See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/7770560
Representation of bioelectric current sources using Whitney elements in the
finite element method
Article in Physics in Medicine & Biology · August 2005
DOI: 10.1088/0031-9155/50/13/004 · Source: PubMed
CITATIONS
READS
36
641
4 authors, including:
Oguz Tanzer
9 PUBLICATIONS 129 CITATIONS
Jukka Nenonen
Elekta
151 PUBLICATIONS 2,747 CITATIONS
SEE PROFILE
SEE PROFILE
Erkki Somersalo
Case Western Reserve University
266 PUBLICATIONS 10,487 CITATIONS
SEE PROFILE
All content following this page was uploaded by Jukka Nenonen on 02 September 2014.
The user has requested enhancement of the downloaded file.
Home
Search
Collections
Journals
About
Contact us
My IOPscience
Representation of bioelectric current sources using Whitney elements in the finite element
method
This content has been downloaded from IOPscience. Please scroll down to see the full text.
2005 Phys. Med. Biol. 50 3023
(http://iopscience.iop.org/0031-9155/50/13/004)
View the table of contents for this issue, or go to the journal homepage for more
Download details:
IP Address: 122.228.66.137
This content was downloaded on 15/10/2013 at 10:32
Please note that terms and conditions apply.
INSTITUTE OF PHYSICS PUBLISHING
Phys. Med. Biol. 50 (2005) 3023–3039
PHYSICS IN MEDICINE AND BIOLOGY
doi:10.1088/0031-9155/50/13/004
Representation of bioelectric current sources using
Whitney elements in the finite element method
I Oğuz Tanzer1,5 , Seppo Järvenpää2, Jukka Nenonen3
and Erkki Somersalo4
1 Laboratory of Biomedical Engineering, PO Box 2200, 02015 HUT and BioMag Laboratory,
Medical Engineering Center, Finland
2 Electromagnetics Laboratory, PO Box 3000, FIN-02015 HUT, Finland
3 Elekta Neuromag Oy, PO Box 68, FIN-00511 Helsinki, Finland
4 Department of Mathematics, PO Box 1100, FIN-02015 HUT, Finland
E-mail: oguz.tanzer@hut.fi
Received 31 December 2004, in final form 20 April 2005
Published 8 June 2005
Online at stacks.iop.org/PMB/50/3023
Abstract
Bioelectric current sources of magneto- and electroencephalograms (MEG,
EEG) are usually modelled with discrete delta-function type current dipoles,
despite the fact that the currents in the brain are naturally continuous throughout
the neuronal tissue. In this study, we represent bioelectric current sources in
terms of Whitney-type elements in the finite element method (FEM) using
a tetrahedral mesh. The aim is to study how well the Whitney elements
can reproduce the potential and magnetic field patterns generated by a point
current dipole in a homogeneous conducting sphere. The electric potential is
solved for a unit sphere model with isotropic conductivity and magnetic fields
are calculated for points located on a cap outside the sphere. The computed
potential and magnetic field are compared with analytical solutions for a current
dipole. Relative difference measures between the FEM and analytical solutions
are less than 1%, suggesting that Whitney elements as bioelectric current
sources are able to produce the same potential and magnetic field patterns as
the point dipole sources.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
In electro- and magnetoencephalographic (EEG, MEG) modelling, analytical solutions of the
forward problem for potential and magnetic field are available if the head is modelled as a
5 Address for correspondence: Low Temperature Laboratory, Brain Research Unit, Helsinki University of
Technology, PO Box 2200, 02015 HUT, Finland
0031-9155/05/133023+17$30.00 © 2005 IOP Publishing Ltd Printed in the UK
3023
3024
I Oğuz Tanzer et al
single-layer sphere (Sarvas 1987, Yao 2000) or with multiple layers (Stok 1987, Berg and
Sherg 1994, Sun 1997) with simple isotropic conductivity inhomogeneity. Also analytical
solutions for multiple spheres with layered anisotropic conductivity have been previously
reported (Zhou and van Oosterom 1992, de Munck and Peters 1993, Zhang 1995).
Analytical calculations also show that spherically symmetric anisotropy does not affect
the magnetic field (Ilmoniemi 1995), while it has a significant impact on surface potentials
(Schmidt and Pilkington 1991, Haueisen et al 2002). For realistic geometries, the boundary
element method (BEM) is a suitable technique for piecewise homogeneous volume conductors
(e.g. Gençer and Tanzer (1999), Gençer et al (2003)). If detailed nonhomogeneous and
anisotropic conductivity information is available, e.g. from techniques such as diffusion tensor
imaging (DTI) (Tuch et al 2001) or magnetic resonance electrical impedance tomography
(MREIT) (Seo et al 2004, Eyuboglu et al 1998), computation of the currents throughout the
volume conductor is required if the finite difference (FDM) or finite element (FEM) method is
used (e.g. Awada et al (1997), Haueisen et al (1997), van den Broek et al (1996)). In the FEM,
the 3D problem domain is partitioned into small volume elements where particular properties
can be assigned for each element.
