The Physics and Mathematics of MRI: PRELIMS: The … Physics and Mathematics of MRI Richard Ansorge University of Cambridge, UK Martin Graves Cambridge University Hospitals NHS Foundation

The Physics and Mathematicsof MRI

The Physics and Mathematicsof MRI

Richard AnsorgeUniversity of Cambridge, UK

Martin GravesCambridge University Hospitals NHS Foundation Trust, UK

Morgan & Claypool Publishers

Copyright ª 2016 Morgan & Claypool Publishers

All rights reserved. No part of this publication may be reproduced, stored in a retrieval systemor transmitted in any form or by any means, electronic, mechanical, photocopying, recordingor otherwise, without the prior permission of the publisher, or as expressly permitted by law orunder terms agreed with the appropriate rights organization. Multiple copying is permitted inaccordance with the terms of licences issued by the Copyright Licensing Agency, the CopyrightClearance Centre and other reproduction rights organisations.

Rights & PermissionsTo obtain permission to re-use copyrighted material from Morgan & Claypool Publishers, pleasecontact [email protected].

ISBN 978-1-6817-4068-3 (ebook)ISBN 978-1-6817-4004-1 (print)ISBN 978-1-6817-4196-3 (mobi)

DOI 10.1088/978-1-6817-4068-3

Version: 20161001

IOP Concise PhysicsISSN 2053-2571 (online)ISSN 2054-7307 (print)

A Morgan & Claypool publication as part of IOP Concise PhysicsPublished by Morgan & Claypool Publishers, 40 Oak Drive, San Rafael, CA, 94903 USA

IOP Publishing, Temple Circus, Temple Way, Bristol BS1 6HG, UK

Richard Ansorge would like to dedicate this book to Catherine and Lydia andMartin Graves would like to dedicate this book to Philippa, Sophie, Katie and Chloe.

