Mathematical Methods in PET and SPECT Imaging

  • Athanasios  S.Fokas
  • George  A.Kastis
Living reference work entry


In this chapter, we present the mathematical formulation of the inverse Radon transform and of the inverse attenuated Radon transform (IART), which are used in PET and SPECT image reconstruction, respectively. Using a new method for deriving transform pairs in one and two dimensions, we derive the inverse Radon transform and the IART. Furthermore, we discuss an alternative approach for computing the Hilbert transform using cubic splines. This new approach, which is referred to as spline reconstruction technique, is formulated in the physical space, in contrast to the well-known filtered backprojection (FBP) algorithm which is formulated in the Fourier space. Finally, we present the results of several rigorous studies comparing FBP with SRT for PET. These studies, which use both simulated and real data and which employ a variety of image quality measures including contrast and bias, indicate that SRT has certain advantages in comparison with FBP.


Positron Emission Tomography Single Photon Emission Compute Tomography Single Photon Emission Compute Tomography Image Order Subset Expectation Maximization Positron Emission Tomography System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    U-King-Im, J.M., Young, V., Gillard, J.H.: Carotid-artery imaging in the diagnosis and management of patients at risk of stroke. Lancet Neurol. 8(6), 569–580 (2009)CrossRefGoogle Scholar
  2. 2.
    Hutchins, G.D., Miller, M.A., Soon, V.C., Receveur, T.: Small animal PET imaging. ILAR J. 49(1), 54–65 (2008)CrossRefGoogle Scholar
  3. 3.
    Cherry, S.R., Gambhir, S.S.: Use of positron emission tomography in animal research. ILAR J. 42(3), 219–232 (2001)CrossRefGoogle Scholar
  4. 4.
    Lucas, A.J., Hawkes, R.C., Ansorge, R.E., Williams, G.B., Nutt, R.E., Clark, J.C., Fryer, T.D., Carpenter, T.A.: Development of a combined microPET-MR system. Technol. Cancer Res. Treat. 5(4), 337–341 (2006)CrossRefGoogle Scholar
  5. 5.
    Cherry, S.R.: Fundamentals of positron emission tomography and applications in preclinical drug development. J. Clin. Pharmacol. 41(5), 482–491 (2001)CrossRefGoogle Scholar
  6. 6.
    Zaidi, H., Hasegawa, B.H.: The problem of photon attenuation in emission tomography. In: Zaidi, H. (ed.) Quantitative Analysis in Nuclear Medicine Imaging, p. 167. Springer, New York (2006)CrossRefGoogle Scholar
  7. 7.
    Kastis, G.A., Kyriakopoulou, D., Gaitanis, A., Fernandez, Y., Hutton, B.F., Fokas, A.S.: Evaluation of the spline reconstruction technique for PET. Med. Phys.,41, 042501 (2014), doi: Scholar
  8. 8.
    Thielemans, K., Tsoumpas, C., Mustafovic, S., Beisel, T., Aguiar, P., Dikaios, N., Jacobson, M.W.: STIR: software for tomographic image reconstruction Release 2. Phys. Med. Biol. 57(4), 867–883 (2012)CrossRefGoogle Scholar
  9. 9.
    Kastis, G.A., Gaitanis, A., Skouras, T., Fokas, A.S.: Evaluation of a spline reconstruction technique for SPECT: comparison with FBP and OSEM. In: Conference Record of the 2011 IEEE Nuclear Science Symposium and Medical Imaging Conference, Oct 2011Google Scholar
  10. 10.
    Fokas, A.S., Gelfand, I.M.: Integrability of linear and nonlinear evolution equations, and the associated nonlinear Fourier transforms. Lett. Math. Phys. 32(3), 189–210 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Pauwels, E.K., Ribeiro, M.J., Stoot, J.H., McCready, V.R., Bourguignon, M., Mazière, B.: FDG accumulation and tumor biology. Nucl. Med. Biol. 25(4), 317–322 (1998)CrossRefGoogle Scholar
  12. 12.
