Inverse problem for reconstruction of components from derivative envelope in ovarian MRS: Citrate quartet as a cancer biomarker with considerably decreased levels in malignant vs benign samples
Authors: Belkić Dževad , Belkić Karen 
Published: December 2022

The harmonic inversion (HI) problem in nuclear magnetic resonance spectroscopy (NMR) is conventionally considered by means of parameter estimations. It consists of extracting the fundamental pairs of complex frequencies and amplitudes from the encoded time signals. This problem is linear in the amplitudes and nonlinear in the frequencies that are entrenched in the complex damped exponentials (harmonics) within the time signal. Nonlinear problems are usually solved approximately by some suitable linearization procedures. However, with the equidistantly sampled time signals, the HI problem can be linearized exactly. The solution is obtained by relying exclusively upon linear algebra, the workhorse of computer science. The fast Padé transform (FPT) can solve the HI problem. The exact analytical solution is obtained uniquely for time signals with at most four complex harmonics (four metabolites in a sample). Moreover, using only the computer linear algebra, the unique numerical solutions, within machine accuracy (the machine epsilon), is obtained for any level of complexity of the chemical composition in the specimen from which the time signals are encoded. The complex frequencies in the fundamental harmonics are recovered by rooting the secular or characteristic polynomial through the equivalent linear operation, which solves the extremely sparse Hessenberg or companion matrix eigenvalue problem. The complex amplitudes are obtained analytically as a closed formula by employing the Cauchy residue calculus. From the frequencies and amplitudes, the components are built and their sum gives the total shape spectrum or envelope. The component spectra in the magnitude mode are described quantitatively by the found peak positions, widths and heights of all the physical resonances. The key question is whether the same components and their said quantifiers can be reconstructed by shape estimations alone. This is uniquely possible with the derivative fast Padé transform (dFPT) applied as a nonparametric processor (shape estimator) at the onset of the analysis. In the end, this signal analyzer can determine all the true components from the input nonparametric envelope. In other words, it can quantify the input time signal. Its performance is presently illustrated utilizing the time signals encoded at a high-field proton NMR spectrometer. The scanned samples are for ovarian cyst fluid from two patients, one histopathologically diagnosed as having a benign lesion and the other with a malignant lesion. These findings are presently correlated with the NMR reconstruction results from the Padé-based solution of the HI problem. Special attention is paid to the citrate metabolites in the benign and malignant samples. The goal of this focus is to see whether the citrates could also be considered as cancer biomarkers as they are now for prostate (low in cancerous, high in normal or benign tissue). Cancer biomarkers are metabolites whose concentration levels can help discriminate between benign and malignant lesions.