of Hebei Province, Hebei University of Technology, Tianjin 300130, China. Tianjin University, Tianjin 300072, China. Florida International University, Miami, FL ...
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3-D Electrical Impedance Tomography Forward Problem With Finite Element Method Guizhi Xu1;2 , Huanli Wu1 , Shuo Yang1 , Shuo Liu3 , Ying Li1 , Qingxin Yang1 , Weili Yan1 , and Mingshi Wang2 Key Laboratory of Electromagnetic Field and Electrical Apparatus Reliability of Hebei Province, Hebei University of Technology, Tianjin 300130, China Tianjin University, Tianjin 300072, China Florida International University, Miami, FL 33174 USA Electrical impedance tomography (EIT) is a newly developed technique by which impedance measurements from the surface of an object are reconstructed into impedance image. The two-dimensional (2-D) EIT problem is regarded as a simplified model. As 2-D model cannot physically represent the three-dimensional (3-D) structure, the spatial information of the place where the impedance is changed by some diseases cannot be detected accurately. Therefore, 3-D EIT is necessary. In this paper, the finite element method (FEM) of 3-D EIT forward problem is presented. A sphere model is studied. The tetrahedron element is used in the meshing. Two types of sphere model, uniform model and multiplayer model are analyzed. The uniform sphere model, to which the analytical solution is applicable, is used to verify the developed FEM as well as examine the accuracy. The comparison between the numerical solution and the analytical solution shows the correctness of the developed FEM for EIT forward problem. The multiplayer model, four-layer model and three-layer model, is used to investigate the physical potential distribution inside the inhomogeneous sphere model. Reasonable potential distributions are obtained. Index Terms—Electrical impedance tomography (EIT), finite element method (FEM), forward problem, inverse problem.
I. INTRODUCTION
forced boundary condition
E
LECTRICAL impedance tomography (EIT) is an important engineering tool. For example, it can be used in biomedical engineering, geological exploration, and chemical engineering, etc. Here it is used to determine the impedance distribution in human’s head or other organum [1]. Achievements have been obtained in the two-dimensional EIT [2], [3]. Since two-dimensional EIT is the simplification of the actual three-dimensional (3-D) problem and it is impractical to be used in clinics, 3-D problem attracts more researcher’s attentions [4], [5]. In this paper, a finite element method to solve the 3-D EIT forward problem, which is the precondition of the inverse problem, is presented. Two types of sphere model, the uniform sphere model and the multilayer sphere model, are studied. The numerical solution of uniform sphere model is compared with the analytical solution. The relative error of the numerical solution is given. The analysis of the multiplayer model provides a better understanding of the potential distribution within the actual 3-D objects.
(2) and the Neumann boundary condition (3) is the boundary potential, and where is the conductivity, is the injected current density. Considering that the finite element method (FEM) has the advantage of a building model for object in any shape and handling inhomogeneous material property, it is adopted here to solve the forward problem of the 3-D EIT. The equivalent variation of Laplace (1) is (4)
II. ESTABLISHMENT OF THE MATHEMATICAL MODEL B. Subdivision of the Sphere Model
A. Mathematical Model The forward problem of EIT is to find the potential arising from the injected current-carrying electrodes onto an object. The mathematical model of such a problem is to solve the following Laplace equation with the given boundary conditions: (1)
Digital Object Identifier 10.1109/TMAG.2005.846503
In this paper, two types of sphere model are used. One is the four-layer model, shown in Fig. 1(a). The other is the three-layer model, shown in Fig. 1(b). The four-layer model has 34 078 elements and 6062 nodes, and the three-layer model has 1260 elements and 277 nodes. Both of them are subdivided with tetrahedron elements. For calculating the node potential by FEM, opposite drive electrodes are used to let drive current enter and leave the sphere. For the four-layer sphere model, the drive electrodes are located at points (0, 0, 1) and (0, 0, ); while for the three-layer model, the drive electrodes are located at points (0, 0, 2) and (0, 0, ).
