1Province-Ministry Joint Key Laboratory of Electromagnetic Field and Electrical Apparatus Reliability,Hebei University of Technology, Tianjin 300130, China
2University of Technology, Sydney 2007, Australia
Electrical impedance tomography (EIT) is a low-cost non-invasive imaging modality. It has the potential to be of great value inclinical diagnosis. One of the major problems in EIT with complex geometry shape is its high demand in computation capability,power, and memory. A generalized finite element method (GFEM) is proposed to calculate the forward problem accurately. Comparedwith the traditional FEM, a smaller number of nodes and elements with the proposed method are required to achieve the sameaccuracy in our numerical computation model. The value of signal-to-noise ratio for two-order GFEM is 47 dB, and 10 dB forconventional FEM. The results demonstrate the efficiency of the GFEM in EIT simulation. In the forward solution, it is capable ofachieving better accuracy using less computational time and memory with GFEM.
Index Terms— Bioimpedance, biomedical computing, computational electromagnetics, conventional finite element method,generalized finite element method.
I. INTRODUCTION
ELECTRICAL impedance tomography (EIT) uses elec-trodes placed on the surface to make measurements and
then an image of the electrical conductivity distribution withinthe body is reconstructed with an algorithm. It is a low-costnon-invasive imaging technique and has evolved over 40 years[1]. It has numerous applications that can be categorized intothree major fields: 1) geophysical [2]; 2) industrial [3]; and3) medical [4]. And EIT shows the potential to be of greatvalue in clinical diagnosis [5].
In EIT, a numerical computation forward problem modelwith capable of predicting the voltages on surface electrodesfor a given conductivity distribution is indispensable for imagereconstruction. The EIT forward problem model is normallybased on the conventional finite element method (FEM)[6]–[8]. One of the major problems of complex geometryshape EIT is its high demand in computation capability powerand memory. High precision both in numerical computationand data acquisition is required for obtaining the reconstruc-tion images for a small anomaly in the computing domain.
To overcome this problem, both versions of FEM includingh-FEM, p-FEM [9], and hp-FEM [10] and spectral elementmethod (SEM) [11] were reported. In the h-FEM which alsoknown as the conventional FEM, the polynomial order of theelement shape functions was constant and the element sizewas decreasing. While in the p-FEM, the polynomial orderwas increasing and the mesh size was constant. Pursiainenet al. [9] used p-version to solve EIT problem and showed
Manuscript received June 29, 2013; revised September 27, 2013; acceptedOctober 1, 2013. Date of current version February 21, 2014. Correspondingauthor: X. Zhang (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMAG.2013.2285161
that the performance of p-version FEM was better than thatof the h-FEM using the unit square with mesh refinement.Saeedizadeh et al. [10] studied the hp-version of FEM in EITcombining both the p-version FEM and h-version FEM, theelement size was decreased and the polynomial order wasincreased, simultaneously. Their numerical results suggestedthat the performance of the hp-version was better than the h-version especially in EIT image reconstruction. Lim et al. [11]adopted SEM to solve the secondary field of the EIT for-ward problem. The basis functions of SEM were based onGauss–Lobatto–Legendre points, whereas the basis functionsof conventional FEM were based on uniform grid points inthe reference domain. These basis functions were orthogonalto one another in both FEM and SEM. Their numerical resultsdemonstrated the efficiency of the SEM with fast convergencefor larger contact impedance in EIT simulation.
In this paper, we address the problem of calculating theforward problem accurately and efficiently. The generalizedFEM (GFEM) [12]–[15] is proposed to overcome the highdemand in computation time and memory. With the introduc-tion of GFEM, a smaller number of nodes and elements andhence less computational cost are required to achieve the sameaccuracy in the forward solution than using the conventionalFEM. The simulation results demonstrate the efficiency of theGFEM in EIT forward solution.
II. METHOD
A. Generalized Finite Element Method
The GFEM comes from manifold method [12] and is also acase of the partition of unity method that has its origins in [13]and [14]. It is a developed general method to analyze materialresponse to external and internal changes in stress originally.And it has been used in electromagnetic fields computation and
7025904 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014
analysis [15]. In GFEM, the node is generalized, and henceit can have more than two or three generalized degrees offreedom and those degrees of freedom are not required tohave their own definite physical meaning. At each generalizednode, we can take a polynomial to define a generalized typeof nodal interpolation function.
