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November  2014, 8(4): 1013-1031. doi: 10.3934/ipi.2014.8.1013

A real-time D-bar algorithm for 2-D electrical impedance tomography data

1. 

Department of Mathematics, Colorado State University, Fort Collins, CO, United States

2. 

Department of Mathematics and School of Biomedical Engineering, Colorado State University, Fort Collins, CO, United States

Received  March 2014 Revised  September 2014 Published  November 2014

The aim of this paper is to show the feasibility of the D-bar method for real-time 2-D EIT reconstructions. A fast implementation of the D-bar method for reconstructing conductivity changes on a 2-D chest-shaped domain is described. Cross-sectional difference images from the chest of a healthy human subject are presented, demonstrating what can be achieved in real time. The images constitute the first D-bar images from EIT data on a human subject collected on a pairwise current injection system.
Citation: Melody Dodd, Jennifer L. Mueller. A real-time D-bar algorithm for 2-D electrical impedance tomography data. Inverse Problems and Imaging, 2014, 8 (4) : 1013-1031. doi: 10.3934/ipi.2014.8.1013
References:
[1]

G. M. Amdahl, Validity of the single-processor approach to achieving large scale computing capabilities, AFIPS Conference Proceedings, 30 (1967), 483-485. doi: 10.1145/1465482.1465560.

[2]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Annals of Math., 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265.

[3]

K. Astala, J. L. Mueller, L. Päivärinta and S. Siltanen, Numerical computation of complex geometrical optics solutions to the conductivity equation, Applied and Computational Harmonic Analysis, 29 (2010), 2-17. doi: 10.1016/j.acha.2009.08.001.

[4]

K. Astala, J. L. Mueller, A. Perämäki, L. Päivärinta and S. Siltanen, Direct electrical impedance tomography for nonsmooth conductivities, Inverse Problems and Imaging, 5 (2011), 531-549. doi: 10.3934/ipi.2011.5.531.

[5]

L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99-R136. doi: 10.1088/0266-5611/18/6/201.

[6]

B. H. Brown, D. C. Barber, A. H. Morice, A. Leathard and A. Sinton, Cardiac and respiratory related electrical impedance changes in the human thorax, IEEE Trans. Biomed. Eng., 41 (1994), 729-734. doi: 10.1109/10.310088.

[7]

R. M. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations, 22 (1997), 1009-1027. doi: 10.1080/03605309708821292.

[8]

A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl, 16 (2008), 19-33. doi: 10.1515/jiip.2008.002.

[9]

A. P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matemàtica, (1980), 65-73.

[10]

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. P. Amato, Real-time detection of pneumothorax using electrical impedance tomography, Crit. Care Med., 36 (2008), 1230-1238. doi: 10.1097/CCM.0b013e31816a0380.

[11]

E. L. V. Costa, R. Gonzalez Lima and M. B. P. Amato, Electrical impedance tomography, Current Opinion in Crit. Care, 15 (2009), 18-24. doi: 10.1097/MCC.0b013e3283220e8c.

[12]

M. DeAngelo and J. L. Mueller, 2-D D-bar reconstructions of human chest and tank data using an improved approximation to the scattering transform, Physiol. Meas., 31 (2010), 221-232. doi: 10.1088/0967-3334/31/2/008.

[13]

L. D. Faddeev, Increasing solutions of the Schrödinger equation, Sov. Phys. Dokl., 10 (1966), 1033-1035.

[14]

E. Francini, Recovering a complex coefficient in a planar domain from the Dirichlet-to-Neumann map, Inverse Problems, 16 (2000), 107-119. doi: 10.1088/0266-5611/16/1/309.

[15]

D. Freimark, M. Arad, R. Sokolover, S. Zlochiver and S. Abboud, Monitoring lung fluid content in CHF patients under intravenous diuretics treatment using bio-impedance measurements, Physiol. Meas., 28 (2007), S269-S277. doi: 10.1088/0967-3334/28/7/S20.