Contents

Preface xi

Acknowledgements xii

Introduction xiii

Symbols and Acronyms xvi

Author biographies xix

1 The basics 1-1

1.1 A brief history of MRI 1-1

1.1.1 Spin and magnetic moments 1-1

1.1.2 NMR 1-3

1.1.3 MRI 1-3

1.1.4 Superconductivity 1-4

1.2 Proton spin 1-4

1.2.1 Precession 1-5

1.3 The Bloch equations 1-7

1.4 Signal generation 1-10

1.4.1 Reversing *T2 effects—spin–echo 1-13

1.4.2 T1 Sensitivity—inversion recovery 1-16

1.4.3 Image contrast 1-17

1.5 Spatial encoding using magnetic field gradients 1-18

1.5.1 Lauterbur’s tomographic method 1-19

1.5.2 Gradients and k-space 1-20

1.6 Spatial image formation 1-22

1.6.1 Pulse sequences 1-22

1.6.2 Slice select 1-23

1.6.3 Phase encode 1-23

1.6.4 Frequency encode 1-23

1.6.5 Rewind and repeat 1-23

1.7 Conclusion 1-25

References 1-25

2 Magnetic field generation 2-1

2.1 Designing the main magnet 2-1

2.1.1 Magnetic field of circular coil 2-1

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2.1.2 Combining coils 2-3

2.1.3 Off-axis fields 2-6

2.2 Designing gradient coils 2-10

2.2.1 Axial gradients 2-11

2.2.2 Transverse gradients 2-11

2.3 Practical issues 2-16

2.3.1 Main magnet 2-16

2.3.2 Keeping cool 2-18

2.3.3 Gradients 2-19

2.3.4 Pre-emphasis 2-20

2.3.5 Shielding and shimming 2-21

2.3.6 Safety 2-22

References 2-24

3 Radio frequency transmission and reception 3-1

3.1 Basic RF pulses 3-1

3.2 The birdcage coil 3-2

3.3 The transmit–receive chain 3-4

3.4 Surface coils 3-7

3.4.1 Phased arrays 3-8

3.4.2 Signal combination 3-8

3.5 Parallel imaging 3-10

3.5.1 Sensitivity encoding (SENSE) 3-11

3.5.2 Parallel k-space methods—SMASH and Grappa 3-13

3.6 Compressed sensing 3-16

3.7 RF pulses 3-19

3.7.1 Pulse shapes 3-19

3.7.2 Specific absorbed radiation (SAR) 3-20

3.8 Multinuclear MRI 3-23

References 3-23

4 Pulse sequences and images 4-1

4.1 Image contrast 4-1

4.2 Pulse sequence overview 4-3

4.2.1 Saturation spoiling and crushing 4-4

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4.3 Sequences in detail 4-4

4.3.1 Spin echo (SE) 4-4

4.3.2 Fast/turbo spin echo (FSE/TSE) 4-5

4.3.3 Echo planar imaging (EPI) 4-6

4.3.4 Gradient echo (GE) 4-7

4.3.5 Steady state imaging 4-9

4.3.6 Inversion recovery (IR) 4-10

4.4 Readout trajectories 4-13

4.4.1 Cartesian trajectories 4-13

4.4.2 Spiral trajectories 4-14

4.4.3 Radial acquisitions 4-17

4.4.4 Hybrid trajectories 4-18

4.5 Magnetic resonance spectrocopy (MRS) 4-18

4.6 k-space sampling in MRI 4-20

4.7 Image reconstruction 4-21

4.8 Conclusion 4-22

References 4-22

5 Applications 5-1

5.1 Introduction 5-1

5.2 Anatomical imaging 5-1

5.3 Chemical shift 5-2

5.4 Blood flow 5-4

5.5 Diffusion-weighted imaging 5-5

5.6 Diffusion tensor imaging 5-9

5.7 Chemical exchange 5-11

5.8 Functional MRI (fMRI) 5-12

5.9 Cerebral perfusion 5-13

5.10 Dynamic contrast enhanced (DCE)-MRI 5-15

5.11 Multinuclear MRI 5-17

5.12 Chemical shift artefact 5-18

References 5-19

6 Conclusion 6-1

References 6-3

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Appendices

A Essential quantum mechanics A-1

B Solutions of Laplace’s equation in spherical polar coordinates B-1

C The Birdcage coil C-1

D Fourier transforms D-1

E Multiple echoes E-1

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Preface

Clinical medical imaging has come a long way since the discovery of X-rays byWilhelm Roentgen in 1895. Conventional planar X-ray imaging is still very muchused, albeit often using more sensitive solid state digital detectors rather than film.X-rays are also used in CT scanners to provide exquisite 3D views of anatomy,particularly bone structure. Positron emission tomography (PET) is a much morerecent tool, capable of 3D imaging of metabolic process, it has applications inoncology and basic research. Ultrasound is also a relatively recent tool, it isrelatively inexpensive, quick and easy to use, it has many applications, includingobstetrics—where it is safe to use because it does not need ionising radiation.Finally, there is magnetic resonance imaging (MRI). Introduced in the mid 1980s atroughly a similar time to PET and CT, it uses dramatically different physicalprinciples to the other modalities and has also turned out to be far more versatile inits range of applications than any other modality.

MRI uses magnetic fields not ionising radiation and hence is just as safe to use asultrasound, however the necessary equipment is certainly more complicated andexpensive. The physical principles involved are also very different, they are subtle,complex and a wonderful example of physics in action.

There are already many books on the subject of MRI—so why another one? Wewould argue that most of the existing books are either aimed at a clinical readershipand emphasise how to use MRI and interpret the resulting images or they are ratherabstract, and while containing a great deal of technical detail they are not goodintroductions to the subject. We see our book as occupying a niche between thesetwo extremes and see it as a useful and rewarding introduction to MRI for readerswith a background in engineering or physical sciences. For example, we hope it willbe useful for students doing a medical physics option as part of their undergraduatecourse.

