# American Institute of Mathematical Sciences

2013, 7(1): 217-242. doi: 10.3934/ipi.2013.7.217

## Recovering boundary shape and conductivity in electrical impedance tomography

 1 University of Eastern Finland, Department of Applied Physics, P.O.Box 1627, 70211 Kuopio, Finland 2 University of Helsinki, Department of Mathematics and Statistics, P.O.Box 68, 00014 University of Helsinki, Finland, Finland 3 Department of Mathematics and Statistics, P.O. Box 68, 00014 University of Helsinki, Finland

Received  November 2011 Revised  May 2012 Published  February 2013

Electrical impedance tomography (EIT) aims to reconstruct the electric conductivity inside a physical body from current-to-voltage measurements at the boundary of the body. In practical EIT one often lacks exact knowledge of the domain boundary, and inaccurate modeling of the boundary causes artifacts in the reconstructions. A novel method is presented for recovering the boundary shape and an isotropic conductivity from EIT data. The first step is to determine the minimally anisotropic conductivity in a model domain reproducing the measured EIT data. Second, a Beltrami equation is solved, providing shape-deforming reconstruction. The algorithm is applied to simulated noisy data from a realistic electrode model, demonstrating that approximate recovery of the boundary shape and conductivity is feasible.
Citation: 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
##### References:
 [1] A. Adler, R. Guardo and Y. Berthiaume, Impedance imaging of lung ventilation: Do we need to account for chest expansion?,, IEEE Trans. Biomed. Eng., 43 (1996), 414. [2] L. Ahlfors, "Lectures on Quasiconformal Mappings,", Van Nostrand Mathematical Studies, (1966). [3] K. Astala, T. Iwaniec and G. Martin, "Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane,", Princeton Mathematical Series, 48 (2009). [4] K. Astala, M. Lassas and L. Päivärinta, Calderón's inverse problem for anisotropic conductivity in the plane,, Comm. Partial Differential Equations, 30 (2005), 207. doi: 10.1081/PDE-200044485. [5] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane,, Ann. of Math. (2), 163 (2006), 265. doi: 10.4007/annals.2006.163.265. [6] K. Astala, J. L. Mueller, L. Paivarinta and S. Siltanen, Numerical computation of complex geometrical optics solutions to the conductivity equation,, Applied and Computational Harmonic Analysis, 29 (2010), 2. doi: 10.1016/j.acha.2009.08.001. [7] D. C. Barber and B. H. Brown, Applied potential tomography,, J. Phys. E: Sci. Instrum., 17 (1984), 723. [8] A. Boyle, W. R. B. Lionheart, C. Gómez-Laberge and A. Adler, Evaluating deformation corrections in electrical impedance tomography,, in, (2008), 16. [9] A. Boyle and A. Adler, The impact of electrode area, contact impedance and boundary shape on EIT images,, Physiological Measurement, 32 (2011), 745. [10] V. Bozin, N. Lakic, V. Markovic and M. Mateljevic, Unique extremality,, J. Anal. Math. 75 (1998), 75 (1998), 299. doi: 10.1007/BF02788704. [11] R. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for non-smooth conductivities in two dimensions,, Comm. Partial Differential Equations, 22 (1997), 1009. doi: 10.1080/03605309708821292. [12] A. Calderón, On an inverse boundary value problem,, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, (1980), 65. [13] M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography,, SIAM Review, 41 (1999), 85. doi: 10.1137/S0036144598333613. [14] K.-S.Cheng, D. Isaacson, J. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography,, IEEE Trans. Biomed. Eng., 36 (1989), 918. [15] E. L. V. Costa, C. N. Chaves, S. Gomes, M. A. Beraldo, M. S. Volpe, M. R. Tucci, I. A. L. Schettino, S. H. Bohm, C. R. R. Carvalho, H. Tanaka, R. G. Lima and M. B. A. Amato, Real-time detection of pneumothorax using electrical impedance tomography,, Crit. Care. Med., 36 (2008), 1230. [16] T. Dai, C. Gomez-Laberge and A. Adler, Reconstruction of conductivity changes and electrode movements based on EIT temporal sequences,, Physiol. Meas., 29 (2008). [17] E. Gersing, B. Hoffman and M. Osypka, Influence of changing peripheral geometry on electrical impedance tomography measurements,, Medical & Biological Engineering & Computing, 34 (1996), 359. [18] A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Full-wave invisibility of active devices at all frequencies,, Comm. Math. Phys., 275 (2007), 749. doi: 10.1007/s00220-007-0311-6. [19] A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Invisibility and inverse problems,, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 55. doi: 10.1090/S0273-0979-08-01232-9. [20] A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Cloaking devices, electromagnetic wormholes