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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 27, NO. 12, DECEMBER 2008
Local Harmonic Bz Algorithm With Domain Decomposition in MREIT: Computer Simulation Study Jin Keun Seo, Sung Wan Kim, Sungwhan Kim, Ji Jun Liu, Eung Je Woo*, Member, IEEE, Kiwan Jeon, and Chang-Ock Lee
Abstract—Magnetic resonance electrical impedance tomography (MREIT) attempts to provide conductivity images of an electrically conducting object with a high spatial resolution. When we inject current into the object, it produces internal distributions of current density J and magnetic flux density B = ( ). By using a magnetic resonance imaging data where is the direction (MRI) scanner, we can measure of the main magnetic field of the scanner. Conductivity images are reconstructed based on the relation between the injection current and data. The harmonic algorithm was the first constructive MREIT imaging method and it has been quite successful in previous numerical and experimental studies. Its performance is, however, degraded when the imaging object contains low-conductivity regions such as bones and lungs. To overcome this difficulty, we carefully analyzed the structure of a current density distribution near such problematic regions and proposed a new technique, called the local harmonic algorithm. We first reconstruct conductivity values in local regions with a low conductivity contrast, separated from those problematic regions. Then, the method of characteristics is employed to find conductivity values in the problematic regions. One of the most interesting observations of the new algorithm is that it can provide a scaled conductivity image in a local region without knowing conductivity values outside the region. We present the performance of the new algorithm by using computer simulation methods.
trical impedance tomography (EIT) [1]. MREIT is expected to provide conductivity images of an electrically conducting object with a high spatial resolution [2]–[5]. We sequentially inject multiple currents through chosen pairs of surface elecand also trodes to produce current density magnetic flux density distributions inside the object [6]. Using an magnetic resonance imaging (MRI) data where is the direction of the scanner, we can measure main magnetic field of the scanner [7]–[9]. The conductivity image reconstruction in MREIT is based on the relationship between the injection current and measured data [10]–[14]. We assume that represents a 3-D domain to be imaged and is an isotropic conductivity distribution in . The current density in subject to an injection current is determined by , electrode configuration, and boundary shape of . In this paper, we assume that electrode configuration and boundary shape are given from MR magnitude images. Meadata in conveys the information about any local sured change of which intermediates the following relationship:
Index Terms—Conductivity image, domain decomposition, harmonic , magnetic resonance electrical impedance tomography (MREIT).
I. INTRODUCTION
with
AGNETIC resonance electrical impedance tomography (MREIT) was motivated to deal with the ill-posed nature of the image reconstruction problem in elec-
M
Manuscript received August 9, 2007; revised April 17, 2008. Current version published November 21, 2008. This work was supported by the SRC/ERC program of MOST/KOSEF (R11-2002-103). The work of J. Liu was supported by JiangSu Natural Science Foundation (BK2007101). The work of E. J. Woo was supported by Kyung Hee University during his sabbatical year. Asterisk indicates corresponding author. J. K. Seo and S. Kim are with the Department of Mathematics, Yonsei University, Seoul 120-749, Korea. S. W. Kim is with the Division of Liberal Arts, Hanbat National University, Daejeon 305-719, Korea. J. J. Liu is with the Department of Mathematics, Southeast University, Nanjing 210096, China. *E. J. Woo is with the Department of Biomedical Engineering, Kyung Hee University, Gyeonggi 446-701, Korea (e-mail:
[email protected]). K. Jeon and C.-O. Lee are with the Department of Mathematical Sciences, KAIST, Daejeon 305-701, Korea. Digital Object Identifier 10.1109/TMI.2008.926055
where and is the magnetic permeability of the free space. Seo et al. proposed the first constructive -based MREIT alalgorithm [10]. Noting the noise gorithm called the harmonic amplification property of the algorithm, various other methods were also suggested to improve the image quality [13]–[17]. Previous studies of numerical simulations and phantom experalgorithm is quite suciments showed that the harmonic cessful in producing high-resolution conductivity images [18], [19]. In those studies, however, imaging objects had homogeneous backgrounds. Lately, we have been trying animal experiments and found that their conductivity distributions are very inhomogeneous with a wide range of contrast [20], [21]. Applying currently available MREIT image reconstruction algorithms, the image quality turned out to be lower than that of
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SEO et al.: LOCAL HARMONIC
ALGORITHM WITH DOMAIN DECOMPOSITION IN MREIT: COMPUTER SIMULATION STUDY
