# 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

show all references

##### 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
 [1] Nuutti Hyvönen, Harri Hakula, Sampsa Pursiainen. Numerical implementation of the factorization method within the complete electrode model of electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (2) : 299-317. doi: 10.3934/ipi.2007.1.299 [2] Fabrice Delbary, Rainer Kress. Electrical impedance tomography using a point electrode inverse scheme for complete electrode data. Inverse Problems & Imaging, 2011, 5 (2) : 355-369. doi: 10.3934/ipi.2011.5.355 [3] Jérémi Dardé, Harri Hakula, Nuutti Hyvönen, Stratos Staboulis. Fine-tuning electrode information in electrical impedance tomography. Inverse Problems & Imaging, 2012, 6 (3) : 399-421. doi: 10.3934/ipi.2012.6.399 [4] Nicolay M. Tanushev, Luminita Vese. A piecewise-constant binary model for electrical impedance tomography. Inverse Problems & Imaging, 2007, 1 (2) : 423-435. doi: 10.3934/ipi.2007.1.423 [5] Bastian Gebauer. Localized potentials in electrical impedance tomography. Inverse Problems & Imaging, 2008, 2 (2) : 251-269. doi: 10.3934/ipi.2008.2.251 [6] Ke Zhang, Maokun Li, Fan Yang, Shenheng Xu, Aria Abubakar. Electrical impedance tomography with multiplicative regularization. Inverse Problems & Imaging, 2019, 13 (6) : 1139-1159. doi: 10.3934/ipi.2019051 [7] Kari Astala, Jennifer L. Mueller, Lassi Päivärinta, Allan Perämäki, Samuli Siltanen. Direct electrical impedance tomography for nonsmooth conductivities. Inverse Problems & Imaging, 2011, 5 (3) : 531-549. doi: 10.3934/ipi.2011.5.531 [8] Ville Kolehmainen, Matti Lassas, Petri Ola, Samuli Siltanen. Recovering boundary shape and conductivity in electrical impedance tomography. Inverse Problems & Imaging, 2013, 7 (1) : 217-242. doi: 10.3934/ipi.2013.7.217 [9] Liliana Borcea, Fernando Guevara Vasquez, Alexander V. Mamonov. Study of noise effects in electrical impedance tomography with resistor networks. Inverse Problems & Imaging, 2013, 7 (2) : 417-443. doi: 10.3934/ipi.2013.7.417 [10] Dong liu, Ville Kolehmainen, Samuli Siltanen, Anne-maria Laukkanen, Aku Seppänen. Estimation of conductivity changes in a region of interest with electrical impedance tomography. Inverse Problems & Imaging, 2015, 9 (1) : 211-229. doi: 10.3934/ipi.2015.9.211 [11] Gen Nakamura, Päivi Ronkanen, Samuli Siltanen, Kazumi Tanuma. Recovering conductivity at the boundary in three-dimensional electrical impedance tomography. Inverse Problems & Imaging, 2011, 5 (2) : 485-510. doi: 10.3934/ipi.2011.5.485 [12] Nuutti Hyvönen, Lassi Päivärinta, Janne P. Tamminen. Enhancing D-bar reconstructions for electrical impedance tomography with conformal maps. Inverse Problems & Imaging, 2018, 12 (2) : 373-400. doi: 10.3934/ipi.2018017 [13] Kimmo Karhunen, Aku Seppänen, Jari P. Kaipio. Adaptive meshing approach to identification of cracks with electrical impedance tomography. Inverse Problems & Imaging, 2014, 8 (1) : 127-148. doi: 10.3934/ipi.2014.8.127 [14] Yuxue Li, Maozhu Jin, Peiyu Ren, Zhixue Liao. Research on the optimal initial shunt strategy of Jiuzhaigou based on the optimization model. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1239-1249. doi: 10.3934/dcdss.2015.8.1239 [15] Melody Dodd, Jennifer L. Mueller. A real-time D-bar algorithm for 2-D electrical impedance tomography data. Inverse Problems & Imaging, 2014, 8 (4) : 1013-1031. doi: 10.3934/ipi.2014.8.1013 [16] Daniela Calvetti, Paul J. Hadwin, Janne M. J. Huttunen, Jari P. Kaipio, Erkki Somersalo. Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Stochastic extension of the boundary map. Inverse Problems & Imaging, 2015, 9 (3) : 767-789. doi: 10.3934/ipi.2015.9.767 [17] Daniela Calvetti, Paul J. Hadwin, Janne M. J. Huttunen, David Isaacson, Jari P. Kaipio, Debra McGivney, Erkki Somersalo, Joseph Volzer. Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results. Inverse Problems & Imaging, 2015, 9 (3) : 749-766. doi: 10.3934/ipi.2015.9.749 [18] Lassi Roininen, Janne M. J. Huttunen, Sari Lasanen. Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography. Inverse Problems & Imaging, 2014, 8 (2) : 561-586. doi: 10.3934/ipi.2014.8.561 [19] Helmut Harbrecht, Thorsten Hohage. A Newton method for reconstructing non star-shaped domains in electrical impedance tomography. Inverse Problems & Imaging, 2009, 3 (2) : 353-371. doi: 10.3934/ipi.2009.3.353 [20] Sarah Jane Hamilton, Andreas Hauptmann, Samuli Siltanen. A data-driven edge-preserving D-bar method for electrical impedance tomography. Inverse Problems & Imaging, 2014, 8 (4) : 1053-1072. doi: 10.3934/ipi.2014.8.1053

2018 Impact Factor: 1.469