and transformation optics,, SIAM Review, 51 (2009), 3. doi: 10.1137/080716827. [21] A. Greenleaf, M. Lassas and G. Uhlmann, On nonuniqueness for Calderón's inverse problem,, Math. Res. Lett., 10 (2003), 685. [22] A. Greenleaf, M. Lassas and G. Uhlmann, Anisotropic conductivities that cannot detected in EIT,, Physiological Measurement, 24 (2003), 413. [23] G. Hahn, A. Just, T. Dudykevych, I. Frerichs, J. Hinz, M. Quintel and G. Hellige, Imaging pathologic pulmonary air and fluid accumulation by functional and absolute EIT,, Physiol. Meas., 27 (2006). [24] D. Isaacson, J. Mueller, J. C. Newell and S. Siltanen, Reconstructions of chest phantoms by the d-bar method for electrical impedance tomography,, IEEE Trans. Med. Im., 23 (2004), 821. [25] D. Isaacson, J. L. Mueller, J. C. Newell and S. Siltanen, Imaging cardiac activity by the D-bar method for electrical impedance tomography,, Physiological Measurement, 27 (2006). [26] H. Jain, D. Isaacson, P. M. Edic and J. C. Newell, Electrical impedance tomography of complex conductivity distributions with noncircular boundary,, IEEE Trans. Biomed. Eng., 44 (1997), 1051. [27] J. P. Kaipio, V. Kolehmainen, E. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography,, Inverse Problems, 16 (2000), 1487. doi: 10.1088/0266-5611/16/5/321. [28] R. Kohn and M. Vogelius, Identification of an unknown conductivity by means of measurements at the boundary,, in, 14 (1984), 113. [29] R. Kohn, H. Shen, M. Vogelius and M. Weinstein, Cloaking via change of variables in electrical impedance tomography,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/1/015016. [30] K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem,, Inverse Problems and Imaging, 3 (2009), 599. doi: 10.3934/ipi.2009.3.599. [31] V. Kolehmainen, M. Vauhkonen, P. A. Karjalainen and J. P. Kaipio, Assessment of errors in static electrical impedance tomography with adjacent and trigonometric current patterns,, Physiological Measurement, 18 (1997), 289. [32] V. Kolehmainen, M. Lassas and P. Ola, The inverse conductivity problem with an imperfectly known boundary,, SIAM J. Appl. Math., 66 (2005), 365. doi: 10.1137/040612737. [33] V. Kolehmainen, M. Lassas and P. Ola, The inverse conductivity problem with an imperfectly known boundary in three dimensions,, SIAM J. Appl. Math., 67 (2007), 1440. doi: 10.1137/060666986. [34] V. Kolehmainen, M. Lassas and P. Ola, Calderón's inverse problem with an imperfectly known boundary and reconstruction up to a conformal deformation,, SIAM J. Math. Anal., 42 (2010), 1371. doi: 10.1137/080716918. [35] V. Kolehmainen, M. Lassas and P. Ola, Electrical impedance tomography problem with inaccurately known boundary and contact impedances,, IEEE Trans. Med. Im., 27 (2008), 1404. [36] M. Lassas and G. Uhlmann, On determining Riemannian manifold from the Dirichlet-to-Neumann map,, Annales Scientifiques de l'Ecole Normale Superieure (4), 34 (2001), 771. doi: 10.1016/S0012-9593(01)01076-X. [37] W. Lionheart, Conformal uniqueness results in anisotropic electrical impedance imaging,, Inverse Problems, 13 (1997), 125. doi: 10.1088/0266-5611/13/1/010. [38] W. Lionheart, Boundary shape and electrical impedance tomography,, Inverse Problems, 14 (1998), 139. doi: 10.1088/0266-5611/14/1/012. [39] A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. Math. (2), 143 (1996), 71. doi: 10.2307/2118653. [40] A. Nissinen, V. Kolehmainen and J. P. Kaipio, Compensation of modelling errors due to unknown domain boundary in electrical impedance tomography,, IEEE Trans. Med. Imag., 30 (2011), 231. [41] A. Nissinen, V. Kolehmainen and J. P. Kaipio, Reconstruction of domain boundary and conductivity in electrical impedance tomography using the approximation error approach,, International Journal for Uncertainty Quantification, 1 (2011), 203. doi: 10.1615/Int.J.UncertaintyQuantification.v1.i3.20. [42] J. Nocedal and S. J. Wright, "Numerical Optimization,", Second edition, (2006). [43] Ch. Pommerenke, "Boundary Behaviour of Conformal Maps,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 299 (1992). [44] S. Siltanen, J. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem,, Inverse Problems, 16 (2000), 681. doi: 10.1088/0266-5611/16/3/310. [45] M. Soleimaini, C. Gómez-Laberge and A. Adler, Imaging of conductivity changes and electrode movement in EIT,, Physiol. Meas., 27 (2006). [46] E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography,, SIAM J. Appl. Math., 52 (1992), 1023. doi: 10.1137/0152060. [47] K. Strebel, On the existence of extremal Teichmüller mappings,, J. Anal. Math., 30 (1976), 464. [48] J. Sylvester, An anisotropic inverse boundary value problem,, Comm. Pure Appl. Math., 43 (1990), 201. doi: 10.1002/cpa.3160430203.