[16]

I. Frerichs, J. Hinz, P. Herrmann, G. Weisser, G. Hahn, T. Dudykevych, M. Quintel and G. Hellige, Detection of local lung air content by electrical impedance tomography compared with electron beam CT, J. Appl. Physiol., 93 (2002), 660-666.

[17]

I. Frerichs, J. Hinz, P. Herrmann, G. Weisser, G. Hahn, M. Quintel and G. Hellige, Regional lung perfusion as determined by electrical impedance tomography in comparison with electron beam CT imaging, IEEE Trans. Med. Imaging, 21 (2002), 646-652. doi: 10.1109/TMI.2002.800585.

[18]

I. Frerichs, G. Schmitz, S. Pulletz, D. Schädler, G. Zick, J. Scholz and N. Weiler, Reproducibility of regional lung ventilation distribution determined by electrical impedance tomography during mechanical ventilation, Physiol. Meas., 28 (2007), S261-S267. doi: 10.1088/0967-3334/28/7/S19.

[19]

S. J. Hamilton and J. L. Mueller, Direct EIT reconstructions of complex conductivities on a chest-shaped domain in 2-D, IEEE Trans. Med. Imaging, 32 (2013), 757-769.

[20]

S. J. Hamilton, C. N. L. Herrera, J. L. Mueller and A. Von Herrmann, A direct D-bar reconstruction algorithm for recovering a complex conductivity in 2-D, Inverse Problems, 28 (2012), 095005, 24pp. doi: 10.1088/0266-5611/28/9/095005.

[21]

C. N. L. Herrera, M. F. M. Vallejo, J. L. Mueller and R. Lima, Direct 2-D reconstructions of conductivity and permittivity from EIT data on a human chest, Medical Imaging, IEEE Transac, pp (2014), p1. doi: 10.1109/TMI.2014.2354333.

[22]

D. Isaacson, J. L. 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), 821-828. doi: 10.1109/TMI.2004.827482.

[23]

D. Isaacson, J. L. Mueller, J. C. Newell and S. Siltanen, Imaging cardiac activity by the D-bar method for electrical impedance tomography, Physiol. Meas., 27 (2006), S43-S50. doi: 10.1088/0967-3334/27/5/S04.

[24]

K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, D-bar method for electrical impedance tomography with discontinuous conductivities, SIAM J. Appl. Math., 67 (2007), 893-913. doi: 10.1137/060656930.

[25]

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), 599-624. doi: 10.3934/ipi.2009.3.599.

[26]

K. Knudsen, On the Inverse Conductivity Problem, Ph.D. thesis, Aalborg University, Denmark, 2002.

[27]

K. Knudsen, A new direct method for reconstructing isotropic conductivities in the plane, Physiol. Meas., 24 (2003), 391-401. doi: 10.1088/0967-3334/24/2/351.

[28]

K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane, Comm. Partial Differential Equations, 29 (2004) 361-381. doi: 10.1081/PDE-120030401.

[29]

K. Knudsen, J. L. Mueller and S. Siltanen, Numerical solution method for the dbar-equation in the plane, J. Comp. Phys., 198 (2004), 500-517. doi: 10.1016/j.jcp.2004.01.028.

[30]

P. W. Kunst, A. Vonk Noordegraaf, O. S. Hoekstra, P. E. Postmus and P. M. de Vries, Ventilation and perfusion imaging by electrical impedance tomography: a comparison with radionuclide scanning, Physiol. Meas., 19 (1998), 481-490. doi: 10.1088/0967-3334/19/4/003.

[31]

P. W. Kunst, A. Vonk Noordegraaf, E. Raaijmakers, J. Bakker, A. B. Groeneveld, P. E. Postmus and P. M. de Vries, Electrical impedance tomography in the assessment of extravascular lung water in noncardiogenic acute respiratory failure, Chest, 116 (1999), 1695-1702. doi: 10.1378/chest.116.6.1695.

[32]

M. Mellenthin, E. L. de Camargo, F. de Moura, T. Santos, J. L. Mueller and R. G. Lima, Complex voltage measurements with active electrodes in electrical impedance tomography, in preparation.

[33]

J. L. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM J. Sci. Comp., 24 (2003), 1232-1266. doi: 10.1137/S1064827501394568.

