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In VivoNMR Spectroscopy

– 2nd EditionPrinciples and

Techniques

ROBIN A. DE GRAAF

Yale University, Connecticut, USA

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In Vivo NMR Spectroscopy – 2nd Edition

i


ii


In VivoNMR Spectroscopy

– 2nd EditionPrinciples and

Techniques

ROBIN A. DE GRAAF

Yale University, Connecticut, USA

iii


Copyright C© 2007 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,West Sussex PO19 8SQ, England

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Library of Congress Cataloging-in-Publication Data

De Graaf, Robin A.In vivo NMR spectroscopy : principles and techniques / Robin de Graaf. – 2nd ed.

p. ; cm.Includes bibliographical references and index.ISBN 978-0-470-02670-0 (cloth : alk. paper)

1. Nuclear magnetic resonance spectroscopy. 2. Magnetic resonance imaging. I. Title.[DNLM: 1. Magnetic Resonance Spectroscopy–diagnostic use. 2. Magnetic Resonance

Spectroscopy–methods. QU 25 D321i 2007]QP519.9.N83D4 2007616.07′548–dc22 2007018548

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 978-0470-026700

Typeset in 10/12pt Times by Aptara Inc., New Delhi, IndiaPrinted and bound in Great Britain by Antony Rowe Ltd, Chippenham, WiltshireThis book is printed on acid-free paper responsibly manufactured from sustainable forestryin which at least two trees are planted for each one used for paper production.

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Contents

Preface xiiiList of Abbreviations and Symbols xv

1 Basic Principles 11.1 Introduction 11.2 Classical Description 21.3 Quantum Mechanical Description 41.4 Macroscopic Magnetization 61.5 Excitation 81.6 Bloch Equations 111.7 Fourier Transform NMR 141.8 Chemical Shift 181.9 Digital Fourier Transform NMR 20

1.9.1 Multi-scan Principle 201.9.2 Time-domain Filtering 201.9.3 Analog-To-Digital Conversion 221.9.4 Zero Filling 25

1.10 Spin–spin Coupling 261.10.1 Spectral Characteristics 30

1.11 T1 Relaxation 331.12 T2 Relaxation and Spin-echoes 361.13 Exercises 38

References 41

2 In Vivo NMR Spectroscopy – Static Aspects 432.1 Introduction 432.2 Proton NMR Spectroscopy 43

2.2.1 Acetate (Ace) 452.2.2 N-Acetyl Aspartate (NAA) 45

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vi Contents

2.2.3 N-Acetyl Aspartyl Glutamate (NAAG) 522.2.4 Adenosine Triphosphate (ATP) 522.2.5 Alanine (Ala) 532.2.6 γ-Aminobutyric Acid (GABA) 542.2.7 Ascorbic Acid (Asc) 542.2.8 Aspartate (Asp) 552.2.9 Choline-containing Compounds (tCho) 552.2.10 Creatine (Cr) and Phosphocreatine (PCr) 572.2.11 Ethanolamine and Phosphorylethanolamine (PE) 582.2.12 Glucose (Glc) 592.2.13 Glutamate (Glu) 592.2.14 Glutamine (Gln) 612.2.15 Glutathione (GSH) 622.2.16 Glycerol 622.2.17 Glycine 632.2.18 Glycogen 632.2.19 Histamine 642.2.20 Histidine 652.2.21 Homocarnosine 652.2.22 β-Hydroxybutyrate (BHB) 662.2.23 Myo-Inositol (mI) and scyllo-Inositol (sI) 662.2.24 Lactate (Lac) 672.2.25 Macromolecules 682.2.26 Phenylalanine 702.2.27 Pyruvate 702.2.28 Serine 712.2.29 Succinate 712.2.30 Taurine (Tau) 722.2.31 Threonine (Thr) 722.2.32 Tryptophan (Trp) 732.2.33 Tyrosine (Tyr) 732.2.34 Valine (Val) 742.2.35 Water 742.2.36 Intra- and Extramyocellular Lipids (IMCL and EMCL) 752.2.37 Deoxymyoglobin (DMb) 762.2.38 Citric Acid 772.2.39 Carnosine 78

2.3 Phosphorus-31 NMR Spectroscopy 782.3.1 Identification of Resonances 792.3.2 Intracellular pH 80

2.4 Carbon-13 NMR Spectroscopy 822.4.1 Identification of Resonances 82

2.5 Sodium-23 and Potassium-39 NMR Spectroscopy 852.6 Fluorine-19 NMR Spectroscopy 89

2.6.1 Identification of Resonances 89


Contents vii

2.6.2 Fluorinated Drugs, Anaesthetics, and FluorodeoxyglucoseMetabolism 90

2.6.3 Fluorinated Probes 922.7 Exercises 93

References 95

3 In Vivo NMR Spectroscopy – Dynamic Aspects 1113.1 Introduction 1113.2 Relaxation 112

3.2.1 General Principles of Dipolar Relaxation 1123.2.2 Nuclear Overhauser Effect 1183.2.3 Alternative Relaxation Mechanisms 1193.2.4 In Vivo Relaxation 122