phantom images especially when imaging slices contained lowconductivity regions. Inside the vertebrate body, there exist local regions with very low conductivity values. Such internal regions may include bones, lungs, and air-filled stomach. Since the air is surrounding the body, the outermost boundary is also facing the insulator. Regardless of a chosen electrode configuration, the internal current density becomes tangential to the boundary of such an insulating or almost insulating region. On the other algorithm requires us to produce at hand, the harmonic least two internal current density distributions that are not parallel in the domain to be imaged [10]–[12]. Therefore, by algorithm, it is difficult to correctly using the harmonic reconstruct conductivity values near the boundary of an internal low-conductivity region and also the outer boundary of the imaging object itself. We realized that, in previous studies of numerical simulation and phantom imaging, we unconsciously took advantage of a homogeneous background which eliminated this difficulty. In the recent experimental study of animal imaging [21], we therefore restricted conductivity image reconstructions only inside the canine brain instead of the head itself and assumed that the periphery of the brain had a constant conductivity value. In order to effectively handle low-conductivity regions inside animal and human bodies, we need to develop a new image reconstruction method that can handle the problem of parallel current densities near boundaries of those regions. Carefully analyzing the structure of internal current density, in this paper, we propose a new MREIT image reconstruction algorithm based on a divide-and-conquer strategy. It consists of three major steps: 1) separate problematic regions where we expect almost parallel current densities, 2) reconstruct conductivity values in regions away from problematic regions, and 3) recover conductivity values near problematic regions. For the first step, we adopt an image segmentation method utilizing the available MR magnitude image. For the second step, we use a modified veralgorithm called the local harmonic sion of the harmonic algorithm. For the third step, we employ the method of characteristics. In this paper, we exclude the low-conductivity region itself in conductivity image reconstructions. A simple method is to set the conductivity value of such a problematic region to be a certain small value. Alternatively, we may use the method proposed by Lee et al. to recover conductivity values inside the low-conductivity region [20]. We will show the performance of the new algorithm using computer simulation methods.
II. HARMONIC A. Setup of
ALGORITHM AND ITS LIMITATION
-Based MREIT
Let the subject to be imaged occupy a 3-D bounded domain and let be a subdomain of occupying low-conductivity regions. The region may include bones, lungs, air-filled stomach, and others. We assume that the conductivity distriin . Let bution in , denoted by , is isotropic and denote the slice of cut by the plane . We denote boundaries of and by and ,
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Fig. 1. Imaging domain and slice. The imaging slice in dashed lines represents
and S is the boundary of . A low-conductivity region is illustrated as R.
respectively. Let and let as shown in Fig. 1. We say that has a conductivity contrast in a region if where and denote the supremum and infimum of in , respectively. and We inject two independent electrical currents through two pairs of surface electrodes and , respecwith gives rise tively. Then, the injection current , which satisfies the to an electrical potential, denoted by following boundary value problem:
(1)
where
unit normal vector, the cross product, and the surface area element. Note that we excluded the region from the imaging domain. The injection current induces a conveys the magnetic flux density whose -component information about any local change of via the Biot-Savart and law: for
for B. Harmonic
is
the
outward
. Algorithm
The harmonic algorithm is based on the -component of which the curl of the Ampere law provides the following identity [10]:
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(2)
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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 27, NO. 12, DECEMBER 2008
for . The above identity leads to the following implicit representation formula for and the iterative process to reconalgorithm: struct is the harmonic
(3) for
where
(4) for
and
(5) Here,
Fig. 2. Comparison of the harmonic B with the local harmonic B algorithms. Two different target conductivity distributions with five insulating regions in black are shown in (a) and (d). The background conductivity is 1 S/m. There are different numbers of three different ellipses whose conductivity values are 2, 0.6, and 0.2 S/m. The target conductivity distribution in (a) has an inhomogeneous background in the entire domain whereas (d) has a homogeneous background near the boundary. Reconstructed conductivity images using the harmonic B algorithm are in (b) and (e). They show that it does not work for the case (a), and it works well for the case (d). Images in (c) and (f) are reconstructed using the one-step local harmonic B algorithm. The one-step local harmonic B algorithm successfully reconstructs scaled conductivity images in a local disk for both cases.
is the 2-D
outward unit normal vector to ment.
