Magnetic Resonance Electrical Impedance Tomography...

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SIAM REVIEW c© 2011 Society for Industrial and Applied MathematicsVol. 53, No. 1, pp. 40–68

Magnetic Resonance ElectricalImpedance Tomography (MREIT)∗

Jin Keun Seo†

Eung Je Woo‡

Abstract. Magnetic resonance electrical impedance tomography (MREIT) is a recently developedmedical imaging modality visualizing conductivity images of an electrically conducting ob-ject. MREIT was motivated by the well-known ill-posedness of the image reconstructionproblem of electrical impedance tomography (EIT). Numerous experiences have shownthat 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 andthe 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 internaldata of magnetic flux density B = (Bx, By, Bz), which is induced by an externally injectedcurrent. MREIT uses an MRI scanner as a tool to measure the z-component Bz of themagnetic flux density, where z is the axial magnetization direction of the MRI scanner. In2001, a constructive Bz-based MREIT algorithm called the harmonic Bz algorithm wasdeveloped and its numerical simulations showed that high-resolution conductivity imagereconstructions are possible. This novel algorithm is based on the key observation thatthe Laplacian ∆Bz probes changes in the log of the conductivity distribution along anyequipotential 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 humanexperiment 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 electricaland mechanical properties of biological tissues have received a great deal of attentionin the biomedical imaging area. In particular, cross-sectional imaging of electricalconductivity and permittivity distributions inside the human body has been an im-portant research topic since these distributions may provide better differentiation oftissues 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) March15, 2010; published electronically February 8, 2011. This work was supported by the WCU programthrough 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

([email protected]).‡Department of Biomedical Engineering, Kyung Hee University, Gyeonggi-do 446-701, Korea

([email protected]). The work of this author was supported by the SRC/ERC program of MOST/KOSEF (R11-2002-103).

40


MAGNETIC RESONANCE ELECTRICAL IMPEDANCE TOMOGRAPHY 41

reveal their physiological and pathological conditions [20, 21, 24, 27]. There are alsovarious biomedical applications such as source imaging of the brain and the heartand electrical stimulators that require in vivo electrical conductivity and permittivityvalues of tissues and organs of the human body.

Since the late 1980s, there have been numerous attempts in electrical impedancetomography (EIT) toward a robust reconstruction of cross-sectional images of a con-ductivity distribution inside the human body [87, 30]. However, static EIT imaginghas not yet reached the stage of clinical applications. Such endeavors during the lastthree decades have led us to admit methodological limitations in terms of practicalfeasibility and to search for a new way to bypass the ill-posedness of the correspondinginverse problem.

This article reviews a recently developed medical imaging modality called mag-netic resonance electrical impedance tomography (MREIT), which aims to visualizeconductivity images of an electrically conducting object using the current injectionMRI technique [74, 75]. In both EIT and MREIT, we inject an electrical current intoa biological imaging object through surface electrodes so that the internal currentflow 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 densityB = (Bx, By, Bz) are dictated by the Maxwell equations with boundary conditionsfor 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 distributionof voltage u, where ∇u = −σ−1J inside the imaging object. For conductivity imagereconstructions, 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 internaldata Bz, the z-component of the induced magnetic flux density B, where z is theaxial magnetization direction of the MRI scanner. In this paper, we will assume thatconductivity 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 knowledgeof discrete NtD (Neumann-to-Dirichlet) data, has a drawback in achieving robustreconstructions of high-resolution conductivity images. This is mainly because theNtD data is very sensitive to various modeling errors including unknown boundarygeometry and electrode positions and other systematic artifacts, while it is insensitiveto any change in a local conductivity distribution in an internal region remote fromthe 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 procedureto attach many electrodes is prone to increased measurement errors in addition toelectronic noise and various systematic artifacts. Within a reasonable level of costand 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 theinverse problem, it seems that measurable information is insufficient for robust recon-structions of high-resolution conductivity images in spite of novel theoretical resultsguaranteeing 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, eventhough its spatial resolution is poor. Noting that common errors may cancel eachother out by a data subtraction method, time-difference EIT imaging has shown itspotential in clinical applications where monitoring temporal changes of a conductivity


42 JIN KEUN SEO AND EUNG JE WOO

distribution is needed [57, 16]. Frequency-difference EIT imaging aims to detect ananomaly 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 humanbody, 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 tobypass the ill-posed nature in EIT, MREIT was proposed in the early 1990s to takeadvantage of an MRI scanner as a tool to capture the internal magnetic flux densitydata B induced by an externally injected current [93, 90, 10, 11]. The magnetic fluxdensity B is stained with the conductivity distribution σ according to the Amperelaw −σ∇u = J = 1

µ0∇× B, where µ0 is the magnetic permeability of the free space

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 withinthe MRI scanner [39, 49, 52, 8, 69]. Assuming knowledge of the full components ofB, we can directly compute the current density J = 1

µ0∇ × B and reconstruct σ

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 conductivityimage reconstruction method called current density impedance imaging (CDII) wassuggested and experimentally verified [29]. Theoretical progress in CDII showed thatconductivity image reconstructions are possible from measurements of one internalcurrent density distribution and one boundary voltage data [63, 64]. These methodsusing B = (Bx, By, Bz) suffer from technical difficulties related to object rotationswithin the main magnet of the MRI scanner.

