Download Effects of Dipolar Fields in NMR and MRI

Effects of Dipolar Fields in
NMR and MRI
by José Pedro R. F. Marques
Sir Peter Mansfield Magnetic Resonance Centre
University of Nottingham
Thesis submitted to The University of Nottingham
for the degree of Doctor of Philosophy, December 2004
Para aqueles que esperava reencontrar no final desta caminhada:
Avó Fina, Avó Rosa, Avô João,
Rita Wemans e Rafael Patrı́cio.
Abstract
The work presented in this thesis was undertaken by the author, except where it is indicated by reference, within the Sir Peter Mansfield
Magnetic Resonance Centre at the University of Nottingham during the
period July 2001 until November 2004.
The initial objective of the PhD was to investigate possible applications
of long range dipolar field effects, but other problems involving the effects
of susceptibility induced magnetisation have also been discussed. The
work on the long range dipolar field involved:
• The study of the evolution at high field of the various orders of
quantum coherences and 2D spectroscopy experiments based on
long range dipolar fields;
• The optimisation of parameters for imaging with the double quantum CRAZED sequence and derivation of phase cycles that allow
the elimination of undesired signals such as stimulated echoes and
refocused spin echoes;
• Simulations of the effects of field inhomogeneities on the long range
dipolar field and signal evolution after a double quantum CRAZED
sequence;
The work relating to the fields generated by susceptibility induced magnetisation involved:
• Application of a methodology originally derived for evaluating the
long range dipolar field to the rapid calculation of field inhomogeneities due to induced magnetisation (dipoles). Studies were performed using a digitised body model. The effect of head movement
and breathing were simulated allowing the evaluation of the consequences of these inhomogeneities for imaging;
• The methodology referred above was used to compare the BOLD
contrast, due to both the intra and extra-vascular compartments,
resulting from infinite cylinders and a realistic vascular model (based
on the rat cortex);
• Development of a methodology for monitoring variation of field inhomogeneities with movement in functional studies, that can be
used to apply an undistortion for each echo planar image acquired.
Acknowledgements
There is a huge list of people to whom I have to thank at the end of
these four years of work, people who have made this thesis possible, or
without whom I might have been driven to insanity. Under this premise
I would like to acknowledge:
- The “Fundação para a Ciência e Tecnologia” for giving me the economical support to start a PhD, and giving me total freedom to choose
what and where to do it...;
- Richard for accepting to be my supervisor, for his knowledge, for his
educative and humble approach to supervising, for giving me freedom
on the development of the various projects without ever loosing track of
what I was doing, always giving extremely valuable inputs (or, always
giving the valuable inputs) and an incredible patience in reading this
thesis (without a doubt more carefully than myself);
- Everybody in the MR Centre for making the work place such a friendly
environment in which working became a pleasant hobby. I will try to
only name some in order to avoid making this respectable section into a
Oscar winning kind of speech... Sarah (for welcoming me into the dipolar
project and introducing me to the English concept of personal space),
Jane (my very close office mate, 50cm, and my almost 100 lengths swimming bud), Kay (for helping me out with the scanner and good music
hints), Davide (for introducing hacky sac in the department, for his great
shimming abilities and welcoming me in his house for the last couple of
months), Gianlo and Benito (for their great friendship in two amazing
years in Janna, making it the closest one can have to home), Wietske (for
the uncountable coffee/tea/cycle breaks...), Jim (for the constant motivation emails for sport activities and help debugging latex), Ben (for the
office charade and the never ending nut stock), Sean (for help with fsl
scripts and juggling tricks), Anthony (for being the greatest moaner),
Dan Green (for his ability to swear in so many languages and help with
fortran and matlab), Silvia (for showing that chicken and chocolate can
go together), Mike and Ian (for making me feel that my work might be
useful at the end of the day), Lesley (for saying good morning every
morning), Ian Thexton (for the help with all the phantoms and bike fiddling), Andy Peters (for making sure that the computers work and being
permissive with my excessive cpu usage)... and as I run out of space, I
will only mention a couple more, again, with no obvious order... Alex,
Adnan, Andy Gibson, Carolyn, Dan Konn, Jiabao, Yas (for having to
share offices with me), Luca, Martin, Bhavana, Alison, Matt, Dinesh,
Arthur (not the computer), Penny, Sue, Walter, Jeff, Steph, Dawn and
many, many more for many and various reasons;
- My examiners, Prof. Peter Morris and Dr. Angelo Bifone, for effectively
having awarded me the PhD after I very pleasant viva voce experience;
- Those whose thesis I used for inspiration, namely: Geoffrey CharlesEdwards and Enrico De Vita;
- À comunidade portuguesa que regularmente se encontra em conferências,
muitos dos quais provenientes do IBEB. Um obrigado muito especial à
Rita (por acumular o estatuto de colega e amiga desde o princı́pio deste
doutoramento), à Patrı́cia (por ter sido a primeira a falar-me de Ressonância Magnética, de um grupo interessante em Nottingham e de um
rapaz esperto que sabia de tudo e mais alguma coisa) e ao Prof. DuclaSoares (que nos atirou para estas andanças por intermédio do programa
doutoral);
- À Joana, pelo carinho com que me aturou e cuidou diariamente durante
estes quase quatro anos, por ter sido sempre companheira, por me ouvir
nos dias em que nada parecia ir dar a lado nenhum e então me ter dado
força, por ter sido fonte de alegria, por me ter feito correr esta ilha de
ponta a ponta, por quase ter percebido como funciona a Ressonância
Magnética e por tantas mais razões que não serão de escrever aqui;
- Aos meus amigos e famı́lia por me fazerem sentir em casa sempre
que volto a Portugal e continuamente mostrarem que a distancia não
nos distanciou. Nomeadamente o Hugo, a Joana Patrı́cio, a Candida, o
João Luı́s, a Ana e o Lus Pinto (por se terem atrevido a vir conhecer
Nottingham), o Alex e o João Miguel (por continuarem a ser bons colegas
de trabalho) e todos os outros que estão sempre lá. Um agradecimento
muito especial para os meus pais por serem presença constante e por me
terem dado todas as condições para chegar até aqui.
5
Contents
i
Contents
Contents
Chapter 1.