, and
is the length ele-
C. Effects of Low-Conductivity Region In the presence of the low-conductivity region , the haralgorithm suffers from the singularity of monic near the boundary since the direction of current density becomes tangential to . This comes on . from the fact that and if does not have a If -component, then and are parallel at and there. This singularity produces spurious spikes fore near and may deteriorate the overall image quality since excessive errors in one region may affect other regions according to (3). This is numerically demonstrated in Section IV [see images in Fig. 2(b) and (e)]. When is homogeneous near the boundary , that is, near , we have near . In such a case, we can eliminate the singularity since
in homogeneous regions. This is the primary reason why the algorithm could produce high-quality conducharmonic tivity images in previous studies of numerical simulations and phantom experiments where we unconsciously took advantage of this background homogeneity providing [18], [19]. In animal or human experiments, we can not expect a homogeneous background conductivity and the imaging domain may contain low-conductivity regions [20], [21]. In this case, not only at we should take into account the singularity of but also at the outermost insulating boundary boundaries of internal low-conductivity regions.
III. LOCAL HARMONIC
ALGORITHM
To deal with the singularity of near , it is important . Let to understand a direct relation between and . We can view the data tional change of
as the
-direc-
since
(6) in
. Observation 3.1: The change of the conductivity along the at each tangential direction of the level set of potential is dictated by the distribution of in the 2-D slice following ways: is convex at is increasing at in the direction • If . is concave at is decreasing at in the direction • If . • If is harmonic at is not changing at in the direc. tion We can, therefore, estimate spatial changes of from measured data, if we could predict the direction of . A. Conductivity Reconstruction Away From Low-Conductivity Region be a potential in (1) subject to a homogeneous Let conductivity . When the conductivity has a low contrast , we have assumed that the direction of is little in influenced by . Instead, it is mostly dictated by the geometry of the boundary and the injection current . Indeed, the direction of the vector field is similar to that of the vector
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SEO et al.: LOCAL HARMONIC
ALGORITHM WITH DOMAIN DECOMPOSITION IN MREIT: COMPUTER SIMULATION STUDY
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field , and the data and hold the major information of the conductivity contrast according to the Observation 3.1. . The formula (2) can be decomposed To be precise, let as
(7) where
and
If
is close to a constant, then
and
. Hence, we have assumed that the term is smaller than in a region with a low conductivity contrast. Observation 3.2: Assume that the conductivity contrast is and . Let be a local region contained in low in . If is not singular in , then we have the following approximation:
(8) Let us call
inside
a scaled conductivity of
in
. It is quite interesting to observe that
contrast of
in
when
satisfies
0 5 (sin 50 sin 50 +1))
= 0 67 ( + = 0 9 (sin100 sin100 +1)+0 2 =1
For the proof of (9), see Appendix. Unlike the harmonic algorithm, the formula (9) does not require inverting the operator defined in (5) and hence the local computation of using (9) is straightforward. In the numerical computation for (9), we use an iteration process in such a way that
reflects spatial
because the term
reflects the direction of
(10) . As numerically demon-
strated in Section IV, a reconstructed scaled conductivity in can provide fine details of the true conduca local region tivity contrast without any knowledge of the scale factor and the conductivity distribution outside (see Figs. 2 and 3). The next observation explains how to reconstruct in a local disk contained in the slice . be an arbitrary disk lying on . Observation 3.3: Let the function defined in (4) with replaced Denote by by . The following identity holds: for :
(9) where
Fig. 3. Robustness of the one-step local harmonic B algorithm against added noise. Three different target conductivity distributions in (a), (d), and (g) with five insulating regions in black and inhomogeneous backgrounds were tried. Denoting the target conductivity distribution of (a) as , its background conductivity is 1 S/m and it includes different numbers of three different ellipses whose conductivity values are 2, 0.6, and 0.2 S/m. Denoting and as the target : 3 conductivity distributions of (d) and (g), respectively, we set : 3 x3 y : 3 x3 y : , reand spectively. The scaled conductivity images inside two circles were reconstructed . using the one-step local harmonic B algorithm with the initial guess We added 10% noise for the cases of (b), (e), and (h). For (c), (f), and (i), the added noise was 20%.