We make no apology for there being a lot of equations in this book. Themathematics is intended to be accessible to people with a background equivalent toan undergraduate course in physical sciences or engineering. Also much of themathematics is presented in a rather pedagogical way—we hope readers will try tofollow the steps and hence appreciate the elegant relationships between the under-lying mathematics and the final clinical images. We have not included anytraditional end of chapter examples—these can be found in existing textbooks forexample (Brown, Cheng et al 2014). We have included many references to journalarticles including both current research and more historical material. We believe youwill be ready to tackle this material with the help of our short book.

Finally, we think it is worth simply celebrating this remarkable contribution ofphysics to clinical practice. Who would have thought that the tiny proton magneticmoment, discovered by Otto Stern in 1933, would have such an important role toplay in modern medicine!

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Acknowledgements

Firsly, we would like to thank the publishers for giving us this opportunity to write abook which we feel will help to bridge a gap in the somewhat crowded field of bookson MRI. We also acknowledge the support of the journal editors, CERN and otherswho have allowed us to reuse their figures in this text.

REA would also like to thank his many colleagues in Cambridge and especially atthe Wolfson Brain Imaging Centre for introducing him to the world of MRI. MJGwould like to thank his colleagues at Cambridge University Hospitals and theUniversity of Cambridge Department of Radiology for providing additional figures.

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Introduction

The MRI experience

Today the experience of an MRI scan using a typical clinical scanner is routine butexceptionally interesting for a physicist. Firstly, as you enter the MR suite, you willbe asked questions to ensure you have no metal implants or other contraindications.If you pass this test you will be asked to remove any metal objects such as yourwallet, watch, any rings and glasses. You will then be allowed to approach thescanner itself. First you pass through the control room from where the scanner isoperated. The control room would not look out of place in a physics lab, having anumber of computer screens displaying multiple windows for instrument control andimage viewing. There will be a window for viewing the scanner itself which is in aseparate adjacent room. The scanner room is actually a Faraday cage, if you lookcarefully you should spot metal sealing strips on the edge of the (unexpectedly thick)door to the scanner room which will be kept closed during your scan. You mightalso notice that the window between the scanner and control room contains a finemetal mesh.

Once inside the scanner room you meet the generally doughnut shaped scanneritself (figure 0.1), superficially it looks similar to other body scanners such as CT orPET but actually the cylindrical bore, within which you are about to be placed, istighter (60 or 70 cm diameter) and longer (between 1.5 and 2.5 m). Most scanners infact contain a very powerful superconducting magnet and precautions are necessaryto prevent accidents caused by ferromagnetic objects flying into the scanner.

The next step is for you to be helped onto the bed associated with the scanner andonce comfortable you will have a ‘coil’ placed over the area to be imaged. This couldeither be a cage placed snugly over your head or a more flexible blanket-likestructure placed around your body. You will also be equipped with ear protectors orheadphones and a panic button. The panic button and headphones are likely to bepneumatic—great care is taken to avoid conducting wires inside the scanner bore.

You are now ready to enter the scanner, the operator, usually a radiographer ortechnician, presses a button and the bed automatically rises and slides gently into thescanner. This is the moment when you find out if you suffer from claustrophobia andunderstand why you have a panic button. Some people might also notice a metallictaste due to eddy currents induced in metal dental work as their head moves into thehigh magnetic field inside the scanner. If you have a brain scan then you will end upwith your head in the geometrical centre of the system and notice that the coilaround you head has a cunningly arranged mirror allowing you to see out throughthe bore.

As you wait for the scan to begin, you notice the operator returns to the controlroom leaving you alone. Actually although leaving would be mandatory for a CTscan this not really necessary for MRI; people can stay if clinically necessary. Theoperator will tell you the scan is starting and after a short time the most appallingbanging noise will suddenly start—you remember you were told not to move duringthe scan and try not to react. The banging continues intermittently to the end of the

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scanning process with interesting variations in rhythm and intensity of the noise. Theprocess continues on and off for quite a time, typically 10–40 min. Finally, at the endthe operator returns and initiates the movement of the bed back out of the scanner.You can now retrieve your possessions and leave the scanning suite.