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Using Clem’s rule, the formulations for calculating coefficients , , , and are obtained
Fig. 1. (a) Four-layer model. (b) Three-layer model.
(10) where is calculated through the determinant expansion. is the volume of the tetrahedron element. Similarly, the formulations for , , and are
Fig. 2. Tetrahedron element used for FEM mesh.
(11)
C. Finite Element Discretization
Then the coefficient matrix
can be obtained by
Expanding the integration term of (4), one has (12)
(5) Performing the extreme value operation on (5), one obtains (6) where is the coefficient matrix, which is the function of mesh and material properties. is the node potential matrix, and is the excitation matrix contributed by the injected current. For element , one has (7) In this paper, the tetrahedron element, shown in Fig. 2, is adopted by the FEM meshing. The linear interpolation function on each given below is adopted to determine the potential element (8) where , , , and are constant, , , and are coordinate. Assuming the coordinates of nodes , , , and are , , , and , one has
(9)
The matrix of (6) is formed by summing subscript in all elements.
with the same
III. SOLUTION OF THE SPHERE MODELS A. Solution of the Uniform Sphere Model The potential distribution is extracted after performing the finite element analysis of the sphere. Fig. 3 shows the result obtained by using the four-layer model in Fig. 1(a), while Fig. 4 shows the result obtained by using the three-layer model in Fig. 1(b). In this case, both the four-layer model and the threelayer model are considered as uniform. The resistivity used is m. And the drive current’s amplitude is 1 mA. As men1 tioned above, the drive electrodes are placed at points (0, 0, 1) ) [Fig. 1(a)] and points (0, 0, 2) and (0, 0, ) and (0, 0, [Fig. 1(b)], respectively. In order to reveal the potential distribution inside the sphere, two different demonstration patterns are used. Fig. 3 shows the potential profile of nodes along the coordinate axis. Fig. 4 shows the potential values of all the nodes. In both Figs. 3 and 4, the abscissa is used to represent the node number and the ordinate is used to represent the potential (V) of the corresponding nodes. From Figs. 3 and 4 we can see that the nodes closer to the drive electrode have higher potential values than the nodes far from it. Such a result accords with the actual potential distribution characteristic of a uniform sphere model.
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Fig. 5.
Uniform sphere model for the analytical solution.
Fig. 3. Potential distribution of the four-layer uniform sphere model.
Fig. 6. Analytical solution of the uniform sphere model.
Fig. 4. Potential distribution of the three-layer uniform sphere model.
In order to examine the precision of FEM numerical calculation, the analytical analysis of the uniform sphere model is performed (Fig. 5). The formulation for evaluating the potential of point is [6]
Fig. 7.
(13) is the where is the conductivity of the uniform sphere. vector from the center of the sphere to point , , and are the vectors from the center of the sphere to point (where the (where the injected injected current enters the sphere) and is the distance becurrent leaves the sphere), respectively. is the distance between tween point and point ; while point and point . When performing the analytical analysis, the current amplitude and the conductivity value are kept the same as the one used in numerical analysis. The obtained potential profile along the axis is given in Fig. 6. From Fig. 6, it can be seen that the potential profile obtained plane symmetric. For the from the analytical operation is
Comparison the analytical solution with the numerical one.
