An Estimation Technique for Spin Echo Electron Paramagnetic Resonance
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An Estimation Technique for Spin Echo Electron Paramagnetic Resonance A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Frank Golub, B.S. Graduate Program in Department of Electrical and Computer Engineering The Ohio State University 2013 Master's Examination Committee: Lee C. Potter, Advisor Bradley Clymer Rizwan Ahmad © Copyright by Frank Golub 2013 Abstract In spin echo electron paramagnetic resonance (SE-EPR) spectroscopy, traditional methods to estimate T2 relaxation time include fitting an exponential to the peaks or the integrated areas of multiple noisy echoes. These methods are suboptimal and result in lower estimation accuracy for a given acquisition time. Here, two data pro- cessing methods to estimate T2 for SE-EPR are proposed. The first method finds the maximum likelihood estimate (MLE) of T2 via parametric modeling of the spin echo and joint least-squares fitting of the collected data. The second method exploits the underlying rank-one structure in SE-EPR data via singular-value decomposition (SVD). The right singular vector corresponding to the largest singular value is then fitted with an exponential to find T2. This method bears strong similarity to a non- parametric MLE-based approach that does not assume a structure of an echo. The methods are validated using simulation and experimental data. The proposed meth- ods provide 41-fold and 3-fold acquisition time savings over the traditional methods of fitting echo peaks and areas, respectively. Interestingly, the results also indicate that the SVD-based approach generates mean squared error nearly identical to that produced by the MLE based on parametric modeling for a wide range of SNR. ii This is dedicated to my parents, my nephews, and their sense of humor. iii Acknowledgments I wish to acknowledge the love and support from my friends and family, and the dedication from my advisors Lee Potter and Rizwan Ahmad. My sister Marjorie deserves much credit for inspiring me to learn math at a young age. iv Vita December 8, 1987 . .Born - Canton, Ohio 2006 . .McKinley Senior High School 2011 . .B.S. Physics, Brandeis University 2011 . .B.S. Mechanical Engineering, Columbia University Fields of Study Major Field: Electrical and Computer Engineering Focus: Signal Processing and Electron Paramagnetic Resonance v Table of Contents Page Abstract . ii Dedication . iii Acknowledgments . iv Vita.........................................v List of Figures . viii 1. Introduction . .1 1.1 Background . .1 1.2 Relevant Trends in EPR . .2 1.3 Approach . .3 1.4 Thesis Organization . .4 2. Spin Echoes . .5 2.1 Resonance . .5 2.2 The Initial Hahn Spin Echo . .5 2.3 Carr and Purcell: The Current Hahn Echo . .6 2.4 Signal Processing Methods . .8 3. Maximum Likelihood Estimation . 10 3.1 Maximum Likelihood Estimation . 10 3.2 Singular Value Decomposition . 12 3.3 Estimators for T2 ............................ 14 3.3.1 Peak-picking and integration . 14 3.3.2 Parametric MLE . 15 3.4 The Relationship Between SVD and MLE . 15 vi 4. The Cram´er-RaoLower Bound . 18 4.1 Derivation for One Non-random Parameter . 18 4.2 Extension to Multiple Parameters . 21 5. Results . 24 5.1 Materials and Methods . 24 5.2 Results . 24 6. Discussion . 29 7. Conclusion . 31 Appendices 32 A. Matlab Code . 32 Bibliography . 42 vii List of Figures Figure Page 4.1 Lower bound on the standard error of Tb2................ 23 5.1 Singular values for the data array formed from the seven acquired echoes. 25 5.2 Left: standard deviation of T2 estimation error for a simulated seven echo train at four noise levels: 17, 24, 30, and 37 dB. Right: relative acquisition time versus SNR to achieve fixed standard deviation of T2 estimation error. 26 5.3 Left: standard deviation of T2 estimation error from simulated data versus number of acquired echoes. Right: relative acquisition time versus number of echoes to achieve fixed standard deviation of T2 esti- mation error. 27 5.4 Left: standard deviation of T2 estimation error for measured data with seven echoes at four synthesized noise levels: 17, 24, 30, and 37 dB. Right: relative acquisition time versus SNR to achieve fixed standard deviation of T2 estimation error. 28 5.5 Left: standard deviation of T2 estimation error from measured data versus number of acquired echoes. Right: relative acquisition time versus the number of echoes to achieve fixed standard deviation of T2 estimation error. 