Journal of Mathematical Chemistry

, Volume 40, Issue 1, pp 85–103 | Cite as

Mathematical Optimization of in vivo NMR Chemistry Through the Fast Padé Transform: Potential Relevance for Early Breast Cancer Detection by Magnetic Resonance Spectroscopy


Mathematical advances in signal processing through the fast Padé transform (FPT) can greatly improve the information extracted via in vivo nuclear magnetic resonance (NMR) chemistry. The FPT is a frequency-dependent, non-linear rational polynomial approximation of the exact Maclaurin series, which dramatically improves resolution and signal-to-noise ratio in a stable manner with robust error analysis and provides precise numerical data for all the peak parameters (position, height, width and phase) for every true resonance including those that are weak and/or overlapping. The concentrations of many of the chemical constituents of tissues can thereby be accurately determined. These advantages of the FPT are particularly germane for in vivo NMR detection and quantification of a number of molecular markers of breast cancer, such as phosphocholine, as well as lactate, which cannot be assessed using standard Fourier data analytical techniques applied to in vivo NMR in the clinical setting.


fast Padé transform NMR chemistry early cancer diagnostics breast cancer 



cytosine diphosphate


CHEmical Shift Selective


confidence interval


fast Fourier transform


fast Padé transform




Padé approximant




magnetic resonance


magnetic resonance imaging


magnetic resonance spectroscopy


magnetic resonance spectroscopic imaging


nuclear magnetic resonance


standard deviation


signal to noise ratio


echo time


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Copyright information

© Springer Science+Business Media, Inc. 2006

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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