In the forward problem computation, the total electric current in the head is usually
divided into the primary current and the Ohmic volume current. In MEG and EEG modelling,
the primary current sources are conventionally described by discrete current dipoles which
are implemented as singular point sources (Sarvas 1987). The volume current is described in
terms of the electric potential satisfying the conductivity equation. In non-trivial geometries,
the volume currents are then solved numerically using the BEM or the FEM. From the point
of view of MEG and EEG, the current dipole presentation has several advantages. Being
localized to a single point, a current dipole represents focal brain activity.
In the case of piecewise homogeneous conductivity, the calculation of lead fields
corresponding to current dipoles is straightforward. However, incorporating singular sources
into the FEM model is not without problems. First, the FEM solution is based on a weak
formulation of the conductivity equation and in order to be convergent, a certain degree of
regularity (typically H 1 ) is required from the source term. The singular sources such as point
dipoles fail to satisfy the regularity requirement. As a consequence, the calculated fields
depend on the location of the dipoles with respect to the finite element mesh (Awada et al
1997). A singular source can be treated as a distribution if the source is inside an element, but
problems arise when we move it to and across the element boundaries where the test functions
fail to be smooth.
Another method of treating the singularities that has been proposed in the literature is
to subtract the analytically computed field of a dipole in homogeneous space from the total
potential (Awada et al 1996, 1997, van den Broek et al 1996, Marin et al 1998, Bertrand
et al 1991). The conductivity of the homogeneous space is matched with the local value of the
conductivity. The scalar potential thus computed has a singularity proportional to the inverse
of the square of the distance. When magnetic fields are calculated, we need the gradient of
the potential that will have a non-integrable singularity, and it is not straightforward how to
treat this case numerically using the Biot–Savart law.
Alternatively, it is desirable to find a regular representation for localized current sources
that allows directly the use of the standard weak form in the FEM. Due to the vector source
term, it is natural to represent the source current using Whitney elements that have proven to
be successful in computational electromagnetics, see, e.g., Bossavit (1998). In particular, we
address the following question in this paper: How do the fields generated by Whitney element
sources compare to the fields generated by singular point sources? In particular, is there a
well-localized Whitney element source that produces approximately the same electromagnetic
Representation of bioelectric current sources using Whitney elements
αL
α2
α1
3025
rL
r2
r1
q
Ω
Figure 1. An arbitrarily shaped body . Points r represent the locations where magnetic field is
measured. A focal source current is represented by q.
field as a singular current dipole? This question seems to be quite relevant, as a large part
of the existing EEG/MEG literature and algorithms are based on the use of Dirac-delta-type
representations of the singular sources. Since about a 2 cm2 patch of cortex needs to be
excited for a detectable MEG signal (Hämäläinen et al 1993), a continuous representation of
the source currents may also reflect the electrophysiology better than point sources.
Our purpose in this work is to demonstrate that the Whitney element basis is capable of
describing local activity, thus providing a reasonable and computationally flexible alternative
for computing bioelectromagnetic fields in inhomogeneous and anisotropic media. In
section 2, we review the equations governing the electric potential and the magnetic field
intensity in non-homogeneous and possibly anisotropic medium, and give the FEM formulation
for the numerical approximation. The Whitney facet elements are introduced in subsection 2.1.
In section 3, we consider the following inverse problem: given the potential or the radial
component of the magnetic field of a single Dirac-type current dipole at multiple sensor
locations, what is the optimal Whitney element-based, spatially localized current source that
corresponds to the dipole data. Observe that while the applicability of the Whitney element
sources is not restricted to any particular geometry, here we consider the inverse problem
only in spherical geometry, since the singular dipole data have an analytic expression in this
setting.
2. Mathematical formulation of the forward problem
Let be a general body of interest as shown in figure 1. The current density within the body
induces a magnetic field outside the body, and this field is observed at given sensor locations
r1 , . . . , rL . Let us denote the current density by J. We can divide the current in two parts, the
primary current and the Ohmic volume current,
J = Jp + J .
(1)
Given the electric conductivity σ of the body, the volume current, within the quasi-static
approximation (Hämäläinen et al 1993), can be written as
J = −σ ∇u,
(2)
where u is the electric potential. From the conservation of charge, ∇ · J = 0, along with
the fact that no current flows out of the body, it follows that the electric potential solves the
boundary value problem
∇ · σ ∇u = ∇ · Jp ,
n · σ ∇u = 0,
on
in (3)
∂.
(4)
3026
I Oğuz Tanzer et al
Note that if u is a solution of equations (3) and (4), then also u + c, where c is a constant,
also fulfils equations (3) and (4). The corresponding weak formulation of this problem is: find
such u ∈ H 1 () that
∇v · σ ∇u dx =
v∇ · Jp dx
∀v ∈ H 1 (),
(5)
−
where v is the test function and H 1 () stands for the standard Sobolev space of square
integrable functions whose first weak derivatives are also square integrable in .
