Journal of Mathematical Chemistry

, Volume 42, Issue 1, pp 1–35 | Cite as

Decisive role of mathematical methods in early cancer diagnostics: optimized Padé-based magnetic resonance spectroscopy

  • Dževad Belkić
  • Karen Belkić

Key to cancer treatment and overall tumor control is early diagnostics. Remarkably, Magnetic Resonance (MR) physics with the underlying mathematics for the reconstruction problems plays a pivotal role not only for early tumor diagnosis, but also for target definition, dose planning systems and therapy. The overall goal of this review is to highlight certain novel mathematical methods for improvement of cancer diagnostics on a quantitative molecular basis by retrieving key information which remains undetected with standard data analysis. We intend to contribute to a large effort aimed at establishing Magnetic Resonance Spectroscopy (MRS) and Magnetic Resonance Spectroscopic Imaging (MRSI) as two standard diagnostic tools for clinical oncology, with their complementary roles relative to anatomical information provided by Magnetic Resonance Imaging (MRI). Crucially, such efforts are within the realm of mathematical descriptions of data measured by the MR methods and the related physical, chemical and bio-medical interpretations. This can be achieved with fidelity by applying the fast Padé transform (FPT) to MRI, MRS and MRSI. Thus far, we have completed the “proof of principle” investigations demonstrating that the FPT is a powerful, stable parametric processor with robust error analysis, which provides unequivocal quantification of in vivo time signals encoded via MRS. These are the most stringent criteria imposed upon MRS and MRSI by clinical oncology. The established overall reliability of the FPT firmly justifies the present suggestion for undertaking further extensive applications of the FPT to a variety of phantom and clinical time signals at vastly different magnetic field strengths, with a broad range of signal-to-noise ratio (SNR). This would enable Padé-based MRI, MRS and MRSI to soon join the standard diagnostic armamentarium for clinical practice, especially in oncology. Of particular importance is to extend the current applications of the FPT to in vivo MRS signals encoded from patients with e.g. breast, prostate and ovarian cancers, so as to compare the obtained results with findings from non-malignant tissue, that have presented differential diagnostic dilemmas, notably benign tumors, infectious or inflammatory lesions. The fact that the FPT is capable of extracting unambiguous quantitative information from tissue via mathematical parametric analysis can be exploited to develop normative data bases for metabolite concentrations versus the corresponding findings seen in malignancy. This would provide the needed standards to aid in cancer diagnostics, identifying malignant versus benign disease with specific patterns of departures from normal metabolite concentrations. Overall, this succinct review focuses on the benefits from a judicious intertwining of spectral analysis from mathematics with quantum-mechanical signal processing from physics as well chemistry, especially when these basic sciences are used synergistically to enhance the diagnostic power of MRI, MRS and MRSI in clinical oncology.


Padé approximant fast Padé transform spectral analysis signal processing quantification early cancer diagnostics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dž. Belkić, Quantum mechanical signal processing and spectral analysis (Institute of Physics Publishing, Bristol, 2004) [and references therein].Google Scholar
  2. 2.
    Dž. Belkić, Principles of Quantum Scattering Theory (Institute of Physics Publishing, Bristol, 2003) [and references therein].Google Scholar
  3. 3.
    Z.-P. Liang and P. Lauturber, Principles of Magnetic Resonance Imaging: A Signal Processing Perspective (IEEE Press Series in Biomedical Engineering, New York, 2000).Google Scholar
  4. 4.
    Howe F., Opstad K.S. (2003) 1H MR spectroscopy of brain tumours and masses. NMR Biomed. 16: 123PubMedCrossRefGoogle Scholar
  5. 5.
    Nelson S. (2003) Multivoxel magnetic resonance spectroscopy of brain tumors. Mol. Cancer Ther. 2: 497PubMedGoogle Scholar
  6. 6.
    Belkić K., Belkić Dž. (2004) Spectroscopic imaging through MR for brain tumour diagnostics. J. Comp. Meth. Sci. Eng. 4: 157Google Scholar
  7. 7.
    K. Belkić, Molecular Imaging through Magnetic Resonance for Clinical Oncology (Cambridge International Science Publishing, Cambridge, 2004) [and references therein].Google Scholar
  8. 8.
    E. Danielsen and B. Ross, Magnetic Resonance Spectroscopy Diagnosis of Neurological Diseases (Marcel Dekker, Inc, New York, 1999).Google Scholar
  9. 9.
    L. Brandão and R. Domingues, MR Spectroscopy of the Brain (Lippincott Williams & Wilkins, Philadelphia, Pennsylvania, 2004).Google Scholar
  10. 10.