[34]

J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, SIAM, 2012. doi: 10.1137/1.9781611972344.

[35]

E. K. Murphy and J. L. Mueller, Effect of domain-shape modeling and measurement errors on the 2-D D-bar method for electrical impedance tomography, IEEE Trans. Med. Im., 28 (2009), 1576-1584.

[36]

A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Math., 143 (1996), 71-96. doi: 10.2307/2118653.

[37]

S. Siltanen, J. L. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem, Inverse Problems, 16 (2000), 681-699. doi: 10.1088/0266-5611/16/3/310.

[38]

H. Smit, A. Vonk Noordegraaf, J. T. Marcus, A. Boonstra, P. M. de Vries and P. E. Postmus, Determinants of pulmonary perfusion measured by electrical impedance tomography, Eur. J. Appl. Physiol., 92 (2004), 45-49. doi: 10.1007/s00421-004-1043-3.

[39]

Y. Saad and M. H. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comp., 7 (1986), 856-869. doi: 10.1137/0907058.

[40]

G. Vainikko, Fast solvers of the Lippmann-Schwinger equation, in Direct and Inverse Problems of Mathematical Physics, (eds. R. P. Gilbert, J. Kajiwara, Y. S. Xu ), Kluwer Academic Publishers (2000), 423-440. doi: 10.1007/978-1-4757-3214-6_25.

[41]

J. A. Victorino, J. B. Borges, V. N. Okamoto, G. F. J. Matos, M. R. Tucci, M. P. R. Caramez, H. Tanaka, D. C. B. Santos, C. S. V. Barbas, C. R. R. Carvalho and M. B. P. Amato, Imbalances in regional lung ventilation: A validation study on electrical impedance tomography, Am. J. Respir. Crit. Care Med., 169 (2004), 791-800. doi: 10.1164/rccm.200301-133OC.

[42]

A. Vonk Noordegraaf, T. J. C. Faes, A. Janse, J. T. Marcus, J. G. F. Bronzwaer, P. E. Postmus and P. M. de Vries, Noninvasive assessment of right ventricular diastolic function by electrical impedance tomography, Chest, 111 (1997), 1222-1228. doi: 10.1378/chest.111.5.1222.

[43]

A. Vonk Noordegraaf, P. W. Kunst, A. Janse, J. T. Marcus, P. E. Postmus, T. J. Faes and P. M. de Vries, Pulmonary perfusion measured by means of electrical impedance tomography, J. Electron. Imaging, 10 (1998), 608-619.

show all references

References:
[1]

G. M. Amdahl, Validity of the single-processor approach to achieving large scale computing capabilities, AFIPS Conference Proceedings, 30 (1967), 483-485. doi: 10.1145/1465482.1465560.

[2]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Annals of Math., 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265.

[3]

K. Astala, J. L. Mueller, L. Päivärinta and S. Siltanen, Numerical computation of complex geometrical optics solutions to the conductivity equation, Applied and Computational Harmonic Analysis, 29 (2010), 2-17. doi: 10.1016/j.acha.2009.08.001.

[4]

K. Astala, J. L. Mueller, A. Perämäki, L. Päivärinta and S. Siltanen, Direct electrical impedance tomography for nonsmooth conductivities, Inverse Problems and Imaging, 5 (2011), 531-549. doi: 10.3934/ipi.2011.5.531.

[5]

L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99-R136. doi: 10.1088/0266-5611/18/6/201.

[6]

B. H. Brown, D. C. Barber, A. H. Morice, A. Leathard and A. Sinton, Cardiac and respiratory related electrical impedance changes in the human thorax, IEEE Trans. Biomed. Eng., 41 (1994), 729-734. doi: 10.1109/10.310088.

[7]

R. M. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations, 22 (1997), 1009-1027. doi: 10.1080/03605309708821292.

[8]

A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl, 16 (2008), 19-33. doi: 10.1515/jiip.2008.002.

[9]

A. P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matemàtica, (1980), 65-73.

[10]

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. P. Amato, Real-time detection of pneumothorax using electrical impedance tomography, Crit. Care Med., 36 (2008), 1230-1238. doi: 10.1097/CCM.0b013e31816a0380.