3.3 Magnetization Transfer 1283.3.1 Creatine Kinase 1303.3.2 Inversion Transfer 1313.3.3 Saturation Transfer 1313.3.4 ATpases 1343.3.5 Fast Magnetization Transfer Methods 1353.3.6 Off-resonance Magnetization Transfer 1363.3.7 Chemical Exchange Dependent Saturation Transfer 141

3.4 Diffusion 1413.4.1 Principles of Diffusion 1413.4.2 Diffusion and NMR 1433.4.3 Anisotropic Diffusion 1523.4.4 Restricted Diffusion 156

3.5 Dynamic Carbon-13 NMR Spectroscopy 1583.5.1 General Set-up 1603.5.2 Metabolic Modeling 1623.5.3 Substrates 1663.5.4 Applications 171

3.6 Hyperpolarization 1713.6.1 “Brute Force” Hyperpolarization 1723.6.2 Optical Pumping of Noble Gases 1733.6.3 Para-hydrogen-induced Polarization (PHIP) 1753.6.4 Dynamic Nuclear Polarization 177

3.7 Exercises 179References 181

4 Magnetic Resonance Imaging 1914.1 Introduction 1914.2 Magnetic Field Gradients 1924.3 Slice Selection 1934.4 Frequency Encoding 1954.5 Phase Encoding 201


viii Contents

4.6 Spatial Frequency Space 2024.7 Fast MRI Sequences 206

4.7.1 Echo-planar Imaging 2084.8 Contrast in MRI 212

4.8.1 T1 and T2 Relaxation Mapping 2134.8.2 Fast T1 and T2 Relaxation Mapping 2144.8.3 Magnetic Field B0 Mapping 2164.8.4 Magnetic Field B1 Mapping 2184.8.5 Alternative Image Contrast Mechanisms 2194.8.6 Functional Imaging 220

4.9 Parallel MRI 2224.10 Exercises 225

References 229

5 Radiofrequency Pulses 2335.1 Introduction 2335.2 Square RF Pulses 2335.3 Selective RF Pulses 239

5.3.1 Sinc Pulses 2395.3.2 Gaussian and Hermitian Pulses 2435.3.3 Multifrequency RF Pulses 246

5.4 Pulse Optimization 2475.4.1 Shinnar–Le Roux Algorithm 248

5.5 DANTE RF Pulses 2545.6 Composite RF Pulses 2555.7 Adiabatic RF Pulses 258

5.7.1 Rotating Frames of Reference 2595.7.2 Adiabatic Half and Full Passage Pulses 2625.7.3 Plane Rotations and Refocused Component 2685.7.4 Adiabatic Full Passage Refocusing 2695.7.5 Adiabatic Plane Rotation Pulses 2695.7.6 Variable Angle Adiabatic Plane Rotation Pulse, BIR-4 2715.7.7 Modulation Functions 2745.7.8 Offset-independent Adiabaticity 275

5.8 Pulse Imperfections and Relaxation 2765.9 Power Deposition 2805.10 Multidimensional RF Pulses 2835.11 Spectral–spatial RF Pulses 2895.12 Exercises 290

References 292

6 Single Volume Localization and Water Suppression 2976.1 Introduction 2976.2 Single Volume Localization 299

6.2.1 Image Selected In Vivo Spectroscopy (ISIS) 2996.2.2 Chemical Shift Displacement 302


Contents ix

6.2.3 Stimulated Echo Acquisition Mode (STEAM) 3066.2.4 Point Resolved Spectroscopy (PRESS) 3106.2.5 Signal Dephasing with Magnetic Field Gradients 3116.2.6 Effects of Imperfect RF Pulses 3176.2.7 Localization by Adiabatic Selective Refocusing

(LASER) 3206.2.8 Chemical Shift Displacement – Scalar-coupled Spins 322

6.3 Water Suppression 3256.3.1 Binomial and Related Pulse Sequences 3266.3.2 Frequency Selective Excitation 3336.3.3 Frequency Selective Refocusing 3376.3.4 Relaxation Based Methods 3386.3.5 Non-water-suppressed NMR Spectroscopy 340


7 Spectroscopic Imaging and Multivolume Localization 3497.1 Introduction 3497.2 Principles of Spectroscopic Imaging 3497.3 Spatial Resolution in MRSI 3547.4 Temporal Resolution in MRSI 357

7.4.1 Conventional Methods 3577.4.2 Methods Based on Fast MRI Sequences 3617.4.3 Methods Based on Prior Knowledge 365

7.5 Lipid Suppression 3677.5.1 Relaxation Based Methods 3687.5.2 Outer Volume Suppression and Volume

Pre-localization 3687.5.3 Methods Utilizing Spatial Prior Knowledge 371

7.6 Spectroscopic Imaging Processing and Display 3737.7 Multivolume Localization 377

7.7.1 Hadamard Localization 3787.7.2 Sequential Multivolume Localization 380


8 Spectral Editing and Two-dimensional NMR 3898.1 Introduction 3898.2 Scalar Evolution 3908.3 J-difference Editing 3928.4 Practical Considerations of J-difference Editing 3978.5 Multiple Quantum Coherence Editing 4028.6 Heteronuclear Spectral Editing 4078.7 Polarization Transfer – INEPT and DEPT 4098.8 Sensitivity 4148.9 Broadband Decoupling 416


x Contents

8.10 Two-dimensional NMR Spectroscopy 4218.10.1 Correlation Spectroscopy (COSY) 4228.10.2 Spin-echo or J-resolved NMR 4328.10.3 Two-dimensional Exchange Spectroscopy 434


