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Phys Med Biol. Author manuscript; available in PMC 2010 July 21.

Published in final edited form as:

Published online 2009 June 26. doi: 10.1088/0031-9155/54/14/012

PMCID: PMC2810423

NIHMSID: NIHMS169755

The publisher's final edited version of this article is available at Phys Med Biol

See other articles in PMC that cite the published article.

We have investigated Induced Current Magnetic Resonance Electrical Impedance Tomography (IC-MREIT) by means of computer simulations. The J-substitution algorithm was implemented to solve the IC-MREIT reconstruction problem. By providing physical insight into the charge accumulating on the interfaces, the convergence characteristics of the reconstruction algorithm were analyzed. The simulation results conducted on different objects were well-correlated with the proposed theoretical analysis. The feasibility of IC-MREIT to reconstruct the conductivity distribution of head-brain tissues was also examined in computer simulations using a multi-compartment realistic head model. The present simulation results suggest that IC-MREIT may have the potential to become a useful conductivity imaging technique.

The recently introduced magnetic resonance electrical impedance tomography (MREIT) is an imaging technique that can noninvasively reconstruct the static image of a conductivity distribution within an object. In MREIT, a small current is injected into the object to be imaged through a pair of surface electrodes, and the magnetic flux density inside the object is measured by a MRI scanner. The current density distribution inside the object can then be obtained according to Ampere's law, $J={\scriptstyle \frac{\nabla \times B}{{\mu}_{0}}}$. The conductivity distribution can be reconstructed through the relation between the conductivity and either the magnetic flux density or the current density.

There has been considerable interest in MREIT in both theoretical and experimental studies. By minimizing the error between the measured and computed current densities, Kwon *et al* (2002) tried to reconstruct cross-sectional conductivity images. In that study, they also suggested that in order to reconstruct the absolute conductivity distribution, at least two current injection patterns were needed along with a single voltage measurement. Lee *et al* (2003) have successfully applied J-substitution algorithm to data acquired from a saline phantom with a cylindrical sausage object inside. When a 28mA current was injected, the relative *L*^{2} –error of the reconstructed image was 25.5%. Under the assumption of the measurement of the internal current density, an equipotential projection-based algorithm for anisotropic conductivity imaging was suggested by Değirmenci *et al* (2007).

Both Seo *et al* (2003) and Oh *et al* (2003) used only one component of the measured magnetic flux density to reconstruct the conductivity distribution of a subject. Since both methods differentiate noisy Bz images twice, they are sensitive to measurement noise. Kim *et al* (2007) have successfully reconstructed the conductivity images of postmortem canine brains using a 3 T MREIT system with 40mA imaging currents. In the most recent experimental study, they demonstrated the feasibility of MREIT to image the conductivity distribution of *in vivo* canine brains using 5mA injection currents (Kim *et al* 2008). It was reported that the transversal J-substitution algorithm (Nam *et al* 2007) could considerably improve the quality of the reconstructed conductivity image under a low injection current. İder *et al* (2004) reported a new reconstruction algorithm by formulating the image reconstruction as the iterative solution of a non-linear matrix equation. Gao *et al* used the MREIT algorithm to reconstruct the conductivity distribution of a piece-wise homogeneous realistic-geometry head model, based on the radial basis function network (2005) and the response surface methodology (2006). Recently, Gao & He (2008) proposed a linear inverse solution of MREIT with a modified J-substitution algorithm.

As an alternative technique to MREIT, induced current magnetic resonance electrical impedance tomography (IC-MREIT) was proposed by Özparlak *et al* (2005). In this technique, an eddy current is induced in the conductive object by the AC primary magnetic field. The secondary magnetic flux density produced by the eddy current can be measured by MRI and used to reconstruct the internal conductivity distribution. With IC-MREIT, Özparlak *et al* (2005) reconstructed a conductivity image of the thorax using the algebraic algorithm.

