Analytical and numerical solutions of the potential and electric field generated by different electrode arrays in a tumor tissue under electrotherapy

Abstract

Electrotherapy is a relatively well established and efficient method of tumor treatment. Circular and elliptical electrode arrays in electrotherapy in voltage mode and in electrochemotherapy are the most used. However, other shapes of electrode arrays have not been explored in order to increase the effectiveness of these therapies.In this paper we focus on analytical and numerical calculations of the potential and electric field distributions inside a tumor tissue in a two-dimensional model (2D-model) generated by means of electrode arrays with shapes of different conic sections (ellipse, parabola and hyperbola). Analytical calculations of the potential and electric field distributions based on 2D-models for different electrode arrays are performed by solving the Laplace equation, meanwhile the numerical solution is solved by means of finite element method in two dimensions. Comparison between the analytical and numerical solutions is carried out. Electrode array with different shapes (circular, elliptic, parabolic and hyperbolic) are used. Unifying principle for the conic sections is also used. The results evidence that both analytical and numerical solutions reveal significant differences between the electric field distributions generated by electrode arrays with shapes of circle and different conic sections (elliptic, parabolic and hyperbolic). Electrode arrays with circular, elliptical and hyperbolic shapes have the advantage of concentrating the electric field lines in the tumor. In conclusion, the mathematical approach presented in this study provides a useful tool for the design of electrode arrays with different shapes of conic sections by means of the use of the unifying principle. At the same time, we verify the good correspondence between the analytical and numerical solutions for the potential and electric field distributions generated by the electrode array with different conic sections.