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Computational aspects of the EEG forward problem solutions for real head model using finite element method
RYTSAR, Romana, PUN, Thierry
Abstract
The real head model has been used for the accurate scalp potential modeling. The realistic shapes of head tissues were derived from a set of 2-D magnetic resonance images (MRI) by extracting surface boundaries for the major tissues such as the scalp, the skull, the cerebrospinal fluid (CSF), the white matter, and the gray matter. From boundary data a 3-D volume generic head model has been constructed and a mesh for an arbitrary complexity head shape has been generated for finite-element method (FEM) modeling. This paper first addresses the use of this realistic finite elements head model to solve the EEG forward problem. The accuracy and computational time of the potential modeling are then examined.
RYTSAR, Romana, PUN, Thierry. Computational aspects of the EEG forward problem solutions for real head model using finite element method. In: Proceedings of the 29th Annual
Internatinal Conference IEEE Engineering in Medicine and Biology Society, EMBS 2007 . IEEE, 2007. p. 829-832
DOI : 10.1109/IEMBS.2007.4352418
Available at:
http://archive-ouverte.unige.ch/unige:47919
Disclaimer: layout of this document may differ from the published version.
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Abstract—The real head model has been used for the accurate scalp potential modeling. The realistic shapes of head tissues were derived from a set of 2-D magnetic resonance images (MRI) by extracting surface boundaries for the major tissues such as the scalp, the skull, the cerebrospinal fluid (CSF), the white matter, and the gray matter. From boundary data a 3-D volume generic head model has been constructed and a mesh for an arbitrary complexity head shape has been generated for finite-element method (FEM) modeling. This paper first addresses the use of this realistic finite elements head model to solve the EEG forward problem. The accuracy and computational time of the potential modeling are then examined.
I. INTRODUCTION
he electrical activity inside the human brain consists of currents generated by biochemical sources at the cellular level. This activity can be measured by an electroencephalograph (EEG). Neurologists have been interested in the so-called EEG inverse problem, that is the determination of the small active brain areas that significantly contribute to the generation of the electric field from the measured potentials on the scalp [1], [2]. Solving the EEG inverse problem requires an appropriate model of the corresponding forward problem [1]-[3]. Moreover, the EEG localization accuracy is highly sensitive to errors in the forward problem [4]. Recent computer simulation studies demonstrated high EEG source localization accuracy once the effects of the forward model errors are removed [5].
Since errors in the modeling of potential will translate into errors in source localization, it is important to have an accurate calculation of the forward solution.
The accuracy of the forward problem is critically determined by the assumptions concerning the shape and conductivity of the volume conductor [6], [7]. The human head as a conductor is often approximated by three or four spherical layers with different electrical conductivities representing the brain, the cerebrospinal fluid (CSF), the skull, and the scalp [2], [8]. It allows reducing the EEG forward problems to closed form analytic solutions [8].
However, spherical head models are geometrically not
Manuscript received April 9, 2007. This work was supported in part by the SNF project (Marie Heim-Vogtlin subsidy PMPD2-110193/1).
R. Rytsar. is with Computer Science Department, University of Geneva, Geneva, Switzerland (corresponding author to provide phone: 41-22-379- 0151; fax: 41-22-379-7780; e-mail: [email protected]).
T. Pun is with Computer Science Department, University of Geneva, Geneva, Switzerland (e-mail: [email protected]).
sufficiently accurate for different applications [9]-[13].
Moreover, significant differences in the current source locations between patient-specific head models and spherical models have been estimated [10]-[12]. In the case of a realistic head model, the analytic solutions do not exist and, therefore, a finite element method that describes the different parts of the head and their properties ought to be used.
Realistic head models can improve the accuracy of forward solutions and inverse source localizations. Head phantom studies show their high accuracy even with a simple three-layers (brain, skull and scalp) model [4]. More recent studies indicate the necessity of highly heterogeneous models of the head for accurate simulations of scalp potentials [6], [7] and for inverse source localization [7].
This paper describes the finite element modeling of the human head and the calculation of the scalp potentials arising from a source in the brain. The paper focuses on the accuracy and computational time of the forward problem solution. The numerical approximation is examined by the comparison with the analytical solution for spherical model.
II. CONSTRUCTION OF THE REALISTIC FEM HEAD MODEL
The generation of the realistic FEM head model implies the segmentation of the MRI and the construction of the mesh for an arbitrary complexity head shape. The segmentation of the head tissues has been performed interactively using the 3D Doctor software package [14].