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##### References:
 [1] A. Adler, R. Guardo and Y. Berthiaume, Impedance imaging of lung ventilation: Do we need to account for chest expansion?,, IEEE Trans. Biomed. Eng., 43 (1996), 414. [2] L. Ahlfors, "Lectures on Quasiconformal Mappings,", Van Nostrand Mathematical Studies, (1966). [3] K. Astala, T. Iwaniec and G. Martin, "Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane,", Princeton Mathematical Series, 48 (2009). [4] K. Astala, M. Lassas and L. Päivärinta, Calderón's inverse problem for anisotropic conductivity in the plane,, Comm. Partial Differential Equations, 30 (2005), 207. doi: 10.1081/PDE-200044485. [5] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane,, Ann. of Math. (2), 163 (2006), 265. doi: 10.4007/annals.2006.163.265. [6] K. Astala, J. L. Mueller, L. Paivarinta and S. Siltanen, Numerical computation of complex geometrical optics solutions to the conductivity equation,, Applied and Computational Harmonic Analysis, 29 (2010), 2. doi: 10.1016/j.acha.2009.08.001. [7] D. C. Barber and B. H. Brown, Applied potential tomography,, J. Phys. E: Sci. Instrum., 17 (1984), 723. [8] A. Boyle, W. R. B. Lionheart, C. Gómez-Laberge and A. Adler, Evaluating deformation corrections in electrical impedance tomography,, in, (2008), 16. [9] A. Boyle and A. Adler, The impact of electrode area, contact impedance and boundary shape on EIT images,, Physiological Measurement, 32 (2011), 745. [10] V. Bozin, N. Lakic, V. Markovic and M. Mateljevic, Unique extremality,, J. Anal. Math. 75 (1998), 75 (1998), 299. doi: 10.1007/BF02788704. [11] R. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for non-smooth conductivities in two dimensions,, Comm. Partial Differential Equations, 22 (1997), 1009. doi: 10.1080/03605309708821292. [12] A. Calderón, On an inverse boundary value problem,, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, (1980), 65. [13] M. Cheney, D. Isaacson and J. C. Newell, Electrical impedance tomography,, SIAM Review, 41 (1999), 85. doi: 10.1137/S0036144598333613. [14] K.-S.Cheng, D. Isaacson, J. C. Newell and D. G. Gisser, Electrode models for electric current computed tomography,, IEEE Trans. Biomed. Eng., 36 (1989), 918. [15] E. L. V. Costa, C. N. Chaves, S. Gomes, M. A. Beraldo, M. S. Volpe, M. R. Tucci, I. A. L. Schettino, S. H. Bohm, C. R. R. Carvalho, H. Tanaka, R. G. Lima and M. B. A. Amato, Real-time detection of pneumothorax using electrical impedance tomography,, Crit. Care. Med., 36 (2008), 1230. [16] T. Dai, C. Gomez-Laberge and A. Adler, Reconstruction of conductivity changes and electrode movements based on EIT temporal sequences,, Physiol. Meas., 29 (2008). [17] E. Gersing, B. Hoffman and M. Osypka, Influence of changing peripheral geometry on electrical impedance tomography measurements,, Medical & Biological Engineering & Computing, 34 (1996), 359. [18] A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Full-wave invisibility of active devices at all frequencies,, Comm. Math. Phys., 275 (2007), 749. doi: 10.1007/s00220-007-0311-6. [19] A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Invisibility and inverse problems,, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 55. doi: 10.1090/S0273-0979-08-01232-9. [20] A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Cloaking devices, electromagnetic wormholes and transformation optics,, SIAM Review, 51 (2009), 3. doi: 10.1137/080716827. [21] A. Greenleaf, M. Lassas and G. Uhlmann, On nonuniqueness for Calderón's inverse problem,, Math. Res. Lett., 10 (2003), 685. [22] A. Greenleaf, M. Lassas and G. Uhlmann, Anisotropic conductivities that cannot detected in EIT,, Physiological Measurement, 24 (2003), 413. [23] G. Hahn, A. Just, T. Dudykevych, I. Frerichs, J. Hinz, M. Quintel and G. Hellige, Imaging pathologic pulmonary air and fluid accumulation by functional and absolute EIT,, Physiol. Meas., 27 (2006). [24] D. Isaacson, J. Mueller, J. C. Newell and S. Siltanen, Reconstructions of chest phantoms by the d-bar method for electrical impedance tomography,, IEEE Trans. Med. Im., 23 (2004), 821. [25] D. Isaacson, J. L. Mueller, J. C. Newell and S. Siltanen, Imaging cardiac activity by the D-bar method for electrical impedance tomography,, Physiological Measurement, 27 (2006). [26] H. Jain, D. Isaacson, P. M. Edic and J. C. Newell, Electrical impedance tomography of complex conductivity distributions with noncircular boundary,, IEEE Trans. Biomed. Eng., 44 (1997), 1051. [27] J. P. Kaipio, V. Kolehmainen, E. