Simulation and Optimal Design of. Nuclear Magnetic Resonance Experiments

Transcription

1 Simulation and Optimal Design of Nuclear Magnetic Resonance Experiments

2 SIMULATION AND OPTIMAL DESIGN OF NUCLEAR MAGNETIC RESONANCE EXPERIMENTS BY ZHENGHUA NIE, B.Sc. a thesis submitted to the School of Graduate Studies of McMaster university in partial fulfilment of the requirements for the degree of Doctor of Philosophy c Copyright by Zhenghua Nie, July 2011 All Rights Reserved

3 Doctor of Philosophy (2011) (School of Computational Engineering and Science) McMaster University Hamilton, Ontario, Canada TITLE: Simulation and Optimal Design of Nuclear Magnetic Resonance Experiments AUTHOR: Zhenghua Nie B.Sc., (Applied Mathematics (including Computer Application)) University of Electronic Science and Technology of China, Chengdu, P. R. China SUPERVISORS: Dr. Christopher Kumar Anand Dr. Alex D. Bain NUMBER OF PAGES: xxi, 258 ii

4 To Qu and Tian with All My Love and In Memory of My Parents and My Brother-in-Law

5 Abstract In this study, we concentrate on spin- 1 2 systems. A series of tools using the Liouville space method have been developed for simulating of NMR of arbitrary pulse sequences. We have calculated one- and two-spin symbolically, and larger systems numerically of steady states. The one-spin calculations show how SSFP converges to continuous wave NMR. A general formula for two-spin systems has been derived for the creation of double-quantum signals as a function of irradiation strength, coupling constant, and chemical shift difference. The formalism is general and can be extended to more complex spin systems. Estimates of transverse relaxation, R 2, are affected by frequency offset and field inhomogeneity. We find that in the presence of expected B 0 inhomogeneity, offresonance effects can be removed from R 2 measurements, when ω 0.5γB 1 in Hahn echo experiments, when ω γb 1 in CPMG experiments with specific phase variations, by fitting exact solutions of the Bloch equations given in the Lagrange form. Approximate solutions of CPMG experiments show the specific phase variations can significantly smooth the dependence of measured intensities on frequency offset in the range of ± 1γB 2 1. The effective R 2 of CPMG experiments when using a phase variation scheme can be expressed as a second-order formula with respect to the ratio iv

6 of offset to π-pulse amplitude. Optimization problems using the exact or approximate solution of the Bloch equations are established for designing optimal broadband universal rotation (OBUR) pulses. OBUR pulses are independent of initial magnetization and can be applied to replace any pulse of the same flip angles in a pulse sequence. We demonstrate the process to exactly and efficiently calculate the first- and second-order derivatives with respect to pulses. Using these exact derivatives, a second-order optimization method is employed to design pulses. Experiments and simulations show that OBUR pulses can provide more uniform spectra in the designed offset range and come up with advantages in CPMG experiments. v

7 Preface This PhD study has eight chapters including four chapters which are based on published peer-reviewed papers and one chapter which may be submitted for publication in future. I performed all analysis, computation and programming works and drafted manuscripts. Dr. Anand and Dr. Bain, my supervisors who are also co-authors of my papers, guided me in these projects, discussed the structure of each paper and revised all papers. Details for each paper (relevant chapter) are outlined below. Publication details are listed at the beginning of each relevant chapter. Chapter 3 introduces the symbolic algebra package to simulate NMR which I wrote. Dr. Anand proposed to use Maple software to develop the package and helped me to understand Maple when I started to write. Dr. Bain taught me about the Liouville space method and validated the results which the package would generate. I gave presentations at the Maple conference, Aug. 2006, and the 15th International Conference Applications of Computer Algebra (ACA), June 2009 to demonstrate the package and its applications. I also applied this package to obtain the optimal delay of a pulse sequence which is used to detect a three-spin system and this work was presented in a poster at Second Mathematical Programming Society International Conference on Continuous Optimization(ICCOPT II)& Modelling and Optimization: Theory and Applications (MOPTA), Aug Chapter 4 demonstrates the application of the algebra package to study the steady vi

8 state. In this work, I presented the derivation of general formals of steady states of continuous wave and pulsed cases of any spin systems and the exact solution of the steady state of continuous wave of a strong coupled-spin system, and also reproduced other known results. Dr. Bain proposed this project to demonstrate the advantages of the symbolic algebra computation, helped me to review the literatures on the steady state and also guided me what we should display in the paper. Dr. Anand continued to help me on the symbolic algebra. I presented a poster to display what we had calculated about the steady state at 19th Annual MOOT Conference, Sep Chapter 5 shows how to fit the experimental data with the exact solution of the Bloch equations. The motivation of this project was that I wanted to examine the performance of optimal pulses which I had developed, so I started to explore the simplest case: measuring the relaxation time using a rectangular pulse. I investigated how to compute the exponential of a matrix symbolically and implemented the Lagrange method which could be applied to fit the NMR experiments. Dr. Anand suggested the method to study the variations of other parameters on the measurements of R 2. Dr. Bain prepared the sample and did the experiments, he also proposed to apply the second-order split methods to explore the approximations and helped me to draft the introduction section of this paper. I presented this work at MOPTA, Aug and Southern Ontario Numerical Analysis Day (SONAD), May We also made a poster to illustrate the solution of the Bloch equations which was displayed at the Small Molecule NMR Conference (SMASH), Sep Chapter 6 investigates CPMG experiments. After we completed the previous work, we extended to investigate more complicated experiments. In this work, I not only developed methods to fit the experiments, but also applied the optimization methods to design phase variations of CPMG. And more importantly, I deduced one simple formula which can provide more reliable results if we do not apply the vii

