- Magnetic resonance electrical impedance. tomography (MREIT): conductivity and current density imaging. Journal of Physics: Conference Series

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1 Journal of Physics: Conference Series Magnetic resonance electrical impedance tomography (MREIT): conductivity and current density imaging To cite this article: Jin Keun Seo et al 2005 J. Phys.: Conf. Ser Related content - Magnetic resonance electrical impedance tomography for determining electric field distribution during electroporation Matej Kranjc, Franci Bajd, Igor Serša et al. - Applications of Electrical Impedance Tomography (EIT): A Short Review Tushar Kanti Bera - A Rotative Electrical Impedance Tomography Reconstruction System Fang-Ming Yu, Chen-Ning Huang, Fang- Wei Chang et al. View the article online for updates and enhancements. Recent citations - bssfp phase correction and its use in magnetic resonance electrical properties tomography Safa Ozdemir and Yusuf Ziya Ider - Toward a magnetic resonance electrical impedance tomography in ultra-low field: A direct magnetic resonance imaging method by an external alternating current Seong-Joo Lee et al - Bruno M. G. Rosa and Guang Z. Yang This content was downloaded from IP address on 30/01/2019 at 14:58

2 Institute of Physics Publishing Journal of Physics: Conference Series 12 (2005) doi: / /12/1/014 Second International Conference on Inverse Problems Magnetic resonance electrical impedance tomography (MREIT): conductivity and current density imaging JinKeunSeo 1,OhinKwon 1 and Eung Je Woo 3,4 1 Department of Mathematics, Yonsei University, Korea 2 Department of Mathematics, Konkuk University, Korea 3 College of Electronics and Information, Kyung Hee University, Korea 4 ejwoo@khu.ac.kr Abstract. This paper reviews the latest impedance imaging technique called Magnetic Resonance Electrical Impedance Tomography (MREIT) providing information on electrical conductivity and current density distributions inside an electrically conducting domain such as the human body. The motivation for this research is explained by discussing conductivity changes related with physiological and pathological events, electromagnetic source imaging and electromagnetic stimulations. We briefly summarize the related technique of Electrical Impedance Tomography (EIT) that deals with cross-sectional image reconstructions of conductivity distributions from boundary measurements of current-voltage data. Noting that EIT suffers from the ill-posed nature of the corresponding inverse problem, we introduce MREIT as a new conductivity imaging modality providing images with better spatial resolution and accuracy. MREIT utilizes internal information on the induced magnetic field in addition to the boundary current-voltage measurements to produce three-dimensional images of conductivity and current density distributions. Mathematical theory, algorithms, and experimental methods of current MREIT research are described. With numerous potential applications in mind, future research directions in MREIT are proposed. 1. Introduction The electrical properties (conductivity and permittivity) of a biological tissue change with ion concentrations in extra- and intra-cellular fluids, cellular structure and density, molecular composition, membrane characteristics and other factors (Grimnes and Martinsen 2000). Consequently, they reflect structural, functional and pathological conditions of the tissue and can provide valuable diagnostic information. In this paper, we focus on conductivity since it is an important physical index that can indicate conditions of tissues or organs at relatively low excitation frequencies. Since the early 1980s, there have been significant efforts to produce cross-sectional images of a conductivity distribution inside the human body using boundary measurements of current-voltage data and this technique is being called Electrical Impedance Tomography (EIT) (Barber and Brown 1984; Webster 1990; Metherall et al 1996; Boone et al 1997; Cheney et al 1999; Saulnier et al 2001; Holder 2005). In EIT, we inject current into the human body and measure the induced boundary voltage data. Boundary measurements of current-voltage data also known as the Neumann-to-Dirichlet data are utilized to produce cross-sectional conductivity images of the subject. Theoretical aspects of EIT are described by Kohn and Vogelius (1984), Sylvester and Uhlmann (1986 and 2005 IOP Publishing Ltd 140