The real head shape is represented as a set of 180 coronal slices. The morphological information about individual head shape is extracted by segmentation of all images. Figure 1 shows a single slice and a segmented slice with five head compartments: the scalp, the skull, the CSF, the gray matter, and the white matter.
Fig. 1. A single coronal slice and a segmented slice with five head compartments.
Computational aspects of the EEG forward problem solution for real head model using finite element method
Romana Rytsar and Thierry Pun
T
Scalp Skull CSF White matter Gray matter
From the boundary data the 3-D volume model was reconstructed using volume-rendering. The surfaces of three segmented tissues are shown in Figure 2.
Fig. 2. Examples of surfaces extracted from MR images: scalp, gray matter and white matter surfaces (from left to right).
The second essential prerequisite of FEM modeling is the construction of a mesh that adequately represents the geometric and electric properties of the head volume conductor. The surface geometry of the head has been depicted by 14964 triangles and 7484 nodes (see Figure 3).
A mesh with first-order tetrahedral elements has been created using the HyperMesh software package [15].
Fig. 3. Surface-meshed and volume-meshed models.
Appropriate conductivity of each voxel was specified knowing the exact coordinates of all the mesh nodes and the type of tissue at each of these points. Isotropic conductivities were assigned to the scalp (0.33 S/m), the skull (0.0042 S/m), the CSF (1.79 S/m), the gray matter (0.33 S/m), and the white matter (0.14 S/m) [16].
III. THE EEG FORWARD PROBLEM
We have used the above-mentioned model for solving the EEG forward problem, which is defined as the potentials determination on the human head surface from a given configuration of the source in the brain, the geometry and the distribution of the electrical conductivity
(r) within the head. The source is assumed to be a dipole of amplituded
I0 in the directionpˆ. An ideal current dipole can be described as two-point sources of opposite polarity with an infinitely large current I0 and an infinitely small separation
d
. Mathematically, the current source density in the element volume
A/m3
is defined by:
lim 0 2 2
0
r d d r
r r I
r o o
d
, (1)where
denotes 3-D Dirac delta function. Theoretically, this forward problem is governed by the Poisson’s equation for the electric potentials U(r) in the head :
r U r
r
( ) ( ) in . (2)
At the outer boundary of the medium there exists a homogeneous Neumann boundary condition:
0
n
r U
on , (3)
where n is the distance measured normal to the boundary.
The numerical solution of the problem is calculated at the nodes using the FEM. Thus, the numerical approximation U~
of the solution U for a mesh consisting of N nodes is given by:
N
i i
iH
U U
1
~ , (4)
where Ui is the solution at node i and Hi is the linear shape function that describes the contribution of the value at node i . The potential values at the nodes Ui are obtained by solving the standard FEM system of equations [1], [2]:
] [ ] ][
[Kji Ui i , (5)
where [Kji] is an NN matrix that depends only on the shape functions and conductivities distribution; [i] is an N-dimensional vector in which the source function is incorporated. The elements of the stiffness matrix [Kji] are defined as:
i e e ej
ij r H H d
K
e
, (6)and the i-th component
i of this vector is given by:
) ( ) (
) (
e e
i
i
r H r d
e
p H M
r d d H
r H I
e i
e i e
d i
ˆ
2 lim 2
) (
0 ) ( 0
) ( 0 0
(7)
where (e) signifies an element subdomain, and M I d
0 is the dipole moment.
Since each tetrahedron directly interacts only with its immediate neighbors, the matrix is quite sparse; the minimum residual iterative method is used for efficient solving of the system equation (6).
IV. RESULTS
Simulations were carried out to evaluate the accuracy and computational time of the scalp potential modeling due to the dipole.
x y z
A. Spherical head model
The accuracy of the surface potential numerical approximation was validated on the homogeneous sphere by comparison of the results with the analytical solution [8].
The relative difference measure (RDM) and the magnitude factor (MAG) were used to measure the similarity between the analytic and numeric solutions:
bb
N
i A i N
i
FEM i A i
U U U RDM
1 2 1
2
,
b b
N
i A i N
i
FEM i
U U MAG
1 2 1
2
(8) where
N
bis a number of nodes on the boundary,U
iA is a exact potential at boundary nodei
, andU
iFEM is thenumerical solution at boundary node
i
.Thus, the influence of the conductivity gradient on the computational error has been neglected. We targeted to examine the calculation precision relatively to the number of the mesh nodes. Three spheres were constructed with the different density of the tetrahedrons. Their surfaces were formed by 844, 1273, 7964 nodes; volume meshes counted 2454, 4526, 65699 nodes and 10455, 24207, 374799 first- order tetrahedral elements, respectively (see Figure 4).
