Strategies for high resolution functional MRI at 7 Tesla: Optimization of Methods

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Strategies for high resolution

functional MRI at 7 Tesla

Optimization of Methods

Catarina Henriques Lopes dos Santos Rua

Supervisor: Prof. M. Tosetti

Prof. A. Del Guerra

Advisor: Dr. M. Costagli

Department of Physics ’E. Fermi’

University of Pisa

This dissertation is submitted for the degree of

Doctor of Philosophy

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For my mom, dad and brother.

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Hora

Sinto que hoje novamente embarco Para as grandes aventuras,

Passam no ar palavras obscuras E o meu desejo canta - por isso marco Nos meus sentidos a imagem desta hora. Sonoro e profundo

Aquele mundo

Que eu sonhara e perdera Espera

O peso dos meus gestos.

E dormem mil gestos nos meus dedos. Desligadas dos círculos funestos Das mentiras alheias,

Finalmente solitárias,

As minhas mãos estão cheias De expectativa e de segredos Como os negros arvoredos

Que baloiçam na noite murmurando. Ao longe por mim oiço chamando A voz das coisas que eu sei amar. E de novo caminho para o mar. Sofia de Mello Breyner Andresen

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Declaration

I hereby declare that except where specific reference is made to the work of others, the contents of this dissertation are original and have not been submitted in whole or in part for consideration for any other degree or qualification in this, or any other university. This dissertation is my own work and contains nothing which is the outcome of work done in collaboration with others, except as specified in the text and Acknowledgements. This dissertation contains fewer than 65,000 words including appendices, bibliography, footnotes, tables and equations and has fewer than 150 figures.

Catarina Henriques Lopes dos Santos Rua April 2017

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Acknowledgements

A doctoral thesis is never a solitary endeavour. It is my opportunity here to openly thank all of those who have helped me during these difficult years of PhD.

First, I would like to thank my supervisors Michela Tosetti, Mauro Costagli and Prof. Alberto del Guerra, each of which helped the project go forward in different ways. Particularly, Mauro has given me a thorough guidance and never gave up on our work, even during the roughest times.

I am grateful for the Marie Curie Training Network for the financial support, allowing me to develop fruitful collaborative work, and friendships, with the AnaBea and GRC group in Munich, Oliver and the BMMR group in Magdeburg, and Stephen and Gareth from King’s College London.

An extremely special thanks to Mark Symms: a true mentor at work and so much more; a life counsellor, friend, best maker of Cat puns that I know of, great moving buddy, etc etc. ’You always had something nice to say to me, never jugging my awkwardness; I cannot thank you enough for what you have done for me.’

I would like to thank my colleagues from Imago7 in Pisa, who made my life easier even if I was always jumping from place to place. To my friends Alessandro, Angela, Chiara and Francesco, who took me in during a complicated time and made me smile again. I have to thank from the bottom of my heart Loretta: ’Mi hai dato quello abbraccio forte, mi hai guidato a fare le scelte giuste, mi hai salvato la vita.’

To Carolina: you are a forever friend for me. I couldn’t have asked for a better partner in this journey. To Joana and Filipa, best friends, who always gave me a call when I was down. Thank you so much for being there.

I need to thank Alison and Wiktor who, over the past seven months, in spite of my endless moans about writing-up, made me laugh and pushed me to finish.

And at last, but most importantly, I would like to thank my family: Mãe, porque és a trave forte na minha vida; Pai, porque desde pequena me convidas à curiosidade; Alex, o meu primeiro amigo, porque me ensinas a viver para além do estudo; e às Avós. . . com saudade.

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Abstract

The great interest in ultra-high field functional magnetic resonance imaging (fMRI) comes from the observation that the functional contrast increases supra-linearly with field strength, allowing imaging to be performed at unprecedented spatial resolutions. However, simply reducing partial volume effects is not sufficient to precisely localize task-induced activation due to the indirect mechanisms that relate brain function and the changes in the measured signal. Also, the increased magnetic field strength increases the effect of problematic image artefacts which arise from local B0inhomogeneities. In the first experimental chapter, a Gradient-Echo Echo Planar Imaging (GRE-EPI) and Spin-Echo EPI (SE-EPI) were evaluated at 7 T at sub-millimetre resolutions in comparison to larger voxel sizes. While similar activation foci were obtained in all acquisitions, SE-EPI acquired with sub-millimetre voxels showed a significant decrease in temporal signal-to-noise ratio which hampered the extent of the activation clusters. At these resolutions, T₂∗-weighted data collected with GRE-EPI provided higher functional contrast and sensitivity. The selection of active voxels within an appropriate gray-matter mask enabled satisfactory specificity to the true site of activation. This work was published in the proceedings of the 24th ISMRM meeting and led to a first-author paper in the Magnetic Resonance Imaging journal.

In a second study, Tailored Radio-Frequency (TRF) pulses optimized to compensate for through-plane field inhomogeneities in GRE-EPI at 7 T were tested with an fMRI experiment targeting the ventral occipito-temporal cortex. The TRF showed BOLD signal recovery in signal dropout areas, highlighting its potential use at 7 T. This work was published in the proceedings of the 23rd ISMRM meeting and is currently under review in the Magnetic Resonance Materials in Physics, Biology and Medicine journal. In a third study, a Multi-Echo EPI sequence was modified to acquire a field map at each fMRI volume by switching the phase encoding polarity of even echoes. Tests on healthy subjects revealed localized physiology-related motion effects captured by the dynamic field maps. In addition, the dynamically distortion corrected fMRI showed sufficient unwarping fidelity and no observable loss of functional sensitivity. This work was published in the proceedings of the 25th ISMRM meeting.

The work presented in this thesis has demonstrated improvements in localization of functional activity at 7 T.

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List of figures

2.1 Geometric representation of the precession of the magnetic moment ⃗µ around an external magnetic field, ⃗B0. Representation of the mi-croscopic magnetic moments of a bulk sample of nuclear spins on a magnetic field, ⃗B₀. . . 7 2.2 Net magnetization vector ⃗Mevolution under the application of an RF

pulse at the frequency ω0in the laboratory frame of reference and the rotating frame of reference. . . 8 2.3 Longitudinal magnetization regrowth plot following a 90° pulse to

equilibrium value, M0. Transverse magnetization decay plot. . . 10 2.4 A free induction decay detection experiment. . . 12 2.5 A spin echo experiment. . . 13 2.6 Sequence of events that occur on a spin echo experiment in the rotating

reference frame. . . 13 2.7 Spatial encoding using a Maxwell pair of coils gradients in an MR

system. . . 15 2.8 Configuration of gradient coils for each of the x, y, and z directions in

an MRI system. . . 15 2.9 Orientation of imaging-planes according to orthogonal axis. . . 16 2.10 Illustration of the slice select gradient. . . 17 2.11 Magnitude image and corresponding k-space representation from a

transverse slice of the head of a healthy volunteer. . . 18 2.12 Illustration of an example Cartesian k-space sampling trajectory. . . . 20 2.13 Pulse sequence diagram for a 2D gradient echo imaging sequence. . . 21 2.14 Example of a pulse sequence diagram for a 2D spin-echo sequence. . 22 2.15 Simulated longitudinal magnetization regrowth curves and transverse

magnetization decay curves for brain matter tissue. The combined choice of TR and TE allows different contrast characteristics. . . 25 2.16 Tissue contrast variations on MRI images of the same subject:

