain the values of the four transport parameters. As far as we know, this is the first study to formulate the solvent and solute transport equations of nondilute solutions in the cartilage matrix. It is shown that the values obtained for the transport parameters are within the ranges reported in the available literature, which confirms the proposed model approach.