J. Biomedical Science and Engineering, 2010, 3, 742-749 JBiSE
doi:10.4236/jbise.2010.37099 Published Online July 2010 (http://www.SciRP.org/journal/jbise/).
Published Online July 2010 in SciRes. http://www.scirp.org/journal/jbise
Reconstruction of conductivity distribution of brain tissue
from two components magnetic flux density
Wenlong Xu, Dandan Yan
College of Information Engineering, China JiLiang University, Hangzhou, China.
Email: dandanyan@cjlu.edu.cn
Received 13 April 2010; revised 25 May 2010; accepted 27 May 2010.
ABSTRACT
In this paper the recent Magnetic resonance electrical
impedance imaging (MREIT) technique is used to im-
age non-invasively the three-dimensional continuous
conductivity distribution of the head tissues. With the
feasibility of the human head being rotated twice in
the magnetic resonance imaging (MRI) system, a con-
tinuous conductivity reconstruction MREIT algorithm
based on two components of the measured magnetic
flux density is introduced. The reconstructed conduc-
tivity image could be obtained through solving iter-
atively a non-linear matrix equation. According to
the present algorithm of using two magnetic flux den-
sity components, numerical simulations were per-
formed on a concentric three-sphere and realistic
human head model (consisting of the scalp, skull and
brain) with the uniform and non-uniform isotropic
target conductivity distributions. Based on the algo-
rithm, the reconstruction of scalp and brain conduc-
tivity ratios could be figured out even under the con-
dition that only one current is injected into the brain.
The present results show that the three-dimensional
continuous conductivity reconstruction method with
two magnetic flux density components for the realis-
tic head could get better results than the method with
only one magnetic flux density component. Given the
skull conductivity ratio, the relative errors of scalp
and brain conductivity values were reduced to less
than 1% with the uniform conductivity distribution
and less than 6.5% with the non-uniform distribution
for different noise levels. Furthermore, the algorithm
also shows fast convergence and improved robustness
against noise.
Keywords: Magnetic Resonance Electrical Impedance
Tomography; Magnetic Flux Density Measurement; Cur-
rent Density; Conductivity Distribution
1. INTRODUCTION
Knowledge of the electrical conductivity distribution in
the human body is of great importance in many bio-
medical applications. The information of the electrical
properties of head tissues is used for electroencephalo-
graphic source location and functional mapping of
brain activity and functions. And it has been proved
that information about the vivo tissue conductivity
values improves the solution accuracy of bioelectrical
field problems [1].
A non-invasive Magnetic Resonance Electrical Im-
pedance Tomography (MREIT) imaging modality has
been developed to reconstruct high-resolution conduc-
tivity distribution images for the biological issues. In
this new imaging modality, electrical impedance tomo-
graphy (EIT) is combined with magnetic resonance-
current density imaging (MR-CDI) techniques to solve
the well-known ill-posedness of the image reconstruc-
tion problem in traditional EIT. Zhang [2] proposed an
image reconstruction algorithm using internal current
density and peripheral voltage measurement to recon-
struct static conductivity images which initiated the
development on the theory of MREIT.
In MREIT, currents are injected into the subject
through pairs of surface electrodes. A Magnetic Reso-
nance Imaging (MRI) scanner is used to measure the
induced magnetic flux density inside the subject and
the current density distribution can be calculated ac-
cording to the Ampere’s law. The conductivity distribu-
tion images can be reconstructed based on the rela-
tionship between the conductivity and the measured
magnetic flux density combined with the current den-
sity [3].
MREIT reconstruction algorithms fall into two cate-
gories: those utilizing internal current density [4-7] and
those making use of only one component of measured
magnetic flux density [3,8-15]. Considering the rota-
tion problem of the object in the MRI system, the latter
has the advantages over the former without the object
rotation dilemma. MREIT algorithms that are based on
current density require knowledge of the magnetic flux
W. L. Xu et al. / J. Biomedical Science and Engineering 3 (2010) 742-749
Copyright © 2010 SciRes. JBiSE
743
density vector B = (Bx, By, Bz). Due to the fact that
only one component of the magnetic flux density which
parallels to the direction of the main magnetic field of
the MRI scanner can be measured once, the rotation of
the object is required, which is impractical for MRI
scanner.
Recently, several MREIT algorithms have been pro-
posed which utilize only one component of magnetic
flux density, such as the harmonic Bz algorithm [8,9],