9 exact complicated fitting method. Dr. Bain prepared the sample and performed the experiments what we displayed in this work. I presented this work at Canadian Operational Research Society (CORS) Annual Conference, June Chapter 7 displays how to design optimal pulses using second-order optimization methods. In order to design a pulse which has the same performance of a hard pulse, I proposed the use of the rotation matrix as the target. These optimal pulses are independent of initial magnetizations and can be applied in the place of pulses which have the same flip angle. Another benefit is that it is easy to construct optimal pulses which have large flip angles without any more efforts. Dr. Anand proposed the design of optimal pulses using second-order optimization methods and developed the efficient algorithm to calculate the first-order and second-order derivatives. My program is based on an initial implementation of Dr. Anand and Andrew Thomas Curtis which only considers the vector model without the effect of relaxation. Dr. Bain also prepared the sample and did all experiments. This work was presented in posters at the 49th Experimental NMR Conference (ENC), March 2008 and SMASH, Sep I also gave a presentation at MOPTA, Aug We plan to submit the manuscript for publication. viii

10 Acknowledgements I would like to express my sincere appreciation to Dr. Christopher K. Anand and Dr. Alex D. Bain, my supervisors, for their encouragement, support, guidance and patience during my study. Without their in-depth guide, in particular the computation techniques from Dr. Anand and the theory and experiments of NMR from Dr. Bain, this work would not have been completed. I would also like to thank them for giving me this opportunity to study at McMaster, as well as their careful reading and editing on the drafts of this thesis. Sincere thanks to Dr. Gillian Goward and Dr. Michael D. Noseworthy, for their agreement to be my supervisory committee members. I would like to thank them for their time taken to read and discuss my works. I am very grateful to my fellow students in my group. Last but not least, I would like to express my greatest thanks to my wife Qu and my wonderful daughter Tian for their support and patience throughout the years of my study. This is dedicated to you! ix

11 Contents Abstract iv Preface vi Acknowledgements ix Notations and Abbreviations xiv 1 Introduction 1 2 Theory and Tools Density Matrix Liouville Space Method A Single Spin- 1 2 System in the Inhomogeneous Form Bloch Equations Computing an Exponential of a Matrix Eigen Decomposition Method Lagrange Interpolation Method Split Operator Method Symbolic Computation of NMR Optimization of NMR x

12 2.7.1 Solving Unconstrained Optimization Problems Solving Nonlinear Constrained Optimization Problems Optimization Problems Described by Systems of ODEs Applications Applying Computer Algebra to Simulate NMR Introduction Using Maple to Simulate the Liouville Space Method Constructions of Elements of Pulsed NMR Computing the Evolution Spin Echo of a Coupled System Optimization of Spin Echo Conclusion Simulation of Steady-State NMR of Coupled Systems Using Liouville Space and Computer Algebra Methods Introduction Liouville Space Formulation General Solution for the Continuous Wave Case General Solution for the Pulsed Case Solutions of a single spin- 1 2 system Single-spin CW Solution Solution of a single spin system of pulsed NMR experiments Ernst Angle of A Single Spin System Double-Quantum Transitions of a 2-spin system Symbolic Solution of Continuous Wave Experiments Extension to multiple spin- 1 2 systems xi

13 4.6 Conclusion Exact Solution of the Bloch Equations and Application to the Hahn Echo Introduction The Exact Symbolic Solution of the Bloch Equations The Exact Solution of the Bloch Equations Approximate Solutions of the Bloch Equations Approximate Solutions via Approximate Eigenvalues Approximate Solutions via Split-Operators Computation of Experiments of the Hahn Echo Measuring R 2 by the Hahn Echo Experiments Effects of Other Parameters on the Measurements of R Conclusion Exact Simulation of the CPMG Pulse Sequence with Phase Variation Down the Echo Train with Application to R 2 Estimation Introduction Overview Computation of CPMG Experiments Oscillations of Conventional CPMG Minimizing Oscillations Estimating R 2 from 0013-phase-cycled CPMG Estimating R 2 using exact CPMG Simulations Conclusion xii

14 7 Design of Universal Rotation Pulses using Large-scale Nonlinear Optimization with Second Derivatives Introduction Designing Pulses Simulating of A Shaped Pulse Establishing Optimization Problems Efficient Derivative Calculation Results and Discussion Experiments of CPMG Using Optimal and Rectangular Pulses Comparison with published refocusing pulses Conclusion Conclusions and Future Work 201 A LSMMP: Liouville Space Method s Maple Package 205 B Solutions of Special Cases of the Bloch Equations 218 B.1 Solution of the Free Evolution B.2 The Solution of the Case R 1 = R C Calculations of the Hahn Echo 224 C.1 Model 0: Exact Solution of Phase Cycling C.2 The Solution of Model C.3 The Solution by Model C.4 The Solution by Model Bibliography 234 xiii

