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1390 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 61, NO. 5, MAY 2014
Electrical Tissue Property Imaging at Low
Frequency Using MREITJin Keun Seo and Eung Je Woo∗ , Senior Member, IEEE
Abstract—The tomographic imaging of tissue’s electricalproperties (e.g., conductivity and permittivity) has been greatlyimproved by recent developments in magnetic resonance (MR)imaging techniques, which include MR electrical impedance to-mography (MREIT) and electrical property tomography. Whenthe biological material is subjected to an external electric field,local changes in its electrical properties become sources of mag-netic field perturbations, which are detectable by the MR signals.Controlling the external excitation and measuring the responsesusing an MRI scanner, we can formulate the imaging problem asan inverse problem in which unknown tissue properties are recov-ered from the acquired MR signals. This inverse problem is non-linear; it involves the incorporation of Maxwell’s equations and
Bloch equations during data acquisition. Each method for visu-alizing internal conductivity and permittivity distributions has itsown methodological limitations, and is restricted to imaging only apart of the ensemble or mean tissue structures or states. Therefore,imaging methods can be improved by developing complementarymethods that can employ the beneficial aspects of various existingtechniques. This paper focuses on recent progress in MREIT anddiscusses its distinct features in comparison with other imagingmethods.
Index Terms—Conductivity, electrical impedance tomogra-phy (EIT), magnetic resonance electrical impedance tomography(MREIT), MRI.
I. INTRODUCTION
IMAGING techniques enable scientists and engineers to visu-
alize various physical phenomena and assess them in detail.
Medical imaging shines a metaphorical light on the internal
structures and states of the human body where no visible light is
present. Various techniques can be used to investigate the body,
exploiting differences in the physical and chemical properties
of tissues. The choice of the observation method depends on the
area of interest, with X-ray, MRI, ultrasound, and gamma rays,
each able to augment simple visual examination. The develop-
ment of any new medical imaging tool should be undertaken
Manuscript received September 7, 2013; revised November 25, 2013; ac-cepted December 29, 2013. Date of publication January 9, 2014; date of current
version April 17, 2014. The work of J. K. Seo was supported by the NationalResearch Foundation of Korea grant funded by the Korean government (MEST)(No. 2011-0028868). The work of E. J. Woo was supported by the NationalResearch Foundation of Korea grant funded by the Korean government (MEST)(No. 20100018275). Asterisk indicates corresponding author.
J. K. Seo is with the Department of Computational Science and Engineering,Yonsei University, Seoul 120-749, Korea (e-mail: [email protected]).
∗E. J. Woo is with the Department of Biomedical Engineering, Kyung HeeUniversity, Gyeonggi-do 446-701, Korea (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TBME.2014.2298859
with consideration of its potential uses and its significance to
the diagnosis of certain diseases.
The electrical current is a promising means of visualizing tis-
sue properties such as conductivity (σ) and permittivity (ε). By
subjecting tissue to the influence of an external electric field (E),
we can assess the conductivity of tissue by observing how the
tissue affects the conduction and movement of charged species
(e.g., ions and molecules) under the influence of the applied E.
Permittivity is associated with the polarizations of charged sub-
stances subject to time-varying E. In biological objects, charge
double layers form across cell membranes, which are close to
insulating; these then act as parallel plate capacitors and in-
crease the effective permittivity of a given sample volume at the
macroscopic scale. The permittivity and conductivity of tissue
are affected by ischemia, hemorrhage, edema, inflammation,
cancers, and neural activities, as well as other functional and
pathological conditions. Imaging of the electrical properties of
tissue has potential in diagnosis and in monitoring of disease
progression, and in the detection of anomalies and neural activ-
ities; it has, therefore, remained an active research topic for the
last three decades.
Reconstruction of the admittivity (κ := σ + iωε) distribution
requires Ohm’s law J = κE and estimation of the 3-D distribu-
tions of the ensemble average values of the current densityJ andelectric fieldE inside a given volume or voxel. Generation of the
internal distributions of J and E requires probing the object by
either injecting the current using surface electrodes or inducing
the current using external coils. While direct and noninvasive
methods to measure E and J inside the body are not available,
it is possible to measure the induced internal magnetic flux den-
sity B. A practically feasible method of acquiring data of the
measurable quantity will employ Maxwell’s equations connect-
ing σ,ε,E,J, and B. Note that images can only be produced of
the effective or apparent conductivity and permittivity distribu-
tions that describe the linear relationship between the ensemble
average current density and the electric field in a voxel of finite
size. The effective quantity of σ + iωε at the macroscopic scale
depends on factors associated with the cellular structure such
as molecular composition, shape, and alignment; cellular mem-
branes, intra- and extracellular fluids; and ion concentrations
and mobilities, as well as temperature and the probing method
itself (e.g., probing frequency and the configuration of current
generation).
