<p>Raw data for Fig 7.</p>

Background
Thermal imaging is a convenient technique for cryoablation and cryotherapy monitoring; however, it does not provide insight into subsurface temperature distribution.
Objectives and methods
Our mathematical model can predict a temperature penetration depth during cryotherapy based on surface thermal imaging. We also generalized this model to use with homogeneous media, such as hydrogels, and multilayered biological structures, including skin, muscle, and subcutaneous fat.
Results
We developed a mathematical model to describe temperature dynamics in non-uniform materials with temperature-dependent thermodynamic properties and a phase-change boundary. We implemented the model with a graphical processing unit (GPU) to simulate the freezing behavior of hydrogels and biological tissues. The model was validated by comparing simulation results with experimental findings from hydrogel cryoapplications and previous in vivo studies on rats during the freezing phase.
Hydrogel, which exhibits thermodynamic properties like those of living tissues and possess optical transparency, enabled direct observation of the freezing front using visual morphometry. This model offers a practical tool for estimating optimal cryotherapy duration, helping to minimize damage to healthy cells. The position of the freezing front in semitransparent hydrogels was quantitatively assessed and found to be consistent with morphometric measurements.
Conclusions
This study provides a useful framework for comparing in vitro and in vivo thermal field dynamics and for estimating optimal timing in cryoapplications. The mathematical model developed here, with its fast and efficient GPU-based implementation, can be extended to investigate thermal behavior in other uniform and non-uniform materials exhibiting temperature-dependent thermodynamic properties during freeze-thawing.