9 Spectral Quantification 4459.1 Introduction 4459.2 Data Acquisition 4469.3 Data Pre-processing 449

9.3.1 Phasing and Frequency Alignment 4509.3.2 Lineshape Correction 4509.3.3 Removal of Residual Water 4519.3.4 Baseline Correction 453

9.4 Data Quantification 4539.4.1 Time- and Frequency-domain Parameters 4549.4.2 Prior Knowledge 4579.4.3 Spectral Fitting Algorithms 4609.4.4 Error Estimation 464

9.5 Data Calibration 4669.5.1 Internal Concentration Reference 4709.5.2 External Concentration Reference 4729.5.3 Phantom Replacement External Concentration

Reference 4739.6 Exercises 473

References 475

10 Hardware 47910.1 Introduction 47910.2 Magnets 48010.3 Magnetic Field Homogeneity 484

10.3.1 Origins and Effects of Magnetic Field InhomogeneityIn Vivo 484

10.3.2 Active Shimming 48910.3.3 Shimming Hardware 49210.3.4 Manual Shimming 49210.3.5 Magnetic Field Map Based Shimming 49410.3.6 Projection Based Shimming 49710.3.7 Dynamic Shim Updating (DSU) 49910.3.8 Passive Shimming 502

10.4 Magnetic Field Gradients 50210.4.1 Eddy Currents 50610.4.2 Pre-Emphasis 50810.4.3 Active Shielding 512


Contents xi

10.5 Radiofrequency Coils 51210.5.1 Resonant LCR Circuits 51310.5.2 RF Coil Performance 51910.5.3 Spatial Field Properties 52110.5.4 Principle of Reciprocity I 52610.5.5 Principle of Reciprocity II 529

10.6 Radiofrequency Coil Types 53010.6.1 Combined Transmit and Receive RF Coils 53110.6.2 Phased-array Coils 53210.6.3 1H-[13C] and 13C-[1H] RF Coils 53310.6.4 Cooled (Superconducting) RF Coils 536

10.7 Complete MR System 53610.7.1 RF Transmission 53610.7.2 Signal Reception 53710.7.3 Quadrature Detection 53910.7.4 Dynamic Range 54010.7.5 Gradient and Shim Systems 541


Appendix 549A1 Matrix Calculations 549A2 Trigonometric Equations 551A3 Fourier Transformation 551

A3.1 Introduction 552A3.2 Properties 553A3.3 Discrete Fourier Transformation 554

A4 Product Operator Formalism 554A4.1 Cartesian Product Operators 555A4.2 Spherical Tensor Product Operators 556References 559Further Reading 560

Index 563


xii


Preface

Since the first edition of this textbook, published in 1998, the field of in vivo NMR spec-troscopy has seen continued development of new techniques and applications, while at thesame time some of the older techniques have become obsolete. One of the driving forces towrite a second edition was to review some of these novel developments, such as hyperpolar-ized NMR, dynamic 13C NMR, automated shimming and parallel acquisitions. To maintainthe flow of the book, several of the older techniques that have limited merits in modern invivo NMR were removed. A second driving force was provided by the need for a textbookto be used in conjunction with a teaching course on in vivo NMR. In order to pursue thisobjective, most techniques are described from an educational point of view, without losingthe practical aspects appreciated by experimental NMR spectroscopists. Furthermore, eachchapter is concluded with a number of exercises designed to review, but often also to extend,the presented NMR principles and techniques.

Many of the ideas and changes that formed the basis for this second edition came fromnumerous discussions with colleagues. I would like to thank Douglas Rothman, TerryNixon, Graeme Mason, Kevin Behar, Peter Brown and Kevin Koch for many fruitful dis-cussions. Special thanks go to Christoph Juchem for his many insightful comments andcareful reading of all chapters.

Finally, I would like to acknowledge the contributions of original data from Dan Greenand Simon Pittard (Magnex Scientific), Andrew Maudsley (University of Miami), GeraldShulman (Yale University) and Graeme Mason (Yale University).