MRI scanners can noninvasively measure the magnetic fields
generated inside a human body that is subjected to injected
or induced internal currents. Current-injection MRI, which
was developed in the late 1980s, can visualize the internal
current density distribution using a conventional MRI scanner
0018-9294 © 2014 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistributionrequires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.
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SEO AND WOO: ELECTRICAL TISSUE PROPERTY IMAGING AT LOW FREQUENCY USING MREIT 1391
Lead Wire
Micro-
Controller
16bit-
DACSwitch
Module
Optical-Rx
(Trigger)Voltmeter
VIC
WG
Electromagnetic Shield Case
MREIT Current Source
DC Power Supply (Battery)
Interface
USB
OpticalTrigger
Interface
Module
Optical-Tx
(Trigger)
Trigger Module
RF Pulse Detector
Coil
Fig. 1. MREIT system. The object to be imaged is placed inside the bore of the MRI scanner with two pairs of electrodes attached to its surface. Currents arethen injected into the object between a chosen pair of electrodes; the z-component of the induced magnetic flux density (Bz ) is measured and used to reconstructcross-sectional conductivity images of the object. To inject currents in a synchronized way with a chosen pulse sequence, one can use the trigger module whichdetects RF pulses using a coil and generates trigger signals to the current source.
augmented by a custom-designed constant-current source[28], [58], [59]. This technique, termed magnetic resonance
current density imaging (MRCDI), motivated early research
into magnetic resonance electrical impedance tomography
(MREIT) in the 1990s, with the aim of reconstructing
conductivity images from acquired current density images.
MRCDI, however, is limited by the requirement of object ro-
tation inside the MRI scanner to acquire all three components
of the induced magnetic flux density vector because the scan-
ner can measure only the z -component when the z -axis is the
axial magnetization direction of the scanner. Serious practical
difficulties arise from this requirement because of the limited
space within the bore, as well as problems associated with tis-sue movement during the rotation. Despite numerous attempts to
overcome these difficulties, drawbacks remain, which seriously
limit the clinical applicability of the method.
The feasibility of this imaging method would be improved
by recovering the conductivity distribution using only the z-
components of the induced magnetic field, thus avoiding the
need for object rotation. According to Maxwell’s equations, the
current density is directly related to the three components of the
induced magnetic flux density B = (Bx , By , Bz ), and the con-
ductivity must be computed from the relationship between the
current density and the electrical field. Therefore, Bz data alone
were considered insufficient for conductivity image reconstruc-
tions, and conductivity imaging using Bz data alone appearedimpossible until 2000.
In 2001, Seo et al. investigated the nonlinear relationship be-
tween conductivity and the measured data via the Biot–Savart
law [61]. Their key observation was that the Laplacian of Bz
data probes changes of the logarithm of the conductivity distri-
bution along any equipotential curve in each imaging slice. In
this method, two different currents are injected into the body
to generate two linearly independent current densities. They
showed that if the area of the parallelogram made by these two
vector fields is nonzero at every position in the imaging slice,
then the spatial change of the conductivity distribution can be
precisely reconstructed. They also rigorously mathematically
proved, using geometric index theory, that the area of the paral-lelogram is nonzero when the two pairs of surface electrodes are
appropriately attached [66]. This proof enabled construction of
a representation formula for the conductivity, which led to the
development of a constructive irrotational MREIT algorithm,
termed the harmonic Bz algorithm [61].
This representation formula exists in an implicit form owing
to the nonlinear relationship between the conductivity and mea-
sured data, but it was designed to use a fixed-point theory. In
other words, the formula has a contraction mapping property
such that an iterative method can be used. This mathematical
analysis formed a basis for subsequent successful tests on an-
imals and humans. MREIT techniques have advanced rapidlyand can now offer high-resolution conductivity images of ani-
mal and human subjects. Fig. 1 depicts the configuration of a
modern MREIT system; in the following sections, we describe
its basic principles and experimental outcomes.
We will start with the basics of electrical tissue property
imaging including brief descriptions on conductivity and per-
mittivity. Introducing two methods to probe the passive material
properties, we will elaborate how to utilize measured boundary
voltage data in electrical impedance tomography (EIT) and in-
ternal magnetic flux density data in MRCDI. Then, we will
focus on MREIT to explain how we can overcome the limi-
tations of EIT and MRCDI. We will restrict the scope of this
paper within these topics, which have been the areas of our maininterests. We will briefly mention other methods such as mag-
netic induction tomography, magnetoacoustic tomography with
magnetic induction (MAT-MI), electrical property tomography
(EPT), and diffusion tensor MRI (DT-MRI), where we need to
make some comparisons.