3 ), Nachman (1988), Isakov (1988), Friedman and Isakov (1989), Alessandrini et al (1995), Nachman (1996) and Uhlmann (1999). When the number of surface electrodes is E, it is well known that there are E (E 1)/2 independent voltage measurements. Since this limits the amount of information available for conductivity image reconstructions, theoretical results in EIT suggest using a large number of electrodes to improve spatial resolution. However, in practice, only a limited number of electrodes can be attached and the amount of boundary information available is restricted. Using a large number of electrodes results in a more complicated instrument. This, and the cumbersome practical problem of attaching many electrodes make this approach prone to bigger measurement errors. When we inject current into a subject, the internal current density distribution is affected nonlinearly by the overall conductivity distribution of the subject. Any change in the conductivity of an internal region alters the current pathway and its effect is reflected in a corresponding change of the boundary voltage. However, these boundary measurements are very insensitive to a local change away from measuring points. For this reason, EIT suffers greatly from the ill-posedness of the corresponding inverse problem. This makes it difficult to reconstruct accurate conductivity images with a high spatial resolution in clinical environments where modeling and measurement errors are unavoidable. With these technical restrictions, it is desirable for EIT to find clinical applications where its portability and high temporal resolution to monitor changes in electrical properties are significant merits even with its poor spatial resolution. There are numerous studies of applying EIT for monitoring physiological events based on the changes of conductivity and permittivity (Holder 2005). On the other hand, there have been strong needs for a new imaging technique capable of providing conductivity images with sufficiently high spatial resolution and accuracy. In the following section, we will discuss why we are interested in high-resolution conductivity and current density imaging beyond the practically achievable performance of EIT techniques. In order to fulfil these needs, Magnetic Resonance Electrical Impedance Tomography (MREIT) has been proposed recently. When we inject current into a subject, it produces distributions of voltage, current density and also magnetic flux density inside the subject. The basic idea of MREIT is to utilize the information on magnetic as well as electric field induced by the injection current. While EIT is limited by the boundary measurements of current-voltage data, MREIT utilizes the internal magnetic flux density data obtained using a Magnetic Resonance Imaging (MRI) scanner. Since the late 1990s, imaging techniques in MREIT have been advanced rapidly and now are at the stage of animal experiments. In this paper, we review these techniques and propose future directions of MREIT research for clinical applications. Before we describe technical details of MREIT, we will briefly review EIT in order to provide the rationale of pursuing MREIT research that requires an expensive MRI scanner. Then, MREIT techniques will be reviewed starting with its mathematical formulation. The formulation will be described with the technical limitations in measuring the induced magnetic flux density using an MRI scanner. Recent progress in MREIT providing conductivity and current density images with a high spatial resolution and accuracy will be presented including image reconstruction algorithms and experimental techniques. 2. Motivation 2.1. A view on bio-electricity and bio-magnetism Our human body is an electrically conducting domain with an anisotropic conductivity distribution determined by the electrical property of numerous biological tissues and organs. This kind of domain is often called a volume conductor. Inside the volume conductor, we have electrical signal sources that can be modelled as current dipoles. The origin of these signals are excitable cells such as nerves and muscles. Since there are numerous nerves and muscles inside the human body, we can consider a distribution of current dipoles. The current dipole

4 (a) (b) (c) (d) (e) (f) Figure 1. Electrically conducting domain with a current dipole (a) at the center and homogeneous conductivity distribution, (c) at the offset position and homogeneous conductivity distribution and (e) at the offset position and inhomogeneous conductivity distribution. (b), (d) and (f) are plots of current streamlines (white) and equipotential lines (black) for the model in (a), (c) and (e), respectively. distribution changes as physiological status of organs or parts of organs including excitable cells. We denote the domain of the volume conductor as Ω R 3 with its boundary Ω. The time-varying current dipole distribution is expressed as f(r; t) where r is a position vector and t is time. Assuming that the time-varying anisotropic conductivity distribution is σ(r; t) = ( ) σ11 σ 12 σ 13 σ 12 σ 22 σ 23, the voltage u(r,t) produced in Ω is a solution of the following Neumann σ 13 σ 23 σ 33 boundary value problem: { [ σ(r; t) u(r; t) ] = f(r; t) inω σ(r; t) u(r; t) n =0on Ω (1) where n is the outward unit normal vector on Ω. The Neumann data is zero on Ω since the air is regarded as an insulator. Setting a reference voltage u(r 0 ; t) =0forr 0 Ω, we obtain a unique solution u of (1). The current density J in Ω is given by J(r; t) = σ(r; t) u(r; t) in Ω. (2) The current density distribution generates a magnetic field and the magnetic flux density B in R 3 is determined by the Biot-Savart law as B(r; t) = µ 0 J(r ; t) r r 4π r r 3 dv (3) Ω where µ 0 is the magnetic permeability of the free space. From the Ampere law, the current density J is also given by J(r; t) = 1 B(r; t) in Ω. (4) µ 0 Figure 1 shows an extremely simplified model of a volume conductor at a certain instant. We can see that the location of the current dipole and also the conductivity distribution significantly alter the distributions of the current density and voltage. As f and σ in (1) change with time, the equipotential lines vary. If we attach a pair of electrodes on the boundary Ω as shown in figure 2(a), we can record a bio-potential signal providing a partial information on f and σ. Figure 2(b) shows a situation of measuring the generated magnetic flux density signal outside the human head using a very sensitive magnetic sensor called SQUID (superconducting quantum interference device). This bio-magnetic signal also provides an incomplete information on f and σ.