    Kurhanewicz J., Swanson M.G., Nelson S.J., Vigneron D.B. (2002) Combined magnetic resonance imaging and spectroscopic imaging approach to molecular imaging of prostate cancer. J. Magn. Reson. Imaging. 16: 451PubMedCrossRefGoogle Scholar
  11. 11.
    Dhingsa R., Qayyum A., Coakley F.V., Lu Y., Jones K.D., Swanson M.G. (2004) Prostate cancer localization with endorectal MR imaging and MR spectroscopic imaging: effect of clinical data on reader accuracy. Radiology 230: 215PubMedCrossRefGoogle Scholar
  12. 12.
    Thompson I., Pauler D.K., Goodman P.J., Tangen C.M., Lucia M.S., Parmes H.L. et al. (2004) Prevalence of prostate cancer among men with prostate-specific antigen level ≤  4.0 ng per ml. N. Engl. J. Med. 350: 2239PubMedCrossRefGoogle Scholar
  13. 13.
    Katz-Brull R., Lavin P.T., Lenkinski R.E. (2002) Clinical utility of proton MR spectroscopy in characterizing breast lesions. J. Natl. Cancer Inst. 94: 1197PubMedGoogle Scholar
  14. 14.
    Griffiths J., Tate A.R., Howe F.A., Stubbs M. (2002) as part of the Multi-Institutional Group on MRS Application to Cancer, Magnetic resonance spectroscopy of cancer—practicalities of multi-centre trials and early results in non-Hodgkin’s lymphoma. Eur. J. Cancer 38: 2085PubMedCrossRefGoogle Scholar
  15. 15.
    Dixon R.H. (1998) NMR studies of phospholipid metabolism in hepatic lymphoma. NMR Biomed. 11: 370PubMedCrossRefGoogle Scholar
  16. 16.
    Mukherji S., Schiro S., Castillo M., Kwock L., Muller K.E., Blackstock W. (1997) Proton MR spectroscopy of squamous cell carcinoma of the extracranial head and neck: in vitro and in vivo studies. Am. J. Neuroradiol. 18: 1057PubMedGoogle Scholar
  17. 17.
    Star-Lack J., Adalsteinsson E., Adam M.F., Terris D.J., Pinto H.A., Brown J.M. et al. (2000) In vivo 1H MR spectroscopy of human head and neck lymph node metastasis and comparison with oxygen tension measurements. Am. J. Neuroradiol. 21: 183PubMedGoogle Scholar
  18. 18.
    Belkić K. (2004) MR spectroscopic imaging in breast cancer detection: possibilities beyond the conventional theoretical framework for data analysis. Nucl. Instr. Meth. Phys. Res. A 525: 313CrossRefADSGoogle Scholar
  19. 19.
    Belkić K. (2004) Current dilemmas and future perspectives for breast cancer screening with a focus upon optimization of MR spectroscopic imaging by advances in signal processing. Isr. Med. Assoc. J. 6: 610PubMedGoogle Scholar
  20. 20.
    Gribbestad I., Sitter B., Lundgren S., Krane J., Axelson D. (1999) Metabolite composition in breast tumors examined by proton nuclear MR spectroscopy. Anticancer Res. 19: 1737PubMedGoogle Scholar
  21. 21.
    Kaminogo M., Ishimaru H., Morikawa M., Ochi M., Ushijima R., Tani M. et al. (2001) Diagnostic potential of short echo time MR spectroscopy of gliomas with single-voxel and point-resolved spatially localised proton spectroscopy of brain. Neuroradiology 43: 353PubMedCrossRefGoogle Scholar
  22. 22.
    Smith I., Blandford D.E. (1998) Diagnosis of cancer in humans by 1H NMR of tissue biopsies. Biochem. Cell Biol. 76: 472PubMedCrossRefGoogle Scholar
  23. 23.
    Wallace J., Raaphorst G.P., Somorjai R.L., Ng C.E., Fung M.F.K. et al. (1997) Classification of 1H MR spectra of biopsies from untreated and recurrent ovarian cancer using linear discriminant analysis. Magn. Reson. Med. 38: 569PubMedCrossRefGoogle Scholar
  24. 24.
    Boss E., Moolenaar S.H., Massuger L.F.A.G., Boonstra H., Engelke U.F.H., de Jong J.G.N. et al. (2000) High-resolution proton nuclear magnetic resonance spectroscopy of ovarian cyst fluid. NMR Biomed. 13: 297PubMedCrossRefGoogle Scholar
  25. 25.