II. BASICS OF ELECTRICAL TISSUE PROPERTY IMAGING
A. Conductivity and Permittivity
We first consider homogeneous saline solution that contains
mobile ions. The ions randomly move at thermal equilibrium
without any external excitation. Because the conduction of a
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1392 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 61, NO. 5, MAY 2014
current requires a net directional flow of mobile ions (or other
charge carriers), this random movement of ions does not con-
stitute a current flow. The application of an external electric
field would exert Coulomb forces on the ions and induce them
to move either parallel to or antiparallel to the direction of the
field. A greater number of ions or the presence of more mobile
ions in the solution would lead to more overall ionic movement
and hence greater conduction. The induced current is, there-
fore, proportional to the applied field intensity and also to the
concentration and mobility of the ions. The property of conduc-
tivity is essentially determined by these properties of the ions in
solution: their concentrations and their mobilities.
Consider the addition of small spherical cells with thin insu-
lating membranes to such a solution. Also, consider the addition
of connective tissues with different compositions and shapes.
The domain becomes heterogeneous and may reveal some direc-
tional properties, i.e., anisotropy. The anisotropy would depend
on the number, shape, and distribution of the cells and tissues.
The application of the same external electrical field would result
in the ions moving differently; hence, the conductivity wouldbe different.
Changes in the direction of the external electric field can
induce polarization phenomena. Polarization can occur with
immobile molecules; this also occurs when mobile ions move
back and forth on either side of an insulating membrane as the
charge double layer formed across the membrane changes its
polarity. The property of permittivity arises from the polarization
of electric charges and contributes to the displacement current
subject to a time-varying electric field.
B. Potential Applications
The conductivities and permittivities of intact wet living bio-
logical tissue are significantly different from those of an ex-
tracted sample. There is also much variation between indi-
viduals, which necessitates the development of new imaging
techniques to visualize these tissue properties in a quantitative
manner.
Brain ischemia causes cells to swell. Enlarged cells occupy
more space and the tissue undergoes microscopic structural
change. As long as the cell membranes remain intact, ions
cannot penetrate because the membranes are electrical insula-
tors. Ions can move around the swollen cells, but their effective
mobility is decreased, and the conductivity of that region is
decreased [22]. If the cells rupture, the extracellular space in-creases, and conductivity thus increases. During the generation
of action potentials in neurons, the ion flux properties of the
membrane change, resulting in a local change in conductivity.
Sadleir et al. investigated the feasibility of conductivity imaging
to detect fast neural activities [57]. Tumors exhibit significantly
higher conductivity values compared with surrounding normal
tissue [17].
Both conductivity and permittivity change with the frequency
[10], [11], [14]. Some physiological events, such as cell swelling
and neural activity, are detectable only at low frequency be-
cause thin cell membranes are transparent at high frequency.
The change in tissue composition can affect observations at
both low and high frequencies. Multifrequency approaches are
desirable for some applications [36].
In addition to the fact that the electrical tissue properties con-
vey useful diagnostic information, there is also strong interest
in obtaining conductivity and permittivity data from individual
subjects. The collection of such bioelectromagnetic data would
aid accurate forward modeling, which is a precursor for seek-
ing a reliable solution of an associated inverse problem. EEG
and MEG source imaging problems are typical examples as
suggested by Gao et al. [12], [13]. Various electric and mag-
netic stimulations, including RF ablation, deep brain stimula-
tion, transcranial electric or magnetic stimulation, and cardiac
defibrillation, require these values either to understand the un-
derlying mechanism or to optimize the protocols.
C. Probing Methods
Electrical conductivity and permittivity are passive tissue
properties, and can be measured only when an electric cur-
rent is applied. When current flows, distributions of the induced
voltage and magnetic field occur. The properties of conductivity
and permittivity determine how the voltage and magnetic flux
density distributions form inside the object. Noninvasive mea-
surement of the voltage and/or magnetic flux density enables
indirect observation of the admittivity distribution inside the
body, using these data and the underlying physical principles.
Assume that a particular object to be imaged occupies a
3-D domain Ω and that its admittivity distribution in Ω is
κ := σ + iωε under the influence of a time-harmonic electric
field E at angular frequency ω . There are two methods of pro-
ducing J,E, andB inside Ω: the injection of the dc or accurrent
into Ω using a pair of surface electrodes, and the use of an ac cur-
rent flowing in a coil outside Ω to induce eddy-currents inside Ω.
The quantities E,J,B, and κ are related by Maxwell’s
equations
J = (σ + iωε)E = 1
µ0∇ × B (1)
∇ × E = −iωB, ∇ · B = 0. (2)
Introduction of the vector magnetic potential A satisfying
B = ∇ × A and ∇ · A = 0 leads the induced time-harmonic
potential u to be governed by the equation
∇ · (κ∇u) = iω∇κ · A in Ω. (3)
Assuming that κ is isotropic, the relation between κ and B can
be obtained from (1) and (2)
−∇2
B = ∇ ln κ × (∇ × B) − iωµ0 κB in Ω (4)
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SEO AND WOO: ELECTRICAL TISSUE PROPERTY IMAGING AT LOW FREQUENCY USING MREIT 1393
Fig. 2. Custom-designed 16-channel multifrequency EIT system and its use in chest and phantom imaging experiments. The right diagram shows an exampleof the current injection and voltage measurement protocol. One can sequentially inject the current between all neighboring pairs of electrodes. For each injectioncurrent, the induced voltages between all neighboring electrode pairs are measured.
where µ0 is the magnetic permeability of free space, which is
approximately equal to that of the human body.