5 143 Electrode SQUID Amplifier ECG MEG (a) (b) Figure 2. Mechanism of bio-potential and bio-magnetic signal generation. (a) Time-varying signal source and conductivity distribution make equipotential lines change with time. The voltage difference between two surface electrodes produces a bio-potential signal. (b) Timevarying signal source and conductivity distribution produce changes in magnetic flux density and a high-sensitive SQUID detects these changes. The primary reason for measuring these kinds of bio-electric and bio-magnetic signals is to extract useful diagnostic information about the status of tissues or organs. Since these biosignals are non-invasively measurable at the surface or vicinity of the human body, they have been extensively used in medicine. However, if there comes a new non-invasive technology providing more direct information on the signal source f and the conductivity distribution σ, there will be numerous biomedical applications Electromagnetic source imaging Neural activity mapping has been actively studied in Electromagnetic Source Imaging (ESI). Excitable tissues such as nerve and muscle produce endogenous current flows along with the generation of action potentials. The current flows produce electromagnetic fields and these can be measured on the surface or outside the human body by using, for example, ECG/MCG for the heart and EEG/MEG for the brain. These signals are often used to map the distribution of the endogenous current sources through an inversion process in ESI (Phillips et al 1997; Mosher et al 1999). Neural activities are the sources for these electric and magnetic signals and they are modelled as current dipoles. Since a solution of the inverse problem is affected by the conductivity distribution, conductivity images will improve the accuracy to find a map of a distribution of the current dipoles It has been also known that conductivity changes are associated with neural activity because of ion migration (Cole and Curtis 1939; Cole 1949). Cortical impedance changes have been observed with an increase of regional cerebral blood volume due to neural activity (Rank 1963; Geddes and Baker 1967). Similar changes due to functional activity and epilepsy have been also observed in cats and rabbits, and recently in humans (Adey et al 1962; Van-Harreveld and Schade 1962; Aladjolova 1964; Tidswell et al 2001). Therefore, visualizing the associated conductivity changes could be a new way of mapping neural activity. In order for a conductivity image itself to be a way of expressing a map of neural activity, the image should be provided with a high spatial resolution and accuracy Design and evaluation of treatment methods using electromagnetic stimulations There are numerous techniques of applying electric or magnetic energy into the human body for therapeutic purposes. Examples include cardiac defibrillation, functional electrical stimulation, magnetic stimulation, radio-frequency ablation and others. In these cases, knowledge of tissue conductivity is necessary to determine deposition of electromagnetic energy. In general, we need to focus current density around a local region of interest to maximize the effects of the treatment.

6 144 With a known conductivity distribution, one can easily determine the internal current density distribution subject to an external excitation. This will enable us to optimize the treatment methods including electrode and coil design and configuration Conductivity-based diagnosis It is well known that tumor tissues have higher conductivity values than normal tissues. Jossinet and Schmitt (1999) and Silva et al (2000) classified three normal and three pathological tissues in the breast based on conductivity and permittivity measurements. Their results suggest that multi-frequency measurements of electrical conductivity and permittivity are promising for the diagnosis of cancerous lesions inside the human breast. Haemmerich et al (2003) measured in vivo hepatic tumor and normal tissue conductivity from rats at different tumor stages. They found that the tumor tissue has a conductivity significantly higher than that of normal tissue especially at the frequency below 1 KHz. They suggested that necrosis within the tumor and the associated membrane breakdown is likely responsible for the observed change in conductivity. Similar experimental findings suggest that conductivity imaging techniques could be potentially useful for early diagnosis of tumors. However, in order to visualize any tumor at its early stage, the reconstructed conductivity image must be accurate and its spatial resolution should be at least comparable to that of MRI or others. 3. Basic idea of MREIT For the static or absolute conductivity imaging, MREIT has been lately proposed to overcome the technical limitations of EIT (Zhang 1992; Woo et al 1994; Ider and Birgul 1998; Eyuboglu et al 2001; Kwon et al 2002a; Oh et al 2003; Birgul et al 2003; Oh et al 2004). Injected current in an electrically conducting subject produces a magnetic field as well as an electric field. Noting that the magnetic field inside the subject can be measured by a non-contact method using an MRI scanner, we may transform the ill-posed problem in EIT into a well-posed one utilizing this additional information. Since late 1980s, measurements of the internal magnetic flux density due to an injection current have been studied in Magnetic Resonance Current Density Imaging (MRCDI) to visualize the internal current density distribution (Joy et al 1989; Scott et al 1991; Scott et al 1992). This requires an MRI scanner as a tool to capture internal magnetic flux density images. Once we obtain the magnetic flux density B =(B x,b y,b z ) due to an injection current I, wecan produce an image of the corresponding internal current density distribution J from the Ampere law J = B/µ 0. The basic concept of MREIT was proposed by combining EIT and MRCDI techniques. In MREIT, we measure the induced magnetic flux density B inside a subject due to an injection current I using an MRI scanner. Then, we may compute the internal current density J as is done in MRCDI. From B and/or J, we can perceive the internal current pathways and produce better conductivity images. However, if we try to utilize J = B/µ 0 by measuring all three components of B, there occurs a technical problem. Since the MRI scanner measures only one component of B that is parallel to the direction of the main magnetic field of the scanner, measuring all three components of B =(B x,b y,b z ) requires subject rotations. These subject rotations are impractical and also cause other problems such as misalignments of pixels and movements of internal organs. Therefore, it is highly desirable to reconstruct conductivity images from only B z instead of B where z is the direction of the main magnetic field. For this reason, most recent MREIT techniques focus on analyzing the information embedded in the measured B z data to extract any constructive relations between B z and the conductivity distribution to be imaged.

