Signal Processing in Magnetic Resonance Spectroscopy with Biomedical Applications 1st Edition by Dzevad Belkic, Karen Belkic ISBN 1439806446 9781439806449

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ISBN 10: 1439806446
ISBN 13: 9781439806449
Author: Dzevad Belkic, Karen Belkic

Uses the FPT to Solve the Quantification Problem in MRS
An invaluable tool in non-invasive clinical oncology diagnostics

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.

Signal Processing in Magnetic Resonance Spectroscopy with Biomedical Applications 1st Edition Table of contents:

1 Basic tasks of signal processing in spectroscopy

1.1 Challenges with quantification of time signals

1.2 The quantum-mechanical concept of resonances in scattering and spectroscopy

1.3 Resonance profiles

1.4 Why is this topic relevant for biomedical researchers and clinical practitioners?

2 The role of quantum mechanics in signal processing

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

2.2 Expansion methods for signal processing

2.2.1 Non-classical polynomials

2.2.2 Classical polynomials

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

2.4 The fast Padé transform for quantum-mechanical spectral analysis and signal processing

2.5 Padé acceleration and analytical continuation of time series

2.6 Description of the background contribution by the off-diagonal fast Padé transform

2.7 Diagonal and para-diagonal fast Padé transform

2.8 Determination of the exact number K of resonances

2.8.1 Exact Shank’s filter for finding K, including the fundamental frequencies and amplitudes: the use of Wynn’s recursion

2.8.2 Exact number K and the existence of the solution of ordinary difference equations

2.8.3 The role of linear dependence as spuriousness in determining K within the state space-based perspective of signal processing

2.8.4 Froissart doublet spuriousness in the frequency domain for finding K

2.8.5 Froissart doublets in exact analytical computations

3 Exact quantum-mechanical, Padé-based recovery of spectral parameters

3.1 Input data (tabular & graphic) and reconstructed tabular data

3.1.1 Input tabular data for the spectral parameters of 25 resonances

3.1.2 Numerical values of the reconstructed spectral parameters at six signal lengths, N/M (N = 1024, M = 1 − 32)

3.1.3 Numerical values of the reconstructed spectral parameters near full convergence for 3 partial signal lengths NP = 180, 220, 260

3.1.4 Graphic presentation of the input data

3.2 Absorption total shape spectra

3.2.1 Absorption total shape spectra or envelopes

3.2.2 Padé and Fourier convergence rates of absorption total shape spectra

3.3 Residual spectra and consecutive difference spectra

3.3.1 Residual or error absorption total shape spectra

3.3.2 Residual or error absorption total shape spectra near full convergence

3.3.3 Consecutive difference spectra for absorption envelope spectra

3.3.4 Consecutive differences for absorption envelope spectra near full convergence

3.4 Absorption component shape spectra of individual resonances

3.4.1 Absorption component spectra and metabolite maps

3.4.2 Absorption component spectra and envelope spectra near full convergence

3.5 Distributions of reconstructed spectral parameters in the complex plane

3.5.1 Distributions of spectral parameters in FPT(+)

3.5.2 Distributions of spectral parameters in FPT(−)

3.5.3 Convergence of Fundamental Frequencies in FPT(−)

3.5.4 Distributions of fundamental frequencies in FPT(±) near full convergence

3.5.5 Convergence of fundamental amplitudes in FPT(−)

3.5.6 Distribution of fundamental amplitudes in FPT(±) near full convergence

3.6 Preview of illustrations for the concept of Froissart doublets

3.7 The importance of exact quantification for MRS

4 Harmonic transients in time signals

4.1 Rational response function to generic external perturbations

4.2 The exact solution for the general harmonic inversion problem

4.3 General time series

4.4 The response or the Green function

4.5 The key prior knowledge: Internal structure of time signals

4.6 The Rutishauser quotient-difference recursive algorithm

4.7 The Gordon product-difference recursive algorithm

4.8 The Lanczos algorithm for continued fractions

4.9 The Padé-Lanczos approximant

4.10 The fast Padé transform FPT(–) outside the unit circle

4.11 The fast Padé transform FPT(+) inside the unit circle

5 Signal-noise separation via Froissart doublets 1

5.1 Critical importance of poles and zeros in generic spectra

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

5.3 Padé canonical spectra

5.4 Signal-noise separation with exclusive reliance upon resonant frequencies

5.5 Model reduction problem via Padé canonical spectra

5.6 Denoising Froissart filter

5.7 Signal-noise separation with exclusive reliance upon resonant amplitudes

5.8 Padé partial fraction spectra

5.9 Model reduction problem via Heaviside or Padé partial fraction spectra

5.10 Disentangling genuine from spurious resonances

6 Machine accurate quantification and illustrated signal-noise separation

6.1 Formulation of the most stringent test for quantification in MRS

6.2 The key factors for high resolution in quantification

6.3 The goals and plan for presentation of results

6.4 Numerical presentation of the spectral parameters

6.4.1 Input spectral parameters with 12-digit accuracy

6.4.2 Exponential convergence rates of Padé reconstructions of spectral parameters with 12-digit accuracy

6.5 Signal-noise separation via Froissart doublets with pole-zero coincidences

6.5.1 Converged Padé genuine resonances and lack of convergence of Froissart doublets in FPT(±) with a quarter of full signal length

6.5.2 Zooming near convergence for Padé genuine resonances and instability of non-converged configurations of Froissart doublets in FPT(±)