Most methods for imaging the electrical properties of tissue
are based on (3) or (4). To recover the admittivity κ, we can
measure either the voltage u in (3) or the magnetic flux density
B in (4). Note that the voltage is measurable noninvasively only
on the boundary, while the magnetic flux density can be detected
either inside or outside of the imaging object.
D. Boundary Measurement Method: EIT
EIT is one of the most studied admittivity imaging meth-
ods using measured voltage data on the boundary. It employs
multiple current sources and voltmeters to inject currents and
measure boundary voltages using multiple surface electrodes
(E j , j = 1, . . . , N ). Its development appears to have been moti-vated by the success of X-ray CT in the late 1970s [1], [4], [8],
[21], [45]. Under the fairly crude assumption that the direction
of J is known roughly, the overall conductivity values along the
lines of current flow can be evaluated from the knowledge of
the current–voltage relation on the boundary.
In the most commonly used EIT technique, N different
currents are injected sequentially using N different adjacent
pairs of electrodes (E j , E j +1 ), j = 1, . . . , N , where we set
E N +1 = E 1 . Using the same or separate N pairs of elec-
trodes (E k , E k + 1 ), k = 1, . . . , N , we measure the voltage dif-
ferences V j k = u j |E k − u j |E k + 1 subject to the jth injection cur-
rent, where u j
denotes the voltage corresponding to the j
th
current that is injected between the electrode pair (E j , E j +1 ).
For simplicity, we ignore the contact impedances underneath
the electrodes in this paper. At low frequency, below 100 KHz,
with the diameter of Ω being less than 1 m, the identity (2) is
approximated as ∇ × E ≈ 0, which leads to −∇u j ≈ E j and
∇ · (κ∇u j ) ≈ 0 in Ω, where E j is the electric field correspond-
ing to the jth current.
The inverse problem of EIT is to provide a cross-sectional
image of κ from the measured current–voltage data V j k for
j, k = 1, . . . , N . Note, the current–voltage reciprocity
V j k = 1
I
Ω
κ∇u j (r) · ∇uk (r)dr = V k j (5)
for j, k = 1, . . . , N , where I = −
E jκ∇u j · dS is the injected
current, dS is the surface element, and r = (x,y,z) is the po-
sition vector. The total number of independent data in an N -channel EIT system is
N (N −1)2 . For N = 32, there are less than
496 independent measurements, which will lead to 496 equa-
tions in a linear system of equations, and thus, we can estimate
at most 496 conductivity values. Such an EIT system could pro-
duce raw images of 22 × 22 pixels. In other words, the effective
pixel size can be as large as 5% of the field of view. Fig. 2
shows a 16-channel multifrequency EIT system developed by
the authors’ group.
For image reconstructions, the domain Ω is discretized into
a computational mesh with elements pm , m = 1, . . . , M . The
value κ on each element pm is assumed to be a constant κm .
The corresponding M × N 2 sensitivity matrix is given by
A = (am n ) , am n :=
pm
∇u j · ∇uk dr (6)
where n = N ( j − 1) + k. The inverse problem of the static
EIT [9] is to find a best fit of the vector κ = (κ1 , . . . , κM )T ,
such that
κ1a1 + · · · + κM aM = V (7)
where am = (am 1 , . . . , am N 2 )T , V = (V 11 , . . . , V N N )T , and
the superscript T denotes a matrix or vector transpose. The sen-
sitivity matrixA is a nonlinearfunction of κ and depends heavily
on the boundary geometry of Ω and the electrode positions. As a
consequence, the current–voltage data V depend mainly on theboundary geometry and the electrode positions, whereas its de-
pendence on a local perturbation of κ is relatively small. Hence,
the problem (7) is nonlinear and severely ill-posed.
Under the strong assumption that the admittivity vector κ is
a small perturbation of a known vector κ0 , we can linearize the
problem (7) by approximating A ≈ A0 , where A0 is the sensitiv-
ity matrix corresponding to κ0 . The computation of A0 requires
the construction of a forward model to compute numerically u j .
In static EIT, it is necessary to have a very accurate knowledge
of the forward model because the computed data V0 = Aκ0 are
very sensitive to modeling errors, including boundary geome-
try errors and electrode position uncertainties. It would be best
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1394 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 61, NO. 5, MAY 2014
Fig. 3. Mathematical framework of MREIT: (top, second from left) real and imaginary parts of k -space data corresponding to the cross-sectional slice Ωz 0 ; (top,second from right) MR magnitude image at the slice Ωz 0 , (top, first from right) four surface electrodes; (bottom, left) typical spin echo pulse sequence for MREIT;(bottom, second from right) conductivity image reconstructed by MREIT; (bottom, right) Bz , 1 and Bz ,2 data.
if the difference V − V0 depends mainly on the perturbation
κ − κ0 , such that V − V0 alleviates the forward modeling er-
rors [25]. However, previous works over the last three decades
have suggested that it is very difficult to eliminate forward mod-
eling errors even when any movement during data acquisition
is neglected. Barber and Brown observed the following [5]: If
electrodes are spaced 10 cm apart around the thorax, variation
in positioning of 1 mm will produce errors of 1% in the data
V. Such a 1% error would likely not allow imaging for clinical
applications.
