Project Summary

In medicine, various thermal therapeutic methods have been widely used to cure tumor tissues, where the objective is to induce thermal injury within the tumor without affecting the surrounding healthy tissue. In biology, Stefan problem encounters in Cryosurgery and Cryopreservation etc. Cryosurgery is the use of extremely low temperature to damage tumor tissues by freezing and thawing. 

Numerical modeling of biological tissues freezing can be very effective tool supporting the cryosurgical treatment. The quality and reliability of result obtained depends on the selection of correct mathematical model and effective solution procedure. 

Due to its simplicity, ease application and effectiveness, Pennes bio heat model is most widely used model for heat transfer in biological tissues. It is based on classical Fourier’s law that depicts infinitely fast propagation of thermal signs in the medium, which is contradictory to physical reality. 

To solve the paradox occurred in Fourier’s law; the lag times referred as ‘a relaxation time’ in heat flux and temperature gradient were introduced and the single and double phase lag heat conduction model were developed. These phase-lagging models are also called as non-Fourier heat conduction models.

Biological tissues form a multidimensional irregular domain, thus the multidimensional simulation of bio-heat transfer model is crucial to analyze the thermal therapies. 

In recent years, mesh-free methods based on radial basis function (RBF) approximation have been proved very attractive tool to solve PDEs due to the fact that these are truly meshes and spatial dimension independent. This makes the implementation of these methods much simpler than e. g. FEM in higher dimensions. Another useful feature of RBFs is their radial symmetry and invariance under Euclidean transformation. Further, some RBFs like (inverse) multi-quadratics, Gaussian gives spectral convergence order in the context of the scattered data interpolation.