7 145 E E-1... E k E I j Ω E i L I j (σ, u, J, B) L i E l... E E Figure 3. Electrically conducting subject Ω with a conductivity σ and voltage u. Surface electrodes E j,j =1,,E are attached on the boundary Ω. Here, we assume that current is injected between the diagonal pair of electrodes E i and E j. 4. Mathematical formulation of MREIT 4.1. Forward problem: magnetic flux density due to an injection current for a given conductivity As shown in figure 3, we assume an electrically conducting domain Ω with its boundary Ω and an anisotropic conductivity distribution σ = σ 11 σ 12 σ 13 σ 12 σ 22 σ 23 σ 13 σ 23 σ 33 where σ is a positive-definite symmetric matrix. We choose a pair of electrodes attached on Ω, for example, E i and E j to inject current I. Lead wires carrying the injection current I are denoted as L i and L j. Then, the voltage u in Ω satisfies the following Neumann boundary value problem: { ] [ σ(r) u(r) =0inΩ (5) σ u n = g on Ω where g is the Neumann boundary data due to the injection current. Knowing the voltage distribution u, the current density J is given by J(r) = σ(r) u(r) in Ω. (6) We now consider the magnetic field produced by the injection current. The induced magnetic flux density B in Ω can be decomposed into three parts as B(r) =B Ω (r)+b E (r)+b L (r) in Ω (7) where B Ω, B E and B L are magnetic flux densities due to J in Ω, J in E = E i E j and I in L = L i L j, respectively. From the Biot-Savart law, B Ω (r) = µ 0 J(r ) r r 4π Ω r r 3 dv, (8) and B E (r) = µ 0 4π B L (r) = µ 0I 4π E L J(r ) r r r r 3 dv (9) a(r ) r r r r 3 dl (10)

8 146 where a(r ) is the unit vector in the direction of the current flow at r L. From the Ampere law, the current density J is also given by J(r) = 1 µ 0 B(r) in Ω. (11) We must have 1 B(r) = σ(r) u(r) and J(r) =0 inω. (12) µ 0 Numerical techniques to solve (5)-(11) are described by Lee et al (2003a) Inverse problem: conductivity from Neumann-to-B z (NtB z )map The problem of interest is to reconstruct an image of σ in Ω from a measured magnetic flux density and boundary voltage. For the uniqueness of a reconstructed isotropic conductivity image, it has been shown that we need to inject at least two currents using more than three electrodes and measure the corresponding magnetic flux densities (Kim et al 2003; Ider et al 2003). In addition, at least one boundary voltage measurement is needed to recover the absolute values of the isotropic conductivity distribution. In this section, we assume that we measure only B z without rotating the subject. The description of the inverse problem in MREIT is based on the following setup. We place a subject Ω inside an MRI scanner and attach surface electrodes. When the number of electrodes is E, we can sequentially select one of N E (E 1)/2 different pairs of electrodes to inject currents into the subject. Let the injection current between the j-th pair of electrodes be I j for j =1,,N with N 2. The current I j produces a current density J j =(Jx,J j y,j j z j )inside the subject. The presence of the internal current density J j and the current I j in external lead wires generates a magnetic flux density B j =(Bx,B j y,b j z)andj j j = B j /µ 0 holds inside the electrically conducting subject. We now assume that we have measured Bz j for j =1,,N. Let u j be the voltage due to the injection current I j for j =1,,N.Sinceσis approximately independent of injection currents, each u j is a solution of the following Neumann boundary value problem: { ( σ(r) uj (r) ) =0 in Ω (13) σ u j n = g j on Ω where g j is the normal component of the current density on Ω for the injection current I j. If σ, I j and electrode configuration are given, we can solve (13) for u j using a numerical method such as FEM (Polydorides and Lionheart 2002; Lee et al 2003a). Now, we introduce a map relating Bz j with the Neumann data g j : Λ σ [g j ](r) =B j z(r), r Ω. We call this map Λ σ by the Neumann-to-B z map (NtB z -map). According to the Biot- Savart law with a given g j,λ σ [g j ] is expressed as Λ σ [g j ](r) = µ 0 σ(r )[(x x ) u j (r ) (y y ) u j x (r )] 4π Ω r r 3 dr (14) where u j is the solution of (13). The inverse problem in MREIT is to reconstruct σ from several NtB z data, Λ σ [g j ],j =1,,N. In order for MREIT to be more practical, N should not be a large number.