Time- and frequency-difference EIT can effectively deal with
the forward modeling errors that prevent static EIT from pro-
ducing useful images. Time-difference EIT provides changes in
κ with time from the time changes of the current–voltage data.
Frequency-difference EIT reconstructs changes in κ with fre-
quency from the corresponding changes in V. The difference
EIT methods require reference data V0 measured at a prede-
termined fixed time or frequency. Assuming that V and V0
are measured from the same domain using the same electrodeconfiguration, the differenceV − V0 effectively eliminates for-
ward modeling errors [63]. Therefore, V − V0 is more directly
related to κ − κ0 and can provide useful images even with a
large amount of total errors and the use of the inaccurate sensi-
tivity matrix A0 [68].
Difference EIT, however, cannot provide high-resolution im-
ages because decreasing the element size will also increase the
coherence between column vectors. With increasing discretiza-
tion, |am ·a |am a ≈ 1 for neighboring elements pm and p distant
from the boundary. Robust imaging requires either decreasing
the mesh size or regularization (grouping elements); either re-
sults in larger effective pixels. If κ − κ
0
is known to be sparse,
sparse sensing allows a reasonable coherence so that the spatial
resolution can be increased by decreasing the mesh size taking
account of the “restricted isometry property” [6], [7].
E. Internal Measurement Method: MRCDI
In the late 1980s and early 1990s, Joy et al. [28], [58], [59]developed MRCDI, an MRI-based imaging technique that can
provide an image of J inside Ω. Low-frequency MRCDI em-
ploys the dc current of I mA injected in pulses, whose timing
is synchronized with an MRI pulse sequence. The current is
usually injected between the 90 and 180 RF pulses and also
between the 180 RF pulse and the read gradient, as shown in
Fig. 3.
The injection current I induces an internal magnetic flux
densityB = (Bx , By , Bz ) to perturb the main magnetic field of
the MRI scanner [54]. This perturbation produces extra phase
shift in the MR phase image in such a way that its accumulation
is proportional to the product of Bz
and the current injection
time T c , where the z -axis is in the direction of the main field.
The acquired complex k -space data include the Bz data on each
imaging slice Ωz0 = Ω ∩ r : z = z0
S (kx , ky ) =
Ω z 0
M (r) eiγ B z (r)T c + iδ (r) ei( xk x +y ky ) dS xy
(8)
where M is theMR magnitudeimage, γ = 26.75 × 107 rad/T · s
is the gyromagnetic ratio of hydrogen, T c is the current pulse
width in seconds, and δ is any systematic phase artifact.
Two-dimensional Fourier transformation of the k -space MR
signal S (kx , ky ) provides the complex MR image M(r) :=
M (r) eiγ B z (r)T c
eiδ (r)
. To extract Bz (r) from M(r), we
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SEO AND WOO: ELECTRICAL TISSUE PROPERTY IMAGING AT LOW FREQUENCY USING MREIT 1395
inject the supplementary current with the opposite polarity to get
M−(r) := M (r) e−iγ B z (r)T c eiδ (r) . Bz can then be calculated
by computing (2γT c )−1 arg (M(r)/M−(r)) and then applying
a phase unwrapping algorithm [15]. For typical examples, see
Fig. 3.
Because an MRI scanner can measure only Bz , acquiring the
other two components Bx and By requires two mechanical ro-
tations while the object to be imaged is inside the bore. With
full knowledge of B = (Bx , By , Bz ), the current density can
be directly computed from Ampere’s law J = ∇ × B/µ0 . Al-
though this current density imaging method is not practicable
due to the requirement of rotations, it provides a noninvasive
method to measure the internal magnetic field at low frequency
in the form of an image with encoded position information.
III. MAGNETIC RESONANCE ELECTRICAL
IMPEDANCE TOMOGRAPHY
To overcome the fundamental limitations of EIT using bound-
ary voltage data only, additional measurements should be incor-
porated in the inverse problem. There are two possibilities for
this: the magnetic field outside the imaging object and the mag-
netic field insidethe object. In MIT, various workshavesought to
utilize theB field outside the domain using coils [16], [37], [77].
However, these methods suffer from low sensitivity—which is
due to the rapid decay of the field strength outside the domain—
and from a tangling of position information due to the mixing
of signals. By contrast, the internal magnetic field measurement
used in MRCDI alleviates both of these technical difficulties at
the expense of using an MRI scanner.
