The time fractional approach for the modeling of thermal therapies: Temperature analysis in laser irradiation

Abstract

In this study we assessed the use of the fractional derivative formulation of the heat transmission equation (FDHTE) as an alternative to the classical or parabolic heat transfer equation (PHTE) in the mathematical modeling of some thermal therapy processes. We obtained the FDHTE analytical solutions in two cases: a general case of heat transfer in a finite bar with different and constant temperature in its extremes (without heat source), and the heating by a laser source of a semi-infinite medium which includes a heat source and it was considered in thermal therapies to destroy or alter biological tissue. Both solutions were obtained analytically and compared with the PHTE results. We also compared the FDHTE solution with the results of the hyperbolic heat transfer equation (HHTE), which is another alternative to the PHTE, but is only used for problems in which intense heat is applied to materials for very short times. The results show that the FDHTE can be used as an alternative to the PHTE in thermal therapy processes in which the PHTE theoretical models underestimate or overestimate the temperatures achieved in tissue. Unlike the HHTE, the FDHTE is not restricted to special problems only. We have thus laid the groundwork for the analytical resolution of the problems considered by the FDHTE.