Skip to main content

Mathematical Problems in Microwave Medical Imaging

  • Chapter
Computational Radiology and Imaging

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 110))

Abstract

The change in tissue structure due to cancer causes the index of refraction of the tissue to change. Hence the question is posed if it is possible to detect the presence of cancer by microwave imagining. In this paper we survey recent results we have obtained on the use of microwaves to detect leukemia in the bone marrow of the leg that suggest that this is possible. Our emphasis in this paper is on the mathematical problems that arise in this investigation.

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

Access this chapter

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. R. Albanese, Wave propagation inverse problems in medicine and environmental health, in Inverse Problems in Wave Propagation, G. Chavent, G. Papanicolaou, P. Sacks, and W. Symes, eds., Springer-Verlag, New York, 1997, pp. 1–12.

    Google Scholar 

  2. R. Albanese, R. Medina, and J. Penn, Mathematics,medicine and microwaves, Inverse Problems, 10 (1994), pp. 995–1007.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), pp. 383–393.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, New York, second edition, 1998.

    MATH  Google Scholar 

  5. D. Colton, R. Kress, and P. Monk, Inverse scattering from an orthotropic medium, J. Comp. Applied Math., 81 (1997), pp. 269–298.

    MathSciNet  MATH  Google Scholar 

  6. D. Colton and P. Monk, A linear sampling method for the detection of leukemia using microwaves, SIAM J. Applied Math., 58 (1998), pp. 926–941.

    MathSciNet  MATH  Google Scholar 

  7. D. Colton and P. Monk, The detection and monitoring of leukemia using electromagnetic waves: Mathematical theory, Inverse Problems, 10 (1994), pp. 1235–1251.

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Colton and P. Monk, The detection and monitoring of leukemia using electromagnetic waves: Numerical analysis, Inverse Problems, 11 (1995), pp. 329–342.

    Article  MathSciNet  MATH  Google Scholar 

  9. D. Colton and P. Monk, A new approach to detecting leukemia: Using computational electromagnetics, IEEE Computational Science and Engineering, 2 (1995), pp. 46–52.

    Article  Google Scholar 

  10. D. Colton and P. Monk, A new algorithm in electromagnetic inverse scattering theory with an application to medical imaging, Math. Methods Applied Science, 20 (1997), pp. 385–401.

    Article  MathSciNet  MATH  Google Scholar 

  11. D. Colton, M. Piana, and R. Potthast, A simple method using Morozov’s discrepancy principle for solving inverse scattering problems, Inverse Problems, 13 (1997), pp. 1477–1493.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Devaney, Current research topics in diffraction tomography, in Inverse Problems in Scattering and Imaging, M. Bertero and E. Pike, eds., Adam Hilger, Bristol, 1992, pp. 47–58.

    Google Scholar 

  13. H. Gray, Anatomy of the Human Body, Lea and Febiger, Philadelphia, 1959.

    Google Scholar 

  14. A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer-Verlag, New York, 1996.

    Book  MATH  Google Scholar 

  15. A. Kirsch and P. Monk, Convergence analysis of a coupled finite element and spectral method in acoustic scattering, IMA J. of Numerical Analysis, 9 (1990), pp. 425–447.

    Article  MathSciNet  Google Scholar 

  16. L. Larsen and J. Jacobi, eds., Medical Applications of Microwave Imaging, IEEE Press, New York, 1986.

    Google Scholar 

  17. H. Lewy, On the reflection laws of second order differential equations in two independent variables, Bull. Amer. Math. Soc., 65 (1959), pp. 37–58.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Penn and E. Bell, Electrical parameter values of some human tissues in the radio frequency radiation range, Tech. Report SAM-TR-78–38, USAF School of Aerospace Medicine, Brooks Air Force Base, San Antonio, Texas, 1978.

    Google Scholar 

  19. B. Rynne and B. Sleeman, The interior transmission problem and inverse scattering from inhomogeneous media, SIAM J. Math. Anal., 22 (1991), pp. 1755–1762.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Stamnes, L. Gelius, I. Johansen, and N. Sponheim, Diffraction tomography applications in seismics and medicine, in Inverse Problems in Scattering and Imaging, M. Bertero and E. Pike, eds., Adam Hilger, Bristol, 1992, pp. 268–292.

    Google Scholar 

  21. Z. Sun and G. Uhlmann, Recovery of singularities for formally determined inverse problems, Comm Math. Physics, 153 (1993), pp. 431–445.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Tikhonov, A. Goncharsky, V. Stepanov, and A. Yagola, Numerical Methods for the solution of ill-posed problems, Kluwer, Dordrecht, 1995.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer Science+Business Media New York

About this chapter

Cite this chapter

Colton, D., Monk, P. (1999). Mathematical Problems in Microwave Medical Imaging. In: Börgers, C., Natterer, F. (eds) Computational Radiology and Imaging. The IMA Volumes in Mathematics and its Applications, vol 110. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1550-9_8

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-1550-9_8

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7189-5

  • Online ISBN: 978-1-4612-1550-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics