1. Finite Element Modelling of the dipole source in EEG
Romana Rytsar and Thierry Pun
Multimodal Interaction Group, Computer Vision and Multimedia Laboratory,
University of Geneva, Switzerland (
Direct method
The principle: to implement directly the dipole source into the head model.
The RHS vector [4]:
Introduction
Electrical potentials generated by the brain can be measured on the scalp of human head using electroencephalograms (EEGs). The estimation of current source location from the EEG waves is known as the inverse problem of electroencephalography. In order to solve this inverse problem, we need to have an appropriate model of the EEG forward problem which has three sub-components: a source model [1], a volume conductor, and a measurement model. Since errors in the modelling of potential will translate into errors in source localization, it is important to have an accurate calculation of the forward solution.
In this poster, we present two finite element method (FEM) approaches for calculating the EEG scalp potentials due to the dipole sources and compare their results with the analytical solution.
the forward solution in the case of quadratic interpolation is sensitive to dipole location within the element in contrast to the linear interpolation;
the quadratic interpolation reduces the potential calculation error in comparison with the linear interpolation;
the NRDM decreases with increasing poles separation of the physical dipole;
the boundary potential is sensitive to the orientation of a dipole in the outer region of the two-layers head model. 2 4 6 8 10
0.135
0.136
0.137
0.138
0.139
0.14
pole separation, mm10-2
NRDM
Direct method, conclusions
EEG forward problem: Known: sources (strength, position, orientation) Wanted: potentials on the head surface
The principle: to subtract from the total potential the potential due to a dipole in an infinite homogeneous space
Forward problem subtraction formulation
- Poisson's equation;
- non-homogeneous Neumann boundary condition.
The RHS vector is calculated in the closed form [4].
Subtraction method 2
Model of the source + -
Current source density distribution:
Dipole moment:
Model of the head
cerebrospinal fluid
brain
scalp
skull
Brain:
CSF:
Skull:
Scalp:
Radial dipole
Tangential dipole
Results
A plot of RDM (left) and MAG (right) of surface potential versus dipole eccentricity
Forward problem formulation
Poisson’s equation for the electric potentials in the volume conductor
with the conductivity , resulting from current source density distribution
Neumann boundary condition: in O R1 R2 R3 R4
Numerical method: FEM
The conducting region is meshed with the
triangular elements. For a mesh with nodes
the potential can be expressed [2]:
is the solution in node
is the shape function
The potential values at the nodes are obtained from the standard FEM system of equations:
is vector of right-hand side (RHS);
is stiffness matrix.
Forward solution CPU time (sec)
Number of nodes
710
2745
10793
Direct method
1.3906
1.7656
2.3125
Subtraction method
1.8438
3.3906
The subtraction approach to the forward problem solution is more accurate in comparison with the
direct method.
Subtraction method, conclusion
Error calculation
Relative difference Normalized relative Magnitude measure (RDM): difference measure (NRDM): factor (MAG):
is the exact potential at boundary node i [3] and is the computed potential by the
FEM at the same node.
[1] P.H. Schimpf, C. Ramon and J. Haneisen, “Dipole Models for EEG and MEG,” IEEE Trans. Biomedical Engineering, 49, (2002).
[2] O. Bertrand, M. Thevenet and F. Perrin, “3-D finite element method in brain electrical activity studies,” in Biomagnetic localization and 3D modeling, ed. J. Nenonen, H.-M.Rajala and T. Katila, Report TKK-F-A689, Helsinki University of Technology, Dept. of Technical Physics, Lab. Of Biomed. Engr. Espoo, Finland (1991).
[3] Z. Zhang “A Fast Method to Compute Surface Potentials Generated by Dipoles within Multilayer Anisotropic Spheres,” Phys. Med. Biol., 40, (1995).
[4] K. A. Awada, D.R. Jackson, J.T. Williams, D.R. Wilton, S.B. Baumann and A.C. Papanicolaou, “Computational aspects of finite element modeling in EEG source localization,” IEEE Trans. Biomed. Engr., 44 (8), (1997).
References