A. MREIT With Subject Rotation
The internal field measurement techniques used in MRCDI
motivated the developers of early MREIT systems to try to
recover σ from J = 1µ 0
∇ × B: Zhang [75] used the line inte-
gral u(P ) − u(Q) =
C 1σ Jdl, where C is a curve joining two
boundary points P and Q, and u is the corresponding potential;
Woo et al. [72] used a least-squares method of finding σ min-
imizing the difference between the measured J and computed
J(σ) = −σ∇u, where the potential u can be viewed as a non-
linear function of σ and the boundary condition; Ider and Bir-
gul [2], [3] used a sensitivity matrix betweenB and σ. However,
none of these early attempts produced high-quality conductivity
images during practical tests.
In the early 2000s, Kwon et al. [38] proposed the J -substitution algorithm based on a 1-Laplacian partial differential
equation
∇ ·
|J|
|∇u|∇u
= 0 in Ω. (9)
This J-based MREIT provided high-resolution σ images by
displaying σ = |J||∇u | [31]. Here, the solution u of (9) with suit-
able boundary data is computed approximately by an iterative
process using ∇ · J
σ n ∇un
= 0 (n = 1, 2 · · ·) with an initial
guess of σ0 = 1. A noniterative conductivity image reconstruc-
tion method termed current density impedance imaging (CDII)
was suggested in [20], [29], and [39]. Some theoretical CDII
studies using a single current density |J| have also been pub-
lished [49]–[51].
All of the J-based MREIT methods have practical difficulties
in imaging experiments because they require rotation of the sub-
ject, which has accompanying technical difficulties such as pixel
misalignment, movement of the internal organs, and distortion
of the boundary geometry. These problems are in addition to the
simple fact that there is no room to rotate a large object inside
the bore of the machine. A new method using only Bz data was
necessary to avoid such problems.
B. MREIT Without Subject Rotation
Bz -basedMREIT aims to reconstruct conductivity imagesus-
ing only Bz data without any rotation of the subject. Before the
creation of the constructive Bz -based MREIT algorithm, most
researchers considered Bz data to be insufficient for conductiv-
ity image reconstructions. Infinitely many conductivities σ can
produce the same B and J values. Considering the structure
σE = ∇ × B and ∇ · B = 0, it appeared that at least two com-
ponents of Bwould be necessary and, therefore, the impractical
rotation of the subject would be unavoidable.
A different view of the inverse problem of recovering σ from
the Bz data was found by Seo et al. [61], who carefully investi-
gated the z-component of the Biot–Savart law
Bz (r) = µ0
4π
Ω
r − r , σ(r)E(r) × z
|r − r|3 dr + B
ex t(r)
(10)
for r ∈ Ω, where Bex t is the magnetic field inside Ω due to
any currents outside Ω, such as external lead wires. In 2001,
they developed the first practically applicable Bz -based MREIT
algorithm based on the following key observation: Bz
data can
probe the change of ln σ in the direction of the vector field
z × J, whereas Bz data cannot probe the change of ln σ in the
direction of J. This observation comes from the identity
−1
µ0∇2 Bz = z × J · ∇ ln σ. (11)
Hence, if we could produce two linearly independent current
densities J1 and J2 , such that (z × J1 ) × (z × J2 ) = 0 in the
image domain Ω, it is possible to probe the change of ln σ in all
transverse directions in each slice Ωz0 = Ω ∩ z = z0 . It would
be best for z × J1 and z × J2 to be orthogonal in the imaging
region [43]. This is the main reason why we usually use two
pairs of surface electrodes E ±1 and E ±2 , as shown in Fig. 3 [52],[53].
C. MREIT Imaging Experiments
The Bz -based MREIT reconstructor, termed as the harmonic
Bz algorithm, was invented based on thefollowing formula [61]:
∇2xy σ =
1
µ0∇ ·
J 1y −J 1x
J 2y −J 2x
−1 ∇2 Bz ,1
∇2 Bz ,2
(12)
where ∇2xy = ∂ 2
∂ x 2 + ∂ 2
∂ y 2 is the transverse Laplacian.
Jeon et al. subsequently developed a noncommercial MREIT
software, the Conductivity Reconstructor using Harmonic
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Fig. 4. MREIT animal and human experiments: (first column) canine head, (second column) canine kidney, (third column) canine prostate, and (fourth column)human leg. The MREIT images shown here are equivalent isotropic conductivity images providing contrast information only. Currently, absolute conductivityimages can be reconstructed from an isotropic object only.
Algorithm (CoReHA), to facilitate experimental MREIT studies
[26], [27], [65]. The Bz -based MREIT technique using the har-
monic Bz algorithm is implemented in CoReHA with the fol-
lowing steps:
1) Attach four surface electrodes in such a way as to maxi-
mize the area of the parallelogram made by the two vectors
J1 × z/J1 and J2 × z/J1 in the region of interest.
