Electrical impedance tomography (EIT) is an imaging technique that reconstructs the conductivity distribution in the interior of an object using electrical measurements from the electrodes that are attached around the boundary. The Complete Electrode Model (CEM) accurately incorporates the electrode size, shape, and effective contact impedance into the forward problem for EIT. In this work, the effect of the conductivity distribution and the electrode contact impedance on the solution of the forward problem is addressed. In particular, the sensitivity of the electric potential with respect to a small-amplitude perturbation in the conductivity, and with respect to some defective electrodes is studied. The Gâteaux derivative is introduced as a tool for the sensitivity analysis and the Gâteaux differentiability of the electric potential with respect to the conductivity and to the contact impedance of the electrodes is proved. The derivative is then expressed as the unique solution to a variational problem and the discretization is performed with Finite Elements of type P1. Numerical simulations for different 2D and 3D configurations are presented. This study illustrates the impact of the presence of perturbations in the parameters of CEM on EIT measurements. Finally, the 2D inverse conductivity problem for EIT is numerically solved for some configurations and the results confirm the conclusions of the numerical sensitivity analysis.
Citation: Marion Darbas, Jérémy Heleine, Renier Mendoza, Arrianne Crystal Velasco. Sensitivity analysis of the complete electrode model for electrical impedance tomography[J]. AIMS Mathematics, 2021, 6(7): 7333-7366. doi: 10.3934/math.2021431
Electrical impedance tomography (EIT) is an imaging technique that reconstructs the conductivity distribution in the interior of an object using electrical measurements from the electrodes that are attached around the boundary. The Complete Electrode Model (CEM) accurately incorporates the electrode size, shape, and effective contact impedance into the forward problem for EIT. In this work, the effect of the conductivity distribution and the electrode contact impedance on the solution of the forward problem is addressed. In particular, the sensitivity of the electric potential with respect to a small-amplitude perturbation in the conductivity, and with respect to some defective electrodes is studied. The Gâteaux derivative is introduced as a tool for the sensitivity analysis and the Gâteaux differentiability of the electric potential with respect to the conductivity and to the contact impedance of the electrodes is proved. The derivative is then expressed as the unique solution to a variational problem and the discretization is performed with Finite Elements of type P1. Numerical simulations for different 2D and 3D configurations are presented. This study illustrates the impact of the presence of perturbations in the parameters of CEM on EIT measurements. Finally, the 2D inverse conductivity problem for EIT is numerically solved for some configurations and the results confirm the conclusions of the numerical sensitivity analysis.
| [1] |
Y. F. Albuquerque, A. Laurain, K. Sturm, A shape optimization approach for electrical impedance tomography with point measurements, Inverse Probl., 36 (2020), 095006. doi: 10.1088/1361-6420/ab9f87
|
| [2] |
L. Andiani, A. Rubiyanto, Endarko, Sensitivity analysis of thorax imaging using two-dimensional electrical impedance tomography (EIT), Journal of Physics: Conference Series, 1248 (2019), 012009. doi: 10.1088/1742-6596/1248/1/012009
|
| [3] |
A. P. Bagshaw, A. D. Liston, R. H. Bayford, A. Tizzard, A. P. Gibson, A. T. Tidswell, et al., Electrical impedance tomography of human brain function using reconstruction algorithms based on the finite element method, Neuroimage, 20 (2003), 752–764. doi: 10.1016/S1053-8119(03)00301-X
|
| [4] |
L. Borcea, Electrical impedance tomography, Inverse Probl., 19 (2003), 997–998. doi: 10.1088/0266-5611/19/4/501
|
| [5] |
L. Bourgeois, A. Recoquillay, A mixed formulation of the Tikhonov regularization and its application to inverse pde problems, ESAIM-Math. Model. Num., 52 (2018), 123–145. doi: 10.1051/m2an/2018008