Accurate human head conductivity imaging can not only enhance the accuracy in EEG/MEG source localization (He 2004, He 2005), but also has potential applications in the diagnosis of neurological diseases (Boone *et al* 1994, Holder 1992). In the present study, the J-substitution algorithm was applied to solve the IC-MREIT inverse problem. A series of computer simulations were conducted to evaluate the feasibility of the IC-MREIT technique to reconstruct the conductivity distribution of head-brain tissues.

Computation of the eddy current density and secondary magnetic flux density for a known conductivity distribution and boundary conditions is defined as the forward problem. Let Ω be a bounded and electrically conductive domain in R^{3} with boundary Γ. The conductivity distribution inside Ω is *σ* and assumed to be positive. Applying a low-frequency magnetic field, an eddy current can be induced in Ω. In an isotropic, linear and conductive medium, the sinusoidal electromagnetic field satisfies the following Poisson's equation and Neumann boundary conditions:

$$\nabla \cdot (\sigma \nabla \varphi )=-j\omega \mathbf{A}\cdot \nabla \sigma \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\text{in}\phantom{\rule{1em}{0ex}}\Omega $$

(1)

$$\frac{\partial \varphi}{\partial \mathbf{n}}=-j\omega \mathbf{A}\cdot \mathbf{n}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\text{on}\phantom{\rule{1em}{0ex}}\Gamma $$

(2)

where **n** is the unit outward normal vector, and *ϕ* and **A** are the electrical scalar potential and total magnetic vector potential in Ω, respectively. The electric field in Ω is given by:

$$\mathbf{E}=-\nabla \varphi -j\omega \mathbf{A}$$

(3)

By recalling the fact that the secondary magnetic vector potential **A*** _{s}* due to the eddy current is much smaller than the primary magnetic vector potential

$$\nabla \cdot (\sigma \nabla \varphi )=-j\omega {\mathbf{A}}_{p}\cdot \nabla \sigma \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\text{in}\phantom{\rule{thickmathspace}{0ex}}\Omega $$

(4)

$$\frac{\partial \varphi}{\partial \mathbf{n}}=-j\omega {\mathbf{A}}_{p}\cdot \mathbf{n}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\text{on}\phantom{\rule{thickmathspace}{0ex}}\Gamma $$

(5)

$$\mathbf{E}=-\nabla \varphi -j\omega {\mathbf{A}}_{p}$$

(6)

The eddy current density induced in the object can be calculated as

$${\mathbf{J}}_{e}=\sigma \mathbf{E}$$

(7)

If the phase of the AC current in the excitation coil is set as the reference phase (zero phase), **A*** _{p}* is purely real. Since

$${\mathbf{B}}_{s}=\frac{{\mu}_{0}}{4\pi}{\int}_{\Omega}{\mathbf{J}}_{e}\times \frac{\mathbf{r}-{\mathbf{r}}^{\prime}}{{\mid \mathbf{r}-{\mathbf{r}}^{\prime}\mid}^{3}}d{v}^{\prime}$$

(8)

where *μ _{0}*=4

$${\mathbf{B}}_{p}=\frac{{\mu}_{0}}{4\pi}{\int}_{V}\mathbf{J}\times \frac{\mathbf{r}-{\mathbf{r}}^{\prime}}{{\mid \mathbf{r}-{\mathbf{r}}^{\prime}\mid}^{3}}d{v}^{\prime}$$

(9)

where **J** (purely real) is the source current density in the excitation coil. As analyzed above, **B*** _{p}* is purely real and

Given the measured secondary magnetic flux density or eddy current density, the calculation of the inner conductivity distribution is called the inverse problem of IC-MREIT. After a brief introduction of the J-substitution algorithm, which was originally proposed for MREIT (Kwon *et al* 2002), we will show that IC-MREIT can reconstruct the absolute conductivity distribution in an object without any boundary voltage measurement. The error function in J-substitution is defined as follows:

$$f\left(\sigma \right)={\int}_{\Omega}{\mid \mid {\mathbf{J}}^{\ast}\mid -\sigma \mid \mathbf{E}\mid \mid}^{2}d{v}^{\prime}$$

