Skip to main content
SpringerLink
Log in

Multi-Frequency Electrical Impedance Tomography and Magnetic Resonance Electrical Impedance Tomography

  • Chapter
  • First Online:
Mathematical Modeling in Biomedical Imaging I

Part of the book series: Lecture Notes in Mathematics ((LNMBIOS,volume 1983))

Abstract

Medical imaging modalities such as computerized tomography (CT) using X-ray and magnetic resonance imaging (MRI) have been well established providing three-dimensional high-resolution images of anatomical structures inside the human body and computer-based mathematical methods have played an essential role for their image reconstructions. However, since each imaging modality has its own limitations, there have been much research efforts to expand our ability to see through the human body in different ways. Lately, biomedical imaging research has been dealing with new imaging techniques to provide knowledge of physiologic functions and pathological conditions in addition to structural information. Electrical impedance tomography (EIT) is one of such attempts for functional imaging and monitoring of physiological events.

EIT is based on numerous experimental findings that different biological tissues inside the human body have different electrical properties of conductivity and permittivity. Viewing the human body as a mixture of distributed resistors and capacitors, we can evaluate its internal electrical properties by injecting a sinusoidal current between a pair of surface electrodes and measuring voltage drops at different positions on the surface. EIT is based on this bioimpedance measurement technique using multiple surface electrodes as many as 8 to 256. See Figs. 1.1a and 1.2. In EIT, we inject linearly independent patterns of sinusoidal currents through all or chosen pairs of electrodes and measure induced boundary voltages on all or selected electrodes. The measured boundary current–voltage data set is used to reconstruct cross-sectional images of the internal conductivity and/or permittivity distribution. The basic idea of the impedance imaging was introduced by Henderson and Webster in 1978 [13], and the first clinical application of a medical EIT system was described by Barber and Brown [7]. Since then, EIT has received considerable attention and several review papers described numerous aspects of the EIT technique [8, 10, 14, 36, 49, 62]. To support the theoretical basis of the EIT system, mathematical theories such as uniqueness and stability were developed [2, 6, 16, 19, 25, 29, 38, 39, 48, 52, 57–59, 61] since Calder´on’s pioneering contribution in 1980 [9].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Alessandrini and R. Magnanini, The index of isolated critical points and solutions of elliptic equations in the plane, Ann. Scoula. Norm. Sup. Pisa Cl Sci. 19 (1992), 567–589.

    MathSciNet  MATH  Google Scholar 

  2. G. Alessandrini, E. Rosset, and J.K. Seo, Optimal size estimates for the inverse conductivity poblem with one measurements, Proc. Amer. Math. Soc 128 (2000), 53–64.

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Ammari and H. Kang, Reconstruction of small inhomogeneities from boundary measurement, Lecture Notes in Mathematics, vol. 1846, Springer, Berlin, 2004.

    Google Scholar 

  4. H. Ammari, O. Kwon, J.K. Seo, and E.J. Woo, T-scan electrical impedance imaging system for anomaly detection, IEEE Trans. Biomed. Eng. 65 (2004), 252–266.

    MathSciNet  MATH  Google Scholar 

  5. H. Ammari and J.K. Seo, An accurate formula for the reconstruction of conductivity inhomogeneity, Advances in Applied Math. 30 (2003), 679–705.

    Article  MathSciNet  MATH  Google Scholar 

  6. K. Astala and L. Paivarinta, Calderon’s inverse conductivity problem in the plane, Annals of Mathematics 163 (2006), 265–299.

    Article  MathSciNet  MATH  Google Scholar 

  7. D.C. Barber and B.H. Brown, Applied potential tomography,. Phys. E: Sci. Instruments 17 (1984), 723.

    Article  Google Scholar 

  8. K. Boone, D. Barber, and B. Brown, Imaging with electricity: report of the european concerted action on impedance tomography, J. Med. Eng. Tech 21 (1997), 201–232.

    Article  Google Scholar 

  9. A.P. Calderon, On an inverse boundary value problem, seminar on numerical analysis and its applications to continuum physics(1980), rio de janeiro, Mat. apl. comput (reprinted) 25 (2006(1980)), 133–138.

    Google Scholar 

  10. M. Cheney, D. Isaacson, and J.C. Newell, Electrical impedance tomography, SIAM Review 41 (1999), 85–101.

    Article  MathSciNet  MATH  Google Scholar 

  11. D.C. Ghiglia and M.D. Pritt, Two-dimensional phase unwrapping: Theory, algorithms and software, Wiley Interscience, New York, 1998.

