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# To extend these toxicity minimization strategies to

To extend these toxicity minimization strategies to 3D tissues, a more complicated mass transfer model is necessary to account diffusion in the interstitial space, as well as transport across cell membranes. Until recently, tissue mass transport in cryobiology was modeled using classic spatially dependent transport models. In these models, some version of the diffusion equation is used to predict either the concentration of CPA or the temperature inside the tissue as a function of the external concentration or temperature field. For example, Han et al. use a radially symmetric diffusion model to find the diffusivity of CPA in rat ovaries [22], Abazari et al. use a more sophisticated triphasic diffusion model that accounts for the biomechanics of articular cartilage as well as the movement of solutes and solvents [1], Manuchehrabadi et al. use heat and mass transport modeling to predict thermal gradient induced stress inside tissues and organs [31]. At the heart of each of these models lies the basic diffusion model (see e.g. Anderson et al. [3]):where u is the quantity transported and D is a diffusion constant. This equation is coupled with initial and boundary conditions in the usual way, and evaluated over a relevant geometry.
In this study, we present a novel approach to quantify the damage due to the accumulation of toxicity as a function of cryoprotectant loading protocol in three tissues. We combine elements of individual cell cryoprotectant loading theory with the diffusion equation that allows the determination of concentration as a function of position. In particular, we expand the cost function Jtox to include spatial dependence while maintaining standard individual cell osmotic tolerance limit constraints as a proxy for the stresses that tissues undergo while equilibrating with high concentrations of cryoprotectants. The model in this study is informed by new diffusivity measurements on three human tissue types (skin, fibroid and myometrium), and existing permeability data from plated endothelial 155 7 synthesis as a proxy for the cells on the exterior of the tissue. These tissue types were chosen because they are readily available and have diverse properties that will allow us to examine the versatility of our methods. In particular, fibroid tissue has a high density of extracellular matrix proteins and is relatively rigid compared with myometrium while skin contains both soft and tough connective tissues.

Methods and models

Results

Discussion
Until this manuscript there has been little guiding theory for the equilibration of cryoprotectants in tissues. There have been many publications prescribing protocols to equilibrate vitrification solutions with minimal damage—many acknowledge that a step-wise approach is appropriate—but most protocols use somewhat arbitrary concentration steps and durations. There have been some approaches that use a mass transfer model to assure that the tissues are suitably equilibrated; for example in the recent publication by Manuchehrabadi et al. [31], the intratissue concentration as a function of time in during exposure is measured using Computed Tomography and informs a similar mass transfer model to the one used in the present manuscript. However, in their approach, no effort is made to optimize the equilibration to and from high concentrations of cryoprotectant solutions.
This lack of modeling to minimize toxicity is surprising in some ways, because CPA toxicity has been highlighted as one of the chief impediments to successful cryopreservation of tissues and organs [28]. This, coupled with the fact that great gains have been made in the area of single cell suspension equilibration protocol optimization, including human and bovine oocytes [9], [24], [32], plated endothelial cells [14], sperm [18], [19] and others, suggests that gains in CPA toxicity minimization may be made in tissues as well. This gain can be seen in Fig. 8, where the accumulated toxicities as a function of protocol are shown for each tissue type. Here we can see that the standard stepwise approach is neither faster nor less toxic than the toxicity optimal protocol, and that, if CPA toxicity is not a concern, time optimal protocols can be 4–6 times shorter than the non-optimized standard stepwise approach and 2–3 times shorter than their respective toxicity optimal protocols.