Mathematical modeling of cryoprotectant addition and removal for the cryopreservation of engineered or natural tissues

Abstract

Long-term storage of natural tissues or tissue-engineered constructs is critical to allow off-the-shelf availability. Vitrification is a method of cryopreservation that eliminates ice formation, as ice may be detrimental to the function of natural or bioartificial tissues. In order to achieve the vitreous state, high concentrations of CPAs must be added and later removed. The high concentrations may be deleterious to cells as the CPAs are cytotoxic and single-step addition or removal will result in excessive osmotic excursions and cell death. A previously described mathematical model accounting for the mass transfer of CPAs through the sample matrix and cell membrane was expanded to incorporate heat transfer and CPA cytotoxicity. Simulations were performed for two systems, an encapsulated system of insulin-secreting cells and articular cartilage, each with different transport properties, geometry and size. Cytotoxicity and mass transfer are dependent on temperature, with a higher temperature allowing more rapid mass transfer but also causing increased cytotoxicity. The effects of temperature are exacerbated for articular cartilage, which has larger dimensions and slower mass transport through the matrix. Simulations indicate that addition and removal at 4°C is preferable to 25°C, as cell death is higher at 25°C due to increased cytotoxicity in spite of the faster mass transport. Additionally, the model indicates that less cytotoxic CPAs, especially at high temperature, would significantly improve the cryopreservation outcome. Overall, the mathematical model allows the design of addition and removal protocols that insure CPA equilibration throughout the sample while still minimizing CPA exposure and maximizing cell survival.

Figure 1

Schematic for A) alginate bead and B) articular cartilage.

Figure 2

Comparison of cytotoxicity equation, equation 17, (lines) with experimentally determined normalized cell viability (n/n₀) (data points) for 2M DMSO (black, diamond) or 6M PD (gray, triangle) with addition steps carried out at 25°C (solid) or 4°C (empty points, dashed line). Experimental data from [28]. n=3 for experimental data, error bars indicate standard deviation. Statistical significance not shown.

Figure 3

Temperature profile at center of slab (z=0), midway through slab (z=0.5mm), and at edge of slab (z=1) in response to temperature step changes at the edge of the slab.

Figure 4

A) Predicted Normalized Viable Cell Number (n/n₀) at edge of bead and B) concentration profile for predominant CPA, PD, at center of bead for 4°C (solid gray) and 25°C (dashed black) addition and concentration in bulk solution (solid black) for 3 min/addition step. All removal steps carried out at 25°C.

Figure 5

Predicted Normalized Viable Cell Number (n/n₀) at edge of bead for different exposure times at 25°C. At this temperature, 1.5 min/addition step is sufficient to achieve equilibrium. 3 min/addition step and 10 min/addition step are overexposures.

Figure 6

A) Concentration Profiles of predominant CPA, PD, throughout slab for two temperatures and B) Total Predicted Normalized Viable Cell Number (n/n₀) in slab for different exposure times and temperatures. All removal steps occurred at 25°C (40 minutes/step).

Figure 7

Predicted Normalized Viable Cell Number (n/n₀) at center of slab (dashed) or edge of slab (solid) for addition steps at 4°C (80 min/step) and removal at 4°C (80 min/step, black) or 25°C (40 min/step, gray).

Figure 8

Predicted Normalized Viable Cell Number (n/n₀) at center of slab (dashed) and at edge of slab (solid) for 25°C, 40 minute addition and removal steps for DMSO values of cytotoxicity parameters (black) vs cocktail parameters (gray).