Continuous wave nuclear magnetic resonance

Continuous wave nuclear magnetic resonance:estimation of spin-system properties from steady-state trajectories

James Christopher Korte

ORCID: 0000-0001-9152-1319

Ph.D Engineering (351AA)

August, 2017

Department of Biomedical Engineering, Melbourne School of Engineering

The University of Melbourne

Submitted in total fulfilment of the degree of

Doctor of Philosophy

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Abstract

Magnetic resonance imaging (MRI) is a powerful imaging modality, widely used in routine clinical

practice and as an investigational tool in basic science. The contrast in MRI is related to both

the underlying tissue properties, which undergo disease or injury related changes, and to the MRI

method and sequence parameters used. It is the latter with which this thesis is concerned: the design

and implementation of novel MRI acquisition paradigms and associated reconstruction methods.

The majority of MRI methods excite the object of interest with a series of short RF pulses, varying

the weaker spatial magnetic field using the gradients, and ensuring the RF transmitter is inactive

while acquiring a series of decaying MR signals. This regime linearises the inherently nonlinear

behaviour of a magnetic resonance spin-system, allowing the acquired signals to be considered in a

spatial frequency space and an image to be reconstructed using the well known Fourier transform.

It is our assertion that nonlinear behaviour of the magnetic spin signal will lead to advantageous

attributes in future MR methods, just as moving beyond conventional linear spatial gradients to

nonlinear encoding fields led to methods for accelerated imaging and variable spatial resolution.

Reconstruction of spin-system properties from nonlinear MR signals requires algorithms beyond

the Fourier transform. In this thesis we propose spectroscopy, radial projection imaging and re-

laxometry methods as optimisation problems which minimise the mismatch between experimental

measurements and predictions from Bloch equation based signal models. The use of continuous

wave (CW) excitation patterns allows the development of signal models which are computationally

efficient as they rely on analytical solutions of the Bloch equations or matrix inversion via harmonic

balancing, rather than numerical integration.

Ultra-short relaxation methods have been applied to a range of applications and demonstrate that

MRI is finding use in areas far beyond traditional soft-tissue imaging. Soft tissues have an easily

observable long duration MR signal, whereas the signal decays rapidly for harder tissues such as

bone, or in regions that distort the magnetic field due to magnetic susceptibility gradients, such as

the lungs. Rabi modulated CW techniques operating in a fully continuous mode have the potential

to measure ultra-short relaxation signals in a similar range to ‘true’ zero echo time techniques.

Work inspired by quantum optics has shown that exciting a spin-system with a long duration Rabi

modulated RF field leads to a significant steady-state MR signal. The steady-state trajectory is

highly nonlinear and can be expressed as a series of harmonics of the amplitude modulation fre-

quency of the RF field. This harmonic response provides a natural decoupling of the excitation

and measurement bandwidth, and the ability to maintain a steady-state response under low power

excitation reduces the isolation requirements between hypothetical transmit and receive chains.

Our experimental investigation of steady-state trajectories makes use of two pseudo-simultaneous

excitation and measurement protocols. Whilst these methods were adequate to explore the proof-of-

concept applications, hardware modifications are suggested to unlock the full potential of continuous

wave excitation patterns.

This thesis demonstrates that CW excitation patterns allow the construction of efficient prediction

models and elicit an information-rich steady-state response from which underlying spin-system prop-

erties can be reconstructed. It is anticipated that further development of these concepts and related

hardware modifications will lead to new continuous wave imaging paradigms.

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Declaration

This is to certify that

(i) the thesis comprises only their original work towards the Ph.D except where indicated in the

preface;

(ii) due acknowledgement has been made in the text to all other material used; and

(iii) the thesis is fewer than 100,000 words in length, exclusive of tables, maps, bibliographies and

appendices.

James Korte

Date

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Publications

The work presented in this thesis has produced the following publications and conference presenta-

tions.

Journal papers

• Korte, J.C., Layton, K.J., Tahayori, B., Farrell, P.M., Moore, S.M. and Johnston, L.A. “NMR

spectroscopy using Rabi modulated continuous wave excitation”, Biomedical Signal Processing

and Control, 2017, 33, p.422-428

Conference abstracts

• Korte, J.C., Tahayori, B., Farrell, P.M., Moore, S.M. and Johnston, L.A. “Relaxometry via

steady-state ring-locked trajectories”, Proceedings of the 25th Annual Meeting of ISMRM,

Honolulu, USA, 2017

• Korte, J.C., Tahayori, B., Farrell, P.M., Moore, S.M. and Johnston, L.A. “Rabi Modulated

Continuous Wave Imaging”, Proceedings of the 24th Annual Meeting of ISMRM, Singapore,

2016 (Power Pitch Presentation)

• Korte, J.C., Tahayori, B., Farrell, P.M., Moore, S.M. and Johnston, L.A. “Gapped measure-

ment of spin system response to periodic continuous wave excitation”, The Australian and New

Zealand Magnetic Resonance Society, Bay of Islands, New Zealand, 2015 (Oral Presentation)

• Korte, J.C., Layton, K.J., Tahayori, B., Farrell, P.M., Moore, S.M. and Johnston, L.A. “En-

coding chemical shift with Rabi modulated continuous wave excitation”, Proceedings of the

22nd Annual Meeting of ISMRM, Milan, Italy, 2014 (Oral Presentation)

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Acknowledgments

Firstly, I would like to express sincere gratitude to my supervisors for their patient guidance through-

out my PhD candidature. Leigh Johnston for the constant enthusiasm and lively debates, Peter

Farrell for the confusing but ultimately insightful comments, Bahman Tahayori for his mathemat-

ical genius and Stephen Moore for imparting his wide knowledge of parallel computation. I have

learnt a great deal from each of you and have no doubt your individual styles of thinking will shape

my research in the years to come. Additional thanks to the members of my advisory committee,

David Grayden and John Wagner, for keeping my candidature on track and offering perspective and

direction.

I am very grateful to my friends and colleagues from the Melbourne School of Engineering, the

Melbourne Brain Centre Imaging Unit and the Florey Neuroscience Institute: Roger Ordidge, Brad

Moffat, Scott Kolbe, Jon Cleary, Sonal Josan, Camille Shanahan, Amanda Ng, Kelvin Layton, David

Wright, Yasmin Blunck, Warda Syeda, Paul Bloembergen, Dhafer Alahmari, Eric Wang, Muhammad

Hanif, Muhammad Usman Khalid, Rosa Shishegar, Edward Green, Myrte Strik, Julia Neugebauer,

Peter Yoo, Annie Shelton, Sanuji Gajamange, Frederique Boonstra, Errol Lloyd and Yamni Mohan.

In particular, a big thank you to Kelvin and David for teaching me how to drive the MRI. Yasmin,

Warda, Rosa and Ed for all the interesting discussions in our weekly “MRI for idiots” meeting.

I have been lucky enough to visit a few labs during my candidature. I learnt a great deal during a

one month lab visit with the MRI group in Freiburg, my thanks to Jurgen Hennig, Maxim Zaitsev,

Frederik Testud and Sebastian Littin. Thanks to Samuel Patz and Mirko Hrovat for your valuable

comments on my experiments and showing me around your laboratory in Boston. I would also like

to thank Steffen Bollmann and the Centre for Advance Imaging in Queensland for your feedback on

my work.

To my friends for reminding me there is more to life than research. Thanks for all the Friday night

sessions at E55 to vent some frustration when things are not going to plan, for helping me stay

active with great company on hiking and climbing trips and keeping me sharp with the occasional

acoustic rap battle.

My family is a great source of inspiration, I would like to thank my parents Chris and Katherine for

instilling in me the importance of knowledge and supporting my scientific curiosity. To my sisters for

teaching and challenging me; Laura for introducing me to scepticism and debate, Anna for reminding

me intelligence has many forms.

Lastly, I’d like to thank all the medical professionals who have patched me up over the years and

continue to inspire me to pursue research in a health related field.

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Contents

1 Introduction 17

2 Theory 23

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Spin Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.1 Single atomic particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.2 Ensemble of atomic particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Bloch equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.1 Steady-state solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.2 Pulse excitation solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4 Measurement Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4.1 Signal detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4.2 Signal formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.4.3 NMR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.4.4 NMR relaxometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.5 Imaging Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.5.1 Slice selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.5.2 Spatial encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.A Bloch equations in the rotating frame . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.B Bloch equations steady-state solution to constant excitation . . . . . . . . . . . . . . 60

3 Methods 63

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 Proton Density Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3 Relaxation Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3.1 RARE-VTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3.2 MSME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.4 Magnetic Field Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.4.1 B0 mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.4.2 B1 mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 Rabi continuous wave spectroscopy 73

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2.1 Rabi modulated excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2.2 Observed NMR signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2.3 Spectroscopy as an inverse problem . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3 Methods and Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3.1 The response of off-resonance spins . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3.2 Rabi modulated spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

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12 CONTENTS

4.4.1 The response of off-resonance spins . . . . . . . . . . . . . . . . . . . . . . . . 83

4.4.2 Rabi modulated spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5 Rabi continuous wave imaging 89

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2.1 Observed NMR signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2.2 Radial projection imaging as an inverse problem . . . . . . . . . . . . . . . . 93

5.3 Methods and Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3.1 Gapped excitation and measurement protocol . . . . . . . . . . . . . . . . . . 94

5.3.2 Gapped measurement of off-resonance response . . . . . . . . . . . . . . . . . 94

5.3.3 Rabi modulated imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.3.4 Rabi modulated imaging contrast . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4.1 Gapped measurement of off-resonance response . . . . . . . . . . . . . . . . . 100

5.4.2 Rabi modulated imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.4.3 Rabi modulated imaging contrast . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6 Ring-lock excitation 111

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2.1 Steady-state ring-lock response . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2.2 Estimation of relaxation constants . . . . . . . . . . . . . . . . . . . . . . . . 114

6.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.3.1 Verification of excitation envelope . . . . . . . . . . . . . . . . . . . . . . . . 115

6.3.2 Estimation of relaxation constants . . . . . . . . . . . . . . . . . . . . . . . . 118

6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.A Ring-lock excitation envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.A.1 Amplitude and phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.B Spin-system response under ring-lock excitation . . . . . . . . . . . . . . . . . . . . . 125

7 Conclusion 129

7.1 Summary of original contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Bibliography 131

List of Figures

2.1 Magnetic dipole precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Laboratory and rotating frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Behaviour of a magnetic dipole in a static magnetic field and an RF field . . . . . . 27

2.4 Static magnetic field leads to bulk magnetisation . . . . . . . . . . . . . . . . . . . . 29

2.5 Random orientation of a precessing nuclear magnetic dipole. . . . . . . . . . . . . . . 29

2.6 Bulk magnetisation response to RF excitation . . . . . . . . . . . . . . . . . . . . . . 30

2.7 Steady-state magnetisation under constant excitation over a range of off-resonance

(low power) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.8 Steady-state magnetisation under constant excitation over a range of off-resonance . 35

2.9 Measured and predicted periodic steady-state magnetisation in the time and frequency

domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.10 Free induction decay signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.11 Spin-echo sequence diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.12 Gradient-echo sequence diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.13 FID relation to spectral width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.14 Inversion recovery sequence diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.15 Progressive saturation sequence diagram . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.16 Spin-echo CPMG sequence diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.17 RF envelope relationship with slice profile . . . . . . . . . . . . . . . . . . . . . . . . 53

2.18 Refocusing a slice selective pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.19 Spin-echo based k-space trajectory diagram . . . . . . . . . . . . . . . . . . . . . . . 57

2.20 Gradient-echo based k-space trajectory diagram . . . . . . . . . . . . . . . . . . . . . 58

3.1 Fast low angle shot imaging (FLASH) sequence diagram. . . . . . . . . . . . . . . . 64

3.2 RARE sequence diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3 RARE sequence diagram for T2 relaxometry . . . . . . . . . . . . . . . . . . . . . . . 66

3.4 MSME sequence diagram for efficient multi-slice T2 relaxometry . . . . . . . . . . . . 68

3.5 B0 mapping sequence diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.6 B1 mapping sequence diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.1 Experimental protocol for the incremental measurement of the steady-state magneti-

sation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2 Measured and predicted periodic steady-state magnetisation in the time and frequency

domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 Feasible and selected excitation parameter pair sets. . . . . . . . . . . . . . . . . . . 80

4.4 Trace of covariance matrix of reconstructed spectra. . . . . . . . . . . . . . . . . . . 82

4.5 Measured distributions of off-resonance and RF power. . . . . . . . . . . . . . . . . . 83

4.6 Measured and predicted frequency coefficients of the steady-state magnetisation. . . 84

4.7 Reconstructed NMR spectra of ethanol. . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.1 Radial coordinate system and the Radon transform. . . . . . . . . . . . . . . . . . . 91

5.2 Comparison of spin-system response under gapped Rabi modulated excitation . . . . 95

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14 LIST OF FIGURES

5.3 Gapped excitation protocol for near-simultaneous transmit and receive . . . . . . . . 96

5.4 Diagram of three test-tube imaging phantom. . . . . . . . . . . . . . . . . . . . . . . 97

5.5 Gapped imaging sequence diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.6 Numerical phantom and simulated contrast of Rabi modulated imaging . . . . . . . 101

5.7 Measured distributions of off-resonance and RF power. . . . . . . . . . . . . . . . . . 102

5.8 Measured and predicted frequency coefficients of the steady-state magnetisation. . . 103

5.9 Reference FLASH image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.10 Rabi modulated radial imaging results . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.11 Line profile of imaging results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.12 Line profiles of simulated contrast of Rabi modulated imaging. . . . . . . . . . . . . 107

6.1 Simulated spin-system response to ring-lock excitations. . . . . . . . . . . . . . . . . 112

6.2 Steady-state ellipsoid for different relaxation parameters. . . . . . . . . . . . . . . . . 113

6.3 Simulated spin-system response and the iterative excitation protocol. . . . . . . . . . 116

6.4 Numerical validation of excitation envelope. . . . . . . . . . . . . . . . . . . . . . . . 117

6.5 Experimental validation of excitation envelope. . . . . . . . . . . . . . . . . . . . . . 119

6.6 The measured and predicted transverse steady-state magnetisation surface. . . . . . 120

List of Tables

4.1 Results from the analysis of the reference and reconstructed ethanol spectra . . . . . 86

6.1 Measured and estimated relaxation constants under ring-lock excitation . . . . . . . 120

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16 LIST OF TABLES

Chapter 1

Introduction

Magnetic resonance imaging (MRI) is a powerful imaging modality, widely used in routine clinical

practice and as an investigational tool in basic science. The contrast in MRI is related to both the

underlying tissue properties, which undergo disease or injury related changes, and to the MRI method

and sequence parameters used. A major advantage of this technology is its ability to non-invasively

image the human body using non-ionising electromagnetic radiation. Comparable medical imaging

techniques, such as X-ray CT, use ionising radiation which under certain conditions can increase

the risk of cancer due to DNA damage (National Toxicology Program, 2016; Leuraud et al., 2015;

Behjati et al., 2016). The use of non-ionising radiation makes MRI an ideal tool for long-term clinical

applications, such as the monitoring of disease progression and longitudinal scientific studies. MRI is

a versatile tool which has the ability to acquire images with structural, functional, diffusion, perfusion

and mechanical information using the same scanner. It has been strongly adopted by neurological

disciplines due to the available contrast in brain images, the dynamic brain activity inferred from

functional MRI data and the micro-structural information provided by diffusion imaging.

A conventional MRI scanner generates a strong static magnetic field, superimposed with weaker spa-

tially varying magnetic fields and an excitatory magnetic field which oscillates in the radio frequency

(RF) range. These electromagnetic fields provide the conditions to form a nuclear magnetic reso-

nance (NMR) signal which is measured by an induced voltage in a coil (Bloch, 1946). Early NMR

experiments used continuous wave RF excitation (Purcell et al., 1946; Bloch et al., 1946) but were

largely replaced by pulsed Fourier transform techniques due to improved efficiency and sensitivity

(Ernst and Anderson, 1966). The spatially varying magnetic field, commonly referred to as the gra-

dient field, creates a spatial frequency dependant NMR signal used to perform imaging (Carr, 1953;

Lauterbur et al., 1973). The majority of MRI methods excite the object of interest with a series of

short RF pulses, varying the weaker spatial magnetic field using the gradients, and ensuring the RF

transmitter is inactive while acquiring a series of decaying MR signals. This regime linearises the

inherently nonlinear behaviour of a magnetic resonance spin-system, allowing the acquired signals

to be considered in a spatial frequency space (k-space) (Brown et al., 1982; Ljunggren, 1983; Twieg,

1983) and an image to be reconstructed using the well known Fourier transform. The efficiency of

any method of this class is related to its efficiency in sampling k-space, and is constrained by the

Nyquist criteria. Alternative methods perform image reconstruction via optimisation of generalised

signal models and reduce the sampling constraints of a full k-space acquisition by incorporating

additional information, such as multiple signals from receiver coil arrays (Pruessmann et al., 1999;

Griswold et al., 2002), or impose sparsity requirements on the sampling pattern (Lustig et al., 2007).

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18 CHAPTER 1. INTRODUCTION

It is often the case that spatial and temporal maps of parameter estimates from the reconstructed

images are desired. Parameter mapping methods often require the measurement of multiple images,

such as images at multiple echo times for relaxation mapping. Accelerated parameter mapping using

model based optimisation methods that utilise Bloch equation simulations, have been demonstrated

to jointly estimate multiple tissue parameter maps from reduced data sets (Block et al., 2009; Sumpf

et al., 2011; Ma et al., 2013).

Nonlinearity of the magnetic resonance spin-system is not typically the focus of any well adopted

spectroscopy or imaging technique. On the contrary, all efforts are made to linearise the observed

spin-system behaviour by use of short RF pulses and acquisition of free-induction decays or echo

signals in the absence of an applied RF field. It is our assertion that nonlinear behaviour of the

magnetic spin signal will lead to advantageous attributes in future MR methods, just as moving be-

yond conventional linear spatial gradients to nonlinear encoding fields led to methods for accelerated

imaging and variable spatial resolution (Hennig et al., 2008; Stockmann et al., 2010).

Work inspired by quantum optics (Cappeller and Muller, 1985) has shown that exciting a spin-

system with a long duration amplitude modulated RF field leads to a significant steady-state MR

signal (Tahayori et al., 2008). The amplitude of the steady-state response is maximised when the

amplitude modulation is set to the average power of the excitation envelope (Layton et al., 2014). We

refer to the excitation pattern necessary to achieve this steady-state behaviour as Rabi modulated

excitation or Rabi excitation. The steady-state trajectory is highly nonlinear and can be expressed

as a series of harmonics of the amplitude modulation frequency of the RF field (Tahayori et al.,

2008). This harmonic response provides a natural decoupling of the excitation and measurement

bandwidth, and the ability to maintain a steady-state response under low power excitation (Tahayori

et al., 2009) reduces the isolation requirements between hypothetical transmit and receive chains.

Rabi modulated excitation is therefore well suited to applications that benefit from the ability

to observe the spin-system during excitation, such as closed-loop control of the magnetisation or

measurement of signals beyond the limits of conventional free induction decay (FID) methods.

In majority of human studies, the magnetic resonance signal comes from water in tissue, or more

specifically the hydrogen proton. The abundance of hydrogen and the influence of the surrounding

chemical environment impact the strength and the duration of the MR signal. Soft tissues have an

easily observable long duration MR signal, whereas the signal decays rapidly for harder tissues such

as bone, or in regions that distort the magnetic field due to magnetic susceptibility gradients, such

as the lungs. An additional fast relaxation application is the imaging of quadrupolar nuclei, such

as sodium and phosphorus, which is becoming more feasible due to stronger signal at higher static

magnetic field strengths (Wang et al., 2009). These rapidly decaying signals lead to poor image

contrast and reduced spatial resolution due to blurring image artefacts. Due to these limitations,

and to relative cost and efficiency, X-ray and X-ray CT are still the dominant clinical tool for many

imaging applications.

The first efforts to image fast relaxing spins was by optimisation of existing techniques based on

gradient or spin-echoes, to reduce the finite echo time, TE, to hardware and safety limits (Robson

and Bydder, 2006). Whilst these optimisations were able to image many tissue types, some with

ultra-short relaxation rates were still unobservable. Free induction decay projection methods such

as ultrashort echo time imaging (Pauly et al., 1989a; Bergin et al., 1991) (UTE) reduced the effective

echo time down to hardware switching limits. This was followed by zero-echo time imaging (Hafner,

1994; Madio and Lowe, 1995) (ZTE), allowing an even shorter effective echo time but at the cost of

a hole in central k-space to be filled by algebraic techniques (Kuethe et al., 1999) or additional data

acquisition (Grodzki et al., 2012). Alternative techniques include single point FID imaging (Nauerth

and Gewiese, 1993), water-and fat-suppressed proton projection imaging (Wu et al., 2003) and

sweep imaging with Fourier transformation (Idiyatullin et al., 2006). As mentioned previously,

19

Rabi modulated excitation drives the magnetisation into a steady-state with favourable properties

for simultaneous excitation and measurement. Rabi modulated techniques operating in a fully

continuous mode have the potential to measure ultra-short relaxation signals in a similar range

to ‘true’ zero echo time techniques such as continuous sweep imaging with Fourier transformation

(Idiyatullin et al., 2012) (cSWIFT).

Sweep imaging with Fourier transformation (Idiyatullin et al., 2006) (SWIFT) is similar to ZTE

imaging in that the gradients are applied during RF excitation. The excitation for SWIFT is a low

power, frequency swept RF pulse rather than a high power hard pulse used in ZTE. SWIFT is based

on early CW NMR techniques such as RapidScan NMR (Gupta et al., 1974) and Correlation NMR

(Dadok and Sprecher, 1974) for fast acquisition of spectra. SWIFT is commonly implemented as a

time-shared excitation and acquisition but is inherently continuous wave, and due to it’s low power

excitation has been demonstrated as a fully continuous method (Idiyatullin et al., 2012) (cSWIFT) .

Continuous SWIFT has zero dead-time allowing the measurement of very short T2 relaxation signals

and broad distributions of chemical shifts. In comparison, the shortest reported dead-time for clinical

ZTE (Weiger et al., 2013a) is 5µs. The effective echo-time of UTE is dominated by ramp-time of

the gradient system and peripheral nerve stimulation constraints. Specific absorption rate concerns

for SWIFT, due to its high duty cycle, have been addressed by gradient modulation (Zhang et al.,

2016) (gSWIFT).

Ultra-short relaxation methods have been demonstrated in a range of applications such as dental

imaging, musculoskeletal imaging, lung imaging, x-nuclei imaging and superparamagnetic iron oxide

nanoparticles (SPION) imaging. SWIFT has been used in studies of bone (Kendi et al., 2011; Luhach

et al., 2014), cartilage (Rautiainen et al., 2013), teeth (Idiyatullin et al., 2011), lungs (Kobayashi

et al., 2015), breast (Nelson et al., 2012), brain calcification (Lehto et al., 2012) and SPIONs (Zhang

et al., 2014). UTE has been used in studies of bone (Robson et al., 2004; Robson and Bydder,

2006), cartilage (Brossmann et al., 1997; Gold et al., 1998), knee (Gatehouse et al., 2004; Rahmer

et al., 2006), achilles tendon (Filho et al., 2009), lung (Bergin et al., 1991), liver (Chappell et al.,

2003), USPIOs (Crowe et al., 2005), sodium (Nielles-Vallespin et al., 2007), phosphorus (Anumula

et al., 2006) and brain calcification (Waldman et al., 2003). ZTE has been used in studies of

teeth (Weiger et al., 2012, 2013b), joints (Weiger et al., 2013a) and the skull (Delso et al., 2015). All

these applications of ultra-short relaxation methods demonstrate that MRI is finding use in areas

far beyond traditional soft-tissue imaging.

Rabi modulated excitation of the hydrogen proton has many exciting properties and potential ap-

plications, but at the commencement of this thesis the existing research was limited to a theoretical

framework and a single experimental study. The theoretical work of Tahayori et al. (2009) chal-

lenged the optimality of existing excitation methods and later developed an efficient framework to

find steady-state trajectories to any periodic RF excitation (Tahayori et al., 2015), using Rabi mod-

ulated excitation as a case study. The work of Layton et al. (2014) derived an analytical solution

of the Bloch equations under on-resonance Rabi modulated excitation and developed experimental

techniques to verify this response in a phantom. This manuscript builds on these existing works and

explores the use of the off-resonance response to Rabi modulated excitation to develop novel mag-

netic resonance methods; analogous to historical nuclear magnetic resonance development, first a

spectroscopy method (Korte et al., 2014, 2017a) followed by an imaging method (Korte et al., 2016).

During the investigation of steady-state trajectories, a known steady-state response to constant RF

excitation was revisited, and with the addition of off-resonance, was used to pose relaxometry as a

Bloch model based optimisation problem (Korte et al., 2017b). The novel contributions in this work

estimate tissue properties via optimisation of Bloch equation models against experimental measure-

ments. The Bloch equation based signal models developed in this manuscript are computationally

efficient as they use analytical solutions or matrix inversion via harmonic balancing (Tahayori et al.,


2015) as detailed in Section 2.3.1.2, rather than numerical integration.

Following this introduction, the thesis is organised into five major chapters, and a final concluding

chapter. Of the five major chapters, the first two provide theory of nuclear magnetic resonance

physics and imaging methods relevant to the following three novel chapters. The final chapter

summarises the major contributions of this work and makes suggestions for future research directions.

The theory chapter, Chapter 2, introduces the requisite knowledge upon which the remainder of

the thesis is constructed, containing a summary of NMR physics followed by fundamental MRI

concepts. The physics section explains the behaviour of a spin-system under a static magnetic field,

an oscillating magnetic field and observed relaxation mechanisms. The combined dynamics of the

spin-system are then expressed with the Bloch equations in a laboratory and rotating frame of

reference. Bloch equation solutions for a steady-state response under constant excitation, and in

response to a short RF pulse are discussed in relation to the shift from continuous wave (CW) to

pulsed free induction decay Fourier transform methods. A method used throughout the thesis for

efficient calculation of steady-state response to any periodic excitation is described. Volume signal

methods for the measurement of NMR spectra and estimation of relaxation constants are discussed.