is the radius of the average of
. This local reconstruction of (9) is robust against for data provided that is not singular. Hence, we noise in can apply it to a region away from the boundary . B. Conductivity Reconstruction Near Low-Conductivity Region We can not apply the formula (9) for the conductivity reconis singular there. Instruction near the boundary since deed, we can apply the formula (9) to the interior region with -distance away from
where is an appropriate positive value. For example, we may is smaller than 10 select so that the condition number of in . Hence, it remains to deal with the region near
the center of , and along the boundary
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Observation 3.4: Assume that for where . Then, for , and we can use the method of characteristics to by solving (6) from the knowledge of determine in in provided that is sufficiently small. This observation is based on the following orthogonality property:
and
This means that both and the normal vector are oralong the surface . Hence, if thogonal to in particular, it is easy to see that the vector along the slice curve is pointing to the normal direction to the itself. Since on , we slice curve get on . as a directional derivative of Viewing in the direction of , the identity provides a change of in the direction which we call the character, we compute istic direction. From the knowledge of in in the region from the following approximation for :
(11) where is a step size. Numerical implementation of this method of characteristics requires a special meshing near the boundary [see Fig. 4(b)] to compute along the characteristic direction . In a 2-D model, the direction of is orthogonal to the insulating boundary, so the method of (11) works well provided that is sufficiently small. We should mention that when is this method can not be applied near close to zero. C. One-Step Local Harmonic
Fig. 4. One-step local harmonic B algorithm and the method of characteristics. (a) is the target conductivity distribution with one insulating region in black and four electrodes. (b) shows how the imaging domain is decomposed into [ ] and [ ]. It also shows a part of a fine mesh to implement the method of characteristics. (c) Reconstructed conductivity image in [ ] using the local harmonic B algorithm. (d) Reconstructed conductivity image in [ ] using the method of characteristics.
5) For each , select a local disk in in which we reconstruct a scaled conductivity image. in the local disk is obtained 6) The scaled conductivity by computing
Algorithm
Based on the key observations, we now explain the one-step local harmonic algorithm which produces a scaled conductivity image in a local region. We may use this one-step algorithm for the cases where we are primarily concerned about the local conductivity contrast. 1) We sequentially inject electrical currents and through two pairs of surface electrodes and , respectively. 2) Using an MR scanner, we get an MR magnitude image and induced magnetic flux density images, for and 2. 3) Using the MR magnitude image , we perform segmen, and internal region with possibly lowtation of conductivity values. in 4) Solve the direct problem (1) with replaced by .
(12) where respectively.
and are the center and radius of
D. Iterative Local Harmonic
,
Algorithm
An iterative process is required to provide conductivity images of the whole domain. In the th iteration, we first apply algorithm for several disks in the one-step local harmonic and get the updated conductivity in the region by selecting disks covering the region . Next, we apply the method of characteristics to compute the updated conducin the remaining region . To be precise, we tivity use the following iteration steps.