    Massuger L., van Vierzen P.B.J., Engelke U., Heerschap A., Wevers R. et al. (1998) 1H-MR spectroscopy. A new technique to discriminate benign from malignant ovarian tumors. Cancer 82: 1726PubMedCrossRefGoogle Scholar
  26. 26.
    Brown T.R., Kincaid B.M., Uğurbil K. (1982) NMR chemical shift imaging in three dimensions. Proc. Natl. Acad. Sci. USA 79: 3523PubMedCrossRefADSGoogle Scholar
  27. 27.
    Belkić Dž., Belkić K. (2005) The fast Padé transform in MR spectroscopy for improvements in early cancer diagnostics. Phys. Med. Biol. 50: 4385PubMedCrossRefGoogle Scholar
  28. 28.
    Bottomley P. (1992) The trouble with spectroscopy papers. J. Magn. Reson. Imaging 2: 1PubMedCrossRefGoogle Scholar
  29. 29.
    Opstad K., Provencher S.W., Bell B.A., Griffiths J.R., Howe F.A. (2003) Detection of elevated glutathione in meningiomas by quantitative in vivo 1H MRS. Magn. Reson. Med. 49: 632PubMedCrossRefGoogle Scholar
  30. 30.
    Cho Y.-D., Choi G-H., Lee S-P., Kim J-K. (2003) 1H-MRS metabolic patterns for distinguishing meningiomas from other brain tumors. Magn. Reson. Imaging 21: 663PubMedCrossRefGoogle Scholar
  31. 31.
    Belkić Dž., Belkić K. (2006) Mathematical optimization of in vivo NMR chemistry through the fast Padé transform: Potential relevance for early breast cancer detection by magnetic resonance spectroscopy. J. Math. Chem. 40: 85CrossRefMathSciNetGoogle Scholar
  32. 32.
    Belkić Dž. (2001) Fast Padé Transform for MRI and computerized tomography. Nucl. Instr. Meth. Phys. Res. A 471: 165CrossRefADSGoogle Scholar
  33. 33.
    Belkić Dž. (2002) Non-Fourier based reconstruction techniques. Magn. Reson. Mater. Phys. Biol. Med. 15: 36CrossRefGoogle Scholar
  34. 34.
    Belkić Dž. (2003) Exact analytical expressions for any Lorentzian spectrum in the fast Padé spectrum. J. Comp. Meth. Sci. Eng. 3: 109Google Scholar
  35. 35.
    Belkić Dž. (2003) Strikingly stable convergence of the fast Padé transform. J. Comp. Meth. Sci. Eng. 3: 299Google Scholar
  36. 36.
    Belkić Dž. (2003) Padé-based magnetic resonance spectroscopy (MRS). J. Comp. Meth. Sci. Eng. 3: 563Google Scholar
  37. 37.
    Belkić Dž. (2004) Strikingly stable convergence of the fast Padé transform (FPT) for high-resolution parametric and non-parametric signal processing of Lorentzian and non-Lorentzian spectra. Nucl. Instr. Meth. Phys. Res. A 525: 366CrossRefADSGoogle Scholar
  38. 38.
    Belkić Dž. (2004) Analytical continuation by numerical means in spectral analysis using the fast Padé transform. Nucl. Instr. Meth. Phys. Res. A 525: 372CrossRefADSGoogle Scholar
  39. 39.
    Belkić Dž. (2004) Error analysis through residual frequency spectra in the fast Padé transform (FPT). Nucl. Instr. Meth. Phys. Res. A 525: 379CrossRefADSGoogle Scholar
  40. 40.
    Belkić Dž., Belkić K. (2005) Fast Padé transform for optimal quantification of time signals from MR spectroscopy. Int. J. Quantum Chem. 105: 493CrossRefGoogle Scholar
  41. 41.
    Belkić Dž., Belkić K. (2006) In vivo magnetic resonance spectroscopy by the fast Padé transform. Phys. Med. Biol. 51: 1049PubMedCrossRefGoogle Scholar
  42. 42.
    Belkić Dž. (2006) Exact quantification of time signals in Padé-based magnetic resonance spectroscopy. Phys. Med. Biol. 51: 2633PubMedCrossRefGoogle Scholar
  43. 43.
    Belkić Dž. (2006) Exponential convergence rate (the spectral convergence) of the fast Padé transform for exact quantification in magnetic resonance spectroscopy. Phys. Med. Biol. 51: 6483PubMedCrossRefGoogle Scholar
  44. 44.
    Belkić Dž. (2006) Fast Padé transform for exact quantification of time signals in magnetic resonance spectroscopy. Adv. Quantum Chem. 51: 157CrossRefGoogle Scholar
  45. 45.
