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.