Fast Pade transform for exact quantification of time signals in Magnetic Resonance Spectroscopy

This work employs the fast Pade transform (FPT) for spectral analysis of theoretically generated time signals. The spectral characteristics of these synthesised signals are reminiscent of the corresponding data that are measured experimentally via encoding digitised free induction decay curves from a healthy human brain using Magnetic Resonance Spectroscopy (MRS). In medicine, in vivo MRS is one of the most promising non-invasive diagnostic tools, especially in oncology, due to the provided biochemical information about functionality of metabolites of the scanned tissue. For success of such diagnostics, it is crucial to carry out the most reliable quantifications of the studied time signals. This quantification problem is the harmonic inversion via the spectral decomposition of the given time signal into its damped harmonic constituents. Such a reconstruction finds the unknown total number of resonances, their complex frequencies and the corresponding complex amplitudes. These spectral parameters of the fundamental harmonics give the peak positions, widths, heights, and phases of all the physical resonances. As per the unified theory of quantum-mechanical spectral analysis and signal processing, the FPT represents the exact solver of the quantification problem, which is mathematically ill-conditioned. The exact and unique solution via the FPT is valid for any noiseless synthesised time signal built from an arbitrary number of damped complex exponentials. These attenuated harmonics can appear as a linear combination with both stationary and non-stationary amplitudes. Such sums produce time signals that yield Lorentzian (non-degenerate) and non-Lorentzian (degenerate) spectra for isolated and overlapped resonances from MRS. We give a convergent validation for these virtues of the FPT. This is achieved through the proof-of-principle investigation by developing an algorithmic feasibility for robust and efficient computations of the exact numerical solution of a typical quantification problem from MRS. The systematics in the methodology designed in the present study represent a veritable paradigm shift for solving the quantification problem in MRS with special ramifications in clinical oncology. This is implied by the explicit demonstration of the remarkable ability of the fast Pade transform to unambiguously quantify all the customary spectral structures, ranging from isolated resonances to those that are tightly overlapped and nearly confluent.