    Fischer, B., Lassen, U., et al.: Preoperative staging of lung cancer with combined PET-CT. N. Engl. J. Med. 361(1), 32–39 (2009)CrossRefGoogle Scholar
  13. 13.
    Armitage, J.O.: Early-stage Hodgkin’s lymphoma. N. Engl. J. Med. 363(7), 653–662 (2010)CrossRefGoogle Scholar
  14. 14.
    Baterman, R.J., Xiong, C., et al.: Clinical and biomarker changes in dominantly inherited Alzheimer’s disease. N. Engl. J. Med. 367(9), 795–804 (2012)CrossRefGoogle Scholar
  15. 15.
    Abrams, J.: Clinical practice. Chronic stable angina. N. Engl. J. Med. 352(24), 2524–2533 (2005)CrossRefGoogle Scholar
  16. 16.
    Gershenwald, J.E., Ross, M.I.: Sentinel-lymph-node biopsy for cutaneous melanoma. N. Engl. J. Med. 364(18), 1738–1745 (2011)CrossRefGoogle Scholar
  17. 17.
    The Parkinson Study Group: Levodopa and the progression of Parkinson’s disease. N. Engl. J. Med. 351(24), 2498–2508 (2004)Google Scholar
  18. 18.
    Fokas, A.S., Novikov, R.G.: Discrete analogues of the \(\overline{\partial }\) equation and of Radon transform. C. R. Acad. Sci. Paris 313(2), 75–80 (1991)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Novikov, R.G.: An inversion formula for the attenuated X-ray transformation. Ark. Mat. 40(1), 145–167 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Miqueles, E.X., De Pierro, A.R.: Exact analytic reconstruction in x-ray fluorescence CT and approximated versions. Phys. Med. Biol. 55(4), 1007–1024 (2010)CrossRefGoogle Scholar
  21. 21.
    Herman, G.T.: Fundamentals of Computerized Tomography: Image Reconstruction from Projections, 2nd edn. Springer, Dordrecht/Heidelberg/London/New York (1980)zbMATHGoogle Scholar
  22. 22.
    Natterer, F.: The Mathematics of Computerized Tomography. Wiley, New York (1986)zbMATHGoogle Scholar
  23. 23.
    Natterer, F., Wübbeling, F.: Mathematical Methods in Image Reconstruction. SIAM, Philadelphia (2001)CrossRefzbMATHGoogle Scholar
  24. 24.
    Parker, J.A.: Image Reconstruction in Radiology. CRC, Boca Raton (1990)Google Scholar
  25. 25.
    Epstein, C.L.: Introduction to the Mathematics of Medical Imaging. SIAM, Philadelphia (2008)CrossRefzbMATHGoogle Scholar
  26. 26.
    Kinahan, P.E., Defrise, M., Clackdoyle, R.: Analytic image reconstruction methods. In: Wernick, M.N., Aarsvold, J.N. (eds.) Emission Tomography: The Fundamentals of PET and SPECT, p. 421. Elsevier Academic, San Diego (2004)Google Scholar
  27. 27.
    Tsui, B.M.W., Frey, E.C.: Analytic image reconstruction methods in emission computed tomography. In: Zaidi, H. (ed.) Quantitative Analysis in Nuclear Medicine Imaging, p. 82. Springer, New York (2006)Google Scholar
  28. 28.
    Cherry, S.R., Dahlbom, M.: PET: physics, instrumentation, and scanners. In: Phelps, M.E. (ed.) Physics, Instrumentation, and Scanners, p. 76. Springer, New York (2010)Google Scholar
  29. 29.
    Hoffman, E.J., Cutler, P.D., Digby, W.M., Mazziotta, J.C.: 3D phantom to simulate cerebral blood flow and metabolic images for PET. IEEE Trans. Nucl. Sci. 37(2), 616–620 (1990)CrossRefGoogle Scholar
  30. 30.
    Bettinardi, V., Danna, M., Savi, A., Lecchi, M., Castiglioni, I., Gilardi, M.K., Bammer, H., Lucignani, G., Fazio, F.: Performance evaluation of the new whole-body PET/CT scanner: discovery ST. Eur. J. Nucl. Med. Mol. Imaging 31(6), 867–881 (2004)CrossRefGoogle Scholar
  31. 31.