9 MREIT image reconstruction techniques When B =(B x,b y,b z ) is available, we may use J from (11) to reconstruct conductivity images using image reconstruction algorithms such as the J-substitution algorithm (Kwon et al 2002a; Khang et al 2002; Lee et al 2003b), current constrained voltage scaled reconstruction (CCVSR) algorithm (Birgul et al 2003) and equipotential line methods (Ider et al 2003; Kwon et al 2002b; Lee 2004). However, since these methods require the impractical subject rotations, we describe algorithms utilizing only B z data Harmonic B z algorithm Assuming the conductivity distribution is isotropic, the harmonic B z algorithm was developed as the first method to produce three-dimensional conductivity images from multi-slice measurements of B z subject to at least two injection currents (Seo et al 2003a and 2003b; Oh et al 2003). Based on the relation of 2 B = µ 0 u σ observed by Scott et al (Scott et al 1991), Seo et al (2003a) derived the following expression that holds for each position in Ω. ( 1 σ 2 Bz j = µ 0 x, σ ) ( ) uj, u j = σ u j x x σ u j, j =1,,N. (15) x Note that the magnetic flux density due to the injection current I j along external lead wires becomes irrelevant by using 2 B j z. Using a matrix form, (15) becomes where U = u 1 u 1 x, s = [ σ x σ Us = b (16) ],andb = 1 µ 0 2 B 1 z... u N u N x 2 Bz N For the case where two injection currents are used (N = 2), we can obtain s provided that two voltages u 1 and u 2 corresponding to two injection currents I 1 and I 2 satisfy u 1 u 2 x + u 1 u 2 x. 0. (17) We can argue that (17) holds for almost all positions within the subject since two current densities J 1 and J 2 due to appropriately chosen I 1 and I 2 will not have the same direction (Kim et al 2002; Kim et al 2003; Ider et al 2003). We use N injection currents to better handle measurement noise in B z and improve the condition number of U T U where U T is the transpose of U. Using the weighted regularized least square method suggested by Oh et al (2003), we can get s as s = (ŨT Ũ + λi) 1 ŨT b (18) where λ is a positive regularization parameter, I is the 2 2 identity matrix, Ũ = WU, b = Wb and W = diag(w 1,,w N )isann N diagonal weight matrix. Oh et al (2003) discussed different ways of determining the values of λ and w j. Computing (18) for each position or pixel, we obtain a distribution of s = [ σ x σ ] T inside the subject. We now tentatively assume that the imaging slice S is lying in the plane {z =0} and the conductivity value at a fixed position r 0 =(x 0,y 0, 0) on its boundary S is 1. For a moment, we denote r =(x, y), r =(x,y )andσ(x, y, 0) = σ(r). In order to compute σ from σ =( σ x, σ ),

10 148 Oh et al (2003) suggested a layer potential technique having a denoising effect. The σ on the fixed plane z = 0 can be expressed as σ(r) = r Ψ ( r r ) σ(r )dr + n r r Ψ(r r ) σ(r )dl r (19) S S where Ψ(r r )= 1 2π log r r and σ denotes the conductivity restricted at the boundary S. Moreover, σ satisfies σ(r) (r r ) n r 2π S r r 2 σ(r )dl r = 1 (r r ) σ(r ) 2π S r r 2 dr. (20) It is well known that the solvability of the integral equation (20) for σ is guaranteed for a given right side of (20) (Folland 1976). Since σ is known in S, so does the right side of (20). This enables us to obtain σ by solving the integral equation (20). Now, we can compute the conductivity σ in S by substituting the boundary conductivity σ into (19). The process of solving (18) for each pixel and (20) for each imaging slice can be repeated for all imaging slices of interest within the subject as long as the measured data B z are available for the slices. As expressed in (13), voltages u j depend on the unknown true isotropic conductivity σ and, therefore, we do not know the matrix U corresponding to σ. This requires us to use the iterative algorithm described below. For j =1,,N, we sequentially inject current I j through a chosen pair of electrodes and measure the z-component of the induced magnetic flux density Bz. j For each I j, we also measure the boundary voltage u j S on electrodes not injecting I j. Then, the harmonic B z algorithm is as follows. 1. Let n = 0 and assume an initial conductivity distribution σ Compute u n j by solving the following Neumann boundary value problems for j =1,,N: { ( ) σ n u n j =0 inω σ n u n j n = g (21) j on Ω. 3. Compute σ n+1 using (18), (19) and (20). Scale σ n+1 using the measured boundary voltages u j S and the corresponding computed ones u n j S. 4. If σ n+1 σ n 2 <ɛ, go to Step 5. Here, ɛ is a given tolerance. Otherwise, set n (n +1) andgotostep2. 5. If needed, compute current density images as J j σ n+1 u M j where u M j is a solution of the boundary value problem in (13) with σ n+1 replacing σ Gradient B z decomposition algorithm After the introduction of the harmonic B z algorithm, there has been an effort to improve its performance especially in terms of the way we numerically differentiate the measured noisy B z data. Based on a novel analysis utilizing the Helmhortz decomposition, Park et al (2004a) suggested the gradient B z decomposition algorithm. Here, we also assume that the conductivity is isotropic. In order to explain the algorithm, let us assume Ω = D [ δ, δ] ={r =(x, y, z) (x, y) D, δ < z < δ} is an electrically conducting subject where D is a two-dimensional smooth simply connected domain. Let u be the solution of the Neumann boundary value problem (5) or (13) with the Neumann data g. We parameterize D as D:= {(x(t),y(t)) : 0 t 1} and define g(x(t),y(t),z):= t 0 g((x(t),y(t),z)) x (t) 2 + y (t) 2 dt for (x, y, z) D ( δ, δ).