the gradient Bz decomposition algorithm [11], the al-
gebraic reconstruction algorithm [12] and an anisot-
ropic conductivity reconstruction algorithm [16]. For
the head tissue conductivity, the relatively novel and
concise RBF (Radial Basis Function) and RSM (Re-
sponse Surface Methodology) MREIT algorithms were
proposed to focus on the piece-wise homogeneous head
tissue conductivity reconstruction [13,14]. However,
the continuous conductivity estimation of the head tis-
sue is more useful to obtain high resolution source lo-
calization and mapping results [17]. The MREIT algo-
rithms recently applied on the head homogeneous and
inhomogeneous conductivities reconstruction [18-20]
show better results. Therefore, in this paper a MREIT
approach is developed to realize the estimation of con-
tinuous conductivity distribution in the human head
tissues based on the three-layer realistic FEM head
model.
For the head tissues, the low skull conductivity is a
dilemma for the conductivity reconstruction, since
much of the current is shunted through the scalp and
does not enter the brain compartment. Nevertheless, the
reconstruction accuracy of MREIT is the same through
the imaging space and the inner part of it will not blur,
which will occur in traditional EIT. In this paper, with
the feasibility of the human head being rotated twice in
the MRI system, a modified continuous conductivity
reconstruction MREIT algorithm which is based on two
components magnetic flux density as [14] did is de-
veloped for the head tissues. With one more rotation,
more information could be gained and used for the re-
construction. Even under the situation that there is only
one current injection used in the simulation, better re-
sults also can be achieved. Herein, the realistic FEM
head model is utilized, as the original hexahedral ele-
ment meshing is unavailable in the finite element mod-
eling. The tetrahedral element meshing is an optimum
choice considering the computational efficiency. Com-
puter simulations are performed on a three-sphere head
model and the results show that the three-dimensional
head model continuous conductivity reconstruction
method with two magnetic flux density components
could get better results than the method with only one
magnetic flux density component used.
For the present approach, some assumptions should
be given. First, generally speaking, it is assumed that
the head geometry is known, since this data can be ob-
tained from the MRI scanner. Second, the skull con-
ductivity of the head tissues is fixed to the constant
value to focus on the continuous conductivity distribu-
tion reconstruction of the scalp and the brain. And in
the following sections, the proposed algorithm based
on two components magnetic flux density is explained
and an iterative conductivity update method is derived.
First, the formulation and numerical solution of the
forward problem and the inverse problem are presented
in Section 2. Then, the performance of the proposed
approach is assessed by a series of computer simula-
tions using a three-layer uniform and non-uniform
conductivity distribution realistic head models in Sec-
tion 3. In Section 4, discussions and conclusion about
present results are given.
2. METHODS
In this paper, reconstruction of conductivity image by
MREIT begins with a numerically relationship between
the conductivity combined with the measured magnetic
flux density due to the injected currents. In the forward
problem, we calculate the peripheral voltage values and
the magnetic flux density for a known conductivity
distribution. And the inverse problem is to reconstruct
conductivity σ from the measured magnetic field dis-
tribution B = (By, Bz) and J as well as physical laws of
electromagnetic.
In MREIT, MR magnitude images provide excellent
structural information, and then different regions (the
scalp, the skull and the brain) of the three-layer realistic
head can be determined. Within each regionthe con-
ductivity values are considered continuous distribution
instead of piece-wise. Such continuous conductivity of
head model could be more practical for forward or in-
verse problem solutions and source reconstruction in
functional brain imaging using magnetoencephalogra-
phy (MEG) and electroencephalography (EEG).
2.1. Forward Problem
Forward problem is defined as the calculation of mag-
netic field generated by the internal current pattern for a
given boundary injected current profile and known con-
ductivity distribution. Forward problem formulation is
used to supply simulation data to test the proposed re-
construction algorithm and it is also used in formulating
the inverse problem. The relation between the conduc-
tivity and the electrical potential U(r) induced by the
injected current is given by Poisson’s equation together
W. L. Xu et al. / J. Biomedical Science and Engineering 3 (2010) 742-749
Copyright © 2010 SciRes. JBiSE
744
with the Neumann boundary conditions as:
 