    Pijnappel W.W.F., van den Boogaart A., de Beer R., van Ormondt D. (1992) SVD-based quantification of magnetic resonance signals. J. Magn. Reson. 97: 122Google Scholar
  46. 46.
    van der Veen J.W.C., de Beer R., Luyten P.R., van Ormondt D. (1988) Accurate quantification of in vivo 31P NMR signals using the variable projection method and prior knowledge. Magn. Reson. Med. 6: 92PubMedCrossRefGoogle Scholar
  47. 47.
    Vanhamme L., van den Boogaart A., van Haffel S. (1997) Improved method for accurate and efficient quantification of MRS data with use of prior knowledge. J. Magn. Reson. 129: 35CrossRefGoogle Scholar
  48. 48.
    Provencher S.W. (1993) Estimation of metabolite concentrations from localized in vivo proton NMR spectra. Magn. Reson. Med. 30: 672PubMedCrossRefGoogle Scholar
  49. 49.
    Williamson D., Hawesa H., Thacker N.A., Williams S.R. (2006) Robust quantification of short echo time 1H MR spectra using the Padé approximant. Magn. Reson. Med. 55: 762PubMedCrossRefGoogle Scholar
  50. 50.
    Belkić Dž., Dando P.A., Main J., Taylor H.S. (2000) Three Novel High-Resolution Nonlinear Methods for Fast Signal Processing. J. Chem. Phys. 113: 6542CrossRefADSGoogle Scholar
  51. 51.
    Govindaraju V., Young K., Maudsley A.A. (2000) Proton NMR chemical shifts and coupling constants for brain metabolites. NMR Biomed. 13: 129PubMedCrossRefGoogle Scholar
  52. 52.
    Swindle P., McCredie S., Russell P., Himmelreich U., Khadra M., Lean C., Mountford C. (2003) Pathologic characterization of human prostate tissue with proton MR spectroscopy. Radiology 228: 144PubMedCrossRefGoogle Scholar
  53. 53.
    McEliece R.J., Shearer J.B. (1978) A property of Euclid’s algorithm and an application to Padé approximation. SIAM J. Appl. Math. 34: 611MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Palmer R.D., Cruz J.R. (1989) An ARMA spectral analysis technique based on a fast Euclidean algorithm. IEEE Trans. Acoust. Speech. Sign. Process. 37: 1532CrossRefMathSciNetGoogle Scholar
  55. 55.
    Frahm J., Bruhn H., Gyngell M.L., Merboldt K.D., Hanicke W., Sauter R. (1989) Localised high-resolution NMR spectroscopy using stimulated echos: initial application to human brain in vivo. Magn. Reson. Med. 9: 79PubMedCrossRefGoogle Scholar
  56. 56.
    Callaghan M., Larkman D.J., Hajnal J.V. (2005) Padé methods for reconstruction and feature extraction in magnetic resonance imaging. Magn. Reson.Med. 54: 1490PubMedCrossRefGoogle Scholar
  57. 57.
    Tkáč I., Andersen P., Adriany G., Merkle H., Uğurbil K., Gruetter R. (2001) In vivo 1H NMR spectroscopy of the human brain at 7 T. Magn. Reson. Med. 46: 451PubMedCrossRefGoogle Scholar
  58. 58.
    Mountford C.E., Lean C.L., Hancock R. (1993) Magnetic resonance spectroscopy detects cancer in draining lymph nodes. Invas. Metast. 13: 57Google Scholar
  59. 59.
    Mountford C.E., Doran S., Lean C., Russell P. (2004) Proton MRS can determine the pathology of human cancers with a high level of accuracy. Chem. Rev., 104: 3677CrossRefGoogle Scholar
  60. 60.
    Malycha P. (2003) Sentinel lymph node biopsy. ANZ J. Surg. 73: 370PubMedCrossRefGoogle Scholar
  61. 61.
    Gluch L. (2005) Magnetic resonance in surgical oncology: I On the origin of the spectrum. ANZ J. Surg. 75: 459PubMedCrossRefGoogle Scholar
  62. 62.
    Gluch L. (2005) Magnetic resonance in surgical oncology: II Literature review. ANZ J. Surg. 75: 464PubMedCrossRefGoogle Scholar
  63. 63.
    Jolesz F. (2005) Future of magnetic resonance imaging and magnetic resonance spectroscopy in oncology. ANZ J. Surg. 75: 372PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Oncology and PathologyKarolinska InstituteStockholmSweden
  2. 2.Institute for Prevention ResearchThe University of Southern California School of MedicineLos AngelesUSA

Personalised recommendations