As you leave you can reflect on the fact that while your scan may have beenuncomfortable, you have not been exposed to any ionising radiation and the softtissue contrast on the 3D images acquired is far superior to that that could beattained using CT.

In fact, the scanning process is a wonderfully complicated physics experimenthaving at its heart the manipulation of small bar magnets. And although these barmagnets are in fact protons—for this purpose they are not so different from theirlarger cousins that you almost certainly played with as a child and which may indeedhave inspired you to choose to study science. A schematic cut section through asuperconducting MRI scanner is shown in figure 0.2

The aim of this book is to explain in detail the physics and mathematics behindthe processes described here.

Finally, if you are now keen to experience MRI for yourself, it may be possible tovolunteer for an MRI research scan as a healthy subject at a nearby university lab orteaching hospital.

Figure 0.1. A typical 1.5T MRI scanner, note the coil over the patient’s head.

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ReferenceBrown R W and Cheng Y-C N et al 2014 Magnetic Resonance Imaging: Physical Principles and

Sequence Design 2nd edn (New York: Wiley)

Figure 0.2. Cut section through a typical superconducting MRI scanner. The current carrying superconduct-ing coils which generate B0 are shown in gold. The B0 screening coils are shown in blue. Both sets of coils areinside a liquid He filled cryostat. Concentric cylinders carrying the three main gradient coil sets are mountedinside the cryostat (the x gradient set is not shown here). The main ‘body’ RF coil of birdcage design ismounted inside the gradient coils, closest to the subject being imaged. Other omitted details include an RFscreen between the body coil and the gradients and possibly warm or superconducting shim coils.

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Symbols and Acronyms

γp Gyromagnetic ratio of the protoμ Magnetic momentα Flip angleω0, f0 Larmor frequencyΔB Non-uniformity (of a magnetic field), expressed in parts-per-million (ppm)χm Magnetic susceptibility180° A RF (radiofrequency) pulse which flips the magnetization

through 180°. May be used either for inversion or refocusing2DFT Two dimensional Fourier transform3DFT Three dimensional Fourier transform90° A RF (radiofrequency) pulse which flips the magnetization

through 90°. Usually used for excitation, but can also act as a reducedflip angle refocusing pulse

AC Alternating CurrentACS Auto-Calibration SignalADC Analogue-to-Digital ConverterAR Anti-RotatingASL Arterial Spin LabellingB Magnetic flux density or induction, measured in tesla (T)I0 The main static magnetic fieldB1 The alternating radiofrequency (RF) magnetic field used to rotate the

net magnetizationBOLD Blood Oxygen Level DependentbSSFP balanced Steady-State-Free-PrecessionRBW Receiver BandWidthCE-MRA Contrast Enhanced Magnetic Resonance AngiographyCEST Chemical Exchange Saturation TransferCHESS CHEmical Shift SelectiveCP-SE Carr Purcell (Spin Echo)CPMG Carr Purcell Meiboom GillCR Co-RotatingCS Compressed sensingCSF CerebroSpinal FluidDCE Dynamic Contrast EnhancedDSC Dynamic Susceptibility ContrastDTI Diffusion Tensor ImagingDWI Diffusion Weighted ImagingEPI Echo Planar ImagingESP Echo SPacingETL Echo Train LengthFA Fractional AnisotropyFE Frequency EncodingFFT Fast Fourier TransformFID Free Induction DecayFLAIR FLuid Attenuated Inversion RecoveryfMRI Functional Magnetic Resonance Imaging

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FOV Field Of ViewFSE Fast Spin EchoGE Gradient EchoGM Grey MatterGRAPPA GeneRAlized Autocalibrating Partial Parallel AcquisitionGx, Gy, Gz Gradients in x, y and z directionsIFFT Inverse Fast Fourier TransformIR Inversion recoveryM0 Equilibrium magnetizationMIP Maximum Intensity ProjectionMRA Magnetic Resonance AngiographyMRI Magnetic Resonance ImagingMRS Magnetic Resonance SpectroscopyMTC Magnetization Transfer ContrastMxy or ⊥M Transverse magnetizationMz or M Longitudinal magnetizationNC-MRA Non-Contrast Magnetic Resonance AngiographyNMR Nuclear Magnetic ResonanceNSA Number of Signal AveragesPC Phase ContrastPD Proton DensityPE Phase EncodingPNS Peripheral Nerve Stimulationppm parts per millionPRESS Point Resolved Spatially localised SpectroscopyPROPELLER Periodically Rotated Overlapping ParallEL Lines with