0,
, on positive current electrode
, negative current electrode
0 . elsewhere
rUr r
J
UJn
n
 


(1)
where σ(r) is the electrical conductivity and is the
imaging subject space. For complex conductivity distri-
butions, analytical solution to the forward problem ex-
pressed in Eq.1 does not exist. Therefore, a numerical
method must be applied. Finite element method (FEM)
is used to calculate the electrical potential and corre-
sponding magnetic flux density distribution for a given
conductivity distribution and boundary conditions.
After obtaining the electric potential distribution U(r)
from solving Eq.1, the electric field E and the interior
current density distribution J are given as:
EU
J
E

(2)
Then the magnetic flux density can be calculated us-
ing the Biot-Savat law:

0
3
() 4
rr
BrJ rdv
rr


(3)
where B(r) is the magnetic flux density at the measure-
ment point, and J(r’) is the current density at the source
point and μ0 is the magnetic permeability of the free
space. In order to avoid the singularity occurring when r
= r’, B(r) is treated as a node variable and J(r’) is used at
the centre of each finite element in (3) (Lee et al. 2003).
The comparison between analysis solution and numeri-
cal solution by FEM method is performed by Nuo Gao
[13] to indicate the feasibility of solving the forward
problem using the FEM.
2.2. Inverse Problem
The inverse problem is the reconstruction of σ(r) using
the measured magnetic flux density combined with the
calculated current density. Since static conditions are
assumed, the equation below can be gained [21]:
SJ J (4)
where S is defined as the natural logarithm of σ(r).
Based on the relation between current density and mag-
netic flux density, one can obtain:
2
0
B
SJ
  (5)
The equation is rearranged by the matrix form:
2
2
02
0
0
0
1
zy
zx
yx
y
z
x
xz
y
z
yx
S
x
JJ
S
JJy
JJ S
z
J
J
yz B
JJB
zx
B
JJ
xy





















 












(6)
With the two components magnetic flux density being
used, the last two rows of the matrix equation can give:
2
0
2
0
y
xz
z
yx
B
SS
JJ
zx
B
SS
JJ
xy






(7)
An iterative algorithm is used to solve the above non-
linear partial differential equation. The two components
Bz and By used promises the equation a unique solution
which is theoretically the same as the situation that two
currents injection are applied.
Discretization equation is obtained by replacing de-
rivatives with finite difference equivalents:


2
(1,) 1, ( ,1)( ,1)
(, )(, )
0(,)
2
(1,) 1, ( ,1)(,1)
(, )(, )
0(, )
22
22
ij ij ij ijy
xij zij
ij
ij ij ij ijz
yijxij
ij
SS SS B
JJ
zx
SS SS B
JJ
xy














(8)
where Δx, Δy and Δz are the distances between adjacent
pixels in the x, y and z directions, respectively. Operator
2
is denoted by the simple three-point difference
scheme as follows:




 

2
11
2
11
2
11
2
,,
,,2 ,,,,
,,2 ,,,,
(, ,)2, ,, ,
iii
iiiiiiiii
ii iiiiii i
iiiiii iii
Bxy z
Bxy zBxy zBxy z
x
Bx yzBxyzBx yz
y
BxyzBxyz Bxyz
z







