Page 1
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
SIAM REVIEW
Vol. 53, No. 1, pp. 40–68
c ? 2011 Society for Industrial and Applied Mathematics
Magnetic Resonance Electrical
Impedance Tomography (MREIT)∗
Jin Keun Seo†
Eung Je Woo‡
Abstract. Magnetic resonance electrical impedance tomography (MREIT) is a recently developed
medical imaging modality visualizing conductivity images of an electrically conducting ob-
ject. MREIT was motivated by the well-known ill-posedness of the image reconstruction
problem of electrical impedance tomography (EIT). Numerous experiences have shown
that practically measurable data sets in an EIT system are insufficient for a robust re-
construction of a high-resolution static conductivity image due to its ill-posed nature and
the influences of errors in forward modeling. To overcome the inherent ill-posed charac-
teristics of EIT, the MREIT system was proposed in the early 1990s to use the internal
data of magnetic flux density B = (Bx,By,Bz), which is induced by an externally injected
current. MREIT uses an MRI scanner as a tool to measure the z-component Bz of the
magnetic flux density, where z is the axial magnetization direction of the MRI scanner. In
2001, a constructive Bz-based MREIT algorithm called the harmonic Bz algorithm was
developed and its numerical simulations showed that high-resolution conductivity image
reconstructions are possible. This novel algorithm is based on the key observation that
the Laplacian ∆Bz probes changes in the log of the conductivity distribution along any
equipotential curve having its tangent to the vector field J×(0, 0,1), where J = (Jx,Jy,Jz)
is the induced current density vector. Since then, imaging techniques in MREIT have ad-
vanced rapidly and have now reached the stage of in vivo animal and human experiments.
This paper reviews MREIT from its mathematical framework to the most recent human
experiment outcomes.
Key words. magnetic resonance EIT, electrical impedance tomography, inverse problems
AMS subject classifications. 35R30, 35J05, 76Q05
DOI. 10.1137/080742932
1. Introduction. Lately, new medical imaging modalities to quantify electrical
and mechanical properties of biological tissues have received a great deal of attention
in the biomedical imaging area. In particular, cross-sectional imaging of electrical
conductivity and permittivity distributions inside the human body has been an im-
portant research topic since these distributions may provide better differentiation of
tissues or organs, resulting in enhanced diagnosis and treatment of numerous diseases
[87, 30]. Indeed, electrical conductivity and permittivity values of biological tissues
∗Received by the editors December 6, 2009; accepted for publication (in revised form) March
15, 2010; published electronically February 8, 2011. This work was supported by the WCU program
through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science,
and Technology R31-2008-000-10049-0.
http://www.siam.org/journals/sirev/53-1/74293.html
†Department of Computational Science and Engineering, Yonsei University, Seoul 120-748, Korea
(seoj@yonsei.ac.kr).
‡Department of Biomedical Engineering, Kyung Hee University, Gyeonggi-do 446-701, Korea
(ejwoo@khu.ac.kr). The work of this author was supported by the SRC/ERC program of MOST/
KOSEF (R11-2002-103).
40
Page 2
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
MAGNETIC RESONANCE ELECTRICAL IMPEDANCE TOMOGRAPHY
41
reveal their physiological and pathological conditions [20, 21, 24, 27]. There are also
various biomedical applications such as source imaging of the brain and the heart
and electrical stimulators that require in vivo electrical conductivity and permittivity
values of tissues and organs of the human body.
Since the late 1980s, there have been numerous attempts in electrical impedance
tomography (EIT) toward a robust reconstruction of cross-sectional images of a con-
ductivity distribution inside the human body [87, 30]. However, static EIT imaging
has not yet reached the stage of clinical applications. Such endeavors during the last
three decades have led us to admit methodological limitations in terms of practical
feasibility and to search for a new way to bypass the ill-posedness of the corresponding
inverse problem.
This article reviews a recently developed medical imaging modality called mag-
netic resonance electrical impedance tomography (MREIT), which aims to visualize
conductivity images of an electrically conducting object using the current injection
MRI technique [74, 75]. In both EIT and MREIT, we inject an electrical current into
a biological imaging object through surface electrodes so that the internal current
flow is stained with the electrical property of the biological medium. More precisely,
the induced internal current density J = (Jx,Jy,Jz) and the magnetic flux density
B = (Bx,By,Bz) are dictated by the Maxwell equations with boundary conditions
for given distributions of conductivity σ and permittivity ε.
Assuming a low frequency of less than a few kHz, we will ignore effects of per-
mittivity and consider only conductivity. The injection current induces a distribution
of voltage u, where ∇u = −σ−1J inside the imaging object. For conductivity image
reconstructions, EIT utilizes a set of voltage data measured on a portion of the bound-
ary using a limited number of surface electrodes. MREIT relies on a set of internal
data Bz, the z-component of the induced magnetic flux density B, where z is the
axial magnetization direction of the MRI scanner. In this paper, we will assume that
conductivity σ is isotropic, although muscle and some neural tissues show anisotropy.
The static EIT system using a set of current-to-voltage data, a rough knowledge
of discrete NtD (Neumann-to-Dirichlet) data, has a drawback in achieving robust
reconstructions of high-resolution conductivity images. This is mainly because the
NtD data is very sensitive to various modeling errors including unknown boundary
geometry and electrode positions and other systematic artifacts, while it is insensitive
to any change in a local conductivity distribution in an internal region remote from
the boundary.
The amount of information in the measured NtD data is limited by the num-
ber of electrodes (usually from 8 to 32). In practice, the cumbersome procedure
to attach many electrodes is prone to increased measurement errors in addition to
electronic noise and various systematic artifacts. Within a reasonable level of cost
and practical applicability, there always exist uncertainties in terms of electrode posi-
tions and boundary shape of the imaging subject. Due to the ill-posed nature of the
inverse problem, it seems that measurable information is insufficient for robust recon-
structions of high-resolution conductivity images in spite of novel theoretical results
guaranteeing a unique identification of σ from the NtD data. For uniqueness refer to
[15, 45, 83, 84, 85, 61, 62, 14, 4, 38] and for EIT image reconstruction algorithms to
[5, 13, 92, 88, 89, 25, 65, 34, 17, 26, 73, 7, 32, 19, 82, 54, 33].
EIT has several merits, such as its portability and high temporal resolution, even
though its spatial resolution is poor. Noting that common errors may cancel each
other out by a data subtraction method, time-difference EIT imaging has shown its
potential in clinical applications where monitoring temporal changes of a conductivity
Page 3
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
42
JIN KEUN SEO AND EUNG JE WOO
distribution is needed [57, 16]. Frequency-difference EIT imaging aims to detect an
anomaly such as bleeding or stroke in the brain and tumor tissue in the breast [79, 44].
For high-resolution static imaging of a conductivity distribution inside the human
body, there have been strong needs for supplementary data to make the problem well-
posed and overcome fundamental limitations of the static EIT imaging. In order to
bypass the ill-posed nature in EIT, MREIT was proposed in the early 1990s to take
advantage of an MRI scanner as a tool to capture the internal magnetic flux density
data B induced by an externally injected current [93, 90, 10, 11]. The magnetic flux
density B is stained with the conductivity distribution σ according to the Amp´ ere
law −σ∇u = J =
and biological tissues.
In early MREIT systems, all three components of B = (Bx,By,Bz) were utilized
as measured data, and this required mechanical rotations of the imaging object within
the MRI scanner [39, 49, 52, 8, 69]. Assuming knowledge of the full components of
B, we can directly compute the current density J =
using an image reconstruction algorithm such as the J-substitution algorithm [49,
39, 52], the current constrained voltage scaled reconstruction (CCVSR) algorithm [8],
and equipotential line methods [46, 69]. Recently, a new noniterative conductivity
image reconstruction method called current density impedance imaging (CDII) was
suggested and experimentally verified [29]. Theoretical progress in CDII showed that
conductivity image reconstructions are possible from measurements of one internal
current density distribution and one boundary voltage data [63, 64]. These methods
using B = (Bx,By,Bz) suffer from technical difficulties related to object rotations
within the main magnet of the MRI scanner.
In order to make the MREIT technique applicable to clinical situations, we should
use only Bzdata to avoid object rotation. In 2001, a constructive Bz-based MREIT
algorithm called the harmonic Bz algorithm was developed and its numerical simu-
lations and phantom experiments showed that high-resolution conductivity imaging
is possible without rotating the object [81, 78, 67, 66, 68]. This novel algorithm is
based on the key observation that the Laplacian of Bz, ∆Bz, probes a change of lnσ
along any curve having its direction tangent to the vector field J × (0,0,1). Since
then, imaging techniques in MREIT have advanced rapidly and have now reached the
stage of in vivo animal and human imaging experiments [70, 71, 80, 47, 68, 55, 56].
In this paper, we review MREIT based on measurements of a single component of
induced magnetic flux density Bz, whose diagram is shown in Figure 1, in terms of its
mathematical framework and modeling, image reconstruction algorithms, and other
theoretical issues of uniqueness and convergence. Experimental results will be shown
as examples.
1
µ0∇ × B, where µ0is the magnetic permeability of the free space
1
µ0∇ × B and reconstruct σ
2. Mathematical Framework. Bearing clinical applications of MREIT in mind,
we set up a mathematical model of MREIT. Let the object to be imaged occupy a
three-dimensional bounded domain Ω ⊂ R3with a smooth boundary ∂Ω. We attach
a pair of surface electrodes E+and E−on the boundary ∂Ω through which we inject
current of I mA at a fixed low angular frequency ω ranging over 0 <
Then the time harmonic current density J, electric field intensity E, and magnetic flux
density B due to the injection current approximately satisfy the following relations:
ω
2π< 500 Hz.
∇ · J = 0 = ∇ · B,
J = σE,
J =
1
µ0∇ × B
in Ω,
in Ω,
(2.1)
∇ × E = 0(2.2)
Page 4
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
MAGNETIC RESONANCE ELECTRICAL IMPEDANCE TOMOGRAPHY
43
Fig. 1
Diagram of an MREIT system. A patient is placed inside the bore of an MRI scanner with
multiple electrodes attached on the surface. Imaging currents are injected into the patient
between a chosen pair of electrodes. We obtain MR magnitude and magnetic flux density
images to reconstruct cross-sectional conductivity images of the patient.
I = −
J · n = 0 on ∂Ω \ E+∪ E−,
?
E+J · n ds =
?
E−J · n ds,
(2.3)
J × n = 0 on E+∪ E−,
(2.4)
where n is the outward unit normal vector on ∂Ω and ds is the surface area ele-
ment. In order to simplify the MREIT problem, we will assume that the conductivity
distribution σ in Ω is isotropic, 0 < σ < ∞, and smooth.