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2.17 Sequence diagram and k-space trajectory for a single-shot GRE-EPI pulse sequence and a single-shot SE-EPI pulse sequence. . . 27 2.18 Diagram of a trapezoidal gradient showing the gradient amplitude,

slew rate and plateau. The rise and decay times are represented by ta and td, respectively, and the gradient plateau by tplateau. Representation of an EPI read-out gradient waveform and associate k-space trajectory. G_x,prepresents the pre-phasing lobe, while the readout gradient lobes are labelled as Gx,1, Gx,2, etc. The solid arrows show the k-space trajectory of the readout gradient lobes. . . 29 2.19 Example of 2-shot EPI k-space coverage with sequential and

inter-leaved forms. Example dataset of a 2-shot EPI acquisition without and with motion. . . 31 2.20 Demonstration of Sensitivity Encoding, SENSE, for image unfolding. 34 2.21 Simulation of image artefacts in EPI. . . 37 2.22 Chemical Shift artefact effect observed on a water and fat phantom

scanned at 7 tesla. . . 39 2.23 Conventional 2D gradient-echo; 2D single-shot gradient-echo EPI;

Conventional 2D spin-echo; 2D single-shot spin-echo EPI. . . 41 2.24 Evaluation of signal dropout in gradient-echo EPI data on two

trans-verse slices of a human brain showing the lower temporal lobes and the orbito-frontal cortex. . . 43

3.1 Schematic illustration of the neurophysiological processes associated with BOLD fMRI. . . 47 3.2 Schematic of the BOLD effect events that result in increased fMRI

signal. . . 48 3.3 Illustration of the BOLD vascular effects across different vessel types.

Extravascular effects from interaction of dHb with extravascular water molecules in a large and a small vessel. . . 49 3.4 Vascular density in the calcarine fissure, where blood vessels appear

in dark in the cortical surface or across the cortical gray matter. . . 50 3.5 Schematic illustration of the effect of acquisition resolution on

GRE-EPI BOLD fMRI in the functional sensitivity and specificity to macro and microvasculature. . . 51 3.6 Examples of columnar functional areas in the visual cortex. . . 52 3.7 Calculated curves for R∗₂(i.e., R∗₂= 1/T₂∗) changes in GRE imaging

and R2(i.e., R2= 1/T2) in SE imaging on extravascular space at 1.5 T as a function of vessel size for two levels of O2saturation (HbO2). . . 53

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List of figures xix

3.8 Simulations for fMRI extravascular (EV) signal contributions in GRE and SE at a TE set for T₂∗and T2, respectively. . . 54 3.9 Example of motion correction on fMRI data. . . 58 3.10 Axial view of the magnitude and phase reconstructed images acquired

with a gradient-echo multi-echo EPI sequence. . . 60 3.11 Reversed phase-encoding EPI on example dataset. . . 63 3.12 Point-Spread Function sequence diagram of a Gradient Echo EPI

sequence where the spin-warp gradient module added and the corre-sponding k-space diagram. . . 64 3.13 Comparison of the distortion correction process in GRE- and SE-EPI

images of an agar phantom with the same contrast properties as the human brain gray matter. . . 66 3.14 Representation of the haemodynamic response function (HRF). . . 69 3.15 Example z-score contrast maps of a human fMRI acquisition with

z-score thresholded at 2.3 without correction for multiple comparisons and cluster-based thresholded. . . 71 3.16 Block design fMRI paradigm with three epochs of alternating blocks

of visual presentation of Faces (F) and objects (O) interleaved with periods of rest. . . 73

4.1 Main components on a 7 T MRI system. . . 76 4.2 Photograph of the currently used coil for brain imaging at 7 tesla in

the GE MR950 scanner fabricated by Nova Medical (Nova Medical, Wilmington,MA,USA). Photograph of an in-house built single loop coil. 77 4.3 Examples of dielectric effects on a agar phantom at 7 T. . . 78 4.4 Examples of EPI volumes at different acquisition resolutions. . . 81

5.1 Registration, ROI definition and segmentation on a representative subject: (A) Sagittal view of EPI coverage at 0.75×0.75×1.5 mm3 res-olution overlaid (top image) and outlined (bottom image) on FSPGR; (B) outline of the V1 and hMT+ ROIs on GRE-EPI (top) and SE-EPI (bottom); (C) gray-matter segmentation in V1 and hMT+ ROIs overlaid on GRE-EPI (top) and SE-EPI (bottom) images. . . 86

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5.2 Output histograms of the automated three-class segmentation (A) and two-class segmentation (B) in the same subject shown in Figure 1. Green and red voxels (C, bottom image) all represent GMfirstvoxels selected by the three-class segmentation. Red voxels represent the OT voxels identified by the two-class segmentation that were excluded from GMfirst. Green voxels represent the final GM mask. For reference, the GRE-EPI image without any overlay is also shown (C, top image). 89 5.3 Cluster threshold ’z-full’ activation maps of GRE- and SE-EPI at 1.5 ×

1.5 mm2and at 0.75 × 0.75 mm2in-plane resolution in a representative subject. The same imaging slice is displayed in all four acquisitions. . 90 5.4 Quantitative measurements of the fMRI time-course for GRE and SE

at different in-plane resolutions, including temporal SNR (A-D), PSC (E-H) and CNR (I-L). . . 94 5.5 (A) Cluster threshold ‘z-full’ activation maps from an example

sub-ject’s high in-plane resolution (0.75×0.75 mm2) GRE dataset. Active voxels from the map shown in (A) are displayed separately for three z-score intervals: 0-20% (B), 40-60% (C) and 80-100% (D). GM acti-vations are shown in red and NGM actiacti-vations in blue. The markedly hypo-intense structures (indicated by green arrows) represent large draining veins, as ascertained by following the course of these struc-tures through slices. . . 96 5.6 Subject average gray-matter specificity as a function of in-plane

res-olution in GRE and SE data in V1 (A) and hMT+ (B). Gray-matter sensitivity as a function of in-plane resolution in GRE and SE data in V1 (C) and hMT+ (D). . . 98 5.7 Subject average ROC curves for varying z-score cluster thresholds in

V1 (A) and hMT+ (B) for the highest in-plane resolution datasets. Blue and red curves refer to GRE-EPI and SE-EPI, respectively. Colour-coded arrowheads indicate the z-score cluster thresholds = 2.3. The AUC in GRE-EPI was significantly larger than in SE-EPI (p=0.001, left bar pair in C), and this difference was more pronounced in V1 than in hMT+ (middle and right bar pairs in C). . . 98 5.8 Plots of the full activated volume, AVz− f ull, (in mm3) as a function of

voxel size (in mm3) for GRE and SE in V1 and hMT+. . . 99

6.1 HS pulse simulation results optimized for 7 tesla. . . 116 6.2 Simulated normalized steady state gray matter voxel signal as a

func-tion of through-slice susceptibility gradient for the 7 tesla optimized HS pulse. . . 116

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List of figures xxi

6.3 Subjects’ dtSNR in percentage in two acquired slices covering the temporal lobe (TL), Orbital Frontal (OF) and Fusiform Cortex (FC). . 117 6.4 Average dPSC (A) and dAV (B) across all subjects in the 10 defined

ROIs (error bars represent inter-subject standard error) for the con-trasts F > B, O > B and I > B. For each subject and in each region, the average regional PSC was obtained from all the active voxels of the corresponding functional contrast. Statistical significance was evalu-ated with the Friedman Test for each ROI and contrast independently. The asterisk indicates a statistically significant result (p < 0.05). . . . 118 6.5 Group average tSNR maps for the STD and TRF runs in seven slices

in the MNI template space. The corresponding dtSNR maps in per-centage are shown on the bottom row. . . 119 6.6 Group level one-sample t-test I > B z-score contrast maps obtained

with the STD and TRF GRE-EPI acquisitions. The green underlay mask represents all voxels showing positive dtSNR on a group level. . 120

