NMR Spectroscopy for Metabolite Quantification in Medical Diagnostics: Exact Solution for Harmonic Inversion Problem and Exact Signal Noise Separation

Magnetic resonance spectroscopy is increasingly recognized as one of the key diagnostic modalities in oncology. This type of spectroscopy can detect early radiation damage to biomolecules as a salient feature of malignancy. Notwithstanding such appealing characteristics, progress in magnetic resonance spectroscopy is currently hampered by the lack of mathematically and clinically reliable spectral analysis for extraction of quantifiable information (e.g., metabolite concentrations, etc.) from scanned tissue. A number of fitting recipes are available, but they are all inadequate due to subjectivity, non-uniqueness and inability to separate physical (genuine) from unphysical (spurious, noisy) information. The fast Padé transform, which is a non-linear, polynomial quotient, bridges this gap by unambiguously resolving and quantifying tightly overlapped and nearly degenerate and/or degenerate resonances that are abundantly present in magnetic resonance spectra generated using encoded in vivo time signals. Spurious resonances appear with all the parametric estimators that must use more than twice the number of unknown frequencies and amplitudes. This leads to an over-determined system of linear equations yielding more resonances than the actual number present in the analyzed signal. The fast Padé transform makes use of pole-zero cancellation, or equivalently, Froissart doublets to unequivocally distinguish true from spurious resonances. Spurious resonances act as noise. The task is to identify such resonances and discard them. The number of spurious resonances is always several times greater than that of the true ones. The computation is carried out by gradually and systematically increasing the degree of the Padé polynomials. As this degree changes, the reconstructed parameters and the associated spectra fluctuate until stabilization occurs. The polynomial degree at which the predetermined level of accuracy is achieved represents the sought exact number of resonances. An illustrative set of results is reported in this work to show how the fast Padé transform unequivocally separates genuine from spurious information. The ensuing PadéFroissart optimization of magnetic resonance spectroscopy is expected to represent a very important modality for early cancer diagnostics.