2.1. Internal Data Bz. Since the late 1980s, measurements of the internal mag-
netic flux density induced by an injection current have been studied in magnetic
resonance current density imaging (MRCDI) to visualize the internal current density
distribution [37, 74, 75]. Assume that z is the coordinate that is parallel to the di-
rection of the main magnetic field B0 of an MRI scanner. Imagine that we try to
measure the induced Bzdata subject to a positive injection current I+in an imaging
slice Ωz0= Ω ∩ {z = z0}. Application of the injection current during an MR imaging
process must be synchronized with a chosen MR pulse sequence, as shown in Figure
2. This generates inhomogeneity of the main magnetic field changing B0to B0+ B,
which alters the MR phase image in such a way that the phase change is proportional
to Bz.
Using the MRI scanner, we obtain the following complex k-space data involving
Bzinformation in the slice Ωz0:
SI+(kx,ky) =
? ?
Ωz0
M(x,y,z0)ei(γBz(x,y,z0)Tc+δ(x,y,z0))ei(xkx+yky)dxdy,
(2.5)
where Bz(x,y,z0) denotes the value of Bz at the position (x,y,z0). Here, M is the
transverse magnetization, δ any systematic phase error, γ = 26.75 × 107rad/T·s the
gyromagnetic ratio of hydrogen, and Tcthe duration of the injection current pulse.
Application of the Fourier transform to the k-space MR signal SI+ yields the following
Page 5
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
44
JIN KEUN SEO AND EUNG JE WOO
RF
Slice
Selection
Phase
Encoding
Reading
Positive
Current I+
Negative
Current I−
90º 180º
Tc/2
Tc/2
I
−I
−I
I
Fig. 2
Typical spin echo pulse sequence for MREIT.
complex MR image, MI+(x,y,z0):
MI+(x,y,z0) := M(x,y,z0) eiγBz(x,y,z0)Tceiδ(x,y,z0).
Similarly, we inject a negative current with the same amplitude and waveform
to obtain MI−(x,y,z0) := M(x,y,z0) e−iγBz(x,y,z0)Tceiδ(x,y,z0). If M(x,y,z0) ?= 0,
dividing MI+(x,y,z0) by MI−(x,y,z0) extracts a wrapped Bz:
1
2γTc?
MI−(x,y,z0)
A standard phase-unwrapping algorithm to restore the continuity of Bzprovides the
Bzdata. Figure 3(a) shows an MR magnitude image M of a cylindrical saline phan-
tom including an agar object whose conductivity is different from that of the saline.
Injection current from the top to the bottom electrodes produced the wrapped phase
image in Figure 3(b). Such phase wrapping may not occur when the amplitude of
the injection current is small. Figure 3(c) is the Bz image after applying a phase-
unwrapping algorithm. We can observe the deflection of Bz across the boundary of
the agar object where conductivity contrast exists.
The relation between the internal Bzdata and the conductivity σ can be expressed
implicitly by the z-component of the Biot–Savart law,
Bz(x,y,z0) =
?
ln
?MI+(x,y,z0)
??
?
modulo
2π
γTc
?
.
(2.6)
Bz(r) =µ0
4π
?
Ω
?r − r?, −σ(r?)∇u(r?) × ez?
|r − r?|3
dr?+ H(r)for r ∈ Ω,
(2.7)
where r = (x,y,z) is a position vector in R3, ez = (0,0,1), H(r) is a harmonic
function in Ω representing a magnetic flux density generated by currents flowing
through external lead wires, and u is a voltage in the Sobolev space H1(Ω) satisfying
the following boundary value problem:
∇ · (σ∇u) = 0
I =?
in Ω,
E+σ∂u
∂nds = −?
E−σ∂u
∂nds,
σ∂u
∇u × n = 0 on E+∪ E−,
∂n= 0 on ∂Ω \ E+∪ E−,
(2.8)
Page 6
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
MAGNETIC RESONANCE ELECTRICAL IMPEDANCE TOMOGRAPHY
45
Recessed
Copper Electrode
Saline
Agar
Recessed
Copper Electrode
(a) (b)(c)
Fig. 3
(a) MR magnitude image M of a cylindrical saline phantom including an agar object. Con-
ductivity values of the saline and agar were different. (b) Wrapped phase image subject to an
injection current from the top to the bottom electrodes. (c) Corresponding image of induced
Bz after applying a phase-unwrapping algorithm.
Bovine Tongue
Chicken Breast
Porcine Muscle
Air Bubble
Recessed
Electrode
Agar
Gelatin
Recessed
Electrode
(a)(b)(c) (d)
Fig. 4
(a) MR magnitude image M of a cylindrical phantom including chunks of three different
biological tissues. Its background was filled with an agar gel. (b) Reconstructed conductivity
image of the same slice using an MREIT conductivity image reconstruction algorithm. (c)
Image of the magnitude of the current density |J| where the thin lines are current stream
lines. Current was injected from the left to the right electrodes. (d) Induced magnetic flux
density Bz image subject to the current density in (c) [68].
where
u|E− = 0, we can obtain a unique solution u of (2.8). In practice, the harmonic
function H is unknown, so we should eliminate its effects in any conductivity image
reconstruction algorithm.
Figure 4(a) shows an MR magnitude image of a cylindrical phantom whose back-
ground was filled with an agar gel. It contains chunks of three different biological
tissues. Its conductivity image is shown in Figure 4(b), where we used an MREIT im-
age reconstruction algorithm described later. From multislice conductivity images of
the three-dimensional phantom, we solved (2.8) for u using the finite element method
(FEM) and computed the internal current density J using J = σE = −σ∇u. Figure
4(c) is a plot of |J| and the thin lines are current stream lines subject to an injection
current from the left to the right electrodes. The induced magnetic flux density Bz
due to the current density in Figure 4(c) is visualized in Figure 4(d).
Let’s assume that an imaging object as shown in Figure 4(a) with a conductivity
distribution as shown in Figure 4(b) is given. In MREIT, we inject current into the
object through a pair of surface electrodes. This produces an internal distribution of
J as in Figure 4(c) that is not directly measurable. Following the relation in (2.7), the
∂u
∂nis the normal derivative of u to the boundary. Setting a reference voltage
Page 7
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
46
JIN KEUN SEO AND EUNG JE WOO
(a)(b)
Fig. 5
(a) and (b) show two different conductivity distributions that produce the same Bz data
subject to Neumann data g(x,y) = δ((x, y) − (0,1)) − δ((x, y) − (0,−1)), x ∈ ∂Ω, where
Ω = (−1,1) × (−1,1).
current density generates an internal distribution of the induced magnetic flux density
Bzas in Figure 4(d) that is measurable by an MRI scanner. The goal in MREIT is
to reconstruct an image of the internal conductivity distribution as in Figure 4(b) by
using the measured data of Bzas in Figure 4(d).
2.2. Three Key Observations. The right-hand side of (2.7) is a sum of a nonlin-
ear function of σ and the harmonic function H, which is independent of σ. We may
consider an inverse problem of recovering the conductivity distribution σ entering the
nonlinear problem (2.7) from knowledge of the measured data Bz, geometry of ∂Ω,
positions of electrodes E±, and the amount of injection current.
First, there exists a scaling uncertainty of σ in the nonlinear problem (2.7) due to
the fact that if σ is a solution of (2.7), so is a scaled conductivity ασ for any scaling
factor α > 0. Hence, we should resolve the scaling uncertainty of σ by measuring a
voltage difference at any two fixed boundary points or by including a piece of elec-
trically conducting material with a known conductivity value as part of the imaging
object [49, 69].
Second, any change of σ in the normal direction ∇u to the equipotential surface
is invisible from Bz data. Assume that a function ϕ : R → R is strictly increasing
and continuously differentiable. Then ϕ(u) is a solution of (2.8) with σ replaced by
σ
ϕ?(u), because
σ(r)∇u(r) =
σ(r)
ϕ?(u(r))∇ϕ(u(r)),
r ∈ Ω.
(2.9)
Noting that this is true for any strictly increasing ϕ ∈ C1(R), we can see that the
data Bz cannot trace a change of σ in the direction ∇u. This means that there are
infinitely many conductivity distributions which satisfy (2.7) and (2.8) for given Bz
data. Figure 5 shows an example of two conductivity distributions producing the
same Bzdata.
Third, Bzdata can trace a change of σ in the tangent direction L∇u to the equi-
potential surface. To see this, we change (2.7) into the following variational form,
where the unknown harmonic term H is eliminated,
?
Ω
∇Bz· ∇η dr =
?
Ω
σ(∇u × ez) · ∇η dr
for all η ∈ C1
0(Ω),
(2.10)
Page 8
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
MAGNETIC RESONANCE ELECTRICAL IMPEDANCE TOMOGRAPHY
47
or, using the smoothness assumption of σ and the fact that ∇ · (∇u × ez) = 0,
∆Bz = ∇lnσ · (σ∇u × ez)
The two expressions (2.10) and (2.11) clearly explain that Bz data probes a change
of lnσ along the vector field flow σ∇u × ez.
Remark 2.1 (about the smoothness assumption of σ in (2.11)). The identity
(2.11) definitely does not make any sense when σ is discontinuous. However, we
can still use (2.11) to develop any MREIT image reconstruction algorithm for a non-
smooth conductivity distribution. To see this, suppose that ˜ σ is a C1-approximation
of a nonsmooth function σ with a finite bounded variation ?σ?BV (Ω)< ∞. An MRI
scanner provides Bz data as a two-dimensional array of Bz intensities inside voxels
of a field of view, and each intensity is affected by the number of protons in each voxel
and an adopted pulse sequence. Hence, any practically available Bz data is always a
blurred version which cannot distinguish σ from ˜ σ. Admitting the obvious fact that
an achievable spatial resolution of a reconstructed conductivity image cannot be better
than the determined voxel size, the Laplacian and gradient in the identity (2.11) should
be understood as discrete differentials at the voxel size of the MR image.
in Ω.
(2.11)
2.3. Data Bz Traces σ∇u × ez-Directional Change of σ. From the formula
∆Bz= ∇lnσ · (σ∇u × ez) in (2.11), the distribution of Bz traces a change of σ in
the direction L∇u in the following ways:
(i) If Bz is superharmonic at r, then lnσ is increasing at r in the direction
σ∇u(r) × ez.
(ii) If Bzis subharmonic at r, then lnσ is decreasing at r in the direction σ∇u(r)×
ez.