Robin A. de GraafNew Haven, USA

January, 2007

Companion website URL: www.spectroscopynow.com/degraaf

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xiv


Abbreviations and Symbols

AC Alternating currentAce AcetateADC Analog-to-digital converterADC Apparent diffusion coefficientADP Adenosine diphosphateAFP Adiabatic full passageAHP Adiabatic half passageAla AlanineAsc Ascorbic acidAsp AspartateATP Adenosine triphosphateBAPTA 1,2-Bis-(o-aminophenoxy)ethane-N ,N ,N ′,N ′-tetraacetic acidBHB β-HydroxybutyrateBIR B1-insensitive rotationBISEP B1-insensitive spectral editing pulseBOLD Blood oxygen level dependentBPP Bloembergen–Purcell–PoundBURP Band-selective pulses with uniform response and pure phaseCHESS Chemical shift selectiveCho Choline containing compoundsCK Creatine kinaseCOSY Correlation spectroscopyCPMG Carr–Purcell–Meiboom–GillCr CreatineCRLB Cramer–Rao lower boundCSF Cerebrospinal fluidCT Constant timeCW Continuous waveCYCLOPS Cyclically ordered phase sequence1D One-dimensional

xv


xvi Abbreviations and Symbols

2D Two-dimensional3D Three-dimensionalDANTE Delays alternating with nutation for tailored excitationdB DecibelDC Direct currentDEFT Driven equilibrium Fourier transformDEPT Distortionless enhancement by polarization transferDMb DeoxymyoblobinDNA Deoxyribonucleic acidDNP Dynamic nuclear polarizationDQC Double quantum coherenceDSS 2,2-Dimethyl-2-silapentane-5-sulfonateEA EthanolamineEMCL Extramyocellular lipidsEMF Electromotive forceEPI Echo planar imagingFDG 2-Fluoro-2-deoxy-glucoseFDG-6P 2-Fluoro-2-deoxy-glucose-6-phosphateFID Free induction decayFLASH Fast low-angle shotFFT Fast Fourier transformationFOCI Frequency offset corrected inversionFOV Field of viewFSW Fourier series windowsFT Fourier transformationFWHM Frequency width at half maximum5-FU 5-Fluorouracil5′dFUrd 5′-Deoxy-fluorouridineGABA γ -Aminobutyric acidGE Gradient echoGlc GlucoseGln GlutamineGlu GlutamateGlx Glutamine and glutamateGly GlycineGPC GlycerophosphorylcholineGPE GlycerophosphorylethanolamineGSH Glutathione (reduced form)HLSVD Hankel Lanczos singular value decompositionHMPT HexamethylphosphorustriamideHMQC Heteronuclear multiple quantum correlationHSQC Heteronuclear single quantum correlationINEPT Insensitive nuclei enhanced by polarization transferIMCL Intramyocellular lipidsIR Inversion recoveryISIS Image-selected in vivo spectroscopy


Abbreviations and Symbols xvii

IT Inversion transferJR Jump-returnLac LactateMb MyoglobinMEOP Metastability exchange optical pumpingmI Myo-inositolMLEV Malcolm LevittMQC Multiple quantum coherenceMRI Magnetic resonance imagingMRS Magnetic resonance spectroscopyMRSI Magnetic resonance spectroscopic imagingMT Magnetization transferMTC Magnetization transfer contrastNAA N -Acetyl aspartateNAAG N -Acetyl aspartyl glutamateNAD(H) Nicotinamide adenine dinucleotide oxidized (reduced)NADP(H) Nicotinamide adenine dinucleotide phosphate oxidized (reduced)NDP Nucleoside diphosphatenOe Nuclear Overhauser effectNOESY Nuclear Overhauser effect spectroscopyNMR Nuclear magnetic resonanceNTP Nucleoside triphosphateOVS Outer volume suppressionPCA Perchloric acidPCr PhosphocreatinePDE PhosphodiestersPE PhosphorylethanolaminePET Positron emission tomographyPHIP Para-hydrogen-induced polarizationPi Inorganic phosphatePME PhosphomonoestersPOCE Proton-observed carbon-editedPPM Parts per millionPRESS Point resolved spectroscopyPSF Point spread functionQUALITY Quantification by converting lineshapes to the Lorentzian typeRAHP Time-reversed adiabatic half passageRMS Root mean squaredRNA Ribonucleic acidROI Region of interestRF RadiofrequencySAR Specific absorption rateSE Spin echoSEOP Spin-exchange optical pumpingSI Spectroscopic imagingsI Scyllo-inositol


xviii Abbreviations and Symbols

SLR Shinnar-Le RouxS/N Signal-to-noise ratioSNR Signal-to-noise ratioSQC Single quantum coherenceSSAP Solvent suppression adiabatic pulseST Saturation transferSTE Stimulated echoSTEAM Stimulated echo acquisition modeSV Single voxel (or volume)SVD Singular value decompositionTau TaurineTCA Tricarboxylic acidtCho Total cholinetCr Total creatineThr ThreonineTMA TrimethylammoniumTMS TetramethylsilaneTOCSY Total correlation spectroscopyTPPI Time proportional phase incrementationTrp TryptophanTSP 3-(Trimethylsilyl)-propionateTyr TyrosineUV UltravioletVal ValineVAPOR Variable pulse powers and optimized relaxation delaysVARPRO Variable projectionVERSE Variable rate selective excitationVOI Volume of interestVNA Variable nutation angleVSE Volume selective excitationWALTZ Wideband alternating phase low-power technique

for zero residue splittingWEFT Water eliminated Fourier transformZQC Zero quantum coherencea Acceleration (in m s−2)A Absorption frequency domain signalAn, Bn Fourier coefficientsb b-value (in s m−2)b b-value matrixB0 External magnetic field (in T)B1 Magnetic radiofrequency field of the transmitter (in T)B1ax Axial amplitude of the irradiating B1 field (in T)B1max Maximum amplitude of the irradiating B1 field (in T)B1rad Radial amplitude of the irradiating B1 field (in T)B1rms Root mean square B1 amplitude of a radiofrequency pulse (in T)B1x, B1y Real and imaginary components of the irradiating B1 field (in T)