15 Notations and Abbreviations NMR: Nuclear Magnetic Resonance...1 T 1 : Spin-lattice relaxation time...3 T 2 : Spin-spin relaxation time D: One Dimensional... 5 INADEQUATE: Incredible Natural Abundance DoublE QUAntum Transfer Experiment...5 S/N: Ratio of the signal-to-noise...5 CW: Continuous-Wave...6 FT: Fourier Transform...6 MRI: Magnetic Resonance Imaging...6 CPMG: Carr-Purcell-Meiboom-Gill...13 ˆP, ˆQ: Operators in the Hilbert space...15 : The right side is the definition of the left side P, Q: Vectors in the Liouville pace L: The Liouvillian matrix encoding information about Larmor frequencies, chemical structure and couplings (if the spin system is multiple)...17 xiv

16 ρ: All of bold symbols will represent a vector or matrix...17 ODE: Ordinary Differential Equations : A unit operator or an identity matrix...18 I x,i y,i z : Three Cartesian angular momentum operators ), 1 +1 ), 1 1 ), 0): Four observables of a single spin- 1 2 system which are defined in Table : Direct product...18 i: ρ eq : The equilibrium state vector in the Liouville space method...20 R: The relaxation matrix ω: Chemical shift or resonance offset ω 0 : The center frequency (or the reference frequency)...23 ω: The offset which is equal to ω ω γ: The gyromagnetic ratio...24 B 1 : The magnitude field associated with a pulse...24 b 1 : The strength of a pulse in the unit Hz which is defined as γb R 1 : Spin-lattice relaxation rate which is defined as 1 T R 2 : Spin-spin relaxation rate which is defined as 1 T t p : The length of the pulse...25 α: The flip angle which is equal to γb 1 t p when B 1 dominates all interactions...25 ρ xy : The total XY magnetization vector which will be applied to obtain the signal of NMR...26 xv

17 M x,m y,m z : The x, y and z component of the bulk magnetization in the Cartesian frame...27 M e : The equilibrium z magnetization which is set as a constant number A: The coefficient matrix of the Bloch equations in the homogeneous form...27 : The two-norm of a vector or a matrix...37 KKT: Karush-Kuhn-Tucker...40 IPOPT: Interior Point OPTimizer...44 BFGS: Broyden-Fletcher-Goldfarb-Shanno...44 SR1: Symmetric Rank LSMMP: Liouville Space Method Maple Package...48 SSFP: Steady-State Free Precession OBUR: Optimal Broadband Universal Rotation pulses OBUR-180-9: A 180 OBUR pulse Henkelman16S: A optimal 180 pulse designed by Poon and Henkelman ref4019: A 180 optimal pulse designed by Skinner, et al HS : A 180 adiabatic pulse designed by Hwang, et al xvi

18 List of Tables 2.1 Spherical tensor basis for Liouville Space of a single spin CPMG experiments with one idential pulse of a given phase CPMG experiments with groups of two pulses CPMG experiments for the top eight phase variation schemes with four pulses Parameters of design of the optimal 180 pulse OBUR List of Refocusing Pulses for Comparison xvii

19 List of Figures 1.1 Illustration of energy difference Illustration of classical NMR Illustration of pulsed NMR experiments Illustration of convex and non-convex functions Angles of rotation matrix Absolute function Local and global minimum Illustration of penalty functions Illustration of barrier functions Illustration of numerical integral Pulsed NMR The relationship of the J coupling constant and the difference of chemical shifts The spin echo NMR experiment with a perfect π pulse The signal corresponding to one of the lines of a weakly coupled system The signal corresponding to one of the lines of a strongly coupled system The spin echo NMR experiment with a α refocusing pulse The objective function of a spin echo of a strong coupled system with a single chemical shift xviii

20 3.8 The objective function of a spin echo of a coupled system with a Gaussian distribution of chemical shifts The objective function of a spin echo of a strongly-coupled system with a Gaussian distribution of chemical shifts The simplest pulsed NMR experiment Illustration of obtaining Ernst angle The intensity of the double quantum transition The intensity of the double quantum transition at infinity δ The spectra of a coupled spin system with different B The spectra of a coupled spin system with different coupling constants Reproducing Worvill s spectrum of a 3-spin system Comparing relative errors of different approximations of a wide range Comparing relative errors of different approximations of a narrow range The pulse sequence of the Hahn echo experiments The approximations of the Hahn echo Contour of the objective function with respect to I 0 and R Fitted curves and measured intensities Fitted curves and measured intensities using denser sampling Estimated R 2 as a function of offset The nth echo in the CPMG experiments One group of four pulses with different phases in CPMG experiments Oscillations with respect to the echo number Illustration of magnetization vectors of the phase variation scheme Approximation of one echo which is called Model Simulated profile of effective decay rates of CPMG experiments with soft 180 pulses xix

