# Cryoprotectant equilibration and the response to ice

Cryoprotectant equilibration and the response to ice concentrated media are the two facets of cellular cryobiological protocols most commonly covered by mathematical modeling. In these models, TMPyP4 tosylate are typically assumed to have uniform, spatially independent extracellular concentrations and temperatures (see Ref. [3] for review). Spatial homogeneity and infinite bath media assumptions mean that ordinary differential equation models can be used to determine the intracellular state as a function of cryoprotocol. The transport equations typically used to describe the intracellular water volume W and moles of solute S as a function of time t:are known as the two-parameter (or 2P) equations in contrast with the three parameter Kedem-Ketchalsky model which is now less used in cryobiological literature (see Ref. [26] for explanation). Here LpRT and Ps are water and solute permeabilities respectively, A the cell surface area (assumed constant), C concentrations with superscript indicating extracellular or intracellular quantities (see Table 1 for a table of parameters). This model is coupled via a relationship between total cell volume and water volume known as the Boyle van\'t Hoff relationship [6]. Several modeling approaches have been used to predict optimal CPA equilibration strategies in single cells. Initially, it was observed that osmotically driven volume changes associated with CPA equilibration might be the cause of cell death in sensitive cells such as sperm [19]. To account for this, cell population specific osmotic tolerance limits were established that defined the limits to which cells could shrink and swell osmotically with no or little damage [17]. These limits are used in conjunction with the 2P model to determine CPA equilibration protocols that equilibrate cells with the desired amount of CPA with minimal osmotic damage. In many cell types, this is achieved with the default, single-step addition and removal protocols as cells are sufficiently permeable to CPA and/or have sufficient abilities to withstand large volume fluctuations with minimal damage [25]. The next step in development of CPA equilibration optimization was to choose among the remaining safe protocols. Levin suggested that one could avoid all osmotic challenges by using a transport equation such as Eq. (1) to determine extracellular concentration functions that facilitate the replacement of water with CPA in cells, effectively holding the cell volume constant [27]. These protocols, to our knowledge, were never put to practice, likely due to the difficulty of predicting the extracellular concentrations that could achieve this safely in one cell (e.g. an oocyte) or in a group of cells. Other investigators chose protocols with the minimal number of steps [18]. Along this vein, Benson described a cost-functional approach defining a protocol as optimal if it was achieved in minimal time [4], [7], and this was implemented independently by Karlsson et al. [24] in a stepwise approach. This cost functional approach has the benefit of formally acknowledging that CPA exposure is associated with cytotoxicity, and that the duration of exposure should be minimized in some way. The cost functional approach to quantify non-osmotically induced CPA exposure damage was expanded by Higgins [23] and then Benson et al. [9] where it was hypothesized that the rate of accumulation of damage during CPA exposure was proportional to a power function of the intracellular CPA concentration:where tf is the time at which the cell reaches a desired intracellular concentration, CCPA is the intracellular CPA concentration, modeled with the 2P equation, and α is a phenomenological rate parameter. Benson et al. fit toxicity kinetics data in the literature and found α = 1.6 [9], a value that was confirmed in plated endothelial cells in a follow up study by Davidson et al. [14]. In fact, the theoretically optimal step-wise CPA equilibration strategies determined by Davidson et al. were implemented in plated endothelial cells [14]. Non-cost functional optimized equilibration strategies were compared with temperature dependent cost functional strategies and untreated controls, and while the classic non-cost-function approach showed dramatic cell death, no significant differences were found in the latter two [14].