(iii) If Bzis harmonic at r, then lnσ is not changing at r in the direction σ∇u(r)×
ez.
According to the above observations, if we could predict the direction of σ∇u × ez,
we could estimate a spatial change of σ in that direction from measured Bz data.
However, the vector field σ∇u×ezis a nonlinear function of the unknown conductivity
σ, and hence estimation of the direction of σ∇u×ezwithout explicit knowledge of σ
appears to be paradoxical.
Assume that the conductivity contrast is reasonably small as ?∇lnσ?L∞(Ω)≤ 1.
The vector flow of the current density J = −σ∇u is mostly dictated by given positions
of electrodes E±, the amount of injection current I, and the geometry of the boundary
∂Ω, while the influence of changes in σ on J is relatively small. This means that
σ∇u ≈ ∇v, where v is a solution of the Laplace equation ∆v = 0 with the same
boundary data as in (2.8). Hence, under the low conductivity contrast assumption,
the change of lnσ along any characteristic curve having its tangent direction J × ez
can be evaluated by using the following approximation:
∇lnσ · (∇v × ez) ≈ ∇lnσ · (σ∇u × ez) = ∆Bz.
2.4. Mathematical Model and Corresponding Inverse Problem. Based on the
observations in previous sections, the harmonic Bzalgorithm, which will be explained
later, was developed. It provides a scaled conductivity image of each transversal slice
Ωz0= Ω ∩ {z = z0}. According to the identity (2.11) and the nonuniqueness result,
we should produce at least two linearly independent currents. With two data Bz,1
and Bz,2 corresponding to two current densities J1 and J2, respectively, satisfying
(J1× J2) · ez?= 0 in Ωz0, we can perceive a transversal change of σ on the slice Ωz0
(2.12)
Page 9
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
48
JIN KEUN SEO AND EUNG JE WOO
using the approximation (2.12). This is the main reason why we usually use two pairs
of surface electrodes E±
We inject two linearly independent currents I1 and I2 into an imaging object
using two pairs of electrodes. In general, one may inject N different currents using
N pairs of electrodes with N ≥ 2, but the data acquisition time of N Bz data sets
is increased byN
is desirable to attach four surface electrodes so that, in the imaging region, the area
of the parallelogram made by two vectors J1×ezand J2× ezis as large as possible.
We may then spend a given fixed data acquisition time to collect Bz,1and Bz,2data
with a sufficient amount of data averaging for a better signal-to-noise ratio (SNR).
1and E±
2, as shown in Figures 3 and 4.
2times. In order to simplify the electrode attachment procedure, it
2.4.1. Model with Two Linearly Independent Currents. Throughout this sec-
tion, we assume that we inject two linearly independent currents through two pairs of
surface electrodes E±
we denote by uj[σ] the induced voltage corresponding to the injection current Ijwith
j = 1,2; that is, uj[σ] is a solution of the following boundary value problem:
1and E±
2. For a given σ ∈ C1
+(Ω) := {σ ∈ C1(Ω) : 0 < σ < ∞},
+(Ω) → H1(Ω) × H1(Ω) × R by
u1[σ]|E+
∇ · (σ∇uj[σ]) = 0
Ij=?
σ∂uj[σ]
∂n
= 0
in Ω,
E+
jσ∂uj[σ]
∂n
ds = −?
E−
jσ∂uj[σ]
j∪ E+
j∪ E−
∂n
ds,
∇uj[σ] × n = 0 on E−
j,
on ∂Ω \ E+
j.
(2.13)
We define a map Λ : C1
Λ[σ](r) =
µ0
4π
µ0
4π
?
?
Ω
?r−r?, σ∇u1[σ](r?)×ez?
|r−r?|3
?r−r?, σ∇u2[σ](r?)×ez?
|r−r?|3
2− u1[σ]|E−
dr?
Ω
dr?
2
,
r ∈ Ω.
(2.14)
We should note that, according to (2.7),
Λ[σ] =?Bz,1− H1, Bz,2− H2, V±
12is the voltage difference u1[σ] between the
2, that is, V±
lead wire effects from the pairs E±
Ω, the first two components of Λ[σ] are available up to harmonic factors.
The inverse problem of MREIT is to identify σ from knowledge of Λ[σ] up to
harmonic factors. In practice, for given data Bz,1,Bz,2, and V±
a robust image reconstruction algorithm to find σ within the admissible class C1
so that such a σ minimizes
12
?,
(2.15)
where Bz,j is the z-component of the magnetic flux density corresponding to the
current density Jj = −σ∇uj[σ] and V±
electrodes E+
1and E±
2and E−
12= u1[σ]|E+
2− u1[σ]|E−
2, respectively. Since we know ∆Hj= 0 in
2. Here, H1and H2are the
12, we should develop
+(Ω)
Φ(σ) =
2
?
j=1
?∆(Λj[σ] − Bz,j)?2
L2(Ω)+ α??Λ3[σ] − V±
12
??2,
(2.16)
where Λ[σ] = (Λ1[σ],Λ2[σ],Λ3[σ]) and α is a positive constant.
Considering the smoothness constraint of σ ∈ C1
again that it is not an important issue in practice since practically available Bzdata
is always a blurred version of a true Bz. See Remark 2.1.
+(Ω), we would like to emphasize
Page 10
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
MAGNETIC RESONANCE ELECTRICAL IMPEDANCE TOMOGRAPHY
49
2.4.2. Uniqueness. For uniqueness, we need to prove that Λ[σ] = Λ[˜ σ] implies
σ = ˜ σ. The following condition is essential for uniqueness:
|(∇u1[σ](r) × ∇u2[σ](r)) · ez| > 0for r ∈ Ω.
(2.17)
However, we still do not have a rigorous theory for the issue related to (2.17) in a
three-dimensional domain. Though there are some two-dimensional results based on
geometric index theory [1, 2, 3, 6, 76], this issue in three dimensions is wide open. In
this section, we briefly explain two-dimensional uniqueness.
Assume that σ, ˜ σ,uj[σ],uj[˜ σ] in a cylindrical domain Ω do not change along the z-
direction, and Λ[σ] = Λ[˜ σ]. This two-dimensional problem has some practical meaning
because many parts of the human body are locally cylindrical in shape. By taking
the Laplacian of Λj[σ] = Λj[˜ σ],j = 1,2, we have
µ0∇ · [σ∇uj× ez] = ∆Λj[σ] = ∆Λj[˜ σ] = µ0∇ · [˜ σ∇˜ uj× ez]
where uj= uj[σ] and ˜ uj= uj[˜ σ].
The above identity leads to ∇ · [σ∇uj× ez− ˜ σ∇˜ uj× ez] = 0, which can be
rewritten as
?
where ∇xy= (
function φj(r) such that
∂x− ˜ σ∂˜ uj
∂x
Then φj satisfies the two-dimensional Laplace equation ∆xyφj = 0 in Ω with zero
Neumann data, and hence φj is a constant function. Using σ∇xyuj− ˜ σ∇xy˜ uj =
∇x,yφj= 0 and (2.11), we can derive
?
∂x∂y
Based on the result of the geometric index theory in [1, 48], we can show that the
matrix
?
∂x
is invertible for all points in Ω. This shows that lnσ
constant c. Due to the fact that u1|E+
have c = 1, which leads to σ = ˜ σ.
Although uniqueness in three dimensions is still an open problem, we can antic-
ipate three-dimensional uniqueness by looking at the roles of the three components
Λ1[σ],Λ2[σ], and Λ3[σ] with appropriate attachments of electrodes. Typical experi-
mental and simulated Bzdata sets are shown in Figures 6 and 7, respectively.
• Comparing Figures 6(a) and 6(c), we can see that the first component Λ1[σ]
probes the vertical change of lnσ where the current density vector field J1
flows mostly in the horizontal direction. Figure 7(b) shows the simulated
Λ1[σ] data with a horizontally oriented current. It is clearer that the Bzdata
subject to the horizontal current flow distinguishes the conductivity contrast
along the vertical direction.
in Ω,
0 = ∇xy×
σ∂uj
∂x− ˜ σ∂˜ uj
∂x
, σ∂uj
∂y
− ˜ σ∂˜ uj
∂y
?
,
∂
∂x,
∂
∂y) is the two-dimensional gradient. Hence, there exists a scalar
∇xyφj:=
?
σ∂uj
, σ∂uj
∂y
− ˜ σ∂˜ uj
∂y
?
in Ω.
(2.18)
σ∂u1
∂x
σ∂u2
−σ∂u1
−σ∂u2
∂y
??
∂
∂ylnσ
∂
∂xlnσ
˜ σ
˜ σ
?
=
?
0
0
?
in Ω.
σ∂u1
∂x
σ∂u2
−σ∂u1
−σ∂u2
∂y
∂y
?
˜ σis constant or σ = c˜ σ for a scaling
2= Λ3[σ] = Λ3[˜ σ] = ˜ u1|E+
2− u1|E−
2− ˜ u1|E−
2, we
Page 11
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
50
JIN KEUN SEO AND EUNG JE WOO
- 6
- 4
- 2
0
2
4
6
x 10
- 8
- 6
- 4
- 2
0
2
4
6
x 10
- 8
[Tesla] [Tesla]
E1+
E1−
E2+
E2−
E1+
E1−
E2+
E2−
Horizontal Current Injection
into Homogeneous Phantom
(a)
E2+
Vertical Current Injection
into Homogeneous Phantom
(b)
- 6
- 4
- 2
0
2
4
6
x 10
- 8
- 6
- 4
- 2
0
2
4
6
x 10
- 8
[Tesla] [Tesla]
E1+
E1−
E2−
E1+
E1−
E2+
E2−
Horizontal Current Injection
into Inhomogeneous Phantom
Vertical Current Injection
into Inhomogeneous Phantom
(c)(d)
Fig. 6
(a) and (b) are measured Bz data from a cylindrical homogeneous saline phantom subject
to current injections along the horizontal and vertical directions, respectively. (c) and (d)
are measured Bz data from the same phantom containing an agar anomaly with a different
conductivity value from the background saline.
Current
Current
Dominant Contrast
along
Horizontal Direction
Dominant Contrast
along
Vertical Direction
(a)(b)(c)
Fig. 7
(a) Conductivity distribution of a model. Electrodes are attached along four sides of the
model. (b) and (c) are simulated Bz data subject to current injections along the horizontal
and vertical directions, respectively.
• Comparing Figures 6(b) and 6(d), the second component Λ2[σ] probes the
horizontal change of lnσ where J2 flows mostly in the vertical direction.
Figure 7(c) shows the simulated Λ2[σ] data with a vertically oriented current.