Supposing Sh is the conventional FEM space and[ϕ1 ϕ2 · · ·ϕn]Tis a Lagrange interpolation function, the fieldvariable Uh can be written as a summation with the conven-tional FEM
Uh =N∑
i=1
�ui �ϕi (1)
where �ui (i = 1, 2, . . . , N) is the vector of degrees of freedom[ui vi ]T on the i th node, which represents the potentialvariation on the node. When the node is generalized, it canhave more degrees of freedom and those degrees of freedomcan be defined as
�ui ={
ui
vi
}=
mi∑
j=1
[fi j (x, y) 0
0 fi j (x, y)
]{di,2 j−1di,2 j
}. (2)
Then, we obtain the following equation from the above:
Uh =N∑
i=1
mi∑
j=1
[fi j (x, y) 0
0 fi j (x, y)
]{di,2 j−1di,2 j
}�ϕi =
N∑
i=1
�Ni �Di
(3)where �Ni is the matrix of interpolation function, and �Di isthe generalized vector of degrees of freedom with the form of
�Di = [di,1 di,2 · · · di,2mi ]T . (4)
When a zero-order generalized nodal interpolation functionis used, the method would be reduced to conventional FEMas
ui ={
ui
vi
}=
[1 00 1
]{di,1di,2
}. (5)
With a one-order generalized type of nodal interpolationfunction
�Fi =[
1 0 x 0 y 00 1 0 x 0 y
](6)
and
�Di = [di,1 di,2 di,3 di,4 di,5 di,6
]T (7)
we can obtain the one-order GFEM as
�ui ={
ui
vi
}=
[1 0 x 0 y 00 1 0 x 0 y
]
⎡⎢⎢⎢⎢⎢⎢⎣
di,1di,2di,3di,4di,5di,6
⎤⎥⎥⎥⎥⎥⎥⎦
. (8)
When a two-order generalized type of nodal interpolationfunction is used, the two-order generalized type of finiteelement formula can be deduced.
B. EIT Forward Problem Model
A low-frequency EIT forward problem is modeled as (9).The electric field is conservative and the conduction currentsdominant with respect to their displacement counterparts leadto the equation
∇ · ρ−1∇φ = 0 in Sh (9)
where ∇ is the gradient operator, ∇φ represents the staticelectric field, ρ is the resistivity of the body, φ is the electricpotential, and Sh represents the body to be imaged. Electrodesare modeled with boundary conditions as the complete elec-trode model [16].
For one triangular element, there are three generalizednodes. Then, the field variable Uh in the element could bewritten as
Uhe =
N∑
i=1
�Nie
�Die (10)
where�Ni
e = �ϕie�Fi , (i = 1, 2, 3). (11)
C. Method of Weighted Residuals
For EIT forward problem, it is difficult to derive thegoverning equations of the GFEM with variational principle.Hence, the method of weighted residuals is used to derive thegoverning equations of the GFEM.
After a discretization, the field variable uh is a combinationof piecewise polynomial interpolation functions, which can beexpressed as
uh =N∑
i=1
Ni Di . (12)
There are errors introduced by using the approximating func-tions. The Laplacian is not zero, hence uh represents afinite approximation of the potential. The weak form of thegoverning equation in the method of weighted residuals couldbe derived through the multiplication of Laplace’s equation(9) with an arbitrary test function v i and integration over thedomain Sh ∫
Sh
v i [∇ · (ρ−1∇uh)]d Sh = 0. (13)
The Galerkin method of weighted residuals is used to solvethe governing equation. The arbitrary test function v i has thesame form as the trial function U
husing the same interpolation
functions �Fi
v i =N∑
i=1
ωi �Fi (14)
where ωi is the coefficient that weigh the interpolation func-tions �Fi . Then, we obtain
∫
Sh
∇ · (ρ−1∇uh)d Sh =∫
Sh
ρ−1∇uh · ∇v i d Sh . (15)
Boundary conditions can be introduced using Gauss’ theorem∫
Sh
ρ−1∇uh · ∇v i d Sh =∫
∂Sh
v iρ−1 ∂ uh
∂ nd� (16)
ZHANG et al.: NUMERICAL COMPUTATION FORWARD PROBLEM MODEL OF EIT 7025904
Fig. 1. Computation model.
Fig. 2. Simulated normalized voltages of electrodes measurement.
The boundary integral is carried out for those elements under-neath electrodes. The left side of (16) is for the entire mesh.When examined for a single 2-D triangular element k, weobtain∫
Sh
ρ−1∇uh · ∇v i d Sh = ρ−1k
3∑
i=1
ui3∑
j=1
ω j∫
Ek
∇Fi · ∇F j d�.