In order to make the MREIT technique applicable to clinical situations, we shoulduse only Bz data to avoid object rotation. In 2001, a constructive Bz-based MREITalgorithm called the harmonic Bz algorithm was developed and its numerical simu-lations and phantom experiments showed that high-resolution conductivity imagingis possible without rotating the object [81, 78, 67, 66, 68]. This novel algorithm isbased on the key observation that the Laplacian of Bz , ∆Bz, probes a change of lnσalong any curve having its direction tangent to the vector field J × (0, 0, 1). Sincethen, imaging techniques in MREIT have advanced rapidly and have now reached thestage 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 ofinduced magnetic flux density Bz, whose diagram is shown in Figure 1, in terms of itsmathematical framework and modeling, image reconstruction algorithms, and othertheoretical issues of uniqueness and convergence. Experimental results will be shownas examples.

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 athree-dimensional bounded domain Ω ⊂ R

3 with a smooth boundary ∂Ω. We attacha pair of surface electrodes E+ and E− on the boundary ∂Ω through which we injectcurrent of I mA at a fixed low angular frequency ω ranging over 0 < ω

2π < 500 Hz.Then the time harmonic current density J, electric field intensity E, and magnetic fluxdensity B due to the injection current approximately satisfy the following relations:

∇ · J = 0 = ∇ ·B, J =1µ0

∇× B in Ω,(2.1)

J = σE, ∇× E = 0 in Ω,(2.2)



Fig. 1 Diagram of an MREIT system. A patient is placed inside the bore of an MRI scanner withmultiple electrodes attached on the surface. Imaging currents are injected into the patientbetween a chosen pair of electrodes. We obtain MR magnitude and magnetic flux densityimages to reconstruct cross-sectional conductivity images of the patient.

I = −∫E+

J · n ds =∫E−

J · n ds,(2.3)

J · n = 0 on ∂Ω \ E+ ∪ E−, 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 conductivitydistribution 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 magneticresonance current density imaging (MRCDI) to visualize the internal current densitydistribution [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 tomeasure the induced Bz data subject to a positive injection current I+ in an imagingslice Ωz0 = Ω ∩ z = z0. Application of the injection current during an MR imagingprocess must be synchronized with a chosen MR pulse sequence, as shown in Figure2. This generates inhomogeneity of the main magnetic field changing B0 to B0 + B,which alters the MR phase image in such a way that the phase change is proportionalto Bz .

Using the MRI scanner, we obtain the following complex k-space data involvingBz information 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 thetransverse magnetization, δ any systematic phase error, γ = 26.75× 107 rad/T·s thegyromagnetic ratio of hydrogen, and Tc the duration of the injection current pulse.Application of the Fourier transform to the k-space MR signal SI+ yields the following



RF

SliceSelection

PhaseEncoding

Reading PositiveCurrent I+ NegativeCurrent I−

90º 180º

Tc/2

Tc/2I

−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)Tc eiδ(x,y,z0).

Similarly, we inject a negative current with the same amplitude and waveformto obtain MI−(x, y, z0) := M(x, y, z0) e−iγBz(x,y,z0)Tc eiδ(x,y,z0). If M(x, y, z0) = 0,dividing MI+(x, y, z0) by MI−(x, y, z0) extracts a wrapped Bz:

Bz(x, y, z0) =1

2γTcln(MI+(x, y, z0)MI−(x, y, z0)

) (modulo 2π

γTc

).(2.6)

A standard phase-unwrapping algorithm to restore the continuity of Bz provides theBz data. 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 phaseimage in Figure 3(b). Such phase wrapping may not occur when the amplitude ofthe 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 ofthe agar object where conductivity contrast exists.

The relation between the internalBz data and the conductivity σ can be expressedimplicitly by the z-component of the Biot–Savart law,

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 flowingthrough external lead wires, and u is a voltage in the Sobolev space H1(Ω) satisfyingthe following boundary value problem:

∇ · (σ∇u) = 0 in Ω,I =

∫E+ σ ∂u∂n ds = −

∫E− σ ∂u∂n ds,

∇u× n = 0 on E+ ∪ E−, σ ∂u∂n = 0 on ∂Ω \ E+ ∪ E−,(2.8)



RecessedCopper Electrode

Saline

Agar

RecessedCopper 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 aninjection current from the top to the bottom electrodes. (c) Corresponding image of inducedBz after applying a phase-unwrapping algorithm.