|
| [6] | G. Boverman, B. S. Kim, D. Isaacson, J. C. Newell, The complete electrode model for imaging and electrode contact compensation in electrical impedance tomography, in 2007 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, (2007), 3462–3465. |
| [7] |
A. Boyle, A. Adler, The impact of electrode area, contact impedance and boundary shape on EIT images, Physiol. Meas., 32 (2011), 745–754. doi: 10.1088/0967-3334/32/7/S02
|
| [8] | A. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Río de Janeiro, 1980), 65–73. |
| [9] | Z. Chen, Reconstruction algorithms for electrical impedance tomography, PhD thesis, University of Wollongong, New South Wales, Australia, 1990. |
| [10] |
M. Cheney, D. Isaacson, J. Newell, Electrical impedance tomography, SIAM REVIEW, 41 (1999), 85–101. doi: 10.1137/S0036144598333613
|
| [11] |
K.-S. Cheng, D. Isaacson, J. Newell, D. Gisser, Electrode models for electric current computed tomography, IEEE T. Biomed. Eng., 36 (1989), 918–924. doi: 10.1109/10.35300
|
| [12] |
M. Crabb, Convergence study of 2D forward problem of electrical impedance tomography with high order finite elements, Inverse Probl. Sci. En., 25 (2017), 1397–1422. doi: 10.1080/17415977.2016.1255739
|
| [13] |
M. Darbas, J. Heleine, S. Lohrengel, Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations, Inverse Probl. Imag., 14 (2020), 1107–1133. doi: 10.3934/ipi.2020056
|
| [14] |
M. Darbas, J. Heleine, S. Lohrengel, Sensitivity analysis for 3D Maxwell's equations and its use in the resolution of an inverse medium problem at fixed frequency, Inverse Probl. Sci. En., 28 (2020), 459–496. doi: 10.1080/17415977.2019.1588896
|
| [15] |
J. Dardé, N. Hyvönen, A. Seppänen, S. Staboulis, Simultaneous recovery of admittivity and body shape in electrical impedance tomography: An experimental evaluation, Inverse Probl., 29 (2013), 085004. doi: 10.1088/0266-5611/29/8/085004
|
| [16] |
J. Dardé, S. Staboulis, Electrode modelling: The effect of contact impedance, ESAIM: M2AN, 50 (2016), 415–431. doi: 10.1051/m2an/2015049
|
| [17] |
J. Dardé, H. Hakula, N. Hyvönen, S. Staboulis, Fine-tuning electrode information in electrical impedance tomography, Inverse Probl. Imag., 6 (2012), 399–421. doi: 10.3934/ipi.2012.6.399
|
| [18] |
J. Dardé, N. Hyvönen, A. Seppänen, S. Staboulis, Simultaneous reconstruction of outer boundary shape and admittivity distribution in electrical impedance tomography, SIAM J. Imaging Sci., 6 (2013), 176–198. doi: 10.1137/120877301
|
| [19] |
M. Dodd, J. Mueller, A real-time D-bar algorithm for 2-D electrical impedance tomography data, Inverse Probl. Imag., 8 (2014), 1013–1031. doi: 10.3934/ipi.2014.8.1013
|
| [20] | M. Fernández-Corazza, S. Turovets, P. Govyadinov, C. Muravchik, D. Tucker, Effects of head hodel inaccuracies on regional scalp and skull conductivity estimation using real EIT measurements, in II Latin American Conference on Bioimpedance, Springer, 2016, 5–8. |
| [21] |
M. Fernández-Corazza, S. Turovets, P. Luu, N. Price, C. Muravchik, D. Tucker, Skull modeling effects in conductivity estimates using parametric electrical impedance tomography, IEEE T. Biomed. Eng., 65 (2018), 1785–1797. doi: 10.1109/TBME.2017.2777143
|
| [22] | M. Fernández-Corazza, N. von Ellenrieder, C. H. Muravchik, Estimation of electrical conductivity of a layered spherical head model using electrical impedance tomography, Journal of Physics: Conference Series, 332 (2011), 12–22. |
| [23] |
L. G. Grassi, R. Santiago, G. Florio, L. Berra, Bedside Evaluation of Pulmonary Embolism by Electrical Impedance Tomography, Anesthesiology, 132 (2020), 896–896. doi: 10.1097/ALN.0000000000003059
|
| [24] |
H. Hakula, N. Hyvönen, T. Tuominen, On the hp-adaptive solution of complete electrode model forward problems of electrical impedance tomography, J. Comput. Appl. Math., 236 (2012), 4645–4659. doi: 10.1016/j.cam.2012.04.005
|
| [25] |
H. Hakula, N. Hyvönen, T. Tuominen, On the hp-adaptive solution of complete electrode model forward problems of electrical impedance tomography, J. Comput. Appl. Math., 236 (2012), 4645–4659. doi: 10.1016/j.cam.2012.04.005