(10)

where |**J***| is the magnitude of measured current density and |**E**| is the magnitude of the calculated electric field when the conductivity is *σ*. Minimizing the above error function yields the updating strategy of conductivity :

$$\sigma =\frac{\mid {\mathbf{J}}^{\ast}\mid}{\mid \mathbf{E}\mid}$$

(11)

In MREIT, if the injected current is kept the same, the internal current density distributions corresponding to two conductivity distributions, *σ* and *kσ*, where *k* is a constant, are the same (Kwon *et al* 2002, Birgül *et al* 2003). Accordingly, the conductivity distribution reconstructed from equation (11) differs from the true distribution by a scale factor. In order to image the absolute conductivity distribution, the boundary voltage data must be introduced into the update formula (Kwon *et al* 2002, İder *et al* 2003, Gao *et al* 2005):

$$\sigma =\frac{\mid {\mathbf{J}}^{\ast}\mid}{\mid \mathbf{E}\mid}\frac{{V}_{\sigma}}{{V}_{{\sigma}^{\ast}}}$$

(12)

where *V*_{σ*} is the measured voltage difference and *V*_{σ} is the calculated voltage difference when the conductivity is *σ*.

For IC-MREIT, the electric field **E** has two sources:

$${\mathbf{E}}_{in}=-j\omega {\mathbf{A}}_{p}$$

(13)

$${\mathbf{E}}_{c}=-\nabla \varphi $$

(14)

The induced electric field **E*** _{in}* is due solely to the current flowing in the excitation coil.

Suppose that the conductive object is composed of two media with different conductivities, *σ*_{1} and *σ*_{2}, respectively. At the interface, the perpendicular component of the current density must be continuous:

$${\mathbf{J}}_{1}\cdot \mathbf{n}={\mathbf{J}}_{2}\cdot \mathbf{n}$$

(15)

From equations (6) and (7), we get

$${\mathbf{J}}_{1}\cdot \mathbf{n}=-j{\sigma}_{1}\omega {\mathbf{A}}_{p1}\cdot \mathbf{n}-{\sigma}_{1}(\nabla {\varphi}_{1})\cdot \mathbf{n}$$

(16)

$${\mathbf{J}}_{2}\cdot \mathbf{n}=-j{\sigma}_{2}\omega {\mathbf{A}}_{p2}\cdot \mathbf{n}-{\sigma}_{2}(\nabla {\varphi}_{2})\cdot \mathbf{n}$$

(17)

Neglecting the secondary magnetic vector potential, we have **A**_{p1} = **A**_{p2} = **A**_{p}. Equations (15)~(17) give

$${\sigma}_{1}\left(\frac{\partial {\varphi}_{1}}{\partial \mathbf{n}}\right)-{\sigma}_{2}\left(\frac{\partial {\varphi}_{2}}{\partial \mathbf{n}}\right)=j({\sigma}_{2}-{\sigma}_{1})\omega {\mathbf{A}}_{p}\cdot \mathbf{n}$$

(18)

If the charge accumulating on the interface is not taken into account, the left-hand side of equation (18) is zero. Assuming that the primary magnetic vector potential **A*** _{p}* has the component perpendicular to the interface, the right-hand side of (18) is nonzero. It can be concluded that the continuous condition of the current density is satisfied because charge builds up on the interface. In the special case where the axis of the circular excitation coil passes through the center of multiple-layer concentric sphere, no charge accumulates on the interface (Grandori

In practice, the current density image is acquired from the measured magnetic flux density via Ampere's law. In the present simulation study, the “measured” current density was simulated. Given the target conductivity distribution, through solving the forward problem, we can get the target current density. Various levels of random noise were then added to the target current density to simulate the “measured” noise-contaminated current density, which was used to reconstruct the conductivity distribution inside the object based on the iterative J-substitution algorithm.