Enhanced Reconstructionr1, r2 T1, T2 relaxivityR1, R2 or R2* T1, T2, T2* relaxation rate (R = 1/T )RF RadioFrequencyRSS Root Sum of SquaresSAR Specific Absorption Rate in W kg−1

SE Spin EchoSENSE SENSitivity EncodingSMASH SiMultaneous Acquisition of Spatial HarmonicsSNR Signal-to-Noise RatioSTEAM STimulated Echo Acquisition ModeSR Slew rate in T m−1 s−1

SS Slice SelectionSTIR Short TI Inversion RecoverySVD Singular Value DecompositionSWI Susceptibility Weighted ImagingT1 Longitudinal or spin-lattice relaxation timeT2 Transverse or spin-spin relaxation timeT2* Apparent spin-spin relaxation timeTA Acquisition or scan timeTE Echo timeTI Inversion timeTOF Time of FlightTR Repetition time

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TS Sampling timeTSE Turbo Spin EchoUTE Ultra-short TEvenc velocity encodingWM White Matter

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Author biographies

Richard Ansorge

Richard Ansorge is a retired senior lecturer at the CavendishLaboratory Cambridge and a former fellow and tutor ofFitzwilliam College Cambridge. He has extensive experience ofexperimental high energy physics, including significantcontributions to the CERN UA5 experiment on the proton–antiproton collider in the 1980s. More recently he has collaboratedwith research groups on the Cambridge Biomedical campus in

several areas including improving 3D medical imaging methods including MRI andPET. He is author of more than 100 scientific publications in these fields.He is particularly interested in applying computers for processing data fromcomplex instrumentation. This has applications which are equally relevant in bothhigh energy physics and medical imaging. He wrote his first computer program in1964 for EDSAC2 and has been coding ever since. Much more recently he hasdeveloped code for 3D medical image registration using GPUs which are probably1010 times more powerful than EDSAC.

He has extensive undergraduate teaching experience and has taught both physicsand mathematics in the first year of the Cambridge Natural Sciences Tripos and alsomore specialised courses in later years. He has also given several popular outreachlectures on Medical Imaging.

He is a Fellow of the Institute of Physics and a member of the IEEE.

Martin Graves

Martin Graves is a Consultant Clinical Scientist and lead of theCambridge University Hospitals MR Physics group. He also holdsan Affiliated Lecturer position with the University of CambridgeClinical School. He is a Fellow of the Institute of Physics andEngineering in Medicine (IPEM), a Fellow of the Higher EducationAcademy (HEA), member of the Institute of Engineering andTechnology (IET) and is an Honorary Member of the Royal

College of Radiologists (RCR). He has served on various national and internationalcommittees including the British Institute of Radiology (BIR), the InternationalSociety of Magnetic Resonance in Medicine (ISMRM) and the European Society ofMagnetic Resonance in Medicine and Biology (ESMRMB). He is a member of theeditorial board of European Radiology.

He has over 30 years’ experience in both clinical and research aspects of MRI andhas published over 170 articles and has been a co-investigator on over £3.2M ofexternal grant funding. In addition to this book he is co-author of the award winningMRI textbookMRI: From Picture to Proton (Cambridge University Press, 2003 and2007) of which the 3rd edition will be published this year. He is a co-author of

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Physics MCQs for the Part 1 FRCR (Cambridge University Press, 2011) andco-editor of Carotid Disease: The Role of Imaging in Diagnosis and Management(Cambridge University Press, 2007). He is a respected MR educator and regularlycontributes to MRI teaching for physicists, radiographers and radiologists locally,nationally and internationally.

