Computational aspects of heat conduction models for biological tissues
The idea that the heat equation is flawed due to the infiniteВ speed of thermal propagation is not new. A hyperbolic model forВ heat conduction was proposed by Cattaneo in 1948 and Vernotte inВ 1958.
The application of femto-second lasers in many fields caused aВ rethink of the parabolic heat conduction model. In thisВ talk we consider the paper by Kim and Guo in 2007 . TheyВ developed a combined radiation and heat transfer model to simulateВ heat transfer in turbid tissues, covering three phases. After theВ irradiation phase they use the hyperbolic model to predict theВ temperature field. The authors quote several other papers toВ support their choice of the hyperbolic model; the argumentsВ against the parabolic model is convincing.В The numerical approximation is by finite differences with so called "errorВ terms correction".
Van Rensburg and Sieberhagen  considered a number of papersВ on the numerical approximation of the hyperbolic heat transferВ model where the authors used various methods to get rid of "spurious" oscillations. They pointed out that the oscillationsВ were due to problems being ill-posed rather than the numericalВ method. However, a well-posed formulation required an extremelyВ fine grid and was not practical. The authors did solve the problem for the one-dimensional case using FEM combined with d'Alembert's method.
We accept that the model in Kim and Guo is realistic. However, theВ model problem for the conduction phase is not well-posed due to theВ boundary conditions. We explain why, and also explain why theВ linearized dual phase lag hyperbolic model is better. Finally, weВ present results obtained with the mixed finite element method.
K. Kim and Z. Guo, Multi-time-scale heat transfer modeling of turbid tissues exposed to short-pulsed irradiations, Computer methods and programs in Biomedicine, 86 (2007), 112-123.
R.H. Sieberhagen and N.F.J. Van Rensburg, Tracking a sharp crested wave front in hyperbolic heat transfer, Applied Mathematical Modelling, 36 (2012), 3399вЂ“3410.