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.
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References
Dž. Belkić, Quantum mechanical signal processing and spectral analysis (Institute of Physics Publishing, Bristol, 2004) [and references therein].
Dž. Belkić, Principles of Quantum Scattering Theory (Institute of Physics Publishing, Bristol, 2003) [and references therein].
Z.-P. Liang and P. Lauturber, Principles of Magnetic Resonance Imaging: A Signal Processing Perspective (IEEE Press Series in Biomedical Engineering, New York, 2000).
Howe F., Opstad K.S. (2003) 1H MR spectroscopy of brain tumours and masses. NMR Biomed. 16: 123
Nelson S. (2003) Multivoxel magnetic resonance spectroscopy of brain tumors. Mol. Cancer Ther. 2: 497
Belkić K., Belkić Dž. (2004) Spectroscopic imaging through MR for brain tumour diagnostics. J. Comp. Meth. Sci. Eng. 4: 157
K. Belkić, Molecular Imaging through Magnetic Resonance for Clinical Oncology (Cambridge International Science Publishing, Cambridge, 2004) [and references therein].
E. Danielsen and B. Ross, Magnetic Resonance Spectroscopy Diagnosis of Neurological Diseases (Marcel Dekker, Inc, New York, 1999).
L. Brandão and R. Domingues, MR Spectroscopy of the Brain (Lippincott Williams & Wilkins, Philadelphia, Pennsylvania, 2004).
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: 451
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: 215
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: 2239
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: 1197
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: 2085
Dixon R.H. (1998) NMR studies of phospholipid metabolism in hepatic lymphoma. NMR Biomed. 11: 370
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: 1057
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: 183
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: 313
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: 610
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: 1737
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: 353
Smith I., Blandford D.E. (1998) Diagnosis of cancer in humans by 1H NMR of tissue biopsies. Biochem. Cell Biol. 76: 472
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: 569
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: 297
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: 1726
Brown T.R., Kincaid B.M., Uğurbil K. (1982) NMR chemical shift imaging in three dimensions. Proc. Natl. Acad. Sci. USA 79: 3523
Belkić Dž., Belkić K. (2005) The fast Padé transform in MR spectroscopy for improvements in early cancer diagnostics. Phys. Med. Biol. 50: 4385
Bottomley P. (1992) The trouble with spectroscopy papers. J. Magn. Reson. Imaging 2: 1
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: 632
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: 663
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: 85
Belkić Dž. (2001) Fast Padé Transform for MRI and computerized tomography. Nucl. Instr. Meth. Phys. Res. A 471: 165
Belkić Dž. (2002) Non-Fourier based reconstruction techniques. Magn. Reson. Mater. Phys. Biol. Med. 15: 36
Belkić Dž. (2003) Exact analytical expressions for any Lorentzian spectrum in the fast Padé spectrum. J. Comp. Meth. Sci. Eng. 3: 109
Belkić Dž. (2003) Strikingly stable convergence of the fast Padé transform. J. Comp. Meth. Sci. Eng. 3: 299
Belkić Dž. (2003) Padé-based magnetic resonance spectroscopy (MRS). J. Comp. Meth. Sci. Eng. 3: 563
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: 366
Belkić Dž. (2004) Analytical continuation by numerical means in spectral analysis using the fast Padé transform. Nucl. Instr. Meth. Phys. Res. A 525: 372
Belkić Dž. (2004) Error analysis through residual frequency spectra in the fast Padé transform (FPT). Nucl. Instr. Meth. Phys. Res. A 525: 379
Belkić Dž., Belkić K. (2005) Fast Padé transform for optimal quantification of time signals from MR spectroscopy. Int. J. Quantum Chem. 105: 493
Belkić Dž., Belkić K. (2006) In vivo magnetic resonance spectroscopy by the fast Padé transform. Phys. Med. Biol. 51: 1049
Belkić Dž. (2006) Exact quantification of time signals in Padé-based magnetic resonance spectroscopy. Phys. Med. Biol. 51: 2633
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: 6483
Belkić Dž. (2006) Fast Padé transform for exact quantification of time signals in magnetic resonance spectroscopy. Adv. Quantum Chem. 51: 157
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: 122
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: 92
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: 35
Provencher S.W. (1993) Estimation of metabolite concentrations from localized in vivo proton NMR spectra. Magn. Reson. Med. 30: 672
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: 762
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: 6542
Govindaraju V., Young K., Maudsley A.A. (2000) Proton NMR chemical shifts and coupling constants for brain metabolites. NMR Biomed. 13: 129
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: 144
McEliece R.J., Shearer J.B. (1978) A property of Euclid’s algorithm and an application to Padé approximation. SIAM J. Appl. Math. 34: 611
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: 1532
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: 79
Callaghan M., Larkman D.J., Hajnal J.V. (2005) Padé methods for reconstruction and feature extraction in magnetic resonance imaging. Magn. Reson.Med. 54: 1490
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: 451
Mountford C.E., Lean C.L., Hancock R. (1993) Magnetic resonance spectroscopy detects cancer in draining lymph nodes. Invas. Metast. 13: 57
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: 3677
Malycha P. (2003) Sentinel lymph node biopsy. ANZ J. Surg. 73: 370
Gluch L. (2005) Magnetic resonance in surgical oncology: I On the origin of the spectrum. ANZ J. Surg. 75: 459
Gluch L. (2005) Magnetic resonance in surgical oncology: II Literature review. ANZ J. Surg. 75: 464
Jolesz F. (2005) Future of magnetic resonance imaging and magnetic resonance spectroscopy in oncology. ANZ J. Surg. 75: 372
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Belkić, D., Belkić, K. Decisive role of mathematical methods in early cancer diagnostics: optimized Padé-based magnetic resonance spectroscopy. J Math Chem 42, 1–35 (2007). https://doi.org/10.1007/s10910-007-9227-9
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DOI: https://doi.org/10.1007/s10910-007-9227-9