(17)In the entire domain, the left side of (16) will be
∫ρ−1∇uh · ∇Fi dV
= ρ−1k
3∑i=1
3∑j=1
ui∫
∂Sh
Fi∇F j · nd�, i, j = 1, . . . , N .
(18)
III. SIMULATION AND RESULTS
A. Numerical Model
To validate the performance of GFEM, a circle forwardmodel with 16 electrodes attached on its boundary is usedas shown in Fig. 1. The radius of the model is 1 m. Thehomogeneous conductivity is 0.08 S/m. The contact impedanceof the electrode is set to 0.01 � ·m2 in this model. The modelcontains 313 generalized nodes, 576 elements. The adjacentpair current patterns and adjacent measurements protocols with1 mA alternative current injection are used in our computersimulation.
B. Comparison Between Different Order GFEM
A numerical experiment is set up to compare the efficiencyof different order GFEM. The model in the experiment is thesame as in Fig. 1. Dividing the maximum value of voltage onelectrode in the same current pattern, the normalized voltage isobtained as shown in Fig. 2. The analytical solution Vanalyticalis computed with the formula in [17].
Fig. 3. Plot of relative error.
TABLE I
SNR OF DIFFERENT ORDER GFEM FOR FORWARD SOLUTION
There are 16 current patterns and each pattern has13 voltage measurements on the electrodes. Therefore, weobtain 16 × 13 = 208 normalized voltage measurements witheach order GFEM. The plot using two-order GFEM agreesbetter with the plot of analytical solution Vanalytical than that ofthe lower order GFEM. The value of the relative error betweendifferent order GFEM and analytical solution is calculated interms of average error defined as (19) and shown in Fig. 3.The average error using the two-order GFEM is 0.00036 thatis much smaller than 0.0029 with the conventional FEM
Eave = 1
N
N∑
i=1
∣∣∣∣Vn−ordernumerical(i) − Vanalytical(i)
Vanalytical(i)
∣∣∣∣. (19)
To compare the different order GFEM forward solutions,the signal-to-noise ratio (SNR) in voltages is defined using(20), and the values of SNR are listed in Table I
SNR = 10 log
L∑i=1
(Vanalytical(i))2
L∑i=1
(Vn−ordernumerical(i) − Vanalytical(i))2
. (20)
The value of SNR for two-order GFEM is 47 dB and 10 dBfor conventional FEM. It indicates the higher order GFEMhas better SNR performance that could bring benefit to bothcomputation accuracy and data acquisition.
The distributions of coputed voltages for one pattern withdifferent order GFEM are shown in Fig. 4. It can be seen thatnear the current injection electrode and the drop out electrode,the variation of the computed voltages with two-order GFEMas shown in Fig. 4(c) is larger than the other two as in Fig. 4(a)and (b) in the field.
A computer is used with 2.8 GHz Intel Pentium processorand 4 GB RAM to evaluate the performance of the method.To obtain an accuracy of 0.0026, two-order GFEM needs only
7025904 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014
Fig. 4. Distributions of computed voltage with different order GFEM forone current pattern. (a) Zero order. (b) One order. (c) Two order.
313 nodes, but conventional FEM needs more than 104 nodes.When given the same accuracy of 0.0026, conventional FEMneeds more than 1800 MB memory, but two-order GFEMneeds ∼90 MB memory which can save 20 times memory.For the same accuracy of 0.0026 with the same computer,the conventional FEM needs more than 120 s to finish thecomputation, whereas the two-order GFEM needs ∼8 s whichmay save 15 times time.
From these above comparisons, the superiority of the GFEMin terms of computational memory and time has been clearlyillustrated. It also can be seen that two-order GFEM is able toachieve better accuracy than the conventional FEM. Therefore,the GFEM is an ideal candidate for overcoming the problemof the high demand in EIT computation capability power andmemory. Our results reveal that the GFEM provides a moreefficient tool for EIT modeling and computation.
IV. CONCLUSION
In this paper, the GFEM has been developed and validatedfor the EIT forward model. Our numerical simulation results
show that GFEM is able to achieve better accuracy andsave more computational cost than the conventional FEM. Insummary, the GFEM is an efficient and promising method inforward problem solution for EIT.