The present simulation study includes two parts. The objective of the first part, conducted on the two-sphere model and prism model with sharp edges and corners, was to verify the theoretical studies given above. To test the feasibility of IC-MREIT to reconstruct the conductivity distribution of head-brain tissues, the second part of the simulation was conducted on the multi-compartment realistic head model. In the present study, the finite element meshing and the forward problem solving were carried out using ANSYS 10.0.

Since the aim of the first part of the simulation study was to verify the theoretical analysis, a one-coil IC-MREIT system was adopted and only the noise-free case was considered. We first carried out the numerical simulation on a two-sphere model. The inner sphere, 5cm in radius, had a conductivity value of 5S/m. The outer sphere, 10cm in radius, had a conductivity value of 1S/m. A 100-turn excitation coil with a 20cm radius was concentric with the two-sphere model, as shown in figure 1. The excitation coil was driven by a 10A current, with a 1k Hz frequency. The finite element mesh of the two-sphere model contained 21075 linear tetrahedral elements and 3788 nodes.

The correlation coefficient (CC) and relative error (RE) between the reconstructed and target conductivity distributions were used to quantitatively assess the performance of the IC-MREIT J-substitution algorithm. The CC and RE were defined as follows:

$$CC[\sigma \left(i\right),{\sigma}^{\ast}\left(i\right)]=\frac{\sum _{i=1}^{N}\sigma \left(i\right)\cdot {\sigma}^{\ast}\left(i\right)}{{\left[\sum _{i=1}^{N}{\sigma}^{2}\left(i\right)\right]}^{1\u22152}\cdot {\left[\sum _{i=1}^{N}{\sigma}^{{\ast}_{2}}\left(i\right)\right]}^{1\u22152}}$$

(19)

$$RE[\sigma ,{\sigma}^{\ast}]=\frac{\Vert {\sigma}^{\ast}-\sigma \Vert}{\Vert {\sigma}^{\ast}\Vert}\times 100\%$$

(20)

where *σ** and *σ* are the target and reconstructed conductivity distributions, respectively.

The initial uniform conductivity distribution was taken to be 0.2 S/m and all of the following simulations began with this initial guess. After the first iteration, the CC and RE between the reconstructed and target conductivity distributions were 0.9999 and 0.1113%, respectively. The target and reconstructed conductivity cross-sectional images of the two-sphere model are shown in figure 2.

As we can see, when the excitation coil is concentric with the two-sphere model, **A*** _{p}* has no component perpendicular to the interfaces and the magnitude of the eddy current is proportional to the conductivity. Therefore, we can reconstruct the conductivity image after the first iteration.

To verify that the IC-MREIT technique does not need any voltage measurement in the reconstruction of the absolute conductivity image, the distributions of the electric field magnitude and the eddy current density vector in the x-y plane for the two-sphere model are presented in figure 3. In the middle column, the conductivity of the inner sphere is set to 5S/m and the conductivity of the outer sphere is set to 1S/m. The conductivity values of the two-sphere model in the left and right columns are 5 times smaller and larger than that of the middle column, respectively. As we can see, the distributions of the electric field corresponding to three conductivity distributions are the same, while the magnitude of the eddy current density increases 5 times from left to right in each column. This result indicates that IC-MREIT is sensitive to the absolute conductivity within an object and does not need any boundary voltage measurement to reconstruct the absolute conductivity distribution.

Distributions of (a) the electric field magnitude and (b) the eddy current density vector in the x-y plane for the two-sphere model.

In order to assess the convergence characteristic of the IC-MREIT J-substitution algorithm when **A*** _{p}* has a component perpendicular to the interfaces, the following numerical simulation was performed on a prism model. The conductivity of the inner regular triangular prism, 5cm in length of each side, was set to 5S/m. The conductivity of the outer regular quadrangular prism was set to 1S/m. The length of each side on the top and bottom faces was 5cm and the height was 10cm. A 10-cm radius excitation coil was placed coaxially with the two prisms as shown in figure 4. The finite element mesh of the prism model included 35302 linear tetrahedral elements and 6442 nodes.