It is clear that the Bzdata subject to the vertical current flow distinguishes
the conductivity contrast along the horizontal direction.
• The third component, Λ3[σ], is used to fix the scaling uncertainty mentioned
in section 2.2.
Page 12
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
MAGNETIC RESONANCE ELECTRICAL IMPEDANCE TOMOGRAPHY
51
Swine LegHuman Leg
Fig. 8
Typical
|(J1× J2) · ez|.
examplesofelectrode attachmenttomaximizetheareaof parallelogram
In general, if we could produce two currents such that J1(r) × ez and J2(r) ×
ez are linearly independent for all r ∈ Ω, we could roughly expect uniqueness by
observing the roles of Λ[σ]. Taking into account the uniqueness and stability, we
carefully attach two pairs of surface electrodes (which determine the two different
Neumann data) as shown in Figure 8 so that the area of parallelogram |(J1× J2) · ez|
is as large as possible in the truncated cylindrical region. However, the proof of
|(J1(r) × J2(r)) · ez| > 0 for r ∈ Ω is difficult due to examples in [12, 50].
2.4.3. Defective Bz Data in a Local Region. One of the most important is-
sues in MREIT is that of developing a robust image reconstruction algorithm that
is applicable to in vivo animal and human experiments. Before developing an image
reconstruction algorithm, we must take account of a possible fundamental defect in
the measured Bz data. Inside the human body, there may exist a region where the
MR magnitude image value is small. Examples may include the outer layer of the
bone, lungs, and gas-filled internal organs. In such a region, M ≈ 0 in (2.6), resulting
in noise amplification. If the MR magnitude image M contains a Gaussian random
noise Z, then the noise standard deviation in measured Bzdata, denoted by sd(Bz),
can be expressed in the following way [75, 72]:
sd(Bz) =
1
√2γTc
sd(Z)
M
.
(2.19)
From the above formula, the data Bz is not reliable inside an internal region
where the MR magnitude image value M is small. It would be desirable to provide a
high-resolution conductivity image in a region having high-quality Bzdata regardless
of the presence of such problematic regions. Fortunately, (2.11) and (2.12) would
provide a local change in lnσ regardless of the global distribution of σ if we could
predict J1and J2in that local region. This is why an MREIT algorithm using (2.11)
and (2.12) can provide a robust conductivity contrast reconstruction in any region
having Bzdata with a sufficient SNR.
For those problematic regions, we can use the harmonic inpainting method [53]
as a process of data restoration. The method is based on the fact that ∆Bz = 0
inside any local region having a homogeneous conductivity. We first segment each
problematic region where the MR magnitude image value M is near zero. Defining
a boundary of the region, we solve ∆Bz= 0 using the measured Bz data along the
boundary where noise is small. Then we replace the original noisy Bz data inside
the problematic region by the computed synthetic data. We must be careful in using
this harmonic inpainting method since the problematic region will appear as a local
homogeneous region in a reconstructed conductivity image. When there exist multiple
Page 13
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
52
JIN KEUN SEO AND EUNG JE WOO
small local regions with large amounts of noise, we may consider using a harmonic
decomposition denoising method [51] instead of harmonic inpainting.
3. Conductivity Image Reconstruction Algorithm.
3.1. Harmonic BzAlgorithm. The harmonic Bz algorithm is based on the fol-
lowing identity:
A[σ](r)
?
∂ lnσ
∂x(r)
∂ lnσ
∂y(r)
?
=
?
∆Λ1[σ](r)
∆Λ2[σ](r)
?
,
r ∈ Ω,
(3.1)
where
A[σ](r) = µ0
?
σ∂u1[σ]
∂y
σ∂u2[σ]
∂y
(r)
(r)
−σ∂u1[σ]
−σ∂u2[σ]
∂x(r)
∂x(r)
?
,
r ∈ Ω.
Noting that ∆Λj[σ] = ∆Bz,jfor j = 1,2 from (2.7), we have
?
∂ lnσ
∂x(r)
∂ lnσ
∂y(r)
?
= (A[σ](r))−1
?
∆Bz,1(r)
∆Bz,2(r)
?
,
r ∈ Ω,
(3.2)
provided that A[σ] is invertible. The above identity (3.2) leads to an implicit repre-
sentation formula for σ on each slice Ωz0:= Ω ∩ {z = z0} in terms of the measured
data set?Bz,1, Bz,2, V±
Lz0lnσ(x) = ΦΩz0[σ](x)
where
12
?. Denoting x = (x,y) and x?= (x?,y?), we have
for all (x,z0) ∈ Ωz0,
(3.3)
ΦΩz0[σ](x) =
1
2π
?
Ωz0
x − x?
|x − x?|2·
?
(A[σ](x?,z0))−1?
∆Bz,1(x?,z0)
∆Bz,2(x?,z0)
??
dsx?
(3.4)
and
Lz0lnσ(x) = lnσ(x,z0) +
1
2π
?
∂Ωz0
(x − x?) · ν(x?)
|x − x?|2
lnσ(x?,z0) d?x?.
(3.5)
Here, ν is the unit outward normal vector to the curve ∂Ωz0and d? is the line element.
From the trace formula for the double layer potential in (3.5), the identity (3.3) on
the boundary ∂Ωz0can be expressed as
Tz0lnσ(x) = ΦΩz0[σ](x)for all (x,z0) ∈ ∂Ωz0,
(3.6)
where
Tz0lnσ(x) =lnσ(x,z0)
2
+
1
2π
?
∂Ωz0
(x − x?) · ν(x?)
|x − x?|2
lnσ(x?,z0) d?x?.
Noting that the operator Tz0is invertible on L2
∂Ωz0φ d? = 0}, from well-known potential theory [18], we might expect that the
following iterative algorithm based on the identities (3.3) and (3.6) can determine σ
0(∂Ωz0) = {φ ∈ L2(∂Ωz0):
?
Page 14
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
MAGNETIC RESONANCE ELECTRICAL IMPEDANCE TOMOGRAPHY
53
up to a scaling factor:
∇xyσn+1(x,z0) =
Lz0lnσn+1(x) = ΦΩz0[σn+1](x)
From the first step in (3.7), we can update ∇xyσn+1for all imaging slices of
interest within the object as long as the measured data Bzare available for the slices.
Next, we obtain σn+1|∂Ω by solving the integral equation (3.6) for the given right-
hand side of the second step in (3.7). Since σn+1|∂Ωz0is known, so is the value of σn+1
inside Ωz0by simple substitutions of σn+1|∂Ωz0and ∇xyσn+1into the corresponding
integrals. This harmonic Bzalgorithm has shown remarkable performance in various
numerical simulations [81, 67] and imaging experiments summarized in section 4.
Early MREIT methods used all three components of the magnetic flux density
B = (Bx,By,Bz), and they required impractical rotations of the imaging object inside
the MRI scanner. The invention of the harmonic Bzalgorithm using only Bzinstead
of B [81] changed the problem of impractical rotations into a mathematical problem
(2.14) with achievable data through application of two linearly independent Neumann
data. This harmonic Bzalgorithm has been widely used in subsequent experimental
studies including the latest in vivo animal and human imaging experiments [42, 44,
43, 40, 41].
We should mention the convergence behavior of (3.7). When σ has a low contrast
in Ω, the direction of the vector field σ∇uj[σ] is mostly dictated by the geometry
of the boundary ∂Ω and the electrode positions E±
instead of the distribution of σ. This ill-posedness was the fundamental drawback of
the corresponding inverse problem of EIT. However, in MREIT we take advantage of
this insensitivity of EIT. This means that the direction of the vector field σ∇uj[σ] is
similar to that of σ0∇uj[σ0] with σ0= 1, and therefore the data Bz,1and Bz,2hold
the major information on the conductivity contrast. Various numerical simulations
show that only one iteration of (3.7) may provide a conductivity image σ1that is
quite similar to the true conductivity σ. Rigorous mathematical theories regarding
its convergence behavior have not yet been proven. In the paper [56] there are some
convergence results on (3.7) under a priori assumptions on the target conductivity.
1
µ0A[σn]−1
?
∆Bz,1
∆Bz,2
?
for (x,z0) ∈ Ωz0,
for (x,z0) ∈ Ωz0.
(3.7)
j
(or Neumann boundary data)
3.2. Gradient BzDecomposition and Variational BzAlgorithm. It would be
better to minimize the amplitude of the injection current. However, the amplitude
of the signal Bz is proportional to the amplitude of the injection current. For a
given noise level of an MREIT system, this means that we have to deal with Bzdata
sets with a low SNR. Numerical implementation methods of an image reconstruction
algorithm affect the quality of a reconstructed conductivity image since noise in Bz
data is transformed into noise in the conductivity image. Depending on a chosen
method, noise could be amplified or weakened.
Since two differentiations of Bz data tend to amplify its noise, the performance
of the harmonic Bz algorithm could deteriorate when the SNR in the measured Bz
data is low. To deal with this noise amplification problem, algorithms to reduce the
number of differentiations were developed. They include the gradient Bz decompo-
sition algorithm [70] and the variational gradient Bz algorithm [71], which need to
differentiate Bzonly once. They show a better performance in some numerical sim-
ulations, but in practical environments these algorithms were fruitless and produced
some artifacts. In this paper, we discuss only one of them for pedagogical purposes.
Page 15
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
54
JIN KEUN SEO AND EUNG JE WOO
We briefly explain the gradient Bzdecomposition algorithm in a special cylindrical
domain Ω = {r = (x,y,z)|(x,y) ∈ D, −δ < z < δ}, where D is a two-dimensional,
smooth, and simply connected domain. Suppose that u is a solution of ∇·(σ∇u) = 0
in Ω with Neumann data g. We parameterize ∂D as ∂D:= {(x(t),y(t)) : 0 ≤ t ≤ 1}
and define
˜ g(x(t),y(t),z) :=
?t
0
g((x(t),y(t),z))
?
|x?(t)|2+ |y?(t)|2dt
for (x,y,z) ∈ ∂Ω \ {z = ±δ}. The gradient Bzdecomposition algorithm is based on
the following implicit reconstruction formula:
σ =
???−
?
∂Υ
∂y+ Θx[u]
?
∂u
∂x+?∂Υ
∂x+ Θy[u]?∂u
∂y
???
(∂u
∂x)2+ (∂u
∂y)2
in Ω,
(3.8)
where
Θx[u] :=∂ψ
∂y−∂Wz
∂x
+∂Wx
∂z
andΘy[u] :=∂ψ
∂x+∂Wz
∂y
−∂Wy
∂z
in Ω
and
Υ = φ +
1
µ0Bz,W(r) :=
?