    MATH  Google Scholar 

  12. H. Griffiths, H.T. Leung, and R.J. Williams, Imaging the complex impedance of the thorax, Clin Phys Physiol Meas. 13 Suppl A (1992), 77–81.

    Google Scholar 

  13. R.P. Henderson and J.G. Webster, An impedance camera for spatially specific measurements of the thorax, IEEE Trans. Biomed. Eng. 25 (1978), 250–254.

    Article  Google Scholar 

  14. D.S. Holder (ed.), Electrical impedance tomography: Methods, history and applications, Series in medical physics and biomedical engineering, Institute of Physics, 2004.

    Google Scholar 

  15. Y.Z. Ider and O. Birgul, Use of the magnetic field generated by the internal distribution of injected currents for electrical impedance tomography (mr-eit), Elektrik 6 (1998), 215–225.

    Google Scholar 

  16. V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficient, Comm.Pure Appl. Math. 41 (1988), 856–877.

    Google Scholar 

  17. M. Joy, G. Scott, and R. Henkelman, In vivo detection of applied electric currents by magnetic resonance imaging, Magn. Reson. Imag. 7 (1989), 89.

    Article  Google Scholar 

  18. H. Kang and J.K. Seo, Identification of domains with near-extreme conductivity: global stability and error estimates, Inverse Problems 15 (1999), 851–867.

    Article  MathSciNet  MATH  Google Scholar 

  19. C.E. Kenig, J. Sjöstrand, and G. Uhlmann, The calderón problem with partial data, Annals of Mathematics 165 (2007), 567–591.

    Article  MathSciNet  MATH  Google Scholar 

  20. H.S. Khang, B.I. Lee, S.H. Oh, E.J. Woo, S.Y. Lee, M.H. Cho, O. Kwon, J.R. Yoon, and J.K. Seo, J-substitution algorithm in magnetic resonance electrical impedance tomography (MREIT): phantom experiments for static resistivity images, IEEE Trans. Med. Imaging 21 (2002), 695–702.

    Article  Google Scholar 

  21. S. Kim, J.K. Seo, E.J. Woo, and H. Zribi, Multi-frequency trans-admittance scanner: mathematical framework and feasibility, SIAM J. Appl. Math. 69(1) (2008), 22–36.

    MathSciNet  MATH  Google Scholar 

  22. S.W. Kim, O. Kwon, J.K. Seo, and J.R. Yoon, On a nonlinear partial differential equation arising in magnetic resonance electrical impedance tomography, SIAM J. Math Anal. 34 (2002), 511–526.

    Article  MathSciNet  MATH  Google Scholar 

  23. S.W. Kim, J.K. Seo, E.J. Woo, and H. Zribi, Frequency difference trans-admittance scanner for breast cancer detection, preprint.

    Google Scholar 

  24. Y.J. Kim, O. Kwon, J.K. Seo, and E.J. Woo, Uniqueness and convergence of conductivity image reconstruction in magnetic resonance electrical impedance tomography, Inv. Prob. 19 (2003), 1213–1225.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Comm.Pure Appl. Math. 37 (1984), 113–123.

    Google Scholar 

  26. O. Kwon, C.J. Park, E.J. Park, J.K. Seo, and E.J. Woo, Electrical conductivity imaging using a variational method in b z -based mreit, Inverse Problems 21 (2005), 969–980.

    Article  MathSciNet  MATH  Google Scholar 

  27. O. Kwon and J.K. Seo, Total size estimation and identification of multiple anomalies in the inverse conductivity problem, Inverse Problems 17 (2001), 59–75.

    Article  MathSciNet  MATH  Google Scholar 

  28. O. Kwon, J.K. Seo, and S.W. Kim, Location search techniques for a grounded conductor, SIAM J. Applied Math. 21, no. 6.

    Google Scholar 

  29. O. Kwon, J.K. Seo, and J.R. Yoon, A real-time algorithm for the location search of discontinuous concuctivites with one measurement, Comm. Pure Appl. Math. 55 (2002), 1–29.

    Article  MathSciNet  MATH  Google Scholar 

  30. O. Kwon, E.J. Woo, J.R. Yoon, and J.K. Seo, Magnetic resonance electrical impedance tomography (mreit): simulation study of j-substitution algorithm, IEEE Trans. Biomed. Eng. 49 (2002), 160–167.

    Article  Google Scholar 

  31. O. Kwon, J. R. Yoon, J.K. Seo, E.J. Woo, and Y.G. Cho, Estimation of anomaly location and size using electrical impedance tomography, IEEE Trans. Biomed. Eng. 50 (2003), 89–96.