7.1 Comparison of the ’distortion-free’ 2D GRE scan with distortion corrected images on the phantom data . . . 134 7.2 Pdiff histograms for the uncorrected i1, ref-echo1-DC, ref-echo2-DC

and MEPI-FB-DC. The top right corner inset shows the non-parametric fits on the Pdiff data. . . 134 7.3 Phantom temporal standard deviation (t-SD) of the detrended ∆B(t)

shown in hot colours on all the acquired slices. The underlay is the 2D GRE scan. The green arrowhead points to a particularly large air-pocket inside the phantom that produced locally high t-SD(∆B(t)). 135 7.4 Histograms of the per-voxel tSNR (A) and t-SD (B) on the phantom

time-series datasets: uncorrected i1, ref-echo1-DC, ref-echo2-DC, st-DC and dy-st-DC. Insets in the upper right corner of each graph display a zoomed view of the histograms, showing the peaks and the ascending and descending lobes. . . 136 7.5 Histograms of the per-voxel tSNR (A) and t-SD (B) on the uncorrected

i₁time-series with five added levels of smoothing. . . 136 7.6 Comparison of uncorrected i1and MEPI-FB-DC images from Subject

1 and Subject 2. Example ∆B maps are also shown for both subjects. . 138 7.7 (A) Subject 1 and (B) Subject 2 (Run #1 and Run #2) t-SD of the

detrended ∆B(t) displayed in hot colours and overlayed on the 2D GRE scan. Green boxes and arrowheads point to where large changes in the field occurring around the eyes (1), brainstem (2), vessels (3), skull (4), and ventricles (5). . . 139

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7.8 Number of voxels that showed positive and negative correlation be-tween the ∆B(t) maps and the physiological data. . . 140 7.10 dPSC (in %) in subject 1 (A) and subject 2 (B) and dz-score (in %) in

subject 1 (C) and subject 2 (D) obtained from the 10 defined ROIs with the contrasts: FF > B, NFF > B and F > B. For each subject and in each region, the average regional PSC and average z-score were obtained from the common active voxels to the dy-DC and to the st-DC z-score maps of the corresponding functional contrast. . . 142

A.1 Schematic and k-space trajectory of the gradients applied in y and x directions of the first two lines of k-space during a single-shot EPI acquisition. . . 177

B.1 Plot of PPG waveform (top), respiration (middle), and scanner triggers (bottom) for a time interval that includes the acquisition of a full 3D volume in 2-shot EPI. . . 183 B.2 Cardiac and respiratory plots of the waveforms after 2-shot

subsam-pling method at the scan TR. . . 183 B.3 Subject 1 and Subject 2 PPG and Respiration sample waveforms. . . . 184 B.4 Subject 1 z-statistic contrast maps on 8 slices for the physiological

EVs from method B and method C. In method B, the most significant z-maps for the cardiac EVs and the two most significant results for the respiration EVs are shown. . . 186 B.5 Subject 2 z-statistic contrast maps on 8 slices for the physiological

EVs from method B and method C. In method B, the most significant z-maps for the cardiac EVs and the two most significant results for the respiration EVs are shown. . . 187 B.6 Same slice comparison of activation maps for the paradigm contrast in

subject 1 and 2 for methods A, B and C. . . 188 B.7 Functional active cluster in the primary visual cortex for subject 1 and

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List of tables

2.1 Representative values of relaxation parameters T1and T2in millisec-onds for hydrogen in various human tissues at B0= 1.5T and 37°C. . 10 2.2 Guidance for choice of TE and TR parameters for tissue-contrast

control in MR images. . . 26 2.3 Effect of imaging parameters that are selected at the console on EPI

images for a sequence with a fixed field of view. . . 41

3.1 Representative values of relaxation parameters T2and T₂∗in millisec-onds for hydrogen in various human tissues at 1.5 T, 3 T and 7 T. . . . 53

5.1 Results from Adonis statistical tests on (A) tSNR, (B) PSC, (C) CNR, (D) Spec_GM, (E) SensGM and (F) AVz− f ull. The statistical models included the factors: ROI (V1 and hMT+), sequence (SE and GRE), resolution (0.75×0.75×1.5 mm3, 1.0×1.0×1.5 mm3, 1.5×1.5×1.5 mm3), z-score percentile interval (0-20%, 20-40%, 40-60%, 60-80%, 80-100%) and tissue type (GM and NGM). The * symbol indicates a statically significant result. . . 93 5.2 Results from Adonis statistical tests on GRE and SE-EPI datasets

on (A) tSNR, (B) PSC, (C) CNR, (D) Spec_GM, (E) SensGM and (F) AVz− f ull. The models included the factors: ROI (V1 and hMT+), resolution (0.75×0.75×1.5 mm3, 1.0×1.0×1.5 mm3, 1.5×1.5×1.5 mm3), z-score interval (0-20%, 20-40%, 40-60%, 60-80%, 80-100%) and tissue-type (GM and NGM). The * symbol indicates a statically significant result. . . 95

6.1 MNI-derived ROIs defined for each subject. . . 114 6.2 Group level cluster sizes and maximum z-scores for the STD and TRF

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Nomenclature

Acronyms / Abbreviations

T₁ Spin-Lattice Relaxation

T₂ Spin-Spin Relaxation

ASSET Array Spatial Sensitivity Encoding Technique

AUC Area Under Curve

AV Activation Volume

B Baseline

BOLD Blood Oxygen Level Dependent

CBF Cerebral Blood Flow

CBV Cerebral Blood Volume

CMR_glc Cerebral Metabolic Rate of Glucose

CMR02 Cerebral Metabolic Rate of Oxygen

CNR Contrast-to-Noise Ratio

CSF Cerebral Spinal Fluid

dHb deoxyHaemoglobin

dPSC Percent Signal Change difference

dy-DC dynamic Distortion Correction

EPI Echo Planar Imaging

EV Extra-Vascular

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F Faces

FC Fusiform Cortex

FDR False Discovery Rate

FF Famous Faces

FFA Fusiform Face Area

FID Free Induction Decay

fMRI functional Magnetic Resonance Imaging

FoV Field of View

FSPGR Fast SPoiled Gradient Recalled

FWER Family-Wise Error Rate

GE General Electric

GLM General Linear Model

GLS Generalized Least Squares

GM Gray Matter

GRAPPA Generalized Autocalibrating Partially Parallel Acquisitions

GRE Gradient Recalled Echo

GRF Gaussian Random Field

HRF Haemodynamic Response Function

HS Hyperbolic Secant

ISMRM International Society for Magnetic Resonance in Medicine

IT Inferior-Temporal

IV Intra-Vascular

MEPI-FB Multi-Echo EPI - Flipped Blips

MNI Montreal National Institute

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Nomenclature xxvii

MT Middle-Temporal

NFF Non-Famous Faces

NGM Non-Gray Matter

NMR Nuclear Magnetic Resonance

OF Orbito-Frontal

OLS Ordinary Least Squares

PE Phase Encoding

PET Positron Emission Tomography

PNS Peripheral Nerve Stimulation

PPG PhotoPlethysmogram

PSC Percent Signal Change

PSF Point Spread Function

PVE Partial Volume Effects

rBW receiver Bandwidth

RF Radio-Frequency

RO Read Out

ROC Receiver Operating Characteristic

ROI Region-Of-Interest

RVT Respiration Volume per Time

SE Spin Echo

SensGM Gray-Matter Sensitivity

SENSE Sensitivity Encoding

SLR Shinnar-Le Roux

SMASH Simultaneous Acquisition of Spatial Harmonics

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SpecGM Gray-Matter Specificity

