# American Institute of Mathematical Sciences

April  2020, 14(2): 385-398. doi: 10.3934/ipi.2020017

## Integral formulation of the complete electrode model of electrical impedance tomography

 Department of Mathematical Sciences, Xi'an Jiaotong-Liverpool University, Laboratory for Intelligent Computing and Financial Technology, 111 Ren'ai Road, Suzhou Industrial Park, Suzhou, Jiangsu Province, 215123, China

Received  August 2019 Revised  November 2019 Published  February 2020

Fund Project: The author is partially supported by KSF in XJTLU.

We model electrical impedance tomography (EIT) based on the minimum energy principle. It results in a constrained minimization problem in terms of current density. The new formulation is proved to have a unique solution within appropriate function spaces. By characterizing its solution with the Lagrange multiplier method, we relate the new formulation to the so-called shunt model and the complete electrode model (CEM) of EIT. Based on the new formulation, we also propose a new numerical method to solve the forward problem of EIT. The new solver is formulated in terms of current. It was shown to give similar results to that of the traditional finite element method, with simulations on a 2D EIT model.

Citation: Erfang Ma. Integral formulation of the complete electrode model of electrical impedance tomography. Inverse Problems & Imaging, 2020, 14 (2) : 385-398. doi: 10.3934/ipi.2020017
##### References:
 [1] K. S. Cheng, D. Isaacson, J. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography, IEEE Transactions on Biomedical Engineering, 36 (1989), 918-924.   Google Scholar [2] P. G. Ciarlet, The Finite Element Method For Elliptic Problems, 2$^{nd}$ edition, SIAM, Philadelphia, 2002.  Google Scholar [3] J. Dardé and S. Staboulis, Electrode modelling: The effect of contact impedance, ESAIM: Mathematical Modelling and Numerical Analysis, 50 (2016), 415-431.  doi: 10.1051/m2an/2015049.  Google Scholar [4] V. Girault and P. Raviart, Finite Element Methods For Navier-Stokes Equations: Theory and Algorithms, 1$^{st}$ edition, Springer-Verlag, Berlin Heidelberg, 1986. doi: 10.1007/978-3-642-61623-5.  Google Scholar [5] N. Hyvönen, Complete electrode model of electrical impedance tomography: Approximation properties and characterization of inclusions, SIAM Journal on Applied Mathematics, 64 (2004), 902-931.  doi: 10.1137/S0036139903423303.  Google Scholar [6] N. Hyvönen and L. Mustonen, Smoothened complete electrode model, SIAM Journal on Applied Mathematics, 77 (2017), 2250-2271.  doi: 10.1137/17M1124292.  Google Scholar [7] P. O. Persson and G. Strang, A simple mesh generator in matlab, SIAM Review, 46 (2004), 329-345.  doi: 10.1137/S0036144503429121.  Google Scholar [8] M. Pidcock, S. Ciulli and S. Ispas, Singuarities of mixed boundary value problems in electrical impedance tomography, Physiological Measurement, 16 (1995), 213-218.   Google Scholar [9] E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM Journal on Applied Mathematics, 52 (1992), 1023-1040.  doi: 10.1137/0152060.  Google Scholar [10] R. Winkler and A. Rieder, Resolution-controlled conductivity discretization in electrical impedance tomography, SIAM Journal on Imaging Sciences, 7 (2014), 2048-2077.  doi: 10.1137/140958955.  Google Scholar [11] A. Zangwill, Modern Electrodynamics, 1$^{st}$ edition, Cambridge University Press, 2013.   Google Scholar [12] E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications, 1$^{st}$ edition, Springer-Verlag, New York, 1995.  Google Scholar

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##### References:
 [1] K. S. Cheng, D. Isaacson, J. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography, IEEE Transactions on Biomedical Engineering, 36 (1989), 918-924.   Google Scholar [2] P. G. Ciarlet, The Finite Element Method For Elliptic Problems, 2$^{nd}$ edition, SIAM, Philadelphia, 2002.  Google Scholar [3] J. Dardé and S. Staboulis, Electrode modelling: The effect of contact impedance, ESAIM: Mathematical Modelling and Numerical Analysis, 50 (2016), 415-431.  doi: 10.1051/m2an/2015049.  Google Scholar [4] V. Girault and P. Raviart, Finite Element Methods For Navier-Stokes Equations: Theory and Algorithms, 1$^{st}$ edition, Springer-Verlag, Berlin Heidelberg, 1986. doi: 10.1007/978-3-642-61623-5.  Google Scholar [5] N. Hyvönen, Complete electrode model of electrical impedance tomography: Approximation properties and characterization of inclusions, SIAM Journal on Applied Mathematics, 64 (2004), 902-931.  doi: 10.1137/S0036139903423303.  Google Scholar [6] N. Hyvönen and L. Mustonen, Smoothened complete electrode model, SIAM Journal on Applied Mathematics, 77 (2017), 2250-2271.  doi: 10.1137/17M1124292.  Google Scholar [7] P. O. Persson and G. Strang, A simple mesh generator in matlab, SIAM Review, 46 (2004), 329-345.  doi: 10.1137/S0036144503429121.  Google Scholar [8] M. Pidcock, S. Ciulli and S. Ispas, Singuarities of mixed boundary value problems in electrical impedance tomography, Physiological Measurement, 16 (1995), 213-218.   Google Scholar [9] E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM Journal on Applied Mathematics, 52 (1992), 1023-1040.  doi: 10.1137/0152060.  Google Scholar [10] R. Winkler and A. Rieder, Resolution-controlled conductivity discretization in electrical impedance tomography, SIAM Journal on Imaging Sciences, 7 (2014), 2048-2077.  doi: 10.1137/140958955.  Google Scholar [11] A. Zangwill, Modern Electrodynamics, 1$^{st}$ edition, Cambridge University Press, 2013.   Google Scholar [12] E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications, 1$^{st}$ edition, Springer-Verlag, New York, 1995.  Google Scholar
The layer of contact impedance for the $i$-th electrode
A toy 2D EIT model
An unstructured mesh over the disk
Potential distributin estimated by the current-based solver (Left) and the traditional potential-based solver (Right)
Non-uniform conductivity distribution. Black region has conductivity 0.1 S/m. Gray region has conductivity 0.3 S/m. Other parts of the region has conductivity 1 S/m
Distribution of current densities estimated by the current-based solver (Left) and the traditional potential-based solver (Right)
The conformal mapping
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