| The localization of areas of excessive electrical activity in the human brain system by multichannel electroencephalography (EEG) recordings is one of the most important problems in Clinical Neurophysiology. This activity can be approximated by equivalent dipole [15], which generates the potential distribution all over the brain system. The essential part of the source localization procedure is the forward problem solution, i.e. computation of the potential on the surface of the head given the location and orientation of the dipole. In this work, the forward problem is solved with the use of the Finite Volume Method (FVM). The implementation of the FVM is done in such a way that the same algorithm can be applied for the realistic head model made by using Magnetic Resonance Imaging (MRI) scans of the head of the real patient. The main objectives of the research are assessment of the errors of the FVM modeling and computational issues such as deflation, properties of deflated matrix and acceleration of computations. The forward problem acceleration is especially important in practice, hence alternative approaches for the solution of the forward problem would be interesting. The Finite Volume Method is implemented in such a way that the deflated matrix of the linear system corresponding to the forward problem is symmetric and positive definite. These properties allow the use of the Conjugate Gradient Method with Polynomial Preconditioning and essential acceleration of the computations. The errors of the numerical solution were studied using analytical solutions for three-shell geometry. The realistic three-shell solution derived in this work allows to separate the source and sink of the equivalent dipole. This is essential for the FVM tests, as with this method the source and sink cannot be infinitely close together. As the radial dipole gets closer to the skull, the error of FVM grows. It can be reduced either with the higher resolution grids or with better domain decomposition algorithms. The idea of analytical matrix inversion developed for one, two and thri-dimensional systems of cubic finite volume elements can have potential for a rapid solution of the forward problem provided that it can be extended for deformed and nonuniform cases. |