SS Slice Selection

st-DC static Distortion Correction

STD Standard

SWAN Susceptibility Weighted ANgiography

t-SD temporal Standard Deviation

TE Echo Time

TLE Temporal Lobe Epilepsy

TR Repetition Time

TRF Tailored Radio-Frequency

tSNR temporal Signal-to-Noise Ratio

UHF Ultra-High Field

VASO VAscular Space Occupancy

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Chapter 1 Introduction

In contrast with the more traditional role of magnetic resonance imaging (MRI) as a tool for studying brain anatomy and pathology, the use of MRI for studying brain function has significantly increased over the last two decades. Moreover, the drive to increase image signal to noise ratio (SNR) led to the development of MR systems operating at increasingly higher field strengths, further benefiting the functional MRI (fMRI) signal [1, 2], allowing research to evolve to unprecedented resolutions [3– 10]. Yet, ultra-high field (UHF) fMRI is not short on new challenges. The spatial accuracy of high-field, high-resolution images is greatly influenced by the magnetic field instability and image-to-image fluctuations.

The blood oxygen level dependent (BOLD) contrast is the most common contrast mechanism to measure brain activity. It relies on dynamic changes of blood oxygena-tion levels in brain vessels occurring during the active state [11–14]. A significant proportion of recent publications describe how gradient-echo (GRE) methods at 7 tesla are greatly sensitive to BOLD allowing the mapping of neuronal activity [15, 16, 2, 17]. However, a major setback for measuring a spatially accurate neuro-vascular coupling is the contribution of large draining veins to the GRE signal. At 7 tesla, BOLD imaging carried out with spin-echo (SE) based acquisition sequences allows, in theory, a signal more spatially localised to the underlying neurological activation.

Echo planar imaging (EPI) is one of the fastest MRI pulse sequences [18] reaching very high signal sampling at relatively high temporal resolutions. Hence, it is the method of choice for BOLD imaging whilst the subject performs a prescribed mental task. However, EPI acquisitions are highly susceptible to off-resonant field effects, that are exacerbated at higher field strengths, and can cause severe geometric distortions, signal modulation and, in extreme cases, signal loss due to complete dephasing.

The aim of this thesis is to explore fMRI methods at 7 tesla in order to perform accurate ultra high-resolution mapping of brain function. The goal is to probe the underlying contrast mechanisms of gradient- and spin-echo signals characterizing the

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functional activation at various spatial resolutions. In this thesis, issues related to echo-planar imaging at ultra-high fields are also investigated in the context of fMRI. A method devised at increasing sensitivity to BOLD signal changes in areas suffering from through-plane signal dropout is tested at 7 tesla, aiming to recover signal and detect brain function in lower-temporal brain regions. In-plane off-resonance effects cause geometrical distortions with both static and dynamic components. In this thesis a technique is developed aiming at measuring and correcting these effects, whilst minimizing any changes to the nature of the functional image acquisition.

1.1 Thesis Overview

In Chapter 2 the fundamental principles of Magnetic Resonance Imaging (MRI) as a versatile clinical tool are described. A basic explanation of the nature of the physical process that occurs when a nucleus is under an external magnetic field is given. This concept is followed by a description of how the spin state system is manipulated to produce detectable radio-frequency signals when irradiated with radio-frequency pulses. The final portion of the chapter focuses on the description of the EPI technique and the fundamental sequences used for functional MRI, and on describing the most important occurring EPI artefacts.

A background on functional MRI is given in detail in Chapter 3. The theory behind the BOLD contrast is presented, with an overview of the factors that influence its measurement with gradient and spin-echo MRI. A description of the acquisition methodologies used in functional MRI is provided, followed by the most common post-processing techniques to analyse the fMRI data.

In Chapter 4 the details regarding imaging with a 7 tesla static magnetic field are described, from hardware changes to changes in the detected signal. Finally, an overview of the effects of field strength on fMRI data is given.

On the first major study of this thesis, the standing issues regarding BOLD contrasts at ultra-high field are addressed. In Chapter 5 the characterization of the functional signal obtained with gradient-echo EPI (GRE-EPI) and spin-echo EPI (SE-EPI) fMRI contrasts is made in the human visual cortex at 7 tesla. In this study, human brain data are optimised at standard and ultra-high resolutions and the signal and spatial distribution of functional activation are evaluated. This work has led to two major contributions (refer to point 1 of subsection "Summary of Contributions").

Off-resonance fields during the slice-selection process on a GRE-EPI acquisition generate localized signal drop-off which, in turn, impairs BOLD signal sensitivity in these regions. The application of tailored radio-frequency (TRF) pulses, shown successful signal recovery at 3 T [19], are tested for the first time at 7 tesla on healthy

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1.2 Summary of Contributions 3

human subjects performing a functional task targeting localized signal-loss areas. This work is described in Chapter 6, was published in the proceedings of ISMRM and is currently under second round of reviews at the Magn. Reson. Mater Phy. journal (refer to point 2 of subsection "Summary of Contributions").

Magnetic off-resonance fields are largely invariant in time. However, dynamic components related to motion or physiological effects might arise during a human functional MRI scan. Chapter 7 describes the novel development and implementation of a multi-echo gradient-echo EPI sequence with opposite phase-encoding directions within the echo-modules. This approach, designated MEPI-FB, allows the estimation of a single magnetic field map per repetition time. The temporal (in)stability of the magnetic field is characterized in healthy volunteers undergoing a visual stimulation task and the performance of a dynamic geometrical distortion correction is evaluated. This work was published in the 2017 proceedings of ISMRM (point 3 of subsection 1.2).

The main results and key-findings of the work described on the last three chapters are included in Chapter 8. Possible future directions for the project are also presented along with suggestions for applications of the work in different settings.

1.2 Summary of Contributions

The research work described in this thesis was presented in the following contributions:

1. A systematic comparison of SE and GRE-EPI fMRI signals at 7 tesla at various acquisition resolutions. This work was presented at the 2016 annual meeting of the International Society of Magnetic Resonance in Medicine (ISMRM) [20] and was accepted as an original article in the Magnetic Resonance Imaging Journal [21].

2. The application for the first time of TRF pulses at 7 tesla in areas sensitive to through-plane signal dropout. This study was presented at the 2015 annual meet-ing of ISMRM [22] and is currently under review at the Magnetic Resonance Materials in Physics, Biology and Medicine Journal.

3. The development of a novel technique for detecting static and dynamic changes in the off-resonance fields, which allows a more appropriate geometrical un-warping of fMRI data at 7 tesla. This work was presented at the 2017 annual meeting of ISMRM [23].

Additional relevant work accepted for presentation to the scientific community includes:

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1. The contribution to a publication regarding the assessment of Silent T1-weighted head imaging at 7 T. This work was accepted as a full article in the European Radiology Journal [24].

2. A study of the PSF distortion correction for ultra-high field BOLD fMRI. This work was presented at the 2017 annual meeting of the ISMRM [25].

3. An oral communication at the ’Congresso Nazionale Società Italiana di Fisica’ entitled ’Functional organization of the primary visual cortex’ [26].