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IOP Concise Physics

The Physics and Mathematics of MRI

Richard Ansorge and Martin Graves

Chapter 1

The basics

In this chapter we introduce magnetic resonance imaging (MRI), starting with abrief history and then explaining the physics of proton spin and the way ensemblesof spins relax back to equilibrium after a perturbation thereby generating the signalsthat are the basis for both nuclear magnetic resonance (NMR) andMRI. Finally, weintroduce the methods used to obtain position dependent signals—or images.

1.1 A brief history of MRIThere are arguably three strands to the story of MRI: firstly, there is the discoveryand basic understanding of nuclear spin, particularly that of the proton. Secondly,there is the development of methods to exploit excitation of nuclear spins forspectroscopy (NMR) and later imaging (MRI). Finally, there is the development ofthe very high field superconducting magnets that are necessary for real worldpractical applications.

Much (but not all) of the story can be found in table 1.1 of Nobel prizes awardedfor relevant work.

1.1.1 Spin and magnetic moments

The iconic Stern–Gerlach experiment performed in 1922 (Gerlach and Stern 1922a,1922b), whereby a beam of silver atoms was split into two components by aninhomogeneous magnetic field, was the first demonstration that atomic electronshad quantized intrinsic angular momentum (spin) and an associated intrinsicmagnetic moment. At the time this was an unexpected result. Stern went on todiscover and measure the proton magnetic moment in 1933. The experimentalmethods of Stern were refined by Rabi who showed that a radiofrequency (RF) fieldwould cause spins to oscillate between energy levels (Rabi 1937). To this day, RFfields are used to excite spins for both NMR and MRI. The introduction of theDirac equation in 1928 (Dirac 1928a, 1928b) provided a sound theoretical frame-work for interpreting these results and correctly predicted the value of the electron

doi:10.1088/978-1-6817-4068-3ch1 1-1 ª Morgan & Claypool Publishers 2016

Table 1.1. Nobel Prize winners important for MRI.

Winner(s) Year Subject Citation

Heike Kamerlingh Onnes 1913 Physics For his investigations on the properties ofmatter at low temperatures, which led, interalia, to the production of liquid 4He, and thediscovery of superconductivity (in 1908).

Erwin Schrödinger andPaul Adrian MauriceDirac1

1933 Physics For the discovery of new productive forms ofatomic theory.

Otto Stern 1943 Physics For his contribution to the development of themolecular ray method and his discovery ofthe magnetic moment of the proton (in1933).

Isidor Isaac Rabi 1944 Physics For his resonance method for recording themagnetic properties of atomic nuclei (in1937).

Felix Bloch and EdwardMills Purcell

1952 Physics For their development of new methods fornuclear magnetic precision measurementsand discoveries in connection therewith.(Established experimental basis for NMR insolids and liquids.)

Pyotr LeonidovichKapitsa2

1978 Physics For his basic inventions and discoveries in thearea of low temperature physics, whichincluded the discovery of superfluidity in He.

Nicolaas Bloembergen3 1981 Physics For contribution to the development of laserspectroscopy.

Norman F Ramsey 1989 Physics For the invention of the separated oscillatoryfields method and its use in the hydrogenmaser and other atomic clocks.

Richard R Ernst 1991 Chemistry For his contributions to the development of themethodology of high resolution nuclearmagnetic resonance (NMR) spectroscopy.

Paul C Lauterbur andPeter Mansfield

2003 Medicine For their discoveries concerning magneticresonance imaging.

Vitaly L Ginsburg4 2003 Physics For pioneering contributions to the theory ofsuperconductors and superfluids.

1Dirac is one of our greatest theoretical physicists but arguably under-recognised, he should be mentionedwhenever possible. The relativistic Dirac equation of 1928 predicted not only the existence of spin ½ andassociated magnetic moments ℏe m/2 but also anti-particles.2Kapitsa is included here for his development of a method for the bulk production of liquid He. This was donewhile working in the Mond Laboratory in Cambridge in 1934. Of course he made many other valuablecontributions to both theoretical and experimental physics.3 For his pioneering work on relaxation mechanisms published in 1948 (Bloembergen et al 1948).4 Specifically for his work on type II superconductors now used in all modern clinical MRI magnets.