i
Introduction
Chapter 2. Introduction to NMR and MRI
2.1 Historical Overview
2.2 Quantum Mechanical Approach
2.2.1 Characterising Matter
2.2.2 Spin 12
2.2.3 Spin Hamiltonian Hypothesis
2.2.4 Zeeman Interaction and the Resonance Condition
2.2.5 Rotating Frame and RF pulses
2.2.6 Net Magnetisation
2.2.7 The Spin Hamiltonian
2.3 Semi-Classical Approach, the Bloch Equations
2.3.1 Static Magnetic Field, Gradients and Chemical Shift
2.3.2 Radio Frequency Pulses
2.3.3 Relaxation times, T1 , T2 and T2∗
2.3.4 Diffusion, D
2.4 Magnetic Resonance Imaging
2.4.1 The Signal Equation and k-space
2.4.2 Imaging Sequences
2.4.3 Image Artifacts
2.5 Basic Pulse Sequences in Quantitative NMR
2.6 The Basic Hardware
2.7 Signal to Noise Ratio in NMR and MRI
2.7.1 SNR as a function of B0 and B1
2.7.2 Signal to Noise Ratio as a function of B0 and B1
2.8 Safety
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Contents
ii
Chapter 3. Long Range Dipolar Field
3.1 The Quantum Description
3.1.1 Hands on the Quantum Approach, the n-CRAZED case
study
3.2 The Classical Approach
3.2.1 Hands on the Analytical Classical Approach,
the n-CRAZED case study
3.2.2 Hands on the Numerical Classical Approach
3.3 High Field Signal Evolution
3.3.1 Methods
3.3.2 Results
3.4 2D-Spectroscopy, the DQ-CRAZED experiment
3.4.1 Classical Approach to Peak Location
3.4.2 Experiments
3.5 Conclusions
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Chapter 4. Optimising the Sequence Parameters for DQC Imaging
4.1 Introduction
4.2 Theory
4.2.1 Single Experiment
4.2.2 Two Experiments
4.2.3 Steady State
4.2.4 Phase Cycling
4.2.5 Echo-Time Dependence
4.3 Methods
4.4 Results
4.5 Discussion
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Chapter 5. Simulation of the DQC in inhomogeneous media
5.1 Introduction
5.2 Numerical Simulations in Inhomogeneous Media
5.2.1 Results
5.3 Sequence Parameter Optimisation
for Maximum Contrast
5.3.1 Results
5.4 Discussions
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Chapter 6. Calculations of Field Inhomogeneity
6.1 Introduction
6.2 Theory
6.3 Method
6.4 Results
6.4.1 Field Inhomogeneity in the Human Head
6.4.2 Respiration Induced Resonance Offsets
6.5 Discussion
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Contents
iii
Chapter 7. BOLD Simulations
7.1 Introduction
7.2 Realistic Vasculature vs Infinite Cylinders
7.3 Methods
7.3.1 Extravascular Contrast
7.3.2 Intravascular Contrast
7.3.3 Draining Veins
7.4 Results
7.4.1 Extravascular Contrast
7.4.2 Intravascular Contrast
7.4.3 Combining the Different Contributions to Contrast
7.4.4 Draining Vein Effect
7.5 Conclusion
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Chapter 8. Field Map Estimation
8.1 Introduction
8.2 Theory
8.3 Methods
8.4 Results
8.5 Conclusions
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Chapter 9.
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Conclusions
Appendix A. Fast Fourier Transforms
A.1 Fourier Transform
A.2 Discrete Fourier Transform
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Appendix B. Dipolar Field. . .
making it easy to use
174
References
177
1
Introduction
1
Chapter 1
Introduction
“The world of nuclear spins is a true paradise for theoretical and experimental physicists”, Richard Ernst, Nobel Lecture, 1992
... in fact the quote above could be rewritten to apply to a range of professionals
besides physicists. ”The world of nuclear spins” and particularly nuclear magnetic
resonance (NMR) has the power of bringing together a variety of researchers from
different backgrounds, bridging the gap between their different disciplines. It provides a truly multidisciplinary environment that joins individuals that arrive with
a wide range of motivations. Some approach NMR for the pure interest of having
a better understanding of this technique, others seek better understanding of the
structure and interactions of molecules both with themselves as well as with the
surroundings. Others look for anatomical information, others want to understand
metabolism and physiology. Others, motivated by all the previously mentioned,
develop new hardware and acquisition schemes in order to enlarge further the applicability of NMR. There are even some that seek oil...
Nevertheless, the most important applications of NMR are related to medical
diagnosis, where thanks to its high spatial resolution and soft tissue contrast, MRI
has had a huge impact, which is achieved with a technique that is so far considered
to be innocuous (unlike most other imaging modalities that rely on ionising radiation). The role of MRI in diagnosis, goes beyond images with different contrasts and
arbitrary slice direction. MRI can provide quantitative information such as relaxation parameters, diffusion, flow, perfusion and, in the case of spectroscopy, it can
give an insight into in vivo metabolite concentrations through the detection of hydrogen nuclei (or other nuclei whose spin is non-zero) that are in different chemical
environments.
1
Introduction
2
The work presented in this thesis was performed by the author during the three
years of his PhD. The initial objective of the PhD was to investigate possible applications of long range dipolar field effects. This aim suffered some drifts that ultimately
took the author to other areas of interest, although the ”long range” might have
disappeared, the concept of dipolar fields was always present:
• Chapter 3 gives an introduction to Long Range Dipolar Fields in both the
quantum and classical pictures. Numerical simulations of the Bloch Equations
(including the long range dipolar field, diffusion and relaxation) are compared
with data acquired on spectrometers operating at different field strengths
(only possible thanks to a collaboration with the University of Florida) for
the n-CRAZED sequence optimized for high field (with n=0,1,...,5). 2D Spectroscopy was explored using different sequences based on the double quantum
(DQ) CRAZED experiment. The results of this work showed that it is possible to obtain CRAZED signals even if multiple quantum coherences are not
present at any time.
• Chapter 4 centres its attention on optimising the parameters of the DQCCRAZED sequence for imaging. By solving the steady state equation it was
possible to: (i) derive the optimal repetition time; (ii) devise a phase cycle
capable of eliminating stimulated echoes; (iii) confirm the values of the rf-pulse
flip-angles and echo times that maximise signal or T2 contrast.
• In Chapter 5 the feasibility of using the DQ-CRAZED sequence to distinguish
sub-pixel field inhomogeneity via the length scale dependence of the long range
dipolar fields was evaluated using numerical simulations.
• Chapter 6, introduces and evaluates a new method for calculating magnetic
fields generated by induced magnetisation, using the theory previously employed to characterise Long Range Dipolar Fields. This method was used to
evaluate the range of field shifts produced in the human head due to the different susceptibilities of its tissues and air. In addition the expected change
of the field shift due to head rotation about different axes and respiration was
assessed.
• Chapter 7 uses the methodology introduced in Chapter 6 to characterise the
pattern of the field perturbation due to the susceptibility difference between
grey matter and blood in the vasculature, using both an infinite cylinders
model and a realistic vasculature model with different blood volume fractions.
1
Introduction
3
Numerical simulations have also been used to calculate the expected NMR signal from such environments, and the resulting signal decays were parametrised
to be able to quantify the different contributions to the BOLD effect and their
function upon blood volume fraction and susceptibility difference.
• Chapter 8 presents work that is motivated by some of the observations of
Chapter 6 regarding field shift variation due to head rotation, and introduces
a methodology for monitoring variation of the field map with movement that
could be applied to the undistortion of echo planar images acquired in functional studies. This method has proved to produce a better correction than is
provided by previously described methods.
Preceding these different chapters there is a background chapter (Chapter 2),
“all you need to know” (and a bit more...) to read this thesis. This spans from the
history of Nuclear Magnetic Resonance (NMR) and Magnetic Resonance Imaging
(MRI), going through a review of the two frameworks in which the phenomenon
of NMR can be described, the presentation of the concept of imaging with magnetic resonance, followed by a description of some basic sequences used to measure
the NMR parameters, what hardware is needed to perform MRI and finally some
discussions of the important issue of safety of magnetic resonance.
I hope you enjoy your reading...
2
Introduction to NMR and MRI
4
Chapter 2
Introduction to NMR and MRI
“There the snow lay around my doorstep - great heaps of protons quietly
precessing in the Earth’s magnetic field. To see the world for a moment
as something rich and strange is the private reward of many a discovery”
Edward M. Purcell, Nobel Lecture, 1952.