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Chapter 2 Magnetic Resonance Imaging

2.1 Introduction

The phenomenon of Nuclear Magnetic Resonance (NMR) as a medical imaging tool has its beginnings in the early 1970’s with work from Lauterbur [27], and Mansfield and Grannell [28]. The theoretical foundations were, nevertheless, described much earlier. In the 1920’s, Stern and Gerlach [29] observed the spin nature of the proton. Rabi et al. [30] followed by providing quantum mechanical concepts to the precession of spins around a magnetic field. Finally, Bloch [31] and Purcell et al. [32] extended his work and were awarded the Nobel Prize in 1952 for the "development of new methods for nuclear magnetic precision measurements and discoveries in connection therewith".

2.2 NMR Theory

NMR has been described using different degrees of complexity and under various formalisms. Yet, most MR imaging processes can be understood by using the fun-damentals of classical physics. In this section a general and brief overview of the principles of NMR theory are described.

2.2.1 Nuclear spin

The basic principle underpinning the NMR phenomenon is the interaction of certain charged particles with a magnetic field. A unique property of protons and neutrons is that they possess an intrinsic angular momentum, often referred as ’spin’. Protons, as neutrons, combine in pairs in the atomic nucleus, forming a net angular momentum that is determined by the number of constituent protons and neutrons. Furthermore,

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the fact that nuclei are positively charged creates a nuclear magnetic moment (i.e. net spin) if the angular momentum is non zero.

It is the net nuclear moment property of the nuclei that is of interest for NMR. Nuclei have a no net spin if they do not possess even atomic mass number and even charge number. Given the abundance of water in the human body, the single proton in hydrogen is the most common nucleus used in Magnetic Resonance Imaging (MRI) and will be the compound considered throughout this thesis. However, even if less abundant,13C,21Na and31P are still utilized in certain research and clinical applications of NMR.

The1H nucleus has an associated magnetic dipole moment, ⃗µ, that is, it behaves like a magnet, with the north-south axis parallel to the net spin axis. When placed in an external magnetic field, ⃗B₀, it experiences a torque, ⃗G, and precesses about the field at the defined frequency (Figure 2.1 (A)). The frequency of precession, ω0, is given by:

ω0= γB0 (2.1)

where γ is designated the gyromagnetic ratio (γ1_H= 2.675 × 108rad s−1T−1). The

above equation is designated the Larmor equation and is a fundamental equation in NMR since it relates the static magnetic field strength with the frequency of the resulting signal.

In quantum physics, the nuclear spins are said to be ’quantized’ in the presence of an external field with defined energy states. One state is aligned with the main magnetic field, known as spin-up or parallel and the state opposite to the external field is known as spin-down or anti-parallel. A proton is in the parallel or anti-parallel direction depending on its internal energy.

2.2.2 Equilibrium magnetization

For a bulk sample of non-interacting nuclear spins, also known as ’spin isochromat’, the spin axes of the individual proton nuclei precess in random directions. In the presence of a magnetic field the nuclear magnetic moments tend to align along the direction of the magnetic field, defined as the z direction, still with a constant precession at the Larmor frequency about the fixed field axis (Figure 2.1 (B)). In accordance with the Boltzmann distribution, the number of spins aligned parallel, N ↑, and anti-parallel, N↓, to the magnetic field is given by:

N↑ N↓= exp _∆E k_BT_s (2.2)

where Ts is the temperature of the spin system and kBis the Boltzmann constant (kB = 1.39×10−23JK−1) and ∆E is the energy difference between the states. The population

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2.2 NMR Theory 7

and energy difference is very small but leads to a preference of the spins for the lower energy state, i.e. the parallel or up state.

In a classical regime in NMR, it is the net magnetization that is of interest rather than the population of protons. The difference (N ↑ −N ↓) originates a net magnetiza-tion, ⃗M₀, which at equilibrium aligns with the magnetic field ⃗B₀. The magnitude of

⃗ M₀is given by: M₀= N_s¯h 2 γ2B0 4kBTa (2.3)

where Ns is the number of all proton spins. The magnitude of ⃗M0is directly propor-tional to the local proton density (also known as spin density). Yet, in the applications of MRI, the ratio in (2.3) is very small. At body temperature (Ta= 37° C) and in a 1.5 tesla scanner, it is of about 1.000005, which means that there are only 5 ppm spins exceeding the spin-down energy state.

Figure 2.1 (A) Geometric representation of the precession of the magnetic moment ⃗µ around an external magnetic field, ⃗B0, which generates a torque force, ⃗G. (B) The behaviour of a

sample of multiple nuclear spins when placed in a strong magnetic field: the nuclear magnetic moments are initially all randomly oriented (left diagram), but gradually the moments align either parallel or anti-parallel with the field (right diagram). (C) The slight energy advantage along the direction of the field acts like a single magnetization vector, ⃗M0.

2.2.3 Excitation via Radio-Frequency Pulses

Since the magnetization in the body is very small (in the order of µT), compared to the main magnetic field (in the range of tesla), it is hard to measure while at an equilibrium state, i.e. lying parallel with ⃗B₀.

It is possible to manipulate the direction of the net magnetization via Radio-Frequency (RF) pulses matched at the Larmor frequency. The created transmit RF field, also known as ⃗B₁, is resonant with the spin precession. A ⃗B₁field applied along the transverse x-y plane causes a change in the angle of precession of the spins, process known as nutation (Figure 2.2 (A)), and the magnetization vector gains a component in the transverse plane. When the ⃗B₁field is turned off, the net magnetization will precess about the main magnetic field with a frequency given by the Larmor equation.

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A more common approach to visualize the phenomenon is in a rotating frame, denoted ( ˆx′, ˆy′, ˆz′) rotating at the Larmor precession frequency about the z axis, which is defined by the direction of ⃗B₀. Since z and z′are aligned, the ’z’ notation will be adopted for this axis in both the rotating and laboratory frames. From the rotating per-spective, the spin axis is stationary, and the RF field appears to ’tip’ the magnetization vector ⃗Mfrom its equilibrium position toward the y′-axis (Figure 2.2 (B)). The angle of rotation, or flip angle θ , by which the magnetization is tipped, depends on the area under the RF pulse envelope:

θ = Z tp

0

γ B1(t′)dt′ (2.4)

where tpis the time the RF pulse is applied. If after the application of an RF pulse, the net magnetization vector ends up exactly in the transverse plane, then the RF pulse is called a 90° pulse. By leaving the RF on for twice as long (or by duplicating the power) it is possible to turn the magnetization vector through exactly 180° aligning it opposite to the main field.

Figure 2.2 Net magnetization vector ⃗Mevolution under the application of a 90° RF pulse at the frequency ω0in (A) the laboratory frame of reference and (B) the rotating frame of reference.

2.2.4 Relaxation

The magnetization vector of a system of spins in a magnetic field after a perturba-tion caused by a 90° RF pulse will tend to return back to its low-energy state. A phenomenological description of the behaviour of the magnetization during an NMR experiment was given by F. Bloch in 1946 [31]. After the RF pulse is turned off, the fundamental equations that describe the system relaxing to its low-energy state are known as the Bloch equations:

dM_z(t)

dt =

M₀− Mz T1

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2.2 NMR Theory 9 dMx(t) dt = ω0My− Mx T₂ (2.6) dMy(t) dt = −ω0Mx− My T₂ (2.7)

where the magnetization vector is separated into x, y and z contributions for clarity. In the above equations the T1and T2define two relaxation time constants: T1is also known as the spin-lattice relaxation and T2the spin-spin relaxation. The origin of these will be described next.

Spin-Lattice Relaxation: T1

Previously, it was observed that the nuclear spins under the effect of a magnetic field do not immediately align, altogether forming M0. Instead, it requires thermal processes to provide an energetic pathway to this state. To this end, the randomly fluctuating magnetic moments of the neighbouring molecules (i.e. the lattice) can affect the orientation of the nuclear spins.