Bovine Tongue

Chicken Breast

Porcine Muscle

Air Bubble

RecessedElectrode

AgarGelatin

RecessedElectrode

(a) (b) (c) (d)

Fig. 4 (a) MR magnitude image M of a cylindrical phantom including chunks of three differentbiological tissues. Its background was filled with an agar gel. (b) Reconstructed conductivityimage 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 streamlines. Current was injected from the left to the right electrodes. (d) Induced magnetic fluxdensity Bz image subject to the current density in (c) [68].

where ∂u∂n is the normal derivative of u to the boundary. Setting a reference voltage

u|E− = 0, we can obtain a unique solution u of (2.8). In practice, the harmonicfunction H is unknown, so we should eliminate its effects in any conductivity imagereconstruction 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 biologicaltissues. Its conductivity image is shown in Figure 4(b), where we used an MREIT im-age reconstruction algorithm described later. From multislice conductivity images ofthe 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. Figure4(c) is a plot of |J| and the thin lines are current stream lines subject to an injectioncurrent from the left to the right electrodes. The induced magnetic flux density Bzdue 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 conductivitydistribution as shown in Figure 4(b) is given. In MREIT, we inject current into theobject through a pair of surface electrodes. This produces an internal distribution ofJ as in Figure 4(c) that is not directly measurable. Following the relation in (2.7), the



(a) (b)

Fig. 5 (a) and (b) show two different conductivity distributions that produce the same Bz datasubject to Neumann data g(x, y) = δ((x, y) − (0, 1)) − δ((x, y) − (0,−1)), x ∈ ∂Ω, whereΩ = (−1, 1)× (−1, 1).

current density generates an internal distribution of the induced magnetic flux densityBz as in Figure 4(d) that is measurable by an MRI scanner. The goal in MREIT isto reconstruct an image of the internal conductivity distribution as in Figure 4(b) byusing the measured data of Bz as 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 mayconsider an inverse problem of recovering the conductivity distribution σ entering thenonlinear 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 tothe fact that if σ is a solution of (2.7), so is a scaled conductivity ασ for any scalingfactor α > 0. Hence, we should resolve the scaling uncertainty of σ by measuring avoltage 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 imagingobject [49, 69].

Second, any change of σ in the normal direction ∇u to the equipotential surfaceis invisible from Bz data. Assume that a function ϕ : R → R is strictly increasingand continuously differentiable. Then ϕ(u) is a solution of (2.8) with σ replaced byσ

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 thedata Bz cannot trace a change of σ in the direction ∇u. This means that there areinfinitely many conductivity distributions which satisfy (2.7) and (2.8) for given Bzdata. Figure 5 shows an example of two conductivity distributions producing thesame Bz data.

Third, Bz data 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 ∈ C10 (Ω),(2.10)



or, using the smoothness assumption of σ and the fact that ∇ · (∇u× ez) = 0,

∆Bz = ∇ lnσ · (σ∇u × ez) in Ω.(2.11)

The two expressions (2.10) and (2.11) clearly explain that Bz data probes a changeof 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, wecan 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-approximationof a nonsmooth function σ with a finite bounded variation ‖σ‖BV (Ω) < ∞. An MRIscanner provides Bz data as a two-dimensional array of Bz intensities inside voxelsof a field of view, and each intensity is affected by the number of protons in each voxeland an adopted pulse sequence. Hence, any practically available Bz data is always ablurred version which cannot distinguish σ from σ. Admitting the obvious fact thatan achievable spatial resolution of a reconstructed conductivity image cannot be betterthan the determined voxel size, the Laplacian and gradient in the identity (2.11) shouldbe understood as discrete differentials at the voxel size of the MR image.

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 inthe direction L∇u in the following ways:

(i) If Bz is superharmonic at r, then lnσ is increasing at r in the directionσ∇u(r)× ez .

(ii) IfBz is subharmonic at r, then lnσ is decreasing at r in the direction σ∇u(r)×ez.

(iii) If Bz is 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×ez is a nonlinear function of the unknown conductivityσ, and hence estimation of the direction of σ∇u× ez without explicit knowledge of σappears 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 positionsof 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 thatσ∇u ≈ ∇v, where v is a solution of the Laplace equation ∆v = 0 with the sameboundary 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 × ezcan be evaluated by using the following approximation:

∇ lnσ · (∇v × ez) ≈ ∇ lnσ · (σ∇u × ez) = ∆Bz.(2.12)

2.4. Mathematical Model and Corresponding Inverse Problem. Based on theobservations in previous sections, the harmonic Bz algorithm, which will be explainedlater, was developed. It provides a scaled conductivity image of each transversal sliceΩz0 = 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,1and 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



using the approximation (2.12). This is the main reason why we usually use two pairsof surface electrodes E±

1 and E±2 , as shown in Figures 3 and 4.

We inject two linearly independent currents I1 and I2 into an imaging objectusing two pairs of electrodes. In general, one may inject N different currents usingN pairs of electrodes with N ≥ 2, but the data acquisition time of N Bz data setsis increased by N

2 times. In order to simplify the electrode attachment procedure, itis desirable to attach four surface electrodes so that, in the imaging region, the areaof the parallelogram made by two vectors J1 × ez and J2 × ez is as large as possible.We may then spend a given fixed data acquisition time to collect Bz,1 and Bz,2 datawith a sufficient amount of data averaging for a better signal-to-noise ratio (SNR).