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography,, Inverse Problems, 16 (2000), 1487. doi: 10.1088/0266-5611/16/5/321. [28] R. Kohn and M. Vogelius, Identification of an unknown conductivity by means of measurements at the boundary,, in, 14 (1984), 113. [29] R. Kohn, H. Shen, M. Vogelius and M. Weinstein, Cloaking via change of variables in electrical impedance tomography,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/1/015016. [30] K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, Regularized D-bar method for the inverse conductivity problem,, Inverse Problems and Imaging, 3 (2009), 599. doi: 10.3934/ipi.2009.3.599. [31] V. Kolehmainen, M. Vauhkonen, P. A. Karjalainen and J. P. Kaipio, Assessment of errors in static electrical impedance tomography with adjacent and trigonometric current patterns,, Physiological Measurement, 18 (1997), 289. [32] V. Kolehmainen, M. Lassas and P. Ola, The inverse conductivity problem with an imperfectly known boundary,, SIAM J. Appl. Math., 66 (2005), 365. doi: 10.1137/040612737. [33] V. Kolehmainen, M. Lassas and P. Ola, The inverse conductivity problem with an imperfectly known boundary in three dimensions,, SIAM J. Appl. Math., 67 (2007), 1440. doi: 10.1137/060666986. [34] V. Kolehmainen, M. Lassas and P. Ola, Calderón's inverse problem with an imperfectly known boundary and reconstruction up to a conformal deformation,, SIAM J. Math. Anal., 42 (2010), 1371. doi: 10.1137/080716918. [35] V. Kolehmainen, M. Lassas and P. Ola, Electrical impedance tomography problem with inaccurately known boundary and contact impedances,, IEEE Trans. Med. Im., 27 (2008), 1404. [36] M. Lassas and G. Uhlmann, On determining Riemannian manifold from the Dirichlet-to-Neumann map,, Annales Scientifiques de l'Ecole Normale Superieure (4), 34 (2001), 771. doi: 10.1016/S0012-9593(01)01076-X. [37] W. Lionheart, Conformal uniqueness results in anisotropic electrical impedance imaging,, Inverse Problems, 13 (1997), 125. doi: 10.1088/0266-5611/13/1/010. [38] W. Lionheart, Boundary shape and electrical impedance tomography,, Inverse Problems, 14 (1998), 139. doi: 10.1088/0266-5611/14/1/012. [39] A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. Math. (2), 143 (1996), 71. doi: 10.2307/2118653. [40] A. Nissinen, V. Kolehmainen and J. P. Kaipio, Compensation of modelling errors due to unknown domain boundary in electrical impedance tomography,, IEEE Trans. Med. Imag., 30 (2011), 231. [41] A. Nissinen, V. Kolehmainen and J. P. Kaipio, Reconstruction of domain boundary and conductivity in electrical impedance tomography using the approximation error approach,, International Journal for Uncertainty Quantification, 1 (2011), 203. doi: 10.1615/Int.J.UncertaintyQuantification.v1.i3.20. [42] J. Nocedal and S. J. Wright, "Numerical Optimization,", Second edition, (2006). [43] Ch. Pommerenke, "Boundary Behaviour of Conformal Maps,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 299 (1992). [44] S. Siltanen, J. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem,, Inverse Problems, 16 (2000), 681. doi: 10.1088/0266-5611/16/3/310. [45] M. Soleimaini, C. Gómez-Laberge and A. Adler, Imaging of conductivity changes and electrode movement in EIT,, Physiol. Meas., 27 (2006). [46] E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography,, SIAM J. Appl. Math., 52 (1992), 1023. doi: 10.1137/0152060. [47] K. Strebel, On the existence of extremal Teichmüller mappings,, J. Anal. Math., 30 (1976), 464. [48] J. Sylvester, An anisotropic inverse boundary value problem,, Comm. Pure Appl. Math., 43 (1990), 201. doi: 10.1002/cpa.3160430203.
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