We then move into basic magnetic resonance imaging concepts: k-space, slice selection and image

reconstruction.

The methods chapter, Chapter 3, details the standard imaging methods used to make reference mea-

surements for the experiments in the novel chapters. In particular, we present the fast proton density

imaging method used to acquire reference images in Chapter 5. Relaxation mapping methods are

described for the measurement of spin-lattice and spin-spin relaxation maps, which are used as prior

knowledge in the model based reconstructions in Chapters 4-5 and as the ground truth in Chapter 6.

Inhomogeneity in the static magnetic field and the RF magnetic field are also incorporated as prior

knowledge in the estimation models of Chapters 4-6. The details of measurement of field maps are

provided.

The first novel chapter, Chapter 4 entitled Rabi continuous wave spectroscopy, presents the first

proof-of-concept application of Rabi modulated excitation (Korte et al., 2017a). The observed

signal from a spin-system under Rabi modulated excitation in the presence of off-resonance and field

inhomogeneities is described. This expression is then used to pose NMR spectroscopy as an inverse

problem, where the spin-system is interrogated with a series of different Rabi modulated excitation

envelopes. An iterative measurement protocol (Layton et al., 2014) is used to experimentally measure

the steady-state trajectory of the spin-system in two experiments: the first experiment to validate

the off-resonance response model under Rabi modulated excitation, the second to demonstrate the

measurement of NMR spectra under a series of Rabi modulated excitations.

The second novel chapter, Chapter 5 entitled Rabi continuous wave imaging, is the progression

of the spectroscopy experiment in Chapter 4. To allow a wider bandwidth reconstruction, the

harmonic estimation model used in the spectroscopy experiment is modified to include an offset to

the RF carrier. Proton projection imaging is then posed as an inverse problem, where a weak radial

gradient is applied and the spin-system is interrogated with a series of different Rabi modulated

excitation envelopes. A series of estimated proton projections are then used to reconstruct a two

dimensional image using filtered back projection. In order to make this method experimentally

feasible, a gapped excitation protocol, inspired by time shared SWIFT (Idiyatullin et al., 2006),

was implemented. The gapped excitation protocol was verified by experimental measurement of the

off-resonance response. A two dimensional imaging experiment was then performed to demonstrate

proton density projections can be reconstructed from the harmonic response to Rabi modulated

excitation. Numerical simulations were performed to explore the relaxation contrast of this new

imaging method.

21

The third and final novel chapter, Chapter 6 entitled Ring-lock excitation, is the exploration of a

different (non-Rabi modulated) steady-state trajectory. The known response to a constant amplitude

excitation is revisited, with the addition of off-resonance effects. The magnetisation spirals down to a

steady-state point in response to an on-resonance constant excitation. The same constant excitation,

applied off-resonance, drives the magnetisation into a steady-state ring trajectory. We derive an

analytical expression for the ring-lock steady-state response in terms of the excitation power and

excitation off-resonance. Two experiments are conducted using the iterative measurement protocol

describe in Chapter 4. The first experiment verifies the derived excitation envelope by demonstrating

control over the ring-lock response, when relaxation constants are known. The second experiment

utilises ring-lock excitation to estimate relaxation constants, through probing a sample with a series

of different ring-lock excitations which are optimised against a theoretical ring-lock model.

This thesis demonstrates that information-rich steady-state trajectories of the spin-system can be

used to reconstruct chemical shift spectra, proton density projections and estimate relaxation rates.

The described psuedo-simultaneous measurement protocols allow the investigation of continuous

wave phenomena on a modern scanner. The novel contributions establish the benefit of contin-

uous wave excitation patterns in conjunction with model based estimation and suggest hardware

modification for a new class of continuous wave steady-state methods.


Chapter 2

Theory

Contents2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Spin Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.1 Single atomic particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.2 Ensemble of atomic particles . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Bloch equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.1 Steady-state solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.2 Pulse excitation solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4 Measurement Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4.1 Signal detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4.2 Signal formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.4.3 NMR spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.4.4 NMR relaxometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.5 Imaging Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.5.1 Slice selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.5.2 Spatial encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.A Bloch equations in the rotating frame . . . . . . . . . . . . . . . . . . . 59

2.B Bloch equations steady-state solution to constant excitation . . . . . 60

2.1 Introduction

This chapter describes the theory which provides the foundation for the methods (Chapter 3) and

the experimental chapters (Chapters 4-6). The physics of nuclear magnetic resonance is presented

from a classical point of view, for a single spin and then for an ensemble of spins to form the bulk

magnetisation. The Bloch equations are defined in the laboratory and rotating frames and some

relevant solutions are discussed. Fundamental measurement concepts such as nuclear induction,

echo formation and techniques for acquiring NMR spectra and estimating relaxation constants are

described. Imaging concepts such as slice selection, the k-space formalism, phase encoding and

23

24 CHAPTER 2. THEORY

frequency encoding are presented with some imaging sequence examples. The contents of this

chapter are derived from a range of sources (Levitt, 2001; Liang et al., 2000; Haacke et al., 1999;

Abragam, 1961; Ernst et al., 1990; Slichter, 1978) where each topic can be found in greater detail.

2.2 Spin Physics

This section describes how a subatomic particle with half-integer spin is observed as a magnetic

dipole, its behaviour in the presence of additional magnetic fields and how the behaviour of a group

of these particles form a macroscopic dipole known as the bulk magnetisation vector.

2.2.1 Single atomic particle

Atoms are composed of particles with known attributes such as mass, electric charge, magnetism

and angular momentum. Angular momentum is a combination of orbital angular momentum, such

as an electron orbiting a nucleus, and spin angular momentum, such as a proton rotating on it’s axis

which is often referred to as spin.

2.2.1.1 Nuclear magnetic dipole

A particle’s magnetic moment, µ, is proportional to it’s spin angular momentum, as stated by a

fundamental symmetry theorem (Levitt, 2001),

µ = γS, (2.1)

where γ is the gyromagnetic ratio in units of rad · s−1 ·T−1. The magnitude of the magnetic dipole

can be expressed as

|µ| = γ~√s(s+ 1), (2.2)

where ~ = h/2π is the reduced Plank’s constant and the spin quantum number, s, can take zero,

positive integer and half integer values (0, 12 , 1,

32 , . . . ,

n2 ). The gyromagnetic ratio is often expressed

in units of Hz ·T−1 and has a positive value for a hydrogen proton but is negative for certain other

particles such as an electron.

2.2.1.2 Behaviour in a static magnetic field

A magnetic dipole in an external magnetic field will rotate to align with that external magnetic

field, such as the bar magnet of a compass aligning with the earth’s magnetic field. A nuclear

magnetic dipole also possesses spin angular momentum and exhibits slightly different dynamics to

a bar magnet as shown in Figure 2.1.

The nuclear magnetic dipole will experience a torque,

τ = µ× b, (2.3)

when in an external magnetic field, b. The bold symbols are vectors in a Cartesian coordinate space

with the orthonormal vectors, i, j,k. An arbitrary vector, v, is expressed in a stationary frame of

reference known as the laboratory frame,

v = vxi+ vyj + vyk. (2.4)

2.2. SPIN PHYSICS 25

Figure 2.1: A single magnetic dipole, µ, under a static magnetic field, B0, precessing at the Larmor frequency, ω0.

We can relate the change of total angular momentum, L, to external torque, τ , using Newton’s

second law of motion,

dL

dt= τ . (2.5)

Here we consider a half-integer spin proton such as a hydrogen proton with a total angular momentum

equal to the spin angular momentum, L = S. Substituting (2.1) into (2.5) gives a differential

equation for a nuclear magnetic dipole in an external magnetic field,

dµ

dt= γµ× b. (2.6)

If we consider a static external magnetic field in the z direction,

b = B0k, (2.7)

the solution of (2.6) gives the dynamics of a nuclear magnetic dipole as,

µxy (t) = µxy (0) e−i γB0 t (2.8a)

µz (t) = µz (0) (2.8b)

where the longitudinal component, µz, is constant and the transverse component is expressed in the

complex plane, µxy = µx + iµy, and precesses around the external field at the Larmor frequency,

ω0 = −γB0. (2.9)

2.2.1.3 Response to an oscillating magnetic field

A nuclear magnetic dipole was shown to precess around a static magnetic field in Section 2.2.1.2.

If an oscillating magnetic field is applied in the xy plane at a similar frequency to the Larmor

frequency (2.9) a secondary precession is observed.


(a) (b)

Figure 2.2: A single magnetic dipole, under a static magnetic field, B0, shown in two frames of reference. In thelaboratory frame (a) the dipole precesses at the Larmor frequency, ω0. In a rotating frame (b) with the rotationalfrequency set to the Larmor frequency, the dipole appears stationary.

Rotating frame

It is useful to consider the spin-system in a rotating frame of reference where transverse rotations,

such as spin precession, appear static if the frequency of the rotating frame and the spin precession

are matched. This is depicted in Figure 2.2. The transformation from the laboratory frame to the

rotating frame is, i′j′k′

=

cos(ωrott) − sin(ωrott) 0

sin(ωrott) cos(ωrott) 0

0 0 1

ijk

(2.10)

= Rz,ccw (ωrot, t)

ijk

(2.11)

where ωrot is the angular frequency of the rotating frame. An arbitrary vector, v′, is expressed in

the rotating frame as,

v′ = v′xi′ + v′yj

′ + v′yk′. (2.12)

Spin nutation

We examine the behavior of a nuclear magnetic dipole under a static magnetic field, b0, and a weaker

oscillating magnetic field, b1, with an oscillation frequency, ωRF, at the nominal Larmor frequency,

ω0. The weaker oscillating field is often referred to as the RF excitation because at common static

field strengths the Larmor frequency (2.9) resides in the radio-frequency range. We define a circularly

polarised RF excitation in the laboratory frame as

b1 (t) = Be1 (t)

[cos(ωRFt+ θRF (t)

)i+ sin

(ωRFt+ θRF (t)

)j], (2.13)


Figure 2.3: A single magnetic dipole, µ, in a Larmor rotating frame, under a static magnetic field, B0, and anoscillating magnetic field, B1, applied in the i′ direction. The oscillating magnetic field causes the magnetic dipole toprecess around i′ at the Rabi frequency, ω1.

where Be1 (t) is the excitation envelope, ωRF is the excitaton carrier frequency and θRF (t) is the

excitation phase offset. The RF excitation appears stationary in the RF rotating frame (2.10) where

ωrot = ωRF and the excitation phase offset is constant θRF (t) = θRF,

b′1 (t) = Be1 (t)

(cos(θRF

)i′ + sin

(θRF

)j′), (2.14)

and for zero excitation phase offset is,

b′1 (t) = Be1 (t) i′. (2.15)

The effect of the weaker oscillating field is to rotate the nuclear magnetic dipole around the excitation

vector, b′1 (t), as shown in Figure 2.3. This secondary rotation has a nutation or Rabi frequency

proportional to the strength of the applied RF excitation,

ω1 = ωe1 (t) = −γBe

1 (t) . (2.16)

See (Levitt, 2001, Chapter 10) for a quantum mechanical derivation of this result.

2.2.2 Ensemble of atomic particles

In the previous section we described the behaviour of a single nuclear magnetic dipole in a static

field and a weaker oscillating field. Experimentally we normally observe the behaviour of a group of

nuclear magnetic dipoles. In this section we introduce the behaviour of an ensemble of spins under

similar condition as considered in Section 2.2.1 where we also observe relaxation phenomena.

To discuss the behaviour of a group of nuclear magnetic dipoles, we define the bulk magnetisation

vector as the vector sum of nuclear magnetic dipoles in a sample,

m =

Ns∑i=1

µi (2.17)


where Ns is the number of spins in a volume of interest. An ensemble of spins in the absence

of a magnetic field has a random distribution of orientations leading to zero bulk magnetisation

(Figure 2.4a).

2.2.2.1 Spin-Lattice relaxation

An ensemble of spins in a static magnetic field is shown in Figure 2.4b. Each spin of the ensem-

ble will precess around the static field at it’s Larmor frequency, but due to a random transverse

phase distribution there is zero transverse bulk magnetisation. The individual spins experience a

fluctuating magnetic field due to thermal energy and their molecular environment which cause a

precessional wandering (Levitt, 2001, Chapter 2) as shown in Figure 2.5. Lower energy orientations,

those with components aligned with the static magnetic field, are slightly favoured by the wandering

spins which leads to a longitudinal bulk magnetisation. The ratio of low and high energy spins is

described by the Boltzmann relationship,

N↑N↓

= exp

(γB0

KTs

), (2.18)

where N↑ is the number of low energy spins, N↓ is the number of high energy spins, B0 is the mag-

nitude of the static magnetic field, K is the Boltzmann constant and Ts is the absolute temperature

of the spin-system. This can also be expressed as a ratio of the difference of spin orientations and

total number of spins,

N↑ −N↓Ns

≈ γB0

2KTs. (2.19)

If we consider a hydrogen sample, γ = 42.58 MHz · T−1, at room temperature, Ts = 300 K, and

a field strength, B0 = 1 T, using (2.19) approximately three in every million spins contribute to

the longitudinal bulk magnetisation at thermal equilibrium. The bulk magnetisation at thermal

equilibrium can be expressed as (Haacke et al., 1999)

M0z =

1

4ρ0γ22

KTsB0, (2.20)

where ρ0 is the spin density.

When the static field, B0, is turned on the bulk magnetisation is not formed instantaneously. The

relaxation rate of the spin-system towards the thermal equilibrium bulk magnetisation, M0z , is depen-

dant on molecular environment, static field strength and temperature. The longitudinal relaxation

of the bulk magnetisation is described by the phenomenological model,

mz (t) =(mz (0)−M0

z

)e−

tT1 +M0

z , (2.21)

where T1 is the longitudinal relaxation constant, also referred to as the spin-latice relaxation constant.

The relaxation model (2.21) can also be expressed as the first order system,

dmz (t)

dt=

1

T1

(M0

z −mz (t)). (2.22)

2.2.2.2 Spin-Spin relaxation

An ensemble of spins in a static magnetic field will relax to a non-zero longitudinal bulk magneti-

sation as discussed in Section 2.2.2.1. The bulk magnetisation has the same response to an RF


(a) (b)

Figure 2.4: An ensemble of randomly oriented magnetic dipoles (a) have zero bulk magnetization. When placed in astatic magnetic field (b) the distribution of spin orientation is slightly skewed, creating a bulk magnetisation, m, inthe direction of the static magnetic field.

Figure 2.5: A nuclear magnetic dipole ‘wandering’ through precessional orientations due to the fluctuating magneticfields caused by thermal energy and the molecular environment.


(a) (b)

Figure 2.6: (a) Bulk magnetisation at thermal equilibrium and (b) the bulk magnetisation after an RF excitation.

excitation as a single spin, rotating around the applied excitation envelope, causing the longitudinal

bulk magnetisation to be rotated into the transverse plane (Figure 2.6). Additionally, the bulk

magnetisation precesses at the Larmor frequency in the same manner as an individual spin.

As the individual spins experience slightly fluctuating magnetic fields, their individual precession

frequencies also vary over time, leading to a dephasing of the transverse bulk magnetisation. We

define the transverse bulk magnetisation in the rotating frame as a complex expression,

m′xy = m′x + im′y. (2.23)

The dephasing of the spins which form the bulk magnetisation leads to a decay of the transverse

signal which is described by a phenomenological model in the rotating frame,

m′xy (t) = m′xy (0) e−tT2 (2.24)

where T2 is the transverse relaxation constant, also referred to as the spin-spin relaxation constant.

When this relationship is expressed in the laboratory frame,

mxy (t) = mxy (0) ei ω0 te−tT2 (2.25)

it is analogous to the precession of a nuclear magnetic dipole (2.8a) with an additional decay term.

The transverse decay (2.24) of the bulk magnetisation can also be expressed as the first order system,

dm′xy (t)

dt= − 1

T2m′xy (t) (2.26)

2.3 Bloch equations

The spin physics in Section 2.2 describe a range of behaviours that can be expressed by a system

of ordinary differential equations. The Bloch equations (Bloch, 1946) are defined in the laboratory

frame as,

d

dtm (t) = γm (t)× b (t)− mxi+myj

T2− (mz −M0

z )

T1k, (2.27)

2.3. BLOCH EQUATIONS 31

where m = [mxmymz]T

is the bulk magnetisation vector, b = [bx by bz]T

is the applied magnetic

field, M0z is the bulk magnetisation at thermal equilibrium, T1 is the longitudinal relaxation constant

and T2 is the transverse relaxation constant.

The Bloch equations (2.27) can be expressed in the rotating frame as,

d

dtm′ (t) = γm′ (t)× b′eff (t)−

m′xi′ +m′yj

′

T2− (m′z −M0

z )

T1k′ (2.28)

in which,

b′eff (t) = b′ (t) +ωrot

γk′ (2.29)

and wherem′ =[m′xm

′ym′z

]Tis the bulk magnetisation vector in the rotating frame, b′ =

[b′x b′y b′z

]Tis the applied magnetic field in the rotating frame. Detail of the transformation into the rotating

frame can be found in Appendix 2.A. The external magnetic field in the rotating frame of reference

is the vector sum of the static and oscillating fields,

b′ (t) = b′0 + b′1 (t) . (2.30)

If we consider the bulk magnetisation in the RF rotating frame, ωrot = ωRF, and substitute the

static field (2.7) and the RF excitation field (2.13) into (2.30), the effective magnetic field (2.29) is,

b′eff (t) = B0k′ +Be

1 (t)(

cos(θRF (t)

)i′ + sin

(θRF (t)

)j′)

+ωRF

γk′

=1

γ

(γBe

1 (t) cos(θRF (t)

)i′ + γBe

1 (t) sin(θRF (t)

)j′ + (ωRF + γB0)k′

)=

1

γ

(− ωe

1 (t) cos(θRF (t)

)i′ − ωe

1 (t) sin(θRF (t)

)j′ + (ωRF − ω0)k′

)=

1

γ

(− ωe

1,x (t) i′ − ωe1,y (t) j′ −∆k′

), (2.31)

where the RF excitation field has been split into orthogonal components,

ωe1,x (t) = ωe

1 (t) cos(θRF (t)

)(2.31a)

ωe1,y (t) = ωe

1 (t) sin(θRF (t)

), (2.31b)

and deviation between an isochromat’s Larmor frequency and the rotating RF frame, is expressed

as off-resonance,

∆ = ω0 − ωRF, (2.32)

which is broken down into individual sources of off-resonance,

∆ = ω0 (1 + δω0)− ωRF (1 + δωrf) , (2.33)

where δω0encompasses chemical shift effects, field inhomogeneities and gradients and δrf is a fre-

quency off-set to the RF transmitter carrier frequency. We can express the Bloch equations in the

RF rotating frame (2.28) in a matrix form by substitution of the effective magnetic field (2.31),

d

dtm′ (t) = Ω (t)m′ (t) +R1m0 (2.34)


where

Ω(t) =

−R2 −∆ ωe1,y (t)

∆ −R2 −ωe1,x (t)

−ωe1,y (t) ωe

1,x (t) −R1

, (2.34a)

m′ (t) =[m′x m′y m′z

]T, (2.34b)

m0 =[

0 0 M0z

]T, (2.34c)

R1 =1

T1, R2 =

1

T2. (2.34d,e)

The Bloch equations can be solved via numerical integration, but by applying certain assumptions

we can derive analytical solutions to gain more insight into the spin-system response. A secondary

advantage of these solutions is computational efficiency, making simulation of more complex spin-

systems tractable.

2.3.1 Steady-state solutions

Here we consider steady-state solutions of the Bloch equations under continuous wave (CW) exci-

tation. First we discuss an analytical expression of spin response under a constant CW excitation.

This is followed by the description of a method for efficient calculation of spin response under any

periodic excitation where we introduce the Rabi modulated CW excitation envelope and response

as a case study particularly relevant to this thesis.

2.3.1.1 Constant excitation

The spin-system response to constant excitation (Bloch, 1946) was important for historic CW NMR

experiments. We define the constant amplitude RF excitation,

ωe1,x (t) = ω1,x (2.35a)

ωe1,y (t) = 0. (2.35b)

which is applied for an adequate duration, > 5T1, driving the bulk magnetisation into a steady-state

which relates to a rate of change of the bulk magnetisation,

d

dtm′ (t) = 0. (2.36)

Substitution of (2.35) and (2.36) into the Bloch equations (2.34) gives the steady-state solution,

m′x =ω1,x ∆ (T2)

2

1 + (∆T2)2

+ (ω1,x)2T1T2

M0z (2.37a)

m′y =−ω1,x T2

1 + (∆T2)2

+ (ω1,x)2T1T2

M0z (2.37b)

m′z =1 + (∆T2)

2

1 + (∆T2)2

+ (ω1,x)2T1T2

M0z , (2.37c)

the full derivation of which is detailed in Appendix 2.B. The set of possible steady-state magnetisa-

tions vectors lie on an elliptical manifold, as noted by (Abragam, 1961, Chapter 3),

1 =

(m′xrx

)2

+

(m′yry

)2

+

(m′z − czrz

)2

(2.38)


where

rx = ry =M0

z

2

√T2

T1(2.38a)

rz =M0

z

2(2.38b)

cz =M0

z

2. (2.38c)

The steady-state ellipsoid (2.38) is located in the upper hemisphere of the Bloch sphere as shown in

Figure 2.7b.

Low power excitation case

If we consider a low power RF limiting case,

(ω1,x)2T1T2 1 (2.39)

then the transverse magnetisations (2.37a) and (2.37b) reduce to,

m′x = M0z ω1,x

∆

∆2 +(

1T2

)2

(2.40a)

m′y = −M0z ω1,x

(

1T2

)∆2 +

(1T2

)2

(2.40b)

where (2.40a) and (2.40b) are commonly referred to as the dispersion and absorption line shapes,

shown in Figure 2.7a. The absorption (2.40b) is a negative Lorentzian peak,

L (δ, µ,Γ, a) =a

π

[Γ

(δ − µ)2 + Γ2

](2.41)

where in this case δ is off-resonance, µ is the peak centre, Γ is the half width at half maximum

(HWHM) and a is the peak area. The dispersion and absorption line shapes are also observed when

applying a higher power excitation but have a broader response (Figure 2.8).

2.3.1.2 Periodic excitation

In the majority of excitation cases an analytical solution to the Bloch equations is not available. A

general approach to calculate the spin-system response to periodic excitation waveforms has been

proposed by Tahayori et al. (2015). This technique is referred to as harmonic balancing.

As in Tahayori et al. (2015) we consider the Bloch equations (2.34) under a periodic excitation with

the same period, Tm, for both ωe1,x and ωe

1,y components. We can re-write the transformation matrix,

Ω(t), as

Ω (t) = B +Rx

∞∑n=−∞

β(n)x einωmt +Ry

∞∑n=−∞

β(n)y einωmt (2.42)


−6 −4 −2 0 2 4 6−0.5

0

0.5

m′ x

∆ (Hz)

−6 −4 −2 0 2 4 6−0.4

−0.2

0m

′ y

∆ (Hz)

−6 −4 −2 0 2 4 6

0.6

0.8

1

m′ z

∆ (Hz)

(a)

(b)

Figure 2.7: Analytical (black line) and numericaly integrated (colored circles) solutions for steady-state magnetisationunder a constant excitation. The excitation was set as, ω1,x = 1.0/

√T1T2, to maximise the absorption peak. (a)

Orthogonal components of the steady-state magnetisation across a range of off-resonance, ∆. (b) Steady-state mag-netisation vector (black line) over a range of off-resonance, ∆. All possible magnetisation vectors are contained by theBloch sphere (light grey) and all steady-state solutions under constant excitation lie on the surface of a steady-stateellipsoid (dark grey).


−6 −4 −2 0 2 4 6−0.5

0

0.5

m′ x

∆ (Hz)

−6 −4 −2 0 2 4 6−0.1

−0.05

0m

′ y

∆ (Hz)

−6 −4 −2 0 2 4 60

0.5

1

m′ z

∆ (Hz)

(a)

(b)

Figure 2.8: Analytical (black line) and numericaly integrated (colored circles) solutions for steady-state magnetisationunder a constant excitation. The excitation was set as, ω1,x = 10.0/

√T1T2, to demonstrate that the Lorentzian

lineshape also exist under higher power excitation. (a) Orthogonal components of the steady-state magnetisationacross a range of off-resonance, ∆. (b) Steady-state magnetisation vector (black line) over a range of off-resonance,∆. All possible magnetisation vectors are contained by the Bloch sphere (light grey) and all steady-state solutionsunder constant excitation lie on the surface of a steady-state ellipsoid (dark grey).


where

B =

−R2 −∆ 0

∆ −R2 0

0 0 −R1

, Rx =

0 0 0

0 0 −1

0 1 0

, Ry =

0 0 1

0 0 0

−1 0 0

, (2.42a-c)

and ωm = 2π/Tm is the fundamental frequency, β(n)x , β

(−n)y are the Fourier series coefficients of the

excitation envelopes, ωe1,x, ω

e1,y. As the excitation envelopes are real signals, β(n) = β∗(n), where

β∗(n) is the complex conjugate.

The magnetisation and its rate of change can be expressed as a Fourier expansion,

m =

∞∑l=−∞

c(l)eilωmt (2.43a)

m =

∞∑l=−∞

ilωmc(l)eilωmt (2.43b)

where the Fourier coefficient vector is,

c(l) =[c(l)x c(l)y c(l)z

]T. (2.43c)

Substitution of (2.43) into the Bloch equations (2.34) gives,

∞∑l=−∞

c(l)eilωmt = Ω (t)

∞∑l=−∞

ilωmc(l)eilωmt +R1m0

= B

∞∑l=−∞

c(l)eilωmt +R1m0 · · ·

+Rx

∞∑l=−∞

∞∑n=−∞

c(l)β(n)x ei(l+n)ωmt · · ·

+Ry

∞∑l=−∞

∞∑n=−∞

c(l)β(n)y ei(l+n)ωmt. (2.44)

We apply the Fourier shift theorem to (2.44) and shuffle the order of summation

∞∑l=−∞

(Bc(l) +Rx

∞∑n=−∞

β(n)x c(l−n) +Ry

∞∑n=−∞

β(n)y c(l−n) − ilωmc

(l)

)eilωmt = −R1m0 (2.45)

Equation (2.45) can then be expanded into an infinite matrix form,

PC = Q, (2.46)


where

P =

. . .