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 ofsurface electrodes E±

1 and E±2 . For a given σ ∈ C1

+(Ω) := σ ∈ C1(Ω) : 0 < σ < ∞,we denote by uj[σ] the induced voltage corresponding to the injection current Ij withj = 1, 2; that is, uj[σ] is a solution of the following boundary value problem:

∇ · (σ∇uj [σ]) = 0 in Ω,

Ij =∫E+

jσ∂uj [σ]∂n ds = −

∫E−

jσ∂uj [σ]∂n ds,

∇uj[σ]× n = 0 on E−j ∪ E+

j ,

σ∂uj [σ]∂n = 0 on ∂Ω \ E+

j ∪ E−j .

(2.13)

We define a map Λ : C1+(Ω) → H1(Ω)×H1(Ω)× R by

Λ[σ](r) =

µ04π

∫Ω

〈r−r′ , σ∇u1 [σ](r′)×ez〉|r−r′|3 dr′

µ04π

∫Ω

〈r−r′ , σ∇u2 [σ](r′)×ez〉|r−r′|3 dr′

u1[σ]|E+2− u1[σ]|E−

2

, r ∈ Ω.(2.14)

We should note that, according to (2.7),

Λ[σ] =(Bz,1 −H1, Bz,2 −H2, V

±12

),(2.15)

where Bz,j is the z-component of the magnetic flux density corresponding to thecurrent density Jj = −σ∇uj [σ] and V ±

12 is the voltage difference u1[σ] between theelectrodes E+

2 and E−2 , that is, V ±

12 = u1[σ]|E+2− u1[σ]|E−

2. Here, H1 and H2 are the

lead wire effects from the pairs E±1 and E±

2 , respectively. Since we know ∆Hj = 0 inΩ, 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 toharmonic factors. In practice, for given data Bz,1, Bz,2, and V ±

12 , we should developa robust image reconstruction algorithm to find σ within the admissible class C1

+(Ω)so that such a σ minimizes

Φ(σ) =2∑j=1

‖∆(Λj [σ]− Bz,j)‖2L2(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

+(Ω), we would like to emphasizeagain that it is not an important issue in practice since practically available Bz datais always a blurred version of a true Bz. See Remark 2.1.



2.4.2. Uniqueness. For uniqueness, we need to prove that Λ[σ] = Λ[σ] impliesσ = σ. The following condition is essential for uniqueness:

|(∇u1[σ](r) ×∇u2[σ](r)) · ez| > 0 for r ∈ Ω.(2.17)

However, we still do not have a rigorous theory for the issue related to (2.17) in athree-dimensional domain. Though there are some two-dimensional results based ongeometric index theory [1, 2, 3, 6, 76], this issue in three dimensions is wide open. Inthis 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 meaningbecause many parts of the human body are locally cylindrical in shape. By takingthe Laplacian of Λj [σ] = Λj[σ], j = 1, 2, we have

µ0∇ · [σ∇uj × ez] = ∆Λj[σ] = ∆Λj[σ] = µ0∇ · [σ∇uj × ez] in Ω,

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

0 = ∇xy ×(σ∂uj∂x

− σ∂uj∂x

, σ∂uj∂y

− σ∂uj∂y

),

where ∇xy = ( ∂∂x , ∂

∂y ) is the two-dimensional gradient. Hence, there exists a scalarfunction φj(r) such that

∇xyφj :=(σ∂uj∂x

− σ∂uj∂x

, σ∂uj∂y

− σ∂uj∂y

)in Ω.(2.18)

Then φj satisfies the two-dimensional Laplace equation ∆xyφj = 0 in Ω with zeroNeumann data, and hence φj is a constant function. Using σ∇xyuj − σ∇xyuj =∇x,yφ

j = 0 and (2.11), we can derive[σ ∂u1∂x −σ ∂u1

∂y

σ ∂u2∂x −σ ∂u2

∂y

][ ∂∂y ln

σσ

∂∂x ln

σσ

]=[

00

]in Ω.

Based on the result of the geometric index theory in [1, 48], we can show that thematrix [

σ ∂u1∂x −σ ∂u1

∂y

σ ∂u2∂x −σ ∂u2

∂y

]

is invertible for all points in Ω. This shows that ln σσ is constant or σ = cσ for a scaling

constant c. Due to the fact that u1|E+2− u1|E−

2= Λ3[σ] = Λ3[σ] = u1|E+

2− u1|E−

2, we

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 componentsΛ1[σ],Λ2[σ], and Λ3[σ] with appropriate attachments of electrodes. Typical experi-mental and simulated Bz data 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 simulatedΛ1[σ] data with a horizontally oriented current. It is clearer that the Bz datasubject to the horizontal current flow distinguishes the conductivity contrastalong the vertical direction.



- 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

Vertical Current Injectioninto Homogeneous Phantom

(a) (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+

E2−

E1+ E1

−

E2+

E2−

Horizontal Current Injection into Inhomogeneous Phantom

Vertical Current Injectioninto Inhomogeneous Phantom

(c) (d)

Fig. 6 (a) and (b) are measured Bz data from a cylindrical homogeneous saline phantom subjectto 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 differentconductivity value from the background saline.