The Physics and Mathematics of MRI

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magnetic moment. The proton magnetic moment, however, was found to be about2.79 times greater than predicted. With the benefit of hindsight, this so-calledanomalous proton magnetic moment can be understood as early evidence that theproton has substructure rather than as any failure of Dirac’s theory. A brieftreatment of the quantum mechanics of spin ½ systems is given in appendix A.

1.1.2 NMR

Bloch and Purcell shared their 1952 Nobel Prize for separate important contribu-tions to the then emerging field of NMR. Bloch also made important contributionsto many areas of solid state physics. His 1946 papers include the introduction of theBloch equations and a description of an experiment to detect RF emission fromexcited protons (Bloch 1946, Bloch et al 1946). Purcell’s contributions includedstudies of the proton spin relaxation mechanisms found in solids (Bloembergen et al1948, Purcell et al 1946).

Richard Ernst introduced major improvements to NMR, in particular he andWes Anderson replaced the slow RF frequency sweep used in previous NMR workby a fast intense RF pulse producing much improved signals from which frequencyspectra could be measured by means of a Fourier transform (Ernst and Anderson1966). Finally, we should also mention Norman Ramsey who made importanttheoretical contributions to the calculation of chemical shifts and spin–spin couplingconstants in the 1950s; the paper by Pyykko (2000) has a nice discussion.

1.1.3 MRI

By 1970, interest was emerging for clinical applications of NMR, for example in1971 Damadian published an ex vivo study showing that the T1 and T2 relaxationtimes differed in healthy tissue and cancers (Damadian 1971); a patent wassubsequently filed for this idea in 1972 and awarded in 1974 (Damadian 1974).Although there has been subsequent controversy over priority, this early patent doesnot appear to have offered any practical method of in vivo imaging.

The crucial first step towards converting NMR spectroscopy of a bulk sampletowards imaging over a distributed sample was made by Paul Lauterbur in 1973; hisseminal experiment (Lauterbur 1973) is described in more detail below. PeterMansfield introduced the use of multiple gradient directions and the k-spaceformalism for rapid image acquisition in 1977 (Mansfield 1977). At this point thestage was set for clinical MRI to take off and indeed this is exactly what happened,with progress from then on depending more on hardware improvements thantheoretical innovations.

A main magnet producing a static magnetic field of sufficient strength anduniformity over useful volumes is the primary hardware requirement of clinicalMRI. We use the symbol B0 to denote the main static magnetic field produced by theMR magnet (or B0 if we are interested in the vector direction). The first clinical MRIscans were performed in the early 1980s using air-cored electromagnets withconventional copper wound coils having =B 0.08 T0 at the University of

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Aberdeen. A nice account can be found online at http://www.hutch73.org.uk/MRIhist/index.html. This system, known commercially as the M&D 800, was alsoused at other sites including St Bartholomew’s Hospital in London where one of theauthors started their career in MRI.

Electromagnets are expensive to run and generate large amounts of heat, thusthey are not very practical in a clinical setting. However, iron-cored resistivemagnets are still used for some purposes. One example is the Fonar 0.6 T openaccess system which allow patients to sit or stand. This can be less stressful for thepatient and may be clinically advantageous in for example spinal studies. In additionthere are systems based on permanent rare-earth magnets that can also providegeometries offering a more open access to patients, and have B0 fields up to 0.3 T.

However, in practice the rapid growth in clinical MR in the mid 1980s was due tothe introduction of reliable superconducting magnets that were cheaper to run andgave higher fields with better uniformity over larger volumes. We will concentrate onthese systems for the rest of this book.