The conductivity σ = σ (x) can be either a positive scalar function or a 3 × 3 symmetric
positive definite matrix. The former case corresponds to an isotropic body, the latter to an
anisotropic one.
In the traditional formulation of the forward problem, the primary current is often modelled
as a sum of current dipoles (Sarvas 1987),
Jp (r) =
N
qs δ(r − rs ),
(6)
s=1
where N is the number of sources. We observe that from the point of view of the weak
formulation above, such a model is problematic since ∇ · Jp is not in L2 () and a point
evaluation makes sense only if the test function is, e.g., continuously differentiable, while in
general the H 1 -functions are not. Therefore, it is desirable to have a less singular model for
the source current. The Whitney basis provides such a model.
2.1. The 2-form Whitney basis
Whitney forms are a family of differential forms on a simplical mesh (i.e. a network of
tetrahedra, see figure 3(b)) as used in the FEM. They are at most first-degree polynomials on
tetrahedra (Bossavit 1988). The Whitney forms provide a hierarchy of basis functions that
can be used to represent the qualitatively different electromagnetic quantities. The 0-forms
represent scalar quantities such as electric potentials; the 1-forms represent field quantities
such as electric and magnetic fields; the 2-forms represent flux quantities such as magnetic
flux densities and current densities; the 3-forms represent volume quantities such as charge
densities. The remarkable property of the Whitney forms is that the physically relevant
continuity conditions across element boundaries are automatically satisfied and need not be
imposed by extra conditions. Here, we are interested only in the current densities, so the
Whitney 2-forms are a natural choice for basis functions.
Let us define λi , λj , λk to be standard first-order nodal basis functions, the 2-form Whitney
shape function for face f of a tetrahedron is given as
ij k
wf = 2(λi ∇λj × λk + λj ∇λk × ∇λi + λk ∇λi × ∇λj ).
(7)
The basis functions wj , j = 1, . . . , F , are defined for two tetrahedra adjacent to each
other and vary linearly inside the volume of both tetrahedra between a vertex and the opposing
face. Outside these two tetrahedra the basis function vanishes. This type of basis function
enables us to model the current density on tetrahedra. Figure 2 depicts a Whitney shape
function in one of the two tetrahedra. The degrees of freedom are fluxes across the faces of
the elements as opposed to nodes. The basis function guarantees that the normal component
of current density is continuous across a face. Physically, the element can be thought to be as
an H 1 -representation of a unit current dipole perpendicular to the ij k-face.
Representation of bioelectric current sources using Whitney elements
3027
j
Fi
Fl
Fk
x
k
l
i
Fj
Figure 2. Whitney shape function for the face Fl opposing the vertex l of a tetrahedron. The
basis function is defined such that the normal component of current density is continuous across
the faces of the elements.
2.2. Calculation of the electric potential
Having defined the basis for the source current, we formulate the weak form for approximating
numerically the electric potential.
Let uh be a solution of the discretized problem
− ∇v h (x) · σ (x)∇uh (x) dx =
v h (x)∇ · Jhp (x) dx
∀v h ∈ H 1 () to .
(8)
The current is written in terms of coefficients and face elements which are 2-form Whitney
basis functions wj as
Jhp (x) =
F
cjh wj (x),
cjh ∈ R,
(9)
j =1
where F is the number of interior faces in the mesh.
The scalar potential and the test functions are expressed as
uh (x) =
N
uhi ϕi (x),
uhi ∈ R
(10)
vih ϕi (x),
vih ∈ R,
(11)
i=1
and
v h (x) =
N
i=1
where N is the number of nodes in the mesh. In this work, the functions ϕi are second-order
nodal polynomial basis functions on each element and are continuous across the element
boundaries.
Let us denote
T
T
T
uh = uh1 , . . . , uhN ,
vh = v1h , . . . , vNh .
ch = c1h , . . . , cFh ,
Now equation (8) can be written in matrix form as
(vh )T Auh = (vh )T Fch
where
∀vh ,
(12)
Ai,j = −
∇ϕi (x) · σ (x)∇ϕj (x) dx,
1 ⩽ i, j ⩽ N
3028
I Oğuz Tanzer et al
and
Fi,j =
ϕi (x)∇ · wj (x) dx,
1 ⩽ i ⩽ N, 1 ⩽ j ⩽ F.
Observe that the electric potential is unique up to an additive constant, and this fact has the
consequence that the matrix A has a one-dimensional null space. To make the solution of
equation (8) unique, we fix the value of the potential to zero in the first node of the mesh.
This can be accomplished by zeroing the first row and column of matrix A and then setting
A1,1 = 1. Further, we must zero the first row of the matrix F. The discretized potential uh is
then obtained as the solution of the system
Auh = Fch .
(13)
2.3. Calculation of the magnetic field
The current source term J is considered to be consisting of Jp , the primary current source, and
Js the secondary current source term due to volume effects as given in equations (1) and (2).