Ωδ
1
4π|r − r?|
∂(σ∇u(r?))
∂z
dr?.
Here, φ is a solution of
∇2φ = 0
φ = ˜ g −
∂φ
∂z= −1
in Ω,
µ0Bz
∂Bz
∂z
1
on ∂Ω \ {z = ±δ},
on ∂Ω ∩ {z = ±δ},
µ0
(3.9)
and ψ is a solution of
∇2ψ = 0
∇ψ · τ = ∇ × W · τ
∂ψ
∂z= −∇ × W · ez
in Ω,
on ∂Ω \ {z = ±δ},
on ∂Ω ∩ {z = ±δ},
(3.10)
where τ := (−νy,νx,0) is the tangent vector on the lateral boundary ∂Ω \ {z = ±δ}.
We may use an iterative reconstruction scheme with multiple Neumann data
gj,j = 1,...,N, to find σ. Denoting by um
Neumann data gj, the reconstructed σ is the limit of a sequence σmthat is obtained
by the following formula:
j a solution of ∇ · (σm∇u) = 0 in Ω with
σm+1=
?N
i=1
???−
?
∂Υi
∂y+ Θx[um
i]
?
i
∂um
∂x+?∂Υi
?2
i
∂x+ Θy[um
?2?
i]?∂um
i
∂y
???
?N
i=1
??∂um
∂x
+
?∂um
i
∂y
.
(3.11)
This method needs to differentiate Bzonly once in contrast to the harmonic Bz
algorithm, where the numerical computation of ∇2Bzis required. It has the advantage
of much improved noise tolerance, and numerical simulations with added random noise
of a realistic quantity showed its feasibility and robustness against measurement noise.
However, in practical environments it shows poorer performance compared with the
harmonic Bzalgorithm and may produce some artifacts.
Page 16
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
MAGNETIC RESONANCE ELECTRICAL IMPEDANCE TOMOGRAPHY
55
The major reason for this is that the conductivity σm+1updated by the iteration
process (3.11) is influenced by the global distribution of σm. We should note that
there always exist some local regions with defective Bzdata in human or animal ex-
periments, and we always deal with a truncated region of the imaging object, which
causes geometry errors. Hence, it would be impossible to reconstruct the conductivity
distribution in the entire region of the human or animal subject with a reasonable accu-
racy, and it would be best to achieve robust reconstruction of σ in local regions where
measured Bzdata are reliable. In order to achieve a stable local reconstruction of con-
ductivity contrast with moderate accuracy, poor conductivity reconstruction at one
local region should not badly influence conductivity reconstructions in other regions.
This means that a conductivity image reconstruction algorithm should not depend
too much on the global distribution of Bz, global structure of σ, and geometry ∂Ω.
3.3. Sensitivity Matrix-Based Algorithm. Using a sensitivity matrix S derived
from (2.7), with the assumption H = 0, we may linearize the relationship between Bz
and σ as follows [10, 11]:
∆Bz= S∆σ,
(3.12)
where ∆Bz is the difference in Bz from the imaging object with homogeneous and
perturbed conductivity distributions, σ0 and σ0+ ∆σ, respectively. Inverting the
sensitivity matrix, one can reconstruct a conductivity image from measured Bzdata.
This approach is similar to those used in time-difference EIT imaging.
Birgul and coworkers elaborated on this method and presented experimental re-
sults using a two-dimensional saline phantom with 20 electrodes [8]. Muftuler et al.
[59] and Birgul et al. [9] studied the sensitivity-based method in terms of image res-
olution and contrast. Hamamura et al. [28] demonstrated that this sensitivity-based
method can image time changes of ion diffusion in agarose. Muftuler et al. [60] per-
formed animal experiments with rats and imaged tumors using an iterative version of
the sensitivity-based method. They showed that conductivity values of tumor areas
are increased in reconstructed conductivity images. This method cannot deal with the
unknown term H, which is not zero unless lead wires are perfectly parallel to the z-axis.
3.4. Anisotropic Conductivity Reconstruction Algorithm. Some biological tis-
sues are known to have anisotropic conductivity values and the ratio of anisotropy de-
pends on the type of tissue. For example, human skeletal muscle shows an anisotropy
of up to 1 to 10 between the longitudinal and transversal directions. Conductivity
image reconstructions in EIT have been mostly based on the assumption of isotropic
conductivity due to the limitations of EIT.
Seo et al. [80] applied the MREIT technique to anisotropic conductivity image
reconstructions. Investigating how an anisotropic conductivity
σ =
σ11
σ12
σ13
σ12
σ22
σ23
σ13
σ23
σ33
affects the internal current density and thereby the magnetic flux density, they found
that at least seven different injection currents are necessary for the anisotropic con-
ductivity image reconstruction algorithm. The algorithm is based on the following
two identities:
Us = b
and
∇ ·
σ11
σ12
σ13
σ12
σ22
σ23
σ13
σ23
σ33
∇uj
= 0,
(3.13)
Page 17
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
56
JIN KEUN SEO AND EUNG JE WOO
where
b =
1
µ0
∇2B1,z
...
∇2BN,z
,
s =
−∂yσ11+ ∂xσ12
−∂yσ12+ ∂xσ22
−∂yσ13+ ∂xσ23
σ12
−σ11+ σ22
σ23
σ13
,
and
U =
u1
...
uN
x
x
u1
...
uN
y
y
u1
...
uN
z
z
u1
xx− u1
...
uN
yy
u1
xy
...
u1
xz
...
−u1
yz
...
xx− uN
yy
uN
xy
uN
xz
−uN
yz
.
Here, ujis the voltage corresponding to the jth injection current, uj
σ is assumed to be a symmetric positive definite matrix. As in the harmonic Bz
algorithm, we may use an iterative procedure to compute s in (3.13). Assuming that
we have computed all seven terms of s, we can immediately determine σ12(r) = s4(r),
σ13(r) = s7(r), and σ23(r) = s6(r). To determine σ11 and σ22 from s, we use the
following relation between s and σ:
x=
∂uj
∂x, and
∂σ11
∂x
= s2−∂s5
∂x+∂s4
∂y
and
∂σ11
∂y
= −s1+∂s4
∂x.
(3.14)
The last component σ33can be obtained by using the physical law ∇ · J = 0.
Numerical simulation results using a relatively simple two-dimensional model
shown in Figure 9 demonstrated that the algorithm can successfully reconstruct im-
ages of an anisotropic conductivity tensor distribution provided the Bz data has a
high SNR. Unfortunately, this algorithm is not successful in the practical environ-
ment since it is very weak against the noise and the matrix U is ill-conditioned in the
interior region.
3.5. Other Algorithms. The algebraic reconstruction method [31] may be con-
sidered as a variation of the harmonic Bz algorithm. The authors of [31] discuss
numerous issues including uniqueness, region-of-interest reconstruction, and the noise
effect. Assuming that Bzdata subject to an injection current into the head is avail-
able, Gao, Zhu, and He [22] developed a method to determine conductivity values
of the brain, skull, and scalp layers using the radial basis function and the simplex
method. This kind of parametric approach may find useful applications in EEG/MEG
source imaging problems. Gao, Zhu, and He [23] also suggested the so-called RSM-
MREIT algorithm, where the total error between measured and calculated magnetic
flux densities is minimized as a function of a model conductivity distribution by using
the response surface methodology algorithm.
4. Summary of Experimental Results.
4.1. ImageReconstruction Procedure Using Harmonic BzAlgorithm. Based
on the harmonic Bz algorithm, the Impedance Imaging Research Center (IIRC) in
Korea developed an MREIT software to offer various computational tools from pre-
processing to reconstruction of conductivity and current density images. Figure 10
shows a screen capture of the MREIT software, CoReHA (conductivity reconstructor
Page 18
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
MAGNETIC RESONANCE ELECTRICAL IMPEDANCE TOMOGRAPHY
57
-2
-1
0
1
2
3
-1.5
-1
-0.5
0
0.5
-2
-1
0
1
2
3
(a)
-2
-1
0
1
2
3
-1.5
-1
-0.5
0
0.5
-2
-1
0
1
2
3
(b)
-2
-1
0
1
2
3
-1.5
-1
-0.5
0
0.5
-2
-1
0
1
2
3
(c)
-2
-1
0
1
2
3
-1.5
-1
-0.5
0
0.5
-2
-1
0
1
2
3
(d)
Fig. 9
Numerical simulation of anisotropic conductivity image reconstruction.
conductivity tensor image. (b), (c), and (d) are reconstructed images when the SNR is
infinity, 300, and 150, respectively. Here, SNR means the SNR of the corresponding MR
magnitude image [80].
(a) is the target
using harmonic algorithms) [35, 36]. It includes the three major tasks of preprocessing,
model construction and data recovery, and conductivity image reconstruction.
• Preprocessing.
Bz,1and Bz,2corresponding to two injection currents I1and I2, respectively,
from the k-space data after applying proper phase unwrapping and unit con-
version. Since in practice the magnetic flux density images could be quite
noisy due to many factors, we may use a PDE-based denoising method such
as harmonic decomposition.
• Model construction and data recovery. In the geometrical modeling of the
conducting domain, identifications of the outermost boundary and electrode
locations are critical to impose boundary conditions. We use a semiautomatic
tool employing a level-set-based segmentation method. There could be an
internal region where an MR signal void occurs. In such a problematic region,
measured Bzdata are defective. We may use the harmonic inpainting method
to recover Bz data assuming that the local region is homogeneous in terms
of the conductivity.
According to (2.6), we obtain magnetic flux density images
Page 19
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
58
JIN KEUN SEO AND EUNG JE WOO
Fig. 10
Screen capture of CoReHA. It provides main menus for image viewing, calibration or coor-
dinate setting, and data processing including data verification, segmentation, meshing, and
image reconstruction [35, 36].
• Conductivity reconstruction. We use the harmonic Bz algorithm as the de-
fault algorithm for three-dimensional conductivity image reconstructions. We
may apply the local harmonic Bz algorithm [77] for conductivity image re-
constructions in chosen regions of interest.
4.2. Conductivity Phantom Imaging. Since Woo and Seo [91] summarized most
of the published results of conductivity phantom imaging experiments [67, 66, 68], we
introduce only one of them in this paper. Figure 11(a) shows a tissue phantom in-
cluding chunks of three different biological tissues in the background of agar gel. Its
MR magnitude and reconstructed conductivity images are shown in Figures 11(b)
and 11(c) [68]. Compared with the MR magnitude image in Figure 11(b), the recon-
structed conductivity image in Figure 11(c) shows excellent structural information as
well as conductivity information. Conductivity values of the tissues were measured
beforehand and it was found that pixel values in the reconstructed conductivity im-
age were close to the measured values. As shown in Figure 11(b), an air bubble was
formed inside the phantom. The MR signal void in the air bubble caused the mea-
sured Bz data to be very noisy there. From Figure 11(c), we can observe that the
reconstructed conductivity image shows spurious spikes inside the region of the air
bubble. Since this kind of technical problem can occur in a living body, the harmonic
inpainting method was proposed [53].