(9)
Then the equation is rearranged into the matrix form as:
W. L. Xu et al. / J. Biomedical Science and Engineering 3 (2010) 742-749
Copyright © 2010 SciRes. JBiSE
745

z
y
z
y
Q
Q
S
P
P
QPS (10)
where P is the (2 KM × M) coefficient matrix, M is the
element number of the FEM head model and K denotes
the injected current number. S is the M × 1 vector which
is the each quadratic tetrahedral element value and Q is a
2 KM × 1 vector. Herein 12
[ ]
TT TT
K
Ppp p
 
is the
vector (KM × M) for internal current density distribution
J = (Jx, Jy, Jz), and T
12
[ q]
TTT
K
Qq q
 
is the (KM × 1)
vector for 2
B/μ0 where represents the x or y
component, respectively. The matrix equation is being
solved iteratively until the following stopping criterion is
satisfied:
1ii


(11)
where σi+1 denotes the i + 1th iteration and σi the ith it-
eration, ε is a given tolerance and is the Euclidian
norm. How to select the ε will affect the accuracy of the
conductivity reconstruction algorithm. The smaller the
selected ε is, the higher the accuracy of the reconstruc-
tion and the more time the convergence would be.
Then the steps of the iterative reconstruction algo-
rithm are as follows:
Step 1: According to the given target uniform and
non-uniform conductivity distributions, the ‘measured’
two components magnetic flux density Bz
* and B
y
* for
one and two currents injection are calculated using the
FEM method.
Step 2: 2
Bz
* and 2
By
* are calculated with the
three-point difference Eq.9, and the vectors Q are
obtained, = x or y.
Step 3: An initial conductivity distribution σm
ini is as-
sumed for m = 1, 2, … , M. This initial conductivity dis-
tribution is randomly selected to be values σm
ini (0, 15)
for the first iteration.
Step 4: With σm
iniσm
i, the current density distribu-
tions J = (Jx, Jy, Jz) are calculated using FEM and coeffi-
cient matrix P
is obtained for = x or y.
Step 5: Rearrange P
and Q as the Eq.10 and
solve the combined system equation, then the solution S
is found and the conductivity distribution σm
i+1 is
achieved.
Step 6: Stop if the stopping criterion (11) is met.
Othewise, set (i + 1)i and go to Step 4 with σm
ini re-
placed by σm
i+1.
3. RESULTS
In order to test the performance of the reconstruction
algorithm using two magnetic flux density components,
numerical simulations were performed on a concentric
three-layer realistic human head model (consisting of the
scalp, skull and brain) to estimate the continuous con-
ductivity values σ = (σscalp, σbrain).
3.1. Preparation of Experiment Data
The three-layer realistic human head model has three
compartments: scalp, skull and brain. Software ANSYS
10.0 was applied to the finite element modeling and
meshing as well as the solving of the forward problem. A
finite element meshing for the 3D three-layer realistic
head model in Figure 1 with 57041 quadratic tetrahedral
elements and 78893 nodes is used. And the conductivity
in each element is assumed to be different values.
In order to get the current density image, a bipolar
rectangular current (5mA) is injected into the human