After scalar potential u is obtained, the magnetic field due to J for a point sensor at
location ri with orientation αi can be calculated using
µ0
ri − r
(Jp − σ ∇u) ×
· αi dr
αi · B(ri ) =
4π |ri − r |3
µ0
ri − r
=
Jp ×
· αi dr
4π |ri − r |3
ri − r
µ0
(σ ∇u) ×
· αi dr .
−
(14)
4π |ri − r |3
Placing Whitney basis function wj in the first term of equation (14) and rewriting in terms
of matrix coefficients we get
µ0
ri − r
p
wj ×
· αi dr .
Bi,j =
(15)
4π |ri − r |3
Similarly, the second term of equation (14) can be written in terms of matrix coefficients as
µ0
ri − r
s
Bi,j = −
(16)
σ ∇ϕj ×
· αi dr ,
4π |ri − r |3
where Bp ∈ RM×F and Bs ∈ RM×N . Here N, F, M denote the number of nodes, the number
of source faces and the number of measurement points, respectively.
The matrix equation for the magnetic field can be written as
p
αi · B(ri ) = Bsi,· uh + Bi,· ch .
Further using the presentation for uh from equation (13) in equation (17) we get
p
h
bhL×1 = Bs L×N A−1
N×N FN×F + B L×F c F ×1 .
(17)
(18)
D
3. Inverse problem: estimation of Whitney coefficients
The question that we address in this section is: how accurately the voltage and magnetic
fields corresponding to the traditional singular current dipoles and localized Whitney sources
Representation of bioelectric current sources using Whitney elements
3029
Figure 3. (a) Unit sphere model for the calculations. The electric field is calculated at points on
the sphere. Magnetic field is calculated at the points on a cap located 1.1 times the radius away
from the sphere centre. (b) Tetrahedral division of the sphere volume for FEM calculation. This
is a linear mesh, 4 nodes per tetrahedron, has a total of 14 804 elements-tetrahedrons, 2945 nodes
and 28 710 faces. There are 900 nodes and 1796 faces on the surface of the sphere.
correspond to each other. This question is particularly relevant if one chooses to use the
Whitney basis in MEG and EEG applications to localize brain activity.
To compare the fields, we choose spherical geometry since closed form expressions to
calculate the magnetic field and potential for a current dipole are readily available (Sarvas
1987, Yao 2000). The closed form expressions used in the calculations are provided in the
appendix. The potential is calculated at point locations on the sphere and the magnetic field is
calculated at point locations on a spherical cap, which is co-centred with the sphere and has
1.1 times the radius. The computation model is shown in figure 3(a).
For the purpose of FEM calculation, the sphere is divided into volume elements. A
representative tetrahedral division of the sphere is shown in figure 3(b). Every tetrahedron
in the volume mesh has four faces, each shared with a neighbouring tetrahedron’s face as
depicted in figure 4(a). A general linear matrix equation relating the outside magnetic field to
current density in terms of Whitney basis functions can be written as
b = Dch + e
(19)
where e is noise and D is defined in equation (18).
The goal is to estimate the coefficient vector ch based on the simulated measurements b
that are calculated here by using a singular dipole as a source in an arbitrary location. This is
an ill-posed problem, due to the fact that the number of observation points is much less than
the number of source coefficients and that numerically, the matrix D is of ill-determined rank,
i.e., some of its singular values are close to or below the working precision. To obtain a useful
estimate for ch , some kind of regularization is needed.
It is well known that the selection of the regularization method has a qualitative effect on
the solution of the biomagnetic inverse problem. By using the truncated SVD regularization
or Tikhonov regularization, the estimated current densities are typically spread over a large
volume (see, e.g., Hämäläinen et al (1993)). A good spatial localization can be achieved by
using, e.g., the minimum current estimate (Uutela et al 1999).
In this work, we do not seek to solve the above inverse problem in all its generality.
We are interested here in evaluating the ability of Whitney elements to represent a locally
confined source such as a point dipole at a known location. Therefore, we only pick certain
faces with an aim to reproduce a well-localized field pattern as observed for magnetic field
and electric potential. In this respect, first we select only one tetrahedron which incorporates
3030
I Oğuz Tanzer et al
c2
c4
c1
c3
(a)
(b)
Figure 4. (a) Whitney source with four coefficients. Inside the volume mesh, two neighbouring
tetrahedrons share a face. The normal vectors for the shared face are parallel and opposite in
direction. (b) Whitney source with 16 coefficients is modelled by one tetrahedron sharing a face
with the four neighbouring tetrahedrons, shown with face normals.
four coefficients, i.e. one coefficient per face. We choose the tetrahedron that contains the
point source producing the simulated data. In this case, the ch vector in equation (19) becomes
ch 4×1 . Then we evaluate how this representation is affected if we take more coefficients by
assigning additional coefficients to faces adjacent to the source tetrahedron. In this case, the
ch vector becomes ch 16×1 .