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SEO et al.: LOCAL HARMONIC
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1) For , solve the direct problem (1) with replaced by . covering 2) For each , select several disks . 3) For each disk , compute
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algorithm was homogeneous. The one-step local harmonic successfully reconstructed scaled conductivity images inside a disk shown in Fig. 2(c) and (f) for both cases. These results support the observation 3.3. Various other numerical simulations also supported the observation provided that the conductivity contrast was reasonably low. In order to test the noise tolerance, we added 10% and 20% . We generated noisy data by random noise to
(13) 4) Choose constants in
so that . Determine
in each slice
by assembling in such a in . This process way that should be done in the -direction. in each region by using the method 5) Compute of characteristics in the observation 3.4. Here we use the on that was obtained in the preknowledge of vious step. , repeat the steps (1)–(5) until 6) For the updated where is a given tolerance. IV. NUMERICAL SIMULATION RESULTS We carried out numerical simulations to test the feasibility algorithm. We considered a of the proposed local harmonic 2-D model of
where numbers are in meter. We used two different target conductivity distributions shown in Fig. 2(a) and (d). They contained an insulating region including five internal insulators marked in black. The first and second target conductivity distributions had an inhomogeneous and homogeneous backgrounds, was 10. We respectively and the conductivity contrast in placed four electrodes and at and in meter on the boundary of the model. The size of each electrode was 15 mm. Two different currents and were sequen. The tially injected between two pairs of opposite electrodes amount of each injection current was 20 mA. We used the PDE toolbox supported by Matlab (The Mathinto a fiworks Inc., Natick, MA) to discretize the model nite element mesh with triangular elements. The maximal edge size of the elements was less than 3 mm. Using the standard finite element method, we solved the forward problem (1) to comcorresponding to each injection current pute the potential . We generated simulated data of from the Ampere law , calculated , and applied (9) to reconstruct an image of the target conductivity . All numerical computations were performed using a PC with a Pentium IV processor, 1 GB RAM, and Window XP Professional operating system. During conductivity image reconstructions, we always as 1. Reconstructed images in fixed the initial guess algoFig. 2(b) and (e) show that the conventional harmonic rithm produced a satisfactory image only when the background
where is the norm of on a random number normally distributed in the interval the area of , and NL a noise level representing the relative -error between the noisy data and the true data , that is, . Fig. 3 shows reconstructed scaled conductivity images of three different target conductivity distributions with inhomogeneous backgrounds. Carefully examining the reconstructed scaled conductivity images, we found that very fine details of conductivity contrast are well recovered even in the presence of added noise. Fig. 4 shows the process to reconstruct conductivity values near an insulating region as explained in the observation 3.4. Fig. 4(a) is a model of with a target conductivity distribution including one internal insulating region marked in black. We divided the imaging domain into and as shown in Fig. 4(b). Using the same numerical method, we first applied the one-step algorithm to reconstruct a scaled conduclocal harmonic shown in Fig. 4(c). In order tivity image of the region to reconstruct conductivity values in the problematic region , we created a fine mesh using triangular elements to apply the method of characteristics. Fig. 4(d) shows a recon. structed conductivity image of the region Fig. 5 shows the performance of the image reconstruction in algorithm and the full domain using both the local harmonic the method of characteristics. We performed only one iteration of the iterative algorithm described in Section III-D. Reconstructed conductivity images in Fig. 5 show that the method of characteristics works very well to produce conductivity images in the entire domain. The conductivity image reconstruction was very robust against added noise though noise effects were relatively bigger near the outermost boundary. V. DISCUSSION Using numerical simulations, we showed that the one-step algorithm can reconstruct a scaled conduclocal harmonic tivity image inside a selected internal region where the condition is small, for example 10 (Fig. 2). We number of the matrix found that this new algorithm is robust against noise (Fig. 3). Although it reconstructs a scaled conductivity image without providing absolute values, it reproduces fine details of conductivity contrast in a specified local region without any knowledge in (8) is of conductivity values outside the region. Since positive, the scaled conductivity image defined in the Observation 3.2 never reverses the true conductivity contrast. How-
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Fig. 5. Noise effects of the local harmonic B algorithm and the method of characteristics. The number of iteration was one and the initial guess was . (a) Target conductivity distribution . Reconstructed images of in the whole imaging domain with (b) 10, (c) 20, and (d) 30% added noise.