Abbreviations and Symbols xix

B2 Magnetic, radiofrequency field of the decoupler (in T)Be Effective magnetic field in the laboratory and frequency frames (in T)B′

e Effective magnetic field in the second rotating frame (in T)Bloc Local magnetic field (in T)C Capacitance (in F)C Correction factor for calculating absolute concentrationsD Dispersion frequency domain signalD (Apparent) diffusion coefficient (in m2 s−1)D (Apparent) diffusion tensorE Energy (in J)Ea Activation energy (in J)f Ratio of immobile spins to mobile spinsf Filling factorfab Fraction of magnetization Ma (a = x or y) converted to

magnetization Mb (b = x or y) by a RF pulsefB(t) Normalized radiofrequency amplitude modulation functionfω(t) Normalized radiofrequency frequency modulation functionF Nyquist frequency (in s−1)F Noise figure (in dB)F Force (in kg m s−2)G(t) Correlation functionG Magnetic field gradient strength (in T m−1)h Planck’s constant (6.626208 × 10−34 J s)H Hadamard matrixI Imaginary time- or frequency domain signalI Refocused componentI Spin quantum numberI0 Boltzmann equilibrium magnetization for spin IInm Shim current for shim coil of order nmJ Spin-spin or scalar coupling constant (in Hz)J0 Zero-order Bessel functionJ(ω) Spectral density functionk Boltzmann equilibrium constant (1.38066 × 10−23 J K−1)k k-space variable (in m−1)kf k-space variable in frequency-encoding direction (in m−1)kp k-space variable in phase-encoding direction (in m−1)kAB, kBA Unidirectional rate constants (in s−1)kfor Forward, unidirectional rate constant (in s−1)krev Reversed, unidirectional rate constant (in s−1)L Inductance (in H)L Angular momentum (in kg m2 s−1)m Mass (in kg)m Magnetic quantum numberM Magnitude-mode frequency domain signalM Mutual inductance (in H)M Macroscopic magnetization


xx Abbreviations and Symbols

M0 Macroscopic equilibrium magnetizationMx, My, Mz Orthogonal components of the macroscopic magnetizationn Total number of nuclei in a macroscopic samplenα, nβ Populations of the α and β spin statesN NoiseN Number of phase-encoding incrementsp Linear momentum (in kg m s−1)p Order of coherencePz Component of angular momentum in z-directionQ Quality factorr Distance (in m)R Composite pulse (sequence)R Product of bandwidth and pulselengthR Real time- or frequency-domain signalR Resistance (in �)R Rotation matrixR1A, R1B Longitudinal relaxation rate constants for spins A and B in the

absence of chemical exchange or cross relaxation (in s−1)R2 Transverse relaxation rate (in s−1)RA, RB Longitudinal relaxation rate constants for spins A and B

in the presence of chemical exchange (in s−1)RH Hydrodynamic radius (in m)S Measured NMR signalS(k) Spatial frequency sampling functiont Time (in s)t1 Incremented time in 2D NMR experiments (in s)t1max Maximum t1 period in constant time 2D NMR experiments (in s)t2 Detection period in 2D NMR experiments (in s)tdiff Diffusion time (in s)tnull Time of zero-crossing (nulling) during an inversion recovery

experiment (in s)T Absolute temperature (in K)T Torque (in kg m2 s−2)T Pulse length (in s)T1 Longitudinal relaxation time constant (in s)

T*1 Apparent longitudinal relaxation time constant (in s)

T1,obs Observed, longitudinal relaxation time constant (in s)T2 Transverse relaxation time constant (in s)

T*2 Apparent transverse relaxation time constant (in s)

T2,obs Observed, transverse relaxation time constant (in s)Tacq Acquisition time (in s)TE Echo time (in s)TECPMG Echo time in a CPMG experiment (in s)TI Inversion time (in s)TI1 First inversion time (in s)TI2 Second inversion time (in s)