In this chapter, an overview of how nuclear magnetic resonance reached the
present stage of development will be given.
“It is a tribute to the inherent harmony and the organic growth of our
branch that every advance in physics is largely due to the developments
that preceded it.” Felix Bloch, Nobel Lecture, 1952.
A description of the basic physics of NMR will be attempted in terms of quantum
mechanical (Section 2.2) and classical approaches (Section 2.3). An insight will
be given into the principles of imaging using magnetic resonance (MR) in Section
2.4. Some of the most relevant sequences used to generate quantitative information
from NMR will be presented in Section 2.5. None of this can be done without
a scanner whose hardware will be described in Section 2.6. Finally some safety
considerations will be discussed in Section 2.8. Although there are many textbooks
written concerning this subject, the aim of this chapter is to give the reader the
required knowledge and tools, so that the work presented in the remainder of this
thesis may be found to be self-explanatory.
2.1
Historical Overview
2.1
5
Historical Overview
Several scientists have already been awarded Nobel Prizes for Nuclear Magnetic
Resonance related work:
• Isidor Rabi (1944) Nobel Prize in Physics “for the resonance method for recording the magnetic properties”
• Felix Bloch and Edward Purcell (1952) Nobel Prize in Physics “for their development of new methods for nuclear magnetic precision measurements and
discoveries therewith”
• Richard Ernst (1992) Nobel Prize in Chemistry “for his contributions to the
development of nuclear magnetic resonance spectroscopy”
• Kurt Würthrich (2002) Nobel Prize in Chemistry “for his developments of
nuclear magnetic resonance spectroscopy for determining the three dimensional
structure of biological macromolecules in solution”
• Paul Lauterbur and Peter Mansfield (2003) Nobel Prize in Medicine or Physiology “for their discoveries concerning magnetic resonance imaging”
This listing of the Nobel Prize laureates gives an idea of how long NMR has been
a cutting edge science topic and how vast are its applications (only the Literature
and the Peace Nobel Prizes are not represented here, but although no bibliographic
evidence was found, it is likely to be true that many of those laureates have already
interacted with a medical MR scanner).
The story though starts a couple of decades earlier. In 1921 Stern and Gerlach passed a beam of silver atoms through an inhomogeneous magnetic field and
recorded the deviation of the atom’s path. The aim was to simply confirm that there
was a continuous distribution pattern due to atoms that should have a continuous
distribution of magnetic moments, as was predicted by Bohr’s quantum theory. The
result however was far more interesting than that; they observed a discrete number
of atomic beams.
In the following years this discretised magnetic moment was measured more and
more carefully by the Zeeman effect and by the hyperfine structure of spectral lines
leading to confirmation of a half-integer angular momentum in units of h̄. The energy
of the interaction of this magnetic moment with the magnetic moment generated by
the orbital electrons at the position of the nuclei depends on the angle between them
and leads to small splittings of energy levels [1]. During the 20’s this was a problem,
2.1
Historical Overview
6
as half-integer angular momentum quantum numbers could not be understood in
terms of the rotation of a charged particle [2]. To get around this problem, Pauli
~ for the
(in 1924), proposed the concept of an intrinsic angular momentum/spin (S)
atomic nucleus, and a parallel magnetic moment (~µ). The proportionality constant
~ is the gyromagnetic
that relates the spin and the magnetic moment via, µ
~ = γ S,
ratio, γ, that is dependent on the type of nucleus (see Table 2.2.4). It was only
four years later that Dirac, by joining quantum mechanics and relativity, predicted
an intrinsic half-integer spin angular momentum for the electron resulting from its
charge [3].
An important step forward was made by Stern who, in 1933, applied the method
of molecular beams to determine the magnetic moment. This involved measuring
the beam deflection due to the interaction of the magnetic moments with an inhomogeneous magnetic field. This approach was further developed by Rabi using an
intricate system with two or three different magnetic field regions. However, the
greatest improvement introduced by Rabi was, in 1937, the use of the magnetic
resonance method jointly with the molecular beam technique. The molecular beam
passed through a region with constant homogeneous magnetic field with a weak perpendicular alternating component. This was analogous to the resonant absorption
of visible light, where the alternating field would satisfy Bohr’s frequency condition
for the energy gap between the two levels. In Rabi’s experiment though, the frequency instead of being in the optical region was in the radio frequency range. In
this experiment, the deflection was merely used as an indicator of the occurrence of
transitions and the accuracy in the calculations raised to one part in a thousand.
This was followed by the simultaneous demonstration of nuclear magnetic resonance in bulk matter by Edward Purcell, Torrey and Pound [4], at Harvard, and by
Felix Bloch, Hansen and Packard [5], at Stanford University1 . Much of our present
understanding of relaxation mechanisms was derived by N. Bloemberg, Purcell, and
Pound and is often called BPP theory [6]. As NMR was more and more understood
it became clear that the main limit to frequency resolution was the inhomogeneity
of the magnetic fields and high resolution began to be pursued. In 1950 an effect
now known as “chemical shift” was simultaneously observed by Dickinson [7] and
Proctor and Yu [8] by noticing that the resonant frequencies of a certain atom (19 F
and 14N respectively) depended on its molecular chemical environment. This effect
had already been observed by experimenters interested in measuring exact ratios of
magnetic moments, but was a nuisance for them. In 1950, R.V. Pound discussed for
1
90 pages separate the publications of these two works in Physical Review, 69 (1946) page 37
for Purcell and page 127 for Bloch
2.1
Historical Overview
7
the first time the effects of nuclear quadrupoles and measured quadrupolar effects in
the nuclear spins of crystals [9]. Until this time all work was done using Continuous
Wave irradiation, of which there were two variants: (i) continuous RF irradiation
of the sample with a fixed frequency while the magnetic field was varied or (ii)
use of fixed field, while the frequency of the RF wave was swept over the desired
spectral region. In both cases, the absorption spectrum was recorded at each field
or frequency point by collecting the signal amplitude. The first studies using pulsed
NMR were carried out by Erwin Hahn, who in 1950 published a paper [10] in which
he showed that the rapid loss of coherence following a short, high-power, RF pulse
could be reversed by simply applying a second RF pulse after a time τ that causes
a signal refocusing at a time 2τ . This is called either a Hahn or spin echo2 . This
observation was kept in the NMR closet until in the mid-1960’s when Richard Ernst
and Wes Anderson conceived Fourier Transform spectroscopy [11]. Wes Anderson
suggested the use of the short RF pulse as a generator of a broad-band frequency
excitation, whilst Ernst had thought of using the Fourier transform of the FID to obtain the frequency response function (somehow inspired [12] by Jean Baptist Joseph
Fourier’s work on heat conduction in 1822 [12]).
As more and more complex substances were studied, more complex spectra
started to arise. The development of two dimensional spectroscopy was fundamental to the treatment of such complexity. In 1971, Correlation Spectroscopy (COSY)
was the first two dimensional NMR experiment to be described. The author of this
work was Jean Jeener, but it was never published in a scientific journal. Fortunately, one of Ernst’s graduate students took adequate notes at a conference and
brought the idea to Ernst’s group. This group subsequently developed the concept,
creating multidimensional spectroscopy (whilst giving credit to Jeener) [2]. Multidimensional spectroscopy opened new doors to NMR; determination of structure of
macromolecules such as proteins is one of the pre-eminent examples. As further
developments in NMR theory are beyond the scope of this thesis, I shall stop this
branch of the NMR overview at this stage.