2) Sequentially pass electric currents I 1 and I 2 through the
two pairs of surface electrodes E ±1 and E ±2 , respectively,
and measure the k-space data S 1 and S 2 using an MRI
scanner.3) Compute Bz ,1 and Bz ,2 from the k-space data S 1 and
S 2 after applying proper phase unwrapping and unit
conversion.
4) Make a geometric model of the imaging domain, which
includes identification of the outermost boundary and the
electrode locations, to impose boundary conditions de-
scribing the current at the boundary. Using this model,
compute J1 and J2 iteratively as functions of σn using the
starting value of σ0 = 1.
5) Segment any problematic regions (such as bones and
lungs), where measured Bz data are defective due to
MR signal void phenomena. Prevent noise propagation
by properly handling such regions.6) Compute σ by solving the Poisson equation (12) with
suitable boundary conditions.
7) Rescale and update the conductivity using additional
information.
As part of the development of MREIT systems, as represented
in Fig. 1, various experiments have been performed post mortem
and in vivo on animal and human subjects [19], [32], [66], [73].
The conductivity images in Fig. 4 clearly show that MREIT
provides unique contrast information that is not available from
conventional MRI or other imaging techniques.
Following the in vivo imaging of the canine brain [33], numer-
ous in vivo animal imaging experiments have been conducted
to image the extremities, abdomen, pelvis, neck, thorax, and
head. Animal models of various diseases are also being investi-
gated. As part of efforts to develop clinical applications of the
technique, in vivo human imaging work is also under way [35].
These trials are expected to generate new diagnostic information
based on in vivo conductivity observations of various biological
tissues [68].
IV. DISCUSSION
MREIT has been developed in an attempt to overcome the
ill-posed nature of the inverse problem in EIT and to providehigh-resolution conductivity images. It provides low-frequency
conductivity images of an electrically conducting object with
a pixel size of about 1 mm. Such high spatial resolution is
achieved by using an MRI scanner to measure internal magnetic
flux density distributions induced by external injected currents.
Theoretical and experimental studies have demonstrated the po-
tential clinical usefulness of MREIT as a bioimaging modality.
Its capability to distinguish the conductivity of different biolog-
ical tissues in vivo is unique.
Future work should aim to overcome two major technical
barriers for the clinical use of the method. First, the amount
of the current passed through an object must be minimized
to a level that does not produce undesirable nerve or musclestimulation. Second, the anisotropy of the conductivity at low
frequency must be properly handled.
The quality of the reconstructed conductivity images depends
greatly on the signal-to-noise ratio (SNR) of the measured Bz
data. The signal, relative to a given amount of random noise
from the MRI scanner, can be enhanced using the current of
higher amplitude because the dynamic range of the induced Bz
is proportional to the current amplitude.
Early experimental studies used current amplitudes as high
as 40 mA to produce clear conductivity images of various parts
of animal subjects, including the head, chest, abdomen, and
pelvis. The development of optimized pulse sequences and the
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SEO AND WOO: ELECTRICAL TISSUE PROPERTY IMAGING AT LOW FREQUENCY USING MREIT 1397
use of highly sensitive multiple RF coils have decreased the
required current amplitude to 1 mA or below [27], [32], [33],
[35], [46]. Algorithmic enhancementshavealso helpedto reduce
the current amplitude by effectively reducing the noise [26],
[40], [41].
Because current injection at low frequency is unavoidable to
probe conductivity, which is a passive material property, the
level of the generated current density should be limited so that
it does not stimulate nerves or muscles. Large flexible surface
electrodes with a hydrogel layer of varying thickness can pro-
duce a more uniform current density underneath them [71]. By
spreading the current density more uniformly inside the imaging
domain, the amplitude of the total current can be increased for
better quality Bz data. The use of internal electrodes for current
injection also deserves more attention.
Forclinical applications, where thespatial resolution andcon-
ductivity contrast are primary concerns, we recommend using
1 to 5 mA injection currents, and 10 to 20 min scan times. For
example, if we inject 2.5 mA through a uniform current density
electrode with 5×5 cm2 contact area, the internal current den-sity can be well below 1 to 23 A/m 2 , which were estimated as
the threshold to stimulate a nerve with 20-µm diameter [55].
For a given range of Bz measured at a chosen current ampli-
tude, increasing the current injection time T c will also increase
the SNR since the amount of accumulated phase changes in the
MR signal is proportional to the product of Bz and T c [56],
[60]. Increasing the current injection time T c within one pulse
repetition time, however, introduces new problems. Decay in
transverse magnetization during the longer echo time that is
necessary to guarantee a longer T c results in weakening of the
MR signal. This causes deterioration of SNRs in both the MR
phase image and the MR magnitude image.For those applications, where fast functional imaging is
needed, we may significantly reduce the scan time at the ex-
pense of an increased voxel size. In certain cases, we may per-
form quantitative detection of conductivity changes instead of its
image reconstructions and utilize image fusion methods for vi-
sualization. Fast MREIT imaging has been investigated with fast
sequencessuch as EPI andSSFP. The most recent MREIT exper-
imentsutilize optimized pulse sequences designed for multiecho
signals from multiple RF coils [34]. These efforts to reduce the
current amplitude and scan time while maximizing SNR for a
given data collection time require innovative data processing
methods based on rigorous mathematical analyses, as well as
improved measurement techniques.To handle anisotropy, Seo et al. [62] developed an anisotropic
conductivity tensor image reconstruction algorithm that was
very sensitive to measurement noise. The equivalent isotropic
conductivity image may be suitable for certain applications such
as tumor imaging, in which image contrast is the primary con-
cern. However, correct representation of the anisotropy of nerve
or muscle tissues requires consideration of the anisotropy issue
in MREIT.