A+ i2ωmI R(β

(1)x , β

(1)y

)R(β

(2)x , β

(2)y

)R(β

(3)x , β

(3)y

)R(β

(4)x , β

(4)y

)R(β

(−1)x , β

(−1)y

)A+ iωmI R

(β

(1)x , β

(1)y

)R(β

(2)x , β

(2)y

)R(β

(3)x , β

(3)y

)R(β

(−2)x , β

(−2)y

)R(β

(−1)x , β

(−1)y

)A R

(β

(1)x , β

(1)y

)R(β

(2)x , β

(2)y

)R(β

(−3)x , β

(−3)y

)R(β

(−2)x , β

(−2)y

)R(β

(−1)x , β

(−1)y

)A− iωmI R

(β

(1)x , β

(1)y

)R(β

(−4)x , β

(−4)y

)R(β

(−3)x , β

(−3)y

)R(β

(−2)x , β

(−2)y

)R(β

(−1)x , β

(−1)y

)A− i2ωmI

. . .

,

(2.46a)

C =[· · · c(−2) c(−1) c(0) c(1) c(2) · · ·

], (2.46b)

Q = [· · · 0 0 −R1m0 0 0 · · · ] , (2.46c)

and we define,

A , B +R(β(0)

x , β(0)y

)(2.46d)

R (βx, βy) , βxRx + βyRy. (2.46e)

If we truncate the coefficients in the infinite matrix, P , we can invert the linear system (2.46) and

solve for the Fourier coefficients of the steady-state response of the spin-system,

C = P−1Q. (2.47)

Rabi modulated continuous wave excitation

Here we introduce the Rabi modulated continuous wave excitation which is a major topic in two ex-

perimental chapters (Chapter 4 and Chapter 5), and a good candidate to demonstrate the harmonic

balancing technique. The Rabi modulated RF envelope is defined as

ωe1 (t) = ω1 (1 + α cosωmt) (2.48)

where ω1 is the average Rabi frequency, ωm is the modulation frequency and α is the modulation

level. When the excitation pattern (2.48) is applied on the i′ axis we have the excitation envelopes,

ωe1,x (t) = ω1 (1 + α cosωmt) (2.49a)

ωe1,y (t) = 0, (2.49b)

which have the related Fourier coefficients,

β(0)x = ω1; β(−1)

x = β(1)x =

αω1

2; β(n)

x = 0, n 6= −1, 0, 1. (2.50)

Substitution of (2.49) into the harmonic balance finite matrix (2.46) gives a tridiagonal matrix,

P rabi =

. . .

A+ i2ωmI D 0 0 0

D A+ iωmI D 0 0

0 D A D 0

0 0 D A− iωmI D

0 0 0 D A− i2ωmI. . .

(2.51)


0 2 4 6 8 10−0.2

0

0.2

0.4

mobs

x

(norm

alised)

time(ms)

−1000 −500 0 500 1000−0.1

−0.05

0

0.05

0.1

frequency(Hz)

Re(

cobs

x

)

(a)0 2 4 6 8 10−0.2

0

0.2

0.4

mobs

x

(norm

alised)

time(ms)

−1000 −500 0 500 1000−0.1

−0.05

0

0.05

0.1

frequency(Hz)

Re(

cobs

x

)

(b)

Figure 2.9: Spin system response of the water phantom under Rabi modulated excitation. (a) Measured (red circles)and theoretical (black line) periodic steady-state magnetisation waveform for excitation parameters α = 1, ω1 =ωm = 100 Hz and δrf = 180 Hz. (b) Measured (coloured circles) and theoretical (black crosses) harmonics of steady-state magnetisation. DC component (blue), first (green), second (purple), third (red) fourth (orange) and fifth (grey)harmonics.

where,

A =

−R2 −∆ 0

∆ −R2 −ω1

0 ω1 −R1

, D =αω1

2Rx. (2.51a,b)

We can solve for the harmonics of the spin-system response using (2.47) and reconstruct the spin-

system time response via (2.43a) with an experimental example shown in Figure 2.9.

It has been shown that the series coefficients can be approximated by CN ≈(α2

)N ( ω1

ωm

)N/N !,

where N is the number of harmonics considered in the solution (Tahayori et al., 2015). Therefore,

the series solution coefficients always converge to zero. When αω1 < 2ωm the convergence rate is

very fast and a few terms are sufficient to obtain accurate results.

2.3.2 Pulse excitation solutions

The focus of this manuscript is CW excitation but the majority of modern NMR methods use RF

pulses, short relatively high power envelopes to tip the bulk magnetisation into the transverse plane.

In this section we explore the on-resonance response to a pulse excitation envelope and compare the

Fourier transform of a free induction decay signal to the spin-system response under constant CW

excitation.


Figure 2.10: Free induction decay (FID) signal diagram. An RF pulse tips the bulk magnetisation into the transverseplane where it can be observed by an induced voltage in a coil. The exponential transverse decay is characterised bythe spin-spin relaxation rate, T2, but experimentally we observe a faster decay, T ∗2 , due to field inhomogeneity.

2.3.2.1 Response to RF Pulse

The response of the bulk magnetisation is often expressed as a tip angle away from thermal equilib-

rium,

α =

∫ωe

1 (t) dt. (2.52)

A time efficient excitation pulse shape is a rectangular function,

ωe1 (t) = Π (t, τ) =

−γB1 0 ≤ t ≤ τ0 otherwise

(2.53)

which has a corresponding flip angle (2.52) for an on-resonance spin,

α = −γB1τ. (2.54)

This flip angle expression (2.52) is useful for considering the effect of excitation field strength on

the bulk magnetisation. The flip angle response for off-resonance isochromats is considered in Sec-

tion 2.5.1.

2.3.2.2 Free induction decay

If we consider a case when the bulk-magnetisation has been flipped into the transverse plane, α =

π/2, under no further excitation we observe a free induction decay (FID) signal (Figure 2.10). The


Fourier transform of a FID signal from a single on-resonance isochromat (2.25) is,

ρxy (ω) = Fmxy (0) ei ω0 te−

tT2

=

∫ ∞0

mxy (0) ei ω0 te−tT2 e−iωt dt

= mxy (0)

∫ ∞0

e

[i(ω0−ω)− 1

T2

]tdt

=mxy (0)

i (ω0 − ω)− 1T2

[e

[i(ω0−ω)− 1

T2

]t

]∞0

=−mxy (0)

i (ω0 − ω)− 1T2

−i (ω0 − ω)− 1T2

−i (ω0 − ω)− 1T2

= mxy (0)

1T2

(ω0 − ω)2

+(

1T2

)2 − i(ω0 − ω)

(ω0 − ω)2

+(

1T2

)2

(2.55)

where ω is the Fourier pair of time. If the thermal equilibrium bulk magnetisation has been rotated

into the transverse plane, mxy (0) = 0− iM0z , then (2.55) becomes,

ρxy (ω) = M0z

(ω0 − ω)

(ω0 − ω)2

+(

1T2

)2 − i1T2

(ω0 − ω)2

+(

1T2

)2

(2.56)

or as transverse components,

ρx (ω) = M0z

(ω0 − ω)

(ω0 − ω)2

+(

1T2

)2

(2.57a)

ρy (ω) = −M0z

1T2

(ω0 − ω)2

+(

1T2

)2

. (2.57b)

The Fourier transform of a free induction decay signal (2.57) gives Lorentzian absorption and disper-

sion lineshapes that are comparable to the constant excitation steady-state response (2.40), shown

again here for clarity,

m′x = M0z ω1,x

(ω0 − ωRF)

(ω0 − ωRF)2

+(

1T2

)2

(2.40a)

m′y = −M0z ω1,x

1T2

(ω0 − ωRF)2

+(

1T2

)2

. (2.40b)

This result is important as it demonstrates that the Fourier transform of a single FID is equivalent

to observing the response to a series of constant power CW excitations with different frequency

offsets.

2.4. MEASUREMENT CONCEPTS 41

2.4 Measurement Concepts

This section covers the experimental measurement of the bulk magnetisation via an induced voltage

in a coil. The effects of field inhomogeneity on the free induction decay signal are considered in

the context of NMR spectroscopy. Echo formation using refocusing pulses and field gradients is

described and applied to volumetric NMR relaxometry.

2.4.1 Signal detection

After excitation by an RF field, the bulk magnetisation generates an oscillating magnetic field which

can be detected via induction with a conductive coil (Bloch et al., 1946). The flux in a coil is

expressed with the reciprocity principle,

Φ (t) =

∫object

br (r) ·m (r, t) dr (2.58)

where r is a spatial vector, br (r) is the magnetic field generated at a point in space by a unit current

in the receive coil and m (r, t) is the bulk magnetisation with spatial dependance in the laboratory

frame.

The induced voltage is proportional to the rate of change of the flux, as per Faraday’s law of

induction,

V (t) = −∂Φ (t)

∂t

= − ∂

∂t

∫object

br (r) ·m (r, t) dr

= − ∂

∂t

∫object

[br,x (r)mx (r, t) + br,y (r)my (r, t) + br,z (r)mz (r, t)

]dr. (2.59)

The transverse components of the bulk magnetisation, mxy, are the primary cause of induced volt-

age as they fluctuate at a much faster rate than the longitudinal components, mz, leading to the

simplification of (2.59),

V (t) = −∫

object

[br,x (r)

∂mx (r, t)

∂t+ br,y (r)

∂my (r, t)

∂t

]dr. (2.60)

Assuming phase sensitive quadrature detection of a FID and ignoring relaxation effects, equation

(2.60) becomes the signal equation,

s (t) =

∫object

b∗r,xy (r) mxy (r, 0) e−i∆(r)t dr (2.61)

where b∗r,xy (r) is the complex conjugate of br,xy (r). If the main static field is spatially and temporally

varying then so is the off-resonance,

∆ (r, t) = ω0 (r, t)− ωRF

= −γB0 (r, t)− ωRF (2.62)

and equation (2.61) is modified to,

s (t) =

∫object

b∗r,xy (r) mxy (r, 0) e−i∫ t0

∆(r,τ)dτ dr. (2.63)

Full derivation of (2.61) and (2.63) is in (Liang et al., 2000, Chapter 3). Here we have defined the

resonance offset with respect to the radio frequency carrier.


2.4.2 Signal formation

Here we introduce the NMR signal types that are observed via nuclear induction on a modern MRI

scanner. Free induction decays, spin echoes and gradient echoes are discussed. Other signals such

as stimulated echoes and rotary echoes are excluded as they are not used in any of the experimental

chapters in this thesis.

2.4.2.1 Free-induction decay

The FID signal (2.25) is the precessing and exponentially decaying transverse bulk magnetisation

(Figure 2.10). The exponential decay of the bulk magnetisation is characterised by a spin-spin

relaxation rate, T2. Inhomogeneity in the B0 field causes a range of off-resonance frequencies which

causes a dephasing of the signal and an observed FID with faster decay than predicted by the

spin-spin relaxation constant. The two decays are shown in Figure 2.13, where the faster decay is

expressed as,

1

T *2

=1

T2+

1

T ′2(2.64)

where T ′2 is the decay attributed to field inhomogeneity.

2.4.2.2 Spin-echo

In the presence of B0 field inhomogeneity the FID signal can become dominated by T ′2 and decay

much faster than the intrinsic T2 decay. The spin-echo or Hahn echo (Hahn, 1950) is described in

this section as per Haacke et al. (1999) and allows us to recover a signal with T2 weighting in the

presence of B0 field inhomogeneity.

Spin echoes are the basis of many standard imaging methods used in this manuscript to take ref-

erence measurements. The T2 weighted signal of a spin echo is useful for measuring relaxation

maps (Section 3.3.1 and Section 3.3.2). The insensitivity to B0 inhomogeneity is beneficial for the

measurement of B1 maps (Section 3.4.2).

Spin-echo formation

A spin-echo is formed via RF excitation with an excitation pulse and a refocusing pulse (Figure 2.11).

The first pulse is an excitation pulse with angle, α = αxi′, the second is a refocusing pulse with

angle, β = βyj′, with an inter-pulse duration, τ . The phase accumulation of an isochromat with

off-resonance, ∆, is

θ (r, t) = ∆ (r) t. (2.65)

In Figure 2.11, we consider a case where the excitation pulse is applied along the i′ axis causing

the magnetisation to rotate by αx = π2 , aligning it with the j′ axis. During the period, τ , between

excitation and refocusing pulses an isochromat will accumulate a phase of

θ (r, τ -) = ∆ (r) τ (2.66)

where τ - is the timepoint directly before the refocusing pulse. We assume the transverse signal

to be completely dephased before the refocusing pulse is applied along the j′ axis, rotating the


Figure 2.11: Spin-echo sequence diagram for excitation angle, αx = π2

, and refocusing angle, βy = π. The responsefor three isochromats at on-resonance (black), positive off-resonance (red) and negative off-resonance (blue) is shown.Isochromats (a) at equilibrium (b) are excited into the transverse plane (c) where they dephase and (d) have theirphase flipped and (e) rephase to form an echo.

dephased transverse magnetisation by βy = π. The refocusing pulse effectively flips the phase of

each isochromat

θ(r, τ+

)= −θ (r, τ -) (2.67)

where τ+ is the timepoint directly after the refocusing point. In the period after the refocusing

pulse, the isochromat will accumulate phase

θ (r, t− τ) = θ(r, τ+

)+ ∆ (r) (t− τ)

= −∆ (r) τ + ∆ (r) (t− τ)

= ∆ (r) (t− 2τ) . (2.68)

It is apparent from (2.68) that all isochromats will refocus to zero phase at time t = 2τ which is

often referred to as the spin echo time,

TE = 2τ, (2.69)

and corresponds to the peak magnitude of the echo.

Spin-echo envelope

The envelope of the transverse magnetisation during a spin-echo can be expressed as,

d

dt|mxy| = −RSE

2 (t) |mxy| , (2.70)


where the effective decay rate is,

RSE2 (t) =

R′2 +R2 0 < t < τ

−R′2 +R2 τ < t < 2τ = TE

R′2 +R2 t > 2τ = TE

. (2.70a)

The solution of (2.70) is,

|mxy| (t) = |mxy| (0)

e− tT∗2 0 < t < τ

e−tT2 e− (TE−t)

T ′2 τ < t < 2τ = TE

e−tT2 e− (t−TE)

T ′2 = e− tT∗2 e

TET ′2 t > 2τ = TE

, (2.71)

and shows that a spin-echo signal is T2 weighted at the echo time, t = TE,

|mxy| (TE) = |mxy| (0) e−TET2 . (2.72)

Spin-echo diffusion effects

The preceding analysis of a spin-echo assumes a spatially varying and time invariant off-resonance,

∆ (r). If we consider diffusion, when an isochromat may wander through space during the echo

period, 2τ , the off-resonance becomes time dependant, ∆ (r, t). This leads to imperfect rephasing

and a reduction of the expected echo magnitude (2.72). Diffusion theory, methods and applications

are covered in detail in Jones (2010).

2.4.2.3 Gradient Echo

A gradient recalled echo is formed via switching the polarity of the linear gradient fields (Fig-

ure 2.12). Gradient echoes are used in fast imaging methods which can be used for proton density

mapping (Section 3.2) and are sensitive to off-resonance effects allowing the acquisition of B0 maps

(Section 3.4.1).

Consider the phase accumulation of off-resonance isochromats (2.65) under a linear gradient in the

x direction,

θ (r, t) = ∆ (r) t

=(ω0 [1 + δω0

(r)] + γ Gx− ωRF

)t

=(

∆B0(r) + γ Gx

)t, (2.73)

where G is the gradient field strength and ∆B0 (r) encompasses remaining sources of off-resonance.

In Figure 2.12, we consider a case where an excitation pulse has caused the magnetisation to rotate

by α = π2 into the transverse plane. For simplicity we ignore any dephasing in the period between

the excitation pulse and the gradient and consider t = 0 to be at the start of the positive gradient

lobe. At the end of the positive gradient lobe an isochromat will have accumulated a phase of

θ (r, τ -) =(

∆B0(r) + γ Gx

)τ. (2.74)


The phase accumulation during the negative gradient is expressed as,

θ (r, t− τ) = θ (r, τ -) +(

∆B0 (r)− γ Gx)

(t− τ)

=(

∆B0 (r) + γ Gx)τ +

(∆B0 (r)− γ Gx

)(t− τ)

= ∆B0 (r) t+ γ Gx (2τ − t) . (2.75)

It is apparent from (2.75) that isochromats will rephase at time, t = 2τ , to form a gradient echo

peak. At the gradient echo time, TE = 2τ , there is a remaining loss of phase coherence,

θ (r, TE) = TE ∆B0(r) (2.76)

causing a loss of intensity and giving the gradient echo a T ∗2 weighting.

46

CHAPTER

2.THEORY

Figure 2.12: Gradient-echo sequence diagram for excitation angle for three groups of isochromats at spatial locations (circles) r, (arrows) 0 and (diamonds) −r. Eachof the three spatial location has the same main field distribution represented by (blue) a slightly slower off-resonance (black) an on-resonance and (red) a slightly fasteroff-resonance. All isochromat groups start (a) at equilibrium (b) are excited into the transverse plane (c) where they dephase due to main field inhomogeneities then (d)dephase due to main field inhomogeneities and the applied gradient (e) partially rephase due to a reversed gradient to form an echo. The loss of phase coherence due tomain field inhomogeneity remains at echo time (e) giving the gradient echo a T ∗2 weighting


2.4.3 NMR spectroscopy

We have seen in previous sections that the precession frequency of a spin is proportional to the

strength of the magnetic field. A circulating electron cloud generates a local magnetic field contri-

bution which can shield or de-shield a nucleus from the B0 field. This perturbation of the magnetic

field alters the precession frequency of spin, creating an offset to the Larmor frequency known as

chemical shift. This relationship between local chemical environment and frequency allows us to

probe the chemical composition of a sample via NMR spectroscopy.

To allow spectral comparison of data across magnetic field strengths, it is common to express chem-

ical shift in parts-per-million (ppm) with respect to a reference compound,

δcs =ωiso − ωref

ωref, (2.77)

where ωiso is the isochromat frequency and ωref is the reference frequency. Chemical shift (2.77) can

be expressed in the Bloch equations as a source of off-resonance (2.33),

∆ = ω0 (1 + δω0 + δcs)− ωRF (1 + δωrf) , (2.78)

where here we set the reference frequency as the Larmor frequency, ωref = ω0.

Historically, NMR spectra were acquired under continuous wave excitation (Arnold et al., 1951),

slowly sweeping the frequency through the spectral range to collect peaks of the spectrum. Modern

techniques use a broadband excitation pulse and the Fourier transform to obtain a spectrum (Ernst

and Anderson, 1966).

2.4.3.1 Continuous Wave NMR spectroscopy

The first CW-NMR spectrum of ethanol (Arnold et al., 1951) demonstrated the ability of NMR to

distinguish chemical groups of methyl, methylene and hydroxyl by their chemical shift and relative

areas. In this type of spectroscopy experiment a CW excitation is swept across the frequency range,

slow enough that the transverse response of the spin-system can be modelled by the steady-state

solution to the Bloch equations (2.40). A limitation of these early methods was a slow spectral

sweep rate to avoid artifacts from previously excited spins. These efficiency issues were addressed

by later improvements to CW spectroscopy (Gupta et al., 1974; Dadok and Sprecher, 1974).

2.4.3.2 Pulsed Fourier Transform NMR spectroscopy

Fourier transform NMR (FT-NMR) using pulsed excitation (Ernst and Anderson, 1966) provided

a large efficiency and sensitivity improvement on the CW-NMR methods. The spectral range is

excited with a single short duration hard RF pulse and the following FID is Fourier transformed

to reconstruct the spectrum as shown in Figure 2.13. This method of acquiring NMR spectra is

equivalent to a slow passage CW-NMR spectra as discussed in Section 2.3.2.2.

Static field inhomogeneity

Inhomogeneity of the B0 field and spin-spin interactions lead to spectral line broadening in both

CW-NMR and FT-NMR techniques. If we assume the field inhomogeneity is Lorentzian (2.41) it

can be related to the FID relaxation rate,

1

T *2

=1

T2+ γ ΓB0

(2.79)


(a) (b)

Figure 2.13: Free induction decay relationship to spectral width. (a) A basic FID experiment showing the envelopefor T2 decay in a homogeneous B0 field (dashed line) and the envelope for T ∗2 decay in an inhomogeneous B0 field(solid line). (b) The spectral width of a Fourier transform of an FID in a homogeneous B0 field (dashed line) and inan inhomogeneous B0 field (solid line).

where ΓB0is the HWHM of the Lorentzian distribution (Abragam, 1961). This relationship (2.79)

describes the importance of shimming the B0 field to achieve a narrow line width when performing

NMR spectroscopy (Figure 2.13b).

2.4.4 NMR relaxometry

The NMR signal is effected by spin-lattice and spin-spin relaxation effects (Section 2.2.2.1 and

Section 2.2.2.2). Here we present common methods for acquiring an appropriately weighted volume

signal and estimating the mean T1 and T2 relaxation constants. The techniques described here are

the basis for the relaxation mapping methods employed in Section 3.3.

2.4.4.1 Spin-lattice relaxometry

T1 relaxation is the exponential decay of the bulk-magnetisation towards thermal equilibrium (Sec-

tion 2.2.2.1). Spin-lattice relaxation rates can be measured using inversion recovery, saturation

recovery and progressive saturation techniques.

Inversion recovery

Vold et al. (1968) measured the spin-lattice relaxation of NMR spectra using an inversion preparation

pulse before a standard FID sequence. Inversion recovery applies a π excitation pulse to invert the

equilibrium magnetisation (Figure 2.14) which then recovers over a period, TIR, before a π2 pulse

excites the magnetisation into the transverse plane for acquisition of an FID. The experiment is

repeated for a range of inversion periods, TIR, and the relaxation rate, T1, is estimated by fitting

the measured decay curve to the expected signal from (2.21),

|mxy| (t) =∣∣∣M0

z

(1− 2 e−

TIRT1

)∣∣∣ e− (t−TIR)T∗2 . (2.80)


Figure 2.14: Inversion recovery sequence diagram. Spin-lattice relaxation rates can be estimated from a series of FIDmeasurement over a range of inversion recovery periods, TIR.

As the inverted magnetisation recovers toward equilibrium is crosses the origin, causing a signal null

with a related inversion time, T nullIR , which can also be used to estimate the relaxation rate,

T1 =T null

IR

ln 2. (2.81)

Additionally, this signal null can be used to suppress signal from a spin population with a known

relaxation rate.

Saturation recovery

Saturation recovery measurements are similar to inversion recovery but with a magnetisation prepa-

ration to saturate the spin-system. The magnetisation can be reliably nulled by a series of π pulses

(Markley et al., 1971) or more efficiently by rotating the magnetisation into the transverse plane

and dephasing with a gradient field (McDonald and Leigh, 1973). Similar to inversion recovery, the

experiment is repeated with a range of saturation recovery times, TSR, and the relaxation rate, T1,

is estimated by fitting the measured decay curve to the expected signal from (2.21),

|mxy| (t) = M0z

(1− e−

TSRT1

)e− (t−TSR)

T∗2 . (2.82)

Progressive saturation

Freeman and Hill (1971) measured T1 relaxation by placing the spin-system in a spin-lattice depen-

dant steady-state using a series of π2 pulses. When the excitation pulses are applied faster than the

longitudinal magnetisation can recover (Figure 2.15) the steady-state longitudinal magnetisation is

dependant on the relaxation rate and the repetition rate,

mz (TR-) = M0

z

(1− e−

TRT1

)(2.83)


Figure 2.15: Progressive saturation sequence diagram. Spin-lattice relaxation rates can be estimated from a series ofpartially saturated FID measurements over a range of repetition times, TR.

giving a measured steady-state transverse signal

|mxy| (t) = M0z

(1− e−

TRT1

)e− tT∗2 . (2.84)

Similar to inversion recovery, the experiment is repeated with a range of repetition times, TR, and the

relaxation rate, T1, is estimated by fitting the measured decay curve to the expected signal (2.84). A

series of excitation pulses will generate spin and stimulated echoes (Hahn, 1950), which in this case

are undesirable and can be suppressed by using a changing field gradient to dephase the transverse

magnetisation on each repetition.

2.4.4.2 Spin-spin relaxometry

T2 relaxation results in the decay of the transverse magnetisation and is often characterised using a

spin-echo (SE) or a multi-spin-echo (MSE) experiments.

Relaxation measurement

Hahn (1950) measured spin-spin relaxation rates by acquiring data from a series of SE experiments

with varied echo times to generate an exponentialy decaying curve. The measurements were then

matched with a decay curve (2.72). This method has limited efficiency due to a long repetition time,

TR ≥ 5T1, and has estimation error due to diffusion effects. Carr and Purcell (1954) demonstrated

that a T2 weighted echo-train is formed by multiple refocusing pulses (Figure 2.16). The MSE

method can measure a T2 decay curve in a single repetition and is less susceptible to diffusion effects

due to shorter echo times. Using coherent pulses and a π/2 phase shift between the excitation and

refocusing pulses compensates for cumulative error from imperfect refocusing pulses (Meiboom and

Gill, 1958).

2.5. IMAGING CONCEPTS 51

Figure 2.16: Carr-Purcell-Meiboom-Gill (CPMG) spin-echo sequence diagram. The spin-echo train is established withan excitation pulse, αx = π

2, and a series of coherent refocusing pulses, βy = π.