21 6.7 The profile of measured effective R 2 using 0013 phase variation with respect to offsets Fitted curves and measured intensities of the conventional CPMG experiments TheprofileofmeasuredR 2 fromconventional CPMGexperiments plotted with respect to ratio of offset over the amplitude of the 180 pulse Fitted curves and measured intensities for phase variation scheme Measured R 2 of the phase variation scheme 0013 plotted against the ratio of offset over the amplitude of the 180 pulse Measured R 2 of the phase variation scheme 0013 plotted against the ratio of offset over the amplitude of th 180 pulse The universal optimal 180 pulse OBUR ZG experiment and simulation Simulated spectrum of inversion-recovery and spin-echo experiments using the pulse OBUR Simulated spectrum of CPMG of 8 refocusing pulses OBUR using different phases Fitted curves and measured intensities of using OBUR with exact solution Measured R 2 of a square pulse and the optimal pulse OBUR plotted against the ratio of offset over the amplitude of the 180 pulse Measured R 2 of using the pulse OBUR plotted against the ratio of offset over the amplitude of the 180 pulse Shape of pulses Simulated CPMG spectrum Amplitude of simulated CPMG spectrum xx

22 7.11 Phases of simulated CPMG spectrum Optimal pulse of using Henkelman16S as the initial pulse xxi

23 Chapter 1 Introduction Nuclear Magnetic Resonance (NMR) experiments detect the interaction of spins (nuclei of 1 H, 13 C, 15 N, andso on) withradio frequency radiationinastrong magneticfield [1]. After electronic spectra had been applied to investigate the atomic and molecular structure and led to the discovery of electron spin [2], in the early 1920 s, Pauli proposed the hypothesis that some atomic nuclei should possess magnetic moments. After 20 years, the interaction of spin with a magnetic field was observed directly. In 1945, two research groups, one group of Purcell, Torrey and Pound, and the other group of Bloch, Hansen and Packard independently observed radio-frequency signals generated by atomic nuclei of different bulk materials [3, 4]. In 1951, Arnold, Dharmatti and Packard first demonstrated the relationship between chemical shifts and chemical structure in a high-resolution 1 H NMR spectrum of ethanol [5]. In 1965, the Fourier transform technique was applied to enhance the sensitivity and allow pulse sequences by Ernst [6, 7]. Since then, NMR techniques have been applied to investigate the structure and dynamics of molecules [8], and also to image nuclei of atoms inside the body which incredibly extends the application of NMR to the medical area [9, 10]. 1

24 Following Pauli s hypothesis, the nuclei which possess magnetic moments are said to have a spin, the nuclear magnetic momentum and energy are quantized and the number of observable spin states is determined by the spin quantum number. When a nucleus is placed in an static magnetic field denoted by B 0, the spin will be split into discrete energy levels which are characterized by the nuclear spin number, the separation of adjacent energy levels are determined by the strength of the magnetic field (see Fig. 1.1), the intensity of the signal depends on the Boltzmann distribution of energy states, and the corresponding observation frequency is called the Larmor frequency [1, 8]. Another external rf field (a pulse) moves the spin from the lower energy level to the higher energy level (changes the distribution of energy states) and induces a transition which can be explained as flipping spins from one orientation to another in the magnetic field. Observable transitions in rf appear in the measured spectrum [11, 12]. Figure 1.1: The energy difference between spin states as a function of the external magnetic field B 0. A line in the spectrum is associated with a transition between two energy levels. In the classical description of NMR, the magnetic moments will experience a torque in a strong external static magnetic field which is set along with the z axis. The bulk magnetic moment is the vector sum of the individual nuclear magnetic 2

25 fields (see Fig. 1.2). An oscillating magnetic field B 1 will cause an absorption of energy by nuclear spins when the frequency of the oscillating magnetic field matches the precessional frequency of the nucleus [1, 8]. Because of the oscillating magnetic field, a net induced magnetization will be raised in the xy plane. Both B 1 and M are normally represented by vectors in the rotating frame. In the rotating frame, the net magnetization vector M can be seen as rotating to the x y plane when B 1 is applied at right angles to B 0. The net oscillating magnetization in the xy plane induces a current in receiver coils which is acquired as the NMR signal [1]. Figure 1.2: Illustration of classical NMR. The left figure shows the vector presentation of population distribution of a spin- 1. At equilibrium,the bulk (net) magnetic 2 moment along the magnetic field B 0 is the sum of nuclear magnetic moment, and it can be represented by a magnetization vector M. The right figure shows the net magnetization M rotating to the x y plane in the rotating frame when B 1 is applied. B 1 and M can be represented by vectors in the rotating frame x y z. When not in the presence of an oscillating magnetic field (i.e., after the rf pulse), the z component of the magnetization slowly relaxes back to the z axis at a rate of 1/T 1. This process is called spin-lattice relaxation or longitudinal relaxation. Similarly, the xy component of the magnetization decays in the xy plane at a rate of 1/T 2. T 2 is called the spin-spin or transverse relaxation time. The relaxation times T 1 and 3

26 T 2 arethe time constants for exponential regrowth/decay. The T 1 valueis characteristic of the movement of spin populations back to their Boltzmann distribution values. The spin-spin relaxation, T 2, is concerned with the decay of coherences because the phase coherence of the xy magnetizations is gradually lost [8]. The study of spin relaxation is important to investigate molecular motion [8]. Accurately measuring the relaxation rates, especially the spin-spin relaxation rates, is the key to performing such studies. Spectroscopy reflects transitions of a set of energy levels of molecules. NMR is a wonderful tool with which to study spectroscopy [1, 12], and, thereby, the quantum mechanics of the underlying molecular system [11]. The energy levels describe nuclei which are parts of atoms which are assembled into molecules, thus, by investigating the transitions of a set of energy levels by NMR it is possible to expose the structure of molecules in detail which are difficult or impossible to obtain by any other methods [13]. The resonance frequency of a nucleus depends on the static magnetic field, but it is also influenced by the local electronic environment because the electrons in the molecules shield the nucleus altering the local magnetic fields [8]. This effect is called the chemical shielding which is an intramolecular interaction and reflects the local electronic environment. The nuclear spins may be coupled together and such coupling affects the resonance frequency of nuclei [8]. This type of nucleus-nucleus coupling is called indirect spin-spin coupling or J-coupling and provides information about the chemical bond. Because the J-coupling is exclusively intramolecular, the magnitudes of the J-coupling constants are independent of the magnetic field strength and are highly dependent on molecular structure [8]. NMR can indeed provide very detailed, highly localized information on molecules 4

