AbstractThe theme of this study is derivative nuclear magnetic resonance (dNMR) spectroscopy. This versatile methodology of peering into the molecular structure of general matter is common to e.g. analytical chemistry and medical diagnostics. Theoretically, the potential of dNMR is huge and the art is putting it into practice. The implementation of dNMR (be it in vitro or in vivo) is wholly dependent on the manner in which the encoded time signals are analyzed. These acquired data contain the entire information which is, however, opaque in the original time domain. Their frequency-dependent dual representation, a spectrum, can be transparent, provided that the appropriate signal processors are used. In signal processing, there are shape and parameter estimators. The former processors are qualitative as they predict only the forms of the lineshape profiles of spectra. The latter processors are quantitative because they can give the peak parameters (positions, widths, heights, phases). Both estimators can produce total shape spectra or envelopes. Additionally, parameter estimators can yield the component spectra, based on the reconstructed peak quantifiers. In principle, only parameter estimators can solve the quantification problem (harmonic inversion) to determine the structure of the time signal and, hence, the quantitative content of the investigated matter. The derivative fast Fourier transform (dFFT) and the derivative fast Padé transform (dFPT) are the two obvious candidates to employ for dNMR spectroscopy. To make fair comparisons between the dFFT and dFPT, the latter should also be applied as a shape estimator. This is what is done in the present study, using the time signals encoded from a patient with brain tumor (glioma) using a 1.5T clinical scanner. Moreover, within the dFPT itself, the shape estimations are compared to the parameter estimations. The goal of these testings is to see whether, for in vivo dNMR spectroscopy, shape estimations by the dFPT could quantify (without fitting), similarly to parameter estimations. We check this key point in two successive steps. First, we compare the envelopes from the shape and parameter estimations in the dFPT. The second comparison is between the envelopes and components from the shape and parameter estimations, respectively, in the dFPT. This plan for benchmarking shape estimations by the dFPT is challenging both on the level of data acquisition and data analysis. The data acquisition reported here provides encoded time signals of short length, only 512 as compared to 2048, which is customarily employed. Moreover, the encoding echo time was long (272 ms) at which most of resonances assigned to metabolites with shorter spin-spin relaxations are likely to be obliterated from the frequency spectra. Yet, in face of such seemingly insurmountable obstacles, we are looking into the possibility to extract diagnostically relevant information, having particularly in focus the resonances for recognized cancer biomarkers, notably lactate, choline and phosphocholine. Further, we want to see how many of the remaining resonances in the spectra could accurately be identified with clinical reliability as some of them could also be diagnostically relevant. From the mathematical stance, we are here shaking the sharp border between shape and parameter estimators. That border stood around for a long time within nonderivative estimations. However, derivative shape estimations have a chance to tear the border down. Recently, shape estimations by the dFPT have been shown to lead such a trend as this processor could quantify using the time signals encoded from a phantom (a test sample of known content). Further, the present task encounters a number of additional challenges, including a low signal-to-noise ratio (SNR) and, of course, the unknown content of the scanned tissue. Nevertheless, we are determined to find out whether the nonparametric dFPT can deliver the unique quantification-equipped shape estimation and, thus, live up to the expectation of derivative processing: a long-sought simultaneous improvement of resolution and SNR. In every facet of in vivo dNMR, we found that shape estimations by the dFPT has successfully passed the outlined most stringent tests. It begins with transforming itself to a parameter estimator (already with the 3rd and 4th derivatives). It ends with reconstructing some 54 well-isolated resonances. These include the peaks assigned to recognized cancer biomarkers. In particular, a clear separation of choline from phosphocholine is evidenced for the first time by reliance upon the dFPT with its shape estimations alone.
On detecting stellar binary black holes via the LISA-Taiji network