We may address this problem through the use of additional
information. Given that the water diffusion and conductivity
tensors share the same directional property [70], diffusion ten-
sor images readily available from most clinical MRI scanners
can be incorporated into the conductivity tensor image recon-
structions. Combining DT-MRI with MREIT, we can set the
eigenvectors of the conductivity tensor as those of the diffu-
sion tensor. Since DT-MRI cannot provide any information on
the eigenvalues of the conductivity tensor, we may utilize the
measured current-induced Bz data in MREIT to completely de-
termine the conductivity tensor. This hybrid approach will be
able to deal with the internal conductivity distribution including
both isotropic and anisotropic regions.
The admittivity values of most biological tissues change with
frequency [10], [11], [17]. Observing conductivity and/or per-
mittivity images at both low and high frequencies can strengthen
the diagnostic power of the MR-based electrical tissue property
imaging method. Here, we briefly introduce MAT-MI and EPT
as complementary methods to provide conductivity images in
megahertz range.
In MAT-MI, both static and time-varying magnetic fields are
used to probe the imaging object. The time-varying magnetic
fields induce an eddy current, which generates the Lorentz force
under the static field. Mechanical vibrations of the tissue pro-duce ultrasonic waves to be detected outside the imaging object.
Since the conductivity distribution affects the induced eddy cur-
rent, one may reconstruct its cross-sectional images from the
acquired ultrasonic signals [74]. The latest experimental re-
sults show absolute conductivity images of relatively simple gel
phantoms and biological tissues [23], [24], [44]. As a noncon-
tact method, MAT-MI has a potential to produce high-resolution
conductivity images at variable frequencies in the megahertz
range. There has been no study yet to implement MAT-MI in an
MRI scanner.
In EPT, which can be implemented simultaneously with
MREIT, the input is the conventional RF excitation used inMRI scanning, and the output is the positive rotating magnetic
field H + = (H x + iH y )/2, which can be measured by a B1-
mapping technique [18], [30], [42], [47], [48], [76]. The inverse
problem of EPT is to recover κ from H + and the governing
equation
−∇2H =
∇κ
κ × [∇ × H] − iωκH.
It is easier to implement EPT than MREIT on a clinical MRI
scanner because EPT does not require any additional instru-
mentation, while MREIT requires the attachment of surface
electrodes and an interface with a constant current source. Dual-
frequency conductivity imaging by combining MREIT and EPT
can be implemented simultaneously with a modified spin-echo
pulse sequence [36]. We refer to [64], [67], and [69] for a review
of EPT.
V. CONCLUSION
Electrical tissue property imaging provides unique diagnos-
tic information that is not available from other medical imag-
ing techniques using ultrasound, MR, visible light, X-rays,
or gamma rays. Considering the abundant information em-
bedded in the electrical properties of tissue, particularly con-
ductivity at low frequency, MREIT may open new areas of
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1398 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 61, NO. 5, MAY 2014
research and clinical applications such as in tumor imaging and
neuroimaging.
As is commonly practiced in the field of medical imaging, the
multimodality approach is expected to form a significant part of
future research. Despite the relatively poor spatial resolution of
current EIT images, their high temporal resolution and porta-
bility are advantageous features of EIT for several biomedical
applications [22]. We consider that MREIT and EIT are com-
plementary, rather than competing techniques. Considering the
high spatial resolution of MREIT, Woo and Seo have discussed
its varied applicability in biomedicine, biology, chemistry, and
material science [73]. It is of note that a current density im-
age can be produced for any electrode configuration once the
conductivity distribution is obtained:
1) EIT (≤1 MHz): boundary current→boundary voltage;
2) MREIT (≤1 KHz): boundary current→internal Bz ;
3) MREPT (128 MHz at 3 T MRI): RF excitation →internal
H + = 12µ0
(H x + iH y ).
As summarized previously, EIT could be used when rapid
monitoring of physiological events is required with a focus onfunctional rather than structural information. MREIT and EPT
should be combined to produce multifrequency high-resolution
images of conductivity and permittivity. As these three imaging
methods begin to produce reliable and meaningful in vivo im-
ages, it will be necessary to accumulate large numbers of case
studies of various disease models to obtain useful diagnostic
information.
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