2.5 Imaging Concepts

This brief history of early imaging techniques is summarised from a general history of MR imag-

ing (Edelman, 2014). The first MR images were acquired under a gradient field and were formed by

back-projecting projections of proton density (Lauterbur et al., 1973). The first human MR image

was of a finger (Mansfield and Maudsley, 1977) and was taken using a line scanning method (Mans-

field et al., 1976) where a line volume is excited then frequency encoded during readout. The first

body MR image was acquired using the field focused nuclear magnetic resonance (FONAR) (Dama-

dian et al., 1977) method which measured an image by moving a small signal volume, created with

a small homogeneous field region, through space. Two dimensional imaging in reasonable time was

possible with the Fourier transform based spin warp imaging (Edelstein et al., 1980) which forms

the basis of the majority of modern MR imaging methods. The efficiency of imaging experiments

was greatly improved by the development of echo-planar imaging (EPI) (Mansfield, 1977), fast low

angle shot imaging (FLASH) (Haase et al., 1986) and rapid acquisition with relaxation enhancement

(RARE) (Hennig et al., 1986).

This section describes image formation from NMR signals with spatial frequency dependence due to

the application of magnetic field gradients. We introduce slice selective pulses, the k-space formalism,

phase encoding and frequency encoding. A spin-echo imaging sequence and a gradient echo imaging

sequence are used as examples to discuss fundamental Cartesian k-space sampling strategies.

2.5.1 Slice selection

Imaging an entire three dimensional volume is time consuming and often a reduced set of two

dimensional slices provide adequate information for clinical applications. A slice of space can be

selectively excited by applying a gradient during RF excitation. The bandwidth and frequency

response of the RF excitation are related to the thickness and spatial profile of the excited slice.

We explore slice selection using the Bloch equations (2.34) and apply an RF excitation, ωe1,x =

−γBe1 (t), a slice selection gradient in the z direction, ∆ = −γGssz, and assume a short pulse

duration to minimise relaxation effects,

d

dtm′ (t) = Ωss (t)m′ (t) (2.85)


where

Ωss(t) =

0 γGssz 0

−γGssz 0 γBe1 (t)

0 −γBe1 (t) 0

. (2.85a)

Assuming a small tip angle m′z = M0z and d

dtm′z = 0 simplifies (2.85) to,

dm′xdt

= γGsszm′y (2.86a)

dm′ydt

= −γGsszm′x + γBe

1 (t)M0z (2.86b)

dm′zdt

= 0. (2.86c)

The transverse components of (2.86) are expressed as a dynamic complex signal

dm′xy (z, t)

dt= −iγGsszm

′xy (z, t) + iγBe

1 (t)M0z (z) , (2.87)

which can be solved using a Laplace transform (Liang et al., 2000) assuming there is no initial trans-

verse magnetisation and the excitation envelope, Be1 (t), is symetric about half the pulse duration

t = τp/2, yielding an expression for the slice excitation profile in the z direction

im′xy (z, τp)

γM0z (z)

eiγGsszτp/2 =

∫ τp/2

−τp/2Be

1

(t+

τp2

)ei2πksst dt (2.88)

where the slice selection frequency is

kss =γ

2πGssz. (2.88a)

2.5.1.1 Slice profile

From the small tip angle approximation result we can state the slice profile (2.88) is proportional to

the inverse Fourier transform of the excitation envelope,∣∣m′xy (z, τp)∣∣ ∝ F -1

Be

1

(t+

τp2

)(2.89)

when relaxation effects are negligible. This relationship has been shown in experiment to be a

reasonable approximation under larger tip angles (Pauly et al., 1989b,c). To design a RF pulse for

a target slice profile we invert the relationship (2.89),

Be1

(t+

τp2

)∝ F

∣∣m′xy (z (kss) , τp)∣∣ . (2.90)

We define an ideal slice profile as a rectangular function,

∣∣m′xy (z (kss) , τp)∣∣ = Π (z,∆z) =

1 |z| > ∆z/2

0 otherwise(2.91)

where ∆z is the slice width in the z direction. Using relationship (2.90) and the rectangular slice

profile (2.91) the slice selective excitation envelope is,

Be1 (t) ∝ sinc

[π∆kss

(t− τp

2

)]. (2.92)


Figure 2.17: The (top right) slice profile is related to shape and duration of the (top left) excitation pulse shape. Thestrength of the gradient field, Gss, affects the spatial width of the slice, ∆z.

The RF pulse envelope has a finite duration, and is commonly truncated to a fixed number of lobes,

nl, which gives a relationship between pulse duration, gradient strength and slice thickness,

τp =2nl

∆kss=

4πnl

γGss∆z, (2.93)

shown in Figure 2.17.

2.5.1.2 Slice rephasing

The transverse magnetisation excited by a slice selective excitation (2.88) suffers from a phase

distortion,

θss (τp, z) =γGsszτp

2. (2.94)

As the spatial phase variation is linear it is possible to rephase the slice using a rephasing gradi-

ent. The area of the rephasing gradient lobe must equal half the slice selection lobe as shown in

Figure 2.18.

2.5.2 Spatial encoding

Spatial encoding is achieved with magnetic gradient fields which vary over space, generally an MRI

scanner will have hardware to generate linear gradients in three orthogonal direction. The gradient


Figure 2.18: A slice selective excitation causes a phase distortion across the slice which can be corrected with arefocusing gradient lobe.

fields are commonly generated by water cooled electro-magnets at a field strength much smaller in

magnitude than the main magnetic field. The gradient fields introduce an audio frequency fluctuation

which is superimposed with the radio-frequency precession due to the B0 field.

2.5.2.1 Spatial frequency k-space

Here we introduce k-space (Brown et al., 1982; Ljunggren, 1983; Twieg, 1983) and for simplicity

consider a one dimensional case. The signal equation (2.63) directly following a π/2 pulse can be

re-written,

s (t) =

∫object

ρ (r) e−i∫ t0

∆(r,τ)dτ dr, (2.95)

where the effictive spin density is,

ρ (r) = b∗r, xy (r)1

4ρ0 (r)

γ22

KTsB0. (2.95a)

When considering a one dimensional case in the z direction then signal (2.95) is,

s (t) =

∫object

ρ (z) e−i∫ t0

∆(z,τ)dτ dz (2.96)

where the projection of spin density in the z direction is,

ρ (z) =

∫ρ (r) dx dy, (2.96a)

and assuming no B0 field inhomogeneity, the off-resonance (2.33) from a linear gradient is,

∆ (z, t) = γGz (t) z (2.96b)

where Gz is the rate of change of the z gradient and is often expressed in units mT/m. The linear

spatial dependence of the gradient allows us to express the signal equation in k-space

s (kz) =

∫object

ρ (z) e−i2πkzz dz (2.97)


where spatial frequency is defined as

kz =γ

2π

∫ t

0

Gz (τ) dτ. (2.97a)

Signal equation (2.97) makes it clear that our observed NMR signal under a linear gradient is a

natural Fourier transform of the pseudo spin density,

s (kz) = F ρ (z) . (2.98)

Furthermore, it is possible to reconstruct the pseudo spin density from the observed signal using an

inverse Fourier transform,

ρ (z) =

∫object

s (kz) ei2πkzz dk (2.99)

= F -1 s (kz) . (2.100)

General k-space

The one dimensional result can be extended to two and three dimensional cases (Liang et al., 2000,

Chapter 5),

s (k) =

∫object

ρ (r) e−i2πk·r dr (2.101)

where spatial frequency is defined as

k =γ

2π

∫ t

0

G (τ) dτ (2.101a)

and

k = [kx ky kz]T

(2.101b)

G = [GxGyGz]T. (2.101c)

A proton density volume is reconstructed by inverse Fourier transforming an adequately sampled

k-space,

ρ (r) =

∫object

s (k) ei2πk·r dk

= F -1 s (k) . (2.102)

2.5.2.2 Phase encoding

A gradient is applied for a duration Tpe to phase encode the NMR signal (2.101),

k =γ

2π

∫ Tpe

0

Gpe (τ) dτ. (2.103)

If the gradients are constant during the phase encode duration then,

k =γ

2πGpeTpe. (2.104)

Phase encoding moves the signal to a new k-space location as shown in Figure 2.19 and Figure 2.20.

Only the net area under the phase encoding gradients is important which gives some flexibility when

designing gradient envelopes.


2.5.2.3 Frequency encoding

A gradient is applied during acquisition to frequency encode the NMR signal (2.101),

k (t) =γ

2π

∫ t

0

Gfe (τ) dτ. (2.105)

If the gradients are constant during signal acquisition then,

k (t) =γ

2πGfe t. (2.106)

Frequency encoding moves the signal through a k-space trajectory during the acquisition period as

shown in Figure 2.19 and Figure 2.20. More complex gradient envelopes can be used to sample

k-space with non-Cartesian trajectories.

2.5.

IMAGIN

GCONCEPTS

57

Figure 2.19: Multiple spin-echo based k-space trajectory diagram. A slice selective pulse is followed by (1) a phase encoding gradient during the slice refocusing lobe. (2)The refocusing pulse causes a phase inversion in the complex plane. (3) A phase encoding gradient shift the starting point before (4) a frequency encoding gradient isapplied during the acquisition of a spin-echo and (5) a phase encoding gradient shifts the k-space position in preparation for the next echo formation and acquisition.

58

CHAPTER

2.THEORY

Figure 2.20: A gradient-echo based k-space trajectory diagram. A slice selective pulse is followed by (1) a phase encoding gradient during the slice refocusing lobe and (2)a frequency encoding gradient during the acquisition of a spin-echo. The phase encoding gradient is changed to acquire a carestian k-space lines to fill the two dimensionk-space.

2.A. BLOCH EQUATIONS IN THE ROTATING FRAME 59

Appendix

2.A Bloch equations in the rotating frame

We derive an expression for the dynamics of the bulk magnetisation as observed in the rotating

frame. The bulk magnetisation in the laboratory frame is previously defined as

m = mxi+myj +mzk (2.A.1)

and in the rotating frame as

m′ = m′xi′ +m′yj

′ +m′zk′. (2.A.2)

The dynamics of a general rotating frame of reference basis, i′, j′,k′, is

di′

dt= ω × i′ (2.A.3a)

dj′

dt= ω × j′ (2.A.3b)

dk′

dt= ω × k′. (2.A.3c)

If we consider a transverse rotation, ω = ωrotk, then (2.A.3) can be expressed as a transform from

the the laboratory basis to the rotating basis,i′j′k′

=

cos(ωrott) − sin(ωrott) 0

sin(ωrott) cos(ωrott) 0

0 0 1

ijk

(2.A.4)

= Rz,ccw (ωrot, t)

ijk

, (2.A.5)

where Rz, ccw is a counter-clockwise rotation in the transverse plane.

The time derivative of the bulk magnetisation observed from the laboratory frame is,

dm

dt=dmx

dti+

dmy

dtj +

dmz

dtk. (2.A.6)

The time derivative of the rotating bulk magnetisation as observed in the rotating frame is,(dm′

dt

)R

=d

dt

(m′x (t) i′ +m′y (t) j′ +m′z (t)k′

)=dm′xdti′ +

dm′ydtj′ +

dm′zdtk′ (2.A.7)

and observed from the laboratory frame,

dm′

dt=

d

dt

(m′x (t) i′ (t) +m′y (t) j′ (t) +m′z (t)k′ (t)

)=dm′xdti′ +

dm′ydtj′ +

dm′zdtk′ +m′x

di′

dt+m′y

dj′

dt+m′z

dk′

dt. (2.A.8)


Substitution of (2.A.3) and (2.A.7) into (2.A.8) gives

dm′

dt=

(dm′

dt

)R

+m′xω × i′ +m′yω × j

′ +m′zω × k′

=

(dm′

dt

)R

+ ω ×(m′xi

′ +m′yj′ +m′zk

′)

=

(dm′

dt

)R

+ ω ×m′. (2.A.9)

Considering these time derivatives we can state,

dm

dt=dm′

dt6=(dm′

dt

)R

(2.A.10)

Substitution of (2.27) into (2.A.9) gives us an expression for the dynamics of the bulk magnetisation

as observed in the rotating frame,(dm′

dt

)R

=dm′

dt− ω ×m′

= γm′ × b′ − ω ×m′ −m′xi

′ +m′yj′

T2− (m′z −M0

z )

T1k′

= γ

(m′ × b′ +m′ × ω

γ

)−m′xi

′ +m′yj′

T2− (m′z −M0

z )

T1k′

= γm′ ×(b′ +

ω

γ

)−m′xi

′ +m′yj′

T2− (m′z −M0

z )

T1k′

= γm′ × b′eff −m′xi

′ +m′yj′

T2− (m′z −M0

z )

T1k′ (2.A.11)

where

b′eff = b′ +ω

γ

= b′ +ωrot

γk′. (2.A.11a)

2.B Bloch equations steady-state solution to constant exci-

tation

We derive the steady-state spin-system response to a constant amplitude and constant frequency

RF excitation,

ωe1,x (t) = ω1,x (2.B.1a)

ωe1,y (t) = 0 (2.B.1b)

which is applied for a long duration. When the magnetisation reaches a steady-state, the rate of

change of the bulk magnetisation is,

d

dtm′ (t) = 0. (2.B.2)

2.B. BLOCH EQUATIONS STEADY-STATE SOLUTION TO CONSTANT EXCITATION 61

Applying these conditions (2.B.1) and (2.B.2) to the Bloch equations in the rotating frame (2.34),0

0

0

=

−R2 −∆ 0

∆ −R2 −ω1,x

0 ω1,x −R1

m′xm′ym′z

+R1

0

0

M0z

(2.B.3)

or,

0 = −R2m′x −∆m′y (2.B.4a)

0 = ∆m′x −R2m′y − ω1,xm

′z (2.B.4b)

0 = ω1,xm′y −R1m

′z +R1M

0z . (2.B.4c)

Arrange (2.B.4a) and (2.B.4c) to be functions of m′y,

m′x = − ∆

R2m′y (2.B.5a)

m′z =ω1,x

R1m′y +M0

z (2.B.5c)

Substitute (2.B.5a) and (2.B.5c) into (2.B.4b)

0 = −∆2

R2m′y −R2m

′y −

ω21,x

R1m′y − ω1,xM

0z (2.B.6)

m′y = −ω1,xM0z

(ω2

1,x

R1+

∆2

R2+R2

)−1

= −ω1,xM0z

R1R2

R1R22 +R1∆2 +R2ω2

1,x

=−ω1,xT2

1 + (T2∆)2

+ T1T2ω21,x

M0z (2.B.7)

gives an expression for the rotating y component as a function of excitation amplitude, off-resonance,

relaxation constants and the magnetisation at thermal equalibrium. Substitution of (2.B.7) into

(2.B.5a) yeilds the rotating x component, giving us transverse components

m′x =ω1,x∆T 2

2

1 + (T2∆)2

+ T1T2ω21,x

M0z (2.B.8a)

m′y =−ω1,xT2

1 + (T2∆)2

+ T1T2ω21,x

M0z (2.B.8b)

m′z =1 + (∆T2)

2

1 + (T2∆)2

+ T1T2ω21,x

M0z (2.B.8c)


Chapter 3

Methods


3.2 Proton Density Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3 Relaxation Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3.1 RARE-VTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3.2 MSME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.4 Magnetic Field Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.4.1 B0 mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.4.2 B1 mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.1 Introduction

This chapter builds on the imaging concepts introduced in Chapter 2 and describes the imaging

methods used to support the developement of novel methods detailed in Chapters 4-6. The pre-

sented imaging methods are used to acquire reference data or to measure parameters for use in

reconstruction models. A fast low shot imaging (FLASH) sequence (Haase et al., 1986) is used to

acquire proton density maps. Relaxation mapping is performed using rapid acquisition with relax-

ation enhancement and variable repetition time (RARE-VTR) and multi-slice multi-echo (MSME)

methods. B0 field maps are measured with a multiple gradient-echo method (Kanayamay et al.,

1996) using the Bruker FieldMap sequence. B1 field maps are acquired using a standard spin-echo

sequence, using a double angle method (Wang et al., 2005). The imaging parameters specific to each

experiment can be found in the Methods section of each chapter.

3.2 Proton Density Imaging

Fast low angle shot imaging (FLASH) (Haase et al., 1986) is an efficient imaging protocol. A basic

FLASH sequence diagram is shown in Figure 3.1. The method uses a low flip angle slice-selective

excitation and measures a phase and frequency encoded gradient echo. These properties allow for a

63

64 CHAPTER 3. METHODS

Figure 3.1: Fast low angle shot imaging (FLASH) sequence diagram.

short repetition time, TR, and the efficiency can be further improved by the use of spoiler or rewinder

gradients. The dynamic steady-state gradient echo intensity can be expressed as (Liang et al., 2000)

|mxy (TE)| = M0z e−TE/T

∗2 sinα

1− e−TR/T1

1− e−TR/T1 cosα. (3.1)

Image contrast can be optimised for a given application by selection of flip angle, α, echo time,

TE and repetition time, TR. A proton density image is measured using FLASH in Chapter 5 as a

reference image to compare the performance to an image reconstructed the steady-state response

under Rabi modulated excitation. The proton density image is acquired by using a short echo time,

TE, to reduce the T ∗2 weighting and a small flip angle, α, to reduce the T1 weighting.

3.3 Relaxation Mapping

Relaxation measurements were acquired in all experiments, with both relaxation constants acquired

using spin-echo based methods. In the Rabi modulated spectroscopy (Chapter 4) and Rabi mod-

ulated imaging experiments (Chapter 5) the measured relaxation rates were used to calculate the

expected off-resonance response and to construct the forward models used to reconstruct spectra and

radial projections of proton density. In Chapter 6, measured relaxation rates were used to calculate

a ring-lock excitation envelope to control the bulk magnetisation on a steady-state manifold, and in

a second experiment to assess the accuracy of a ring-lock relaxometry technique.

The signal magnitude in a multiple spin-echo experiment is weighted by T1 and T2 which allows the

estimation of relaxation constants from measurements with appropriately selected repetition times,

TR, and effective echo times, kTE. A multiple echo spin-echo experiment with an initial excitation

angle, α, followed by Nse refocusing pulses with a fixed refocusing angle, π, has a signal intensity at

3.3. RELAXATION MAPPING 65

Figure 3.2: RARE sequence diagram. Multiple spin-echoes are acquired to fill multiple lines of k-space in a singlerepetition time. RARE has a higher efficiency in comparison to a standard spin-echo sequence which acquires a singleline of k-space per repetition.

the kth echo (Mills et al., 1986),

|mxy (kTE)| = M0z sinα

[1 + (−1)

Nse+1e−TR/T1 − ase

] e−kTE/T2

bse, (3.2)

where

ase =

Nse∑n=1

(−1)Nse+n

2e−(TR−(2n−1)τ)/T1 (3.2a)

bse = 1 + (−1)Nse+1

e−TR/T1 cosα. (3.2b)

If the excitation angle is α = π/2, the echo intensity becomes dependant on the sequence parameters,

TR and TE ,

|mxy (kTE)| = M0z

[1 + (−1)

Nse+1e−TR/T1 − ase

]e−kTE/T2 , (3.3)

where,

ase =

Nse∑n=1

(−1)Nse+n

2e−(TR−(2n−1)τ)/T1 (3.3a)

This expression (3.3) is used to select sequence parameters and describe the signals acquired by the

RARE-VTR and MSME methods.

3.3.1 RARE-VTR

Rapid acquisition with relaxation enhancement (RARE) (Hennig et al., 1986) improves the efficiency

of a basic spin-echo sequence by acquiring multiple spin-echoes to fill multiple lines of k-space in a


Figure 3.3: RARE sequence diagram for T2 relaxometry. To estimate T2, multiple k-spaces are acquired over a rangeof effective echo times. This diagram shows a RARE factor = 2, where during each repetition two k-space lines areacquired for each effective echo time.

3.3. RELAXATION MAPPING 67

single repetition time (Figure 3.2). The number of k-space lines acquired per repetition, which is

also the scan acceleration factor, is referred to as the RARE factor. This acquisition strategy gives

a non-uniform T2 weighting across k-space which is related to the point spread function (PSF) in

the image domain. Lower RARE factors should be used measuring samples with fast T2 decay to

shorten the effective echo time and avoid image artifacts.

Spin-lattice relaxation, T1, maps can be measured by repeating a RARE experiment with a variable

repetition time (RARE-VTR). The spin-spin weighting of the echoes is minimised by using a short

echo-time, TE, reducing equation (3.3) to,

|mxy (TR)| = M0z

(1− e−

TRT1

), (3.4)

which is a similar result to progressive saturation in Section 2.4.4.1.

Spin-spin relaxation, T2, maps can be measured from a RARE-VTR experiment by using the spin-

echoes to fill multiple k-spaces at different effective echo times rather than multiple lines of the same

k-space (Figure 3.3). Details of the T2 fitting are the same as for MSME discussed in Section 3.3.2.

3.3.2 MSME

The multi-slice multi-echo (MSME) method can be used for efficient T2 mapping. Spin-spin relax-

ation, T2, maps can be measured with a multiple spin-echo experiment. The spin-lattice weighting

of the echoes is minimised by using a long repetition time, TR, reducing equation (3.3) to,

|mxy (kTE)| = M0z e− kTET2 (3.5)

which is similar to 2.72 and can be used to estimate T2 as discussed in Section 2.4.4.2.

The necessity for a long repetition time, TR, causes a long experimental time, where the majority of

time is waiting for the magnetisation to return to thermal equilibrium. To improve efficiency, multiple

slices can be interleaved (Figure 3.4), acquiring data from a single slice while the magnetisation of

other slices return to equilibrium.

68

CHAPTER

3.METHODS

Figure 3.4: Multi-slice multi-echo (MSME) sequence diagram for efficient multi-slice T2 relaxometry. A long repetition is required to reduce T1 weighting of the spin-echoeswhich causes a long experiment time in a single slice experiment. MSME reduces this inefficiency but acquiring data from other spatial locations as previously excitedslices return to equilibrium.

3.4. MAGNETIC FIELD MAPPING 69

Figure 3.5: B0 mapping sequence diagram.

3.4 Magnetic Field Mapping

Field maps were measured in the novel experiments (Chapters 4-6) to inform the magnetisation

prediction models. In the Rabi modulated spectroscopy experiments (Chapter 4) a B0 and a B1

map were used to predict the off-resonance harmonic response, and a B1 map was incorporated in

the forward model used to reconstruct NMR spectra. In the Rabi modulated imaging experiments

(Chapter 5) a B0 and a B1 map were used to predict the off-resonance harmonic response, and a B0

map was incorporated in the forward model used to reconstruct projections of proton density. In

Chapter 6 both B0 and B1 maps were acquired for two ring-lock experiments; in the first experiment

to predict the steady-state trajectory, in the second to inform a model used to estimate relaxation

constants.

3.4.1 B0 mapping

The B0 field near the isocentre of a MR spectrometer is designed to provide a homogeneous field

over a specified spatial volume. When a phantom or patient is placed in the scanner, the field is

distorted. Field mapping methods allow the measurement of the spatial variation of the B0 field

and allow correction using the system’s shim coils or during reconstruction.

Early field mapping measurements (Maudsley et al., 1979, 1984) were taken with chemical shift

imaging which was time consuming, but much faster than manual movement and measurement with

a small NMR probe. A spin-echo based method (Sekihara et al., 1985) derived a field map from the

phase evolution between two spin-echo experiments with different echo times. The efficiency of this

method was improved by Prammer et al. (1988) who used the first and third echoes in a Carr-Purcell

echo train. A gradient recalled echo method (Schneider and Glover, 1991) calculates B0 strength

from the phase difference between two gradient echo experiments with different echo times. The

efficiency was improved by Kanayamay et al. (1996) who proposed acquiring both gradient echoes

in a single repetition.

In our experiments the B0 mapping sequence (Kanayamay et al., 1996) is used to excite a slice and

generate three gradient echo images (Figure 3.5). The phase difference between the first and third


echo images,

δφ (r) = φ3 (r)− φ1 (r)

= ∆ (r) δT (3.6)

is proportional to the off-resonance, ∆ (r), and the duration between the first and third echo, δT .

The field map is the expressed as a deviation from the Larmor frequency and is related to the phase

difference (3.6) by

δω0(r) = ∆ (r) =

δφ (r)

δT. (3.7)

The echo spacing δT can be selected to ensure any large spin populations with a chemical shift, such

as fat, undergo a 2π phase shift and don’t corrupt the B0 map.

3.4.2 B1 mapping

Inhomogeneity in the B1 field is caused by coil geometry and dielectric effects resulting in spatially

non-uniform flip angles and image intensity artifacts. Early mapping of the excitation field involved

moving a small pickup coil and sample to different spatial locations and making multiple measure-

ments. These methods were replaced by imaging methods which are more efficient and reflect the

actual experimental conditions.

B1 field maps can be calculated from the intensity of a single gradient-echo or spin-echo experiment

(Hornak et al., 1988) but suffer from intensity changes due coil sensitivity, proton density and

relaxation effects. Theses confounding factors are cancelled out by calculating the ratio of two

intensity images acquired from two experiments (Stollberger et al., 1988) where the excitation angle

is often doubled in the second experiment. The sensitivity of these double angle methods (DAM) can

be improved when using a spin-echo sequence by setting the refocusing excitation angle to twice the

excitation angle (Insko and Bolinger, 1993). Efficiency improvements to double angle methods for

application in-vivo are the use of compensation pulses (Stollberger and Wach, 1996) and saturation

pulses (Cunningham et al., 2006). Alternative intensity based methods are actual flip angle imaging

(Yarnykh, 2007) and a stimulated echo approach (Akoka et al., 1993). An recent phase based method

makes use of the Bloch-Seigert shift (Sacolick et al., 2010).

In our experiments we use the spin-echo double angle method as described by Stollberger et al.