In the early 1970’s a new branch of NMR was founded with the invention of
nuclear magnetic resonance imaging. In the imaging community the term “nuclear”
was eventually dropped, mainly due to public relations concerns and the technique is
now known as magnetic resonance imaging (MRI). In the early 1970’s, it was already
obvious that different tissues had different NMR characteristic parameters, but how
to transform the information carried by a wave with a wave length of the order of
2
The spin echo is commonly associated to the refocusing generated by an 1800 pulse, but that
referred in the original paper was equal to that used for excitation
2.2
Quantum Mechanical Approach
8
meters into something that is spatially localised on a much smaller length scale was
an unsolved problem. The first 2D MR images, or proton spin density maps, were
produced and published in 1973 by Lauterbur [13] using gradients applied in different radial directions to obtain several 1D profiles. Using a reconstruction analogous
to that developed in 1963 for Computerised Tomography (CT) by Cormack [14],
the 1D profiles were transformed into a 2D image. This methodology was baptised as Zeugmatography. In the same year, Mansfield, coming from a completely
different perspective, introduced the mathematical basis of k-space (also based on
gradient encoding) inspired by work on optical diffraction [15]. In 1974 Garroway,
Grannell and Mansfield noticed that magnetic field gradients combined with the
known frequency selectivity of RF pulses could be used for slice selection [16]. In
1975 Kumar, Welti and Ernst [17] devised a two-dimensional Cartesian technique
they called Fourier Imaging or Fourier Zeugmatography. This technique is essentially what is used today in most NMR scanners for structural imaging (it is to a
large extent equivalent to spin warp imaging, that was introduced by Edelstein et
al [18] working in the Aberdeen group in 1980). Until 1976 many test tubes and
vegetables were imaged, but the first in vivo image was of a student’s finger and was
published in 1976 by Mansfield and Maudsley [19]. In 1977 [20] Mansfield proposed
a method where all k-space could be sampled in a single shot, reducing significantly
the acquisition times, and christened it as Echo Planar Imaging (EPI). This technique, due to its large demands on hardware remained out of the commercial market
until the early 90’s. Its greatest advantage relates to do with its ability to perform
functional studies. As the demand for high-speed acquisition keeps growing some
work towards shortening the sampling time even further has been carried out. By
using several RF coils to acquire the image the era of parallel imaging was born.
This approach allows a reduction of the acquisition time that is proportional to
the number of coils used. The use of various RF receivers was first introduced by
J. Carlson in 1987 [21]. The most successful implementations are SMASH [22] and
SENSE [23] that generally represent two alternatives approaches of parallel imaging,
in the k-space and in the image domain respectively.
2.2
Quantum Mechanical Approach
Most NMR phenomena can be understood using the semiclassical approach but,
because NMR deals with the non-classical concept of spin, some interactions can
only be fully described in a quantum picture. Quantum Mechanics gives three
major theoretical tools:
2.2
Quantum Mechanical Approach
9
• the wavefunction, |ψ > that describes the state of the particle and is necessarily
normalised.
• the prediction of either the probability of a single experimental result, qn ,
related to the observable Q̂ that is given by P (qn ) = |hn|ψi|2 (where |ni is the
eigenstate of Q̂ with eigenvalue qn ) or the average value when dealing with a
large ensemble of particles hQ̂i = hψ|Q̂|ψi = T r{Q̂|ψihψ|} = T r{Q̂ρ̂}. The
method of the density operator, ρ̂, is an elegant approach to describing the
quantum state of the entire sample without ever referring to individual spin
states.
• the equation of motion or time dependent Shrödinger equation
ih̄
∂
|ψ >= Ĥ|ψ >
∂t
(2.1)
where Ĥ is the Hamiltonian operator that includes different terms for the
potential and kinetics of the particle. From the Schrödinger Equation it is
also possible to compute the evolution of the density matrix as ih̄ ∂∂tρ̂ = [Ĥ, ρ̂]
What has so far been introduced can be called “spinless” quantum mechanics.
The quantum picture introduced in this thesis is non-relativistic, meaning that the
concept of spin is postulated and has a framework of its own.
Before going any further a minimal overview of the structure of matter is due.
2.2.1
Characterising Matter
Matter is made of atoms, that contain a nucleus, composed to a “first order” of protons and neutrons. Surrounding the nucleus there are orbiting electrons. Atoms are
characterised by their mass (mostly due to nucleons), electrical charge, magnetism
and spin. An atomic nucleus can be specified by three numbers:
• the atomic number, Z, that is equal to the number of protons of the nucleus
and specifies the chemical properties of the atom of which the nucleus is a
part.
• the mass number, N, that is equal to the number of nucleons of the nucleus
and specifies the isotope (nuclei with the same Z, but different N are isotopes).
• the spin quantum number, I,3 refers to the combined spin of the protons (Ip )
and neutrons (In ) and can take one of the values I = |Ip − In |, |Ip − In | +
3
The notation S = h̄I is used throughout this thesis
2.2
Quantum Mechanical Approach
10
1, ..., |Ip + In | − 1, |Ip + In |. The lowest energy nuclear state is called the
ground nuclear state. To know the spin quantum number of a nucleus in its
ground nuclear state, models of the nuclear structure have to be used4 . Nuclei
with
a spin number I have a spin angular momentum magnitude of the form
q
h̄ I(I + 1) and in the absence of an external field have 2I + 1 degenerate
sub-levels that correspond to the discretised values of the projection of the
spin along any direction (−I, −I + 1...I − 1, I).
2.2.2
Spin
1
2
In this thesis all we are interested in is nuclei with spin 12 , which implies that the
projection of the spin along z, Iz , can either be 12 h̄ or − 12 h̄. These two Zeeman
eigenstates are often written as
"
|αi =
| 12 , 21 i
1
0
=
#
"
(spin up) |βi =
| 21 , − 21 i
=
0
1
#
(spin down).