From this particular example of a phantom experiment, one may find no signif-
icant difference between the two images in Figures 11(b) and 11(c). We emphasize
that pixel values in Figure 11(c) provide totally different information about electrical
Page 20
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
MAGNETIC RESONANCE ELECTRICAL IMPEDANCE TOMOGRAPHY
59
Bovine Tongue
Porcine Muscle
Chicken Breast
Agar Gelatin
Recessed
Electrode
Air
Bubble
140mm
140mm
(1) Chicken
breast
(2) Porcine
muscle
(3) Bovine
tongue
(3)
(2)
(1)
(a) (b)(c)
Fig. 11
Biological tissue phantom imaging using a 3 T MRI scanner [68]. (a) Photo of the phantom.
(b) Its MR magnitude image. (c) Reconstructed conductivity image using the harmonic Bz
algorithm.
Electrode
(a)
Lead Wire
RF Coil
MRI Bore
B0
x
y
z
(b)
Fig. 12
(a) Attachment of electrodes around a chosen imaging region and (b) placement of an
imaging object inside an MRI scanner. B0 is the main magnetic field of the MRI scanner
[42].
conductivity values, whereas pixel values in Figure 11(b) are basically related to pro-
ton densities. There are enough examples showing that a conductivity image clearly
distinguishes two objects, whereas they are indistinguishable in the corresponding con-
ventional MR image. This happens, for example, when two objects have almost same
proton densities but significantly different numbers of mobile ions. This important
point will be demonstrated later.
4.3. Animal Imaging. Figure 12 shows the experimental setup for postmortem
canine brain imaging experiments. Figure 13 shows reconstructed multislice conduc-
tivity images of a postmortem canine brain [42]. These high-resolution conductivity
images with a pixel size of 1.4 mm were obtained by using a 3 T MRI scanner and
40 mA injection currents. Restricting the conductivity image reconstruction to within
the brain region to avoid technical difficulties related to the skull, these conductivity
images of the intact canine brain clearly distinguish white and gray matter. Since
the harmonic Bz algorithm cannot handle the tissue anisotropy, the concept of the
equivalent isotropic conductivity should be adopted to interpret the reconstructed
conductivity images. Figure 14 compares an MR magnitude image in (a), the con-
ductivity image of the brain region only in (b), and the conductivity image of the
entire head in (c) obtained from a postmortem canine head.
The image quality can be improved by using flexible electrodes with a larger con-
tact area. Recently, Minhas et al. [58] proposed a thin and flexible carbon-hydrogel
electrode for MREIT imaging experiments. Using a pair of carbon-hydrogel electrodes
Page 21
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
60
JIN KEUN SEO AND EUNG JE WOO
M
M
σ
σ
Slice #1 Slice #2 Slice #3
Slice #4 Slice #5 Slice #6
Fig. 13
Postmortem animal imaging of a canine head using a 3 T MRI scanner [42]. Multislice MR
magnitude images of a canine head are shown in the top rows and reconstructed equivalent
isotropic conductivity images of its brain are in the bottom two rows.
(a) (b) (c)
Fig. 14
Comparison of (a) MR magnitude image, (b) conductivity image of the brain only, and (c)
conductivity image of the entire head from a postmortem canine head.
with a large contact area, the amplitude of an injection current can be increased pri-
marily due to a reduced average current density underneath the electrodes. Using two
pairs of such electrodes, they reconstructed equivalent isotropic conductivity images
Page 22
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
MAGNETIC RESONANCE ELECTRICAL IMPEDANCE TOMOGRAPHY
61
40
30
20
10
0
0.6
0.4
0.2
0
0.6
0.4
0.2
0
[S/m]
[S/m]
Fig. 15
Postmortem animal imaging of a swine leg using a 3 T MRI scanner [58]. Multislice MR
magnitude (top), conductivity (middle), and color-coded conductivity (bottom) images.
of a swine leg in Figure 15, which shows a good contrast among different muscles
and bones [58]. From the reconstructed images, we can observe spurious spikes in the
outer layers of bones, primarily due to the MR signal void there.
Figures 16(a) and 16(b) are MR magnitude and reconstructed conductivity images
of a postmortem canine abdomen [36]. Since the abdomen includes a complicated
mixture of different organs, interpretation of a reconstructed conductivity image needs
further investigation. It was found that conductivity image contrast in the canine
kidney is quite different from that of an MR magnitude image, clearly distinguishing
the cortex, internal medulla, renal pelvis, and urethra.
Figure 17 compares in vivo and postmortem conductivity images of the same ca-
nine brain [43]. Though the in vivo conductivity image is noisier than the postmortem
image, primarily due to the reduced amplitude of injection currents, the in vivo image
shows a good contrast among white matter, gray matter, and other brain tissues. Fig-
ure 18 shows in vivo imaging experiments of canine brains with and without a regional
brain ischemia. As shown in Figure 18, the ischemia produced noticeable conductiv-
ity changes in reconstructed images. Accumulated results of these postmortem and
in vivo animal imaging experiments will guide us to properly design in vivo human
imaging experiments.
4.4. Human Imaging. For an in vivo human imaging experiment, Kim et al.
chose the lower extremity as the imaging region [40, 41]. After a review of the in-
stitutional review board, they performed an MREIT experiment of a human subject
using a 3 T MRI scanner. They adopted thin and flexible carbon-hydrogel electrodes
with conductive adhesive for current injections [58]. Due to their large surface area
Page 23
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
62
JIN KEUN SEO AND EUNG JE WOO
Kidney
Spinal Cord
Liver
Spleen
Stomach
Intestines
Carbon-hydrogel
Electrode
(a) (b)
Fig. 16
(a) MR magnitude image and (b) reconstructed conductivity image from a postmortem
canine abdomen [36]. The conductivity image in (b) shows a significantly different image
contrast compared with the MR magnitude image in (a).
RightLeft
Dorsal
Ventral
White Matter Gray Matter
(a)
(c)
(b)
(d)
Fig. 17
(a) In vivo and (c) postmortem MR magnitude images of a canine head. (b) In vivo and
(d) postmortem equivalent isotropic conductivity images of the brain. The same animal was
used for both in vivo and postmortem experiments [43]. The image in (b) was obtained by
using 5 mA injection currents, whereas 40 mA was used in (d).
of 80×60 mm2and good contact with the skin, they could inject pulse-type currents
with amplitude of as much as 9 mA into the lower extremity without producing a
painful sensation. Sequential injections of two currents in orthogonal directions were
used to produce cross-sectional equivalent isotropic conductivity images in Figure 19
with 1.7 mm pixel size and 4 mm slice gap. The conductivity images well distinguished
different parts of muscles and bones. The outermost fatty layer was also clearly shown
in each conductivity image. We could observe excessive noise in the outer layers of
two bones due to the MR signal void phenomenon there. Further human imaging
experiments have been planned and are being conducted to produce high-resolution
conductivity images from different parts of the human body.
5. Future Directions and Conclusion. MREIT provides conductivity images of
an electrically conducting object with a pixel size of about 1 mm. It achieves such a
high spatial resolution by adopting an MRI scanner to measure internal magnetic flux
density distributions induced by externally injected imaging currents. Theoretical
Page 24
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
MAGNETIC RESONANCE ELECTRICAL IMPEDANCE TOMOGRAPHY
63
ROI
Counter-ROI
ROI
(Ischemic Region)
Counter-ROI
(a)
(c)
(b)
(d)
Fig. 18
T2-weighted MR images of a canine head (a) before and (c) after the embolization. (b)
and (d) are corresponding equivalent isotropic conductivity images. The region of interest
(ROI) defines the ischemic region and counter-ROI defines the symmetrical region in the
other side of the brain [43].
Fig. 19
In vivo MREIT imaging experiment of a human leg using a 3 T MRI scanner [40, 41].
Multislice MR magnitude images, reconstructed equivalent isotropic conductivity images,
and color-coded conductivity images of a human leg are shown in the top, middle, and
bottom rows, respectively.
and experimental studies in MREIT demonstrate that it is expected to be a new
clinically useful bioimaging modality. Its capability to distinguish conductivity values
of different biological tissues in their living wetted states is unique.
Following the in vivo imaging experiment of the canine brain [43], numerous
in vivo animal imaging experiments are being conducted for extremities, abdomen,
pelvis, neck, thorax, and head. Animal models of various diseases are also being tried.
To reach the stage of clinical applications, in vivo human imaging experiments are also
in progress [41]. These trials are expected to accumulate new diagnostic information
based on in vivo conductivity values of numerous biological tissues.
Page 25
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
64
JIN KEUN SEO AND EUNG JE WOO
MREIT was developed to overcome the ill-posed nature of the inverse problem in
EIT and provide high-resolution conductivity images. Even though current EIT im-
ages have a relatively poor spatial resolution, high temporal resolution and portability
in EIT could be advantageous in several biomedical application areas [30]. Instead of
competing in a certain application area, MREIT and EIT will be complementary to
each other. Taking advantage of the high spatial resolution in MREIT, Woo and Seo
discussed numerous application areas of MREIT in biomedicine, biology, chemistry,
and material science [91]. We should note that it is possible to produce a current
density image for any electrode configuration once the conductivity distribution is
obtained.
Future studies should overcome a few technical barriers to advance the method
to the stage of routine clinical use. The biggest hurdle at present is the amount of
injection current that may stimulate muscle and nerve. Reducing the injection current
down to a level that does not produce undesirable side effects is the key to the success
of this new bioimaging modality. This demands innovative data processing methods
based on rigorous mathematical analysis as well as improved measurement techniques
to maximize SNRs for a given data collection time.
Acknowledgment. The authors thank collaborators at the Impedance Imaging
Research Center for their invaluable contributions.
REFERENCES
[1] G. Alessandrini and R. Magnanini, The index of isolated critical points and solutions of
elliptic equations in the plane, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 19 (1992), pp.
567–589.
[2] G. Alessandrini and V. Nesi, Univalent σ-harmonic mappings, Arch. Rational Mech. Anal.,
50 (2001), pp. 747–757.