1
=
ever, future work is needed to quantitatively analyze how much distortion of the true conductivity contrast can occur in a reconstructed scaled conductivity image. Under the assumption of low conductivity contrast, we found algorithm is satthat the performance of the local harmonic isfactory. We note that conductivity values away from a local region has relatively little influence on voltage, current density, and also magnetic flux density inside the region. This wellknown property causes the insensitivity problem in EIT limiting the spatial resolution. In MREIT, we speculate that it helps us to reconstruct scaled conductivity images with a high spatial resodata. We will not go into the detail of this lution using only complicated issue since it is beyond the scope of this paper. We should mention that Observations 3.3 and 3.4 are not rigorously proven in this paper. Their quantitative analysis requires a study of an extremely difficult mathematical problem to es. Even for the timate the condition number of the matrix homogeneous conductivity case, it is quite difficult to estimate , which has been the lower bound for the determinant of an open problem in a 3-D case. However, in practice, we can based on our exroughly estimate the condition number of perience of many numerical simulations. In the numerical imalgorithm, we found that plementation of the local harmonic this kind of rough estimates are good enough. For applications where we are primarily interested in the conductivity contrast inside a local region, we may use the local algorithm. However, some applications request harmonic conductivity images of the entire imaging domain. As shown in Figs. 4 and 5, the method of characteristics must be employed for these cases. Once properly implemented, the method of characteristics is quite robust against added noise. However, the most difficult part here is the generation of a special mesh as illustrated in Fig. 4 since it requires a very laborious work for a
general geometry. For this reason, we tested the method of characteristics using a simpler model in Figs. 4 and 5 with just one insulating region instead of using the more complicated models in Figs. 2 and 3. When we need to use the iterative algorithm in Section III.D, we face the problem of the convergence issue. Liu et al. lately studied the convergence characteristics of the harmonic algorithm assuming that the imaging domain has a low conductivity contrast [12]. We need to investigate the convergence property of the iterative version of the local harmonic algorithm in our future study. In order to apply the local harmonic algorithm, we should perform a segmentation of an MR magnitude image. Extraction of the outermost boundary is relative easy and can be almost fully automated. However, segmentation of internal problematic regions having low conductivity values is not a trivial problem. Apart from the numerous technical issues in the general biomedical image segmentation problem, we face the question about which regions have possibly low conductivity values. Since we should determine the regions before trying a conductivity image reconstruction, we have no choice but to rely on our a priori knowledge. From animal experiments [20], [21], we found that there exist three problematic regions including the outer layer of the bone, the lung, and any gas-filled internal organ. They usually appear as dark regions in an MR magnitude image due to weak MR signals from them. We are currently using a semi-automatic image segmentation method to extract those internal regions. By performing more animal imaging experiments, we should accumulate more of our a priori knowledge on other problematic regions. Future work should include a development of a 3-D image reconstruction software with a graphical user interface that algofacilitates the interactive use of the local harmonic rithm. We need to develop an interactive segmentation tool that can easily incorporate our a priori knowledge into the semi-automatic segmentation process. Mesh generation tool must be imbedded in the software to be able to implement the method of characteristics for more general cases. Once the segmentation and mesh generation are performed, we should be able to interactively choose local imaging regions. algorithm is fast enough to show Then, the local harmonic conductivity images in the local regions within an acceptable time delay. We are currently developing such a software and plan to investigate its performance. We also plan to apply the new method proposed in this paper to measured data from postmortem and in vivo animal experiments. VI. CONCLUSION MREIT has not yet reached the stage of clinical applications primarily due to two technical difficulties. One is the limited capability of its image reconstruction algorithm to correctly recover an inhomogeneous conductivity distribution in the presence of a high conductivity contrast region including bones, lungs, and any air-filled organs. The other is the required amount of injection current that is higher than a physiologically acceptalable level. Addressing the first issue, the local harmonic gorithm proposed in this paper is very promising since it may
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SEO et al.: LOCAL HARMONIC
ALGORITHM WITH DOMAIN DECOMPOSITION IN MREIT: COMPUTER SIMULATION STUDY
provide a scaled conductivity image inside a specified local region of low conductivity contrast without knowledge of an inhomogeneous conductivity distribution outside the local region. algorithm may include Applications of the local harmonic high-resolution conductivity imaging of the brain, breast, liver, and other parts of the body. Reduction of injection currents must be pursued utilizing all the latest developments in experimental MREIT techniques including improvements in RF coil, pulse sequence optimization, signal averaging, and denosing. We plan to perform in vivo anesthetized animal experiments using 5 mA injection current. Our short term goal is to provide high-resolution conductivity images of a canine brain followed by other parts. Preliminary human experiments are also being planned using 2 or 3 mA injection currents. Limbs will be the initial imaging areas of the human experiment.
APPENDIX I DERIVATION OF THE IDENTITY (9) According to the trace formula of the double layer potential [22], [23], we have
(14) Since
is the disk with the radius , a direct computation yields
Hence, the trace formula (14) becomes 15) From (3) and (15), we have the expression of boundary:
along the
(16) Now, we substitute (16) into the integral term in (5) and combine with (3) to get
(17) for
. Using the fact that for all
, the identity (17) becomes
(18) for identity.
. The formula (9) follows from (3) and the above
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