Analytical Solution for Electrical Problem Forced by a Finite-Length Needle Electrode: Implications in Electrostimulation

Needle electrodes, widely used in clinical procedures, are responsible for creating an electric field in the treated biological tissue. This is achieved by setting a constant voltage along the length of their metallic section. In accordance with Laplace’s equation, the electric field is spatially non-uniform around the electrode surface. Mathematical modelling can provide useful information on the spatial distribution of electrical fields. Indeed, exact solutions for the electrical problem are indispensable for validating numerical codes. All the analytical models developed to date to solve the needle electrode electrical problem have been one-dimensional models, which assumed an electrode of infinite length. We here propose the first analytical solution based on a two-dimensional model that considers the real length of the electrode in which the Laplace equation is solved through the method of separation of variables, dealing with the nonhomogeneous source term and boundary conditions by Green’s functions. On assuming a needle electrode of given length, the problem combines boundary conditions on the electrode boundary (of the first and second kind). Since this rules out using the Sturm-Liouville Theorem, the problem is decomposed into two different problems and the principle of superposition is used. The solution obtained can reproduce a reasonable electric field around the electrode, especially the edge effect characterized by an extremely high gradient around the electrode tip.