Current

Curre

nt Dominant Contrast

along Horizontal Direction

Dominant Contrastalong

Vertical Direction

(a) (b) (c)

Fig. 7 (a) Conductivity distribution of a model. Electrodes are attached along four sides of themodel. (b) and (c) are simulated Bz data subject to current injections along the horizontaland vertical directions, respectively.

• Comparing Figures 6(b) and 6(d), the second component Λ2[σ] probes thehorizontal 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 Bz data subject to the vertical current flow distinguishesthe conductivity contrast along the horizontal direction.

• The third component, Λ3[σ], is used to fix the scaling uncertainty mentionedin section 2.2.



Swine Leg Human Leg

Fig. 8 Typical examples of electrode attachment to maximize the area of parallelogram|(J1 × J2) · ez |.

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 byobserving the roles of Λ[σ]. Taking into account the uniqueness and stability, wecarefully attach two pairs of surface electrodes (which determine the two differentNeumann 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 thatis applicable to in vivo animal and human experiments. Before developing an imagereconstruction algorithm, we must take account of a possible fundamental defect inthe measured Bz data. Inside the human body, there may exist a region where theMR magnitude image value is small. Examples may include the outer layer of thebone, lungs, and gas-filled internal organs. In such a region, M ≈ 0 in (2.6), resultingin noise amplification. If the MR magnitude image M contains a Gaussian randomnoise Z, then the noise standard deviation in measured Bz data, 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 regionwhere the MR magnitude image value M is small. It would be desirable to provide ahigh-resolution conductivity image in a region having high-quality Bz data regardlessof the presence of such problematic regions. Fortunately, (2.11) and (2.12) wouldprovide a local change in lnσ regardless of the global distribution of σ if we couldpredict J1 and J2 in that local region. This is why an MREIT algorithm using (2.11)and (2.12) can provide a robust conductivity contrast reconstruction in any regionhaving Bz data 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 = 0inside any local region having a homogeneous conductivity. We first segment eachproblematic region where the MR magnitude image value M is near zero. Defininga boundary of the region, we solve ∆Bz = 0 using the measured Bz data along theboundary where noise is small. Then we replace the original noisy Bz data insidethe problematic region by the computed synthetic data. We must be careful in usingthis harmonic inpainting method since the problematic region will appear as a localhomogeneous region in a reconstructed conductivity image. When there exist multiple



small local regions with large amounts of noise, we may consider using a harmonicdecomposition denoising method [51] instead of harmonic inpainting.

3. Conductivity Image Reconstruction Algorithm.

3.1. Harmonic Bz Algorithm. 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 (r) −σ ∂u1[σ]∂x (r)

σ ∂u2 [σ]∂y (r) −σ ∂u2[σ]

∂x (r)

], r ∈ Ω.

Noting that ∆Λj [σ] = ∆Bz,j for 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 measureddata set

(Bz,1, Bz,2, V

±12

). Denoting x = (x, y) and x′ = (x′, y′), we have

Lz0 lnσ(x) = ΦΩz0[σ](x) for all (x, z0) ∈ Ωz0 ,(3.3)

where

ΦΩz0[σ](x) =

12π

∫Ωz0

x− x′

|x − x′|2 ·((A[σ](x′, z0))

−1[

∆Bz,1(x′, z0)∆Bz,2(x′, z0)

])dsx′(3.4)

and

Lz0 lnσ(x) = lnσ(x, z0) +12π

∫∂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 ∂Ωz0 and d' is the line element.From the trace formula for the double layer potential in (3.5), the identity (3.3) onthe boundary ∂Ωz0 can be expressed as

Tz0 lnσ(x) = ΦΩz0[σ](x) for all (x, z0) ∈ ∂Ωz0 ,(3.6)

where

Tz0 lnσ(x) =lnσ(x, z0)

2+

12π

∫∂Ωz0

(x − x′) · ν(x′)|x − x′|2 lnσ(x′, z0) d'x′ .

Noting that the operator Tz0 is invertible on L20(∂Ωz0) = φ ∈ L2(∂Ωz0) :∫

∂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



up to a scaling factor:

∇xyσn+1(x, z0) = 1

µ0A[σn]−1

[∆Bz,1∆Bz,2

]for (x, z0) ∈ Ωz0 ,

Lz0 lnσn+1(x) = ΦΩz0[σn+1](x) for (x, z0) ∈ Ωz0 .

(3.7)

From the first step in (3.7), we can update ∇xyσn+1 for all imaging slices of

interest within the object as long as the measured data Bz are 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|∂Ωz0

is known, so is the value of σn+1

inside Ωz0 by simple substitutions of σn+1|∂Ωz0and ∇xyσ

n+1 into the correspondingintegrals. This harmonic Bz algorithm has shown remarkable performance in variousnumerical simulations [81, 67] and imaging experiments summarized in section 4.

