THE COMPLETE ELLIPSOIDAL SHELL-MODEL IN EEG IMAGING S. N. GIAPALAKI AND F. KARIOTOU Received 5 December 2004; Accepted 16 December 2004 This work provides the solution of the direct Electroencephalography (EEG) problem for the complete ellipsoidal shell-model of the human head. The model involves four confocal ellipsoids that represent the successive interfaces between the brain tissue, the cerebrospinal fluid, the skull, and the skin characterized by different conductivities. The electric excitation of the brain is due to an equivalent electric dipole, which is located within the inner ellipsoid. The proposed model is considered to be physically complete, since the effect of the substance surrounding the brain is taken into account. The direct EEG problem consists in finding the electric potential inside each conductive space, as well as at the nonconductive exterior space. The solution of this multitransmission prob- lem is given analytically in terms of elliptic integrals and ellipsoidal harmonics, in such way that makes clear the effect that each shell has on the next one and outside of the head. It is remarkable that the dependence on the observation point is not affected by the presence of the conductive shells. Reduction to simpler ellipsoidal models and to the corresponding spherical models is included. Copyright © 2006 S. N. Giapalaki and F. Kariotou. This is an open access article distrib- uted under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction The method of Electroencephalography (EEG) is the most widely used, noninvasive method for studying the human brain in vivo. The data of an Electroencephalogram are obtained by measuring the electric potentials in the exterior of the head. The in- verse EEG problem consists in determining the location of the electrochemical source inside the brain that produces the externally measured electric potential field. The results obtained from the solution of the forward EEG problem, namely the electric potential field that a given source produces, are of major importance for the inverse problem. The Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2006, Article ID 57429, Pages 1–18 DOI 10.1155/AAA/2006/57429