|
| [26] |
R. J. Halter, A. Hartov, K. D. Paulsen, A broadband high-frequency electrical impedance tomography system for breast imaging, IEEE T. Biomed. Eng., 55 (2008), 650–659. doi: 10.1109/TBME.2007.903516
|
| [27] |
S. J. Hamilton, A. Hauptmann, Deep D-bar: Real-time Electrical Impedance Tomography imaging with deep neural networks, IEEE T. Med. Imaging, 37 (2018), 2367–2377. doi: 10.1109/TMI.2018.2828303
|
| [28] | S. Hamilton, D. Isaacson, V. Kolehmainen, P. Muller, J. Toivanen, P. Bray, 3d EIT reconstructions from electrode data using direct inversion D-bar and Calderón methods, arXiv preprint arXiv: 2007.03018. |
| [29] | F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251–265. |
| [30] |
L. Heikkinen, T. Vilhunen, R. West, M. Vauhkonen, Simultaneous reconstruction of electrode contact impedances and internal electrical properties: Ii. laboratory experiments, Meas. Sci. Technol., 13 (2002), 1855–1861. doi: 10.1088/0957-0233/13/12/308
|
| [31] | M. Hintermüller, A. Laurain, Electrical impedance tomography: from topology to shape, Control Cybern., 37 (2008), 913–933. |
| [32] |
N. Hyvönen, Complete electrode model of electrical impedance tomography: Approximation properties and characterization of inclusions, SIAM J. Appl. Math., 64 (2004), 902–931. doi: 10.1137/S0036139903423303
|
| [33] |
N. Hyvönen, L. Mustonen, Smoothened complete electrode model, SIAM J. Appl. Math., 77 (2017), 2250–2271. doi: 10.1137/17M1124292
|
| [34] |
N. Hyvönen, P. Piiroinen, O. Seiskari, Point measurements for a Neumann-to-Dirichlet map and the Calderón problem in the plane, SIAM J. Math. Anal., 44 (2012), 3526–3536. doi: 10.1137/120872164
|
| [35] |
O. Y. Imanuvilov, G. Uhlmann, M. Yamamoto, The Calderón problem with partial data in two dimensions, J. Am. Math. Soc., 23 (2010), 655–691. doi: 10.1090/S0894-0347-10-00656-9
|
| [36] |
J. P. Kaipio, V. Kolehmainen, E. Somersalo, M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography, Inverse Probl., 16 (2000), 1487–1522. doi: 10.1088/0266-5611/16/5/321
|
| [37] | P. Kauppinen, J. Hyttinen, J. Malmivuo, Sensitivity distribution visualizations of impedance tomography measurement strategies, International Journal of Bioelectromagnetism, 8 (2006), 1–9. |
| [38] |
C. Kenig, J. Sjostraa, G. Uhlmann, The Calderón problem with partial data, Ann. Math., 165 (2007), 567–591. doi: 10.4007/annals.2007.165.567
|
| [39] | R. Kohn, M. Vogelius, Determining conductivity by boundary measurements, Commun. Pure Appl. Math., 37 (1984), 113–123. |
| [40] |
V. Kolehmainen, M. Lassas, P. Ola, The inverse conductivity problem with an imperfectly known boundary in three dimensions, SIAM J. Appl. Math., 67 (2007), 1440–1452. doi: 10.1137/060666986
|
| [41] |
V. Kolehmainen, M. Lassas, P. Ola, Electrical impedance tomography problem with inaccurately known boundary and contact impedances, IEEE T. Med. Imaging, 27 (2008), 1404–1414. doi: 10.1109/TMI.2008.920600
|
| [42] |
A. Lechleiter, A. Rieder, Newton regularizations for impedance tomography: A numerical study, Inverse Probl., 22 (2006), 1967–1987. doi: 10.1088/0266-5611/22/6/004
|
| [43] |
X. Li, F. Yang, J. Ming, A. Jadoon, S. Han, Imaging the corrosion in grounding grid branch with inner-source electrical impedance tomography, Energies, 11 (2018), 1739. doi: 10.3390/en11071739
|
| [44] | D. Miklavcic, N. Pavselj, F. Hart, Electric Properties of Tissues, vol. 6, 2006. |
| [45] |
C. C. A. Morais, B. S. Fakhr, R. R. de Santis Santiago, R. D. Fenza, E. Marutani, S. Gianni, et al., Bedside electrical impedance tomography unveils respiratory chimera in covid-19, Am. J. Resp. Crit. Care, 203 (2021), 120–121. doi: 10.1164/rccm.202005-1801IM
|
| [46] |
A. Nissinen, V. Kolehmainen, J. Kaipio, Compensation of modelling errors due to unknown boundary in electrical impedance tomography, IEEE T. Med. Imaging, 30 (2011), 231–242. doi: 10.1109/TMI.2010.2073716
|
| [47] | R. Parker, The inverse problem of resistivity sounding, Geophysics, 142 (1984), 2143–2158. |
| [48] |
