Mathematical modeling of surface deformation during vitrification

Large surface deformation due to thermo-mechanical effects during cryopreservation by vitrification – mathematical model and experimental validation

“…and where E is Young's Modulus, ν is Poisson's ratio, σ is the stress, tr is the trace of the stress tensor, S is the deviatoric stress tensor, and α is the thermal expansion coefficient. Finally, consistent with current and prior experimental observations [7,11,24,28], it is assumed that the CPA adheres to the container walls as it vitrifies.…”

“…The overall heat transfer coefficient between the cuvette wall and the cooling chamber environment, h, was measured experimentally in a previous study [24]. Continuity in temperature and heat flux is assumed on all interfaces between the container wall and CPA [11,35].…”

“…While cryopreservation has successfully been implemented for preservation of small-sized specimens (in the scale of μm to mm), preservation of large-sized tissues and organs (in a scale of cm and above) requires overcoming challenges related to toxicity of cryoprotective agents (CPA) [4], ice formation [5], chilling and ischemic injuries [6], and thermomechanical stress [7][8][9][10][11]. Cryopreservation of tissues and organs by means of vitrification (vitreous in Latin means glassy) is a promising ice-free preservation alternative for large size biological specimens [12,13].…”

“…The viscosity of the CPA increases exponentially by fifteen orders of magnitude during the vitrification process [22], from a water-like viscosity level at room temperature, to such a high value around the glass transition temperature, that flow is practically arrested and the material behaves like an amorphous solid. Due to the combined effect of temperature gradients during the inward cooling process of the specimen, and variations in thermal expansion and viscosity with temperature, dramatic deformation at free surfaces may be observed [11,[23][24][25]. However, since these surface deformations dissipate during rewarming, as the CPA regains fluid-like viscosity at higher temperatures, and since the specimens are most frequently evaluated only after recovery from cryogenic storage, such large surface deformations has eluded cryobiologists for decades.…”

“…Such large deformation can increase the average stress level within the specimen and cause stress concentrations along its surfaces [11,[23][24][25][26]. The stress developed during cooling might intensify when the temperature distribution across the specimen equilibrates during cryogenic storage (i.e., residual stress), and might even further intensify at the onset of rewarming [11,27,28]. Previous thermomechanical stress analysis relied on extracting the deformed surface geometry from cryomacroscopy images [26].…”