Mathematical optimization of in vivo NMR chemistry through the fast Pade transform:: potential relevance for early breast cancer detection by magnetic resonance spectroscopy

Mathematical advances in signal processing through the fast Pade 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.