The spin-lattice relaxation refers to the process in which the nuclear spin system gives the energy they obtained from the RF pulse back to the surrounding lattice, which provides the fluctuating magnetic fields needed so that the spins can alter their energy state in order to orient their magnetic moment along the longitudinal (z) direction leading to restoration of the Boltzmann equilibrium.

The relaxation time T1 (average lifetime of nuclei in the higher energy state) is dependent on the gyromagnetic ratio of the nucleus and the mobility of the lattice. At high mobility rates (translational, rotational, and vibrational modes of motion) it is more likely for a component of the lattice field to be able to stimulate the transition from high to low energy states. Since there is a broad spectrum of molecular motions over a wide frequency band, it is possible to have very different T1relaxation times for different tissues of the human body.

Whenever the magnetization vector is disturbed from its low energy state, i.e. aligned with B0, the z-component of the magnetization will recover in an exponential fashion as shown in Figure 2.3 (A). The T1 characterizes the rate at which the lon-gitudinal component of the magnetization vector, Mz, recovers towards its thermal equilibrium as shown in equation (2.5). If the component of the magnetization along z is zero at t = 0, then the solution of equation (2.5) is:

M_z(t) = M0(1 − e−t/T1) (2.8)

This equation shows the gradual exponential recovery to equilibrium magnetization M₀ with time constant T1. Table 2.1 shows typical T1 values for various tissues in humans.

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Figure 2.3 (A) Longitudinal magnetization regrowth plot following a 90° pulse to equilibrium value, M0. (B) Transverse magnetization decay plot from initial Mxy(0). Simulations of

relaxation curves were made according to gray-matter T1= 950 ms and T2= 100 ms values at

1.5 tesla (Table 2.1) with MATLAB (The MathWorks Inc., Natick, MA.).

Tissue T₁[ms] T₂[ms] Gray matter (GM) 950 100 White matter (WM) 600 80 Muscle 900 50 Cerebrospinal Fluid (CSF) 4500 2200 Fat 250 60 Blood 1200 100-200

Table 2.1 Representative values of relaxation parameters T1and T2in milliseconds for

hydro-gen in various human tissues at B0= 1.5T and 37°C. For blood T2 values, the lower limit

corresponds to venous blood whereas the higher value to arterial blood [33].

Spin-Spin Relaxation: T2

Whereas T1describes a recovery process of longitudinal magnetization, the T2 relax-ation describes the decay process of the transverse component of the magnetizrelax-ation, M_x,y[34]. Immediately after an RF pulse, the magnetization gains a component along a direction in the x-y plane. In an ideal situation, with a perfectly homogeneous magnetic field, the nuclei would experience the same applied magnetic field, and hence the transverse magnetization would remain coherent, rotating at the Larmor frequency until the T1 relaxation process would occur. It is the nuclei spin interactions that create low frequency random fluctuations in the local magnetic field. Over time, these fluctuations of the Larmor frequencies lead to a loss of bulk transverse magnetization and to a loss of signal. The greater the field inhomogeneity, the faster the dephasing and, hence, the faster the magnetization decay.

The characterization of the overall reduction rate gives rise to the T2parameter, or spin-spin relaxation time, described by the Bloch equations (2.6) and (2.7), which can

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2.2 NMR Theory 11 be re-written as: dMx,y dt = − Mx,y T₂ (2.9)

The solution of the differential in equation (2.9) is:

Mx,y(t) = Mx,y(0)e−t/T2 (2.10)

Figure 2.3 (B) shows a plot of the decay curve of equation (2.10). Typical values of T2for various tissues are given in Table 2.1.

T₂∗Relaxation

As it discussed above, spspin interactions create internal local magnetic field in-homogeneities that are characterized by the time constant T2. However, in practice, additional dephasing of the magnetization is introduced by field inhomogeneities external to the nuclei such as intrinsic defects in the main magnetic field or from susceptibility-field distortions produced by the sample when placed within the field. The additional inhomogeneities are described by the time constant T₂′. The total relaxation time, T₂∗, can be defined as:

1 T₂∗ = 1 T₂+ 1 T₂′ (2.11)

In the human head there is a large variability in the local magnetic susceptibility (χbone= −8.9 × 10−6, χwater= −9.05 × 10−6, χair= 0.36 × 10−6[35]), which means that the tissue responds differently to the applied magnetic field. Tissue-air boundaries near sinuses are particularly problematic and can lead to very rapid T₂∗relaxation.

2.2.5 The free induction decay (FID)

The simplest NMR experiment is to perturb the magnetization into the transverse plane using a brief ⃗B₁pulse, and to detect the voltage induced in a receiver coil placed perpendicular to ⃗B₀. In the laboratory frame, the net magnetization will precess about ⃗

B₀ inducing a corresponding oscillating voltage in the detection coil: the signal is oscillating at a frequency ω0and decaying with a time constant T₂∗. Thus the NMR signal is proportional to the precessing magnetization. This experiment is known as the ’Free Induction Decay’ (FID) (Figure 2.4).

2.2.6 The spin echo

Consider adding a 180° RF pulse to an FID experiment of Figure 2.4. While it was observed that on an FID experiment an RF pulse with 180° tips the magnetization

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Figure 2.4 A free induction decay detection experiment. A brief radio-frequency pulse is applied at the Larmor frequency and a receiver coil is detecting signal after the RF pulse is turned off. The detected signal has a high frequency component oscillating at the Larmor frequency, ω0, and an exponential decay component.

vector opposite to the ⃗B₀field, in this case, the goal of the 180° pulse has a different purpose.

After the 90° pulse (Figure 2.6 (A,B)), the magnetization vector will precess in the transverse plane about the z-axis, accruing a phase angle, φ , proportional to the duration of the free precession (Figure 2.6 (C)). If a coil would be recording the signal in the y-axis at this point, it would record an FID signal. At a time t = τ, the 180° RF pulse is applied, which flips the magnetization vector to a phase −φ (Figure 2.6 (D)). The magnetization vector will continue to accrue phase, but now in opposite way (Figure 2.6 (E)), such that at time t = 2τ the phase of the magnetization vector is zero (Figure 2.6 (F)) and a spin-echo occurs [36]. The 180° pulse is designated a refocusing pulse, because it refocuses any dephasing that occurs after the 90° pulse. The refocusing time is often known as ’echo time’ or TE, which is also t = 2τ.

An advantage of a spin echo experiment is that the T₂′ relaxation processes are refocused by the 180° refocusing pulse. The spin echo experiment is, thus, only sensitive to the molecular and random field inhomogeneity mechanisms that lead to the pure T2 relaxation. The particular benefit of the spin-echo lies in cases when imperfections in the static magnetic field exist and will be subject of discussion in Chapter 5.

2.3 Principles of Magnetic Resonance Imaging

In section (2.2) the behaviour of the magnetization is observed with the application of a ⃗B1 field allowing the detection of a particular species in a sample. However, in MRI it is possible to determine the spatial distribution of a given species within the sample. In 1973, Lauterbur [27] proposed the use of a field gradient to localize the NMR signal. Using the back-projection method, a linear gradient was applied

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2.3 Principles of Magnetic Resonance Imaging 13

Figure 2.5 A spin echo experiment. The 90° pulse is followed a time τ later by a 180° refocusing pulse. This acts to refocus the magnetization at time TE after the initial pulse.