(2.2)
The operator that has such eigenvectors in the Zeeman eigenbasis is Sˆz , other
relevant operators give the projection of the spin along x and y. and are given by
"
Sˆz =
h̄
2
1 0
0 −1
#
"
Sˆx =
h̄
2
0 1
1 0
#
"
Sˆy =
h̄
2i
0 1
−1 0
#
(2.3)
The important shift operators, Sˆ+ = Sˆx + iSˆy and Sˆ− = Sˆx − iSˆy , can be derived
from Sˆx and Sˆy . Some important results of these operators are shown:
Sˆz |αi = 21 h̄|αi;
T r{Sˆz |αihα|} = 12 h̄;
T r{Sˆx |αihα|} = 0;
T r{Sˆy |αihα|} = 0;
Sˆ+ |αi = 0;
Sˆ− |αi = h̄|βi;
2.2.3
Sˆz |βi = − 21 h̄|βi;
T r{Sˆz |βihβ|} = − 12 h̄;
T r{Sˆx |βihβ|} = 0;
T r{Sˆy |βihβ|} = 0;
Sˆ+ |βi = h̄|αi;
Sˆ− |βi = 0;
(2.4)
Spin Hamiltonian Hypothesis
Each nuclear spin is not alone, since any real sample contains thousands of spins as
well as electrons. To describe fully such a sample “all is needed” is a wave function
|ψf ull i, that can be calculated using the Schrödinger equation whose Hamiltonian
4
But some straightforward conclusions can be drawn, even N implies integer I, while odd N
implies half-integer I, furthermore if a nucleus has an even number of protons and neutrons then
the ground state will have I=0
2.2
Quantum Mechanical Approach
11
describes all the interactions, Ĥf ull . Although complete, the resulting equation is
useless as it is not realistically solvable. For this reason the Schrödinger equation is
often only solved for the spin state of the nuclei,|ψspin i,
ih̄
d
|ψspin (x, t)i = Ĥspin |ψ(x, t)i
dt
(2.5)
where Ĥspin is the Nuclear Spin Hamiltonian. This assumption relies on the
assumption that the effects of the rapidly moving electrons occur on a much shorter
timescale, so that the nucleus only senses a time averaged magnetic or electric field,
and on the other hand nuclear spin energies are assumed to be too small to affect
the motions of the electrons within the molecule or the motions of other molecules.
This is the spin Hamiltonian Hypothesis
5
[2].
If the Hamiltonian happens to be time independent (there is no external time
dependent disturbance), then Eq. 2.5 yields the result,
|ψspin (x, t)i = U (t)|ψspin (x, 0)i = e−iĤt/h̄ |ψspin (x, 0)i
(2.6)
where U (t) is the evolution operator (the same concept can be applied to the
density matrix formalism, ρ(t) = U (t)ρ(0)U −1 (t)).
-31
1
x 10
C spin β
C spin α
13
Nuclear energy levels (J)
13
H spin β
H spin α
1
1
∆ E=hγHB0 /2π
∆ E=hγCB0 /2π
0
-1
0
2
4
6
8
10
B (T)
0
12
14
16
18
20
Figure 2.1: Nuclear energy levels dependence on the static magnetic field, B0 , for 1 H and
5
13
C
Longitudinal relaxation conflicts with this hypothesis, because a nuclear spin system relaxes
towards an anisotropic equilibrium state (responsible for the net magnetisation), such relaxation
is impossible if there was not a small contribution from the nuclear spin states to the motion of
molecules [2].
2.2
Quantum Mechanical Approach
2.2.4
12
Zeeman Interaction and the Resonance Condition
As Pauli stated, atomic nuclei have a magnetic dipole moment proportional to the
~ Although in the quantum description, an
intrinsic angular momentum, ~µ = γ S.
electromagnetic field appears as a collection of photons, this approach is complex
and difficult to use for analysing the interaction with nuclear spin and has negligible
~ and B
~ are the vectors
differences when compared to the classical description where E
that represent the electric and magnetic fields of an electromagnetic wave. The
energy associated with the presence of a magnetic moment in stationary magnetic
field, B~0 , is given by −~µ · B~0 . If B~0 is parallel to the z-axis, the Hamiltonian can
be written as Ĥ = −γB0 Sˆz , implying that the degeneracy is raised and the discrete
levels of energy are given by −γB0 Sz . Consequently the energy gap is proportional
to the externally applied magnetic field and depends on the specific nuclei, see Table
2.2.4. For the case of 1 H and 13 C that have spin 21 and positive gyromagnetic ratio,
there will be two energy levels (the lower of which will be for Sz = 12 h̄), but the
energy gap will be four times smaller for
13
C when compared to that of 1 H. This
form of magnetic interaction is known as the Zeeman interaction and generally is the
dominant interaction of a spin with the environment [24](when there is an external
magnetic field).
At this stage the evolution operator introduced in Eq. 2.6 is given by U (t) =
−iγB0 Sˆz t
e
. This can be interpreted as a rotation by an angle γB0 Sz t around the z-axis
(see Fig. 2.2)6 , meaning that the spins will be precessing at the Larmor frequency,
ω0 , given by
ω0 = −γB0 .
(2.7)
The angle of the precession cone and the main magnetic field will be given by
arccos √ Sz
which for half integer spins is the magic angle ' 54.73o
S(S+1)
The term Nuclear Magnetic Resonance refers to the resonance phenomenon that
takes place when this ensemble of spins precessing with the Larmor frequency is
irradiated with photons with the same frequency and therefore an energy equal to
that of the gap between the two states. The magnetic field component of the photons
has to be in the plane perpendicular to the static field (this argument will become
clear as the rotating frame is introduced).
6
Rotations of spins around an arbitrary axis z’ by an angle φ can be represented as eiφSz0 /h̄
2.2
Quantum Mechanical Approach
Ground
Natural
Isotope state spin abundance
1
1
H
∼ 100%
2
2
H
1
0.15%
1
3
H
0%
2
12
C
0
98.9%
1
13
C
1.1%
2
14
N
1
99.6%
1
15
N
0.37%
2
16
O
0
∼ 100%
5
17
O
0.04%
2
1
19
F
∼
100%
2
3
23
Na
∼ 100%
2
5
27
Al
∼ 100%
2
1
31
P
∼ 100%
2
1
19
F
∼ 100%
2
3
35
Cl
75.7%
2
3
37
Cl
24.3%
2
1
129
Xe
24.4%
2
13
Gyromagnetic ratio NMR frequency at
(rad.s−1 T −1 )
11.7433 T (MHz)
6
267.522 × 10
−500.000
6
41.066 × 10
−76.753
285.349 × 106
−533.320
67.283 × 106
−125.725
19.338 × 106
−36.132
6
−27.126 × 10
50.684
6
−36.281 × 10
67.782
251.815 × 106
−470.470
70.808 × 106
−132.259
6
69.763 × 10
−130.285
108.394 × 106
−202.606
6
251.815 × 10
−470.470
10.610 × 106
−48.990
6
8.832 × 10
−40.779
6
−74.521 × 10
139.045
Table 2.1:
2.2.5
Rotating Frame and RF pulses
The equation of motion is more complex in the presence of radio frequency pulses
described by an oscillating magnetic field of amplitude 2B1 transverse to the equilibrium polarisation, since the Hamiltonian becomes time dependent
Ĥlab = −γB0 Sˆz − 2γB1 cos(−ωt)Sˆx
(2.8)
and a solution of the form shown in Eq. 2.6 is no longer valid. A small trick is
needed to cancel out the time dependence of the Hamiltonian. If we consider a frame
rotating at a frequency ω, and decompose the oscillatory field into two circularly
polarised components, one rotating in the same sense as the precessing spin and one
counter-rotating, then the Hamiltonian in the frame rotating with the precessing
spins is given by,
Ĥrot = −γ(B0 −
ω ˆ
)Sz − γB1 Sˆx − γB1 cos(2ωt)Sˆx
γ
(2.9)
The last term can be considered to average to zero, as it is varying very rapidly
when compared to all the other terms in the Hamiltonian. This transformation can
be demonstrated mathematically, but here an intuitive view of why “it couldn’t be
2.2
Quantum Mechanical Approach
Null static magnetic field
14
Static magnetic field B 0
B0
Null net magnetisation
Net magnetisation=
N γ 2 h2 B 0
16 π k BT
Figure 2.2: (a)Schematic diagram of nuclear spins in the absence of an external magnetic field
showing random orientations and (b) in the presence of a static magnetic field around which they
precess at the Larmor frequency and add up to create a net magnetisation (this is not visible in
the figure as for the field range present in most scanners at room temperature 105 spins would be
needed before an excess of spins in the up-state could be shown).
any other way” will be given. As the reference frame is now rotating, the precession
will be slower, and the field has to be corrected so that the Schrödinger Equation
gives the desired frequency of precession in the new frame. If ω = ω0 , the apparent
longitudinal field will vanish, leaving only the B1 term, which is now constant in time.
