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

Monitoring Aqueous Sucrose Solutions Using Droplet Microfluidics: Ice Nucleation, Growth, Glass Transition, and Melting

Freezing and freeze-drying processes are commonly used to extend the shelf life of drug products and to ensure their safety and efficacy upon use. When designing a freezing process, it is beneficial to characterize multiple physicochemical properties of the formulation, such as nucleation rate, crystal growth rate, temperature and concentration of the maximally freeze-concentrated solution, and melting point. Differential scanning calorimetry has predominantly been used in this context but does have practical limitations and is unable to quantify the kinetics of crystal growth and nucleation. In this work, we introduce a microfluidic technique capable of quantifying the properties of interest and use it to investigate aqueous sucrose solutions of varying concentration. Three freeze-thaw cycles were performed on droplets with 75-μm diameters at cooling and warming rates of 1 °C/min. During each cycle, the visual appearance of the droplets was optically monitored as they experienced nucleation, crystal growth, formation of the maximally freeze-concentrated solution, and melting. Nucleation and crystal growth manifested as increases in droplet brightness during the cooling phase. Heating was associated with a further increase as the temperature associated with the maximally freeze-concentrated solution was approached. Heating beyond the melting point corresponded to a decrease in brightness. Comparison with the literature confirmed the accuracy of the new technique while offering new visual data on the maximally freeze-concentrated solution. Thus, the microfluidic technique presented here may serve as a complement to differential scanning calorimetry in the context of freezing and freeze-drying. In the future, it could be applied to a plethora of mixtures that undergo such processing, whether in pharmaceutics, food production, or beyond.

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.

Predicting freezing points of ternary salt solutions with the multisolute osmotic virial equation

Previously, the multisolute osmotic virial equation with the combining rules of Elliott et al. has been shown to make accurate predictions for multisolute solutions with only single-solute osmotic virial coefficients as inputs. The original combining rules take the form of an arithmetic average for the second-order mixed coefficients and a geometric average for the third-order mixed coefficients. Recently, we derived generalized combining rules from a first principles solution theory, where all mixed coefficients could be expressed as arithmetic averages of suitable binary coefficients. In this work, we empirically extended the new model to account for electrolyte effects, including solute dissociation, and demonstrated its usefulness for calculating the properties of multielectrolyte solutions. First, the osmotic virial coefficients of 31 common salts in water were tabulated based on the available freezing point depression (FPD) data. This was achieved by polynomial fitting, where the degree of the polynomial was determined using a special criterion that accounts for the confidence intervals of the coefficients. Then, the multisolute model was used to predict the FPD of 11 ternary electrolyte solutions. Furthermore, models with the new combining rules and the original combining rules of Elliott et al. were compared using both mole fraction and molality as concentration units. We find that the mole-fraction-based model with the new combining rules performs the best and that the results agree well with independent experimental measurements with an all-system root-mean-square error of 0.24 osmoles/kg (0.45 °C) and close to zero mean bias for the entire dataset (371 data points).

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.

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.

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.

Effect of cryoprotectant concentration on bovine oocyte permeability and comparison of two membrane permeability modelling approaches

The plasma membrane permeability to water and cryoprotectant (CPA) significantly impacts vitrification efficiency of bovine oocytes. Our study was designed to determine the concentration-dependent permeability characteristics for immature (GV) and mature (MII) bovine oocytes in the presence of ethylene glycol (EG) and dimethyl sulphoxide (Me₂SO), and to compare two different modeling approaches: the two parameter (2P) model and a nondilute transport model. Membrane permeability parameters were determined by consecutively exposing oocytes to increasing concentrations of Me₂SO or EG. Higher water permeability was observed for MII oocytes than GV oocytes in the presence of both Me₂SO and EG, and in all cases the water permeability was observed to decrease as CPA concentration increased. At high CPA concentrations, the CPA permeability was similar for Me₂SO and EG, for both MII and GV oocytes, but at low concentrations the EG permeability of GV oocytes was substantially higher. Predictions of cell volume changes during CPA addition and removal indicate that accounting for the concentration dependence of permeability only has a modest effect, but there were substantial differences between the 2P model and the nondilute model during CPA removal, which may have implications for design of improved methods for bovine oocyte vitrification.

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.

Gibbsian Surface Thermodynamics

Gibbsian composite-system thermodynamics is the framework governing the equilibrium of composite systems, including systems that at equilibrium have more than one value of pressure because of the action of surface tension, semipermeable membranes, or fields, and thus cannot be treated as simple systems. J. W. Gibbs's paper that lays out composite-system thermodynamics, "On the Equilibrium of Heterogeneous Substances", communicated in two parts in 1876 and 1878, is widely regarded as one of the most important pieces of scientific literature of its century. Many scientists adopted and stressed the importance of Gibbsian thermodynamics. In 1960, H. B. Callen wrote a textbook that made Gibbsian composite-system thermodynamics more accessible to thermodynamicists. However, Callen's book left out Gibbs's work on curved fluid interfaces and did not treat the complicated nonideal systems of interest to today's thermodynamicists. In this Feature Article, I have attempted to convey in a comprehensive manner the framework of Gibbsian composite-system thermodynamics including in detail the treatment of systems with interface effects and with nonideal, multicomponent phases. This work lays out the relationships between important equilibrium equations including the following: the Gibbs-Duhem equation, the Gibbs adsorption equation, the Young-Laplace equation, the Young equation, the Cassie-Baxter equation, the Wenzel equation, the Kelvin equation, the Gibbs-Thompson equation, and the Ostwald-Freundlich equation, including nonideal and multicomponent forms. Equations of state that are often useful for Gibbsian composite-system thermodynamics are reviewed including adsorption isotherms and our own work on two semiempirical equations of state: the Elliott et al. form of the osmotic virial equation and the Shardt-Elliott-Connors-Wright equation for the temperature and composition dependence of surface tension. I summarize the work of our group developing Gibbisan composite-system thermodynamics including new equations for such things as the curvature-induced depression of the eutectic temperature or the removal of azeotropes by nanoscale fluid interface curvature. Gibbsian composite-system thermodynamics has broad applications in biotechnology, nanostructured materials, surface textures and coatings, microfluidics, nanoscience, atmospheric and environmental physics, among others.

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.

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.

Comparison of the Fitting Validity Between the 2P Model and the Nondilute Solution Model Using Statistical Methods in Modeling Cell Membrane Permeabilities

In cryopreservation, the two-parameter (2P) model and the nondilute solution model have been developed to study the membrane transport properties of cells. However, to our knowledge, comparison of the fitting validity has never been made between the two models. In this study, to make this comparison, the permeability parameters of porcine adipose-derived stem cells (pADSCs) were first determined with the two models, and then the errors between the predictions and the experimental data were tested using the Lilliefors test. The results indicate that the 2P model is slightly better than the nondilute solution model in predicting the mass transport across cell membrane. The causes for this phenomenon are discussed and suggestions on using these two models are given.