Skip to main content
SpringerLink
Log in

On the error estimate of the harmonic B z algorithm in MREIT from noisy magnetic flux field

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

Magnetic resonance electrical impedance tomography (MREIT, for short) is a new medical imaging technique developed recently to visualize the cross-section conductivity of biologic tissues. A new MREIT image reconstruction method called harmonic B z algorithm was proposed in 2002 with the measurement of B z that is a single component of an induced magnetic flux density subject to an injection current. The key idea is to solve a nonlinear integral equation by some iteration process. This paper deals with the convergence analysis as well as the error estimate for noisy input data B z , which is the practical situation for MREIT. By analyzing the iteration process containing the Laplacian operation on the input magnetic field rigorously, the authors give the error estimate for the iterative solution in terms of the noisy level δ and the regularizing scheme for determining ΔB z approximately from the noisy input data. The regularizing scheme for computing the Laplacian from noisy input data is proposed with error analysis. Our results provide both the theoretical basis and the implementable scheme for evaluating the reconstruction accuracy using harmonic B z algorithm with practical measurement data containing noise.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cheney, M., Isaacson, D. and Newell, J. C., Electrical impedance tomography, SIAM Rev., 41, 1999, 85–101.

    Article  MATH  MathSciNet  Google Scholar 

  2. Kwon, O., Park, C. J., Park, E. J., et al., Electrical conductivity imaging using a variational method in B z-based MREIT, Inverse Problems, 21, 2005, 969–980.

    Article  MATH  MathSciNet  Google Scholar 

  3. Kwon, O., Pyo, H. C., Seo, J. K. and Woo, E. J., Mathematical framework for Bz-Based MREIT model in electrical impedance imaging, Comput. and Math. with Appl., 51(5), 2006, 817–828.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Liu, J. J., Seo, J. K. and Woo, E. J., A posteriori error estimate and convergence analysis for conductivity image reconstruction in MREIT, SIAM J. Appl. Math., 70, 2011, 2883–2903.

    Article  MathSciNet  Google Scholar 

  6. Liu, J. J. and Xu, H. L., Reconstruction of biologic tissue conductivity from noisy magnetic field by integral equation method, Appl. Math. and Comput., 218, 2011, 2647–2660.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  8. OH, S. H., Lee, B. I., Woo, E. J., et al., 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 

  9. OH, S. H., Lee, B. I., Woo, E. J., et al., Electrical conductivity images of biological tissue phantoms in MREIT, Physiol. Meas., 26, 2005, 279–288.

    Article  Google Scholar 

  10. Park, C., Kwon, O., Woo, E. J. and Seo, J. K., 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 

  11. Park, C., Park, E. J., Woo, E. J., et al., Static conductivity imaging using variational gradient B z algorithm in magnetic resonance electrical impedance tomography, Physiol. Meas., 25, 2004, 257–269.

    Article  Google Scholar 

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

    Article  Google Scholar 

  13. Renardy, M. and Rogers, R. C., An Introduction to Partial Differential Equations, TAM, Vol.13, Springer-Verlag, New York, 2004.

    Google Scholar 

  14. Seo, J. K., A uniqueness result on inverse conductivity problems with two measurements, J. Fourier Anal. Appl., 2, 1996, 515–524.

    Google Scholar 

  15. Seo, J. K., Kwon, O. and Woo, E. J., Magnetic resonance electrical impedance tomography (MREIT): conductivity and current density imaging, J. Phys.: Conf. Ser., 12, 2005, 140–155.

    Google Scholar 

  16. Seo, J. K., Pyo, H. C., Park, C., et al., Image reconstruction of anisotropic conductivity tensor distribution in MREIT: Computer simulation study, Phys. Med. Biol., 49, 2004, 4371–4382.

    Article  Google Scholar 

  17. Seo, J. K., Yoon, J. R. and Woo, E. J., Reconstruction of conductivity and current density images using only one component of magnetic field measurements, IEEE Trans. Biomed. Eng., 50, 2003, 1121–1124.

    Article  Google Scholar 

  18. Webster, J. G., Electrical Impedance Tomography, Adam Hilger, Bristol, 1990.

    Google Scholar 

  19. Xu, H. L. and Liu, J. J., On the Laplacian operation with applications in magnetic resonance electrical impedance imaging, Inverse Problems in Science and Engineering, 21, 2013, 251–268.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Qun Chen

  2. Department of Mathematics, Southeast University, Nanjing, 210096, China

    Jijun Liu

Authors
  1. Qun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jijun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jijun Liu.

Additional information

This work was supported by the National Natural Science Foundation of China (No. 91330109) and the Research Found for the Doctoral Program of Higher Education of China (No. 20110092110018).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Q., Liu, J. On the error estimate of the harmonic B z algorithm in MREIT from noisy magnetic flux field. Chin. Ann. Math. Ser. B 35, 319–336 (2014). https://doi.org/10.1007/s11401-014-0838-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-014-0838-8

Keywords

2000 MR Subject Classification

Search

Navigation