Signal Processing in Magnetic Resonance Spectroscopy with Biomedical Applications

Addressing the critical need in clinical oncology for robust and stable signal processing in magnetic resonance spectroscopy (MRS), Signal Processing in Magnetic Resonance Spectroscopy with Biomedical Applications explores cutting-edge theory-based innovations for obtaining reliable quantitative information from MR signals for cancer diagnostics. By defining the natural framework of signal processing using the well-established theory of quantum physics, the book illustrates how advances in signal processing can optimize MRS.

The authors employ the fast Padé transform (FPT) as the unique polynomial quotient for the spectral analysis of MR time signals. They prove that residual spectra are necessary but not sufficient criteria to estimate the error invoked in quantification. Instead, they provide a more comprehensive strategy that monitors constancy of spectral parameters as one of the most reliable signatures of stability and robustness of quantification. The authors also use Froissart doublets to unequivocally distinguish between genuine and spurious resonances in both noise-free and noise-corrupted time signals, enabling the exact reconstruction of all the genuine spectral parameters. They show how the FPT resolves and quantifies tightly overlapped resonances that are abundantly seen in MR spectra generated using data from encoded time signals from the brain, breast, ovary, and prostate.

Written by a mathematical physicist and a clinical scientist, this book captures the multidisciplinary nature of biomedicine. It examines the remarkable ability of the FPT to unambiguously quantify isolated, tightly overlapped, and nearly confluent resonances.

Basic Tasks of Signal Processing in Spectroscopy

Challenges with quantification of time signals

The quantum-mechanical concept of resonances in scattering and spectroscopy

Resonance profiles

Why is this topic relevant to biomedical researchers and clinical practitioners?

The Role of Quantum Mechanics in Signal Processing

Direct link of quantum-mechanical spectral analysis with rational response functions

Expansion methods for signal processing

Recurrent time signals and their generating fractions as spectra with no recourse to Fourier integrals

Fast Padé transform (FPT) for quantum-mechanical spectral analysis and signal processing

Padé acceleration and analytical continuation of time series

Description of the background contribution by the off-diagonal FPT

Diagonal and para-diagonal FPT

Froissart doublets and the exact number of resonances

Harmonic Transients in Time Signals

Rational response function to generic external perturbations

The exact solution for the general harmonic inversion problem

General time series

Response or Green function

The key prior knowledge: internal structure of time signals

The Rutishauser quotient-difference recursive algorithm

The Gordon product-difference recursive algorithm

The Lanczos continued fractions

The Padé–Lanczos approximant

FPT(−) outside the unit circle

FPT(+) inside the unit circle

Signal-Noise Separation via Froissart Doublets

Critical importance of poles and zeros in generic spectra

Spectral representations via Padé poles and zeros: pFPT(±) and zFPT(±)

Padé canonical spectra

Signal-noise separation: exclusive reliance upon resonant frequencies

Model reduction problem via Padé canonical spectra

Denoising Froissart filter

Signal-noise separation: exclusive reliance upon resonant amplitudes

Padé partial fraction spectra

Model reduction problem via Padé partial fraction spectra

Disentangling genuine from spurious resonances

Padé Processing for Magnetic Resonance (MR) Total Shape Spectra from in vivo Free Induction Decays (FIDs)

Comparison of the performances of the FPT and fast Fourier transform (FFT) for total shape spectra

The FIDs, convergence regions, and absorption spectra at full signal length for 4T and 7T

Convergence patterns of the FPT(−) and FFT for absorption spectra at 4T and 7T

Error analysis

Prospects for comprehensive applications of the FPT to in vivo MR time signals for brain tumor diagnostics

Exact Reconstructions of Spectral Parameters by FPT

Tabular data

Absorption total shape spectra

Residual spectra and consecutive difference spectra

Absorption component shape spectra of individual resonances

Distributions of reconstructed spectral parameters in the complex plane

Discussion

Relevance of exact quantification in brain tumor diagnostics

Machine Accurate Padé Quantification and Exact Signal-Noise Separation

Numerical presentation of the spectral parameters

Direct comparison of the performance of the FFT and the FPT

Convergence of total shape spectra versus component spectra in the FPT

Signal-noise separation through the concept of Froissart doublets/pole-zero cancellation

Diagnostic significance of the Froissart filter for exact signal-noise separation

Magnetic Resonance Spectroscopy (MRS) and Magnetic Resonance Spectroscopic Imaging (MRSI) in Neuro-Oncology: Achievements and Challenges

MRS and MRSI as a key non-invasive diagnostic modality for neuro-oncology

Major limitations and dilemmas with MRS and MRSI in neuro-oncology related to reliance upon conventional Fourier-based data analysis

Accurate extraction of clinically relevant metabolite concentrations for neurodiagnostics via MRS

Padé Quantification of Malignant and Benign Ovarian MRS Data

Studies to date using in vivo proton MRS to evaluate benign and malignant ovarian lesions

Insights for ovarian cancer diagnostics from in vitro MRS

Performance of the FPT for in vitro MRS data derived from benign and malignant ovarian cyst fluid, and comparisons with the FFT

Prospects for Padé-optimized MRS for ovarian cancer diagnostics

Breast Cancer and Non-Malignant Breast Data: Quantification by FPT

Current challenges in breast cancer diagnostics

In vivo MR-based modalities for breast cancer diagnostics and clinical assessment

Insights for breast cancer diagnostics from in vitro MRS

Performance of the FPT for MRS data from breast tissue

Prospects for Padé-optimized MRS for breast cancer diagnostics

Multiplet Resonances in MRS Data from Normal and Cancerous Prostate

Dilemmas in prostate cancer diagnostics and screening

Insights for prostate cancer diagnostics by means of 2D in vivo MRS and in vitro MRS

Performance of the FPT for MRS data from prostate tissue

Prospects for Padé-optimized MRSI within prostate cancer diagnostics

General Discussion

Why the FPT for signal processing?

The two variants of the FPT converging inside and outside the unit circle

Computation of the complex frequencies and amplitudes by FPT

Interpolation and extrapolation by the FPT

Determination of the exact number of metabolites

Lorentzian and non-Lorentzian spectra both computed by FPT

Validity assessment of the FPT

Error analysis

Clinical ramifications of implementing Padé-based in vivo MRS: special importance for cancer diagnostics

Conclusions and Outlooks

Prediction and extrapolation for resolution improvement

Dževad Belkic is a professor of mathematical radiation physics at the Karolinska Institute in Stockholm, Sweden. Dr. Belkic’s current research activities encompass atomic collision physics, radiation physics, radiobiology, magnetic resonance physics and mathematical physics.

Karen Belkic is a physician specialist in internal medicine, adjunct professor of preventive medicine at the Keck School of Medicine of the University of Southern California in Los Angeles, and senior scientist in the oncology and pathology department at the Karolinska Institute in Stockholm, Sweden.