Fig. 4. The volume meshed sphere.
The dipole is placed along the z-axis with the different eccentricities pointing in the radial and tangential directions (see Figure 4 for the coordinate orientation). The electrical potentials were calculated at the nodes on the spherical surface. Figure 5 shows the RDM and MAG for radial and tangential sources as a function of dipole eccentricity. The RDM increases with source eccentricity and is lower than 5% for eccentricities less than 0.7 (
N
=65699). However, RDM increases exponentially while eccentricity is greater 0.7. RDM reaches 17% at the point where eccentricity equal to 0.85. The MAG remains very close to 1 for eccentricities less than 0.7.The RDM of the surface potential due to the radial dipole for the different number of the mesh nodes (2454, 4526, and 65699) is shown on Figure 6. This plot highlights the important role of the mesh density on the accuracy of the scalp potentials modeling. However, the increasing of the number of the nodes leads to the increasing of the
computational time. The creation of the stiffness matrix (6) has been performed in 29.06 min. (
N
=65699) on a Pentium 4 CPU at 3 GHz, with 2 GB of RAM.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0 10 20 30 40 50 60 70 80 90 100
dipole ec c entric ity
RDM
radial tangential
a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0 0.2 0.4 0.6 0.8 1 1.2
dipole ec c entric ity
MAG
radial tangential
b)
Fig. 5. A plot of RDM (a) and MAG (b) of the surface potential due to the radial (solid line) and tangential dipole (dashed line).
To solve the large and sparse system of linear equations (5) different iterative methods were applied: biconjugate gradients stabilized method (BCGS), conjugate gradients squared method (CGS), generalized minimum residual method (GMR), and minimum residual method (MR). Table 1 details the computational time for each method.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0 10 20 30 40 50 60 70 80 90 100
dipole ec c entric ity
RDM
2454 4526 65699
Fig. 6. RDM of the surface potential due to the radial dipole for the different number of the mesh nodes (N=2454; 4526; 65699).
The potential modeling accuracy has been investigated with usage of unit sphere relatively to the mesh density.
x y z
Obviously the calculation precision of surface potential on the multi-layer sphere decreases due to conductivity inhomogeneities.
Tab. 1. CPU time for the solution of linear system of equations (5) using biconjugate gradients stabilized method (BCGS), conjugate gradients squared method (CGS), Generalized Minimum residual method (GMR), and Minimum residual method (MR).
BCGS CGS GMR MR
Time,
min 23.42 23.49 24.79 0.572
B. Real head model
The accuracy of the potential calculations cannot be easily verified in the case of an arbitrarily shaped volume conductor. The obvious way to proceed would be to increase the density of the tetrahedrons gradually and to show that obtained result approaches some limiting value. However, this method can only verify how the FEM works.
The electrical potentials have been calculated on the scalp of the head. The generated realistic finite element head model includes 212439 tetrahedrons altogether, and after being labeled, 46040 tetrahedrons belong to the scalp, 37679 to the skull, 40531 to the cerebrospinal fluid, 41214 to the gray matter, and 27270 to the white matter. The total number of the nodes is 39575. The dipole is located in the cortex at the distance of 35.8mm from the scalp along the z- axis. The electrical potentials were calculated for 1814 nodes on the scalp (see Figure 7).
-0.172 1.953 4.078 6.203 8.328 10.453 12.578 14.703 16.828
-1 0 1
-0.5 -1 0.5 0
1 -1
-0.5 0 0.5 1
Fig. 7. Electrical potentials on the scalp of the real head model due to the dipole in the cortex.
V. CONCLUSIONS
Defining a realistic head model is a key element in obtaining more accurate source distributions that represent the neural activity in the human brain. The purpose of this study was to improve the accuracy in the forward problem solutions using the finite element method. In general, the numerical accuracy is directly related to the number of elements used in the discretization procedure, and errors always decrease with increasing the number of elements.
However, the practical usage of realistic head models for solving of the EEG forward and inverse problems is limited by the time-consuming amount of numerical calculations.
Often, a tradeoff should thus be accepted between numerical accuracy and computational time.
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