Numerical calculation of the potential distribution due to dipole sources in a spherical model of the head

A three-dimensional spherical model of the head was investigated numerically. The model consists of four conductive layers representing the scalp, the skull, the cerebrospinal fluid, and the cortex with a dipole current source. The potential created by the dipole was calculated using quasistatic formulation and a linear medium. The volume conduction equation was discretized by the finite volume method to ensure the conservation of fluxes and efficient solution method. The large set of algebraic equations for the electric potential was solved iteratively by the successive over relaxation method. The new formulation of the volume conduction problem was validated by comparing the numerical results with two analytical solutions. The first test-case considers a homogeneous spherical model with a dipole in the center. The potential on the outer surface, as well as within the volume conductor, was calculated and very good agreement was obtained with the analytical solution. In the second test-case, the scalp potential due to a radially oriented eccentric dipole in a four concentric spheres model was compared with an analytic solution. It was found that a grid of 90 x 90 x 90 volume elements yielded accurate results on the scalp surface with errors on the order of 1%. The present numerical model can be extended to general cases with any volume conductor shape or with any distribution or orientation of the current dipoles. Compared to other numerical methods, this approach offers enhanced accuracy for given computational resources (both in CPU time and memory). The gain might be more than one order of magnitude, allowing simulation with considerably larger meshes.