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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
Article

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.

Keywords

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

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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

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