Early MREIT methods used all three components of the magnetic flux densityB = (Bx, By, Bz), and they required impractical rotations of the imaging object insidethe MRI scanner. The invention of the harmonic Bz algorithm using only Bz insteadof B [81] changed the problem of impractical rotations into a mathematical problem(2.14) with achievable data through application of two linearly independent Neumanndata. This harmonic Bz algorithm has been widely used in subsequent experimentalstudies 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 contrastin Ω, the direction of the vector field σ∇uj [σ] is mostly dictated by the geometryof the boundary ∂Ω and the electrode positions E±

j (or Neumann boundary data)instead of the distribution of σ. This ill-posedness was the fundamental drawback ofthe corresponding inverse problem of EIT. However, in MREIT we take advantage ofthis insensitivity of EIT. This means that the direction of the vector field σ∇uj [σ] issimilar to that of σ0∇uj [σ0] with σ0 = 1, and therefore the data Bz,1 and Bz,2 holdthe major information on the conductivity contrast. Various numerical simulationsshow that only one iteration of (3.7) may provide a conductivity image σ1 that isquite similar to the true conductivity σ. Rigorous mathematical theories regardingits convergence behavior have not yet been proven. In the paper [56] there are someconvergence results on (3.7) under a priori assumptions on the target conductivity.

3.2. Gradient Bz Decomposition and Variational Bz Algorithm. It would bebetter to minimize the amplitude of the injection current. However, the amplitudeof the signal Bz is proportional to the amplitude of the injection current. For agiven noise level of an MREIT system, this means that we have to deal with Bz datasets with a low SNR. Numerical implementation methods of an image reconstructionalgorithm affect the quality of a reconstructed conductivity image since noise in Bzdata is transformed into noise in the conductivity image. Depending on a chosenmethod, noise could be amplified or weakened.

Since two differentiations of Bz data tend to amplify its noise, the performanceof the harmonic Bz algorithm could deteriorate when the SNR in the measured Bzdata is low. To deal with this noise amplification problem, algorithms to reduce thenumber of differentiations were developed. They include the gradient Bz decompo-sition algorithm [70] and the variational gradient Bz algorithm [71], which need todifferentiate Bz only once. They show a better performance in some numerical sim-ulations, but in practical environments these algorithms were fruitless and producedsome artifacts. In this paper, we discuss only one of them for pedagogical purposes.



We briefly explain the gradientBz decomposition algorithm in a special cylindricaldomain 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) = 0in Ω with Neumann data g. We parameterize ∂D as ∂D:= (x(t), y(t)) : 0 ≤ t ≤ 1and 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 Bz decomposition algorithm is based onthe 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

∂zand Θy[u] :=

∂ψ

∂x+

∂Wz

∂y− ∂Wy

∂zin Ω

and

Υ = φ+1µ0

Bz, W (r) :=∫

Ωδ

14π|r− r′|

∂(σ∇u(r′))∂z

dr′.

Here, φ is a solution of

∇2φ = 0 in Ω,φ = g − 1

µ0Bz on ∂Ω \ z = ±δ,

∂φ∂z = − 1

µ0

∂Bz

∂z on ∂Ω ∩ z = ±δ,(3.9)

and ψ is a solution of

∇2ψ = 0 in Ω,∇ψ · τ = ∇×W · τ on ∂Ω \ z = ±δ,∂ψ∂z = −∇×W · ez 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 umj a solution of ∇ · (σm∇u) = 0 in Ω withNeumann data gj , the reconstructed σ is the limit of a sequence σm that is obtainedby the following formula:

σm+1 =

∑Ni=1

∣∣∣−(∂Υi

∂y +Θx[umi ])∂um

i

∂x +(∂Υi

∂x +Θy[umi ]) ∂um

i

∂y

∣∣∣∑N

i=1

[(∂um

i

∂x

)2

+(∂um

i

∂y

)2] .(3.11)

This method needs to differentiate Bz only once in contrast to the harmonic Bzalgorithm, where the numerical computation of∇2Bz is required. It has the advantageof much improved noise tolerance, and numerical simulations with added random noiseof a realistic quantity showed its feasibility and robustness against measurement noise.However, in practical environments it shows poorer performance compared with theharmonic Bz algorithm and may produce some artifacts.



The major reason for this is that the conductivity σm+1 updated by the iterationprocess (3.11) is influenced by the global distribution of σm. We should note thatthere always exist some local regions with defective Bz data in human or animal ex-periments, and we always deal with a truncated region of the imaging object, whichcauses geometry errors. Hence, it would be impossible to reconstruct the conductivitydistribution 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 wheremeasured Bz data are reliable. In order to achieve a stable local reconstruction of con-ductivity contrast with moderate accuracy, poor conductivity reconstruction at onelocal region should not badly influence conductivity reconstructions in other regions.This means that a conductivity image reconstruction algorithm should not dependtoo 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 derivedfrom (2.7), with the assumption H = 0, we may linearize the relationship between Bzand σ as follows [10, 11]:

∆Bz = S∆σ,(3.12)

where ∆Bz is the difference in Bz from the imaging object with homogeneous andperturbed conductivity distributions, σ0 and σ0 + ∆σ, respectively. Inverting thesensitivity matrix, one can reconstruct a conductivity image from measured Bz data.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-basedmethod 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 ofthe sensitivity-based method. They showed that conductivity values of tumor areasare increased in reconstructed conductivity images. This method cannot deal with theunknown termH, 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 anisotropyof up to 1 to 10 between the longitudinal and transversal directions. Conductivityimage reconstructions in EIT have been mostly based on the assumption of isotropicconductivity due to the limitations of EIT.

