Signal Processing in Magnetic Resonance Spectroscopy with Biomedical Applications

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ISBN: 9781439806449 | 1439806446
Cover: Hardcover
Copyright: 1/29/2010

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Uses the FPT to Solve the Quantification Problem in MRS An invaluable tool in non-invasive clinical oncology diagnosticsAddressing the critical need in clinical oncology 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.

About the Authors	p. viii
Preface	p. x
Acknowledgments	p. xv
Basic tasks of signal processing in spectroscopy	p. 1
Challenges with quantification of time signals	p. 11
The quantum-mechanical concept of resonances in scattering and spectroscopy	p. 20
Resonance profiles	p. 24
Why is this topic relevant for biomedical researchers and clinical practitioners?	p. 26
The role of quantum mechanics in signal processing	p. 29
Direct link of quantum-mechanical spectral analysis with rational response functions	p. 31
Expansion methods for signal processing	p. 40
Non-classical polynomials	p. 40
Classical polynomials	p. 53
Recurrent time signals and their generating fractions as spectra with no recourse to Fourier integrals	p. 56
The fast Padé transform for quantum-mechanical spectral analysis and signal processing	p. 63
Padé acceleration and analytical continuation of time series	p. 65
Description of the background contribution by the off-diagonal fast Padé transform	p. 67
Diagonal and para-diagonal fast Padé transform	p. 69
Determination of the exact number K of resonances	p. 74
Exact Shank's filter for finding K, including the fundamental frequencies and amplitudes: the use of Wynn's recursion	p. 74
Exact number K and the existence of the solution of ordinary difference equations	p. 77
The role of linear dependence as spuriousness in determining K within the state space-based perspective of signal processing	p. 78
Froissart doublet spuriousness in the frequency domain for finding K	p. 81
Froissart doublets in exact analytical computations	p. 82
Exact quantum-mechanical, Padé-based recovery of spectral parameters	p. 85
Input data (tabular & graphic) and reconstructed tabular data	p. 91
Input tabular data for the spectral parameters of 25 resonances	p. 91
Numerical values of the reconstructed spectral parameters at six signal lengths, N/M (N = 1024, M = 1-32)	p. 95
Numerical values of the reconstructed spectral parameters near full convergence for 3 partial signal lengths N_P = 180, 220, 260	p. 97
Graphic presentation of the input data	p. 99
Absorption total shape spectra	p. 104
Absorption total shape spectra or envelopes	p. 104
Padé and Fourier convergence rates of absorption total shape spectra	p. 107
Residual spectra and consecutive difference spectra	p. 110
Residual or error absorption total shape spectra	p. 110
Residual or error absorption total shape spectra near full convergence	p. 113
Consecutive difference spectra for absorption envelope spectra	p. 113
Consecutive differences for absorption envelope spectra near full convergence	p. 116
Absorption component shape spectra of individual resonances	p. 116
Absorption component spectra and metabolite maps	p. 116
Absorption component spectra and envelope spectra near full convergence	p. 118
Distributions of reconstructed spectral parameters in the complex plane	p. 121
Distributions of spectral parameters in FPT⁽⁺⁾	p. 121
Distributions of spectral parameters in FPT^(-)	p. 124
Convergence of fundamental frequencies in FPT^(-)	p. 126
Distributions of fundamental frequencies in FPT^(±) near full convergence	p. 126
Convergence of fundamental amplitudes in FPT^(-)	p. 129
Distribution of fundamental amplitudes in FPT^(±) near full convergence	p. 131
Preview of illustrations for the concept of Froissart doublets	p. 133
The importance of exact quantification for MRS	p. 138
Harmonic transients in time signals	p. 149
Rational response function to generic external perturbations	p. 149
The exact solution for the general harmonic inversion problem	p. 151
General time series	p. 152
The response or the Green function	p. 154
The key prior knowledge: Internal structure of time signals	p. 155
The Rutishauser quotient-difference recursive algorithm	p. 159
The Gordon product-difference recursive algorithm	p. 161
The Lanczos algorithm for continued fractions	p. 167
The Padé-Lanczos approximant	p. 169
The fast Padé transform FPT^(-) outside the unit circle	p. 170
The fast Padé transform FPT⁽⁺⁾ inside the unit circle	p. 173
Signal-noise separation via Froissart doublets	p. 177
Critical importance of poles and zeros in generic spectra	p. 178
Spectral representations via Padé poles and zeros as pFPT^(±) and zFPT^(±)	p. 178
Padé canonical spectra	p. 180
Signal-noise separation with exclusive reliance upon resonant frequencies	p. 181
Model reduction problem via Padé canonical spectra	p. 183
Denoising Froissart filter	p. 184
Signal-noise separation with exclusive reliance upon resonant amplitudes	p. 185
Padé partial fraction spectra	p. 189
Model reduction problem via Heaviside or Padé partial fraction spectra	p. 190
Disentangling genuine from spurious resonances	p. 192
Machine accurate quantification and illustrated signal-noise separation	p. 193
Formulation of the most stringent test for quantification in MRS	p. 193
The key factors for high resolution in quantification	p. 195
The goals and plan for presentation of results	p. 196
Numerical presentation of the spectral parameters	p. 200
Input spectral parameters with 12-digit accuracy	p. 200
Exponential convergence rates of Padé reconstructions of spectral parameters with 12-digit accuracy	p. 202
Signal-noise separation via Froissart doublets with pole-zero coincidences	p. 205
Converged Padé genuine resonances and lack of convergence of Froissart doublets in FPT^(±) with a quarter of full signal length	p. 205
