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

Abstract

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.

Differences in the Electric Field Distribution Predicted with a Mathematical Model of Cylindrical Electrodes of Finite Length vs. Infinite Length: A Comparison Based on Analytical Solution

“…Our main finding confirms what has already been observed by other authors [10,12]: The infinite-length model does not allow us to reproduce the edge effect, i.e., the extremely high values of the electric field induced in the tissue located just at the ends of the finitelength electrode (joining zones with the plastic). However, these "hot spots" in electrical terms, which also appear on the periphery of circular disk-type electrodes [25], appear to be only relevant in areas very close to the electrode (<1 mm for the specific conditions considered in our study, i.e., electrode dimensions and applied voltage value).…”

“…The solution for Equation ( 8) is very similar to what was previously presented in [12]. In essence, it was obtained through the method of separation of variables, dealing with the nonhomogeneous source term and boundary conditions by Green's functions.…”

“…The cylindrical structure of these electrodes suggests an axial symmetry that can significantly simplify the mathematical resolution of the problem by assuming a onedimensional problem, at least when the surrounding tissue is homogeneous. While the problem is very simple to solve for the case of an electrode of infinite length [3], the analytical solution is relatively complex for the case of an electrode of finite length, as occurs in reality [12].…”

“…Our objective was to quantify the differences in the electric field distributions predicted with the mathematical models considering finite vs. infinite length of cylindrical electrodes rounded by biological tissue and hence assess the adequacy of the mentioned approximation (i.e., model based on infinite length) for different values of electrode diameter and length. To this end, we used analytical solutions for the electric field previously obtained in the case of homogeneous tissue, specifically, a 1D model that assumes an infinite-length electrode [11] and a 2D model that assumes a finite-length electrode [12]. In contrast to the use of numerical techniques (such as the finite element method and finite difference method), the use of exact solutions is indispensable to validate numerical codes, since the mesh refinements used in the codes should be able to capture the large gradients of voltage and electric field around the cylindrical electrode, which should be accurately determined using the analytical solution.…”