To represent the locally confined source with Whitney elements, the task is to find the
coefficients cih which describe a field with a minimum error produced by a dipole, i.e. we are
looking for some ch that satisfies equation (19) in the least square sense.
The solution of this equation is given as
−1
ch = ((D.,J )T (D.,J )) (D.,J )T b.
(20)
Here, J is a set {i1 , i2 , i3 , i4 } or {i1 , . . . , i16 }, where ij are the numbers of the corresponding
faces of the chosen tetrahedron that we aim to represent our confined source. These two face
configurations are shown in figure 4. The vector b consists of given components of the
magnetic field at the observation points. We remark that reducing the size of the biomagnetic
inverse problem does not render it well-posed, and the solution ch may be sensitive to errors in
b. However, in this work we are not concerned about the stability of the solution but only the
ability of the Whitney elements to reproduce magnetic fields calculated with the point source
model.
Equation (20) represents the least square solution for the coefficients that best fit the
measured field produced by the FEM solver employing Whitney elements and the analytical
solution for the magnetic field calculated using a dipole. In a similar fashion, the best fit is
found for the electric potential.
4. Results
In order to evaluate the correspondence between the FEM formulation and the traditional
current dipole model, numerical results for magnetic field and potential are compared to the
results obtained using analytical expressions provided in the appendix. For the analytical
calculations a single dipole source with moment (1, 0, 0) is used. Sources are placed on the zaxis with depths 0.20, 0.40, 0.60, 0.70, 0.80, 0.85 and 0.89. Numerical calculations are carried
Representation of bioelectric current sources using Whitney elements
3031
Table 1. Properties of tetrahedral meshes used in the study.
Mesh number
Seed nodes
Nodes
Elements
Faces
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
1100
1300
1500
1700
1900
2100
2300
2500
2700
2900
3100
3300
3500
3700
4000
4400
4800
29 431
38 591
45 663
53 784
60 994
70 461
79 006
88 571
96 769
106 206
115 548
126 249
136 110
146 495
152 235
183 518
205 757
20 404
26 960
31 982
37 741
42 891
49 680
55 781
62 680
68 504
75 371
82 093
89 838
96 947
104 458
115 286
131 219
147 366
39 710
52 622
62 466
73 784
83 884
97 262
109 264
122 862
134 310
147 844
161 088
176 378
190 396
205 218
227 654
258 040
289 934
out for 33 different FEM grids with varying element density to determine the convergence of
the method. Grids consisting of uniform tetrahedra are generated from a surface triangulation
which serves as a seed for the volume grid generation. The seed surface triangulation itself is
obtained from a uniform distribution of points on the sphere. The mesh parameters used in
this study are provided in table 1.
The spherical geometry we used has a radius of 1.0 and a homogeneous isotropic
conductivity of 0.2 S m−1 . In the numerical model, the conductivity value is assigned to
each tetrahedron. The potential is calculated for all mesh nodes in the FEM, while only
surface nodes of the sphere are used in the analytical calculations. The magnetic field is
calculated at 122 points located on a cap which is co-centred with the sphere model with a
radius of 1.1 times the sphere radius as shown in figure 3.
The relative difference measure (RDM) (Meijs and Peters 1987) is used in the comparison
of potential and magnetic field patterns with the analytical solutions for dipole sources at the
given source depths and for each grid. The per cent RDM is defined as
RDM = 100 ×
(va − vn )T (va − vn )/va T va ,
(21)
where va is the vector of potential or magnetic field intensity values calculated using the
analytical formulae and vn is the vector of potential or magnetic field intensity values calculated
numerically.
Figure 5 shows the obtained per cent RDM results for the potential and magnetic field
using a source depth of 0.85 using all grids. In this example, 16 facet elements were fitted in
the least squares sense to the dipole data. No artificial noise was added, i.e., the error term
e in (19) represents only the discrepancy between the dipole model and the Whitney element
model. The trend of the convergence is obtained using a linear fit to the RDM values for the
simulation. It is observed that the FEM formulation converges to the analytically calculated
results when a finer grid is used.
Figure 6 shows the RDM values for potential which is calculated using different dipole
depths. Similarly the y-component (tangential) of the magnetic field and the z-component
(radial) of the magnetic field are presented in figures 7 and 8, respectively. It is observed that
3032
I Oğuz Tanzer et al
RDM for source at 85% for 16–Face Whitney Source, Potential
4
Source at 0.85, sd=0.68397
3.5
3
RDM %
2.5
2
1.5
1
0.5
0
0
5
10
15
20
25
30
35
Mesh Number
(a)
RDM, source at 85% for 16–face Whitney Source, Magnetic Total z–component
1.4
Source at 0.85, sd=0.16726
1.2
RDM %
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
Mesh Number
(b)
Figure 5. (a) The simulation results for 16-coefficient source centred at (0, 0, 0.85) for potential
depicting the change in per cent RDM with different mesh sizes. (b) The simulation results for
z-component of magnetic field in the same fashion. The straight lines in both cases show the linear
fit to the obtained RDM values with respect to different mesh sizes. Standard deviations from the
linear fit are given in the figure legend.
the per cent RDM value increases when the dipole is close to the boundary of the spherical
model.