This equation has strong similarities with the Zeeman interaction and the evolution
ˆ
operator will be given by U (t) = e−iγB1 tSx that is a rotation of the magnetisation
around the x-axis at a frequency γB1 . This rotation can be understood in another
perspective by considering the shift operator, Sx = 21 (S + + S − ), all that will be
happening is the inter-exchange of up and down spin states at a rate γB1 .
2.2.6
Net Magnetisation
Crucial for the NMR signal is how much magnetisation exists in a pool of identical
independent spins in the presence of a constant magnetic field. If such an ensemble
is characterised by a wave function in the Zeeman basis of the form |Ψ >= a 1 |α >
2
+a− 1 |β >, then the averaged expectation value of Sz will be given by
2
´
1 ³
T r{Sˆz |ΨihΨ|} = h̄ |a 1 |2 − |a− 1 |2
2
2
2
(2.10)
which states that the expectation value of Sz , or polarisation of the ensemble is
determined by the population difference between the upper and lower energy states.
The population of each energy level in thermal equilibrium is given by the Boltzman
probability distribution. In equilibrium where there are two possible levels of energy
(± 12 h̄γB0 )
2.2
Quantum Mechanical Approach
|a± 1 |2 =
15
h̄γB0
BT
± 2k
e
(2.11)
h̄γB0
h̄γB0
−
e 2kB T + e 2kB T
As the energy gap (h̄γB0 ) at room temperature is much smaller than the Boltz2
mann energy (kB T ), the exponentials can be expanded to first order, leading to
a population difference of
magnetisation of
2.2.7
h̄2 γ 2 B0 7
.
4kB T
h̄γB0
2kB T
that once substituted in Eq. 2.10 leads to a net
The Spin Hamiltonian
The nuclear spins of an ensemble interact with electric and magnetic fields they experience locally, which may either be generated by the external apparatus (external
spin interactions or couplings) or by the sample itself (internal spin interactions).
The external couplings are the Zeeman interaction with the static magnetic field and
the RF pulses, while the internal couplings are Chemical Shift, Dipolar Coupling,
Scalar or J-coupling and Quadrupolar Coupling 8 .
Chemical Shift The electrons in molecules cause the magnetic fields to vary on
a submolecular distance scale thus, different sites in a molecule experience different
fields if their surrounding electronic environment is different. This is called the
chemical shift. The induced field is linear in B0 , Bjind = δ j · B0 where δ j is a 3 × 3
matrix, and is responsible for creating different directions in the local field. In the
case of isotropic liquids, due to the secular approximation (the Zeeman interaction
is greater then any other interaction) and motional narrowing, chemical shift effects
can be simplified and defined in terms of a chemically shifted Larmor frequency,
j
j
j
ωj0 = −γj0 B0 (1 + δjiso ) where δjiso = 13 (δxx
+ δyy
+ δzz
) is the isotropic chemical shift
and is an average of the induced components in the three cartesian directions, when
the field is in that same direction. The major contribution to chemical shifts comes
from the low lying electronic states and the more electrons the atom has, the larger
the chemical shift is. δ is also correlated with the electronegativity of neighbouring
atoms, since high values diminish the electron density in the atom of interest, thus
increasing the local fields and δ.
7
Note that 13 C NMR will be penalised with respect to 1 H by a factor of 16 only on the net
magnetisation available for equal concentrations
8
This is the only electric interaction and only occurs for nuclei with S > 21 h̄.
2.2
Quantum Mechanical Approach
16
Dipole-dipole coupling Direct dipole-dipole coupling is a consequence of each
dipole feeling the field generated by the neighbouring dipoles and also generating a
field that can be felt by its neighbours. The Hamiltonian will “simply” describe the
sum of the effects of magnetic fields generated by each dipole on every other dipole,
B0
k
j
Θjk
Figure 2.3: Schematic diagram of nuclear spins showing the direct dipole-dipole interaction
between two spins.
³
´
X
X k−1
X
dip−dip
dip−dip
1 X k−1
ˆ
ˆ
Hjk
Hall
=
bjk 3(Ŝj · ejk )(Ŝk · ejk ) − Ŝj · Ŝk (2.12)
=
h̄ k j
j
k
µ0 γi γj h̄
where bjk = − 4π
and ejk is the unit vector that connects the nuclei j and k.
3
rjk
Note that the energy for atoms with the same γ will be minimised for a position
when the spins are parallel to one another. Probing this dipole coupling can give a
very important insight into structure [2]. As the static magnetic field increases, the
Zeeman energy becomes much larger than that due to the dipolar interaction and a
secular approximation can be made,
³
Hˆjk
dip−dip
´
3Sˆjz Sˆkz − Ŝj · Sˆk
1 1³
= bjk 3 cos2 Θjk − 1 { ³ ˆ ˆ ´
h̄ 2
2Sjz Skz
´
homonuclear case
heteronuclear case
(2.13)
where Θjk is the angle between the external static field and ejk . Note that the
difference between the homo and heteronuclear cases is a consequence of transverse
terms of the vector dot product averaging to zero due to the different rates of
precession of different spins.
In the case of isotropic liquids, the secular part will average to zero as on the time
scale of the coupling, the liquid can be considered to be a homogeneous distribution
2.2
Quantum Mechanical Approach
17
R
of spins and 0π dΘjk sinΘjk (3 cos2 Θjk − 1) = 0. In the case of solids, dipole-dipole
effects are not averaged out and the dipole-dipole interaction (both intra and intermolecular) is the main cause of the large line width of the spectra of solids9 .
J-coupling In contrast to the previous interaction this effect arises from indirect
dipole-dipole coupling and is known as indirect spin-spin coupling or more commonly
J-coupling. This internuclear interaction is said to be indirect as it is mediated
by the bonding electrons in the molecular orbital (making it an exclusively intramolecular coupling). In a simple two-atom molecule, the J-coupling mechanism
can be understood as resulting from a three step process: the magnetic hyperfine
interaction between the spin of the nucleus 1 and the “closest” orbital electron leads
to a reduction of energy in the states, where the nucleus and electron spins are
antiparallel; by Pauli’s principle the other electron in the molecular orbital will have
opposite spin direction; due to hyperfine coupling the spin orientation of nucleus
2 that will minimise energy is antiparallel to both the second electron spin and,
consequently, the spin of nucleus 1. The indirect nature of the coupling and its
independence of the applied magnetic field leads to the possibility of the effects of
this term being observable in isotropic liquids. The Hamiltonian of the interaction
between spin j and k can be written as,
J,f ull
Ĥjk
=
2π
Ŝj · Jjk · Ŝk
h̄
(2.14)
where Jjk is a 3 × 3 real matrix. In the case of liquids, all the anisotropic terms
are cancelled as a result of rapid molecular motion, leaving only the diagonal terms
ˆ = 2π J Ŝ · Sˆ 10 .
