Explicit extraction of absorption peak positions, widths and heights using higher order derivatives of total shape spectra by nonparametric processing of time signals as complex damped multi-exponentials

Abstract

Sole lineshape estimation of non-parametrically computed higher-order derivatives of spectral envelopes in different modes (complex, real, imaginary, magnitude) is investigated. The processed time signals are sums of complex attenuated exponentials (harmonics). The peak parameters of the derivative spectra are directly connected to those of the customary (non-derivative) absorption lineshapes. Crucially, this permits the reconstruction of the latter from the former parameters (the latter are sought since they are unknown, whereas the former are extractable from the derivative envelopes). The cross-checking relationships of the lineshapes for the magnitude modes with the real and imaginary parts of the complex-valued envelopes (total shape spectra) are established. The explicit procedure and the necessary analytical expressions are presented for reconstruction of the exact locations, widths and heights of all the retrieved physical resonances (spectral peaks). These facets are illuminated in the derivative fast Padé transform (dFPT) using its non-parametric version, i.e. without solving the quantification problem (no polynomial rooting, no tackling of eigen-value problems, etc.). Two kinds of illustrations for derivative spectra are reported. One deals with the general Breit-Wigner resonance formula and its first three derivatives. The other is concerned with the dFPT in clinical diagnostics of relevance to breast cancer detection by magnetic resonance spectroscopy. A systematic parallel between these two examples is drawn to highlight, in a stepwise manner, the role of paramount importance played by derivative lineshapes, especially for disentangling overlapping resonances that invariably plague all quantitative analyses of spectra.

Feasibility study for applying the lower-order derivative fast Padé transform to measured time signals

“…The introduction of the dFPT took place in Refs. [15][16][17] for synthesized MRS time signals with zero-valued phases. Therein, the primary focus was on the critical challenge of identifying and quantifying phosphocholine, PC, in a manner which would be the most amenable to unambiguous interpretation.…”

“…One of such mathematical transforms in MRS is the fast Padé transform (FPT) [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. It can perform both shape and parameter estimations.…”

“…The most recent upgrade of the FPT is the derivative fast Padé transform (dFPT) [6,[15][16][17]. This processor can provide quantification by exclusive reliance upon shape estimation.…”

“…First introduced and benchmarked on synthesized MRS time signals, the dFPT showed its outstanding performance by simultaneously eliminating noise and separating tightly overlapping resonances. For instance, despite being merely 0.001 ppm apart, metabolites phosphocholine (PC) and phosphoethanolamine (PE), have been separated and exactly quantified by the non-parametric dFPT [15][16][17] (hereafter, ppm denotes parts per million). In the reconstructed total shape spectrum or envelope, these two resonances were overlapped to such an extent that they appeared as a single perfectly symmetrical Lorentzian.…”

“…In Refs. [15][16][17], this goal was accomplished, with both noiseless and noisy simulated MRS time signals. After a thorough benchmarking of the dFPT for synthesized input data, it would be important to extend the analysis to encoded MRS time signals.…”