Figure 5 shows the target and reconstructed conductivity cross-sectional images of the prism model. It took twelve iterations to reconstruct this image. The CC and RE between the reconstructed and target conductivity distributions were 0.9980 and 6.3858%, respectively.

In this case, **A*** _{p}* is no longer parallel to the interfaces on which a large amount of charge builds up. This results in the non-linear relationship between the magnitude of the eddy current density and the conductivity. Satisfactory reconstruction results can be achieved by multiple iterations.

In figure 6, the distributions of the electric field magnitude and the eddy current density vector in the x-y plane for the prism model corresponding to three conductivity distributions are illustrated. The conductivity values of the prism model in the left and right columns are 5 times smaller and larger than that of the middle column, respectively. The simulation results demonstrate that distributions of the electric field corresponding to three conductivity distributions are the same, while the magnitude of the eddy current density increases 5 times from left to right in each column. The present simulation results provide a further confirmation that the IC-MREIT technique can reconstruct the absolute conductivity image without any voltage measurement.

In order to evaluate the feasibility of the IC-MREIT technique to reconstruct the human head conductivity distribution, a series of simulations were conducted on a realistic head model constructed from the MRI data of a human subject. This head model included five compartments: scalp, skull, cerebrospinal fluid (CSF), gray and white matter. We assigned the mean value from multiple references to the tissue conductivities; scalp at 0.33S/m, skull at 0.0165S/m, CSF at 1S/m, gray at 0.276S/m and white matter at 0.126S/m (Zhang *et al* 2006a,b, Wagner *et al* 2004). The finite element mesh of the realistic head model, as shown in figure 7, contained 100790 linear hexahedra elements and 108953 nodes. Four circular coils, 18cm in radii, were horizontally placed in the plane 6.5cm below the vertex, and their centers were 3cm away from the vertical axis of the head toward the anterior, posterior, left and right of the subject. A 35A 1k Hz AC current was directed into each excitation coil in turn. The relative position of the first coil to the head is presented in figure 8.

Finite element meshes of the head model in (a) axial and (b) coronal view. The color labels correspond to red - scalp, blue - skull, purple - CSF, green - gray matter, yellow - white matter.

Relative position of the first coil to the head. The coil is placed in the plane 6.5cm below the vertex and shifted 3cm toward the anterior of the subject.

To examine the imaging quality at different levels of SNR, we added Gaussian random noise to the target current density. The standard deviations in *J _{x}*,

$$\begin{array}{cc}\hfill {\sigma}_{Jx}& =\frac{1}{2{\mu}_{0}\gamma {t}_{curr}\mathrm{SNR}}\sqrt{{\left(\frac{{F}_{y}}{\Delta y}\right)}^{2}+{\left(\frac{{F}_{z}}{\Delta z}\right)}^{2}}\hfill \\ \hfill {\sigma}_{Jy}& =\frac{1}{2{\mu}_{0}\gamma {t}_{curr}\mathrm{SNR}}\sqrt{{\left(\frac{{F}_{x}}{\Delta x}\right)}^{2}+{\left(\frac{{F}_{z}}{\Delta z}\right)}^{2}}\hfill \\ \hfill {\sigma}_{Jz}& =\frac{1}{2{\mu}_{0}\gamma {t}_{curr}\mathrm{SNR}}\sqrt{{\left(\frac{{F}_{x}}{\Delta x}\right)}^{2}+{\left(\frac{{F}_{y}}{\Delta y}\right)}^{2}}\hfill \end{array}$$

where *γ* = 26.75×10^{7} rads^{−1}T^{−1} is the gyromagnetic ratio of the hydrogen, *t _{curr}* = 48 ms is the duration of the applied current, SNR is the signal-to-noise ratio of the MR magnitude image, and Δ

Table 1 lists the reconstruction results of the realistic head model at different SNR levels after six iterations. From these simulation results, we could see that the SNR of the MR magnitude images should be greater than 180 for achieving less than 20% relative error in reconstructed conductivity images. Since 200 SNR is possible for body coils and higher values can be obtained for head coils (Özparlak *et al* 2005), the IC-MEIT has promise in imaging the conductivity distribution of human head. Figure 9 shows the target and reconstructed conductivity distributions of the realistic head model in the plane of coils. It is shown that the IC-MREIT technique based on J-substitution algorithm can accurately reconstruct the conductivity distribution of head-brain tissues for SNR of infinity. Even with the SNR of 90, different types of brain tissues can be successfully distinguished.