11 149 The gradient B z decomposition algorithm is based on the following key identity: ( ) H +Λ u x[u] x + ( H x +Λ y[u] ) u σ = ( u x )2 +( u in Ω (22) )2 where Λ x [u] := ψ W z x + W x and Λ y [u] := ψ z x + W z W y in Ω z and H = φ + 1 µ 0 B z, W(r) := Here, φ and ψ are solutions of the following equations: 2 φ =0 inω φ = g 1 µ 0 B z on Ω side and φ z = 1 B z µ 0 z on Ω tb 1 Ω δ 4π r r 2 ψ =0 inω ψ τ = W τ ψ z = W ɛ z (σ u(r )) dr. z on Ω side on Ω tb where ɛ z =(0, 0, 1), Ω side = D ( δ, δ), Ω tb is the top and bottom surfaces of Ω, and τ := ( n y,n x, 0) is the tangent vector on the lateral boundary D ( δ, δ). Since the term u in (22) is a highly nonlinear function of σ, the identity (22) can be viewed as an implicit reconstruction formula for σ. It should be noticed that we can not identify σ with a single g using (22). Hence, we may use an iterative reconstruction scheme with multiple Neumann data g j,j =1,,N to find σ. Let u m j be the solution of (13) with σ = σ m and g j. Then, the reconstructed σ is the limit of a sequence σ m that is obtained by the following formula: ( ) ( ) N i=1 Hi +Λ x[u m i ] u m i x + Hi x +Λ y[u m i ] u m i σ m+1 = [ ( ) N u m 2 ( ) ] i u m 2. i=1 x + i 5.3. Anisotropic conductivity image reconstruction algorithm In the previous two sections, we assumed an isotropic conductivity distribution to simplify the underlying mathematical theory of the image reconstruction problem. However, most biological tissues are known to have anisotropic conductivity values. The ratio of the anisotropy depends on the type of tissue and the human skeletal muscle, for example, shows an anisotropy of up to one to ten between the longitudinal and transversal direction. Hence, clinical applications of MREIT require us to develop an anisotropic conductivity image reconstruction method. Lately, a new algorithm that can handle anisotropic conductivity distributions has been suggested by Seo et al (2004). This algorithm requires at least seven different currents, while the previous two algorithms for isotropic cases require at least two different currents. The reconstruction algorithm begins with 1 µ 0 2 B z = y J x x J y to get where b = 1 2 Bz 1 µ. 0 2 Bz N Us = b (23), s = y σ 11 + x σ 12 y σ 12 + x σ 22 y σ 13 + x σ 23 σ 12 σ 11 + σ 22 σ 23 σ 13