Figure 2.6 Sequence of events that occur on a spin echo experiment in the rotating reference frame. (A) All of the spins are aligned with the static magnetic field in the minimum energy state. (B) A 90° pulse rotates the spins in to the x′-y′plane. (C) Spins start to dephase within the transverse plane. (D) A 180° pulse is applied which will invert the accumulated phases. (E) The inverted phases rephase at the same rate as they were dephasing before the second pulse. (G) All of the spins return to the y′-axis together forming an echo. Figure adapted from Haacke et al. [33].

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at different angles to obtain a 2D image of the sample. At the same time Mansfield and Grannell [28] presented a method to determine spatial structures in solids based on NMR ’diffraction’, using a sequence of multiple pulses and gradients. A year later, in 1974, Mansfield and colleagues proposed the slice-selection technique still used today [37]. Following this, Ernst published work on the first Fourier imaging method [38], using non-selective excitation and orthogonal linear gradients to generate 2D Fourier encoded images. These techniques were the groundwork of the MRI as is done today.

2.3.1 The concept of spatial encoding and Fourier transform

If a magnetic field is varied in the z-direction, the Larmor frequency of the spins will acquire a z dependency. A linear spatial variation on the main magnetic field can be achieved by passing current through a ’Maxwell Pair’ of coils (Figure 2.7 (A)) [39].

Consider two spins, P1 and P2, as shown in Figure 2.7 (A), placed inside the magnet in different z positions. If one would pass current through the ’Maxwell Pair’ in order to introduce a linear variation in z, the two spins would start to precess at two different frequencies. The signal detected by the receiver coil will be the the summation of the signals from the two spins. By performing a Fourier transform it is possible to unfold these frequencies, and, therefore, to plot the number of spins that are contributing to the detected signal at each z (Figure 2.7 (B)).

In MRI three sets of coils are positioned to linearly encode the three orthogonal directions of space (x, y and z) and are designated gradient coils. These are shown for a typical MRI system configuration in Figure 2.8. The x and y gradient sets have a different coil winding than the original ’Maxwell Pair’, and are based on a ’double-saddle’ coil configuration [39].

The three gradient coils produce a linear variation in the field as a function of the position,⃗r, of the sample:

ω (t) = ω0+ γ ⃗G(t) ·⃗r (2.12)

The combination of all field gradients on the sample, ⃗G(t), expressed in mT/m, is:

⃗ G(t) ≡ ∇⃗B0= _{∂ ⃗}B_z ∂ x , ∂ ⃗Bz ∂ y , ∂ ⃗Bz ∂ z (2.13)

Note that following an excitation pulse, it is possible to generate one dimensional profiles in turn along the x, y, and z directions by collecting signal with the appropriate gradient.

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Figure 2.7 (A) The current I passing through the Maxwell pair of coils generates a linearly varying magnetic field along the z direction. (B) Acquisition of signal in the presence of a field gradient and signal and frequency profile of the sample in (A) along the gradient direction. In (B) the frequency plot is the Fourier transform of the acquired signal.

Figure 2.8 Configuration of gradient coils for each of the x, y, and z directions in an MRI system.

2.3.2 Multi-Dimensional Imaging

The most conventional way to acquire a three dimensional volume is designated ’2D multi-slice imaging’, where firstly a slice-selection gradient is applied to localize a two dimensional imaging plane. The orientation of the slice can be varied by using physically different gradient axis. The selected slice is always perpendicular to the gradient applied. Transverse slices are chosen with the slice-select gradient applied along the z-axis, while for coronal and sagittal slices the slice-select gradient is applied along the y-axis and x-axis, respectively (Figure 2.9). If one wishes to choose an oblique acquisition, the effective slice-selection gradient will consist of the combination of the three orthogonal gradients.

Slice-selection is then followed by the manipulation of frequency and phase of the spins to localize the NMR signal in a three dimensional space.

Slice Selection

Slice-selection, also known as selective excitation, is a method that consists on the application of a gradient at the same time as the RF pulse used for excitation. The use

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Figure 2.9 Orientation of imaging-planes according to orthogonal axis. In the most standard acquisition, the subject is placed head-first. Common abbreviations for vector orientations in MRI are: Anterior->Posterior (A->P), Left-Right (L->R), Superior->Inferior (S->I) and the opposite directions respectively. On GE HDx and Discovery MR950 systems the ⃗B0vector is

conventionally chosen in the I->S direction. Figure adapted from Haacke et al. [33].

of the RF alone excites all the spins in the volume within a certain bandwidth, ∆ω. This is known as a non-selective RF pulse. However, if an RF pulse is applied in the presence of a field gradient, Gz, the gradient spatially isolates the effect of the pulse by selecting only a thin slab where the spins are rotating at a frequency lying on the bandwidth, ∆ω. The spins lying outside the slab are rotating either faster or slower than the RF frequency and will not be tuned. As a result, the thickness of the slice, ∆z, is a function of both the RF bandwidth, ∆ω, and the applied gradient, Gz (Figure 2.10 (A)):

∆z = ∆ω γ Gz

(2.14)

In order to obtain a uniform flip angle across the entire slice, the frequency profile of the RF excitation should ideally be proportional to a boxcar function, rect(ω/∆ω) (Figure 2.10 (B)) [40]. In the low flip-angle regime (θ < 90◦), the inverse Fourier transform is enough to characterize the temporal RF envelope, implying, therefore, that the RF pulse in temporal domain will ideally be a sinc function. In practice, the ⃗_B₁_{pulse can simply be a windowed version of the sinc pulse, in which the greater} number of lobes included, the better the approximation to the ideal frequency profile. Adding more lobes increases the duration of the pulse, which has its adversities for sequence design.

In case of large flip-angles, the Fourier methods fail due to the non-linearity of the Bloch equations. Other more robust approaches such as RF profiles produced with the Shinnar-Le Roux (SLR) algorithm have been introduced [41–43].

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Figure 2.10 Slice Selection gradient: (A) Illustration of how to selectively excite spins in a slice of thickness ∆z with a range of frequency ∆ω and with a slice selection gradient Gz(red

line). With the same frequency bandwidth, but with higher gradient strength it is possible to excite a thinner slab (light blue line). (B) Boxcar frequency profile of an RF pulse. The ideal temporal profile of the boxcar RF pulse corresponds to a infinitely long sinc pulse.

k-space

Having selected a slice plane in a 3D object, it is now necessary to decode the information on a two dimensional plane, and the method of choice in modern scanners is based on the Fourier transform. Subsection (2.3.1) shows how to make use of encoding gradients to form a signal with encoded amplitudes and frequencies. In order to correctly represent a profile consisting of N points it is necessary to acquire a signal with N frequencies, given by n2π/FoV, where n takes the values −N/2 and N/2, and FoV is the field of view of the object, i.e. the selected size of the imaging sample.

The concept of k-space was introduced in 1983 [44] as a way of visualizing the spins phases under the influence of spin gradients. k-space is thus, an array of complex numbers that represent the spatial frequencies coded by the gradients.

It is possible to extend the principle of the Fourier transform to any number of dimensions. Figure 2.11 (A) shows a two dimensional image together with its two dimensional Fourier coefficients (Figure 2.11 (B)). If the matrix size is N × M, in this case, one requires the sampling of an N × M array of Fourier coefficients.

Having assumed that a plane at some location z has been excited, the Larmor equation for the frequency of an elemental volume in the presence of the static and Gx and Gylinear magnetic fields is given by:

ωG(x, y) = γB0+ γGxx+ γGyy (2.15) Thus, after RF excitation, the accumulated phase of the spins of the elemental volume is:

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Figure 2.11 Magnitude image and corresponding k-space representation from a transverse slice of the head of a healthy volunteer. A two dimensional magnitude image (A) and its corresponding Fourier coefficients (B). Filtered magnitude image (C) where the high frequency components were zero-padded (D). Filtered magnitude image (E) where the low frequency components were zero-padded (F).