27 [14 16]. The simplest one-dimensional (1D) spectroscopy experiment provides information on the ratio of nuclei by integrating the area of each line. Based on the ratio of areas, the groups of nuclei are easily identified [5]. In complex molecules, spectroscopy may have complex splitting patterns, thus, more complicated experiments or more powerful magnets are required to deal with overlaps in the spectra of complex molecules. Because increasing the strength of the magnetic field is limited and it does not adequately resolve all overlapped multiplets, other techniques have to be developed, for example, higher-dimensional spectroscopy, used to determine connectivity and structure of complex molecules [1, 12]. The 2D Incredible Natural Abundance DoublE QUAntum Transfer Experiment (INADEQUATE) has been applied to map out explicitly the carbon-carbon connectivity of a molecule, using the scalar coupling between two 13 C nuclei [17]. Another core application of NMR spectroscopy is the study of the dynamics of molecules such as chemical exchange [18, 19] and protein dynamics [20, 21]. NMR provides an extremely powerful and convenient method for monitoring the dynamics of molecules because the dynamical process will take nuclei from one magnetic environment to another one and back [14, 19, 20]. Relaxation experiments are the basis of NMR dynamics measurements used to study both chemical exchange [18, 19] and nano-pico and milli-micro second dynamics [20 22]. The major disadvantage of NMR is the inherently low sensitivity [6]. Applying Fourier transform techniques to improve the sensitivity of experiments has the following advantages to NMR experiments. Firstly, pulsed NMR can increase the signal-to-noise ratio by measuring all the signal all the time. Signal averaging increases the signal-to-noise ratio (S/N) of the spectrum by a factor of n 1/2, n being the number of transients making up the average [6]. Secondly, pulsed NMR can be used for multiple-dimensional experiments so as to overcome the overlap in complex 5

28 molecules [1]. Thirdly, pulsed NMR is capable of polarization transfer [23]. Moreover, pulsed NMR can select the coherence pathway so as to suppress the signal we do not want [24]. One more advantage is that pulsed NMR can optimize the pulse sequence so as to increase the efficiency of spectroscopy [25 28]. Because of these advantages of Fourier transform techniques, pulsed FT-NMR instruments have largely replaced continuouswave (CW) NMR spectrometers and pulse sequences are the core of modern pulsed FT-NMR spectrometers which have led to the development of a library of hundreds of NMR and Magnetic Resonance Imaging (MRI) experiments [1, 16, 29, 30]. Pulse sequences consist of bursts of radiofrequency (rf) irradiation (pulses) followed by periods of free evolution (delays) (see Fig. 1.3) [1, 8, 12]. Basically, pulses are required to do a number of things: take magnetization from +z to z (inversion pulses), take magnetization from +z to x or y (excitation pulses) or reverse the direction of precession (refocusing pulses). Sophisticated combinations of these manipulations allow us to extract almost all possible information about a spin system. Figure 1.3: Illustration of pulsed NMR experiments. If pulses dominate all other interactions, which means that the strength of the rf field is much bigger than other terms, especially offsets, these pulses are called hard or non-selective. Otherwise, the pulses are called soft or selective. The effect of a hard pulse is equivalent to a rotation of the frame of reference which is relatively easy to calculate, but the effect of a soft pulse is more complicated because we should 6

29 not ignore other effects within pulses. In this study, we will apply hard pulses to explore coupled spin systems, but we will focus on investigating soft pulses acting on single spin systems. For most spin systems, the static energy levels are well-understood, so it is their evolution in time during a pulse sequence that most interests us. But the most basic pulsed NMR experiments cannot be described via the energy level approach, some more tools are needed to understand NMR experiments [12]. As we see in Fig. 1.2, vectors are used to represent the oscillating magnetic field (called a pulse) and the net bulk magnetization in the rotating frame, the effect of a pulse may be represented by a rotation of the magnetization vector. The Bloch equations exactly describe the motion of the magnetization [31]. This method is called the vector model which has been around as long as NMR itself [12]. The vector model is sufficient to understand NMR experiments involving isolated single spin- 1 systems. 2 The vector model is not sufficient for us to interpret large or coupled spin systems, so we need another tool to present accurate theoretical descriptions of the dynamics of these spin systems. Since we have a finite number of spin states for any spin systems, the density matrix [1, 8, 12] is a tractable possibility. We are able to have an exact picture of spin states from the density matrix, so it is a vital tool in describing and analyzing magnetic resonance experiments for any spin systems [32, 33]. Operator formalisms are methods of acting on the density matrix so as to give reasonable solutions to show the effect of a pulse and a delay in NMR [8, 16, 32]. The Liouville space method, one of the operator formalisms, works directly on the density matrix via linear algebra [32 37]. Theoretically, experimental spectroscopy will reproduce the results given by theory if errors are ignored, thus, the simulation or computation of NMR is a critical tool to investigate and develop NMR [37]. In the Liouville space method, a density matrix is represented by a vector, and 7