purpose of discerning the numerical solution and the analytical solution, the potential profiles in Figs. 3 and 6 are drawn in Fig. 7. In this figure, the numerical result is denoted with a dashed–dotted line, and the analytical result is denoted the solid line. The relative error of the numerical solution is drawn in Fig. 8. It can be seen that the error cure is asymmetrical. This is because that the subdivision of the sphere model is not symmetrical. Mapping meshing needs to be developed in the future. B. Solution of the Four-Layer Sphere Model The resistivities of the four-layer model, which represents scalp, skull, cerebrospinal fluid, and cerebrum, are 1/0.33, 1/0.0024, 1.0, 1/0.33 m , respectively. The amplitude of the drive current is kept as 1 mA. The node potentials on the
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the outermost layer because the resistivity of the outer layer is much higher than that of the inner layers. IV. DISCUSSION Up to now, the research on the 3-D EIT has attracted more and more scientific researchers. There is no doubt that the 3-D EIT has become the focus of the image reconstruction in the biomedical field. As known, there are many difficulties in the image reconstruction. For example, how to give a visual image of the impedance distribution in the 3-D space and the effect of the 3-D distribution of electrodes in the object are the two tough problems to handle. Though researchers in this field are facing these difficulties, their hard work has given out satisfactory successes and these successes inspire us devote ourselves to this research passionately. Fig. 8.
Relative error of the uniform sphere.
V. CONCLUSION TABLE I POTENTIAL OF THE NODES ON z COORDINATE AXIS
In this paper, finite element analysis of 3-D EIT forward problem is presented. The uniform model and the multiplayer model are studied. In addition, the analytical analysis of the uniform model is also performed to examine the accuracy of the developed numerical algorithm. The comparison between the analytical and numerical solution of the uniform model verifies the correctness of the 3-D finite element analysis. The solutions of the multiplayer model give the overall picture of the potential distribution of the sphere model. These results support the conclusion that the FEM can be used to solve the 3-D EIT problem. The future works are to improve the accuracy of the 3-D finite element numerical analysis of the EIT forward problem and implement it to impedance image reconstructions. ACKNOWLEDGMENT This work was supported in part by the Natural Science Foundation of Hebei Province under Grant E2004000054 and in part by the Specialized Research Fund for the Doctoral Program of Higher Education. REFERENCES
Fig. 9. Surface node potential of the three-layer sphere model. O is the position where the electrode is placed.
axis are listed in Table I. This table shows that the nodes in the external layer, e.g., nos. 1 and 8, have higher potential values than nodes in the inner layer. And the outside layer has sharp voltage drop due to the very low conductivity of the second layer. It makes the node potential on the inner layer is lower. C. Solution of the Three-Layer Sphere Model m , respecThe resistivities of the three layers are 1, 1, 80 tively. The amplitude of the drive current is the same, 1 mA. The potential of all the nodes is shown in Fig. 9. From this figure, the same conclusion can be drawn: the potential of nodes near the electrode is higher and the sharp voltage drop happens in
[1] M. Cheney, D. Isaacson, and J. C. Newell, “Electrical impedance tomography,” SIAM Rev., vol. 41, no. 1, pp. 85–101, 1999. [2] G. Z. Xu, Q. X. Yang, Y. Li, Q. Wu, and W. L. Yan, “The electrical properties of real head model based on electrical impedance,” in Proc. Engineering in Medicine and Biology Soc. 25th Annu. (Silver Anniversary) Int. Conf., Cancun, Mexico, Sep. 17–21, 2003, pp. 994–997. [3] Y. Li, L. Y. Rao, R. J. He, G. Z. Xu, Q. Wu, and M. L. Ge, “Image reconstruction of EIT using differential evolution algorithm,” in Proc. Engineering in Medicine and Biology Soc. 25th Annu. (Silver Anniversary) Int. Conf., Cancun, Mexico, Sep. 17–21, 2003, pp. 1011–1014. [4] M. Molinari et al., “Efficient nonlinear 3-D electrical tomography reconstruction,” in Proc. 1st World Congress on Industrial Process Tomography, 2001, pp. 29–31. [5] F. Kleinermann, N. J. Avis, and Alhargan, “Analytical solution to the three dimensional electrical forward problem for a circular cylinder,” in Proc. 1st World Congress on Industrial Process Tomography, 14–17, 1999, pp. 189–194. [6] A. Gibson, “Electrical impedance tomography of human brain function,” Ph.D. thesis, Univ. College, London, U.K., 2000. Manucript received June 6, 2004.