φG(x, y,t) = Z t

0

ωG(x, y,t′)dt′ (2.16)

where t is the time after the excitation. The signal contribution that is introduced in the receiver coil is a vector whose magnitude is the spin density at position (x, y), ρ (x, y), multiplied by the size of the elemental pixel dxdy, and whose phase is equal to φG(x, y,t).

The demodulated signal can be written as [33]:

s(t) = Z Z

ρ (x, y){cos[2π (γ Gxx+ γGyy)t] + i sin[2π(γGxx+ γGyy)t]}dxdy (2.17)

Applying Euler’s formula, the signal equation becomes:

s(t) = Z Z

ρ (x, y)e−i2πγ(Gxx+Gyy)tdxdy (2.18)

It is possible to define quantities that describe the area under the curve of the gradient versus time in the coordinates of the k-space. These terms are:

kx(t) = γGxt (2.19)

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Considering these k-space terms, equation (2.18) simplifies to:

s(kx, ky) = Z Z

ρ (x, y)e−2πi(kxx+kyy)dxdy (2.21)

From equation (2.21) it is recognizable that the signal s(kx, ky) and the spin density ρ (x, y) are a Fourier pair. As k-space represents the spatial frequency distribution of the MR image, low spatial frequencies (coarse image structures) appear close to the origin (Figure 2.11 (C,D)) while higher spatial frequencies (edges and fine detail) appear at the outer edges of k-space (Figure 2.11 (E,F)).

The goal of Fourier imaging is to collect the necessary information to fill up the s(kx, ky) matrix. In general the MR signal is collected over a set of uniformly spaced points in k-space (spaced by ∆kxand ∆kyin the x and y directions, respectively). An inversely proportional relationship exists between the Field of View (FoV ) and the distance between data points in k-space according to [33]:

1 Lx ∝ ∆kx (2.22) and: 1 Ly ∝ ∆ky (2.23)

where Lxand Lyare the FoV dimensions in the x and y directions respectively.

k-space sampling with phase and frequency encoding gradients

As was implied in the previous subsection, the magnitude and duration of magnetic field gradients are manipulated in order to navigate through k-space. In the most straightforward method, the Gx and Gy gradients are employed in order to produce a line-by-line Cartesian k-space trajectory (Figure 2.12). Many other methods have since been introduced based on spiral [45] or radial [46] oriented trajectories.

In a Cartesian method each digitized echo completely fills a line of k-space. In the example of Figure 2.12, after a slice selective excitation, the signal has a k-space coordinate of (0, 0). A negative Gxand a negative Gygradient are then applied for a determined time which moves the k-space coordinate to (−kmax_x , −k_ymax). Afterwards, the Gygradient is switched off and the Gx gradient is switched to positive for a given time driving the k-space coordinate in the +kxdirection, during which the receiver is turned to collect data. After the collection of the last data point, a new excitation pulse is applied. The time between two consecutive excitation pulses is known as repetition time, or TR.

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On the following TR period, the Gyis applied with a smaller amplitude compared to the first TR and the Gxis applied in the same manner. This allows the acquisition of the consecutive k-space line (Figure 2.12).

Figure 2.12 Illustration of an example Cartesian k-space sampling trajectory.

On this example, the short-lived pulse of the Gygradient will modify the resonant frequency of the spins which were all initially in phase in the y-direction. When the gradient is turned off, the spins will return to their Larmor frequency precession, but the phase shifts along the y-direction maintain. Hence, the Gy gradient sets a specific phase dependency on the spins and this is why it is also designated as the ’phase-encode’ or ’PE’ gradient. On the other hand, the Gxgradient is applied during the read-out of the signal, hence described as the ’read-out’ or ’RO’ gradient. In addition, since the read-out gradient is encoding the frequency component of the signal it also often referred to as the ’frequency-encoding’ gradient. To note, however, that switching between the phase- and frequency-encoding directions, i.e. switching the Gx and Gygradients, is commonly used in MR image, in order to avoid Nyquist wrapping 1_.

1_{Nyquist wrapping or aliasing in MRI is an artefact that occurs when the FoV is smaller than the}

body part being imaged. According to the relationships in (2.22) and (2.23) and to the Nyquist criterion, if the sampling rate is less than twice the highest frequency component, the signal is undersampled and wrap-around occurs.

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2.3.3 Imaging Sequences

By manipulating the gradients and RF pulses, it is possible to acquire images of various parts of the sample with numerous contrasts and sensitivities. These are widely known as imaging sequences. In this section, the two most classical imaging approaches are described, namely the two-dimensional gradient and spin echo methodologies.

Two-Dimensional Gradient Recalled Echo Imaging

The basic mechanism underlying a 2D Gradient Recalled Echo (GRE) sequence is based on the FID experiment explained in subsection (2.2.5). Its sequence diagram is shown in Figure (2.13). The RF excitation pulse is first applied together with the slice selection, Gz, gradient. The gradient will cause spins to dephase from each other. This results in accelerated dephasing of the FID. The signal will then be refocused by a negative lobe of Gz. The area under the rephase lobe counterbalances half of the area under the slice select lobe. Note that the rephasing gradient has only refocused spins dephased by the positive lobe of Gzbut does not affect T2or T₂∗relaxation processes. The signal is localized in-plane by the use of phase- and frequency-encoding gradients, Gyand Gx, respectively. The signal data that are collected during the positive lobe of the frequency-encoding gradient is decaying with a T₂∗relaxation rate. In this sequence scheme, the echo time is defined as the time from the application of the RF pulse to the center of the positive Gx gradient.

Figure 2.13 Pulse sequence diagram for a 2D gradient echo imaging sequence. The RF pulse is represented as a sinc pulse with a flip-angle θ . The step-wise phase encoding gradient is represented with multiple horizontal lines. Gradient areas are given by A and B. ∆Gy is the

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After the collection of data for the first k-space line during a period of TR, the sequence is repeated with a different phase-encoding gradient, such that the signal from a different k-space line is acquired. The time to acquire a full 2D slice with this sequence scheme is: T = TR × NPE, where NPE is the number of phase-encoding steps.

Two-Dimensional Spin Echo Imaging

In a spin echo (SE) scheme [36] two RF pulses are applied: a 90° pulse followed by the 180° refocusing pulse as was observed in subsection (2.2.6). The pulse sequence timing for a 2D spin-echo sequence is shown in Figure (2.14) [40]. Both RF pulses are played together with slice-select, Gz, gradients. The slice selection gradient applied for the 180° pulse does not require a rephasing lobe. As for the GRE sequence, the echo is both phase- and frequency-encoded as explained in subsection (2.3.2). In this schematics, the phase encoding steps read-out prephaser are placed before the 180° pulse. However, either can be placed before or after the 180° pulse. The time to cover a full k-space in a spin echo sequence is still T = T R × NPE (as for GRE), but there are differences on how the k-space is covered, depending on when the gradients are played (see Figure 2.13).

Figure 2.14 Example of a pulse sequence diagram for a 2D spin-echo sequence. The phase encoding polarity is indicated by the arrow on the step-wise gradient Gy, with a step size of

∆Gy. The k-space diagrams at time points t1and t2are shown on the boxes on the right.

2.3.4 2D Multi-slice Imaging

The combined acquisition of multiple two-dimensional slices across a volume of, for example, the human head, forms a 2D multi-slice acquisition. Because of an imperfect boxcar shape of the excitation, the immediate neighbourhood of the slice can

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