Abbreviations and Symbols xxi

TM Delay time between the second and third 90◦ pulses in STEAM (in s)TR Repetition time (in s)v Velocity (in m s−1)W Transition probability (in s−1)Wnm Angular function of spherical polar coordinatesW(k) Spatial frequency weighting functionx Molar fractionXC Capacitive reactance (in �)XL Inductive reactance (in �)Z Impedance (in �)α Nutation angle (in rad)β Precession angle of magnetization perpendicular to the effective

magnetic field Be (in rad)γ Gyromagnetic ratio (in rad T−1 s−1)δ Chemical shift (in ppm)δ Gradient duration (in s)� Separation between a pair of gradients (in s)�B0 Magnetic field shift (in T)�ν1/2 Full width at half maximum of an absorption line (in Hz)�ω Frequency offset (in Hz)�ωmax Maximum frequency modulation of an adiabatic radiofrequency

pulse (in Hz)�� Frequency offset (in Hz)ε Gradient risetime for a trapezoidal magnetic field gradient (in s)η Nuclear Overhauser enhancementη Viscosity (in N s m−2)θ Nutation angle (in rad)μ Magnetic moment (in A m2)μ0 Permeability constant in vacuum (4π 10−7 kg m s−2 A−2)μe Electronic magnetic moment (in A m2)μz Component of magnetic moment in z directionν0 Larmor frequency (in Hz)νA Frequency of a unprotonated compound A (in Hz)νHA Frequency of a protonated compound HA (in Hz)νref Reference frequency (in Hz)ξ Electromotive force (in V)σ Magnetic shielding constant (in ppm)σ Density matrixτc Rotation correlation time (in s)τm Mixing time in 2D NMR experiments (in s)φ Phase (in rad)φc Phase correction (in rad)φ0 Zero-order (constant) phase (in rad)φ1 First-order (linear) phase (in rad)χ Magnetic susceptibilityω0 Larmor frequency (in rad s−1)[ ] Concentration (in M)


xxii

c01 JWBK188/Degraff September 18, 2007 18:24 Char Count=

1Basic Principles

1.1 Introduction

The field of spectroscopy is in general concerned with the interaction between matter andelectromagnetic radiation. Atoms and molecules have a range of discrete energy levels cor-responding to different electronic, vibrational or rotational states. The interaction betweenatoms and electromagnetic radiation is characterized by the absorption and emission ofphotons, such that the energy of the photons exactly matches an energy level differencein the atom. Since the energy of a photon is proportional to the frequency, the differentforms of spectroscopy are often distinguished on the basis of the frequencies involved. Forinstance, absorption and emission between electronic states of the outer electrons typicallyrequire frequencies in the ultraviolet (UV) range, hence giving rise to UV spectroscopy.Molecular vibrational modes are characterized by frequencies just below visible red lightand are thus studied with infrared (IR) spectroscopy. Nuclear magnetic resonance (NMR)spectroscopy uses radiofrequencies, which are typically in the range of 10–800 MHz.

NMR is the study of the magnetic properties (and energies) of nuclei. The absorptionand emission of electromagnetic radiation can be observed when the nuclei are placed ina (strong) external magnetic field. Purcell, Torrey and Pound [1] at MIT, Cambridge andBloch, Hansen and Packard [2] at Stanford simultaneously, but independently discoveredNMR in 1946. In 1952 Bloch and Purcell shared the Nobel Prize for physics in recognitionof their pioneering achievements [1–4]. At this stage, NMR was purely an experimentfor physicists to determine the nuclear magnetic moments of nuclei. NMR could onlydevelop into one of the most versatile forms of spectroscopy after the discovery that nucleiwithin the same molecule absorb energy at different resonance frequencies. These so-calledchemical shift effects, which are directly related to the chemical environment of the nuclei,were first observed in 1950 by Proctor and Yu [5], and independently by Dickinson [6].

In the first two decades, NMR spectra were recorded in a continuous wave mode in whichthe magnetic field strength or the radiofrequency (RF) was swept through the spectral area

In Vivo NMR Spectroscopy – 2nd Edition: Principles and Techniques Robin A. de GraafC© 2007 by John Wiley & Sons, Ltd

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2 In Vivo NMR Spectroscopy

of interest, whilst keeping the other fixed. In 1966, NMR was revolutionized by Ernst andAnderson [7] who introduced pulsed NMR in combination with Fourier transformation.Pulsed or Fourier transform NMR is at the heart of all modern NMR experiments.

The induced energy level difference of nuclei in an external magnetic field is very smallwhen compared with the thermal energy, making it that the energy levels are almost equallypopulated. As a result the absorption of photons is very low, making NMR a very insensitivetechnique when compared with the other forms of spectroscopy. However, the low energyabsorption makes NMR also a noninvasive and nondestructive technique, ideally suited forin vivo measurements. It is believed that, by observing the water signal from his own finger,Bloch was the first to use NMR on a living system. Soon after the discovery of NMR, othersshowed the utility of using NMR to study living objects. In 1950, Shaw and Elsken [8] usedproton NMR to investigate the water content of vegetable material. Odebald and Lindstrom[9] obtained proton NMR signals from a number of mammalian preparations in 1955.Continued interest in defining and explaining the properties of water in biological tissuesled to the promising report of Damadian in 1971 [10] that NMR properties (relaxationtimes) of malignant tumorous tissue significantly differs from normal tissue, suggestingthat (proton) NMR may have diagnostic value. In the early 1970s, the first experiments ofNMR spectroscopy on intact living tissues were reported. Moon and Richards [11] used31P NMR on intact red blood cells and showed how the intracellular pH can be determinedfrom chemical shift differences. In 1974, Hoult et al. [12] reported the first study of 31PNMR to study intact, excised rat hind leg.