12 150 and u 1 x u 1 y u 1 z u 1 xx u 1 yy u 1 xy u 1 xz u 1 yz U = u N x u N y u N z u N xx u N yy u N xy u N xz u N yz We do not know the true σ and therefore the matrix U is unknown. This requires us to use an iterative procedure to compute s in (23). For now, let us assume that we have computed all seven terms of s. From s, we can immediately determine σ 12 (r) =s 4 (r), σ 13 (r) =s 7 (r) and σ 23 (r) =s 6 (r). To determine σ 11 and σ 22 from s, we use the relation between s and σ; σ 11 x = s 2 s 5 x + s 4 and Using the above relation, we can derive the following expression of σ 11 : σ 11 (a, b, z) = Ω z [( s 2 + s 5 x s 4 σ 11 = s 1 + s 4 x. (24) ) x Ψ(x a, y b) ] dxdy + ) ] Ω z [(s 1 s 4 x Ψ(x a, y b) dxdy + (25) Ω z σ 11 (x, y, z)[ñ x,y Ψ(x a, y b)] dl x,y where Ω z = {(x, y) :(x, y, z) Ω}, Ψ(x, y) = 1 2π log x 2 + y 2 and σ 11 is σ 11 restricted at the boundary Ω z. We can compute σ 11 on Ω z by solving the corresponding singular integral and therefore we obtain the representation of the internal σ 11. Similarly, we can have the representation for the component σ 22. The last component σ 33 whose information is missing in (23) can be obtained by using he physical law J =0. 6. Measurement techniques in MREIT Let z be the coordinate that is parallel to the direction of the main magnetic field B 0 of an MRI scanner. Using a constant current source and a pair of surface electrodes, we sequentially inject two current pulses of I ± and I synchronized with a standard spin echo or gradient echo pulse sequence. The application of the injection current during MR imaging induces a magnetic flux density B =(B x,b y,b z ). Since the magnetic flux density B produces inhomogeneity of the main magnetic field changing B 0 to (B 0 + B), it causes phase changes that are proportional to the z-component of B, that is B z. Then, the corresponding MRI signals are S I± (m, n) = M(x, y)e jδ(x,y) e jγbz(x,y)tc e j(xm kx+yn ky) dxdy (26) and S I (m, n) = M(x, y)e jδ(x,y) e jγbz(x,y)tc e j(xm kx+yn ky) dxdy. (27) Here, M is the transverse magnetization, δ is any systematic phase error, γ = rad/tesla s is the gyromagnetic ratio of the hydrogen, T c is the duration of current pulses. Two-dimensional discrete Fourier transformations of S I± (m, n) and S I (m, n) result in complex images of M ± c (x, y) andm c (x, y). Dividing them, we get ( ) M ± Arg c (x, y) ) M =Arg (e j2γbz(x,y)tc = Φ z (x, y) c (x, y) where Arg(ω) is the principal value of the argument of the complex number ω. Since Φ z is wrapped in π < Φ z π, wemustunwrap Φ z to obtain Φ z (Ghiglia and Pritt 1998). Finally, we get B z (x, y) = 1 Φ z (x, y). (28) 2γT c

13 151 (a) (b) (c) (d) (e) Figure 4. MREIT experiment using a simple saline phantom with an agar object. (a) Saline phantom with a size of the human head. It is equipped with four recessed electrodes for current injections. (b) and (c) are measured magnetic flux density images of the phantom at the middle imaging slice for the horizontal and vertical injection current, respectively. (d) Reconstructed conductivity image at the middle imaging slice and (e) corresponding MR magnitude image. Details of the measurement techniques can be found in experimental works (Scott et al 1991 and 1992; Khang et al 2002; Lee et al 2003b; Oh et al 2003 and 2004). 7. Progress in MREIT images Since MREIT is still at its early stage of development, most of the published MREIT conductivity images are from numerical simulations and phantom experiments. In this section, we describe experimental results obtained by using an MREIT system based on a 3 Tesla MRI scanner. Figure 4(a) shows a cylindrical saline phantom with a size of the human head. It was filled with an electrolyte solution with 2 S/m conductivity. At the center of the phantom, an agar object with 0.5 S/m conductivity was placed. Figure 4(b) and (c) are measured images of B z at the middle imaging slice of the phantom for the horizontal and vertical injection current of ma ms, respectively. Figure 4(d) is the reconstructed conductivity image and (e) is the corresponding MR magnitude image at the same imaging slice. In order to verify the spatial resolution and accuracy of the reconstructed conductivity image, Oh et al (2004) constructed a resolution phantom of which design is shown in figure 5(a). Inside the phantom, six cotton threads with diameters of 2, 3 and 4 mm are vertically placed. With the background solution of 0.63 S/m, threads had a different conductivity value than the solution. They also put two wedge-shaped sponges facing each other. Since the sponges had different densities, the conductivity of each sponge was also different from the solution. Figure 5(b) is an MR magnitude image of the phantom at the middle imaging slice. Figure 5(c), (d) and (e) are reconstructed conductivity images at three imaging slices with different vertical positions. We can observe that six threads are clearly distinguished suggesting the spatial resolution of about 2 mm with injection currents of a sufficient amount. Depending on the imaging slice, areas of two rectangular sponge regions change and this indicates that the three-dimensional conductivity distribution with vertical variations could be reconstructed. Lately, Woo et al (2004) used a 3 Tesla MREIT system to produce conductivity images of a biological tissue phantom shown in figure 6. They injected current pulses of ma ms. Measured magnetic flux density images are shown in figure 7(a) and (b). Figure 7(c) is the MR magnitude image at the middle imaging slice and (d) is the corresponding reconstructed conductivity image. The pixel size is mm 2 and there are pixels in the reconstructed conductivity image. They found that the reconstructed conductivity values of the image in figure 7(d) are very close to the measured ones using an impedance analyzer after the experiment. This result demonstrates the feasibility of the MREIT technique in producing conductivity images of different biological tissues with a high spatial resolution and accuracy as long as we use a sufficient amount of injection current.