[11]

E. L. V. Costa, R. Gonzalez Lima and M. B. P. Amato, Electrical impedance tomography, Current Opinion in Crit. Care, 15 (2009), 18-24. doi: 10.1097/MCC.0b013e3283220e8c.

[12]

M. DeAngelo and J. L. Mueller, 2-D D-bar reconstructions of human chest and tank data using an improved approximation to the scattering transform, Physiol. Meas., 31 (2010), 221-232. doi: 10.1088/0967-3334/31/2/008.

[13]

L. D. Faddeev, Increasing solutions of the Schrödinger equation, Sov. Phys. Dokl., 10 (1966), 1033-1035.

[14]

E. Francini, Recovering a complex coefficient in a planar domain from the Dirichlet-to-Neumann map, Inverse Problems, 16 (2000), 107-119. doi: 10.1088/0266-5611/16/1/309.

[15]

D. Freimark, M. Arad, R. Sokolover, S. Zlochiver and S. Abboud, Monitoring lung fluid content in CHF patients under intravenous diuretics treatment using bio-impedance measurements, Physiol. Meas., 28 (2007), S269-S277. doi: 10.1088/0967-3334/28/7/S20.

[16]

I. Frerichs, J. Hinz, P. Herrmann, G. Weisser, G. Hahn, T. Dudykevych, M. Quintel and G. Hellige, Detection of local lung air content by electrical impedance tomography compared with electron beam CT, J. Appl. Physiol., 93 (2002), 660-666.

[17]

I. Frerichs, J. Hinz, P. Herrmann, G. Weisser, G. Hahn, M. Quintel and G. Hellige, Regional lung perfusion as determined by electrical impedance tomography in comparison with electron beam CT imaging, IEEE Trans. Med. Imaging, 21 (2002), 646-652. doi: 10.1109/TMI.2002.800585.

[18]

I. Frerichs, G. Schmitz, S. Pulletz, D. Schädler, G. Zick, J. Scholz and N. Weiler, Reproducibility of regional lung ventilation distribution determined by electrical impedance tomography during mechanical ventilation, Physiol. Meas., 28 (2007), S261-S267. doi: 10.1088/0967-3334/28/7/S19.

[19]

S. J. Hamilton and J. L. Mueller, Direct EIT reconstructions of complex conductivities on a chest-shaped domain in 2-D, IEEE Trans. Med. Imaging, 32 (2013), 757-769.

[20]

S. J. Hamilton, C. N. L. Herrera, J. L. Mueller and A. Von Herrmann, A direct D-bar reconstruction algorithm for recovering a complex conductivity in 2-D, Inverse Problems, 28 (2012), 095005, 24pp. doi: 10.1088/0266-5611/28/9/095005.

[21]

C. N. L. Herrera, M. F. M. Vallejo, J. L. Mueller and R. Lima, Direct 2-D reconstructions of conductivity and permittivity from EIT data on a human chest, Medical Imaging, IEEE Transac, pp (2014), p1. doi: 10.1109/TMI.2014.2354333.

[22]

D. Isaacson, J. L. 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), 821-828. doi: 10.1109/TMI.2004.827482.

[23]

D. Isaacson, J. L. Mueller, J. C. Newell and S. Siltanen, Imaging cardiac activity by the D-bar method for electrical impedance tomography, Physiol. Meas., 27 (2006), S43-S50. doi: 10.1088/0967-3334/27/5/S04.

[24]

K. Knudsen, M. Lassas, J. L. Mueller and S. Siltanen, D-bar method for electrical impedance tomography with discontinuous conductivities, SIAM J. Appl. Math., 67 (2007), 893-913. doi: 10.1137/060656930.

[25]

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), 599-624. doi: 10.3934/ipi.2009.3.599.

[26]

K. Knudsen, On the Inverse Conductivity Problem, Ph.D. thesis, Aalborg University, Denmark, 2002.

[27]

K. Knudsen, A new direct method for reconstructing isotropic conductivities in the plane, Physiol. Meas., 24 (2003), 391-401. doi: 10.1088/0967-3334/24/2/351.