Around the same time reports on in vivo NMR spectroscopy appeared, Lauterbur [13]and Mansfield and Grannell [14] described the first reports of a major application of modernNMR, namely in vivo NMR imaging or magnetic resonance imaging (MRI). By applyingposition dependent magnetic fields in addition to the static magnetic field, they were ableto reconstruct the spatial distribution of the spins in the form of an image. Lauterbur andMansfield shared the 2003 Nobel Prize in medicine. In vivo NMR spectroscopy or magneticresonance spectroscopy (MRS) and MRI have evolved from relatively simple one or twoRF pulse sequences to complex techniques involving spatial localization, water and lipidsuppression and spectral editing for MRS and time-varying magnetic field gradients, ultrafast and multiparametric acquisition schemes for MRI.

In this chapter the basic phenomenon of NMR is considered. After establishing theLarmor resonance condition with a combination of classical and quantum mechanicalarguments, the NMR phenomenon is approached from a more practical point of viewwith the aid of the macroscopic Bloch equations. The phenomena of chemical shift, scalarcoupling and spin echoes will be described, as well as some elementary processing of theNMR signal.

1.2 Classical Description

NMR is based on the concept of nuclear spin. Before discussing the properties of nuclearspins, some relations from classical physics will be introduced which will simplify furtherdiscussions. Although classical physics is incapable of describing the quantum mechanicalspin, it can be used to create a familiar frame of reference in which the existence of a spinangular momentum can be visualized.


Basic Principles 3

Motion (linear or rotational) always has a corresponding momentum (linear or angular).For an object of mass m and velocity v, the linear momentum p is given by:

p = mv (1.1)

Conceptually, momentum can be thought of as the tendency for an object to continue itsmotion. The momentum only changes when an external force F is applied, in accordancewith Newton’s second law:

F =(

dpdt

)= ma (1.2)

where a is the acceleration. In the absence of external forces, the object does not accelerate(or decelerate) and the linear momentum and hence the speed is constant.

Now consider an object rotating with constant velocity about a fixed point at a distancer. This motion is described with an angular momentum vector L, defined as:

L = r × p (1.3)

Therefore, the magnitude of L is mvr and its direction is perpendicular to the plane ofmotion. Angular momentum can only be changed when an external torque is applied, inanalogy with the application of force on a linear momentum. Torque T (or rotational force)is defined as the cross product of force and the distance over which the force has to bedelivered:

T = r × F = r ×(

dpdt

)=

(dLdt

)(1.4)

Now suppose that the rotating object carries an electrical charge so that a current loopis created. According to classical physics this current generates a magnetic field, whichis characterized by the magnetic dipole moment, µ, a fundamental magnetic quantityassociated with the current. In general the magnetic moment µ is given by:

µ = [current][area] (1.5)

For an object of mass m and charge e rotating at constant rotational velocity v about a fixedpoint at distance r, the magnetic moment µ is given by:

µ =[ ev

2�r

]�r2 (1.6)

Using L = mvr, a fundamental relation between magnetic moment and angular momentis obtained:

µ =( e

2m

)L = �L (1.7)

where � is the (classical) gyromagnetic ratio. It turns out that relation (1.7) is valid forany periodic, orbital motion, including microscopic motion of elementary particles. In thenext section it is shown that relation (1.7) is also obtained when using quantum mechanicalarguments. When the rotating object is placed in an external magnetic field B0, the loopwill feel a torque given by:

T = µ × B0 (1.8)



Combining Equations (1.4), (1.7) and (1.8) gives:(dµ

dt

)= �µ × B0 (1.9)

Since the amplitude of µ is constant, the differential equation in Equation (1.9) expressesthe fact that µ changes its orientation relative to B0, i.e. µ rotates (precesses) about B0.Alternatively, a precession of µ about B0 can be described by:(

dµ

dt

)= µ × ω0 (1.10)

Combining Equations (1.9) and (1.10) results in the famous Larmor equation:

ω0 = γB0

or

ν0 =(ω0

2�

)=

( �

2�

)B0 (1.11)

The precession (or Larmor) frequency �0 is thus directly proportional to the appliedmagnetic field B0 and also to the gyromagnetic ratio � (or µ), which is characteristic forthe nucleus under investigation.

A magnetic moment in an external magnetic field also has an associated magnetic energydefined as:

E = −µ · B0 = −µB0cos � (1.12)

where θ is the angle between the magnetic moment µ and the external magnetic field B0.Equation (1.12) indicates that the magnetic energy is minimized when µ is parallel withB0 (θ = 0◦) and maximized when µ is antiparallel with B0 (θ = 180◦). According toEquation (1.12), the classical magnetic moment may assume any orientation (0◦ ≤ θ ≤180◦), with energy varying between +µB0 and −µB0. Therefore, even though classicalmechanics can create a familiar picture of the relation between angular momentum, mag-netic moment and Larmor frequency, it cannot explain how the general resonance conditionfor spectroscopy, �E = h�, relates to the magnetic energy associated with the magneticmoment. A quantum mechanical treatment is necessary to obtain information about theinteraction of electromagnetic waves and nuclear spins. In the next section basic quantummechanical concepts are introduced, after which the NMR resonance condition is derived.

