AbstractTime signals are measured experimentally throughout sciences, technologies and industries. Of particular interest here is the focus on time signals encoded by means of magnetic resonance spectroscopy (MRS). The great majority of generic time signals are equivalent to auto-correlation functions from quantum physics. Therefore, a quantum-mechanical theory of measurements of encoded MRS time signals is achievable by performing quantum-mechanical spectral analysis. When time signals are measured, such an analysis becomes an inverse problem (harmonic inversion) with the task of reconstruction of the fundamental frequencies and the corresponding amplitudes. These complex-valued nodal parameters are the building blocks of the associated resonances in the frequency spectrum. Customarily, the MRS literature reports on fitting some ad hoc mathematical expressions to a set of resonances in a Fourier spectrum to extract their positions, widths and heights. Instead, an alternative would be to diagonalize the so-called data matrix with the signal points as its elements and to extract the resonance parameters without varying any adjusting, free constants as these would be absent altogether. Such a data matrix (the Hankel matrix) is from the category of the evolution matrix in the Schrödinger picture of quantum mechanics. Therefore, the spectrum of this matrix, i.e. the eigenvalues and the corresponding amplitudes, as the Cauchy residues (that are the squared projections of the full wave functions of the system onto the initial state) are equivalent to the sought resonance parameters, just mentioned. The lineshape profile of the frequency-dependent quantum-mechanical spectral envelope is given by the Heaviside partial fraction sum. Each term (i.e. every partial fraction) in this summation represents a component lineshape to be assigned to a given molecule (metabolite) in the tissue scanned by MRS. This is far reaching, since such a procedure allows reconstruction of the most basic quantum-mechanical entities, e.g. the total wave function of the investigated system and its ’Hamiltonian’ (a generator of the dynamics), directly from the encoded time signals. Since quantum mechanics operates with abstract objects, it can be applied to any system including living species. For example, time signals measured from the brain of a human being can be analyzed along these lines, as has actually been done e.g. by own our research. In this way, one can arrive at a quantum-mechanical description of the dynamics of vital organs of the patient by retrieving the interactions as the most important parts of various pathways of the tissue functions and metabolism. Of practical importance is that the outlined quantum-mechanical prediction of the frequency spectrum coincides with the Padé approximant, which is in signal processing alternatively called the fast Padé transform (FPT) for nonderivative estimations. Further, there is a novelty called the derivative fast Padé transform (dFPT). The FPT and dFPT passed the test of time with three fundamentally different time signals, synthesized (noise-free, noise-contaminated) as well as encoded from phantoms and from patients. Such systematics are necessary as they permit robust and reliable benchmarkings of the theory in a manner which can build confidence of the physician, while interpreting the patient’s data and making the appropriate diagnosis. In the present study, we pursue further this road paved earlier by applying the FPT and dFPT (both as shape and parameter estimators) to time signals encoded by in vivo proton MRS from an ovarian tumor. A clinical 3T scanner is used for encoding at a short echo time (30 ms) at which most resonances have not reached yet their decay mode and, as such, could be detected to assist with diagnostics. We have two goals, mathematical and clinical. First, we want to find out whether particularly the nonparametric dFPT, as a shape estimator, can accurately quantify. Secondly, we want to determine whether this processor can provide reliable information for evaluating an ovarian tumor. From the obtained results, it follows that both goals have met with success. The nonparametric dFPT, from its onset as a shape estimator, transformed itself into a parameter estimator. Its quantification capabilities are confirmed by reproducing the components reconstructed by the parametric dFPT. Thereby, fully quantified information is provided to such a precise extent that a large number of sharp resonances (more than 160) appear as being well isolated and, thus, assignable to the known metabolites with no ambiguities. Importantly, some of these metabolites are recognized cancer biomarkers (e.g. choline, phosphocholine, lactate). Also, broader resonances assigned to macromolecules are quantifiable by a sequential estimation (after subtracting the formerly quantified sharp resonances and processing the residual spectrum by the nonparametric dFPT). This is essential too as the presence of macromolecules in nonoderivative envelopes deceptively exaggerates the intensities of sharper resonances and, hence, can be misleading for diagnostics. The dFPT, as the quantification-equipped shape estimator, rules out such possibilities as wider resonances can be separately quantified. This, in turn, helps make adequate assessment of the true yield from sharp resonances assigned to metabolites of recognized diagnostic relevance.
In vitro proton magnetic resonance spectroscopy at 14T for benign and malignant ovary: Part I, signal processing by the nonparametric fast Padé transform