The calculated Whitney representation of the neuronal current calculated from the
reference dipole is shown in figure 9.
Representation of bioelectric current sources using Whitney elements
3033
Linear fit RDM for different dipole eccentricity
4–Face Whitney Source, Potential
30
Source 0.20, sd=0.65127
Source 0.40, sd=0.73301
Source 0.60, sd=1.275
Source 0.70, sd=1.4967
Source 0.80, sd=2.533
Source 0.85, sd=3.0709
Source 0.89, sd=4.006
Source 0.92, sd=5.526
25
RDM %
20
15
10
5
0
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
Mesh Number
(a)
Linear fit RDM for different dipole eccentricity
16–Face Whitney Source, Potential
3.5
Source 0.20, sd=0.024939
Source 0.40, sd=0.031921
Source 0.60, sd=0.075865
Source 0.70, sd=0.090563
Source 0.80, sd=0.3556
Source 0.85, sd=0.67571
Source 0.89, sd=0.81179
3
RDM %
2.5
2
1.5
1
0.5
0
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
Mesh Number
(b)
Figure 6. (a) RDM results for potential modelled by 4-coefficient source evaluated for all source
positions and all grids. (b) RDM results for potential modelled by 16-coefficient source.
5. Discussion and conclusions
The present study focuses on accurate numerical simulation in bioelectromagnetic calculations
using the FEM, with the purpose of improving inverse source modelling in the human head.
For realistic simulations of the bioelectromagnetic forward problems, one should include all
3034
I Oğuz Tanzer et al
Linear fit RDM for different dipole eccentricity
4–face Whitney Source, Magnetic y–component
14
Source 0.20, sd=1.9245
Source 0.40, sd=1.0034
Source 0.60, sd=1.0247
Source 0.70, sd=1.0379
Source 0.80, sd=1.8149
Source 0.85, sd=1.4378
Source 0.89, sd=4.2379
Source 0.92, sd=6.1274
12
RDM %
10
8
6
4
2
0
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
Mesh Number
(a)
Linear fit RDM for different dipole eccentricity
16–Face Whitney Source, Magnetic y–component
0.4
Source 0.20, sd=0.030866
Source 0.40, sd=0.056264
Source 0.60, sd=0.052172
Source 0.80, sd=0.080632
Source 0.85, sd=0.076543
Source 0.89, sd=0.11117
0.35
0.3
RDM %
0.25
0.2
0.15
0.1
0.05
0
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
Mesh Number
(b)
Figure 7. (a) RDM results for y-component of magnetic field modelled by 4-coefficient source.
(b) RDM results for y-component of magnetic field modelled by 16-coefficient source.
available and pertinent information on the object in the numerical model. This study explores
the performance of Whitney-type finite elements in representing bioelectric current sources
compared to methods that use a point dipole. The present FEM formulation allows us to
specify a conductivity tensor for each tetrahedral element in the mesh. Thus, if available,
realistic conductivity information on the head can be directly included in the computation of
the forward problem.
Representation of bioelectric current sources using Whitney elements
3035
Linear fit RDM for different dipole eccentricity
4–face Whitney Source, Magnetic z–component
30
Source 0.20, sd=3.0082
Source 0.40, sd=2.0982
Source 0.60, sd=2.0926
Source 0.70, sd=1.9802
Source 0.80, sd=3.2773
Source 0.85, sd=2.9785
Source 0.89, sd=6.8391
Source 0.92, sd=7.818
25
RDM %
20
15
10
5
0
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
Mesh Number
(a)
Linear fit RDM for different dipole eccentricity
16–Face Whitney Source, Magnetic z–component
0.8
Source 0.20, sd=0.028584
Source 0.40, sd=0.075326
Source 0.60, sd=0.076406
Source 0.80, sd=0.11297
Source 0.85, sd=0.17182
Source 0.89, sd=0.20939
0.7
0.6
RDM %
0.5
0.4
0.3
0.2
0.1
0
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
Mesh Number
(b)
Figure 8. (a) RDM results for z-component of magnetic field modelled by 4-coefficient source.
(b) RDM results for z-component of magnetic field modelled by 16-coefficient source.
Here we did not attempt to solve the problem with full complexity of an anisotropic wholehead model. Instead, numerical simulations for electric and magnetic fields were carried out
using an isotropic single sphere model.
Biological currents are continuous in the neural tissue. The intracellular currents in the
white matter axons can be modelled with tail-to-tail dipoles (Wikswo et al 1980). Such axonal
sources do not have sufficient spatial and temporal summation so fields due to white matter
3036
I Oğuz Tanzer et al
Figure 9. (a) Current dipole in tetrahedron directed to (1, 0, 0). (b) Whitney representation of
current dipole modelled using four coefficients.
current sources are generally buried under the measurement noise. Thus, the primary currents
in MEG and EEG arise mainly in the cortex from postsynaptic intracellular currents. Rather
than discrete current dipoles, the physical continuity of Whitney elements might serve better
in modelling the primary currents in the neural tissue.
