Numerical Mathematics of the Subtraction Method for the Modeling of a Current Dipole in EEG Source Reconstruction Using Finite Element Head Models

Abstract

In electroencephalography (EEG) source analysis, a dipole is widely used as the model of the current source. The dipole introduces a singularity on the right-hand side of the governing Poisson-type differential equation that has to be treated specifically when solving the equation toward the electric potential. In this paper, we give a proof for existence and uniqueness of the weak solution in the function space of zero-mean potential functions, using a subtraction approach. The method divides the total potential into a singularity and a correction potential. The singularity potential is due to a dipole in an infinite region of homogeneous conductivity. We then state convergence properties of the finite element (FE) method for the numerical solution to the correction potential. We validate our approach using tetrahedra and regular and geometry-conforming node-shifted hexahedra elements in an isotropic three-layer sphere model and a model with anisotropic middle compartment. Validation is carried out using sophisticated visualization techniques, correlation coefficient (CC), and magnification factor (MAG) for a comparison of the numerical results with analytical series expansion formulas at the surface and within the volume conductor. For the subtraction approach, with regard to the accuracy in the anisotropic three-layer sphere model (CC of 0.998 or better and MAG of 4.3% or better over the whole range of realistic eccentricities) and to the computational complexity, 2mm node-shifted hexahedra achieve the best results. A relative FE solver accuracy of $10^{-4}$ is sufficient for the used algebraic multigrid preconditioned conjugate gradient approach. Finally, we visualize the computed potentials of the subtraction method in realistically shaped FE head volume conductor models with anisotropic skull compartments.