Eliminating osmotic stress during cryoprotectant loading: A mathematical investigation of solute-solvent transport

Osmotic stresses during cryoprotectant loading induce changes in cellular volume, leading to membrane damage or even cell death. Appropriate model-guided mitigation of these osmotic gradients during cryoprotectant loading is currently lacking, but would be highly beneficial in reducing viability loss during the loading process. To address this need, we reformulate the two-parameter formalism described by Jacobs and Stewart for cryoprotectant loading under the constraint of constant cell volume. We then derive simple, concise, analytic solutions to these equations, showing the transient extracellular permeating and nonpermeating cryoprotectant concentrations required to load a cell at constant volume, thus eliminating osmotic stresses during cryoprotectant loading. Additionally, we show analytic approximations of both ramp (linear) as well as step-wise loading and how one can use the hydraulic conductivity L_p, membrane permeability P_s, cell volume V_o, and osmotically inactive fraction to derive cryoprotectant loading protocols that minimize osmotic stress. We also present timescales for water and cryoprotectant transport which can be used to estimate loading times as well as L_p and P_s. We discuss how previous optimized loading strategies are inherently sensitive to parameter uncertainties and biological variability, increasing the likelihood of exceeding critical osmotic limits. By contrast, the proposed protocol provides a larger buffer against deviations, offering a safer and more robust solution to CPA loading. Importantly, we demonstrate that the volume-loss-free CPA loading protocols outlined in this paper occur on the same timescale as conventional and step-loading methods, suggesting that these protocols could be a safer alternative for CPA loading.