Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 9021616, 12 pages

https://doi.org/10.1155/2017/9021616

## An Analytical Solution for Radiofrequency Ablation with a Cooled Cylindrical Electrode

^{1}Facultad de Ingeniería, Universidad Autónoma de San Luis Potosí, Manuel Nava No. 8, Zona Universitaria, 78290 San Luis Potosí, SLP, Mexico^{2}Biomedical Synergy, Department of Electronic Engineering, Universitat Politècnica de València, Camí de Vera, 46022 Valencia, Spain

Correspondence should be addressed to Ricardo Romero-Méndez; xm.plsau@moremorr

Received 19 December 2016; Accepted 26 January 2017; Published 6 March 2017

Academic Editor: Alessio Gizzi

Copyright © 2017 Ricardo Romero-Méndez and Enrique Berjano. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We present an analytical solution to the electrothermal mathematical model of radiofrequency ablation of biological tissue using a cooled cylindrical electrode. The solution presented here makes use of the method of separation of variables to solve the problem. Green’s functions are used for the handling of nonhomogeneous terms, such as effect of electrical currents circulation and the nonhomogeneous boundary condition due to cooling at the electrode surface. The transcendental equation for determination of eigenvalues of this problem is solved using Newton’s method, and the integrals that appear in the solution of the problem are obtained by Simpson’s rule. The solution obtained here has the possibility of handling different functional dependencies of the source term and nonhomogeneous boundary condition. The solution provides a tool to understand the physics of the problem, as it shows how the solution depends on different parameters, to provide mathematical tools for the design of surgical procedures and to validate other modeling techniques, such as the numerical methods that are frequently used to solve the problem.

#### 1. Introduction

Radiofrequency (RF) ablation is a surgical technique considered as marginally invasive, used, for instance, to destroy malignant tissue in bland organs, such as liver, kidney, and lungs. The objective of this treatment is to heat the tumor to a temperature above a threshold where the tissue is damaged. Initially, this procedure was limited to small volumes of tissue damage. Rossi et al. [1] and Goldberg [2] reported that the diameter of tissue damage attained by a single electrode was of only 1.5 cm, reason for which RF ablation was only considered viable for treatment of small tumors. A problem of RF ablation is tissue charring around the electrode, which produces a phenomenon called roll-off, in which the tissue surrounding the electrode increases its electrical resistance and with it the circuit impedance, with which current circulation stops and no further damage to tissue is produced [3]. Several alternatives have been used to avoid early appearance of roll-off and to increase the volume of tissue damaged: internally cooled electrodes, expandable electrodes, hybrid electrodes, and so forth [4]. The first alternative is to use internal cooling of the active electrode. Internally cooled electrode (known as cool-tip electrode) is one of the most frequently used alternatives for hepatic tissue damage [5, 6]; it consists of a needle shaped electrode with an active tip, inside which there is a circulation of chilled water, which cools down the tissue that surrounds the electrode and avoids carbonization of the tissue [7].

Much of the studies that have been performed in relation to RF current ablation have relied on experimental techniques and numerical methods. Very few analytical solutions have been reported in relation to this phenomenon. Haemmerich et al. [8] presented analytical results of the modeling of RF ablation for steady state conditions, where blood perfusion and metabolic heat generation were neglected. López Molina et al. [9] presented a solution to the transient problem for an infinite cylindrical geometry that was solved by Laplace transform. Here we present an alternative solution to the time dependent RF ablation heating process from a cooled cylindrical RF electrode in a finite domain using Green’s function. This solution allows including any kind of source term (time dependent and with any spatial dependence) and boundary conditions that may be time dependent. The techniques presented here provide a closed form solution of this phenomenon, with which it is possible to understand better the influence of the different factors influencing tissue heating, and it is useful also as a tool for the design of surgical protocols and for validating numerical solutions obtained for the purpose of modeling RF ablation.

#### 2. Methods

##### 2.1. Problem Description and Mathematical Model

Figure 1 shows how the geometry of the problem was defined. RF ablation is performed by placing the active electrode in the middle of the tumor. The problem is illustrated by Figure 1(a), where the tissue region is modeled as a cylindrical volume, with the active electrode placed inside the tissue and the dispersive electrode being placed on the external surface. The tissue consists of a region of cancerous tissue (tumor) of spherical geometry, surrounded by healthy tissue, represented by the cylindrical geometry. The active electrode is illustrated in Figure 1(a) as the part of the RF applicator placed inside the sphere that represents the cancerous tissue. The axial mid-plane of the conductive part of the active electrode, represented by the circle placed at the mid-height of the domain, is modeled as a symmetry plane, where the electric and thermal problems behave as axisymmetric one dimensional problems, depending only on the radial coordinate. For simplicity, the region to be analyzed is this plane: the annulus of inner radius and outer radius . In some studied cases, here we also consider a two-compartment model, with cancerous tissue located in the zone , while the healthy tissue is in the region , represented by Figure 1(b). Inside this annular section of tissue is an active RF electrode of radius and the dispersive electrode is considered to be at the periphery of the tissue.