J,iso
whose average is written as Jjk , leaving the Hamiltonian as Hjk
k
h̄ jk j
Note that this mechanism provides an explanation for the positive values of Jjk for
one-bound molecules between nuclei with the same sign of gyromagnetic ratio (the
energy is minimised for antiparallel spins).
Electric Quadrupole coupling This term is only present for nuclei with spin
greater than 12 h̄, for which there is a strong interaction between the electric quadrupole
moment of the nucleus and the electric field generated by surrounding electrons. The
secular interaction term is given by
Q
Ĥj =
9
1 Q ˆ2
ω (3Sjz − Ŝj · Ŝj )
h̄ j
(2.15)
There are techniques to manipulate the Hamiltonian in order to make the dipolar Hamiltonian
disappear, the most popular and effective is Magic Angle Spining (MAS)
10
The same simplification as in Eq. 2.13 occurs in the case of heteronuclear interactions
2.3
Semi-Classical Approach, the Bloch Equations
18
where ωjQ is the quadrupolar frequency (which is zero for liquids and is described by
a more complex expression that depends on the quadrupolar moment of the nucleus
and the gradient of the electric field in the region of the nucleus for solids).
^
H
^
H
Static
Field
spin
^
H
ext
RF
Field
Chemical
Shift
Dipoledipole
(short range)
int
Dipoledipole
(long range)
Scalar JCoupling
Quadrupole
Coupling
SOLIDS
LIQUIDS
Figure 2.4: Summary of the motionally averaged spin Hamiltonian terms and their relative
importance in different phases of matter. Note that the Quadrupolar coupling disappears for
S = 12 h̄, and the liquid state represented is the one relating to isotropic liquids (in the case of
anisotropic liquids, the short range dipolar coupling doesn’t completely average to zero and may
be more important then the chemical shift). This scheme is based on reference [2].
Figure 2.4 gives an overview of all the terms involved in the spin Hamiltonian
and their relative importance. Chapter 3 gives a further discussion of the relevance
of the long range dipolar field in liquid NMR.
As has been shown, the interaction of a single spin S = 12 nuclei with the surrounding environment can be described in terms of two spin states and the Hamiltonian is a rank 1 tensor in the spin operator involving only Zeeman like interactions
with the surroundings (static B0 field, RF pulses when considered in the rotating
frame and Chemical Shift) while for dipole-dipole and J-coupling more complex
products of spin operators are needed. The meaning of such operators is described
in Table 2.2.7. This is the theoretical justification for using a vector magnetisation
to describe the evolution of an ensemble of independent spin- 21 nuclei, as is the case
in the next section.
2.3
Semi-Classical Approach, the Bloch Equations
Quantum mechanics allows an exact description of NMR that goes to the root of
the observed effects, while the classical approach gives a more phenomenological
2.3
Semi-Classical Approach, the Bloch Equations
one spin operators
Ŝ1z , Ŝ2z
Ŝ1+ , Ŝ2+
Ŝ1− , Ŝ2−
19
name
polarisation of spins 1 and 2
in phase +1-quantum coherence spins 1 and 2
in phase -1-quantum coherence spins 1 and 2
two spin operators name
Ŝ1+ Ŝ2z
+1-quantum coherence of spin 1 in antiphase with spin 2
Ŝ1− Ŝ2z
-1-quantum coherence of spin 1 in antiphase with spin 2
Ŝ1+ Ŝ2+
in-phase +2-quantum coherence (+DQC) of spins 1 and 2
Ŝ1− Ŝ2−
in-phase -2-quantum coherence (-DQC) of spins 1 and 2
Ŝ1− Ŝ2+ , Ŝ1+ Ŝ2−
in-phase +0-quantum coherence (+ZQC) of spins 1 and 2
Ŝ1z Ŝ2z
longitudinal two-spin order of spins 1 and 2
Table 2.2: One and Two-Spin basis operators of coupled spins. This table is based on reference
[24].
description, and tends to be easier to handle. In this section the main equations
describing the behaviour of spins will be presented and discussed. Processes such
as relaxation and diffusion will be introduced, whilst others, such as chemical shift
and RF pulses will be revisited in this new framework.
As was discussed in the previous section, for nuclei of spin 12 , in the absence
of couplings between spins, the magnetisation vector (also called the macroscopic
~ , can be used to describe the evolution of spins. This
or bulk magnetisation), M
magnetisation is defined as the magnetic moment per unit of volume which, as
2 2
described in Section 2.2.6, has an amplitude of ργ4kh̄b TB0 , where ρ is the number of
nuclei per unit volume.
The equations used to predict the evolution of the magnetisation are the Bloch
Equations,
Ã
!
~ (~r, t)
dM
~ (~r, t) × B~0 + ∆B
~ 0 (~r) + (G(t)
~ · ~r)ẑ + B
~ 1 (t) + δωCS ẑ
= γM
dt
γ
M0 − Mz (~r, t)
Mx (~r, t)
My (~r, t)
~ (~r, t) (2.16)
+
ẑ −
x̂ −
ŷ + D∇2 M
T1
T2
T2
where the evolution of the magnetisation is dependent on several terms such as:
the external static magnetic field B~0 , magnetic field inhomogeneities ∆B0 (~r), linear
~
magnetic field gradients G(t),
RF pulses B~1 (t), chemical shift δωCS , spin-lattice
relaxation T1 , spin-spin relaxation T2 and self diffusion coefficient D. In the following
subsections each term will be discussed and its meaning will be described.11
11
In chapter refchap:LongRangeDipolar another magnetic field contribution to the Bloch Equa-
2.3
Semi-Classical Approach, the Bloch Equations
2.3.1
20
Static Magnetic Field, Gradients and Chemical Shift
The interaction of magnetisation with a static magnetic field is classically described
by
dM~(t)
~ × B~0
= γM
(2.17)
dt
if B~0 is applied in the z-direction, the solution will be Mx (t) = Mxy cos(ω0 t + φ),
My (t) = Mxy sin(ω0 t+φ) and Mz (t) = Mz (0), where Mxy is a constant describing the
total transverse magnetisation and φ is an arbitrary phase angle. This is a precessing
motion of frequency, ω0 , proportional to the applied magnetic field, B0 , and the
gyromagnetic ratio, γ as was demonstrated in Eq. 2.7 (ω0 = −γB0 ). Because the
vector is precessing at a frequency corresponding to the region of the electromagnetic
spectrum of radio waves, it is intuitive that an antenna (RF coil), such as those used
to detect radio signals, can be used to record the signal.
A complex notation is introduced here for simplicity. Instead of dealing with
the x- and y-components of the magnetisation individually, M + is defined as M + =
Mx + iMy = Mxy eiω0 t+φ . As in Section 2.2, transformation to the rotating frame
facilitates the calculations,
~
dM
~ × (B~0 − ωrot )
= γM
(2.18)
dt
γ
if ωrot is at the Larmor frequency, the right hand side will be zero. In this rotating
frame the magnetisation will remain “mostly” still. “Mostly” because there are
other terms that affect the frequency of precession:
• Localised magnetic field inhomogeneities can arise either from imperfections
of the applied static magnetic field or from magnetic fields generated by magnetisation of the sample.