    Wang, Y., Seidel, J., Tsui, B.M.W., Vaquero, J.J., Pomper, M.G.: Performance evaluation of the GE Healthcare eXplore VISTA dual-ring small-animal PET scanner. J. Nucl. Med. 47(11), 1891–1900 (2006)Google Scholar
  32. 32.
    NEMA: NEMA NU 4-2008: Performance Measurements of Small Animal Positron Emission Tomographs. National Electrical Manufacturers Association, Rosslyn (2008)Google Scholar
  33. 33.
    Fokas, A.S., Iserles, A., Marinakis, V.: Reconstruction algorithm for single photon emission computed tomography and its numerical implementation. J. R. Soc. Interface 3(6), 45–54 (2006)CrossRefGoogle Scholar
  34. 34.
    Shepp, L.A., Vardi, Y.: Maximum likelihood reconstruction for emission tomography. IEEE Trans. Med. Imaging 1(2), 113–121 (1982)CrossRefGoogle Scholar
  35. 35.
    Hudson, M., Larkin, R.S.: Accelerated image reconstruction using ordered subsets of projection data. IEEE Trans. Med. Imaging 13(4), 601–609 (1994)CrossRefGoogle Scholar
  36. 36.
    Carson, R.E., Yan, Y., Chodkowski, B., Yap, T.K., Daube-Witherspoon, M.E.: Precision and accuracy of regional radioactivity quantitation using the maximum likelihood EM reconstruction algorithm. IEEE Trans. Med. Imag. 13(3), 526–537 (1994)CrossRefGoogle Scholar
  37. 37.
    Johnson, C.A., Yan, Y., Carson, R.E., Martino, R.L., Daube-Witherspoon, M.E.: A system for the 3D reconstruction of retracted-septa PET data using the EM algorithm. IEEE Trans. Nucl. Sci. 42(4), 1223–1227 (1995)CrossRefGoogle Scholar
  38. 38.
    De Jonge, F.A.A., Blokland, K.A.K.: Statistical tomographic reconstruction: How many more iterations to go? Eur. J. Nucl. Med. 26(10), 1247–1250 (1999)CrossRefGoogle Scholar
  39. 39.
    Bettinardi, V., Pagani, E., Gilardi, M., Alenius, S., Thielemans, K., Teras, M., Fazio, F.: Implementation and evaluation of a 3D one-step late reconstruction algorithm for 3D positron emission tomography brain studies using median root prior. Eur. J. Nucl. Med. Mol. Imaging 29(1), 7–18 (2002)CrossRefGoogle Scholar
  40. 40.
    Reilhac, A., Tomei, S., Buvat, I., Michel, C., Keheren, F., Costes, N.: Simulation-based evaluation of OSEM iterative reconstruction methods in dynamic brain PET studies. Neuroimage 39(1), 359–368 (2008)CrossRefGoogle Scholar
  41. 41.
    Verhaeghe, J., Reader, A.J.: AB-OSEM reconstruction for improved Patlak kinetic parameter estimation: a simulation study. Phys. Med. Biol. 55(22), 6739–6757 (2010)CrossRefGoogle Scholar
  42. 42.
    Bélanger, M.J., Mann, J.J., Parsey, R.V.: OS-EM and FBP reconstructions at low count rates: effect on 3D PET studies of [11C] WAY-100635. Neuroimage 21(1), 244–250 (2004)CrossRefGoogle Scholar
  43. 43.
    Brenner, D.J.: Medical imaging in the 21st century-getting the best bang for the rad. N. Engl. J. Med. 362(10), 943–945 (2010)CrossRefGoogle Scholar
  44. 44.
    Kovalchik, S.A., Tammemagi, M., Berg, C.D., Caporaso, N.E., Riley, T.L., Korch, M., Silvestri, G.A., Chaturvedi, A.K., Katki, H.A.: Targeting of low-dose CT screening according to the risk of lung-cancer death. N. Engl. J. Med. 369(3), 245–254 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Theoretical Physics (DAMTP)University of CambridgeCambridgeUK

Personalised recommendations