Game theory; Cournot game; Incomplete information; Oligopoly market; Estimation; Nash equilibrium

The knowledge of the electric current density distribution generated by an electrode array is very useful in electrotherapy for tumor treatment. We propose an innovative mathematical approach that takes into account planar solid tumor elliptic geometry, electrical differences between it and its surrounding healthy tissue, and positioning of the electrodes with respect to tumor-surrounding healthy tissue interface. We show the distributions of the electric current density in leading order and first correction terms in a heterogeneous planar medium formed by two regions (tumor and its surrounding healthy tissue) in function of these parameters. The results show that when electrodes are completely inserted in tumor and/or its conductivity is higher than that of its surrounding healthy tissue, the electric current density lines concentrate more in tumor and its tumor-surrounding healthy tissue interface. No significant differences are reported between the electric current density distributions in leading-order and first-order correction for each parameter investigated. However, norm of this physical magnitude reveals that these distributions are different when the ratio between radius of the electrodes and radius of the tumor is less than 0.8. We conclude that the analytical modeling presented in this study is of practical interest because it provides a convenient way to visualize the electric current density distributions generated by an electrode elliptic array in order to efficiently destroy the localized planar tumors with the minimum damage to organism, through an increase of the potential applied to the electrodes, the tumor conductivity with respect to its surrounding healthy tissue and insertion of all electrodes into tumor.