    Article  Google Scholar 

  32. B.I. Lee, S.H. Oh, E.J. Woo, S.Y. Lee, M.H. Cho, O. Kwon, J.K. Seo, and W.S. Baek, Static resistivity image of a cubic saline phantom in magnetic resonance electrical impedance tomography (mreit), Physiol. Meas. 24 (2003), 579–589.

    Article  Google Scholar 

  33. J. Lee, E.J. Woo, H. Zribi, S.W. Kim, and J.K. Seo, Multi-frequency electrical impedance tomography (mfeit): frequency-difference imaging and its feasibility, preprint.

    Google Scholar 

  34. J.J. Liu, H.C. Pyo, J.K. Seo, and E.J. Woo, Convergence properties and stability issues in mreit algorithm, Contemporary Mathematics 408 (2006), 201–216.

    Article  MathSciNet  Google Scholar 

  35. J.J. Liu, J.K. Seo, M. Sini, and E.J. Woo, On the convergence of the harmonic b z algorithm in magnetic resonance electrical impedance tomography, SIAM J. Appl. Math. 67(5) (2007), 1259–1282.

    MathSciNet  MATH  Google Scholar 

  36. P. Metherall, Three dimensional electrical impedance tomography of the human thorax, University of Sheffield, Dept. of Med. Phys and Clin. Eng., 1998.

    Google Scholar 

  37. P. Metherall, D.C. Barber, R.H. Smallwood, and B.H. Brown, Three-dimensional electrical imepdance tomography, Nature 380 (1996), 509–512.

    Article  Google Scholar 

  38. A. Nachman, Reconstructions from boundary measurements, Ann. Math. 128 (1988), 531–577.

    Article  MathSciNet  MATH  Google Scholar 

  39. A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. Math. 142 (1996), 71–96.

    Article  MathSciNet  Google Scholar 

  40. B.M. Eyuboglu O. Birgul and Y.Z. Ider, Current constrained voltage scaled reconstruction (ccvsr) algorithm for mr-eit and its performance with different probing current patterns, Phys. Med. Biol. 48 (2003), 653–671.

    Article  Google Scholar 

  41. S.H. Oh, B.I. Lee, T.S. Park, S.Y. Lee, E.J. Woo, M.H. Cho, J.K. Seo, and O. Kwon, Magnetic resonance electrical impedance tomography at 3 tesla field strength, Magn. Reson. Med. 51 (2004), 1292–1296.

    Article  Google Scholar 

  42. S.H. Oh, B.I. Lee, E.J. Woo, S.Y. Lee, M.H. Cho, O. Kwon, and J.K. Seo, Conductivity and current density image reconstruction using harmonic b z algorithm in magnetic resonance electrical impedance tomography, Phys. Med. Biol. 48 (2003), 3101–3116.

    Article  Google Scholar 

  43. T.I. Oh, J. Lee, J.K. Seo, S.W. Kim, and E.J. Woo, Feasibility of breast cancer lesion detection using multi-frequency trans-admittance scanner (tas) with 10hz to 500khz bandwidth, Physiological Measurement 28 (2007), S71–S84.

    Article  Google Scholar 

  44. C. Park, O. Kwon, E.J. Woo, and J.K. Seo, Electrical conductivity imaging using gradient b z decomposition algorithm in magnetic resonance electrical impedance tomography (mreit), IEEE Trans. Med. Imag. 23 (2004), 388–394.

    Article  Google Scholar 

  45. C. Park, E.J. Park, E.J. Woo, O. Kwon, and J.K. Seo, Static conductivity imaging using variational gradient b z algorithm in magnetic resonance electrical impedance tomography, Physiol.Meas. 25 (2004), 257–269.

    Article  Google Scholar 

  46. H.C. Pyo, O. Kwon, J.K. Seo, and E.J. Woo, Mathematical framework for bz-based mreit model in electrical impedance imaging, Compter and Mathematics with Applications 51 (2006), 817–828.

    Article  MathSciNet  MATH  Google Scholar 

  47. R. Sadleir, S. Grant, S.U. Zhang, B.I. Lee, H.C. Pyo, S.H. Oh, C. Park, E.J. Woo, S.Y. Lee, O. Kwon, and J.K. Seo, Noise analysis in magnetic resonance electrical impedance tomography at 3 and 11 t field strengths, Physiol. Meas. 26 (2005), 875–884.

    Article  Google Scholar 

  48. F. Santosa and M. Vogelius, A backprojection algorithm for electrical impedance imaging, SIAM J. Appl. Math. 50 (1990), 216–243.

    Article  MathSciNet  MATH  Google Scholar 

  49. G.J. Saulnier, R.S. Blue, J.C. Newell, D. Isaacson, and P.M. Edic, Electrical impedance tomography, IEEE Sig. Proc. Mag. 18 (2001), 31–43.