(1988) and shown in Figure 3.6. The signal intensity for a spin-echo with arbitrary excitation angle,

α, and refocusing angle, β, is (Wang et al., 2005)

|mxy (r)| = M0z e−TE/T2S (r)

sinα[1− e−TR/T1 cosβ − (1− cosβ) e−TR/T1eTE/2T1

]1− e−TR/T1 cosα cosβ

, (3.8)

where S (r) is the coil sensitivity. If we ensure, TR T1, TE T1, and, β = 2α, then (3.8) simplifies

to,

|mxy (r)| = C (r)S (r) sin3 α (r) , (3.9)

where C (r) is the signal weighting due to relaxation constants and proton density. The intensity

ratio of two spin-echo experiments, with flip angles, α1, α2, can be expressed as

λ (r) =|mxy,2 (r)||mxy,1 (r)|

=sin3 α2 (r)

sin3 α1 (r), (3.10)

3.4. MAGNETIC FIELD MAPPING 71

Figure 3.6: B1 mapping sequence diagram. Two spin-echo experiments are conducted where the refocusing angle isset to twice the excitation angle, that is β = 2α. The excitation and refocusing angles are doubled in the secondexperiment, α2 = 2α1 and β2 = 2β1. The B1 map is calculated from the ratio of the two reconstructed intensityimages using equation (3.12).

which removes the dependance on tissue and coil parameters. If the excitation angle in the second

experiment is set as α2 = 2α1, then the flip angle can be calculated from the intensity ratio (3.10),

α1 (r) = arccos

(λ (r)

8

)1/3

. (3.11)

The B1 map can be calculated from (3.11) and (2.52) assuming a rectangular pulse of duration τ1,

B1 (r) =α1 (r)

γτ1=

1

γτ1arccos

(λ (r)

8

)1/3

. (3.12)

It is often more useful to express the excitation field inhomogeneity as a ratio of measured and

calibrated field strengths,

B1 (r)

B1=α1 (r)

α1, (3.13)

as this removes the excitation field strength dependence.

This chapter has outlined the standard MRI sequences employed in this thesis to support the devel-

opment of novel methods. Chapters 4-6 now proceed to detail these contributions.


Chapter 4

Rabi continuous wave spectroscopy


4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2.1 Rabi modulated excitation . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2.2 Observed NMR signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2.3 Spectroscopy as an inverse problem . . . . . . . . . . . . . . . . . . . . . . 76

4.3 Methods and Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3.1 The response of off-resonance spins . . . . . . . . . . . . . . . . . . . . . . 79

4.3.2 Rabi modulated spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.4.1 The response of off-resonance spins . . . . . . . . . . . . . . . . . . . . . . 83

4.4.2 Rabi modulated spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.1 Introduction

This chapter explores the magnetic resonance response to an amplitude modulated RF excitation

envelope, which we refer to as Rabi modulated excitation. The experiments presented in this chapter

have been published in (Korte et al., 2014, 2017a). They were conducted to demonstrate that

the steady-state spin system response to Rabi modulated CW excitation contains off-resonance

information, in this case chemical shift information. We extend the original experiment (Layton

et al., 2014), which was limited to on-resonance excitation, to investigate the response of the spin

system to off-resonance Rabi modulated CW excitation. We propose a CW method by which to

perform NMR spectroscopy, posed as an inverse problem. Chemical shift information is encoded in

a series of Rabi modulated CW excitations. A forward model is constructed from a periodic solution

of the Bloch equation, and is used to reconstruct a simple NMR spectrum of ethanol. An algorithm

based on the A-optimality criteria (Chaloner and Verdinelli, 1995) is used to select a theoretically

optimal set of excitation parameter pairs.

73

74 CHAPTER 4. RABI CONTINUOUS WAVE SPECTROSCOPY

Historically, NMR spectra were obtained under continuous wave RF excitation by sweeping either

the strength of the main magnetic field or the carrier frequency of the excitation magnetic field, with

the spectrum generated by plotting the magnitude of the resulting NMR signal against the sweep

range (Bloch et al., 1946; Purcell et al., 1946). These methods were overshadowed by the more

efficient technique in which a sample is excited by a powerful, short duration, RF pulse (Ernst and

Anderson, 1966). The spectrum is obtained directly from a Fourier transform of the free induction

decay signal which is a broadband combination of all precessing isochromats.

A resurgence of interest in CW NMR spectroscopy methods (Blumich, 1984; Dadok and Sprecher,

1974) has influenced the development of new magnetic resonance imaging techniques capable of

imaging samples with ultra-fast spin-spin relaxation (Idiyatullin et al., 2006, 2012). The optimality

of pulse excitation sequences has also been challenged (Tahayori et al., 2008, 2009) with the sug-

gestion that CW excitation may improve signal intensity for lower energy excitation. Recent work

(Layton et al., 2014), inspired by quantum optics (Cappeller and Muller, 1985), has experimentally

demonstrated that a spin system excited by a Rabi modulated CW achieves substantial periodic

steady-state magnetisation. The frequency components of this steady-state magnetisation are re-

stricted to harmonics of the excitation modulation frequency (Tahayori et al., 2009) and a maximum

harmonic magnitude is achieved when a secondary resonance condition is met (Layton et al., 2014).

The Rabi resonance condition has also been investigated in CW electron paramagnetic resonance

(EPR) (Saiko et al., 2015).

We anticipate that the results we present here will find application in the acquisition of MR informa-

tion from ultra-fast relaxation samples. The proposed excitation method can maintain an observ-

able steady-state magnetisation for such samples and provides an alternative to existing ultra-fast

relaxation methods such as Sweep Imaging with Fourier Transformation (Idiyatullin et al., 2006),

Ultra-short echo time imaging (Bergin et al., 1991) and Zero-echo time imaging (Weiger et al., 2011).

Such methods have beneficial clinical applications such as the assessment of bone and connective

tissue (Robson and Bydder, 2006), measurement of sodium concentration in brain tissue (Ouwerkerk

et al., 2003) and the detection of iron oxide nano-particles delivered to tumours (Wang et al., 2014).

4.2 Theory

4.2.1 Rabi modulated excitation

As introduced in Chapter 2, Rabi modulated excitation is an amplitude modulated RF field with

the previously defined envelope function,

ωe1 (t) = ω1 (1 + α cosωmt) , (2.48)

where ω1 = −γB1 is the average of the excitation envelope, B1 is the average excitation field

strength, α is the modulation level and ωm is the modulation frequency. It is known that under this

excitation the spin system achieves a significant periodic steady-state magnetisation and that the

magnitude of this steady-state is maximised when the Rabi resonance condition, ωm = ω1, is met

(Layton et al., 2014).

4.2. THEORY 75

4.2.2 Observed NMR signal

The observed noise-free NMR signal under Rabi modulated excitation is,

mobs (t, α, ωm, δrf, T1, T2) = . . . (4.1)∫∫m (t, α, ωm, ω1, δω0

, δrf, T1, T2) p (ω1) p (δω0) dω1 dδω0

where δω0is any deviation from the Larmor frequency, ω0, due to main field inhomogeneities,

chemical shift effects or applied gradients. The p (δω0) distribution represents deviations from the

Larmor frequency, p(ω1) represents inhomogeneities in the strength of the excitation field and δrf is

an offset of the excitation carrier frequency from the Larmor frequency.

The signal equation can be approximated by numerical integration over a regular grid of J excitation

amplitudes and K off-resonances

mobs (t, α, ωm, δrf, T1, T2) ≈ . . . (4.2)

J∑j=1

K∑k=1

m(t, α, ωm, ω

(j)1 , δ(k)

ω0, δrf, T1, T2

)p(ω

(j)1

)p(δ(k)ω0

).

The steady-state magnetisation can be described by a complex Fourier series, restricted to harmonics

of the modulation frequency, ωm,

m =

∞∑l=−∞

c(l)eilωmt, (4.3)

m =

∞∑l=−∞

ilωmc(l)eilωmt, (4.4)

where c(l) =[c(l)x c

(l)y c

(l)z

]Tare the Fourier coefficients. In matrix notation, the relationship

becomes

m = eTC (4.5)

where

e =[. . . e−2iωmt e−iωmt 1 eiωmt e2iωmt . . .

]T(4.5a)

C =[. . . c(−2) c(−1) c(0) c(1) c(2) . . .

]T. (4.5b)

The bulk magnetisation in the frequency domain, C, is predicted by a harmonic balance solution of

the Bloch equations (2.47) as described in Section 2.3.1.2. The observed bulk magnetisation (4.2) is

transformed into the frequency domain by substitution of (4.5) and (2.47) into (4.2)

Cobs (α, ωm, δrf, T1, T2) ≈ . . . (4.6)

J∑j=1

K∑k=1

C(α, ωm, ω

(j)1 , δ(k)

ω0, δrf, T1, T2

)p(ω

(j)1

)p(δ(k)ω0

)

=

K∑k=1

S(α, ωm, δ

(k)ω0, δrf, T1, T2

)p(δ(k)ω0

)where

S (α, ωm, δω0 , δrf, T1, T2) =

J∑j=1

C(α, ωm, ω

(j)1 , δω0 , δrf, T1, T2

)p(ω

(j)1

). (4.6a)


4.2.3 Spectroscopy as an inverse problem

It is possible to reconstruct a NMR spectrum under Rabi modulated excitation, as chemical shift

effects are encoded in the observed NMR signal. A linear system can be constructed from (4.6) by

exciting the spin system with a set of N excitation parameter pairs

(α(1), ω(1)m ) . . . (α(N), ω

(N)m )

and subsequent measurement of the transverse magnetisation,

Hx = z (4.7)

where

H =

S(α(1), ω

(1)m , δ

(1)ω0

)· · · S

(α(1), ω

(1)m , δ

(K)ω0

)S(α(2), ω

(2)m , δ

(1)ω0

)· · · S

(α(2), ω

(2)m , δ

(K)ω0

)...

. . ....

S(α(N), ω

(N)m , δ

(1)ω0

)· · · S

(α(N), ω

(N)m , δ

(K)ω0

)

(4.7a)

x =[p(δ(1)ω0

). . . p

(δ(K)ω0

) ]T(4.7b)

z =[Cobs

(α(1), ω(1)

m

)Cobs

(α(2), ω(2)

m

). . . Cobs

(α(N), ω(N)

m

) ]T. (4.7c)

The discrete NMR spectrum, x, is reconstructed from the known forward model, H, and an observa-

tion vector, z, by solving the linear system (4.7). In this formulation (4.7), the relaxation constants,

T1 and T2, and the RF offset, δrf, are assumed to be known constants and are omitted for notational

simplicity.

4.3 Methods and Materials

Two experiments were performed on a 4.7T Bruker BioSpec small bore MRI scanner, using the

experimental protocol developed in (Layton et al., 2014) and summarised in Figure 4.1. Rabi mod-

ulated excitation (2.48) was applied for an initial duration of T=1000 ms to allow the magnetisation

to reach a steady-state, after which the free induction decay (FID) was measured. A steady-state

magnetisation waveform, mobs, was incrementally acquired by selecting the first FID point, and

repeating the process with an increase, ε, to the excitation duration on each repetition. Whilst this

method is inefficient, it allows the measurement of the steady-state magnetisation waveform, such

as Figure 4.2a, without hardware modification.

The incremental increase to the excitation duration, ε, is used to set the sampling frequency of

the measured steady-state magnetisation. The experimental sampling frequency is determined by

the modulation frequency of the Rabi excitation envelope and the desired harmonic content of

the measured steady-state response. The finite switching time between RF excitation and signal

acquisition is fast enough (∼5µs) that relaxation effects on the first points of the FID are considered

negligible.

In the first experiment, the steady-state harmonics, Cobs, are measured over a range of off-resonance,

δrf, and compared to the predicted steady-state harmonics by integrating the periodic solution of the

Bloch equations over the measured p(δω0) and p(ω1) field distributions. In the second experiment,

ethanol spectra, p(δω0), are reconstructed from a series of steady-state harmonics measured under a

range of Rabi modulated excitation envelopes.

4.3. METHODS AND MATERIALS 77

Figure 4.1: Experimental protocol for the incremental measurement of the steady-state transverse magnetisationwaveform. The spin system is perturbed by an excitation, ωe

1 (t), causing a transverse response mxy. By repeatedlyperturbing the spin system with an increasing excitation duration it is possible to construct an observed transversemagnetisation mobs

xy .


0 2 4 6 8 10−0.2

0

0.2

0.4

mobs

x

(norm

alised)

time(ms)

−1000 −500 0 500 1000−0.1

−0.05

0

0.05

0.1

frequency(Hz)

Re(

cobs

x

)

(a)0 2 4 6 8 10−0.2

0

0.2

0.4

mobs

x

(norm

alised)

time(ms)

−1000 −500 0 500 1000−0.1

−0.05

0

0.05

0.1

frequency(Hz)

Re(

cobs

x

)

(b)

Figure 4.2: Spin system response of the water phantom (T1 = 287 ms, T2 = 150 ms) under Rabi modulated excitation.(a) Measured (red circles) and theoretical (black line) periodic steady-state magnetisation waveform for excitationparameters α = 1, ω1 = ωm = 100 Hz and δrf = 180 Hz. (b) Measured (coloured circles) and theoretical (blackcrosses) harmonics of steady-state magnetisation. DC component (blue), first (green), second (purple), third (red)fourth (orange) and fifth (grey) harmonics.


Two spherical phantoms were used in this study, one filled with tap water and the other with ethanol.

Both phantoms were doped with Magnevist R© to reduce the experimental time.

4.3.1 The response of off-resonance spins

We investigated the effects of off-resonance on the observed steady-state magnetisation waveform by

offsetting the excitation carrier frequency from the Larmor frequency. A spherical phantom of Gd-

doped water (T1 = 287 ms, T2 = 150 ms) was selected for its narrow off-resonance distribution. Off-

resonance measurements were taken from δrf = −400 Hz to δrf = 400 Hz with 20 Hz increments. The

excitation envelope had parameters α = 1 and ω1 = ωm = 100 Hz. A discrete Fourier transform was

applied to the measured steady-state magnetisation waveform to extract its harmonic components.

A theoretical curve was generated from the periodic solution of the Bloch equations, numerically

integrated over the measured p (δω0) and p(ω1) distributions using (4.6).

The distribution of off-resonances, p(δω0), was measured using a B0 field mapping sequence (Kanaya-

may et al., 1996) described in Section 3.4.1. Data was acquired using a multiple gradient echo se-

quence with effective echo times (TE = 1.95, 6.22 ms), FA = 20, TR = 20 ms, FOV = 4.5 cm and

128×128×128 matrix. The distribution of excitation field strengths, p(ω1), was measured with a B1

mapping sequence (Stollberger et al., 1988) described in Section 3.4.2. Data was acquired with a

MSME sequence over 9 slices, 1 mm thickness, FOV = 4 cm, 64×64 matrix, TE = 12 ms, TR = 2.5

s. The MSME scans were taken for two excitation angle, α, and refocusing angle, β, configurations;

α/β = 45/90 in the first scan, α/β = 90/180 in the second scan.

The average T1 relaxation rate was calculated from the spatial average of T1 maps, which were

measured as described in Section 3.3.1. Data was acquired with a RARE-VTR sequence over 9

slices, 1 mm thickness, FOV = 4 cm, 32×32 matrix, TE = 12 ms, TR = 4500, 1500, 800, 500, 400,

300, 250, 215 ms, RARE factor 2. The average T2 relaxation rate was calculated from the spatial

average of T2 maps, which were measured as described in Section 3.3.2. Data was acquired with a

MSME sequence using 32 echoes with a 12 ms echo spacing, 9 slices, 1 mm thickness, FOV = 4 cm,

64×64 matrix, TR = 2500 ms.

4.3.2 Rabi modulated spectroscopy

A proof of concept experiment was undertaken to demonstrate the ability to encode chemical shift

information in a series of steady-state magnetisation trajectories. Our objective was to reconstruct

the spectrum of a spherical phantom of gadolinium doped ethanol (T1 = 120 ms, T2 = 43 ms) from

the response to a series of CW excitations. A reference spectrum from a single, 2048-point, FID was

acquired with a dwell time of 100µs. Spectra and variance are plotted on a ppm scale,

δ =δω0× 106

ω0+ δtms, (4.8)

where δtms = 1.25 ppm is an offset commonly observed when using Tetramethylsilane (TMS) as a

reference compound. No reference compound was added to the doped ethanol sample used in our

experiments.

4.3.2.1 Excitation parameter selection

Experimental parameters were selected from a set of candidate parameter pairs, Ω = (α, ωm),defined by a dense grid adhering to the power limitation of the excitation coil, as illustrated in


0 5 10 150

100

200

300

400

500

600

α

ωm(H

z)

Figure 4.3: Feasible (light grey) excitation parameter pairs Ω, set of gridded (dark grey) excitation parameter pairsΩgrid, set of optimal (black) excitation parameter pairs Ωopti for a 101 point reconstruction from -600 Hz to 600 Hzand a homogeneous B1 field.

Figure 4.3. Two reduced sets of 500 parameter pairs were then selected for experiments. The first

set, Ωgrid, was selected as a coarse grid of the candidate parameter pairs. The second set, Ωopti,

was constructed iteratively to minimise the theoretical variance of the reconstructed spectrum, as

detailed in Algorithm 1.

At each measurement selection step, k, in Algorithm 1, the covariance, Xk, for every candidate

parameter pair in the candidate set was calculated using,

Xk = Xk−1 −Xk−1HTk

(HkXk−1H

Tk +W

)−1

HkXk−1 , (4.9)

where W is the covariance of observation noise. The optimal parameter pair was selected from the

candidate set using the A-optimality criteria (Dette, 1997),

minimise(α,ωm)∈Ω

Tr (Xk) , (4.10)

where Tr denotes the matrix trace.

The selected excitation parameter pair was then added to Ωopti and used to update Xk. The

selection cycle was repeated until the required number of excitation parameter pairs were selected.

The variance of the optimal set is shown in Figure 4.4.

The initial covariance was set to X0 = σ2I where the variance, σ2 = 10,000, and I is an identity

matrix, providing an initial condition with negligible regularisation. The observation noise covariance

matrix was set to W = σ2obsI, with a standard deviation σobs = 7.2× 10−3, which is approximately

5% of the mean signal.

4.3.2.2 Measurement

Steady-state magnetisation waveforms were acquired for every CW excitation parameter pair in

Ωgrid and Ωopti. The excitation carrier frequency was offset from the system frequency by δrf = 400


Algorithm 1 Select the optimal set, Ωopti, of N excitation parameter pairs from a candidate set,Ω.

function OptimalSet(Ω, N)X0 ← σ2IΩopti ← . an empty setk ← 1 . measurement counterwhile k ≤ N do

(α′, ω′m)← argmin(α,ωm)∈Ω

Tr (Xk(α, ωm))

add (α′, ω′m) to Ωopti

remove (α′, ω′m) from ΩXk ←Xk(α′, ω′m) . update covariancek ← k + 1

end whilereturn Ωopti

end function

Hz to centre the spectrum and reduce the required bandwidth. The DC component and first five

harmonics were extracted from each steady-state magnetisation and recorded in a measurement

vector z.

Relaxation constants T1 and T2 were measured before each experiment, using a rapid acquisition

with relaxation enhancement with variable repetition time (RARE-VTR) scan as described in Sec-

tion 3.3.1, to ensure accuracy of the encoding matrix H. Scans were taken with RARE factor 2

over a single 1 mm slice, FOV = 4 cm, 64×64 matrix, TR = 200, 400, 800, 1500, 3000, 4500 ms, TE

= 11, 33, 55, 77, 99 ms. The measured relaxation maps were spatially averaged to produce a single

T1 and T2 constant value per phantom.

The distribution of excitation field strengths, p(ω1), was measured with a B1 mapping sequence

(Stollberger et al., 1988) described in Section 3.4.2. Data was acquired with a MSME sequence over

9 slices, 1 mm thickness, FOV = 4 cm, 64×64 matrix, TE = 12 ms, TR = 2.5 s. The MSME scans

were taken for two excitation angle, α, and refocusing angle, β, configurations; α/β = 45/90 in

the first scan, α/β = 90/180 in the second scan.

Reference spectra were measured with a single FID (FA = 90, 2048 complex points, spectral width

5.0 kHz). The Fourier transform of the FID was manually phase corrected and the real component

used as a reference spectrum. All measurements were duplicated to verify the reproducibility of

results.

4.3.2.3 Reconstruction

The spectrum, x, was reconstructed by least squares optimisation with a non-negative constraint:

minimisex∈[0,∞)

‖Hx− z‖2. (4.11)

The forward model, H, was constructed from the Fourier series approximation of the Bloch equation,

numerically integrated over the measured p(ω1) distribution using (4.7). The optimisation algorithm

was initialised with the spectrum, x, set as a vector of zeros.


1234560

1

2

3

x 10−4

δ (ppm)

Tr(X)

Figure 4.4: Trace of covariance matrix of reconstructed spectra (4.10) for the gridded (grey triangles) set of excitation

parameter pairs Ωgrid and the optimal (black circles) set of excitation parameter pairs Ωopti. Mean variance for the

optimal set is 1.4× 10−4 and 2.7× 10−4 for the gridded set.

4.3.2.4 Spectrum analysis

Let x be the area normalised version of the reconstructed spectra, x. Lorentzian peaks were fitted

to x, to assess the accuracy of peak locations and the ratio of peak areas. A Lorentzian peak is

defined as

L (δ, µ,Γ, a) =a

π

[Γ

(δ − µ)2 + Γ2

](4.12)

where δ is a discrete vector of off-resonance, µ is the peak centre, Γ is the half width at half maximum

(HWHM) and a is the peak area.

As ethanol has three distinct peaks, the objective function is

minimiseµ,Γ,a

∥∥∥∥∥x−3∑i=1

L (δ, µi,Γi, ai)

∥∥∥∥∥2

+ g

∥∥∥∥∥1−3∑i=1

ai

∥∥∥∥∥2

(4.13)

where

µ =

µ1

µ2

µ3

, Γ =

Γ1

Γ2

Γ3

, a =

a1

a2

a3

(4.13a-c)

and g is a weighting factor for the unity area term. The constrained minimisation function, fmincon,

from MATLAB R© was used to solve optimisation problems (4.11) and (4.13).

4.4. RESULTS 83

−100 −50 0 50 1000

0.01

0.02

0.03

0.04

0.05

off-resonance, δω0(Hz)

p(δ

ω0)

(a)

0.7 0.8 0.9 1 1.1 1.2 1.30

0.02

0.04

0.06

0.08

0.1

0.12

0.14

RF amplitude scaling (×ω1)

p(ω

1)

(b)

Figure 4.5: Measured distributions of the water phantom. (a) Off-resonance distribution p (δω0 ) extracted from the

histogram of a B0 field map. (b) RF power distribution p (ω1) extracted from the histogram of a B1 field map.

4.4 Results

4.4.1 The response of off-resonance spins

The measured and theoretical steady-state magnetisation waveform, for a single measurement, δrf= 180 Hz, is shown in Figure 4.2. The spectrum of this steady-state magnetisation (Figure 4.2b)

shows that information is restricted to harmonics of the Rabi frequency, ω1 = 100 Hz. The measured

p (δω0) and p (ω1) distributions, used in the generation of the predicted harmonic curves, are shown

in Figure 4.5a and Figure 4.5b, respectively. The variation in the harmonics is shown over a range of

off-resonances (Figure 4.6). The experimental measurements agree with the theoretical curves and

show that the relative strengths of the harmonic components are influenced by off-resonance effects.

4.4.2 Rabi modulated spectroscopy

Reconstructed ethanol spectra from the reference FID, gridded parameter set Ωgrid and optimal

parameter set Ωopti are shown in Figure 4.7. These reconstructions demonstrate that under Rabi

modulated CW excitation, chemical shift information can be encoded in the steady-state magneti-

sation.


−400 −300 −200 −100 0 100 200 300 400

−0.1

−0.05

0

0.05

0.1

0.15

δrf (Hz)

Re(

cobs

x

)

(a)

−400 −300 −200 −100 0 100 200 300 400

−0.1

−0.05

0

0.05

0.1

0.15

δrf (Hz)

Im(

cobs

y

)

(b)

Figure 4.6: Harmonic curves of the water phantom under Rabi modulated CW excitation with parameters α = 1 and

ω1 = ωm = 100 Hz. Measured (circles) and theoretical (solid line) frequency coefficients of steady state magnetisation.

DC component (blue), first (green), second (purple), third (red) fourth (orange) and fifth (grey) harmonics.

4.4.

RESULTS

85

1234560.00

0.20

0.40

δ (ppm)

Ωgrid

spectru

m(areanorm

alised)

1234560.00

0.20

0.40

δ (ppm)

Ωopti

spectru

m(areanorm

alised)

(a)

1234560.00

0.20

0.40

δ (ppm)

Ωgrid

spectru

m(areanorm

alised)

1234560.00

0.20

0.40

δ (ppm)

Ωopti

spectru

m(areanorm

alised)

(b)

Figure 4.7: NMR spectrum of doped ethanol phantom from reference FID (grey) and reconstruction from Rabi modulated CW excitation (black). (a) Reconstruction from

the Ωgrid measurement set. (b) Reconstruction from the Ωopti measurement set.


Table 4.1: Results from the spectrum analysis, with a g = 0.01, of the reference and reconstructed ethanol spectra.

The parameters of three fitted Lorentzian peaks are centre µ, half width half max Γ and area a. The chemical shift δ1

is between CH3 and CH2 and the chemical shift δ2 is between CH2 and OH. Peak centre, peak width and chemical

shifts are listed in parts per million (ppm).

CH3 Peak CH2 Peak OH Peak Summary

µ Γ a µ Γ a µ Γ a δ1 δ2 aratio

REFERENCE

Literature (Arnold, 1956) 1.250 0.500 3.709 0.333 5.365 0.167 2.459 1.656 3.0 : 2.0 : 1.0

FID 1.305 0.07 0.524 3.737 0.07 0.341 5.525 0.08 0.156 2.432 1.788 3.1 : 2.0 : 0.9

MEASUREMENT 1

Optimal 1.272 0.05 0.533 3.751 0.04 0.332 5.589 0.07 0.135 2.479 1.838 3.2 : 2.0 : 0.8

Grid 1.270 0.05 0.527 3.753 0.04 0.334 5.567 0.06 0.138 2.483 1.815 3.2 : 2.0 : 0.8

MEASUREMENT 2

Optimal 1.266 0.05 0.534 3.746 0.05 0.336 5.575 0.07 0.130 2.480 1.829 3.2 : 2.0 : 0.8

Grid 1.268 0.05 0.521 3.744 0.04 0.333 5.552 0.04 0.146 2.476 1.808 3.1 : 2.0 : 0.9

The results of the Lorentzian fitting are shown in Table 4.1 where the reference entries for relative

peak shifts (δ1, δ2) and area ratios (aratio) are experimental results from (Arnold, 1956). The

results for both parameter sets, for both measurements, and the result from the reference FID

are comparable. The measured and reconstructed spectra differ slightly from those observed in

(Arnold, 1956) with larger relative peak shifts, enlarged CH3 peak areas and reduced OH peak

areas. Spectra reconstructed from the optimal parameter set, Ωopti, are slightly more accurate than

those reconstructed from the gridded parameter set, Ωgrid, which suffer from an artifact near 0.25

and 6.25 ppm.