head model through a pair of opposite electrodes along
the equator of the realistic head model (seeing Figure 2).
A MRI scanner is used to measure the component of the
induced magnetic flux density parallel to the main mag-
netic field of the MRI system (along the z direction),
denoted as Bz
*. Then the head model is rotated once, the
other component of the induced magnetic flux density
denoted as By
* (along the y direction) is measured.
The two components magnetic flux density is calcu-
lated with the given target human head conductivity dis-
tribution in the forward problem. To prove the noise tol-
erance of the algorithm, Gaussian White Noise (GWN)
at different levels is added to the magnetic flux density
to simulate the ‘measured’ noise-contaminated MR ma-
gnitude image B
z
* and By
*. The standard deviation of
noise SB in ‘measured’ magnetic flux density is intro-
duced by [22]:
1
2
B
c
STSNR
(12)
where γ = 26.75 × 107radT-1s-1 is the gyromagnetic ratio
of hydrogen, Tc is the duration of injection current pulse
and SNR is the signal-to-noise ratio of the MR magni-
tude image. The SNR of the GWN is set to be infinite,
80, 60, 40, 20 and 10 with Tc = 50ms, SB is obtained to
be 2.9356 × 10-9T, 3.9142 × 10-9T, 5.8713 × 10-9T,
1.1742 × 10-8T, and 1.5657 × 10-8T, respectively.
Given the skull conductivity, the continuous conduc-
tivity of the scalp and the brain is reconstructed with the
proposed algorithm based on the measured two compo-
nents magnetic flux density when one or two currents
are injected, respectively. Two types of numerical simu-
lations were performed on the three-layer realistic head
model: uniform and non-uniform isotropic target con-
tinuous conductivity distribution reconstructions. The
parameters of the head model are listed in Table 1,
where σm denotes the conductivity value of the mth ele-
ment of the finite element realistic head model.
For uniform case, the conductivity distribution in each
W. L. Xu et al. / J. Biomedical Science and Engineering 3 (2010) 742-749
Copyright © 2010 SciRes. JBiSE
746
(a) (b)
Figure 1. Three-layer realistic finite element head model: (a) a tetrahedral element is used to mesh the three-layer realistic head
model (b) meshed model of the finite element realistic head model.
Figure 2. Current injection and electrodes locations of the
realistic finite element head model.
Table 1. Uniform and non-uniform conductivity head model
parameters.
Uniform Conductivity
(s/m)
Non-uniform Conductivity
(s/m)
Scalp Skull Brain Scalp Skull Brain
σmtarget
1 1/15 1 (0.9,1.1) 1/15 (0.9,1.1)
element of the head FEM model is assumed to be iden-
tical in the scalp, skull and brain compartments, respec-
tively. For non-uniform case, the conductivity distribu-
tion in each element is assumed to be different value in
each compartment. In this study, the relative error (RE)
between the estimated and the target conductivity distri-
bution is used to quantitatively assess the performance of
the MREIT reconstruction algorithm. The relative error
is defined as follows:
%100),(


