European Journal of Nuclear Medicine

, Volume 23, Issue 6, pp 705–719 | Cite as

Nuclear medicine and mathematics

  • J. J. Pedroso de Lima
Review Article


The purpose of this review is not to present a comprehensive description of all the mathematical tools used in nuclear medicine, but to emphasize the importance of the mathematical method in nuclear medicine and to elucidate some of the mathematical concepts currently used. We can distinguish three different areas in which mathematical support has been offered to nuclear medicine: physiology, methodology and data processing. Nevertheless, the boundaries between these areas can be indistinct. It is impossible in a single article to give even an idea of the extent and complexity of the procedures currently used in nuclear medicine, such as image processing, reconstruction from projections and artificial intelligence. These disciplines do not belong to nuclear medicine: they are already branches of engineering, and my interest will reside simply in revealing a little of the elegance and the fantastic potential of these new “allies” of nuclear medicine. In this review the mathematics of physiological interpretation and methodology are considered together in the same section. General aspects of data-processing methods, including image processing and artificial intelligence, are briefly analysed. The mathematical tools that are most often used to assist the interpretation of biological phenomena in nuclear medicine are considered; these include convolution and deconvolution methods, Fourier analysis, factorial analysis and neural networking.

Key words

Mathematics Mathematical methods 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Sharp PF, Dendy PP, Keyes WI.Radionuclide imaging techniques. New York: Academic Press, 1985.Google Scholar
  2. 2.
    Bellman R.Mathematical methods in medicine. Singapore: World Scientific, 1983.Google Scholar
  3. 3.
    Nimon CC, Lee TY, Britton KE, Granowska M, Gruenewald S. Practical applications of deconvolution techniques in dy namic studies. In:Medical radionuclide imaging. Vienna: IAEA-SM-247/26, 1981; 1: 367–388.Google Scholar
  4. 4.
    Alderson PO, Kenneth HD, Mendenhall KG, Guadiani VA, Watson DC, Links JM, Wagner HN Jr. Deconvolution analysis in radionuclide quantitation of left-to-right cardiac shunts.J Nucl Med1979; 20: 502–506.Google Scholar
  5. 5.
    Brendel AJ, Commenges D, Salamon R, Ducassou D, Blanquet P. Deconvolution analysis of radionuclide angiography curves: problems arising from fragmented bolus injections.Eur J Nucl Med 1983; 8: 93–98.Google Scholar
  6. 6.
    Parker JA. Image reconstruction in radiology. Boca Raton, FL.: CRC Press, 1990.Google Scholar
  7. 7.
    De Lima JJP. Image processing in nuclear pneumology.Comput Methods Programs biomed 1995; 48: 7–14.Google Scholar
  8. 8.
    Britton KE, Granowska M, Nimon CC. Total and regional cerebral blood flow - a new quantitative non-invasive method for cerebrovascular disease. In:Medical radionuclide imaging. Vienna: IAEA-SM-247/24, 1981;11: 315.Google Scholar
  9. 9.
    Brown HP, Juni JJ, Lieberman DA, Krishnamurthy GT. Hepatocyte versus biliary disease: a distinction by deconvolution analysis of 99-TcIDA time-activity curves.J Nucl Med 1988; 29:623–630.Google Scholar
  10. 10.
    Britton KE, Nimon CC, Whitfield HN, Fry IK, Hendry WF, Wickham JEA. The evaluation of obstructive nephropathy by means of parenchymal retention functions. In: Hollenberg NK and Lange S, eds. Radionuclides nephrology. Stuttgart: Thieme; 1980: 164–172.Google Scholar
  11. 11.
    Axelsson B, Msaki P, Israelsson. A subtraction of Comptonscattered photons in single photon emisson computorized tomography.J Nucl Med 1984; 25: 490–494.Google Scholar
  12. 12.
    Floyd CE, Jasczack RJ, Greer KJ, Coleman RE. Deconvolution of Compton scatter in SPELT.J Nucl Med 1985; 26: 403–408.Google Scholar
  13. 13.
    Russ JC. The image processing handbook. Boca Raton, FL.: CRC Press, 1992.Google Scholar
  14. 14.