[28]

K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane, Comm. Partial Differential Equations, 29 (2004) 361-381. doi: 10.1081/PDE-120030401.

[29]

K. Knudsen, J. L. Mueller and S. Siltanen, Numerical solution method for the dbar-equation in the plane, J. Comp. Phys., 198 (2004), 500-517. doi: 10.1016/j.jcp.2004.01.028.

[30]

P. W. Kunst, A. Vonk Noordegraaf, O. S. Hoekstra, P. E. Postmus and P. M. de Vries, Ventilation and perfusion imaging by electrical impedance tomography: a comparison with radionuclide scanning, Physiol. Meas., 19 (1998), 481-490. doi: 10.1088/0967-3334/19/4/003.

[31]

P. W. Kunst, A. Vonk Noordegraaf, E. Raaijmakers, J. Bakker, A. B. Groeneveld, P. E. Postmus and P. M. de Vries, Electrical impedance tomography in the assessment of extravascular lung water in noncardiogenic acute respiratory failure, Chest, 116 (1999), 1695-1702. doi: 10.1378/chest.116.6.1695.

[32]

M. Mellenthin, E. L. de Camargo, F. de Moura, T. Santos, J. L. Mueller and R. G. Lima, Complex voltage measurements with active electrodes in electrical impedance tomography, in preparation.

[33]

J. L. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM J. Sci. Comp., 24 (2003), 1232-1266. doi: 10.1137/S1064827501394568.

[34]

J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, SIAM, 2012. doi: 10.1137/1.9781611972344.

[35]

E. K. Murphy and J. L. Mueller, Effect of domain-shape modeling and measurement errors on the 2-D D-bar method for electrical impedance tomography, IEEE Trans. Med. Im., 28 (2009), 1576-1584.

[36]

A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals of Math., 143 (1996), 71-96. doi: 10.2307/2118653.

[37]

S. Siltanen, J. L. Mueller and D. Isaacson, An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem, Inverse Problems, 16 (2000), 681-699. doi: 10.1088/0266-5611/16/3/310.

[38]

H. Smit, A. Vonk Noordegraaf, J. T. Marcus, A. Boonstra, P. M. de Vries and P. E. Postmus, Determinants of pulmonary perfusion measured by electrical impedance tomography, Eur. J. Appl. Physiol., 92 (2004), 45-49. doi: 10.1007/s00421-004-1043-3.

[39]

Y. Saad and M. H. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comp., 7 (1986), 856-869. doi: 10.1137/0907058.

[40]

G. Vainikko, Fast solvers of the Lippmann-Schwinger equation, in Direct and Inverse Problems of Mathematical Physics, (eds. R. P. Gilbert, J. Kajiwara, Y. S. Xu ), Kluwer Academic Publishers (2000), 423-440. doi: 10.1007/978-1-4757-3214-6_25.

[41]

J. A. Victorino, J. B. Borges, V. N. Okamoto, G. F. J. Matos, M. R. Tucci, M. P. R. Caramez, H. Tanaka, D. C. B. Santos, C. S. V. Barbas, C. R. R. Carvalho and M. B. P. Amato, Imbalances in regional lung ventilation: A validation study on electrical impedance tomography, Am. J. Respir. Crit. Care Med., 169 (2004), 791-800. doi: 10.1164/rccm.200301-133OC.

[42]

A. Vonk Noordegraaf, T. J. C. Faes, A. Janse, J. T. Marcus, J. G. F. Bronzwaer, P. E. Postmus and P. M. de Vries, Noninvasive assessment of right ventricular diastolic function by electrical impedance tomography, Chest, 111 (1997), 1222-1228. doi: 10.1378/chest.111.5.1222.

[43]

A. Vonk Noordegraaf, P. W. Kunst, A. Janse, J. T. Marcus, P. E. Postmus, T. J. Faes and P. M. de Vries, Pulmonary perfusion measured by means of electrical impedance tomography, J. Electron. Imaging, 10 (1998), 608-619.

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