RE (13)
where σ* is the target conductivity distribution and σ is
the estimation conductivity distribution.
3.2. Conductivity Image Reconstruction
The MREIT reconstruction algorithm based on two com-
ponents magnetic flux density for the two different target
conductivity distribution cases were tested with six noise
levels, and some compared results are given below. With
the tolerance ε = 0.01, it took seven iterations to recon-
struct these continuous conductivity distributions. In
each situation (uniform conductivity or non-uniform
conductivity) with one current injection, simulation re-
sults are listed at Table 2. But with one current injection,
meaningful reconstructions are not achieved even for
noise-free simulations based on the only one component
magnetic flux density algorithm. And given the two cur-
rents injection, the compared results between the algo-
rithms based on two components and only one compo-
nent magnetic flux density are given at Table 2.
For uniform simulation in Table 2, with different
noise levels, all the REs are less than 1% for one current
injection and two currents injection simulations, which
are reduced considerably compared with the algorithm
based on only one magnetic flux density. It also suggests
that the RE is insensitive to the added noise. The data
also show that with the noise level of SNR = 10, one
current injection results are more influenced than two
injection ones. The case is same to the non-uniform
simulation.
For non-uniform simulation, the REs are less than
6.5% for one current injection and two currents injection
simulations, which are not much less compared with the
only one magnetic flux density algorithm. The reason is
that the non-uniform distribution influences the per-
formance of the algorithm. However, the non-uniform
range of the real head conductivity distribution in one
tissue changes not so severely, the feasibility of the algo-
rithm on human head is confirmed.
The reconstruction conductivity images are shown in
Figure 3 and Figure 4, respectively. Figures 3(a) and (b)
demonstrate the reconstruction conductivity images un-
der the condition that without adding any noise, SNR =
Infinite, SNR = 40 and SNR = 10 for uniform distribu-
tion. Figure 4 is for the non-uniform case, respectively.
W. L. Xu et al. / J. Biomedical Science and Engineering 3 (2010) 742-749
Copyright © 2010 SciRes. JBiSE
747
Table 2. Results of the conductivity reconstruction.
SNR of the added GWN
RE (%)
Infinite 80 60 40 20 10
Brain Based on By and Bz 0.0189 0.0243 0.0352 0.0510 0.0636 0.1848
K 1
Scalp Based on By and Bz 0.0394 0.0420 0.0528 0.1050 0.1665 0.4264
Brain Based on By and Bz 0.0209 0.0251 0.0354 0.0468 0.0568 0.0830
Uniform
K 2
Scalp Based on By and Bz 0.0484 0.0513 0.0586 0.0737 0.1227 0.1519
Brain Based on By and Bz 5.7802 5.7808 5.7811 5.7822 5.8315 6.0814
K 1
Scalp Based on By and Bz 5.7889 5.7894 5.7923 5.9451 6.1307 6.3492
Brain Based on By and Bz 5.7806 5.7807 5.7808 5.7812 5.8274 5.9026
Non-niform
K 2
Scalp Based on By and Bz 5.7884 5.7890 5.7892 5.7898 5.9152 6.1483
Figure 3. Uniform conductivity reconstruction images with different noise levels: (a) corresponding to one current injection, (b)
corresponding to two currents injection. (a) current injection k = 1; (b) current injection k = 2.
4. DISCUSSION
In the present study, the continuous conductivity recon-
struction MREIT algorithm for human head tissues util-
izes measured two components magnetic flux density
without using the impractical subject rotation procedure.
The proposed approach is based on the algebraic recon-
struction algorithm [12] and is extended to detect con-
ductivity on object as human head model. The recon-
structed image could be obtained through iteratively
solving a non-linear matrix equation. Due to the two
components magnetic flux density data available, the
scalp and brain conductivity reconstruction which is
important parameters for brain inverse problems could
be figured out even with only one current injection into
the brain.
The present study suggests that the full three dimen-
sional conductivity reconstruction images over the entire
subject instead of the image slice is obtained with the
satisfied accuracy and spatial resolution. For uniform
and non-uniform cases, the REs between the target and
the estimated conductivity distribution are less than 1%
and 6.5% with different noise levels, respectively. The
W. L. Xu et al. / J. Biomedical Science and Engineering 3 (2010) 742-749
Copyright © 2010 SciRes. JBiSE
748
Figure 4. Non-uniform conductivity reconstruction images with different noise levels: (a) corresponding to one current injec-
tion, (b) corresponding to two currents injection. (a) current injection k = 1; (b) current injection k = 2.
present simulation results demonstrate the excellence
performance of the algorithm for conductivity recon-
struction of human head tissue.
In the procedure of the computation of 2
B, three-
point difference method is utilized which will arise er-
rors and vulnerable to measurement noise. So the means
of finite difference modeling correction to the finite
element modeling is a good choice to lessen the errors
and improve the robustness against noise. A more so-
phisticated head model such as real geometry head
model would be utilized to enhance the accuracy and
resolution of reconstruction image at the cost of compu-
tation time. It is also found that increasing the number of
injection currents does not obviously improve the solu-
tion of the reconstruction conductivity image when three
or four currents are injected in the simulation.
In the present simulation, a current of 5 mA was used,
which is thought to be the upper safe limit for human
beings (IEC criterion). And for human head, it is a little
higher. So it would be desirable to utilize a better MRI
scanner, some denoising techniques and improved ap-
proaches.
In summary, we proposed MREIT approach based on
two components measured magnetic flux density for
noninvasive imaging of the three-dimensional continu-
ous conductivity distribution of the head tissues. A series
of computer simulations demonstrate the feasibility of
the algorithm and fast convergence ability combined
with improved robustness against noise. In our future
studies, researches should focus on real geometry head
model study and the experimental validation the algo-
rithm on the human head phantom experiment, as well as
ways to reduce the amount of the injection current down
to less than 1 mA for human security consideration in
clinical experiment and setting.
5. ACKNOWLEDGEMENTS
This research was possible thanks to the projects (No. Y1080215 and
No. Y2090966) supported by the Zhejiang Provincial Natural Science
Foundation of China.
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