Seo et al. [80] applied the MREIT technique to anisotropic conductivity imagereconstructions. 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 foundthat at least seven different injection currents are necessary for the anisotropic con-ductivity image reconstruction algorithm. The algorithm is based on the followingtwo identities:

Us = b and ∇ ·

σ11 σ12 σ13

σ12 σ22 σ23

σ13 σ23 σ33

∇uj

= 0,(3.13)



where

b =1µ0

∇2B1,z

...∇2BN,z

, s =

−∂yσ11 + ∂xσ12

−∂yσ12 + ∂xσ22

−∂yσ13 + ∂xσ23

σ12

−σ11 + σ22

σ23

σ13

,

and

U =

u1x u1

y u1z u1

xx − u1yy u1

xy u1xz −u1

yz...

......

......

......

uNx uNy uNz uNxx − uNyy uNxy uNxz −uNyz

.

Here, uj is the voltage corresponding to the jth injection current, ujx = ∂uj

∂x , andσ is assumed to be a symmetric positive definite matrix. As in the harmonic Bzalgorithm, we may use an iterative procedure to compute s in (3.13). Assuming thatwe 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 thefollowing relation between s and σ:

∂σ11

∂x= s2 −

∂s5

∂x+

∂s4

∂yand

∂σ11

∂y= −s1 +

∂s4

∂x.(3.14)

The last component σ33 can 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 ahigh 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 theinterior 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] discussnumerous issues including uniqueness, region-of-interest reconstruction, and the noiseeffect. Assuming that Bz data subject to an injection current into the head is avail-able, Gao, Zhu, and He [22] developed a method to determine conductivity valuesof the brain, skull, and scalp layers using the radial basis function and the simplexmethod. This kind of parametric approach may find useful applications in EEG/MEGsource imaging problems. Gao, Zhu, and He [23] also suggested the so-called RSM-MREIT algorithm, where the total error between measured and calculated magneticflux densities is minimized as a function of a model conductivity distribution by usingthe response surface methodology algorithm.

4. Summary of Experimental Results.

4.1. Image Reconstruction Procedure Using Harmonic Bz Algorithm. Basedon the harmonic Bz algorithm, the Impedance Imaging Research Center (IIRC) inKorea developed an MREIT software to offer various computational tools from pre-processing to reconstruction of conductivity and current density images. Figure 10shows a screen capture of the MREIT software, CoReHA (conductivity reconstructor



-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. (a) is the targetconductivity tensor image. (b), (c), and (d) are reconstructed images when the SNR isinfinity, 300, and 150, respectively. Here, SNR means the SNR of the corresponding MRmagnitude image [80].

using harmonic algorithms) [35, 36]. It includes the three major tasks of preprocessing,model construction and data recovery, and conductivity image reconstruction.

• Preprocessing. According to (2.6), we obtain magnetic flux density imagesBz,1 and Bz,2 corresponding to two injection currents I1 and 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 quitenoisy due to many factors, we may use a PDE-based denoising method suchas harmonic decomposition.

• Model construction and data recovery. In the geometrical modeling of theconducting domain, identifications of the outermost boundary and electrodelocations are critical to impose boundary conditions. We use a semiautomatictool employing a level-set-based segmentation method. There could be aninternal region where an MR signal void occurs. In such a problematic region,measured Bz data are defective. We may use the harmonic inpainting methodto recover Bz data assuming that the local region is homogeneous in termsof the conductivity.



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, andimage reconstruction [35, 36].

• Conductivity reconstruction. We use the harmonic Bz algorithm as the de-fault algorithm for three-dimensional conductivity image reconstructions. Wemay 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 mostof the published results of conductivity phantom imaging experiments [67, 66, 68], weintroduce 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. ItsMR 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 aswell as conductivity information. Conductivity values of the tissues were measuredbeforehand 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 wasformed 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 thereconstructed conductivity image shows spurious spikes inside the region of the airbubble. Since this kind of technical problem can occur in a living body, the harmonicinpainting 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 emphasizethat pixel values in Figure 11(c) provide totally different information about electrical



Bovine Tongue Porcine Muscle

Chicken BreastAgar Gelatin

RecessedElectrode

AirBubble

140mm

140mm

(1) Chickenbreast

(2) Porcinemuscle

(3) Bovinetongue

(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

Lead Wire RF Coil

MRI Bore

B0x

y

z

(a) (b)

Fig. 12 (a) Attachment of electrodes around a chosen imaging region and (b) placement of animaging 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 clearlydistinguishes two objects, whereas they are indistinguishable in the corresponding con-ventional MR image. This happens, for example, when two objects have almost sameproton densities but significantly different numbers of mobile ions. This importantpoint will be demonstrated later.