1.1.4 Superconductivity

The rapid uptake of clinical MRI in the 1980s was directly linked the availability ofreliable superconducting magnets having uniform high magnetic fields over aclinically useful field of view (FOV). These developments were actually some 70years after the original, Nobel Prize winning, discovery by Kamerlingh Onnes in1911 of superconductivity in mercury cooled by liquid He to a temperature of 4.2 K(Onnes 1911). The path from the original discovery to practical high field magnets isa long and interesting story involving both a deeper theoretical understanding of thephenomenon itself, the development of practical superconducting wire, and methodsof producing and containing large volumes of liquid He. Accounts of the theoreticalbasis for superconductivity can be found in many textbooks, for example Kittel(2005). The development of large superconducting solenoids for high energy physicsdetectors began in the 1960s with the development of effcient conductors based ontype II superconductivity in 1961 by Kunzler et al (1961). This topic is discussedfurther in chapter 2.

Currently most clinical MRI systems have magnets with fields between 0.5 and3.0 T. Higher field human capable systems with fields of up to 11 T also exist, but arecurrently used only for research.

1.2 Proton spinThe proton is the nucleus of the hydrogen atom and is an elementary particle withpositive electrical charge =e 1.60217 C, spin angular momentum = ℏS 1

2and

magnetic moment μ μ= 2.792847p N where μN is the nuclear magneton. Actually,both the magnetic moment and the spin are vector quantities so it is better towrite:

μμ = ℏS5.5857 / . (1.1)p N

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When measured1 ℏ = ±S / 12and the value of + 1

2is used for the standard value of μ .p

Thus the proton is a tiny bar magnet (magnetic dipole) and gyroscope as well asbeing electrically charged. It is the proton magnetic moment that is the source of thesignals measured by MRI. In this context it is helpful that human tissue is typically70% water, with two hydrogen atoms per molecule. Thus in clinical MRI there areusually plenty of available protons to generate the required signals2.

1.2.1 Precession

A magnetic dipole μ placed in a magnetic field B0 experiences a torque μ × B0 andalso a forceμ ⋅ ∇B0 if the field is non-uniform. The torque causes the dipole to acquirea potential energy μ− ⋅ B0 which is a minimum when μ and B0 are parallel. Anyrotation induced by the torque causes a change of angular momentum in the directionperpendicular to bothμ and B0. As indicated in equation (1.1) the proton spin angularmomentum S is parallel to μp causing the proton magnetic moment to precess at rateγ Bp in the magnetic field. (The proton-gyromagnetic ratio γp is defined in appendixA.13.) The proton precession is analogous to the precession of a gyroscope in theEarth’s gravitational field and figure 1.1 shows the classical calculation of theprecession rate for both cases. Appendix A gives an alternative quantum mechanicalderivation of the proton precession rate (happily both methods agree).

In the absence of a magnetic field the proton spins have no preferred direction andno macroscopic effects due to the magnetic moments are observable. If the sample isplaced in a uniform magnetic field B0 each proton acquires energy μ− ⋅ B0 andthermal equilibrium will quickly be reached. The actual distribution of spinorientation is most properly described using the density matrix formulation, seefor example Levitt (2008) chapter 11, but for a simple two-state problem it issufficient to just consider individual proton spins quantized to be either parallel (↑)or anti-parallel (↓) to the magnetic field. These protons have energies of μ− B and

μ+ B, respectively, and in thermal equilibrium for a sample of N protons, where= +↑ ↓N N N we have:

μ

μ μ

= ≈ +

− = ≈

μ↑ ↓

↑ ↓ ↓

N NB

kT

N NB

kTN

BkT

N

/ e 12

,

2,

B kT2 / 0

0 0

μ = =−BkT

B Twhere 3.3010 for 310 K.0 60

Thus there is a net magnetization μ= −↑ ↓N NM ( ) of parts per million at room orbody temperature and, although this is small, protons are abundant in many samples

1The caveat about measurement is because quantized angular momentum vectors have a maximummeasurable component less than their actual length, see appendix A for more discussion.2One exception is the air in your lungs, which typically produces tiny MR signals, and indeed special tricksincluding inhaling hyperpolarized gases are sometimes used to overcome this problem.

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