5.1. Results using RDM measure
The RDM provides us with an overall measure of differences in fields between numerical
and analytical calculations for different source locations. Using this measure, results are
provided to evaluate the effect of source location, grid size and the size of the Whitney
Representation of bioelectric current sources using Whitney elements
3037
Volume of tetrahedra where source is located inside
3.5
Mean volume of all tetrahedra
3
Tetrahedron Volume
2.5
2
1.5
1
0.5
0
0
5
10
15
20
Mesh Number
25
30
35
Figure 10. Individual points represent the volume of one tetrahedron where the source is located.
The full line represents the mean volume of all tetrahedra for the same grid. The computations
were done for a 1 m radius sphere, but here the radius was scaled to 10 cm for illustrative purposes.
Thus, the vertical scale shows the tetrahedron volume in mm3 .
element coefficient vector in the simulations. Generally, accuracy of the numerical method is
related to the number of elements used in the discretization. It is expected that the numerical
errors are lower with increasing number of elements, i.e. decreasing volumes of the elements.
In our simulations, we observed this behaviour as presented in figures 5–8. Sources which
are very near to the sphere surface gave worse RDM results compared to deeper ones due to
steep field change. Using more coefficients, i.e., more elements adjacent to the point source
location provided better RDM values.
The oscillatory behaviour in RDM results observed in figure 5 are attributed to the
properties of the tetrahedral volume grid for which the simulation is performed, i.e., how the
forward model dipole is located with respect to the FEM faces. The final quality of the mesh
depends on the positions of the nodes inside the sphere volume which are automatically added
by the mesh generator program during the generation process. As a result, the solutions we
obtained depended especially on the quality of the tetrahedra around our source point, given
the fact that the difference in number of surface seed nodes for generating the volume grids
was small. We attribute the volume of the tetrahedron to be a quality factor in our results. As
an example, figure 10 shows the volume of one tetrahedron in which the source is located at
0.80. Even though the mean volume of all tetrahedra in the grid decreases with increasing
number of elements, the volume of the tetrahedron inside which our source is located could be
larger than the mean volume for that grid. This in turn affects the RDM result. However, for
all source locations we obtained a reduction of error when grid size is increased as observed in
figures 6–8. Also the source locations closer to the surface benefited most from the increase
in element density compared to sources at deeper locations.
5.2. Future directions
With the method developed in this study, it is possible to include detailed conductivity
information in the volume conductor. It is well known that biological tissue shows conductivity
3038
I Oğuz Tanzer et al
anisotropy. This is especially observed in the grey matter, fibre tracts of the white matter
and the skull. It was previously reported that conductivity has a large effect on the scalp
EEG distribution (Marin et al 1998, van den Broek et al 1998). Also a large influence was
observed on the amplitude of EEG and MEG due to conductivity anisotropy in the single
dipole modelling case (Haueisen et al 1997, 2002). A previous finding also suggested that the
inclusion of anisotropy information would improve the accuracy of source estimation (Gevins
et al 1999). In the light of this information, the use of Whitney elements can provide a flexible
alternative for evaluating the effect of anisotropy in the forward and inverse modelling of EEG
and MEG.
Acknowledgments
This work is supported by the Academy of Finland, MaDaMe Project, the Graduate School
Functional Research in Medicine and Foundation for Technology Advancement (TES),
Finland.
Appendix. Analytical expressions for magnetic field and potential
The magnetic field due to a dipole source jp (r) = qδ(r − rs ) inside a spherically symmetric
conductor can be calculated using the formula (Sarvas 1987)
µ0
(F q × rs − q × rs · r∇F ),
(A.1)
b(r) =
4π F 2
where a = r − rs , a = |a|, r = |r|, F = a(ra + r 2 − rs · r) and ∇F = (r −1 a 2 + a −1 a · r +
2a + 2r)r − (a + 2r + a −1 a · r)rs . Here q is the dipole moment, the vectors r and rs denote the
location of field and source points, respectively.
In the case of a dipole inside a sphere with homogeneous conductivity (Yao 2000), the
potential on the surface of the sphere due to a dipole at rs can be calculated as
1 q · (r − rs )
1 1
q · r|rs | cos(θ ) − q · rs |r|
rp (q · r) +
.
(A.2)
φ(r) =
+
2π σ
rp3
4π σ |r|2
|r| + rp − |rs | cos(θ )
Here, rp is the length of displacement given by rp = |r2 | + |r2s | − 2|r||rs | cos(θ ) and
r·rs
. The conductivity of the isotropic and homogeneous sphere is represented
cos(θ ) = |r||r
s|
by σ .