6.6 Practical significance of the Froissart filter for exact signal-noise separation

7 Padé processing for MR spectra from in vivo time signals

7.1 Relative performance of the FPT and FFT for total shape spectra for encoded FIDs

7.2 The FIDs, convergence regions and absorption spectra at full signal length encoded at high magnetic field strengths

7.3 Convergence patterns of FPT(-) and FFT for absorption total shape spectra

7.4 Error Analysis for encoded in vivo time signals

7.4.1 Residual spectra as the difference between the fully converged Fourier and Padé spectra at various partial signal lengths

7.4.2 Self-contained Padé error analysis: Consecutive difference spectra

7.5 Prospects for comprehensive applications of the fast Padé transform to in vivo MR time signals encoded from the human brain

8 Magnetic resonance in neuro-oncology: Achievements and challenges

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

8.1.1 MRI for brain tumor diagnostics

8.1.2 Primary diagnosis of brain tumors by MRS & MRSI

8.1.3 Grading of primary brain tumors by MRS & MRSI

8.1.4 Characterization of brain tumors by MRS & MRSI

8.1.5 MRSI for target planning for brain tumors

8.1.6 Assessing response of brain tumors to therapy and prognosis via MRSI

8.2 Major limitations and dilemmas in MRS & MRSI for neuro-oncology due to FFT envelopes and fittings

8.2.1 Poor resolution and SNR

8.2.2 Unreliable quantifications by fitting FFT spectra

8.2.3 Fitting estimates for concentrations of a small number of metabolites

8.2.4 Lack of component spectra of clinically important overlapping resonances for brain tumor diagnostics

8.2.5 The number of metabolites & non-uniqueness of fitting

8.3 Accurate extraction of clinically-relevant metabolite concentrations for neuro-diagnostics via MRS

8.3.1 Methodological strategy: The need for standards in quantification

8.3.2 High-resolution quantification of brain MR signals in a clinician-friendly format

8.3.3 Padé-reconstructed lipids in the MR brain spectrum

8.3.4 Padé reconstruction of the components of total choline at 3.2 ppm to 3.3 ppm on the MR brain spectrum

8.3.5 Padé reconstruction in the region between 3.6 ppm and 4.0 ppm on the MR brain spectrum

9 Padé quantification of malignant and benign ovarian MRS data

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

9.2 Insights for ovarian cancer diagnostics from in vitro MRS

9.3 Padé versus Fourier for in vitro MRS data derived from benign and malignant ovarian cyst fluid

9.3.1 Padé versus Fourier for MRS data derived from benign ovarian cyst fluid

9.3.2 Padé versus Fourier for MRS data derived from malignant ovarian cyst fluid

9.3.3 Summary comparisons of the performance of FPT and FFT for MRS data derived from benign and malignant ovarian cyst fluid

9.4 Prospects for Padé-optimized MRS for ovarian cancer diagnostics

10 Breast cancer and non-malignant breast data: Quantification by FPT

10.1 Current challenges in breast cancer diagnostics

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

10.2.1 Magnetic resonance imaging applied to detection of breast cancer

10.2.2 Studies to date using in vivo MRS for distinguishing between benign and malignant breast lesions

10.2.3 In vivo MRS to assess response of breast cancer to therapy

10.2.4 Special challenges of in vivo MRS for breast cancer diagnostics

10.3 Insights for breast cancer diagnostics from in vitro MRS

10.4 Performance of the FPT for MRS data from breast tissue

10.4.1 Padé-reconstruction of MRS data for normal breast tissue

10.4.2 Padé-reconstruction of MRS data from fibroadenoma

10.4.3 Padé-reconstruction of MRS data from breast cancer

10.4.4 Comparison of the Padé findings for normal breast, fibroadenoma and breast cancer

10.5 Prospects for Padé-optimized MRS for breast cancer diagnostics

11 Multiplet resonances in MRS data from normal and cancerous prostate 3

11.1 Dilemmas and difficulties in prostate cancer diagnostics and screening

11.1.1 Initial detection of prostate cancer with in vivo MRS and MRSI

11.1.2 Distinguishing high from low risk prostate cancer

11.1.3 Surveillance for residual disease or local recurrence after therapy

11.1.4 Treatment planning and other aspects of clinical management

11.1.5 Limitations of current applications of in vivo MRSI directly relevant to prostate cancer

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

11.3 Performance of the fast Padé transform for MRS data from prostate tissue

11.3.1 Normal glandular prostate tissue: MR spectral data reconstructed by FPT

11.3.2 Normal stromal prostate tissue: MR spectral data reconstructed by FPT

11.3.3 Malignant prostate tissue: MR spectral information reconstructed by FPT

11.3.4 Comparison of MRS retrievals from prostate tissue: Normal glandular, normal stromal and cancerous

11.4 Prospects for Padé-optimized MRSI within prostate cancer diagnostics

12 Recapitulation of Padé-optimized processing of biomedical time signals 3

12.1 The central role of rational functions in the theory of approximations

12.2 The dominant role of Padé approximant among all rational functions

12.3 Relevance of Padé-optimized MRS for diagnostics in clinical oncology

13 Conclusion and outlooks

13.1 Leading role of Padé approximants in the theory of rational functions and in MRS

13.2 Outlooks for Padé-optimized MRS and MRSI from a clinical perspective

List of acronyms

References

Index

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