[3] G. Alessandrini, E. Rosset, and J. K. Seo, Optimal size estimates for the inverse conduc-
tivity problem with one measurement, Proc. Amer. Math. Soc., 128 (2000), pp. 53–64.
[4] K. Astala and L. P¨ aiv¨ arinta, Calderon’s inverse conductivity problem in the plane, Ann. of
Math. (2), 163 (2006), pp. 265–299.
[5] D. C. Barber and B. H. Brown, Applied potential tomography, J. Phys. E. Sci. Instrum., 17
(1984), pp. 723–733.
[6] P. Bauman, A. Marini, and V. Nesi, Univalent solutions of an elliptic system of partial
differential equations arising in homogenization, Indiana Univ. Math. J., 128 (2000), pp.
53–64.
[7] C. Berenstein and E. C. Tarabusi, Inversion formulas for the k-dimensional Radon trans-
form in real hyperbolic spaces, Duke Math. J., 62 (1991), pp. 1–19.
[8] O. Birgul, B. E. Eyuboglu, and Y. Z. Ider, Experimental results for 2D magnetic resonance
electrical impedance tomography (MREIT) using magnetic flux density in one direction,
Phys. Med. Biol., 48 (2003), pp. 3485–3504.
[9] O. Birgul, M. J. Hamamura, T. Muftuler, and O. Nalcioglu, Contrast and spatial reso-
lution in MREIT using low amplitude current, Phys. Med. Biol., 51 (2006), pp. 5035–5049.
[10] O. Birgul and Y. Z. Ider, Use of the magnetic field generated by the internal distribution of
injected currents for electrical impedance tomography, in Proceedings of the 9th Interna-
tional Conference on Electrical Bio-impedance, Heidelberg, Germany, 1995, pp. 418–419.
[11] O. Birgul and Y. Z. Ider, Electrical impedance tomography using the magnetic field generated
by injected currents, in Proceedings of the 18th Annual International Conference of the
IEEE Engineering in Medicine and Biology Society, Amsterdam, The Netherlands, 1996,
pp. 784–785.
[12] M. Briane, G. W. Milton, and V. Nesi, Change of sign of the corrector’s determinant for
homogenization in three-dimensional conductivity, Arch. Rational Mech. Anal., 173 (2004),
pp. 133–150.
[13] B. H. Brown, D. C. Barber, and A. D. Seagar, Applied potential tomography: Possible
clinical applications, Clin. Phys. Physiol. Meas., 6 (1985), pp. 109–121.
Page 26
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
MAGNETIC RESONANCE ELECTRICAL IMPEDANCE TOMOGRAPHY
65
[14] R. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem with less regular
conductivities in two dimensions, Comm. Partial Differential Equations, 22 (1997), pp.
1009–1027.
[15] A. P. Calder´ on, On an inverse boundary value problem, in Seminar on Numerical Analysis
and Its Applications to Continuum Physics, Soc. Brasileira de Matem` atica, 1980, pp. 65–73.
[16] M. Cheney, D. Isaacson, and J. C. Newell, Electrical impedance tomography, SIAM Rev.,
41 (1999), pp. 85–101.
[17] M. Cheney, D. Isaacson, J. Newell, J. Goble, and S. Simske, NOSER: An algorithm for
solving the inverse conductivity problem, Internat. J. Imaging Systems Technol., 2 (1990),
pp. 66–75.
[18] G. B. Folland, Introduction to Partial Differential Equations, Princeton University Press,
Princeton, NJ, 1976.
[19] L. F. Fuks, M. Cheney, D. Isaacson, D. G. Gisser, and J. C. Newell, Detection and
imaging of electric conductivity and permittivity at low frequency, IEEE Trans. Biomed.
Eng., 3 (1991), pp. 1106–1110.
[20] C. Gabriel, S. Gabriel, and E. Corthout, The dielectric properties of biological tissues: I.
Literature survey, Phys. Med. Biol., 41 (1996), pp. 2231–2249.
[21] S. Gabriel, R. W. Lau, and C. Gabriel, The dielectric properties of biological tissues: II.
Measurements in the frequency range 10Hz to 20GHz, Phys. Med. Biol., 41 (1996), pp.
2251–2269.
[22] G. Gao, S. A. Zhu, and B. He, Estimation of electrical conductivity distribution within the
human head from magnetic flux density measurement, Phys. Med. Biol., 50 (2005), pp.
2675–2687.
[23] N. Gao, S. A. Zhu, and B. He, New magnetic resonance electrical impedance tomography
(MREIT) algorithm: The RSM-MREIT algorithm with applications to estimation of hu-
man head conductivity, Phys. Med. Biol., 51 (2006), pp. 3067–3083.
[24] L. A. Geddes and L. E. Baker, The specific resistance of biological material: A compendium
of data for the biomedical engineer and physiologist, Med. Biol. Eng., 5 (1967), pp. 271–293.
[25] D. G. Gisser, D. Isaacson, and J. C. Newell, Theory and performance of an adaptive
current tomography system, Clin. Phys. Physiol. Meas., 9, Suppl. A (1988), pp. 35–41.
[26] D. G. Gisser, D. Isaacson, and J. C. Newell, Electric current computed tomography and
eigenvalues, SIAM J. Appl. Math., 50 (1990), pp. 1623–1634.
[27] S. Grimnes and O. G. Martinsen, Bioimpedance and Bioelectricity Basics, Academic Press,
London, 2000.
[28] M. J. Hamamura, L. T. Muftuler, O. Birgul, and O. Nalcioglu, Measurement of ion
diffusion using magnetic resonance electrical impedance tomography, Phys. Med. Biol., 51
(2006), pp. 2753–2762.
[29] K. F. Hasanov, A. W. Ma, A. I. Nachman, and M. L. G. Joy, Current density impedance
imaging, IEEE Trans. Med. Imag., 27 (2008), pp. 1301–1309.
[30] D. Holder, Electrical Impedance Tomography: Methods, History and Applications, IOP Pub-
lishing, Bristol, UK, 2005.
[31] Y. Z. Ider and S. Onart, Algebraic reconstruction for 3D magnetic resonance electrical
impedance tomography (MREIT) using one component of magnetic flux density, Phys-
iol. Meas., 25 (2004), pp. 281–294.
[32] D. Isaacson and M. Cheney, Effects of measurement precision and finite numbers of electrodes
on linear impedance imaging algorithms, SIAM J. Appl. Math., 51 (1991), pp. 1705–1731.
[33] D. Isaacson and M. Cheney, Process for Producing Optimal Current Patterns for Electrical
Impedance Tomography, U.S. Patent 5,588,429, 1996.
[34] D. Isaacson and E. Isaacson, Comment on Calderon’s paper: “On an inverse boundary value
problem,” Math. Comp., 52 (1989), pp. 553–559.
[35] K. Jeon, C.-O. Lee, H. J. Kim, E. J. Woo, and J. K. Seo, CoReHA: Conductivity reconstruc-
tor using harmonic algorithms for magnetic resonance electrical impedance tomography
(MREIT), J. Biomed. Eng. Res., 30 (2009), pp. 279–287.
[36] K. Jeon, A. S. Minhas, T. T. Kim, W. C. Jeong, H. J. Kim, B. T. Kang, H. M. Park,
C.-O. Lee, J. K. Seo, and E. J. Woo, MREIT conductivity imaging of the postmortem
canine abdomen using CoReHA, Physiol. Meas., 30 (2009), pp. 957–966.
[37] M. L. Joy, G. C. Scott, and R. M. Henkelman, In-vivo detection of applied electric currents
by magnetic resonance imaging, Mag. Res. Imaging, 7 (1989), pp. 89–94.
[38] C. Kenig, J. Sjostrand, and G. Uhlmann, The Calderon problem with partial data, Ann. of
Math. (2), 165 (2007), pp. 567–591.
[39] H. S. Khang, B. I. Lee, S. H. Oh, E. J. Woo, S. Y. Lee, M. H. Cho, O. I. Kwon, J. R.
Yoon, and J. K. Seo, J-substitution algorithm in magnetic resonance electrical impedance
Page 27
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
66
JIN KEUN SEO AND EUNG JE WOO
tomography (MREIT): Phantom experiments for static resistivity images, IEEE Trans.
Med. Imag., 21 (2002), pp. 695–702.
[40] H. J. Kim, Y. T. Kim, W. C. Jeong, A. S. Minhas, E. J. Woo, O. J. Kwon, and J. K. Seo,
In vivo conductivity imaging of a human leg using a 3 T MREIT system, in Proceedings of
the 9th Conference on Electrical Impedance Tomography, Dartmouth, NH, 2008, pp. 59–62.
[41] H. J. Kim, Y. T. Kim, A. S. Minhas, W. C. Jeong, E. J. Woo, J. K. Seo, and O. J. Kwon,
In vivo high-resolution conductivity imaging of the human leg using MREIT: The first
human experiment, IEEE Trans. Med. Imag., 28 (2009), pp. 1681–1687.
[42] H. J. Kim, B. I. Lee, Y. Cho, Y. T. Kim, B. T. Kang, H. M. Park, S. L. Lee, J. K.
Seo, and E. J. Woo, Conductivity imaging of canine brain using a 3 T MREIT system:
Postmortem experiments, Physiol. Meas., 28 (2007), pp. 1341–1353.
[43] H. J. Kim, T. I. Oh, Y. T. Kim, B. I. Lee, E. J. Woo, J. K. Seo, S. Y. Lee, O. Kwon, C.
Park, B. T. Kang, and H. M. Park, In vivo electrical conductivity imaging of a canine
brain using a 3 T MREIT system, Physiol. Meas., 29 (2008), pp. 1145–1155.
[44] S. Kim, J. Lee, J. K. Seo, E. J. Woo, and H. Zribi, Multifrequency trans-admittance scanner:
Mathematical framework and feasibility, SIAM J. Appl. Math., 69 (2008), pp. 22–36.
[45] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Comm.
Pure Appl. Math., 37 (1984), pp. 113–123.
[46] O. Kwon, J. Y. Lee, and J. R. Yoon, Equipotential line method for magnetic resonance
electrical impedance tomography (MREIT), Inverse Problems, 18 (2002), pp. 1089–1100.
[47] O. Kwon, C. J. Park, E. J. Park, J. K. Seo, and E. J. Woo, Electrical conductivity imaging
using a variational method in Bz-based MREIT, Inverse Problems, 21 (2005), pp. 969–980.
