The function of Green for the bioheat equation of Pennes in an axisymmetric unbounded domain

Abstract

[EN] The function of Green associated to a linear partial differential operator P(D) in a domain ¿ acting at point x0 of the domain, is a distribution G(x,x0) such that P(D)G(x,x0) = ¿(x¿x0), where ¿is the Dirac¿s delta distribution. The property P(D)G(x,x0) = ¿(x¿x0) of a Green¿s function can be exploited to solve differential equations of the form P(D)u = f, because Hence which implies that u = G(x,x0)f(x0)dx0. Not every operator P(D) admits a Green¿s function. And the Green¿s function, if it exists, is not unique, but adding boundary conditions it will be unique. In regular Sturm-Liouville problems, there is an standard way to obtain the corresponding Green¿s function, and after that, as the domain is bounded, to incorporate the initial and boundary conditions using also the Green¿s function. But the method doesn¿t work if the domain is not bounded, because the justification is based in the use of the Green¿s Theorem. In this paper we find the Green¿s function for the Pennes¿s bioheat equation, see [1], in a unbounded domain consisting in the space R3 with an infinite cylindrical hole. This type of problems appears in radiofrequency (RF) ablation with needle-like electrodes, which is widely used for medical techniques such as tumor ablation or cardiac ablation to cure arrhythmias. We recall that theoretical modeling is a rapid and inexpensive way of studying different aspects of the RF process.