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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
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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
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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)
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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
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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)
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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
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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)
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MAGNETIC RESONANCE ELECTRICAL IMPEDANCE TOMOGRAPHY
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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)
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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
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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
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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
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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
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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):
?
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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
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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.
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