References
Awada K, Jackson D, Williams J, Wilton D, Baumann S and Papanicolau A 1997 Computational aspects of finite
element modeling in EEG source localization IEEE Trans. Biomed. Eng. 44 736–52
Awada K, Jackson D, Wilton D and Williams J 1996 Closed-form evaluation of flux integrals appearing in a FEM
solution of the 2D Poisson equation with dipole sources Electromagnetics 16 75–90
Berg P and Scherg M 1994 A fast method for forward computation of multiple-shell spherical head models
Electroencephalogr. Clin. Neurophysiol. 90 58–64
Bertrand O, Thevenet M and Perrin F 1991 3-D finite element method in brain electrical activity studies Biomag. Loc.
3D Modelling TKK-F-A689 154–71
Bossavit A 1988 Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism
IEEE Proc. A 135 493–500
Bossavit A 1998 Computational Electromagnetism (San Diego, CA: Academic)
de Munck J and Peters M 1993 A fast method to compute the potential in the multi-sphere model IEEE Trans. Biomed.
Eng. 40 1166–74
Eyuboglu M, Reddy R and Leigh J 1998 Imaging electrical current density using nuclear magnetic resonance Turk.
J. Electr. Eng. 6 201–14
Representation of bioelectric current sources using Whitney elements
3039
Gençer N, Acar C and Tanzer I O 2003 Forward problem solution of magnetic source imaging Magnetic Source
Imaging of the Human Brain ed Z Lu and L Kaufman (Hillsdale, NJ: Lawrence Erlbaum Associates)
Gençer N and Tanzer I O 1999 Forward problem solution of electromagnetic source imaging using a new BEM
formulation with high-order elements Phys. Med. Biol. 44 2275–87
Gevins A, Le J, Leong H, McEvoy L and Smith M 1999 Deblurring J. Clin. Neurophysiol. 16 204–13
Hämäläinen M, Hari R, Ilmoniemi R, Knuutila J and Lounasmaa O V 1993 Rev. Mod. Phys. 65 413–97
Haueisen J, Ramon C, Eiselt M, Brauer H and Nowak H 1997 Influence of tissue resistivities on neuromagnetic fields
and electric potentials studied with a finite element model of the head IEEE Trans. Biomed. Eng. 44 727–35
Haueisen J, Tuch D, Ramon C, Schimpf P, Wedeen V, George J and Belliveau J W 2002 The influence of brain tissue
anisotropy on human EEG and MEG NeuroImage 15 159–66
Ilmoniemi R 1995 Radial anisotropy added to a spherically symmetric conductor does not affect the external magnetic
field due to internal sources Europhys. Lett. 30 313–6
Marin G, Guerin C, Baillet S, Garnero L and Meunier G 1998 Influence of skull anisotropy for the forward and inverse
problem in EEG: simulation studies using FEM on realistic head models Hum. Brain Mapp. 6 250–69
Meijs J and Peters M 1987 The EEG and MEG using a model of eccentric spheres to describe the head IEEE Trans.
Biomed. Eng. 34 913–20
Sarvas J 1987 Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem Phys. Med. Biol.
32 11–22
Schmidt J and Pilkington T 1991 The volume conductor effects of anisotropic muscle on body surface potentials
using an eccentric spheres model IEEE Trans. Biomed. Eng. 38 300–3
Seo J, Pyo H, Park C, Kwon O and Woo E 2004 Image reconstruction of anisotropic conductivity tensor distribution
in MREIT: computer simulation study Phys. Med. Biol. 49 4371–82
Stok C J 1987 The influence of model parameters on EEG/MEG single dipole source estimation IEEE Trans. Biomed.
Eng. 34 289–96
Sun M 1997 An efficient algorithm for computing multishell spherical volume conductor models in EEG dipole
source localization IEEE Trans. Biomed. Eng. 44 1243–52
Tuch D, Wedeen V, Dale A, George J and Belliveau J 2001 Conductivity tensor mapping of the human brain using
diffusion tensor MRI Proc. Natl Acad. Sci. 20 11697–701
Uutela K, Hämäläinen M and Somersalo E 1999 Visualization of magnetoencephalographic data using minimum
current estmates NeuroImage 10 173–80
van den Broek S, Reinders F, Donderwinkel M and Peters M 1998 Volume conduction effects in EEG and MEG
Electroencephalogr. Clin. Neurophysiol. 106 522–34
van den Broek S, Zhou H and Peters M 1996 Computation of neuromagnetic fields using finite-element method and
Biot–Savart law Med. Biol. Eng. Comput. 34 21–6
Wikswo J, Barach P and Freeman J 1980 Magnetic field of a nerve impulse: first measurements Science 208 53–5
Yao D 2000 Electric potential produced by a dipole in a homogeneous conducting sphere IEEE Trans. Biomed. Eng.
47 964–6
Zhang Z 1995 A fast method to compute surface potentials generated by dipoles within multilayer anisotropic spheres
Phys. Med. Biol. 40 335–49
Zhou H and van Oosterom A 1992 Computation of the potential distribution in a four layer anisotropic concentric
spherical volume conductor IEEE Trans. Biomed. Eng. 39 154–8
View publication stats