30 matrices are used to represent effects of pulses and delays. The Liouvillian matrix has the information of a spin system including structure (strong or weak coupled), chemical shifts, J-coupling constants. A pulse sequence can be exactly described by a series of matrix operations in Liouville space, whereas the vector model in a Hilbert space is inadequate [32, 33]. The simulation and computation of NMR not only help us to understand and explore NMR experiments, but also allow us to improve the performance of experiments such as sensitivity enhancement [6], coherence pathway [24], optimal delays [26, 28] and accurate measurements of relaxation times [38, 39]. We need to distinguish between ideal experiments (which we can often understand) and real experiments (where we need computation). For example, in dynamics research, we need to measure the relaxation time, the theory tells us that the infinitely powerful, instantaneous pulses will minimize other effects resulting in perfect measurements of relaxation time, but because of the limits of spectrometers or samples, we cannot make the power of pulses infinite. When the delay time is comparable to the pulse length, we cannot ignore the effects of relaxation and chemical shift offsets during pulses, so we need to understand what happens within pulses, and that understanding will bring advantages to measuring relaxation times more accurately [38]. If we know everything about a spin system and pulse sequence, which means that we are able to describe the spin system and pulse sequence using numeric values, numerical computation is able to simulate large spin systems and more complicated pulse sequences to help us to explore some NMR phenomena. The spectra which are obtained by the numerical simulation of coupled spin systems may help us to observe how the power of the magnetic field or the coupling constants affect the spectra. Many software programs and packages have been developed to numerically simulate NMR using different computer languages such as C, C++, Matlab, Mathematica, 8

31 Maple [40 54]. Currently, numerical computation is often the sole or best way to deal with large spin systems [38, 54 56]. Model fitting[38], designing optimal pulses or pulse sequences [27, 39], and observing how the parameters affect experiments [39] require analytical solutions of NMR. The spin systems are governed by the equations of motion which are ordinary differential equations with respect to the time. When we design pulses or fit experiments, we have to integrate the equations of motion and compute the partial derivatives with respect to the pulse or relaxation times. We need derivatives to solve the optimization problems using second-order methods which will increase the convergence rate and provide more stable solutions. Exact analytical solutions which can provide exact derivatives improve the performance of designing pulses [57] and the accuracy of measured parameters [38, 39]. Maple and Mathematica are the principal computer tools used to construct spin systems, display pulse sequences, and operate on the density matrix in NMR analytically using symbolic computation [38, 39, 55, 56, 58 73]. Generally speaking, it is easy to obtain numerical solutions, but it is more difficult to deal with symbolic solutions even using approximate methods [56] or working on the simplest system [38]. The most powerful symbolic approximation in spin dynamics is to ignore relaxation, or at least to restrict it to operating only during delays. This is slightly stronger than the common assumption that pulses are hard. An rf pulse is hard if the rf terms dominate all other interactions, such as off-resonance effects and relaxation. With this approximation, a pulse becomes equivalent to a rotation of the frame of reference, whose effect is relatively easy to calculate. In the standard product operator formalism [74, 75], this is done by exploiting the simple commutator rules for spin This hard pulse approximation implies the pulse is instantaneous, so there is no time 9

32 for evolution or relaxation to occur. With these symbolic approximations we can understand the vast majority of pulse NMR experiments. Accurate theoretical descriptions of the dynamics of a spin system are required for us to understand current experiments and design new experiments. It is not trivial to obtain full and exact solutions even dealing with the simplest single spin- 1 system 2 [76 80]. The simplest system is described by the well-known Bloch equations which contain the effects of precession, rf irradiation and relaxation [4, 31, 81 83]. Under some simplifying approximations, solutions to the Bloch equations are well-known, but some methodologyisneeded to provide anexact solution which ismoreinkeeping with current theory, and which can easily be extended to more complex systems [38]. Because NMR experiments can be exactly described by functions of parameters of experiments, we can optimize these parameters or extract these parameters from the experimental data. Optimization has been applied in different areas of NMR including fitting the relaxation times [38, 39, 84], designing pulses and pulse sequences [27, 85 88], and reconstructing spectra or obtaining other parameters of spectra [89 93]. The least square is mostly used to model the optimization problems of NMR experiments. Since the equations of motion of spin dynamics are ordinary differential equations, the objective function often has a term which is the integral of the ordinary differential equations. The approximations used and the type of problem (design or parameter estimation) determine the complexity of the optimization problem, for example convex versus non-convex (see Fig. 1.4), linear versus non-linear objective functions. A convex function has a unique global minimum point which may be obtained in one step no matter the starting point, but a non-convex function may have many local minima and it is a challenge to obtain the global minimum point [94]. A linear function has a constant slope or rate of change (the graph of a linear function is a line), a quadratic function has a linearly varying slope, but the second derivative, or 10