4.3. Animal Imaging. Figure 12 shows the experimental setup for postmortemcanine brain imaging experiments. Figure 13 shows reconstructed multislice conduc-tivity images of a postmortem canine brain [42]. These high-resolution conductivityimages with a pixel size of 1.4 mm were obtained by using a 3 T MRI scanner and40 mA injection currents. Restricting the conductivity image reconstruction to withinthe brain region to avoid technical difficulties related to the skull, these conductivityimages of the intact canine brain clearly distinguish white and gray matter. Sincethe harmonic Bz algorithm cannot handle the tissue anisotropy, the concept of theequivalent isotropic conductivity should be adopted to interpret the reconstructedconductivity 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 theentire 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-hydrogelelectrode for MREIT imaging experiments. Using a pair of carbon-hydrogel electrodes



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 MRmagnitude images of a canine head are shown in the top rows and reconstructed equivalentisotropic 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 twopairs of such electrodes, they reconstructed equivalent isotropic conductivity images



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 MRmagnitude (top), conductivity (middle), and color-coded conductivity (bottom) images.

of a swine leg in Figure 15, which shows a good contrast among different musclesand bones [58]. From the reconstructed images, we can observe spurious spikes in theouter layers of bones, primarily due to the MR signal void there.

Figures 16(a) and 16(b) are MRmagnitude and reconstructed conductivity imagesof a postmortem canine abdomen [36]. Since the abdomen includes a complicatedmixture of different organs, interpretation of a reconstructed conductivity image needsfurther investigation. It was found that conductivity image contrast in the caninekidney is quite different from that of an MR magnitude image, clearly distinguishingthe 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 postmortemimage, primarily due to the reduced amplitude of injection currents, the in vivo imageshows 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 regionalbrain ischemia. As shown in Figure 18, the ischemia produced noticeable conductiv-ity changes in reconstructed images. Accumulated results of these postmortem andin vivo animal imaging experiments will guide us to properly design in vivo humanimaging 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 subjectusing a 3 T MRI scanner. They adopted thin and flexible carbon-hydrogel electrodeswith conductive adhesive for current injections [58]. Due to their large surface area



Kidney

Spinal Cord

Liver

Spleen

Stomach

IntestinesCarbon-hydrogel

Electrode

(a) (b)

Fig. 16 (a) MR magnitude image and (b) reconstructed conductivity image from a postmortemcanine abdomen [36]. The conductivity image in (b) shows a significantly different imagecontrast compared with the MR magnitude image in (a).

Right Left

Dorsal

Ventral

White MatterGray 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 wasused for both in vivo and postmortem experiments [43]. The image in (b) was obtained byusing 5 mA injection currents, whereas 40 mA was used in (d).

of 80× 60 mm2 and good contact with the skin, they could inject pulse-type currentswith amplitude of as much as 9 mA into the lower extremity without producing apainful sensation. Sequential injections of two currents in orthogonal directions wereused to produce cross-sectional equivalent isotropic conductivity images in Figure 19with 1.7 mm pixel size and 4 mm slice gap. The conductivity images well distinguisheddifferent parts of muscles and bones. The outermost fatty layer was also clearly shownin each conductivity image. We could observe excessive noise in the outer layers oftwo bones due to the MR signal void phenomenon there. Further human imagingexperiments have been planned and are being conducted to produce high-resolutionconductivity images from different parts of the human body.

5. Future Directions and Conclusion. MREIT provides conductivity images ofan electrically conducting object with a pixel size of about 1 mm. It achieves such ahigh spatial resolution by adopting an MRI scanner to measure internal magnetic fluxdensity distributions induced by externally injected imaging currents. Theoretical



ROICounter-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 theother 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, andbottom rows, respectively.

and experimental studies in MREIT demonstrate that it is expected to be a newclinically useful bioimaging modality. Its capability to distinguish conductivity valuesof different biological tissues in their living wetted states is unique.

Following the in vivo imaging experiment of the canine brain [43], numerousin 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 alsoin progress [41]. These trials are expected to accumulate new diagnostic informationbased on in vivo conductivity values of numerous biological tissues.



MREIT was developed to overcome the ill-posed nature of the inverse problem inEIT and provide high-resolution conductivity images. Even though current EIT im-ages have a relatively poor spatial resolution, high temporal resolution and portabilityin EIT could be advantageous in several biomedical application areas [30]. Instead ofcompeting in a certain application area, MREIT and EIT will be complementary toeach other. Taking advantage of the high spatial resolution in MREIT, Woo and Seodiscussed numerous application areas of MREIT in biomedicine, biology, chemistry,and material science [91]. We should note that it is possible to produce a currentdensity image for any electrode configuration once the conductivity distribution isobtained.

Future studies should overcome a few technical barriers to advance the methodto the stage of routine clinical use. The biggest hurdle at present is the amount ofinjection current that may stimulate muscle and nerve. Reducing the injection currentdown to a level that does not produce undesirable side effects is the key to the successof this new bioimaging modality. This demands innovative data processing methodsbased on rigorous mathematical analysis as well as improved measurement techniquesto maximize SNRs for a given data collection time.

Acknowledgment. The authors thank collaborators at the Impedance ImagingResearch Center for their invaluable contributions.

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