    Pedersen F, Bergstbm M, Bengtsson, Langström B. Principal component analysis of dynamic positron emisson tomography images.Eur J Nucl Med 1994; 21: 1285–1292.Google Scholar
  15. 15.
    Benali H, Buvat I, Frouin F, Bazin JP, Di Paola R. Foundations of factor analysis of medical image sequences: a unified approach and some practical implications. Image and Vision Computing 1994; 12: 375–385.Google Scholar
  16. 16.
    Sámal M. Factor analysis revisited: a potential key for clinicians.Eur J Nucl Med 1993; 20: 562–564.Google Scholar
  17. 17.
    Di Paola R, Bazin JP, Aubry F, Aurengo A, Cavailloles F, Berry JY, Kahn E. Handling of dynamic sequences.Nucl Med IEEE Trans Nucl Sci 1982; NS-29 NoA: 1310-1321.Google Scholar
  18. 18.
    Sámal M, Karnt' M, Surová H, Penicka P, Mariková E, Dienstbier Z. On the existence of an unambiguous solution in factor analysis of dynamic studies.Phys Med Biol 1989; 34: 223–228.Google Scholar
  19. 19.
    Raeside DE. Monte Carlo principles and applications.Phys Med Biol 1976; 21: 181–197.Google Scholar
  20. 20.
    De Valk JPJ. Diagnostic processing and analysis of medical images. In: Dendy PP, ernest DW, Sengün A, eds.Technical advances in biomedical physics. NATO ASI Series; 1984: 271-285.Google Scholar
  21. 21.
    Natterer F.The mathematics of computorized tomography. New York: John Wiley/Stuttgart: Teubner, 1986.Google Scholar
  22. 22.
    Houston AS. Signal processing. In: Moores BM, Parker RP, Pullan BR, eds. Practical aspects of medical imaging. New York: Wiley; 1981: 203–215.Google Scholar
  23. 23.
    Houston A, Craig A. Perceptual and statistical analysis of cardiac phase and amplitude images.Eur J Nucl Med 1991; 18: 720–724.Google Scholar
  24. 24.
    Jain AK.Fundamentals of digital imaging processing. Englewood Cliffs, N.J.: Prentice Hall, 1989.Google Scholar
  25. 25.
    Rusinek H.Reconstruction technique. In: Kramer EL, Sanger JJ, eds.Clinical Spect imaging. New York: Raven Press; 1994: 43–67.Google Scholar
  26. 26.
    Todd-Pokropek A. Image processing with special reference to ECT. In: Albery G, Bajzer Z, Baxa P, eds.Proc. Int. Conf. on applications of physics to medicine and biology. Trieste: World Scientific; 1982: 391–429.Google Scholar
  27. 27.
    Spiegel MR.Theory and problems of advanced calculus. Schaum's outline series. New York: Schaum, 1963.Google Scholar
  28. 28.
    Ginsberg M.Essentials of artificial intelligence. Hove, East Sussex Morgan Kaufman, 1993.Google Scholar
  29. 29.
    Miller PL ed.Selected topics in medical artificial intelligence. Berlin Heidelberg New York: Springer (Series Computers and Medicine), 1988.Google Scholar
  30. 30.
    Cooke DC, Faber TL, Garcia EV Advanced Computer methods in cardiac SPECT. In: DePuey EG, Berman DS, Garcia EV, eds.Cardiac SPECT imaging. New York: Raven Press; 1994:75–89.Google Scholar
  31. 31.
    Kulkarni AD.Artificial neural networks for image understanding. VNR Computer Library, 1994.Google Scholar
  32. 32.
    Yoshiyasu T.Neural network parallel computing. London: Kluwer Academic, 1992.Google Scholar
  33. 33.
    Coppens A, Sibomana M, Bol A, Michel C. Mediman: an object oriented programming approach for medical image analysis.1993 Ecat technical user's meeting. Köln, Germany: Siemens.Google Scholar
  34. 34.
    Levin SG. Understanding and using statistics in nuclear medicine.J Nucl Med 1979; 20: 550–558.Google Scholar
  35. 35.
    Kuikka JT, Bassingthwaighte JB, Henrich MM, Feinendegen LE. Mathematical modelling in nuclear medicine.Eur J Nucl Med 1991; 18: 351–362.Google Scholar
  36. 36.
    Rescigno A, Lambrecht Rm, Duncan CC. Stochastic modelling of physiologic processes with radiotracers and positron emisson tomography. In: Albery G, Bajzer Z, Baxa P, eds.Proc Int. Conf on applications of physics to medicine and biology. Trieste: World Scientific; 1982: 303–318.Google Scholar

Copyright information

© Springer-Verlag 1996 1996

Authors and Affiliations

  • J. J. Pedroso de Lima
    • 1
  1. 1.Departamento de Biofísica e Proc. de ImagemIBILI - Faculdade de MedicinaCoimbraPortugal

Personalised recommendations