33 curvature, of a nonquadratic function may vary from point to point [95]. If a function is nonquadratic and nonconvex, as we see in Fig. 1.4, it is more difficult to find out the global minimum point. Different solution methods must be applied, according to the complexity of the optimization problem. For example, if we apply hard pulses to measure relaxation times, the theoretical solution will be a single exponential decay, thenthefitting model issimple andcanbesolved efficiently [94]. Butif weuse shaped pulses, the problem may become extremely non-linear, non-convex and large-scale, then more complicated optimization solvers are required [27]. Figure 1.4: Illustration of convex and non-convex functions. Both of these two curves are nonlinear functions. We can efficiently find the global minimum if the objective function is convex. If the objective function is non-convex, there may be more than one local minimum, and it is difficult to find the global minimum. Since pulsed NMR was invented, many pulses or pulse sequences have been developed [1, 16, 29, 30]. New spectrometers with strong magnetic fields improve the performance of experiments and provide opportunities to do more complicated experiments. NMR is also applied to explore not only structures and functions of isolated large complex biological macromolecules, but also in vivo NMR studies [26, 96 98]. Most new experiments, new instruments and new applications require more accurate descriptions of NMR to explain the experiments so as to extract the information more 11

34 accurately. For example, the power of the pulses may be restricted in biological NMR experiments, so off-resonance and relaxation effects during rf pulses will become significant, so we need to quantify these effects and try to achieve optimal results. This was our original motivation for doing this research: to use modern computational tools to simulate NMR and apply optimization methods to extract information from experiments more accurately and to design new experiments so as to obtain optimal NMR spectroscopy. Following this motivation, this thesis will discuss the following topics: 1. Chapter 2 introduces the basic theories and methods including the density matrix, the Liouville space method, Bloch equations, computing an exponential of a matrix, symbolic computation of NMR and optimization. 2. Chapter 3 introduces a computer algebra package Liouville Space Method Maple Package (LSMMP) for simulating NMR experiments of n-spins, validates simulation of LSMMP with published results and demonstrates challenges of optimizing NMR experiments; 3. Chapter 4 provides a general formalism for dealing with the steady state of any spin system subject to any pulse sequence, and presents the symbolic solution of single and coupled-spin systems; 4. Chapter 5 displays the full and exact solution of the Bloch equations and demonstrates the method for accurately measuring the relaxation times using Hahn echo experiments in the presence of expected B 0 inhomogeneity, and off-resonance effects when ω 1γB 2 1; 5. Chapter 6 presents an implicit exact solution of the Carr-Purcell-Meiboom-Gill 12

35 (CPMG) experiment. We also apply approximate solutions to explore oscillations and effective decay rates, and optimize the phase variations of CPMG. Applying the optimal phase variations of CPMG can remove oscillation and field-inhomogeneity effects, provide reliable R 2 s with offsets ± 1γB 2 1 using the second-order approximation of CPMG, and using a more complicated fitting model can provide reliable R 2 s within ±γb 1 ; 6. Chapter 7 demonstrates a model for designing universal rotation optimal pulses which can be applied as pseudo hard pulses when hard pulses are not available. We solve the model with efficient algorithms and solvers. We also display that the optimal refocusing pulses designed by our method are better than any other published refocusing pulses of which we are aware. 13

36 Chapter 2 Theory and Tools 2.1 Density Matrix In quantum mechanics, since two basis wavefunctions α> and β> form a basis set for all possible wavefunctions for the single spin- 1, any wavefunction can be expressed 2 as Ψ>= c α α> +c β β> (2.1) where c α and c β are complex coefficients. Its adjoint which is also a wavefunction is written as <Ψ = c α <α +c β <β (2.2) where c α and c β are the complex conjugate of c α and c β, respectively. Physical observables including energy, angular momentum and magnetization are represented by operators. These are linear operators acting on wavefunctions, which 14

37 are matrices in Hilbert space and are denoted with hat symbols. For any such operator, ˆP, the expectation value of the observable is given by < ˆP>=<Ψ ˆP Ψ> (2.3) For a single spin- 1 2 system, the x-, y- and z-components of the magnetization, which are three of the physical observables, are represented by the spin angular momentum operators Îx, Î y, and Îz respectively. The total number of spins is also observable, but never changes, so we ignore it usually. This plants the idea that there are four elements of the density matrix. The density operator which is the state of the single spin- 1 2 system can be represented by a linear combination of these three operators with different coefficients computed by Eq. (2.3) at any time. The amounts of the three operators will vary with time during pulses and delays. Since there are three vectors, it is isomorphic to a vector in 3 dimensional space. Thus, this method is also called the vector model. We are able to use the density matrix to explain it. If we expand Eq. (2.3), we have < ˆP > = (c α <α +c β <β )ˆP(c α α> +c β β>) = c αc α <α ˆP α> +c αc β <α ˆP β> +c βc α <β ˆP α> +c βc β <β ˆP β> = trace c αc α c β c α <α ˆP α> <α ˆP β> c α c β c β c β <β ˆP α> <β ˆP β> trace(ˆρ T ˆP) (2.4) 15