1.3 Quantum Mechanical Description

One of the fundamental postulates in quantum mechanics is that the angular momentumof elementary particles (be it protons, neutrons, or electrons) is limited to discrete values,i.e. the angular momentum L is quantized and its amplitude is given by:

L =(

h

2�

) √I (I + 1) (1.13)

where I is the spin quantum number, which can only be integral or half-integral and his Planck’s constant. Since angular momentum is a vector, the full description of L must


Basic Principles 5

involve its amplitude, given by Equation (1.13), and its direction. In quantum mechanicsthe direction of angular momentum is specified by a second quantum number m, which canonly have certain discrete orientations with respect to a given direction. The component ofangular momentum in the z direction, Lz, is given by:

Lz =(

h

2�

)m (1.14)

Quantum mechanics shows that m can have 2I+1 values, given by:

m = I, I − 1, I − 2, . . . ,−I (1.15)

For protons, neutrons and electrons, the spin quantum number I equals 1/2. For nuclei,I cannot simply be calculated by summation of its individual components. However, byusing the atomic mass and the charge number, I can be deduced from the following rules:

1. For nuclei with an odd mass number, I is half-integral (1/2, 3/2, 5/2, . . . , e.g. 1H, 13C,15N, 23Na, 31P).

2. For nuclei with an even mass number and an even charge number, I is zero (e.g. 12C,16O, 32S).

3. For nuclei with an even mass and an odd charge number I is an integral number (1, 2,. . . , e.g. 2H, 14N).

By analogy with Equation (1.7), elementary particles also have a magnetic moment µ

which is related to the angular momentum L through:

µ = �L (1.16)

where � is again the gyromagnetic ratio. Since the angular momentum is quantized, themagnetic moment will also be quantized. The component of the magnetic moment alongthe longitudinal z axis is given [by analogy with Equation (1.14)] by:

µz = �

(h

2�

)m (1.17)

where m is given by Equation (1.15). In an external magnetic field B0, the particle acquiresa magnetic energy given by Equation (1.12). Combining this classical description of themagnetic energy with the quantum mechanical formulation of magnetic moment gives:

E = −µzB0 = −�

(h

2�

)mB0 (1.18)

Since m is a discrete quantum number [see Equation (1.15)], the energy levels are alsoquantized. For a particle of spin I = 1/2, there are only two energy levels (m = –1/2 and+1/2) and the energy difference �E is given by (see Figure 1.1):

�E = �

(h

2�

)B0 (1.19)

The resonance phenomenon in NMR is achieved by applying an oscillating magnetic fieldperpendicular to �z with a frequency �0, such that the energy equals the magnetic energy



E

0

E = h B0

B0

A B

Figure 1.1 (A) The nuclear spin energy for a spin-1/2 nucleus as a function of the externalmagnetic field strength B0. (B) The lower energy level (α spin state) corresponds to magneticmoments parallel with B0, while spins in the higher energy level (β spin state) have anantiparallel alignment with B0. For all currently available magnets, the energy level differencebetween the two spin states corresponds to electromagnetic radiation in the RF range.

given by Equation (1.19), i.e. the energy of the electromagnetic wave is given by:

�E = h�0 (1.20)

Combining Equations (1.19) and (1.20) will give the earlier derived Larmor equation:

�0 =( �

2�

)B0 (1.21)

Even though the classical and quantum mechanical descriptions of NMR lead to thesame result, they play a different role in the understanding of the technique. Quantummechanics is the only theory which can quantitatively describe the NMR phenomenon.Classical principles are mainly used to visualize the effects of RF pulses on macroscopicmagnetization vectors.

1.4 Macroscopic Magnetization

Figure 1.2A shows the precession (at the Larmor frequency) of a magnetic moment aroundan external magnetic field according to classical principles. Quantization of magneticmoment (and magnetic energy) can readily be incorporated in this picture. For elementaryparticles the angle θ between µ and B0 can no longer be arbitrary as in Section 1.2 but isgiven by:

cos θ = m√I(I + 1)

(1.22)

For a nucleus of spin I = 1/2, m = +1/2 or −1/2 yielding an angle θ = 54.74◦ relativeto the +z or −z axis, respectively. Therefore, the nuclei of spin I = 1/2 are distributed onthe surface of two cones, and rotate about B0 at the Larmor frequency (Figure 1.2B). Inthe general case of a spin I nucleus, the magnetic moments will be distributed on 2I+1cones at discrete angles θ as defined by Equation (1.22). For a spin 1/2 nucleus the twospin states m = +1/2 (µ parallel with B0) and m = –1/2 (µ antiparallel to B0) are oftenreferred to as the � and � spin states, respectively.

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