    Article  Google Scholar 

  50. G. Scott, R. Armstrong M. Joy, and R. Henkelman, Measurement of nonuniform current density by magnetic resonance, IEEE Trans. Med. Imag. 10 (1991), 362–374.

    Article  Google Scholar 

  51. G. Scott, R. Armstrong M. Joy, and R. Henkelman, Sensitivity of magnetic-resonance current density imaging, J. Mag. Res. 97 (1992), 235–254.

    Google Scholar 

  52. J.K. Seo, On the uniqueness in the inverse conductivity problem, J. Fourier Anal. Appl. 2 (1996), 227–235.

    Article  MathSciNet  MATH  Google Scholar 

  53. J.K. Seo, O. Kwon, H. Ammari, and E.J. Woo, Mathematical framework and anomaly estimation algorithm for breast cancer detection: electrical impedance technique using ts2000 configuration, IEEE Trans. Biomed. Eng. 51 (2004), 1898–1906.

    Article  Google Scholar 

  54. J.K. Seo, O. Kwon, B.I. Lee, and E.J. Woo, Reconstruction of current density distributions in axially symmetric cylindrical sections using one component of magnetic flux density: computer simulation study, Physiol. Meas. 24 (2003), 565–577.

    Article  Google Scholar 

  55. J.K. Seo, H.C. Pyo, C.J. Park, O. Kwon, and E.J. Woo, Image reconstruction of anisotropic conductivity tensor distribution in mreit: computer simulation study, Phys. Med. Biol. 49(18) (2004), 4371–4382.

    Article  Google Scholar 

  56. J.K. Seo, J.R. Yoon, E.J. Woo, and O. Kwon, Reconstruction of conductivity and current density images using only one component of magnetic field measurements, 2003, pp. 1121–1124.

    Google Scholar 

  57. E. Somersalo, D. Isaacson, and M. Cheney, Existence and uniqueness for electrode models for electric current computed tomography, SIAM J. Appl. Math. 52 (1992), 1023–40.

    Article  MathSciNet  MATH  Google Scholar 

  58. J. Sylvester and G. Uhlmann, A uniqueness theorem for an inverse boundary value problem in electrical prospection, Comm. Pure Appl. Math. 39 (1986), 91–112.

    Article  MathSciNet  MATH  Google Scholar 

  59. J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. Math. 125 (1987), 153–169.

    Article  MathSciNet  MATH  Google Scholar 

  60. A.T. Tidswell, A. Gibson, R.H. Bayford, and D.S. Holder, Three-dimensional electrical impedance tomography of human brain activity, Physiol. Meas. 22 (2001), 177–186.

    Article  Google Scholar 

  61. G. Uhlmann, Developments in inverse problems since calderdon’s foundational paper, Harmonic analysis and partial differential equations (Chicago, IL, 1996), Univ. Chicago Press (1999), 295–345.

    Google Scholar 

  62. J.G. Webster, Electrical impedance tomography, Adam Hilger, Bristol, UK, 1990.

    Google Scholar 

  63. E.J. Woo, S.Y. Lee, and C.W. Mun, Impedance tomography using internal current density distribution measured by nuclear magnetic resonance, SPIE 2299 (1994), 377–385.

    Article  Google Scholar 

  64. S. Onart Y.Z. Ider and W. Lionheart, Uniqueness and reconstruction in magnetic resonance-electrical impedance tomography (mr-eit) physiol. meas., Physiol. Meas. 24 (2003), 591–604.

    Google Scholar 

  65. N. Zhang, Electrical impedance tomography based on current density imaging, MS Thesis Dept. of Elec. Eng. Univ. of Toronto Toronto Canada (1992).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Yonsei University, 262 Seongsanno, Seodaemun, Seoul, 120-749, Korea

    Jin Keun Seo

  2. College of Electronics and Information, Kyung Hee University, Seoul, 130-701, Korea

    Eung Je Woo

Authors
  1. Jin Keun Seo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eung Je Woo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jin Keun Seo .

Editor information

Editors and Affiliations

  1. CNRS UMR 7641, Centre de Mathématiques Appliquées, École Polytechnique, Palaiseau CX, 91128, France

    Habib Ammari

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Seo, J.K., Woo, E.J. (2009). Multi-Frequency Electrical Impedance Tomography and Magnetic Resonance Electrical Impedance Tomography. In: Ammari, H. (eds) Mathematical Modeling in Biomedical Imaging I. Lecture Notes in Mathematics(), vol 1983. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03444-2_1

Download citation

Publish with us

Policies and ethics

Search

Navigation