4.5 Discussion

The results from the investigation of off-resonance spins verify that our periodic solution of the Bloch

equations (4.6) can predict off-resonance behaviour of the spin system under Rabi continuous wave

excitation. Furthermore, the hypothesis that spectral information is restricted to harmonics of the

Rabi frequency (Layton et al., 2014) for the resonant case, ∆ = 0 Hz, has now been extended to

off-resonance cases.

Variations between prediction and measurement can be attributed to a number of factors such as

measurement error of relaxation time constants, measurement error of field distributions or phase

error between the prediction model and measurements. Whilst the error introduced by each of these

factors should be minimal, the combined effect of these errors on the prediction model, H, is the

subject of ongoing investigation.

Results from the Rabi modulated spectroscopy experiment demonstrate that it is possible to encode

chemical shift in a series of CW excitations and reconstruct a spectrum. This method requires the

measurement of relaxation constants to ensure an accurate forward model, H, this requirement may

be removed in future experiments by joint estimation (Bretthorst et al., 2005).

The reconstructions from the optimal and gridded parameter sets are comparable and the lack of

an artifact around 0.25 and 6.25 ppm in the optimal reconstruction may be attributed to the lower

theoretical variance in these regions as shown in Figure 4.4. The reconstruction routine (4.11) also

converges in less iterations when reconstructing with measurements from the optimal parameter set.

4.6. CONCLUSION 87

Imperfections in the spectra reconstructed from measurements under Rabi modulated excitation can

be attributed to error in the forward model, H. The receiver attenuation was fixed during each

experiment which may have introduced error when measuring magnetisation waveforms with a lower

relative power, where experimental imperfections such as low SNR begin to dominate.

In this chapter spectroscopy is posed as an inverse problem thus it is important to consider the

stability of the reconstruction. The low theoretical variance of the estimated spectra, shown in

Figure 4.4, provides an exact quantification of the predicted accuracy and demonstrates that the

inverse problem is well conditioned. The reconstruction algorithm was tested with multiple starting

points and converged to the same solution as presented. Currently, this method has the potential

to produce non-physical spectra, such as the small spurious peaks around 4.5 ppm in Figure 4.7.

This is likely due to minor imperfections in experimental setup and with an improved acquisition

strategy this should not be an issue.

The proposed method is not time efficient and in Chapter 5 we improve efficiency with a gapped

excitation and measurement protocol, similar to (Idiyatullin et al., 2006). Future work will con-

sider hardware modification (Brunner et al., 2011, 2012) to allow simultaneous transmit and receive.

In (Brunner et al., 2011, 2012), sideband modulation was used to spectrally isolate the excitation

and induced NMR signal. Similar filtering could be used here but no sideband modulation would

be required as the Rabi modulated response contains harmonic information outside the excitation

bandwidth. Alternative strategies for decoupling the transmit and NMR signal are analog cancela-

tion (Ozen et al., 2017b) and signal isolation in the digital domain under low power excitation (Ozen

et al., 2017a).

With a gradient applied during the Rabi spectroscopy experiment, the reconstruction vector, x, is a

projection of proton density along the gradient direction rather than a spectrum. Imaging will thus

result from the acquisition of a sufficient number of projections, to be reconstructed using a filtered

back-projection algorithm or an iterative algorithm, such as the conjugate gradient method. This

idea is explored in Chapter 5.

The ability to maintain an observable steady-state magnetisation under Rabi modulated excitation

may provide better SNR than methods that measure a single fast decaying FID signal. Rabi modu-

lated steady-state methods are true zero echo time techniques, if hardware modifications are made

to allow simultaneous excitation and observation. Measurement of ultra-short relaxation samples

such as sodium, bone or connective tissue will be explored in future work.

4.6 Conclusion

Rabi modulated CW excitation causes an observable periodic steady-state magnetisation with spec-

tral information at harmonics of the Rabi frequency. This steady-state magnetisation is influenced

by off-resonance effects, in a manner that can be accurately predicted by a periodic solution of the

Bloch equations. In this chapter, we have verified the existence of the off-resonance response, by

experimental measurement of the spin-system under Rabi modulated excitation. We have further

demonstrated the utility of the off-resonance harmonic response by encoding chemical shift infor-

mation in a series of measurements and the reconstruction of doped ethanol spectra. In Chapter 5,

using a more efficient measurement protocol, we extend this idea to encode radial projections under

Rabi modulated excitation and reconstruct a proton density image.


Chapter 5

Rabi continuous wave imaging


5.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2.1 Observed NMR signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2.2 Radial projection imaging as an inverse problem . . . . . . . . . . . . . . 93

5.3 Methods and Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3.1 Gapped excitation and measurement protocol . . . . . . . . . . . . . . . . 94

5.3.2 Gapped measurement of off-resonance response . . . . . . . . . . . . . . . 94

5.3.3 Rabi modulated imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.3.4 Rabi modulated imaging contrast . . . . . . . . . . . . . . . . . . . . . . . 100

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4.1 Gapped measurement of off-resonance response . . . . . . . . . . . . . . . 100

5.4.2 Rabi modulated imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.4.3 Rabi modulated imaging contrast . . . . . . . . . . . . . . . . . . . . . . . 102

5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.1 Introduction

In Chapter 4 the off-resonance response to a series of Rabi modulated RF excitations was used to

encode chemical shift information and reconstruct NMR spectra. In this chapter we extend the idea

to the imaging problem, and implement a proof of concept continuous wave (CW) imaging technique

(Korte et al., 2016). Again, we exploit the response to Rabi modulated excitation, where the spin-

system achieves an observable periodic magnetisation (Layton et al., 2014) and the steady-state

magnetisation can be fully described as harmonics of the excitation envelope modulation frequency

(Tahayori et al., 2015).

It is known that off-resonance effects influence the steady-state harmonics, as shown in the previous

chapter to encode chemical shift information and reconstruct ethanol spectra. Here, we focus a

different source of off-resonance, linear gradient fields, which are applied during a Rabi modulated

89

90 CHAPTER 5. RABI CONTINUOUS WAVE IMAGING

excitation envelope to encode spatial projections of proton density. Harmonics of the steady-state

response are acquired under a series of Rabi excitation parameters and a range of radial gradients.

For each gradient direction a model based optimisation is performed to reconstruct radial projections

of proton density, followed by filtered back projection (FBP) of all the reconstructed projections to

form a proton density image.

Rabi modulated excitation has potential to image ultra-short T2 tissues, which existing techniques

such as UTE (Bergin et al., 1991), ZTE (Weiger et al., 2011) and SWIFT (Idiyatullin et al., 2006)

have shown is clinically valuable in the diagnosis of muscular skeletal injury and disease (Robson

and Bydder, 2006; Weiger et al., 2013a; Luhach et al., 2014). A time-shared measurement protocol,

which rapidly switches between excitation and data acquisition, is used to observe the response to

Rabi modulated excitation in this chapter. This gapped protocol is an order of magnitude faster

than the iterative protocol used in Chapter 4, but still suffers from the finite switching time between

the transmitter and receiver. Rabi modulated excitation and the spin-system response exhibit

properties which may allow simultaneous excitation and measurement, but require non-standard

hardware. Observations made with the gapped excitation protocol should allow measurement of

ultra-fast relaxation signal in the same order as UTE, ZTE and time shared SWIFT methods;

a simultaneous excitation and measurement implementation would allow measurement of signals

similar to continuous SWIFT (Idiyatullin et al., 2012).

5.2 Theory

As introduced in Chapter 2, Rabi modulated excitation is an amplitude modulated RF field with

the previously defined envelope function

ωe1 (t) = ω1 (1 + α cosωmt) (2.48)

where ω1 = −γB1 is the average of the excitation envelope, B1 is the average excitation field

strength, α is the modulation level and ωm is the modulation frequency. It is known that under this

excitation the spin system achieves a significant periodic steady-state magnetisation and that the

magnitude of this steady-state is maximised when the Rabi resonance condition, ωm = ω1, is met

(Layton et al., 2014).

5.2.1 Observed NMR signal

The observed noise-free NMR signal under Rabi excitation, here with radial 2D spatial dependance,

is,

mobs (α, ω1, ωm,∆, T1, T2) = . . . (5.1)∫∫m (α, ωm, ω1 (r, s) ,∆ (r, s) , T1 (r, s) , T2 (r, s)) ρ (r, s) dr ds,

5.2. THEORY 91

Figure 5.1: Radial coordinate system and the Radon transform.

where there are projections in the z direction for spin density, ρ, excitation field strength, ω1, and

relaxation constants, T1 and T2,

ρ (r, s) =

∫object

ρ (r) dz (5.1a)

ω1 (r, s) =

∫object

ω1 (r) dz (5.1b)

T1 (r, s) =

∫object

T1 (r) dz (5.1c)

T2 (r, s) =

∫object

T2 (r) dz. (5.1d)

The projection of off-resonance (2.33) in the z direction is,

∆ (r, s) =

∫object

[ω0

(1 + δω0

(r) + g (r))− ωrf

(1 + δrf (r)

)]dz, (5.1e)

where δrf is an RF carrier frequency offset δω0encompasses B0 field inhomogeneity and chemical

shift and g is the gradient field contribution. The radial coordinate system is defined for an abitrary

projection angle, θ, [r

s

]=

[cos θ sin θ

− sin θ cos θ

] [x

y

], (5.2)

and the Radon transform,

ρθ (r) =

∫ρθ (r, s) ds, (5.3)

is shown graphically in Figure 5.1.

To simplify (5.1), we assume a homogenous RF excitation field, ω1 = ωm, and approximate the

spatially dependant relaxation constants as two average relaxation constants, T av1 and T av

2 . We


assume the B0 field inhomogeneity and chemical shift is adequately represented by the distribution,

p (δω0), for all voxels. These assumptions reduce equation (5.1) to,

mobs

(α, ωm, δrf, T

av1 , T av

2

)= . . . (5.4)∫∫∫m(α, ωm, δrf, δω0

, g (r, s) , T av1 , T av

2

)p (δω0

) ρ (r, s) dδω0dr ds.

We re-order the integration in the simplified observed NMR signal (5.4),

mobs

(α, ωm, δrf, T

av1 , T av

2

)= . . . (5.5)∫∫m(α, ωm, δrf, δω0

, g (r, s) , T av1 , T av

2

)p (δω0

) dδω0

∫ρ (r, s) ds dr.

A radial gradient is defined as,

g (r) = Gr r, (5.6)

where Gr is a constant gradient strength. The Radon transform (5.3) and radial gradient (5.6) are

applied to (5.5) which gives,

mobs

(α, ωm, δrf, T

av1 , T av

2

)= . . . (5.7)∫∫m(α, ωm, δrf, δω0 , g (r) , T av

1 , T av2

)p (δω0) dδω0 ρθ (r) dr.

The integral (5.7) can be numerically approximated over a discrete grid,

mobs (α, ωm, δrf, Tav1 , T av

2 ) ≈ . . . (5.8)

J∑j=1

K∑k=1

m(α, ωm, δrf, δ

(j)ω0, g(r(k)

), T av

1 , T av2

)p(δ(j)ω0

)ρθ

(r(k)

)and expressed in the frequency domain by substitution of (4.5),

Cobs (α, ωm, δrf, Tav1 , T av

2 , Gr) ≈ . . . (5.9)

J∑j=1

K∑k=1

C(α, ωm, δrf, δ

(j)ω0, g(r(k)

), T av

1 , T av2

)p(δ(j)ω0

)ρθ

(r(k)

)

=

K∑k=1

S(α, ωm, δrf, r

(k), T av1 , T av

2 , Gr

)ρθ

(r(k)

)where

S (α, ωm, δrf, r, Tav1 , T av

2 , Gr) =

J∑j=1

C(α, ωm, δrf, δ

(j)ω0, Grr, T

av1 , T av

2

)p(δ(j)ω0

). (5.9a)

The expression for predicting the observed Rabi harmonics under a gradient field (5.9) has a sim-

ilar form to the expression (4.6) used in the Rabi spectroscopy inverse formulation. The proton

projection expression (5.9) is different from (4.6) as it integrates out the dependance on main field

inhomogeneity and chemical shift effects, δω0, reducing the off-resonance dependance to gradient

sources, g (r).


5.2.2 Radial projection imaging as an inverse problem

It is possible to reconstruct a radial projection of proton density under Rabi modulated excitation,

as the off-resonance information from the gradient field is encoded in the observed NMR signal.

A linear system can be constructed from (5.9) by exciting the spin system with a set of N exci-

tation parameter tuples

(α(1), ω(1)m , δ

(1)rf ) . . . (α(N), ω

(N)m , δ

(N)rf )

and subsequent measurement of

the transverse magnetisation,

Hxθ = zθ (5.10)

where:

H =

S(α(1), ω

(1)m , δ

(1)rf , r

(1))

· · · S(α(1), ω

(1)m , δ

(1)rf , r

(K))

S(α(2), ω

(2)m , δ

(2)rf , r

(1))

· · · S(α(2), ω

(2)m , δ

(2)rf , r

(K))

.... . .

...

S(α(N), ω

(N)m , δ

(N)rf , r(1)

)· · · S

(α(N), ω

(N)m , δ

(N)rf , r(K)

)

(5.10a)

xθ =[ρθ

(r(1))

. . . ρθ

(r(K)

) ]T(5.10b)

zθ =[Cobs,θ

(α(1), ω(1)

m , δ(1)rf

)Cobs,θ

(α(2), ω(2)

m , δ(2)rf

). . . Cobs,θ

(α(N), ω(N)

m , δ(N)rf

) ]T(5.10c)

A discrete projection of proton density, xθ, with K points is reconstructed from the known forward

model, H, and an observation vector, zθ, by solving the linear system (5.10). If multiple projec-

tions are acquired with the same excitation envelope parameters and gradient strength, the forward

model (5.10a) only needs to be computed once, and can be used to reconstruct all proton density

projections. In this formulation (5.10), the average relaxation constants, T av1 and T av

2 , and gradient

amplitude, Gr, are assumed to be known constants and are omitted as functional arguments for

notational simplicity.

5.2.2.1 Image reconstruction

If we encode and reconstruct the radial proton density, xθ, for a range of projections, θ = [0, π), we

can construct a sinogram. For a given projection angle, θ, the radial gradient (5.6) is related to the

linear gradients by,

Gx = Gr cos θ (5.11a)

Gy = Gr sin θ. (5.11b)

A proton density image, ρ (r, s), can then be reconstructed from the sinogram using a standard

filtered back-projection algorithm, such as ‘iradon’ in MATLAB R©.

5.3 Methods and Materials

Two experiments were conducted on a 4.7T Bruker Biospec small bore scanner with an AVANCE III

console. In the first experiment the off-resonance response of a spin-system under Rabi modulated


excitation is measured using a gapped measurement protocol, developed to improve experimental

efficiency from the Rabi modulated experiments in Chapter 4. In the second experiment a series

of proton density projections are encoded under Rabi modulated excitation and used to recon-

struct a two dimensional proton density image. A Rabi modulated projection imaging simulation

was conducted on a numerical phantom with spatially varying relaxation constants to investigate

reconstructed image contrast.

5.3.1 Gapped excitation and measurement protocol

A gapped excitation measurement protocol (Korte et al., 2015) (Figure 5.3), similar to that used in

SWIFT (Idiyatullin et al., 2006), was used to achieve near-simultaneous transmit and receive. To

maintain a similar steady-state magnetisation trajectory as that observed under CW excitation, the

gapped Rabi modulated excitation envelope must be power corrected,

ωe1,gap (α, ω1, ωm, t, dcycle, Tgap) =

100dcycle

ωe1(α, ω1, ωm, t) tloc (t, Tgap) ≤ dcycleTgap

0 otherwise, (5.12)

where

tloc (t, Tgap) = t− floor

(t

Tgap

)Tgap (5.12a)

where dcycle is duty cycle and Tgap is the gapped sampling period. A comparison of the spin-system

steady-state response under continuous wave, ωe1, and gapped excitation envelope, ωe

1,gap, is shown

in Figure 5.2. The gapped excitation response oscillates around the continuous wave response, due

to the increased power during the excitation period, and relaxation during the acquisition period.

The gapped response intercepts the continuous wave response, approximately in the centre of the

acquisition period.

5.3.2 Gapped measurement of off-resonance response

To verify the gapped excitation protocol, the harmonic off-resonance response was experimentally

measured using a spherical phantom of Gd-doped water (T1 = 44 ms, T2 = 38 ms), selected for

its narrow off-resonance distribution. We investigated the effects of off-resonance on the observed

steady-state magnetisation waveform by offsetting the excitation carrier frequency from the Larmor

frequency. Off-resonance measurements were taken from δrf = −400 Hz to δrf = 400 Hz with an 8 Hz

increment. The gapped excitation envelope had parameters, α = 1, ω1 = ωm = 50 Hz, duty cycle,

dcycle = 90% and sampling period, Tgap = 1.67 ms. A discrete Fourier transform was applied to the

measured steady-state magnetisation waveform to extract its harmonic components. A theoretical

curve was generated from the periodic solution of the Bloch equations, numerically integrated over

the measured p (δω0) and p(ω1) distributions using (4.6).



quence with effective echo times (TE = 1.58, 4.43 ms), FA = 30, TR = 35 ms, FOV = 6 cm and

64×64×64 matrix. The distribution of excitation field strengths, p(ω1), was measured with a B1


MSME sequence over 10 slices, 2 mm thickness, FOV = 6 cm, 128×128 matrix, TE = 20 ms, TR = 1




2 4 6 8 10

−0.2

0

0.2

time (ms)

my

2 4 6 8 100

159

time (ms)

ωe 1(H

z)

(a)

2 4 6 8 10

−0.2

0

0.2

time (ms)

my

2 4 6 8 100

159

time (ms)

ωe 1(H

z)

(b)

Figure 5.2: Comparison of spin-system response under Rabi modulated excitation, numerical simulation of Blochequation. (a) Rabi excitation envelope in continuous wave (grey line) and gapped (black area) mode. (b) Steady-statetrajectory in response to continuous wave excitation (grey line) and gapped excitation (black lines), relaxation duringthe measurement period (red lines) to acquire a single sample (red dots).


Figure 5.3: Gapped excitation protocol for near-simultaneous transmit and receive. The sample is excited by a Rabimodulated CW envelope, ωe

1 (t), causing the magnetisation to reach a periodic trajectory, mxy . The Fourier transformof this magnetisation, cxy , has information restricted to harmonics of the excitation envelope modulation frequency,ω1.


Figure 5.4: Diagram of three test-tube phantom.



slice, 1 mm thickness, FOV = 6 cm, 128×128 matrix, TE = 80 ms, TR = 200, 400, 800, 1500, 3000,

4500 ms, RARE factor 8. The average T2 relaxation rate was calculated from the spatial average

of T2 maps, which were measured as described in Section 3.3.2. Data was acquired with a MSME

sequence using 9 echoes with a 22 ms echo spacing, 1 slice, 1 mm thickness, FOV = 6 cm, 128×128

matrix, TR = 1 s.

5.3.3 Rabi modulated imaging

To demonstrate Rabi modulated imaging, an imaging phantom of three test tubes of Gadolinium

doped water (T1 = 41 ms, T2 = 33 ms) was aligned in the longitudinal axis (Figure 5.4). A coil insert

was 3D printed to hold the three test tubes in position, and ensure their longitudinal alignment.

This relatively accurate positioning allowed the three dimensional phantom to be considered as a

two dimensional object. An alternative strategy is to use a phantom which is thin in one dimension,

our method provides a larger proton signal.

5.3.3.1 Reference measurements

A reference image was acquired with the FLASH protocol as described in as described in Section 3.2

for 10 axial slices with a 2 mm thickness, FA = 30, TE = 4 ms, TR = 100 ms, FOV = 6 cm,

Matrix=128x128. The axial slices were averaged to construct a single two dimensional proton density

image. A Radon transform of the reference proton density image was used to generate projections


along the same projection angles, θ, as used in the Rabi CW experiment. From the reference image

projections, an under-projected reference image was reconstructed using a standard FBP algorithm.



quence with effective echo times (TE = 1.58, 4.43 ms), FA = 30, TR = 35 ms, FOV = 6 cm and

64×64×64 matrix.




4500 ms, RARE factor 8. The average T2 relaxation rate was calculated from the spatial average

of T2 maps, which were measured as described in Section 3.3.2. Data was acquired with a MSME

sequence using 9 echoes with a 22 ms echo spacing, 1 slice, 1 mm thickness, FOV = 6 cm, 128×128

matrix, TR = 1 s.

5.3.3.2 Rabi modulated measurements

Using the gapped excitation measurement protocol, the phantom was excited by a set of N = 4030

Rabi modulated CW excitations(α(1), ω

(1)1 , δ

(1)rf

), · · · ,

(α(N), ω

(N)1 , δ

(N)rf

)where δrf is an offset to

the RF carrier frequency. The modulation level, α, ranged from 0.5 to 5.0, the modulation frequency,

ωm = ω1, ranged from 30 Hz to 90 Hz and the offset to RF carrier, δrf, ranged from -4.5 kHz to 4.5

kHz. For each CW excitation the phantom was measured over 18 projections angles, θ, as shown

in the sequence diagram (Figure 5.5). A low gradient strength of 102.1 Hz/mm was used to reduce

experimental time, and is 0.5% of the maximum available.

5.3.

METHODSAND

MATERIA

LS

99

Figure 5.5: Sequence diagram for the pseudo-continuous wave imaging experiment. The phantom is excited by a series of gapped Rabi modulated excitations. The lineargradients in the x and y directions are applied to take measurements over a range of 2D radial projections.


5.3.3.3 Reconstruction

The DC component and first five harmonics of the envelope modulation frequency, ω1, were extracted

from the measured steady-state magnetisation, mxy, and used to construct a measurement vector,

zθ, for each projection. The forward model matrix, H, was constructed from a Fourier series

approximation of the Bloch equation, numerically integrated over a voxel distribution, ρ (δω0). Each

proton density projection, xθ, was solved by least squares optimisation with a nonnegative and

smoothness constraint,

minimisexθ∈[0,∞)

(1− g) ‖Hxθ − zθ ‖2 + g ‖ ∆fd xθ ‖2, (5.13)

where g is a smoothing factor and ∆fd is a finite difference matrix. A two dimensional proton density

image was formed from the radial projections using a standard FBP algorithm.

5.3.4 Rabi modulated imaging contrast

The imaging experiment (Section 5.3.3) used a phantom with homogeneous relaxation, to match the

assumption of average relaxation constants, T av1 and T av

2 , used in the formulation of imaging as an

inverse problem (5.10). An imaging simulation was conducted to test if proton density projections

would be accurately reconstructed in a numerical phantom with multiple relaxation constants, and

to explore the imaging contrast under Rabi modulated excitation.

A two-dimensional numerical phantom of sixteen test tubes (Figure 5.6) has constant proton density

(Figure 5.6c) and a range of relaxation times; T1 relaxation rates of 1000, 2000, 3000, 4000 ms

(Figure 5.6a), T2 relaxation rates of 250, 500, 750, 1000 ms (Figure 5.6b). Imaging signals were

simulated over a 256x256 voxel grid via harmonic balancing of the Bloch equations (2.47) integrated

over a Gaussian voxel distribution, p (δω0), using (4.6). Rabi modulated excitation was simulated

for a grid of N = 500 excitation parameter pairs

(α(1), ω(1)m ) . . . (α(N), ω

(N)m )

as used in the

Rabi spectroscopy experiment (Figure 4.3). The spin system response was generated for 36 radial

projections with five degree increments of the projection angle, θ. Image reconstruction was identical

to the imaging experiment (Section 5.3.3.3) with a discrete proton density reconstruction vector,

xθ, of K = 64 points. The forward model was constructed with the average relaxation constants,

T av1 = 2500 ms and T av

2 = 600 ms. Projections of proton density were reconstructed via optimisation

of objective function (5.13) with no regularisation, g = 0.

5.4 Results

5.4.1 Gapped measurement of off-resonance response

The measured p (δω0) and p (ω1) distributions, used in the generation of the predicted harmonic

curves, are shown in Figure 5.7a and Figure 5.7b, respectively. The variation in the harmonics of

the response to the Rabi modulation frequency are shown over a range of off-resonances (Figure 5.8).

The experimental measurements are very similar to the theoretical curves, with some minor phase

error visible in the measurements at δrf = −320 Hz and δrf = 152 Hz.

5.4. RESULTS 101

x (mm)

y (

mm

)

0 5 10 15 20

0

5

10

15

20 0

1

2

3

4

(a)

x (mm)y

(m

m)

0 5 10 15 20

0

5

10

15

20 0

0.2

0.4

0.6

0.8

1

(b)

x (mm)

y (

mm

)

0 5 10 15 20

0

5

10

15

20 0

0.2

0.4

0.6

0.8

1

(c)

x (mm)

y(m

m)

0 5 10 15 20

0

5

10

15

20 0

0.2

0.4

0.6

0.8

1

(d)

Figure 5.6: Numerical phantom and simulated contrast of Rabi modulated imaging. Numerical phantom of 16 testtubes with spin system properties defined by (a) T1 map (b) T2 map and (c) proton density map. (d) is the FBP ofa sinogram reconstructed from a series of simulated Rabi modulated responses.


−400 −200 0 200 4000

1

2

3

4

5

6x 10

−3

p(δ

ω0)

off-resonance, δω 0(Hz)

(a)

0.8 0.9 1 1.1 1.20

0.05

0.1

0.15

0.2

0.25

p(ω

1)

RF amplitude scaling, (×ω1)

(b)

Figure 5.7: Measured distributions of the water phantom used in the gapped measurement of off-resonance experiment.(a) Off-resonance distribution p (δω0 ) extracted from the histogram of a B0 field map. (b) RF power distributionp (ω1) extracted from the histogram of a B1 field map.

