Perturbative analytical solutions of the electric forward problem for realistic volume conductors

Exact analytical solutions for the surface potential due to a current sourc e are available only for special volume conductors. Here we derive approxim ate analytical solutions exploiting the fact that for the upper convexity o f the head corrections to the spherical approximation are small compared to its radius. First we approximate the real surface, defined in terms of an angular dependent "radius," by a finite sum of spherical harmonics, regardi ng everything but the zeroth component (the sphere) as a small perturbation . Inserting this formulation into the standard surface integral equation al lows us to analytically construct and invert the integral operator as a Tay lor expansion with respect to the perturbation. Remarkably, for finite orde r of perturbation the integral operator and its inverse, expressed as a mat rix in the basis of spherical harmonics, is sparse. Furthermore, without lo ss of generality the solution due to a current dipole can be expressed as a set of sums over a single index. Explicit formulas and examples will be pr esented for one shell, for an approximation of the surface up to second ord er of spherical harmonics, and up to first order of the perturbation. By co mparing the results to the exact solution for a prolate spheroid we estimat e the performance for realistic deformations. (C) 1999 American Institute o f Physics. [S0021-8979(99)06017-X].