(a) Target and (b)–(f) reconstructed conductivity images of realistic head model in the coil plane (6.5cm below the vertex) corresponding to SNR=∞, 200, 180, 150 and 90.

In the present study, the J-substitution algorithm (Kwon *et al*, 2002) has been applied to solve the IC-MREIT (Özparlak *et al* 2005) inverse problem. Using basic electromagnetic equations, we have shown that there is no need for voltage measurement in IC-MREIT to reconstruct the absolute conductivity distribution. Moreover, we have provided a physical insight into the charge accumulating on the interfaces, and suggest that both the relative position of the excitation coil to the object to be imaged and the medium distribution inside the object have significant effects on the convergence behavior of the reconstruction algorithm. The present simulation results obtained on a two-sphere model and prism model provide a meaningful verification of the theoretical analysis. To test the feasibility of IC-MREIT to reconstruct the conductivity distribution of the head-brain tissues, computer simulations were conducted on a multiple-compartment realistic head model. The present simulation results suggest the potential capability of IC-MREIT to image the conductivity distribution of head-brain tissues.

Özparlak *et al* (2005) suggested that since the combined system matrix is nonsingular, IC-MREIT can reconstruct the absolute conductivity image without the need for any additional peripheral voltage measurement. In the present study, we have derived this conclusion based on completely different and more straightforward grounds.

In clinical MR imaging, fast-switching gradients induce a time-varying electric field that may cause undesirable peripheral nerve stimulation (Vogt *et al* 2004, Schaefer *et al* 2000). According to IEC standard 60601-2-33, for the gradient system sinusoidally switched at a frequency of 1k Hz, the limits for the induced electric field in the normal operating mode and in the first-level controlled operating mode are 3.8V/m and 4.7V/m, respectively (Brand *et al* 2002). In the present simulation studies conducted on a multiple-compartment realistic head model, the maximum induced electric field and eddy current density were 8.98V/m and 3.52 A/m^{2}, respectively. Since the induced electric field of 8.98V/m is beyond the safety limit prescribed by IEC standard, further efforts must be made to reduce the strength of induced electric field. The number, position and size of the excitation coils should be optimized. Using an efficient denoising technique to preprocess the noisy magnetic flux density images will be also helpful.

In practice, some methods of effectively acquiring the secondary magnetic flux density should be further investigated. The approach presented by Gao & He (2008) can be used to overcome the rotation problem. In that approach, they firstly calculated the current density distribution from one component of the magnetic flux density, then reconstructed the conductivity image using the J-substitution algorithm. Other reconstruction algorithms based on one component of the magnetic flux density should be studied in the future.

It is worth noting that while the present simulation results are promising, the quality of the reconstructed image and the convergence rate of the IC-MREIT J-substitution algorithm depend on the coil configuration and conductivity distribution inside the object. We will work on a method to overcome this adverse sensitivity.

In summary, we applied the J-substitution algorithm to solve the IC-MREIT reconstruction problem, and verified in a straightforward manner that IC-MREIT could reconstruct the absolute conductivity distribution without the need for any boundary voltage measurement. In addition, the convergence characteristics of the reconstruction algorithm were analyzed by providing insight into the charge accumulating on the interfaces. The simulation results performed on the realistic head model demonstrated the potential feasibility of IC-MREIT in reconstructing conductivity distributions of head-brain tissues, which may provide an important alternative to noninvasive conductivity imaging of the human head.

The authors would like to thank Dandan Yan and Xiaotong Zhang for helpful discussions. This work was supported in part by NSFC 5057705, NSF BES-0602957, NIH R21006070, and R01EB007920.

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