A meshless method for solving the EEG forward problem

We present a numerical method to solve the quasi-static Maxwell equations and compute the electroencephalography (EEG) forward problem solution. More generally, we develop a computationally efficient method to obtain the electric potential distribution generated by a source of electric activity inside a three-dimensional body of arbitrary shape and layers of different electric conductivities. The method needs only a set of nodes on the surface and inside the head, but not a mesh connecting the nodes. This represents an advantage over traditional methods like boundary elements or finite elements since the generation of the mesh is typically computationally intensive. The performance of the proposed method is compared with the boundary element method (BEM) by numerically solving some EEG forward problems examples. For a large number of nodes and the same precision, our method has lower computational load than BEM due to a faster convergence rate and to the sparsity of the linear system to be solved.

[1]  Mark A Fleming,et al.  Meshless methods: An overview and recent developments , 1996 .

[2]  H. Spekreijse,et al.  Mathematical dipoles are adequate to describe realistic generators of human brain activity , 1988, IEEE Transactions on Biomedical Engineering.

[3]  J. Oden,et al.  H‐p clouds—an h‐p meshless method , 1996 .

[4]  I. Duff,et al.  Sparse numerical linear algebra: Direct methods and preconditioning , 1996 .

[5]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[6]  M. Peters,et al.  Computation of neuromagnetic fields using finite-element method and Biot-Savart law , 2007, Medical and Biological Engineering and Computing.

[7]  A. Nehorai,et al.  Effect of perturbations in the geometry on the electroencephalography inverse problem , 2003, IEEE Workshop on Statistical Signal Processing, 2003.

[8]  E. Oñate,et al.  A FINITE POINT METHOD IN COMPUTATIONAL MECHANICS. APPLICATIONS TO CONVECTIVE TRANSPORT AND FLUID FLOW , 1996 .

[9]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .

[10]  Gregory E. Fasshauer,et al.  Solving differential equations with radial basis functions: multilevel methods and smoothing , 1999, Adv. Comput. Math..

[11]  J. D. Munck.  A linear discretization of the volume conductor boundary integral equation using analytically integrated elements (electrophysiology application) , 1992 .

[12]  Amanda L. Kaas,et al.  A fast method to derive realistic BEM models for E/MEG source reconstruction , 2001, IEEE Trans. Biomed. Eng..

[13]  B. Jiang.  The Least-Squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics , 1998 .

[14]  S. Mukherjee,et al.  THE BOUNDARY NODE METHOD FOR POTENTIAL PROBLEMS , 1997 .

[15]  M. Hämäläinen,et al.  Realistic conductivity geometry model of the human head for interpretation of neuromagnetic data , 1989, IEEE Transactions on Biomedical Engineering.

[16]  R. T. Hart,et al.  Finite-element model of the human head: scalp potentials due to dipole sources , 1991, Medical and Biological Engineering and Computing.

[17]  B. Nayroles,et al.  Generalizing the finite element method: Diffuse approximation and diffuse elements , 1992 .

[18]  Guillaume Alléon,et al.  Sparse approximate inverse preconditioning for dense linear systems arising in computational electromagnetics , 1997, Numerical Algorithms.

[19]  J. Jackson,et al.  Classical Electrodynamics, 2nd Edition , 1975 .

[20]  Y. Marechal,et al.  Boundary and interface conditions meshless methods [for EM field analysis] , 1999 .

[21]  T. Belytschko,et al.  A three-dimensional explicit element-free galerkin method , 1997 .

[22]  M. Brereton.  Classical Electrodynamics (2nd edn) , 1976 .

[23]  Stefan A. Sauter,et al.  Fast cluster techniques for BEM , 2003 .

[24]  S. Atluri,et al.  A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach , 1998 .

[25]  A. van Oosterom,et al.  The Solid Angle of a Plane Triangle , 1983, IEEE Transactions on Biomedical Engineering.

[26]  S. Atluri,et al.  The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple \& Less-costly Alternative to the Finite Element and Boundary Element Methods , 2002 .

[27]  J. D. Munck,et al.  A fast method to compute the potential in the multisphere model (EEG application) , 1993, IEEE Transactions on Biomedical Engineering.

[28]  Onno W. Weier,et al.  On the numerical accuracy of the boundary element method (EEG application) , 1989, IEEE Transactions on Biomedical Engineering.