38 where ˆρ = c α c α c α c β c β c α c β c β (2.5) and ˆP = <α ˆP α> <α ˆP β> <β ˆP α> <β ˆP β> (2.6) ˆρ T is the transpose of ˆρ in the Hilbert space. ˆρ is the density matrix in the Hilbert space which can also be seen as the representation of the density operator, ˆρ = <α ˆρ α> <α ˆρ β> <β ˆρ α> <β ˆρ β> (2.7) and its elements encapsulate all the observables of the spin system, and the result of any particular measurement is computed by Eq. (2.4) [32, 33]. Since systems of higher spins or coupled spin- 1 2 systems have many more observables, we cannot apply the vector model to handle the complete dynamics [99]. However, the density matrix provides an exact treatment [32, 33]. When we do experiments of NMR, we often start the experiment at the equilibrium state, which means that the density operator is proportional to the static Hamiltonian Ĥ 0 attimezero. TheLiouville-vonNeumannequationwilldescribethedensitymatrix with different Hamiltonian operators in the Hilbert space during the process of the experiment [32, 33], d = i[ĥ, ˆρ(t)] (2.8) dtˆρ(t) where the square bracket indicates the commutator, i = 1. In quantum mechanics, any physical observable is represented by an operator, and the density operator is represented by the density matrix which gives all possible 16

39 information about a system. Operator formalisms are methods to act on the density matrix so as to give reasonable solutions to show the effect of a pulse and a delay in NMR [32, 37]. In this thesis, we use the Liouville space method for pulsed NMR of spin- 1 2 systems to compute solutions of the equation of motion Eq. (2.8), rather than working directly with the density matrix [32 37]. 2.2 Liouville Space Method In Liouville space, we detect observables by taking the dot (scalar) product with the density matrix. A scalar product between two Liouville space vectors is defined as the trace of their product as operators in spin space (see Eq. (2.4)). This product gives the projection of one vector on the other vector which is equivalent to the trace product in the operator formalism [33, 34, 100], i.e. (P Q) = trace(ˆp T ˆQ), (2.9) where on the left hand side, P and Q are Liouville Space vectors, whereas on the right hand side ˆP and ˆQ denote operators in Hilbert space. The Liouville-von Neumann equation in Liouville space is [32, 33], d ρ(t) = ilρ(t) (2.10) dt where L is the Liouvillian matrix encoding information about Larmor frequencies, chemical structure and couplings (if the spin system is multiple) in Liouville space, and ρ representing the density operator is a vector in Liouville space with respect to the given basis. In fact, the Liouville-von Neumann equation in Liouville space is a set of first-order ordinary differential equations (ODE). 17

40 In Liouville space method, each spin- 1 2 has four observables 1 0), 1 +1 ), 1 1 ) and 0) which are defined in Table 2.2, following the notation of Bain and Martin [34]. 0) = ) = (I x +ii y ) 1 0 ) = 2I z 1 1 ) = (I x ii y ) Table 2.1: Spherical tensor basis for Liouville Space of a single spin is a unit 2 operator which can be seen as a constant number to represent the total number of spins, I x,i y,i z stand for three Cartesian angular momentum operators, respectively. These observables form a basis for a single spin system, 0) 1 +1 ) 1 0 ) 1 1 ) (2.11) and combine to form bases for multiple spin- 1 2 systems by using direct products of the single spin system basis, for example, the basis of two coupled spins in the direct product form is given by Basis (2.12). 18

41 0) 1 +1 ) 1 0 ) 1 1 ) 0) 1 +1 ) 1 0 ) 1 1 ) = 0) 0) 0) 1 +1 ) 0) 1 0 ) 0) 1 1 ) 1 +1 ) 0) 1 +1 ) 1 +1 ) 1 +1 ) 1 0 ) 1 +1 ) 1 1 ) 1 0 ) 0) 1 0 ) 1 +1 ) 1 0 ) 1 0 ) 1 0 ) 1 1 ) 1 1 ) 0) 1 1 ) 1 +1 ) 1 1 ) 1 0 ) 1 1 ) 1 1 ) (2.12) The behaviour of a single spin is straightforward, but coupled spins offer significant challenges because the size of the matrix will be and the matrix will have more information including the structure and coupling. We can only detect the single quantum signal, for example, 1 +1 ) 0) and 0) 1 +1 ) in the basis of a coupled-spin system, representing the total xy magnetization precessing in the positive direction. When the relaxations are considered, the equation of motion for the density matrix 19

42 vectors of the inhomogeneous form is expressed as d dt ρ = ilρ R(ρ ρ eq), (2.13) where ρ eq is the equilibrium state, and R is the relaxation matrix. In this study, we use a simple random-field mechanism, but more sophisticated relaxation can be incorporated easily. Because ilρ eq = 0 for any spin systems [32, 33], we have d dt (ρ ρ eq) = (il+r)(ρ ρ eq ), (2.14) Define x ρ ρ eq, we get d x = (il+r)x, (2.15) dt We see that in Eq. (2.15), the coefficient of the zeroth order of the variable x is zero, the power of x and its derivatives are 1 and there are no products of x and its derivatives. The differential equations like Eq. (2.15) are called the homogeneous linear differential equations, whereas the differential equations like Eq. (2.13) are called the inhomogeneous linear differential equations because the coefficient of ρ 0 is Rρ eq 0. The homogeneous linear differential equations have a useful property that the set of the solutions is a vector linear space which satisfies the rules of addition and scalar multiplication [101]. When the coefficients of the homogeneous linear differential equations are constants, the analytical solution can be computed in terms of the exponential of the coefficient matrix with the given initial values [ ]. The solution of Eq. (2.15) is x(t) = e (il+r)t x(0) (2.16) 20