Zooming near convergence for Padé genuine resonances and instability of non-converged configurations of Froissart doublets in FPT^(±)	p. 216
Practical significance of the Froissart filter for exact signal-noise separation	p. 224
Padé processing for MR spectra from in vivo time signals	p. 227
Relative performance of the FPT and FFT for total shape spectra for encoded FIDs	p. 227
The FIDs, convergence regions and absorption spectra at full signal length encoded at high magnetic field strengths	p. 228
Convergence patterns of FPT^(-) and FFT for absorption total shape spectra	p. 231
Error Analysis for encoded in vivo time signals	p. 241
Residual spectra as the difference between the fully converged Fourier and Padé spectra at various partial signal lengths	p. 242
Self-contained Padé error analysis: Consecutive difference spectra	p. 245
Prospects for comprehensive applications of the fast Padé transform to in vivo MR time signals encoded from the human brain	p. 250
Magnetic resonance in neuro-oncology: Achievements and challenges	p. 251
MRS and MRSI as a key non-invasive diagnostic modality for neuro-oncology	p. 251
MRI for brain tumor diagnostics	p. 251
Primary diagnosis of brain tumors by MRS & MRSI	p. 253
Grading of primary brain tumors by MRS & MRSI	p. 256
Characterization of brain tumors by MRS & MRSI	p. 258
MRSI for target planning for brain tumors	p. 260
Assessing response of brain tumors to therapy and prognosis via MRSI	p. 260
Major limitations and dilemmas in MRS & MRSI for neuro-oncology due to FFT envelopes and fittings	p. 262
Poor resolution and SNR	p. 263
Unreliable quantifications by fitting FFT spectra	p. 267
Fitting estimates for concentrations of a small number of metabolites	p. 269
Lack of component spectra of clinically important over-lapping resonances for brain tumor diagnostics	p. 269
The number of metabolites & non-uniqueness of fitting	p. 270
Accurate extraction of clinically-relevant metabolite concentrations for neuro-diagnostics via MRS	p. 272
Methodological strategy: The need for standards in quantification	p. 272
High-resolution quantification of brain MR signals in a clinician-friendly format	p. 272
Padé-reconstructed lipids in the MR brain spectrum	p. 275
Padé reconstruction of the components of total choline at 3.2 ppm to 3.3 ppm on the MR brain spectrum	p. 276
Padé reconstruction in the region between 3.6 ppm and 4.0 ppm on the MR brain spectrum	p. 276
Padé quantification of malignant and benign ovarian MRS data	p. 277
Studies to date using in vivo proton MRS to evaluate benign and malignant ovarian lesions	p. 278
Insights for ovarian cancer diagnostics from in vitro MRS	p. 280
Padé versus Fourier for in vitro MRS data derived from benign and malignant ovarian cyst fluid	p. 282
Padé versus Fourier for MRS data derived from benign ovarian cyst fluid	p. 284
Padé versus Fourier for MRS data derived from malignant ovarian cyst fluid	p. 289
Summary comparisons of the performance of FPT and FFT for MRS data derived from benign and malignant ovarian cyst fluid	p. 293
Prospects for Padé-optimized MRS for ovarian cancer diagnostics	p. 302
Breast cancer and non-malignant breast data: Quantification by FPT	p. 303
Current challenges in breast cancer diagnostics	p. 303
In vivo MR-based modalities for breast cancer diagnostics and clinical assessment	p. 304
Magnetic resonance imaging applied to detection of breast cancer	p. 304
Studies to date using in vivo MRS for distinguishing between benign and malignant breast lesions	p. 305
In vivo MRS to assess response of breast cancer to therapy	p. 307
Special challenges of in vivo MRS for breast cancer diagnostics	p. 308
Insights for breast cancer diagnostics from in vitro MRS	p. 309
Performance of the FPT for MRS data from breast tissue	p. 311
Padé-reconstruction of MRS data for normal breast tissue	p. 312
Padé-reconstruction of MRS data from fibroadenoma	p. 318
Padé-reconstruction of MRS data from breast cancer	p. 322
Comparison of the Padé findings for normal breast, fibroadenoma and breast cancer	p. 326
Prospects for Padé-optimized MRS for breast cancer diagnostics	p. 330
Multiplet resonances in MRS data from normal and cancerous prostate	p. 331
Dilemmas and difficulties in prostate cancer diagnostics and screening	p. 331
Initial detection of prostate cancer with in vivo MRS and MRSI	p. 332
Distinguishing high from low risk prostate cancer	p. 334
Surveillance for residual disease or local recurrence after therapy	p. 334
Treatment planning and other aspects of clinical management	p. 335
Limitations of current applications of in vivo MRSI directly relevant to prostate cancer	p. 335
Insights for prostate cancer diagnostics by means of 2D in vivo MRS and in vitro MRS	p. 336
Performance of the fast Padé transform for MRS data from prostate tissue	p. 339
Normal glandular prostate tissue: MR spectral data reconstructed by FPT	p. 344
Normal stromal prostate tissue: MR spectral data reconstructed by FPT	p. 350
Malignant prostate tissue: MR spectral information reconstructed by FPT	p. 356
Comparison of MRS retrievals from prostate tissue: Normal glandular, normal stromal and cancerous	p. 362
Prospects for Padé-optimized MRSI within prostate cancer diagnostics	p. 370
Recapitulation of Padé-optimized processing of biomedical time signals	p. 371
The central role of rational functions in the theory of approximations	p. 371
The dominant role of Padé approximant among all rational functions	p. 372
Relevance of Padé-optimized MRS for diagnostics in clinical oncology	p. 382
Conclusion and outlooks	p. 389
Leading role of Padé approximants in the theory of rational functions and in MRS	p. 390
Outlooks for Padé-optimized MRS and MRSI from a clinical perspective	p. 395
List of acronyms	p. 399
References	p. 401
Index	p. 441
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Signal Processing in Magnetic Resonance Spectroscopy with Biomedical Applications

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