28 viii Chapter 1: Introduction 1.1 Background Electron paramagnetic resonance (EPR) is a spectroscopic method capable of detecting and quantifying free radicals. The physics that governs nuclear magnetic resonance (NMR) technology also governs electron paramagnetic resonance (EPR). NMR focuses on the spins of protons, whereas EPR revolves around the spins of unpaired electrons. Felix Bloch won the 1952 Nobel Prize for discovering how nucleons move in a magnetic field. Using the Bloch equations, Hahn invented the concept of spin echoes, later refined by Carr and Purcell, that is prevalent in both nuclear imaging domains. But whereas NMR can successfully map the tissue densities in the body, EPR has failed to gain widespread clinical practice. EPR promises to measure the oxygen concentration in tumors, but it is hindered by the need to inject a patient with a spin label. Internal biochemistry quickly degrades signals from spin labels, highlighting the need to assess these signals with minimal acquisition time. The proposed processing techniques improve upon the traditional methods by offering estimates with smaller variances of the transverse decay constant T2. Because linewidth is linearly related to partial pressure of oxygen, the T2 parameter provides a sensitive indicator of oxygen 1 concentration. Thus, for a given desired estimation accuracy, the proposed methods allow for a reduction in the amount of noise averaging and hence the acquisition time. 1.2 Relevant Trends in EPR Over the past several decades, EPR has found numerous applications in biology, chemistry, physics, and medicine [16]. Among in vivo applications of EPR, oximetry has been arguably the most investigated area of research, with emphasis on quanti- tative assessment of tumor hypoxia [13, 19]. Continuous wave (CW) EPR and pulsed EPR are competing yet often complemen- tary modes of data acquisition. At present, CW EPR remains the most widespread technique for in vivo oximetry [2] because of its simple equipment design and ability to utilize a wide variety of oxygen sensitive spin probes. However, the data acquisi- tion in CW EPR is generally slow, resulting in long acquisition times. One way to accelerate data acquisition is to use pulsed EPR methods. With recent technologi- cal advances and the development of EPR oximetry probes with narrow linewidth, pulsed EPR oximetry has become an attractive option, especially for studying tumor hypoxia [10]. In pulsed EPR oximetry, a spin echo (SE) approach allows measuring intrinsic T2 directly, which can be readily converted to the homogeneous broadening component of the EPR lineshape. Since the homogeneous broadening is proportional to the oxygen concentration, T2 can thus be used to estimate oxygen concentration via a precom- puted calibration curve. The SE-EPR oximetry can be used in both spectroscopic and imaging modes. 2 In SE-EPR, the data are generally collected using conventional π=2 − τ − π−echo pulse sequence [11]. Since the echo amplitude decays with exp(−2τ=T2), collecting and processing multiple echoes with different τ values enables estimation of T2. The traditional methods of processing SE-EPR data include fitting an exponential to echo peaks (peak-picking) or echo areas (echo-integration). The peak-picking method is inefficient, because it only uses one data point, the peak, from each echo. Additionally, echo-integration is likewise shown to be an inefficient use of the acquired data. 1.3 Approach The general approach towards selecting an estimation technique is to fully charac- terize all samples in a data set to better ascertain T2. In this study, a measured data set consists of a sequence of echoes. The amplitude of each echo decays exponentially with rate constant T2. Traditional methods such as peak picking and integration fail to represent the underlying structure of each echo. Here, we propose two methods to process SE-EPR data for estimating T2. The first approach parametrically models the spin echo and finds T2, along with other nuisance parameters, by nonlinear least-squares to generate the maximum likelihood estimate (MLE). A similar approach has been used for pH measurement using CW EPR [1]. The second method is an SVD-based estimator for an unknown echo shape. We note the presence of a similar MLE that also makes no prior assumption of an echo shape. A Cram´er-Raobound investigation demonstrates the similarity in the proposed techniques in a typical SNR region. Furthermore, the two maximum likelihood estimators are consistent estimators for the SE-EPR signal model. All 3 techniques were compared using simulated and measured data with synthetic noise. A small vial of activated charcoal was excited at X-band. 1.4 Thesis Organization Chapter 2 introduces the concept of spin echoes and the early signal models. Chapter 3 presents the two traditional and two proposed signal models in the con- text of maximum likelihood estimation. Chapter 4 begins with a derivation of the Cram´er-Raolower bound and discusses the asymptotic variance qualities of maximum likelihood estimation. Chapter 5 details the experimental conditions and presents the results for the simulated and measured data set, respectively. A performance com- parison of the four methods is discussed in Chapter 6. Chapter 7 concludes the findings. 4 Chapter 2: Spin Echoes 2.1 Resonance Resonance may be more familiar in the context of an operatic soprano shattering glassware. Glass has a tendency to vibrate at its natural frequency.