14 152 (a) (b) (c) (d) (e) Figure 5. MREIT experiment using a resolution phantom. (a) Phantom design including six cotton threads with 2, 3 and 4 mm diameters and two wedge-shaped sponges facing each other. (b) MR magnitude image of the phantom at the middle imaging slice. (c), (d) and (e) are reconstructed conductivity images of the phantom at three different imaging slices with different vertical positions. (b) (a) Chicken breast (b) Porcine muscle (c) Bovine tongue Figure 6. Biological tissue phantom. Bovine Tongue Porcine Muscle Recessed Electrode Air Bubble Agar Gelatin Chicken Breast (a) (b) (c) (d) Figure 7. Magnetic flux density images of the tissue phantom in figure 6 at the middle imaging slice for (a) horizontal and (b) vertical injection current. (c) MR magnitude image and (d) reconstructed conductivity image at the middle imaging slice. 8. Future directions and conclusions Struggling to overcome the ill-posed nature of the inverse problem in EIT, numerous techniques have been suggested. Even though current EIT images have a relatively poor spatial resolution, the high temporal resolution and portability could be a big advantage in numerous biomedical application areas. Three-dimensional dynamic EIT imaging with a wireless miniaturized EIT system is believed to make the next breakthrough in EIT technology. Obtaining and utilizing the accurate data for the shape and size of a subject with electrode positions, static EIT imaging could be improved. However, since the ill-posedness in EIT still remains anyway, we should not expect EIT to compete with other medical imaging modalities such as MRI and X-ray CT in terms of spatial resolution. The significance of EIT should be emphasized based on the fact that it provides the information on electrical properties of biological tissues. Since this kind of information is not available from any other imaging modality, EIT should keep finding unique

15 153 application areas especially in dynamic functional imaging. Based on the frequency dependent characteristics of tissue conductivity and permittivity, multi-frequency three-dimensional EIT imaging is also quite promising. Further improvements in the conductivity image quality are needed for many other clinical applications not covered by EIT. MREIT has been suggested for this goal and recently there have been great advances in developing MREIT techniques including theory, algorithms and experimental methods. Experimental studies in MREIT showed that much improved contrast resolution can be achievable. However, these successful outcomes in the engineering point of view have not been supported yet by rigorous mathematical theories such as uniqueness, stability and convergence of the numerical algorithms. Theoretical as well as experimental studies in MREIT are required for the progress of the technique until it becomes clinically useful. Clinical applications of MREIT have not been tried yet since it is still in its early stage of development. Once we could reconstruct cross-sectional conductivity and current density images with improved spatial resolution and accuracy, MREIT will find numerous clinical applications as described in the section on the motivation of the research. These include most clinical application areas of EIT where static or absolute values of conductivity and current density are needed. However, the temporal resolution of MREIT is expected to be much worse than that of EIT and MREIT lacks the portability. Therefore, MREIT will never replace EIT in application areas where monitoring of fast physiological events is requested. For this kind of applications, MREIT may provide static conductivity images to be utilized as aprioriinformation in subsequent dynamic EIT image reconstructions With many possible clinical and biological applications in mind, future research direction in MREIT should follow the way to reduce the amount of the injection current. Anisotropic conductivity imaging is believed to be pursuable in MREIT. MREIT should always include the current density imaging to provide more information from the same measured data. The performance of the MRI system itself has been greatly enhanced to make 3 Tesla systems available to clinical settings. The progress in MREIT techniques will follow this trend. Acknowledgments This work was supported by grant R from Korea Science and Engineering Foundation. References [1] Adey W, Kado R and Didio J 1962 Impedance measurements in brain tissue of animals using microvolt signals Exp. Neruol [2] Aladjolova N A 1964 Slow electrical processes in the brain Prog. in Brain Res [3] Alessandrini G, Isakov V and Powell J 1995 Local uniqueness in the inverse problem with one measurement Trans. Amer. Math. Soc [4] Barber D C and Brown B H 1984 Applied potential tomography J. Phys. E:Sci. Instrum [5] Birgul O, Eyuboglu B M and Ider Y Z 2003 Current constrained voltage scaled reconstruction (CCVSR) algorithm for MR-EIT and its performance with different probing current patterns Phys. Med. Biol [6] Boone K, Barber D and Brown B 1997 Imaging with electricity: report of the European concerted action on impedance tomography J. Med. Eng. Tech [7] Cheney M, Isaacson D and Newell J C 1999 Electrical impedance tomography SIAM Review [8] Cole K S and Curtis H J 1939 Electrical impedance of the squid giant axon during activity J. Gen. Physiol [9] Cole K S 1949 Dynamic electrical characteristics of squid axon membrane Arch. Sci. Physiol [10] Eyuboglu M, Birgul O and Ider Y Z 2001 A dual modality system for high resolution-true conductivity imaging Proc. XI Int. Conf. Elec. Bioimpedance (ICEBI) [11] Folland G 1976 Introduction to Partial Differential Equations (Princeton, NJ, USA: Princeton University Press)

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