5.4.2 Rabi modulated imaging

The reference image (Figure 5.9) is a z projection of proton density and displays three defined test

tubes, aligned along the longitudinal axis. The reference sinogram (Figure 5.10a) and FBP image

(Figure 5.10b) show an expected decrease in image quality due artifacts from the reduced number

of projection angles. Two reconstruction cases are considered for the measurements taken under

Rabi modulated CW excitation; the first assuming B0 homogeneity and the second incorporating

a measured voxel distribution in the forward model. The results assuming B0 homogeneity show

a sinogram (Figure 5.10c) with a consistent artifact through all projections, and a FBP image

(Figure 5.10d) with three clearly defined test tubes but have some error in shape and size. Including

knowledge of the voxel distribution in the forward model removes the artifact from the sinogram

(Figure 5.10e) and leads to a blurred FBP image (Figure 5.10f).

A line profile over one test tube (Figure 5.11), is used to assess the quality of the FBP images

against the reference FLASH image. The line profile for the reference FLASH image (Figure 5.11a)

has excellent SNR, the small signals at the edge of the test-tube are due to water trapped in the

thread of the test-tube cap. The line profile for the FBP of an under-projected reference image

(Figure 5.11b) has lower SNR than the reference image but maintains a strong transition between

the background and the test-tube. The line profile of the Rabi modulated FBP image (Figure 5.11c)

has the lowest SNR and a blurred transition between the background and the test-tube.

5.4.3 Rabi modulated imaging contrast

The imaging results from the simulated Rabi modulated response from a numerical phantom of 16

test tubes are shown in Figure 5.6d and as line profiles in Figure 5.12. The FBP image (Figure 5.6d)

shows 16 test tubes with a range of intensities, there is an artifact in the centre of the image and a

circular streaking artifact. The vertical line profiles (Figure 5.12a) over test tubes with a decreasing

T2 in the positive y direction (Figures 5.12c,5.12e,5.12g and 5.12i) show a related decrease in image

intensity. The horizontal line profiles (Figure 5.12b) over test tubes with a increasing T1 in the

positive x direction (Figures 5.12d,5.12f,5.12h and 5.12j) show a related increase in image intensity.

5.4. RESULTS 103

−400 −200 0 200 400−15

−10

−5

0

5

10

15

δrf(Hz)

Re(cobs

x)

(a)

−400 −200 0 200 400−15

−10

−5

0

5

10

15

δrf(Hz)

Im(c

obs

y)

(b)

Figure 5.8: Harmonic curves of the water phantom under Rabi modulated CW excitation with parameters α = 1and ω1 = ωm = 50 Hz, acquired with a gapped excitation protocol of duty cycle, dcycle = 90% and sampling period,Tgap = 1.67 ms. Measured (circles) and theoretical (solid line) frequency coefficients of steady state magnetisation.DC component (blue), first (green), second (purple), third (red) fourth (orange) and fifth (grey) harmonics.


x (mm)

y(m

m)

−20 0 20

−30

−20

−10

0

10

20

30 0

0.2

0.4

0.6

0.8

1

Figure 5.9: Reference image is the average of 10 axial slices acquired with a FLASH protocol. The ring around eachtest tube is due to water trapped in the test tube cap thread.

All line profiles (Figure 5.12) have a non-zero background signal, in the fastest T2 case (Figure 5.12j)

the magnitude of test tube signal is similar to the background signal.

5.5 Discussion

The measurement of off-resonance spins under power corrected gapped Rabi modulated excitation

(Figure 5.8) verifies that the CW spin system response can be observed under a gapped excitation

and measurement protocol. Variations between prediction and measurement can be attributed to

cumulative error in the prediction model, H, from relaxation and field distribution measurements.

There is also a noticeable phase jump in the measurements at δrf = −320, 152 Hz which may indicate

some mild instability in phase coherence between measurements. This discrepancy between model

and measurement may also be an artifact related to the gapped measurement protocol (Idiyatullin

et al., 2008).

The gapped protocol offers a significant efficiency improvement over the iterative protocol used in

Chapter 4 and Chapter 6 which makes observation of the transient response feasible for future

studies. For an experiment where steady-state is assumed after 5T1 and the same time is allowed

for relaxation to equilibrium in each repetition, the approximate experimental duration for iterative

and gapped protocols is,

T iterN = 20N T1Nharm (5.14)

T gapN = 10N T1 (5.15)

where N is the number of steady-states to be measured and Nharm is the number of measured steady-

state harmonics. In our steady-state experiments, the efficiency is improved by a factor of twelve

when acquiring enough gapped data for six harmonics. The theoretical limit, assuming steady-state

after 5T1 and smooth transitions between different steady-state trajectories can be achieved, gives

5.5. DISCUSSION 105

θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

(a)

θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

(b)θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1(c)

θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1(d)

θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

(e)

θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

θ()

r(m

m)

0 50 100 150

−40

−20

0

20

400

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

(f)

Figure 5.10: (a) Reference FLASH sinogram (b) FBP of reference FLASH sinogram (c) Rabi CW sinogram (d) FBPof Rabi CW sinogram (e) Rabi CW sinogram with voxel distribution in forward model (f) FBP of Rabi CW sinogramwith voxel distribution in forward model.


(a) (b) (c)

−30 −20 −10 0 10 20 300

0.5

1

x (mm)

ρ(δ

ω0)

(d)

Figure 5.11: Line profile (red line) marked on (a) Reference FLASH image (b) FBP of under-projected referenceFLASH sinogram (c) FBP of Rabi CW sinogram. (d) Line profile of proton density from reference FLASH image(thick line), FBP of under-projected reference FLASH sinogram (thin line) and FBP of Rabi CW sinogram (dashedline).

5.5. DISCUSSION 107

y(m

m)

x (mm)0 5 10 15 20

0

5

10

15

20

(a)

y(m

m)

x (mm)0 5 10 15 20

0

5

10

15

20

(b)

0 2 4 6 8 10 12 14 16 18 200

0.5

1

y (mm)

ρ(δ

ω0)

(c)

0 2 4 6 8 10 12 14 16 18 200

0.5

1

x (mm)

ρ(δ

ω0)

(d)

0 2 4 6 8 10 12 14 16 18 200

0.5

1

y (mm)

ρ(δ

ω0)

(e)

0 2 4 6 8 10 12 14 16 18 200

0.5

1

x (mm)

ρ(δ

ω0)

(f)

0 2 4 6 8 10 12 14 16 18 200

0.5

1

y (mm)

ρ(δ

ω0)

(g)

0 2 4 6 8 10 12 14 16 18 200

0.5

1

x (mm)

ρ(δ

ω0)

(h)

0 2 4 6 8 10 12 14 16 18 200

0.5

1

y (mm)

ρ(δ

ω0)

(i)

0 2 4 6 8 10 12 14 16 18 200

0.5

1

x (mm)

ρ(δ

ω0)

(j)

Figure 5.12: Line profiles of simulated contrast of Rabi modulated imaging. (a) Vertical line profiles (red lines) alongtest-tubes with constant T1 and a decreasing T2 in the positive y direction. The vertical red lines shown in (a) areplotted from left to right as (c),(e),(g) and (i). (b) Horizontal line profiles (red lines) along test-tubes with constantT2 and a increasing T1 in the positive x direction. The horizontal red lines shown in (b) are plotted from top tobottom as (d),(f),(h) and (j).


a best case experimental time

T limN = 10T1 +N

2π

ωavm

(5.16)

where ωavm is the average Rabi modulation of the set of N excitation parameters. Using the imaging

experiment parameters, data acquisition for a single projection is, T iterN = 5.5 hours, using the

iterative protocol, T gapN = 28 minutes, using the gapped protocol and T lim

N = 1.1 minutes, using

a theoretical ideal protocol. These experimental times are to encode a chemical shift spectrum or

a single projection of proton density, for a radial imaging experiment these are multiplied by the

number of projections to calculate the total experimental time.

The results of the radial imaging experiment (Figure 5.10 and Figure 5.11) demonstrate that Rabi

modulated excitation can encode off-resonant information, in this case off-resonance induced by

linear field gradients. The gradients used in the imaging experiment lead to a larger off-resonance

bandwidth to be encoded than the spectroscopy experiment of Chapter 4. This was achieved by

shifting the encoding function across the off-resonance range using the RF carrier offset δrf.

The image (Figure 5.10b) reconstructed from the reference sinogram (Figure 5.10a) shows an ex-

pected streak artifact from the reduced number of projections. Rabi modulated proton density

projections reconstructed with a homogeneous voxel form the sinogram (Figure 5.10c) which is

comparable to the reference sinogram (Figure 5.10a). The horizontal lines in the Rabi modulated

sinogram (Figure 5.10c) may be due to uneven coverage of the encoding function, this may be im-

proved by a finer increment of δrf. Another potential source of the horizontal line artifact in the

sinogram (Figure 5.10c) is related to the use of a gapped excitation envelope, which causes a bullseye

artifact in SWIFT (Idiyatullin et al., 2008) and can be removed by a gap cycling strategy during

acquisition (Corum et al., 2015). The FBP image (Figure 5.10d) from homogeneous voxel sinogram

(Figure 5.10c), shows three test tubes with a similar back projection artifact to the reference image

(Figure 5.10b) and error in the diameter of the test tubes. Rabi modulated proton density projec-

tions reconstructed with a measured voxel distribution in the forward model, H, form a smooth

sinogram (Figure 5.10e) without the horizontal artifact seen in reconstructions assuming a homoge-

neous voxel (Figure 5.10c) and mildly improves the size of the test tubes in the reconstructed image

(Figure 5.10f).

The image line profiles (Figure 5.11d) show an expected loss of SNR in the FBP cases (Figure 5.11b

and 5.11c) due to the under projected streak artifact. The line profile of the Rabi modulated FBP

(Figure 5.11c) fails to reconstruct the sharp edge of the test tube as shown in the reference image

(Figure 5.11a). This loss of high frequency information may be an effect of regularisation in the

reconstruction objective function (5.13) or due to uncorrected B0 field inhomogeneity effects.

The contrast simulation results (Figure 5.12) demonstrate that an object with variable relaxation

rates can be reconstructed using a forward model computed with two average relaxation rates, T av1

and T av2 . In addition, the intensity of the FBP image with be weighted by the actual relaxation

rates. The line profiles showed two general image intensity trends; a faster T1 constant gives a higher

image intensity, a slower T2 constant gives a higher image intensity.

Filtered back projection was an adequate method for this experiment, better reconstruction tech-

niques such as the iterative methods used in CT may lead to a better reconstructed image quality.

We anticipate that further improvements to experimental efficiency will allow the acquisition of more

excitation parameters under a higher gradient and lead to more accurate reconstruction, without

prior knowledge of a voxel distribution.

5.6. CONCLUSION 109

5.6 Conclusion

This chapter introduces a gapped excitation and measurement protocol which can be used to study

spin-system response to periodic excitation. The greater efficiency of the gapped protocol made an

imaging experiment feasible and will allow future investigation of the transient spin-system response

under Rabi modulated excitation. The proof of concept imaging experiment demonstrates that

under Rabi modulated CW excitation, gradient localisation information can be encoded in the

steady-state magnetisation and used to reconstruct proton density images. Rabi modulated imaging

was simulated for a numerical phantom with variable relaxation rates and reconstructed images show

T1 and T2 contrast. Future work will focus on improving the efficiency of acquisition, the accuracy

of reconstruction and evaluation of the methods ability to image samples with ultra-short spin-spin

relaxation.


Chapter 6

Ring-lock excitation


6.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2.1 Steady-state ring-lock response . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2.2 Estimation of relaxation constants . . . . . . . . . . . . . . . . . . . . . . 114

6.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.3.1 Verification of excitation envelope . . . . . . . . . . . . . . . . . . . . . . 115

6.3.2 Estimation of relaxation constants . . . . . . . . . . . . . . . . . . . . . . 118

6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.A Ring-lock excitation envelope . . . . . . . . . . . . . . . . . . . . . . . . 122

6.A.1 Amplitude and phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.B Spin-system response under ring-lock excitation . . . . . . . . . . . . . 125

6.1 Introduction

In previous chapters 4 and 5, we explored potential applications of the steady-state magnetisation

under continuous wave periodic RF excitation, in particular Rabi modulated excitation. In this

chapter, we revisit a well known steady-state response of the Bloch equations, with the addition

of off-resonance. Under continuous wave, constant amplitude RF excitation, the Bloch equations

predict that the bulk magnetisation will reach a steady-state magnetisation on an ellipsoid (Bloch,

1946). The steady-state ellipsoid is located in the upper hemisphere of the Bloch sphere (Figure 6.1).

The power of the constant amplitude excitation determines the location of the steady-state magneti-

sation on this manifold. A very low power excitation will perturb the magnetisation only slightly

from thermal equilibrium, as power is increased the steady-state magnetisation will move down the

ellipsoid, approaching saturation at the centre of the Bloch sphere. The ability to saturate the

NMR signal can be used to estimate spin-spin relaxation rates with saturation recovery (Markley

111

112 CHAPTER 6. RING-LOCK EXCITATION

(a) (b) (c)

Figure 6.1: Steady-state ring trajectory (black ring) on steady-state manifold (purple ellipsoid) in the Bloch sphere(grey sphere). Trajectories are shown for a (a) low power case (b) slightly higher power case (c) slower ring-lockfrequency with the same power as (b).

et al., 1971) and partial saturation (Freeman and Hill, 1971) methods. Unwanted solvent peaks

are saturated in NMR spectroscopy (Jesson et al., 1973) and similar frequency selective saturation

methods (Bottomley et al., 1984; Haase et al., 1985) are used to saturate water and lipid signals in

chemical shift imaging.

When a constant amplitude RF excitation is applied off-resonance, the magnetisation is driven into

a steady-state ring trajectory (Figure 6.1), on the previously described ellipse. The off-resonance is

achieved by amplitude modulating the excitation envelope, or by frequency offset of the transmitter

RF carrier. An alternative way to consider the response, is viewing a spin-lock (Redfield, 1955;

Hartmann and Hahn, 1962) response from an off-resonance frame of reference, with the magnetisation

decaying to a ring rather than a point. Not a completely fair comparison, as spin-lock applications

focus on the transient dynamics for T1ρ estimation or contrast (Sepponen et al., 1985; Santyr et al.,

1989, 1994), and are generally not applied for a long enough duration to reach steady-state. In this

chapter, we refer to an off-resonance constant amplitude envelope as ring-lock excitation, named

for the ring shaped steady-state trajectory of the spin-system; and in recognition of similarities to

spin-lock excitation. It is most useful though to consider the excitation as an off-resonance partial

saturation pulse, where the saturation level is influenced by both the amplitude and the frequency

offset.

The ring-lock excitation envelope can control the steady-state magnetisation on an ellipsoid, with

adequate knowledge of the spin-system properties, such as relaxation rates. We demonstrate this

control experimentally and develop a model based method for estimating the ratio of relaxation

constants (Korte et al., 2017b). In our method, we apply off-resonance partial saturation pulses and

derive an analytical steady-state signal model, which is dependent on the excitation off-resonance,

in addition to the relaxation constants.

The shape of the steady-state manifold is dependant on relaxation constants, as shown in Fig-

ure 6.2. We exploit the relationship between excitation parameters and unknown relaxation con-

stants to develop a volume relaxometry technique. Existing steady-state free precession (Carr,

1958) (SSFP) methods such as IR-TrueFISP (Scheffler and Hennig, 2001; Schmitt et al., 2004),

DESPOT (Homer and Beevers, 1985; Deoni et al., 2003), DESS (Bruder et al., 1988; Welsch et al.,

2009) and TESS (Heule et al., 2014) generate relaxation maps via optimisation of signal models,

dependent on relaxation constants T1 and T2. We plan to develop our volumetric ring-lock relaxom-

etry technique into a relaxation mapping method, utilising the elliptical nature (Hargreaves, 2012)

of balanced steady-state free precession (Oppelt et al., 1986) (bSSFP).

6.2. THEORY 113

(a) (b) (c)

Figure 6.2: Steady-state ring trajectory (black ring) on steady-state manifold (purple ellipsoid) in the Bloch sphere(grey sphere). Steady-state ellipsoids are shown for three different tissue relaxation parameters (a) white matter (b)lipid (c) cerebrospinal fluid.

6.2 Theory

A spin-system under constant amplitude RF excitation will reach a steady-state magnetisation on

an ellipsoid (Bloch, 1946) whose shape is defined by the relaxation constants and is influenced by

off-resonance effects and imperfect excitation (Abragam, 1961). We introduce a constant frequency

modulation, ωrl, to the constant amplitude excitation, |ωe1|, which induces a ring-locked steady-state

trajectory on the surface of the ellipsoid (Figure 6.3a). The ring-lock excitation envelope is defined

as,

γBe1 (t) = |ωe

1| eiφ(t) (6.1)

where

|ωe1| =

√T2

T1

(1− a1 + a

)[(ωrl + ∆)

2+

(1

T2

)2 ](6.1a)

φ (t) = ωrlt+ θ − tan−1

(1

T2 (ωrl + ∆)

)+π

2. (6.1b)

The amplitude and phase of the envelope are expressing in terms of the estimated spin parameters

and the desired response parameters. The estimated spin parameters are relaxation, T1, T2, and

off-resonance, ∆. The desired ring response parameters are phase offset, θ, and elevation on the

steady-state ellipsoid, a ∈ (−1, 1]. The desired elevation, a, is related to the longitudinal component

of the steady-state magnetisation (6.2c); at thermal equilibrium, a = 1, the magnetisation becomes

saturated as, lima→−1

. When the estimated spin parameters used in the ring-lock envelope are accurate,

the spin-system response is,

mx =M0

z

2

√T2

T1(1− a2) sin (ωrlt+ θ) (6.2a)

my =M0

z

2

√T2

T1(1− a2) cos (ωrlt+ θ) (6.2b)

mz =M0

z

2(1 + a) . (6.2c)


6.2.1 Steady-state ring-lock response

The estimated tissue parameters may differ from the actual tissue parameters for relaxation T1, T2,

and off-resonance ∆, which leads to a different ring-lock response with parameters for elevation, a,

and phase offset, θ. The difference between the desired steady-state trajectory parameters, (a, θ),

and the observed steady-state trajectory parameters,(a, θ), is also influenced by an excitation

field inhomogeneity factor, Bmod1 . The observed steady-state trajectory, in response to ring-lock

excitation (6.1), is defined as,

mx =M0

z

2

√T2

T1(1− a2) sin

(ωrlt+ θ

)(6.3a)

my =M0

z

2

√T2

T1(1− a2) cos

(ωrlt+ θ

)(6.3b)

mz =M0

z

2(1 + a) (6.3c)

where observed elevation is,

a(a, ωrl, T1, T2,∆, T1, T2, ∆, B

mod1

)=

1− β1 + β

(6.3d)

β = Bmod1

2 T1 T2

T1 T2

(1− a1 + a

) (ωrl + ∆)2 +(

1T2

)2

(ωrl + ∆)2 +(

1T2

)2

(6.3e)

and observed phase is,

θ = θ − tan−1

(1

T2 (ωrl + ∆)

)+ tan−1

(1

T2

(ωrl + ∆

)). (6.3f)

For derivation details of the ring-lock excitation envelope and spin-system steady-state response

see Appendix 6.A and Appendix 6.B, respectively. We have verified this analytical solution using

numerical simulations (Figure 6.4) and harmonic balancing (Tahayori et al., 2015) of the Bloch

equations, an experimental verification is detailed in the methods and results sections.

6.2.2 Estimation of relaxation constants

If we consider a case when parameters for the ring-lock envelope amplitude (6.1a) are not known, we

can treat the amplitude, |ωe1|, as an adjustable power. The elevation on the steady-state ellipsoid,

a, can then be expressed in terms of the power, |ωe1|, and frequency, ωrl, of the ring-lock envelope,

a(|ωe

1|, ωrl, T1, T2, ∆, Bmod1

)=

1− β1 + β

(6.4a)

β =T1

T2

|ωe1|2Bmod

12(

ωrl + ∆)2

+(

1T2

)2

. (6.4b)

This relationship (6.4) gives an observed magnetisation,

mmodelxy

(|ωe

1|, ωrl, T1, T2, t)

= . . . (6.5)∫∫mxy

(|ωe

1|, ωrl, T1, T2, ∆ (δω0) , Bmod

1 , t)p(Bmod

1

)p (δω0

) dBmod1 dδω0

.

6.3. METHODS 115

which is influenced by unknown tissue parameters, T1, T2, and excitation parameters, |ωe1|, ωrl. Re-

laxation constants can therefore be estimated from a series of ring-lock responses, acquired under a

range of excitation parameters, |ωe1|, ωrl, using the objective function,

minimiseT1, T2∈[0,∞)

‖ ∣∣mmodelxy

(|ωe

1|, ωrl, T1, T2

)∣∣− ∣∣mobsxy (|ωe

1|, ωrl)∣∣ ‖

2. (6.6)

6.3 Methods

Two experiments were conducted on a 4.7T Bruker Biospec scanner with an AVANCE III console, to

test the ring-lock excitation envelope. The first to experimentally verify the analytical response of the

spin-system (6.2) under ring-lock excitation; using measured spin-system properties to control the

magnetisation on the surface of the steady-state ellipsoid. A second, proof-of-concept experiment,

demonstrates the estimation of relaxation constants from a series of steady-state measurements,

acquired under a range of ring-lock excitation envelopes.

In both experiments, an iterative measurement protocol (Layton et al., 2014; Korte et al., 2017a)

(Figure 6.3b) was used to incrementally measure the steady-state magnetisation. Rapid acquisition

with relaxation enhancement with variable repetition time (RARE-VTR) scans and multi-slice multi-

echo (MSME) scans were used to measure T1 and T2 maps respectively. The measured relaxation

maps were spatially averaged to produce a single T1 and T2 constant per phantom. The distri-

bution of off-resonances, p (δω0), was measured using a field-mapping sequence (Kanayamay et al.,

1996). The distribution of excitation field strength, p(Bmod

1

), was measured with a B1 mapping

sequence (Stollberger et al., 1988). Distributions were extracted via a histogram of non-background

voxels.

6.3.1 Verification of excitation envelope

The first experiment applied an excitation with the ring-lock envelope (6.1) to a spherical phantom

of water to demonstrate control of the steady-state magnetisation. Ring-locked trajectories were

measured over a range of elevations, a, from -0.95 to 1.0 with a 0.05 increment, using a constant

excitation frequency, ωrl = 50 Hz, measured relaxation constants, T1 = 3.20 s and T2 = 2.02

s, and off-resonance ∆ = 0 Hz. For each measurement point, two incremental acquisitions were

required, the first as shown in Figure 6.3b to acquire the transverse magnetisation, the second with

a hard pulse directly before the FID to measure the longitudinal magnetisation. The predicted

response was generated by numerically integrating the analytical response (6.5) over the measured

field distributions, incorporating the measured relaxation constants.



quence with effective echo times (TE = 1.94, 6.22 ms), FA = 20, TR = 20 ms, FOV = 4.5 cm and

64×64×64 matrix. The distribution of excitation field strengths, p(Bmod

1

), was measured with a B1


MSME sequence over 18 slices, 2 mm thickness, FOV = 4.5 cm, 64×64 matrix, TE = 12 ms, TR = 2.5




measured as described in Section 3.3.1. Data was acquired with a RARE-VTR sequence over 1 slice,


(a)

(b)

Figure 6.3: (a) Simulated spin-system response of the magnetisation (dashed line) approaching a steady-state ring-trajectory on an ellipse (purple) in the upper half of the Bloch sphere (grey). (b) The iterative excitation protocolapplies the excitation envelope, γBe

1 (t), for 5T1 until the spin-system has reached steady-state, after which a freeinduction decay (FID) is measured. A steady-state magnetisation waveform, mobs

xy , can be acquired by selecting thefirst FID point, and repeating the process with an incremental increase to the excitation duration on each repetition.The longitudinal magnetisation is measured in the same manner, with the addition of a hard pulse directly beforeeach FID.

6.3. METHODS 117

0 1 2 3 4 5 6−1

0

1

t(s)

mx

(a)

5.96 5.97 5.98 5.99 6−0.2

0

0.2

t(s)

mx

(b)

0 1 2 3 4 5 6−1

0

1

t(s)

my

(c)

5.96 5.97 5.98 5.99 6−0.2

0

0.2

t(s)

my

(d)

0 1 2 3 4 5 6−1

0

1

t(s)

mz

(e)

5.96 5.97 5.98 5.99 60

0.5

1

t(s)

mz

(f)

Figure 6.4: Numerical validation of excitation envelope with (a),(c),(e) simulated magnetisation from numericalintegration of Bloch equations. (b),(d),(f) steady-state magnetisation under ring-lock excitation with a comparisonof numerically integrated magnetisation (line) and magnetisation predicted by analytical solution of Bloch equations(circles). The excitation parameters used were ring-lock frequency, ωrl = 50Hz, desired elevation, a = 0.2, estimatedrelaxation constants, T1 = 500ms, T2 = 250ms, and estimated off-resonance, ∆ = 20Hz. Spin parameters used wererelaxation constants, T1 = 1200ms, T2 = 300ms, and off-resonance, ∆ = −10Hz.


4 cm thickness, FOV = 4 cm, 32×32 matrix, TE = 11 ms, TR = 50, 100, 200, 400, 800, 1500, 3000,

6000, 9000, 12000, 15000 ms, RARE factor 1. The average T2 relaxation rate was calculated from the

spatial average of T2 maps, which were measured as described in Section 3.3.2. Data was acquired

with a MSME sequence using 128 echoes with a 50 ms echo spacing, 1 slice, 8 mm thickness, FOV

= 4 cm, 32×32 matrix, TR = 13.2 s.