S. Ren, M. Soleimani, Y. Xu, F. Dong, Inclusion boundary reconstruction and sensitivity analysis in electrical impedance tomography, Inverse Probl. Sci. Eng., 26 (2018), 1037–1061. doi: 10.1080/17415977.2017.1378195
|
| [49] |
F. Santosa, M. Vogelius, A computational algorithm to determine cracks from electrostatic boundary measurements, Int. J. Eng. Sci., 29 (1991), 917–937. doi: 10.1016/0020-7225(91)90166-Z
|
| [50] |
G. Saulnier, A. Ross, N. Liu, A high-precision voltage source for EIT, Physiol. meas., 27 (2006), S221–S236. doi: 10.1088/0967-3334/27/5/S19
|
| [51] | O. Shuvo, M. Islam, Sensitivity analysis of the tetrapolar electrical impedance measurement systems using comsol multiphysics for the non-uniform and inhomogeneous medium, Dhaka University Journal of Science, 1 (2016), 7–12. |
| [52] |
M. Soleimani, C. Gómez-Laberge, A. Adler, Imaging of conductivity changes and electrode movement in eit, Physiol. Meas., 27 (2006), S103–S113. doi: 10.1088/0967-3334/27/5/S09
|
| [53] |
E. Somersalo, M. Cheney, D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math., 52 (1992), 1023–1040. doi: 10.1137/0152055
|
| [54] |
V. Tomicic, R. Cornejo, Lung monitoring with electrical impedance tomography: technical considerations and clinical applications, J. Thorac. Dis., 11 (2019), 3122. doi: 10.21037/jtd.2019.06.27
|
| [55] |
O.-P. Tossavainen, M. Vauhkonen, L. M. Heikkinen, T. Savolainen, Estimating shapes and free surfaces with electrical impedance tomography, Meas. Sci. Technol., 15 (2004), 1402–1411. doi: 10.1088/0957-0233/15/7/024
|
| [56] |
P. J. Vauhkonen, M. Vauhkonen, T. Savolainen, J. P. Kaipio, Three-dimensional electrical impedance tomography based on the complete electrode model, IEEE T. Biomed. Eng., 46 (1999), 1150–1160. doi: 10.1109/10.784147
|
| [57] | A. Velasco, M. Darbas, R. Mendoza, M. Bacon, J. de Leon, Comparative study of heuristic algorithms for electrical impedance tomography, Philippine Journal of Science, 149 (2020), 747–761. |
| [58] |
C. Venkatratnam, F. Nagi, Spatial resolution in electrical impedance tomography: A topical review, Journal of Electrical Bioimpedance, 8 (2017), 66–78. doi: 10.5617/jeb.3350
|
| [59] |
T. Vilhunen, J. Kaipio, P. Vauhkonen, T. Savolainen, M. Vauhkonen, Simultaneous reconstruction of electrode contact impedances and internal electrical properties, part i: theory, Meas. Sci. Technol., 13 (2002), 1848–1854. doi: 10.1088/0957-0233/13/12/307
|
| [60] |
H. Wang, K. Liu, Y. Wu, S. Wang, Z. Zhang, F. Li, et al., Image reconstruction for electrical impedance tomography using radial basis function neural network based on hybrid particle swarm optimization algorithm, IEEE Sens. J., 21 (2021), 1926–1934. doi: 10.1109/JSEN.2020.3019309
|
| [61] |
Z. Wei, D. Liu, X. Chen, Dominant-current deep learning scheme for electrical impedance tomography, IEEE T. Biomed. Eng., 66 (2019), 2546–2555. doi: 10.1109/TBME.2019.2891676
|
| [62] |
R. Winkler, A. Rieder, Resolution-controlled conductivity discretization in electrical impedance tomography, SIAM J. Imaging Sci., 7 (2014), 2048–2077. doi: 10.1137/140958955
|
| [63] |
Y. Wu, B. Chen, K. Liu, C. Zhu, H. Pan, J. Jia, et al., Shape reconstruction with multiphase conductivity for electrical impedance tomography using improved convolutional neural network method, IEEE Sens. J., 21 (2021), 9277–9287. doi: 10.1109/JSEN.2021.3050845
|
| [64] |
Y. Zhang, H. Chen, L. Yang, K. Liu, F. Li, C. Bai, et al., A proportional genetic algorithm for image reconstruction of static electrical impedance tomography, IEEE Sens. J., 20 (2020), 15026–15033. doi: 10.1109/JSEN.2020.3012544
|
| [65] |
T. Zhu, R. Feng, J.-Q. Hao, J.-G. Zhou, H.-L. Wang, S.-Q. Wang, The application of electrical resistivity tomography to detecting a buried fault: A case study, J. Environ. Eng. Geoph., 14 (2009), 145–151. doi: 10.2113/JEEG14.3.145
|
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Marion Darbas, Jérémy Heleine, Renier Mendoza, Arrianne Crystal Velasco. Sensitivity analysis of the complete electrode model for electrical impedance tomography[J]. AIMS Mathematics, 2021, 6(7): 7333-7366. doi: 10.3934/math.2021431
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