ACKNOWLEDGMENT
This work was supported in part by the National NaturalScience Foundation of China under Grant 50937005 and Grant51077040, in part by the Natural Science Foundation ofHebei Province under Grant E2011202026, in part by TianjinResearch Program of Application Foundation and AdvancedTechnology under Grant 13JCQNJC14100, and in part by theScience and Technology Research Project for Universities ofEducation Department Hebei Province, China under GrantZ2010125.
REFERENCES
[1] R. H. Bayford, “Bioimpedance tomography (electrical impedancetomography),” Ann. Biomed. Eng., vol. 8, pp. 63–91, Aug. 2006.
[2] F. Nicollin, D. Gibert, F. Beauducel, G. Boudon, and J. C. Komorowski,“Electrical tomography of La Soufrière of Guadeloupe Volcano: Fieldexperiments, 1D inversion and qualitative interpretation,” Earth Planet.Sci. Lett., vol. 244, nos. 3–4, pp. 709–724, 2006.
[3] F. Dickin and M. Wang, “Electrical resistance tomography for processapplications,” Meas. Sci. Technol., vol. 7, no. 3, pp. 247–260, 1996.
[4] B. Brandstatter, “Jacobian calculation for electrical impedance tomog-raphy based on the reciprocity principle,” IEEE Trans. Magn., vol. 39,no. 3, pp. 1309–1312, May 2003.
[5] K. Y. Kim, B. S. Kim, M. C. Kim, S. Kim, Y. J. Lee, H. J. Jeon,et al., “Electrical impedance imaging of two-phase fields with anadaptive mesh grouping scheme,” IEEE Trans. Magn., vol. 40, no. 2,pp. 1124–1127, Mar. 2004.
[6] L. Gong, K. Zhang, and R. Unbehauen, “3-D anisotropic electricalimpedance imaging,” IEEE Trans. Magn., vol. 33, no. 2, pp. 2120–2122,Mar. 1997.
[7] K. Y. Kim, S. I. Kang, M. C. Kim, S. Kim, Y. J. Lee, and M. Vauhkonen,“Dynamic image reconstruction in electrical impedance tomographywith known internal structures,” IEEE Trans. Magn., vol. 38, no. 2,pp. 1301–1304, Mar. 2002.
[8] S. Zhang, G. Xu, X. Zhang, B. Zhang, H. Wang, Y. Xu, et al., “Com-putation of a 3-D model for lung imaging with electrical impedancetomography,” IEEE Trans. Magn., vol. 48, no. 2, pp. 651–654, Feb. 2012.
[9] S. Pursiainen and H. Hakula, “A high-order finite element method forelectrical impedance tomography,” in Proc. Progr. Electromagn. Res.Symp., Cambridge, MA, USA, Mar. 2006, pp. 260–264.
[10] N. Saeedizadeh, S. Kermani, and H. Rabbani, “A comparison betweenthe hp-version of finite element method with EIDORS for electri-cal impedance tomography,” J. Med. Signals Sensors, vol. 1, no. 3,pp. 200–205, Sep./Dec. 2011.
[11] K. H. Lim, J. H. Lee, G. Ye, and Q. H. Liu, “An efficient forward solverin electrical impedance tomography by spectral element method,” IEEETrans. Med. Imag., vol. 25, no. 8, pp. 1044–1051, Aug. 2006.
[12] G. H. Shi, “Manifold method of material analysis,” in Proc. Trans. 9thConf. Appl. Math Comput., 1991, pp. 51–76.
[13] I. Babuska and J. M. Melenk, “The partition of unity method,” Int.J. Numer. Methods Eng., vol. 40, no. 4, pp. 727–758, 1997.
[14] I. Babuska, G. Caloz, and J. E. Osborn, “Special finite element methodsfor a class of second order elliptic problems with rough coefficients,”SIAM J. Numer. Anal., vol. 31, no. 4, pp. 745–981, 1994.
[15] C. Lu and B. Shanker, “Generalized finite element method for vectorelectromagnetic problems,” IEEE Trans. Antennas Propag., vol. 55,no. 5, pp. 1369–1381, May 2007.
[16] E. Somersalo, M. Cheney, and D. Isaacson, “Existence and uniquenessfor electrode models for electric current computed tomography,” SIAMJ. Appl. Math., vol. 52, no. 4, pp. 1023–1040. Aug. 1992.
[17] M. K. Pidcock, M. Kuzuoglu, and K. Leblebicioglu, “Analytic andsemi-analytic solutions in electrical impedance tomography. I. Twodimensional problems,” Physiol. Meas., vol. 16, no. 2, pp. 77–90, 1995.