6.3.2 Estimation of relaxation constants

In a second experiment, we measured the transverse steady-state magnetisation, mobsxy , of a spherical

phantom of Gadolinium doped water (T1 = 360 ms, T2 = 200 ms). Steady-state measurements were

acquired under RF excitation with the ring-lock envelope (6.1), over a grid of excitation parameters;

excitation amplitude, |ωe1|, from 100 Hz to 1800 Hz with an increment of 100 Hz and ring-lock

frequency, ωrl, from 25 Hz to 400 Hz with a 25 Hz increment. Relaxation constants were estimated

from the grid of observations, by solving the objective function (6.6) with a constrained non-linear

optimisation algorithm, fmincon, from MATLAB R©. In addition to the estimation from experimental

measurements, the relaxation constants were estimated from a “perfect” observation, generated from

the observation model (6.5) using the relaxation constants measured with standard methods. The

relaxation constants estimated from experimental measurement were used to generate a predicted

response surface, by numerically integrating the analytical response (6.5) over the measured field

distributions.


may et al., 1996) described in Section 3.4.1. Data was acquired using a multiple gradient echo

sequence with effective echo times (TE = 1.94, 6.22 ms), FA = 20, TR = 20 ms, FOV = 4.5 cm

and 128×128×128 matrix. The distribution of excitation field strengths, p(Bmod

1

), was measured

with a B1 mapping sequence (Wang et al., 2005) described in Section 3.4.2. Data was acquired

with a MSME sequence over 18 slices, 2 mm thickness, FOV = 4.5 cm, 128×128 matrix, TE = 12

ms, TR = 2.5 s. The MSME scans were taken for two excitation angle, α, and refocusing angle, β,

configurations; α/β = 45/90 in the first scan, α/β = 90/180 in the second scan.




1600, 3200, 6400 ms, RARE factor 2. The average T2 relaxation rate was calculated from the spatial

average of T2 maps, which were measured as described in Section 3.3.2. Data was acquired with a

MSME sequence using 64 echoes with a 10 ms echo spacing, 1 slice, 2 mm thickness, FOV = 3 cm,

256×256 matrix, TR = 5.0 s.

6.4 Results

Verification of the excitation envelope and steady-state response are the shown in Figure 6.5. The

steady-state measurements are purple and the analytically predicted steady-state trajectories are

shown in black. The steady-state trajectories (Figure 6.5a-6.5c) form ellipsoids viewed from a range

of angles. In Figure 6.5b the elevation, or longitudinal magnetisation, mz, of measured steady-state

ring trajectories is consistent with the model prediction. In Figure 6.5c the phase of measured

steady-state ring trajectories is consistent with the model prediction. The average magnitude of the

transverse magnetisation, or each ring-lock response is shown in Figure 6.5d, with minor error (3.1%

M0z ) between measurement and model prediction.

6.4. RESULTS 119

−0.5

0

0.5 −0.5

0

0.50

0.2

0.4

0.6

0.8

1

mymx

mz

(a)

−0.5 0 0.50

0.2

0.4

0.6

0.8

1

mx

mz

(b)

−0.5 0 0.5−0.5

0

0.5

mx

my

(c)

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

mean(|mxy|)

mean(m

z)

(d)

Figure 6.5: Experimental validation of excitation envelope using predicted (black) and measured (purple) steady-statering-trajectory response, displayed in (a) three-dimensional (b) side-on and (c) top-down views. The mz and mxy

components of each ring trajectory are averaged to generate (d) a mean profile of the ellipse.


(a) (b)

Figure 6.6: (a) The transverse steady-state magnetisation, predicted (black) and measured (purple) over the excitationparameter space. (b) The error surface of the difference between measured and predicted transverse magnetisation(purple) and a reference plane at zero (black).

Table 6.1: Measured and estimated relaxation constants.

T1 (ms) T2 (ms) T1/T2 ratio

REFERENCE

RARE-VTR 360.23±11.84 -1.8057

MSME - 199.50±2.60

METHOD

Ideal surface 360.20 199.48 1.8057

Measured surface 97.19 53.81 1.8059

The estimation of relaxation constants results are visualised in Figure 6.6. The magnitude of trans-

verse steady-state magnetisation, |mxy|, is shown as a surface across a grid of ring-lock envelope

parameters (Figure 6.6a), where measurements are purple and the black surface is generated using

the relaxation parameters estimated via an optimisation routine. Figure 6.6b is a surface represent-

ing error between the transverse magnetisation and the predicted transverse magnetisation. The flat

black surface can be considered a zero error reference, the purple surface is the error between mea-

sured and predicted magnitude of the transverse magnetisation; maximum error is approximately

4.1% M0z .

The relaxation constants from standard measurements and estimations from ring-lock response,

are shown in Table 6.1. The reference measurements for spin-lattice relaxation, T1 = 360 ms,

and for spin-spin relaxation, T2 = 200 ms, giving a relaxation ratio, T1/T2 = 1.8. Relaxation

constants estimated from the simulated “perfect” observation, T1 = 360 ms, T2 = 200 ms, match

those measured with standard methods. Relaxation constants estimated from the experimental

observations, T1 = 97 ms, T2 = 54 ms, do not match those measured with standard methods, but

the relaxation ratio is accurate, T1/T2 = 1.8.

6.5. DISCUSSION 121

6.5 Discussion

The results of the first experiment (Figure 6.5), to control the magnetisation on the steady-state

ellipsoid are generally in good agreement with the analytic prediction integrated over the measured

field distributions. The linearly increasing phase of the measurements shows the steady-state ring

trajectory is locked to the envelope frequency. The similarity between measured and predicted

elevation, demonstrates that with known spin-system parameters the ring-lock envelope can drive

the steady-state response to a desired elevation. The minor error in the transverse steady-state

remains an open question; the measured transverse response is very similar to what we would expect

from a single on-resonance isochromat, as opposed to our prediction model which is integrated over

measured field distributions. Exploratory measurements, not shown here, also show unexpected

change in the ellipse profile dependant on the ring-lock frequency used, which could possibly indicate

magnetisation transfer effects. This will be investigated in future work, using distilled rather than

tap water.

The results from the relaxometry experiment (Table 6.1) demonstrate that with an ideal mea-

surement that both relaxation constants can be accurately estimated. The optimisation algorithm

estimates an accurate relaxation ratio from experimental measurements, but inaccurately estimates

the individual relaxation rates. If we consider a term used to calculate the elevation response of the

spin-system,

β =T1

T2

|ωe1|2Bmod

12(

ωrl + ∆)2

+(

1T2

)2

, (6.4b)

which is dominated by the relaxation ratio, T1/T2, when sources of off-resonance,(ωrl + ∆

), are

larger than the inverse spin-spin relaxation, 1/T2. The ring-lock frequencies used for this experiment

ranged from 25 Hz to 400 Hz, which were selected to suppress effects of sample off-resonance, ∆, and

are larger than the inverse spin-spin relaxation, 1/T2 = 5 Hz. This leads to an optimisation surface

with multiple, objectively similar local minima, for the optimisation algorithm to converge to. The

error between measured and optimised surfaces (Figure 6.6b), which is similar to that observed in

the first experiment, may also have contributed to the optimisation algorithm not converging on the

global minimum.

Hargreaves (2012) depicts the bSSFP trajectory on an ellipse, where elevation is dependant on the

flip angle and repetition time. If we consider the pulsed bSSFP excitation envelope as the power

corrected gapped equivalent of a continuous wave constant amplitude excitation, then we can relate

the flip angle and repetition time to the analytical response model developed in this chapter; with

the ring-lock frequency, ωrl, set to zero. If we can maintain a bSSFP signal using an off-resonance

gapped ring-lock excitation, then the presented relaxometry method can also be implemented for

relaxation mapping. The off-resonance excitation properties of the ring-lock envelope may make slice

selection challenging, so a three dimensional bSSFP sequence will be aim for a first implementation.

6.6 Conclusion

The proposed excitation envelope is able to control the steady-state ring-locked magnetisation tra-

jectories. A model for steady-state magnetisation response under this excitation was applied as an

alternate technique for estimation of the ratio of relaxation constants. Future work will focus on

reducing measurement error and application of this excitation envelope to a balanced steady-state

free precession relaxation parameter mapping method.

122

CHAPTER

6.RIN

G-LOCK

EXCITATIO

N

Appendix

6.A Ring-lock excitation envelope

A possible trajectory on the surface of the steady-state elipsoid (2.38) is a ring on a transverse plane anywhere on the ellipsoid,

mx =M0

z

2

√R1

R2(1− a2) sin (ωrlt+ θ) (6.A.1a)

my =M0

z

2

√R1

R2(1− a2) cos (ωrlt+ θ) (6.A.1b)

mz =M0

z

2(1 + a) (6.A.1c)

where, a ∈ (−1, 1], represents elevation in the top hemisphere of the Bloch sphere, ωrl, is rotational frequency of the trajectory and θ is a phase

offset. The derivatives of the trajectory (6.A.1) are,

mx = ωrlM0

z

2

√R1

R2(1− a2) cos (ωrlt+ θ) (6.A.2a)

my = −ωrlM0

z

2

√R1

R2(1− a2) sin (ωrlt+ θ) (6.A.2b)

mz = 0. (6.A.2c)

Substitution of the trajectory (6.A.1) and its derivative (6.A.2) into the Bloch equations (2.34) yelids

ωrlM0

z

2

√R1

R2(1− a2) cos (ωrlt+ θ) = −R2

M0z

2

√R1

R2(1− a2) sin (ωrlt+ θ)−∆

M0z

2

√R1

R2(1− a2) cos (ωrlt+ θ) + ωe

1,y

M0z

2(1 + a)

(6.A.3a)

−ωrlM0

z

2

√R1

R2(1− a2) sin (ωrlt+ θ) = ∆

M0z

2

√R1

R2(1− a2) sin (ωrlt+ θ)−R2

M0z

2

√R1

R2(1− a2) cos (ωrlt+ θ)− ωe

1,x

M0z

2(1 + a)

(6.A.3b)

0 = −ωe1,y

M0z

2

√R1

R2(1− a2) sin (ωrlt+ θ) + ωe

1,x

M0z

2

√R1

R2(1− a2) cos (ωrlt+ θ) . . .

−R1M0

z

2(1 + a) +R1M

0z (6.A.3c)

6.A

.RIN

G-LOCK

EXCIT

ATIO

NENVELOPE

123

re-arranging (6.A.3a),(6.A.3b) for the excitation envelope,

ωe1,x =

1

1 + a

√R1

R2(1− a2)

([∆ + ωrl] sin (ωrlt+ θ)−R2 cos (ωrlt+ θ)

)(6.A.4a)

ωe1,y =

1

1 + a

√R1

R2(1− a2)

([∆ + ωrl] cos (ωrlt+ θ) +R2 sin (ωrlt+ θ)

). (6.A.4b)

Substitution of the the excitation envelope (6.A.4) into the longitudinal trajectory (6.A.3c) gives,

0 = − 1

1 + a

√R1

R2(1− a2)

([∆ + ωrl] cos (ωrlt+ θ) +R2 sin (ωrlt+ θ)

)M0

z

2

√R1

R2(1− a2) sin (ωrlt+ θ) . . .

+1

1 + a

√R1

R2(1− a2)

([∆ + ωrl] sin (ωrlt+ θ)−R2 cos (ωrlt+ θ)

)M0

z

2

√R1

R2(1− a2) cos (ωrlt+ θ) . . .

−R1M0

z

2(1 + a) +R1M

0z (6.A.5)

0 = − 1

1 + a

(R1

R2

(1− a2

))M0z

2

([∆ + ωrl] cos (ωrlt+ θ) sin (ωrlt+ θ) +R2 sin2 (ωrlt+ θ)

). . .

+1

1 + a

(R1

R2

(1− a2

))M0z

2

([∆ + ωrl] sin (ωrlt+ θ) cos (ωrlt+ θ)−R2 cos2 (ωrlt+ θ)

). . .

−R1M0z

(1

2(1 + a)− 1

)(6.A.6)

R1M0

z

2(a− 1) =

−R2

1 + a

(R1

R2

(1− a2

))M0z

2

(sin2 (ωrlt+ θ) + cos2 (ωrlt+ θ)

)(6.A.7)

(1− a) =1

1 + a

(1− a2

)(cos2 (ωrlt+ θ) + sin2 (ωrlt+ θ)

)(6.A.8)

(1− a) (1 + a)

(1− a2)= cos2 (ωrlt+ θ) + sin2 (ωrlt+ θ) (6.A.9)

1 = cos2 (ωrlt+ θ) + sin2 (ωrlt+ θ) (6.A.10)

leads to the well known Pythagorean trigonometric identity (6.A.10). This demonstrates that the excitation envelope (6.A.4) derived from the

transverse magnetisation components (6.A.3a), (6.A.3b) also meet the constraint of the longitudinal magnetisation (6.A.3c).

124

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6.RIN

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EXCITATIO

N

6.A.1 Amplitude and phase

We express the excitation envelope in amplitude and phase terms, for simplicity in power calculations and sequence programing. Using iden-

tity (6.A.11),

A cos (ωrlt+ θ) +B sin (ωrlt+ θ) =√A2 +B2 cos

(ωrlt+ θ − tan−1 B

A

)(6.A.11)

we can rewrite equations (6.A.4a) and (6.A.4b) as

ωe1,x = β

√R2

2 + (∆ + ωrl)2 cos

(ωrlt+ θ − tan−1

(∆ + ωrl

−R2

))(6.A.12a)

ωe1,y = β

√(∆ + ωrl)2 +R2

2 cos


(R2

∆ + ωrl

))(6.A.12b)

where,

β =

√R1

R2

(1− a1 + a

). (6.A.13)

Apply the identities,

tan−1

(1

x

)= − tan−1 (x) +

π

2for: x > 0 (6.A.14a)

tan−1

(1

x

)= − tan−1 (x)− π

2for:x < 0 (6.A.14b)

and,

tan−1 (−x) = − tan−1 (x) (6.A.15)

to (6.A.12) gives,

ωe1,x = β

√(ωrl + ∆)2 +R2

2 cos


(R2

∆ + ωrl

)+π

2

)(6.A.16a)

ωe1,y = β

√(ωrl + ∆)2 +R2

2 cos


(R2

∆ + ωrl

)). (6.A.16b)

6.B.SPIN

-SYSTEM

RESPONSE

UNDER

RIN

G-LOCK

EXCIT

ATIO

N125

We can also express (6.A.16) as,

ωe1,x = β

√(ωrl + ∆)2 +R2

2 cos


(R2

∆ + ωrl

)+π

2

)(6.A.17a)

ωe1,y = β

√(ωrl + ∆)2 +R2

2 sin


(R2

∆ + ωrl

)+π

2

)(6.A.17b)

allowing us to express the ring-lock excitation envelope as a complex exponential

|ωe1|eiφ = |ωe

1| (cosφ+ i sinφ) (6.A.18)

where,

|ωe1| =

√R1

R2

(1− a1 + a

)[(∆ + ωrl)2 +R2

2

](6.A.18a)

φ = ωrlt+ θ − tan−1

(R2

∆ + ωrl

)+π

2(6.A.18b)

6.B Spin-system response under ring-lock excitation

The ring-lock excitation envelope (6.A.17) will drive a spin into the target trajectory (6.A.1) only when the excitation parameters are matched to

the spin properties. Here we consider the case where there is a mismatch between the excitation parameters, R1, R2,∆, and the spin parameters,

R1, R2, ∆, leading to a deviation from the desired trajectory, a, θ, the spin will respond with a trajectory a, θ. The mismatched spin trajectory

is,

mx =M0

z

2

√R1

R2(1− a2) sin

(ωrlt+ θ

)(6.B.1a)

my =M0

z

2

√R1

R2(1− a2) cos

(ωrlt+ θ

)(6.B.1b)

mz =M0

z

2(1 + a) (6.B.1c)

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CHAPTER

6.RIN

G-LOCK

EXCITATIO

N

where a ∈ (−1, 1] represents elevation. The derivative of the trajectory (6.B.1) is,

mx = ωrlM0

z

2

√R1

R2(1− a2) cos

(ωrlt+ θ

)(6.B.2a)

my = −ωrlM0

z

2

√R1

R2(1− a2) sin

(ωrlt+ θ

)(6.B.2b)

mz = 0. (6.B.2c)

Substitution of the trajectory (6.B.1) and its derivative (6.B.2) into the Bloch equations (2.34) yelids

ωrlM0

z

2

√R1

R2(1− a2) cos

(ωrlt+ θ

)= −R2

M0z

2

√R1

R2(1− a2) sin

(ωrlt+ θ

)− ∆

M0z

2

√R1

R2(1− a2) cos

(ωrlt+ θ

)+ ωe

1,y

M0z

2(1 + a)

(6.B.3a)

−ωrlM0

z

2

√R1

R2(1− a2) sin

(ωrlt+ θ

)= +∆

M0z

2

√R1

R2(1− a2) sin

(ωrlt+ θ

)− R2

M0z

2

√R1

R2(1− a2) cos

(ωrlt+ θ

)− ωe

1,x

M0z

2(1 + a)

(6.B.3b)

0 = −ωe1,y

M0z

2

√R1

R2(1− a2) sin

(ωrlt+ θ

)+ ωe

1,x

M0z

2

√R1

R2(1− a2) cos

(ωrlt+ θ

). . .

− R1M0

z

2(1 + a) + R1M

0z (6.B.3c)

Consider equation (6.B.3a) and substitute the ring-lock envelope pattern (6.A.17b),

ωrlM0

z

2

√R1

R2(1− a2) cos

(ωrlt+ θ

)= −R2

M0z

2

√R1

R2(1− a2) sin

(ωrlt+ θ

)− ∆

M0z

2

√R1

R2(1− a2) cos

(ωrlt+ θ

). . . (6.B.4)

+

√R1

R2

(1− a1 + a

)[(∆ + ωrl)2 +R2

2

]sin


(R2

∆ + ωrl

)+π

2

)M0

z

2(1 + a)

6.B.SPIN

-SYSTEM

RESPONSE

UNDER

RIN

G-LOCK

EXCIT

ATIO

N127

Rearrange (6.B.4) to seperate the excitation and spin parameters,

1

(1 + a)

√R1

R2(1− a2)

[ (∆ + ωrl

)cos(ωrlt+ θ

)+ R2 sin

(ωrlt+ θ

)]= . . . (6.B.5)√

R1

R2

(1− a1 + a

)[(∆ + ωrl)2 +R2

2

]sin


(R2

∆ + ωrl

)+π

2

)√R1

R2

(1− a1 + a

)[ (∆ + ωrl

)cos(ωrlt+ θ

)+ R2 sin

(ωrlt+ θ

)]= . . . (6.B.6)√

R1

R2

(1− a1 + a

)[(∆ + ωrl)2 +R2

2

]cos


(R2

∆ + ωrl

)).

Apply (6.A.11) to the LHS of (6.B.6)√R1

R2

(1− a1 + a

)[(∆ + ωrl)2 + R2

2]

cos


(R2

∆ + ωrl

))= . . . (6.B.7)√

R1

R2

(1− a1 + a

)[(∆ + ωrl)2 +R2

2

]cos


(R2

∆ + ωrl

)).

Equate the amplitude of sinusoids in (6.B.7),√R1

R2

(1− a1 + a

)[(∆ + ωrl)2 + R2

2]

=

√R1

R2

(1− a1 + a

)[(∆ + ωrl)2 +R2

2

]1− a1 + a

=R1 R2

R1R2

(1− a1 + a

)((∆ + ωrl)

2 +R22

(∆ + ωrl)2 + R22

)to give an expression of the spin elevation,

a(a, ωrl, R1, R2,∆, R1, R2, ∆

)=

1− β1 + β

, (6.B.8)

where

β =R1R2

R1R2

(1− a1 + a

)((∆ + ωrl)

2 +R22

(∆ + ωrl)2 + R22

). (6.B.9)

128

CHAPTER

6.RIN

G-LOCK

EXCITATIO

N

The spin elevation (6.B.8) can also be expressed as,

a(a, ωrl, R1, R2,∆, R1, R2, ∆

)=R1R2 (1 + a)

[(∆ + ωrl

)2 − R22]−R1R2 (1− a)

[(∆ + ωrl)

2+R2

2

]R1R2 (1 + a)

[(∆ + ωrl

)2+ R2

2]

+R1R2 (1− a)[(∆ + ωrl)

2+R2

2

] . (6.B.10)

Equate the phase of sinusoids in (6.B.7), to give an expression for the spin phase,

ωrlt+ θ − tan−1

(R2

∆ + ωrl

)= ωrlt+ θ − tan−1

(R2

∆ + ωrl

)θ = θ − tan−1

(R2

∆ + ωrl

)+ tan−1

(R2

∆ + ωrl

). (6.B.11)

Chapter 7

Conclusion

Magnetic resonance imaging encompasses a wide selection of well developed methods for gaining

insight into the properties of a physical object. In this thesis, we have explored interrogating the spin

system with complex continuous wave excitation envelopes as an alternative method for estimating

spin system properties. The use of periodic excitation envelopes that drive the magnetisation into

steady-state trajectories allows for efficient model construction when optimising objective functions

to estimate system parameters. Using a series of Rabi modulated excitations we pose spectroscopy

and proton projection imaging as inverse problems. In a similar manner, using a series of ring-

lock excitations, we estimate relaxation constants using an analytical model derived from the Bloch

equations. Rabi modulated excitation has favourable properties for simultaneous excitation and

measurement, opening possibilities for observation of ultra-short relaxation spin systems and closed-

loop control. A current limitation of the proposed methods is measurement efficiency. A suggested

solution is to make controlled transitions between steady-state trajectories, and online monitoring of

the magnetisation to detect a steady-state condition. Our experimental investigation of steady-state

trajectories makes use of two pseudo-simultaneous excitation and measurement protocols. Whilst

these methods were adequate to explore the proof-of-concept applications, hardware modifications

are suggested to unlock the full potential of continuous wave excitation patterns.

7.1 Summary of original contributions

Chapter 4 investigated the off-resonance response to Rabi modulated continuous wave excitation.

The off-resonance response is fully expressed by harmonics of the excitation modulation frequency

and varies dependant on the excitation envelope parameters. The accuracy of the harmonic pre-

diction model of the off-resonance steady-state response was measured experimentally, then used to

construct a forward model, posing spectroscopy as an inverse problem. Using the proposed method,

ethanol spectra were reconstructed from a series of measured steady-state harmonics, acquired un-

der Rabi modulated excitation. Improvement to the reconstructed spectra was achieved by using

an A-optimality based algorithm to select experimental excitation parameters.

In Chapter 5, we took the idea of encoding off-resonance information via a series of Rabi modulated

excitations and applied it to imaging. Under a radial gradient, spatial information rather than

chemical shift information is encoded in the harmonic response to Rabi modulated excitation. A

129

130 CHAPTER 7. CONCLUSION

more efficient gapped measurement protocol was implemented and verified by measurement of the off-

resonance response. Projections of proton density were reconstructed from multiple measurements

acquired using the gapped protocol, and these were back-projected to form a two dimensional image,

whose structure was in line with reference data. Preliminary simulations of the imaging method,

applied to a numerical phantom with mixed relaxation rates, showed contrast between different spin

regions.

Chapter 6 revisited the known response to a constant amplitude excitation (Bloch, 1946; Abragam,

1961), applying a frequency offset which generates a steady-state ring trajectory. An analytical

solution of the Bloch equations was derived for the spin-system response under this ring-lock excita-

tion. Power and frequency of the ring-lock excitation envelope are expressed in terms of the system

parameters and a target response for the ring-lock trajectory. To verify the excitation and response

model, a basic spin-system was accurately controlled over the steady-state ellipsoid. Relaxometry

was posed as an optimisation problem, where relaxation rates are estimated from the response to

a series of ring-lock excitations. Simulations of this method were able to accurately estimate spin-

lattice and spin-spin relaxation rates, inferences from experimental data were limited to accurate

prediction of the ratio of the two rates.

7.2 Future work

The methods proposed in this thesis exploit information encoded in steady-state trajectories. The

described measurement techniques assume the spin-system is initially at equilibrium and apply the

excitation for a duration adequate to ensure a steady-state has been achieved. The efficiency of

the methods is limited, as only a small portion of the total experimental time is used for data

acquisition. If the magnetisation could be smoothly transitioned from one steady-state to the next,

the experimental efficiency would be greatly improved. This is a non-trivial problem, even in the

unrealistic case of a single spin; knowledge of the instantaneous spin orientation and an estimate

of relaxation constants would likely be required to maintain a controlled transition. The potential

to observe the spin-system during Rabi modulated excitation could find application here; first to

maintain an estimate of the spin position and secondly to monitor the steady-state transition and test

for a steady-state condition such as periodicity. If the data collection were then considered streaming

from an active experiment, model based estimation could be run throughout data acquisition, and

the experiment terminated once the parameter estimates converged.

One advantage of a fully simultaneous excitation and measurement technique is the absence of an

echo time, which allows the observation of ultra-short relaxation signals. Rabi modulated excitation

has two properties, low power and harmonic decoupling, that may be exploited to allow simultaneous

continuous wave excitation and measurement. It is possible to maintain a significant steady-state

trajectory with low power Rabi modulated excitation envelopes. Other groups have demonstrated

simultaneous excitation and measurement with custom hardware, when the excitation power is kept

to a minimum (Idiyatullin et al., 2012; Ozen et al., 2017a). The frequency decoupling between

Rabi modulated excitation and the harmonic response of the spin-system make it ideal for fre-

quency filtering hardware (Brunner et al., 2012; Zheng et al., 2011). There are certain similarities

to Magnetic Particle Imaging where a mixture of frequency filtering and active analog cancellation

has been suggested (Graeser et al., 2013). It was beyond the scope of this thesis to implement a

hardware solution for simultaneous continuous wave transmission and signal reception. All of the

afore-mentioned techniques, however, are potential candidates for future hardware realisation of the

methods developed in this thesis.

This thesis demonstrates the ability to investigate continuous wave steady-state phenomena with

7.2. FUTURE WORK 131

pseudo-simultaneous excitation and measurement protocols. Continuous wave excitation patterns

allow the construction of efficient prediction models and elicit an information-rich steady-state re-

sponse from which underlying spin-system properties can be reconstructed. It is anticipated that

further development of these concepts and related hardware modifications will lead to new continuous

wave imaging paradigms.

132 CHAPTER 7. CONCLUSION

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