[48] O. Kwon, H. C. Pyo, J. K. Seo, and E. J. Woo, Mathematical framework for Bz-based
MREIT model in electrical impedance imaging, Comput. Math. Appl., 51 (2006), pp. 817–
828
[49] O. Kwon, E. J. Woo, J. R. Yoon, and J. K. Seo, Magnetic resonance electrical impedance
tomography (MREIT): Simulation study of J-substitution algorithm, IEEE Trans. Biomed.
Eng., 49 (2002), pp. 160–167.
[50] R. S. Laugesen, Injectivity can fail for higher-dimensional harmonic extensions, Complex
Variables Theory Appl., 28 (1996), pp. 357–369.
[51] B. I. Lee, S. H. Lee, T. S. Kim, O. Kwon, E. J. Woo, and J. K. Seo, Harmonic decom-
position in PDE-based denoising technique for magnetic resonance electrical impedance
tomography, IEEE Trans. Biomed. Eng., 52 (2005), pp. 1912–1920.
[52] B. I. Lee, S. H. Oh, E. J. Woo, S. Y. Lee, M. H. Cho, O. Kwon, J. K. Seo, J. Y. Lee, and
W. S. Baek, Three-dimensional forward solver and its performance analysis in magnetic
resonance electrical impedance tomography (MREIT) using recessed electrodes, Phys. Med.
Biol., 48 (2003), pp. 1971–1986.
[53] S. Lee, J. K. Seo, C. Park, B. I. Lee, E. J. Woo, S. Y. Lee, O. Kwon, and J. Hahn,
Conductivity image reconstruction from defective data in MREIT: Numerical simulation
and animal experiment, IEEE Trans. Med. Imag., 25 (2006), pp. 168–176.
[54] W. Lionheart, W. Polydorides, and A. Borsic, The reconstruction problem, in Electrical
Impedance Tomography: Methods, History and Applications, IOP Publishing, Bristol, UK,
2005, pp. 3–64.
[55] J. J. Liu, H. C. Pyo, J. K. Seo, and E. J. Woo, Convergence properties and stability issues
in MREIT algorithm, Contemp. Math., 25 (2006), pp. 168–176.
[56] J. J. Liu, J. K. Seo, M. Sini, and E. J. Woo, On the convergence of the harmonic Bz
algorithm in magnetic resonance electrical impedance tomography, SIAM J. Appl. Math.,
67 (2007), pp. 1259–1282.
[57] P. Metherall, D. C. Barber, R. H. Smallwood, and B. H. Brown, Three-dimensional
electrical impedance tomography, Nature, 380 (1996), pp. 509–512.
[58] A. S. Minhas, W. C. Jeong, Y. T. Kim, H. J. Kim, T. H. Lee, and E. J. Woo, MREIT of
postmortem swine legs using carbon-hydrogel electrodes, J. Biomed. Eng. Res., 29 (2008),
pp. 436–442.
[59] L. T. Muftuler, M. J. Hamamura, O. Birgul, and O. Nalcioglu, Resolution and contrast
in magnetic resonance electrical impedance tomography (MREIT) and its application to
cancer imaging, Technol. Cancer Res. Treat., 3 (2004), pp. 599–609.
[60] L. T. Muftuler, M. J. Hamamura, O. Birgul, and O. Nalcioglu, In vivo MRI electrical
impedance tomography (MREIT) of tumors, Technol. Cancer Res. Treat., 5 (2006), pp.
381–387.
[61] A. Nachman, Reconstructions from boundary measurements, Ann. of Math. (2), 128 (1988),
pp. 531–576.
[62] A. Nachman, Global uniqueness for a two-dimensional inverse boundary problem, Ann. of
Math. (2), 143 (1996), pp. 71–96.
Page 28
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
MAGNETIC RESONANCE ELECTRICAL IMPEDANCE TOMOGRAPHY
67
[63] A. Nachman, A. Tamasan, and A. Timonov, Conductivity imaging with a single measurement
of boundary and interior data, Inverse Problems, 23 (2007), pp. 2551–2563.
[64] A. Nachman, A. Tamasan, and A. Timonov, Recovering the conductivity from a single mea-
surement of interior data, Inverse Problems, 25 (2009), article 035014.
[65] J. C. Newell, D. G. Gisser, and D. Isaacson, An electric current tomograph, IEEE Trans.
Biomed. Eng., 35 (1988), pp. 828–833.
[66] S. H. Oh, B. I. Lee, T. S. Park, S. Y. Lee, E. J. Woo, M. H. Cho, O. Kwon, and J. K.
Seo, Magnetic resonance electrical impedance tomography at 3 Tesla field strength, Mag.
Reson. Med., 51 (2004), pp. 1292–1296.
[67] S. H. Oh, B. I. Lee, E. J. Woo, S. Y. Lee, M. H. Cho, O. Kwon, and J. K. Seo, Conduc-
tivity and current density image reconstruction using harmonic Bz algorithm in magnetic
resonance electrical impedance tomography, Phys. Med. Biol., 48 (2003), pp. 3101–3016.
[68] S. H. Oh, B. I. Lee, E. J. Woo, S. Y. Lee, T. S. Kim, O. Kwon, and J. K. Seo, Electrical
conductivity images of biological tissue phantoms in MREIT, Physiol. Meas., 26 (2005),
pp. S279–S288.
[69] S. Onart, Y. Z. Ider, and W. Lionheart, Uniqueness and reconstruction in magnetic
resonance-electrical impedance tomography (MR-EIT), Physiol. Meas., 24 (2003), pp. 591–
604.
[70] C. Park, O. Kwon, E. J. Woo, and J. K. Seo, Electrical conductivity imaging using gradi-
ent Bz decomposition algorithm in magnetic resonance electrical impedance tomography
(MREIT), IEEE Trans. Med. Imag., 23 (2004), pp. 388–394.
[71] C. Park, E. J. Park, E. J. Woo, O. Kwon, and J. K. Seo, Static conductivity imaging using
variational gradient Bz algorithm in magnetic resonance electrical impedance tomography,
Physiol. Meas., 25 (2004), pp. 257–269.
[72] R. Sadleir, S. Grant, S. U. Zhang, B. I. Lee, H. C. Pyo, S. H. Oh, C. Park, E. J. Woo,
S. Y. Lee, O. Kwon, and J. K. Seo, Noise analysis in magnetic resonance electrical
impedance tomography at 3 and 11 T field strengths, Physiol. Meas., 26 (2005), pp. 875–
884.
[73] F. Santosa and M. Vogelius, A backprojection algorithm for electrical impedance imaging,
SIAM J. Appl. Math., 50 (1990), pp. 216–243.
[74] G. C. Scott, M. L. G. Joy, R. L. Armstrong, and R. M. Henkelman, Measurement of
nonuniform current density by magnetic resonance, IEEE Trans. Med. Imag., 10 (1991),
pp. 362–374.
[75] G. C. Scott, M. L. G. Joy, R. L. Armstrong, and R. M. Henkelman, Sensitivity of
magnetic-resonance current density imaging, J. Mag. Res., 97 (1992), pp. 235–254.
[76] J. K. Seo, On the uniqueness in the inverse conductivity problem, J. Fourier Anal. Appl., 2
(1996), pp. 227–235.
[77] J. K. Seo, S. W. Kim, S. Kim, J. J. Liu, E. J. Woo, K. Jeon, and C.-O. Lee, Local harmonic
Bz algorithm with domain decomposition in MREIT: Computer simulation study, IEEE
Trans. Med. Imag., 27 (2008), pp. 1754–1761.
[78] J. K. Seo, O. Kwon, B. I. Lee, and E. J. Woo, Reconstruction of current density distributions
in axially symmetric cylindrical sections using one component of magnetic flux density:
Computer simulation study, Physiol. Meas., 24 (2003), pp. 565–577.
[79] J. K. Seo, J. Lee, S. W. Kim, H. Zribi, and E. J. Woo, Frequency-difference electri-
cal impedance tomography (fdEIT): Algorithm development and feasibility study, Physiol.
Meas., 29 (2008), pp. 929–944.
[80] J. K. Seo, H. C. Pyo, C. Park, O. Kwon, and E. J. Woo, Image reconstruction of anisotropic
conductivity tensor distribution in MREIT: Computer simulation study, Phys. Med. Biol.,
49 (2004), pp. 4371–4382.
[81] J. K. Seo, J. R. Yoon, E. J. Woo, and O. Kwon, Reconstruction of conductivity and current
density images using only one component of magnetic field measurements, IEEE Trans.
Biomed. Eng., 50 (2003), pp. 1121–1124.
[82] E. Somersalo, M. Cheney, and D. Isaacson, Existence and uniqueness for electrode models
for electric current computed tomography, SIAM J. Appl. Math., 52 (1992), pp. 1023–1040.
[83] J. Sylvester and G. Uhlmann, A uniqueness theorem for an inverse boundary value problem
in electrical prospection, Comm. Pure Appl. Math., 39 (1986), pp. 92–112.
[84] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value
problem, Ann. of Math. (2), 125 (1987), pp. 153–169.
[85] J. Sylvester and G. Uhlmann, Inverse boundary value problems at the boundary-continuous
dependence, Comm. Pure Appl. Math., 21 (1988), pp 197–221.
[86] G. Verchota, Layer potentials and boundary value problems for Laplace’s equation in Lipschitz
domains, J. Funct. Anal., 59 (1984), pp. 572–611.
Page 29
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
68
JIN KEUN SEO AND EUNG JE WOO
[87] J. Webster, Electrical Impedance Tomography, Adam Hilger, Bristol, UK, 1990.
[88] A. Wexler, B. Fry, and M. R. Neiman, Impedance-computed tomography algorithm and
system, Appl. Opt., 24 (1985), pp. 3985–3992.
[89] E. J. Woo, P. Hua, J. Webster, and W. Tompkins, A robust image reconstruction algorithm
and its parallel implementation in electrical impedance tomography, IEEE Trans. Med.
Imag., 12 (1993), pp. 137–146.
[90] E. J. Woo, S. Y. Lee, and C. W. Mun, Impedance tomography using internal current density
distribution measured by nuclear magnetic resonance, SPIE, 2299 (1994), pp. 377–385.
[91] E. J. Woo and J. K. Seo, Magnetic resonance electrical impedance tomography (MREIT) for
high-resolution conductivity imaging, Physiol. Meas., 29 (2008), pp. R1–R26.
[92] T. Yorkey, J. Webster, and W. Tompkins, Comparing reconstruction algorithms for elec-
trical impedance tomography, IEEE Trans. Biomed. Engr., 34 (1987), pp. 843–852.
[93] N. Zhang, Electrical Impedance Tomography Based on Current Density Imaging, M.S. thesis,
Department of Electrical Engineering, University of Toronto, Toronto, Canada, 1992.
Download full-text