• Linear magnetic field gradients are extremely important in NMR and MRI (as
~ r, t) =
will be seen in Section 2.4). In the presence of a gradient such as G(~
~ r, t) · ẑ) the frequency of precession has an offset from the Larmor fre∇(B(~
~ r, t) · ~r that varies with position. As a consequency of ∆ω(~r, t) = −γ G(~
quence of the varying frequency, the signal has a varying phase given by
R
~ r, τ ) · ~rdτ .
φ = −γ 0t G(~
• Chemical Shift was already discussed in Section 2.2.7, where it was concluded
that for liquids this causes a simple frequency shift, δωCS .
tions that can be extremely important in predicting the evolution of signal in some sequences will
be introduced.
2.3
Semi-Classical Approach, the Bloch Equations
2.3.2
21
Radio Frequency Pulses
Time-dependent magnetic fields are used to transfer the net magnetisation that is
directed along z into the transverse plane, so that its precession can be detected
by an RF coil. In Section 2.2.5 it was shown that RF pulses could be used to
induce transitions between two spin states leading to changes in the components
of magnetisation. The same can be demonstrated in the classical framework of the
Bloch Equations. In the case of an on-resonance RF pulse, the Bloch Equations in
the Rotating Frame neglecting relaxation and diffusion will be:
~ (~r, t)
dM
~ (~r, t) × (∆Bz ẑ + B~1 (t))
= γM
dt
(2.19)
~
r
where ∆Bz is a combination of effects ∆Bz = γ(ω0 + G(t)·~
+ δωCS − ωrot ). If
γ
∆Bz = 0, the resonance condition occurs and the solution in the rotating frame for
B~1 (t) = B1 x̂0 (notation as in Fig 2.6) is a precession around the x-axis at a frequency
ω1 = −γB1 . Depending on the duration of the RF pulse, various rotations can be
accomplished... and pulsed NMR is born!
For various reasons it is impossible for ∆Bz to be nulled everywhere within
the sample (in imaging this is not even desired) and B1 is also not necessarily a
constant. The easiest way to deal with reality is to break it into smaller “better
known” realities. In RF pulse design this is called the low flip angle approximation.
If ω1 (t) = −γB1 (t), the B1 field is directed along the x axis in the rotating frame,
and ω3 = −γ∆Bz is defined and the flip angle is so small that, given an initial
~ (~r, 0) = M0 ẑ, dMz ≈ 0 leading to the assumption that Mz (t) ≈ M0 ,
condition of M
dt
Eq. 2.19 can be rewritten in the complex notation as
dM~ + (~r, t)
− iω3 M + (~r, t) = −iω1 (t)M0
dt
(2.20)
This equation can be easily integrated if both sides are multiplied by e−iω3 t . The
integration gives an interesting result:
+
M = −iM0 e
+iω3 t
Z t
0
0
ω1 (t0 )e−iω3 t dt0
(2.21)
where B1 (t0 ) is is switched on at time t0 = 0 and off at time t. In other words,
the transverse magnetisation as a function of the resonance off-set depends on the
Fourier Transform of the pulse B1 (t). It is intuitive that pulses are frequency selective unless they are infinitely narrow. This property is extremely useful, as pulses
can be designed to have a narrower or broader frequency response. They are referred
to as soft pulses (as opposed to hard pulses which are very narrow pulses leading to
2.3
Semi-Classical Approach, the Bloch Equations
z
(a)
z'
(b)
M
B1
y
x
22
M
y'
x'
z'
(c)
z'
(d)
Bz
Beff
Beff
Bz
B1
x'
M
M
y'
B1
y'
x'
~ ef f . In (a) the magnetisation
Figure 2.5: Precession of the Bulk Magnetisation vector around B
M along the z-axis, is seen in the laboratory frame to follow a nutation motion with frequency ω0
around the z-axis due to the static magnetic field and ω1 around x due to a RF pulse with frequency
ω0 and amplitude B1 = ω1 /γ. In (b),(c) and (d) the behaviour of magnetisation is followed in
the rotating frame. The rotating frame is chosen so that the field B1 appears to be static. Three
different cases are represented: (b) the rotating frame is on resonance with the precessing spins,
so that no field exists along z ; the rotating frame is not on resonance with the precessing spins
and “residual” field exists along z which is equal (c) or 5 times greater than B1 (d) leading to a
~ ef f = B
~z + B
~ 1 with a frequency proportional to
precession around an axis in the direction of B
Bef f . A positive value of γ was used in this illustration
a very broad frequency response). In the case where ∆Bz is dominated by chemical
shift contributions, depending on its carrier frequency, the pulse may only rotate
protons in a certain chemical environment, and leave all other protons with different chemical shifts undisturbed. In the case where ∆Bz is dominated by a linear
gradient, for example along z, so that ω3 = −γGz z, it is possible to excite only a
certain region of the sample. The extent of this region can be decreased either by
increasing the gradient strength or by stretching (in duration) and weakening (in
amplitude) the pulse. This is extremely relevant for imaging purposes as it allows a
slice to be defined. Note that in such a case, due to the term e−iω3 t , it is important
to use a reversed gradient for a time t/2 to ensure that all the excited signal is in
phase.12
12
In environments dominated by chemical shift this is not possible and there is a class of pulses
2.3
Semi-Classical Approach, the Bloch Equations
2.3.3
23
Relaxation times, T1 , T2 and T2∗
The spin-lattice relaxation time, T1 , describes the return to equilibrium of the longitudinal component of the magnetisation of an ensemble of spins (in an environment
with a static magnetic field applied along z ) after having been perturbed by an RF
pulse. The Bloch equation describing the longitudinal component of magnetisation,
when only the T1 term is considered, is dMz (t)/dt = (M0 −Mz (t))/T1 whose solution
is
µ
Mz (t) = M0 1 − e
−t
T1
¶
−t
+ Mz (0)e T1
(2.22)
In a very naive view this equation can be interpreted in the following way: the
first term on the right hand side represents the recovery of net magnetisation of most
of the ensemble which has no initial net magnetisation, as the populations of spin
up and down are the same; the second term represents an opposite effect, a small
ensemble which is totaly polarised tends to decay into an equilibrium in which, to
a first order, the populations are the same. The term, spin-lattice, is an heritage
left by solid state NMR, as the spin system is driven to the minimum energy state
it deposits the energy in the surrounding solid lattice.
The spin-spin relaxation time, T2 , describes the return to equilibrium of the
precessing transverse components of the magnetisation. The Bloch equations in
an on-resonance rotating frame have only one term affecting the evolution of the
transverse magnetisation (when diffusion is neglected), dM + (t)/dt = −M + (t)/T2 ,
whose solution is given by,
−t
M + (t) = M + (0)e T2
(2.23)
However, this is not necessarily the decay observed in an NMR free induction
decay (FID). The FID in vivo often decays in much shorter times than T2 because
of field inhomogeneity and diffusion effects. The rate of decay of the FID in the
rotating frame is given by
−t
∗
M + (t) = M + (0)e T2
(2.24)
where T2∗ is called the apparent transverse relaxation time. These relaxation
times can be ordered in magnitude, the shortest being the apparent transverse relaxation time, T2∗ , that has as a limiting maximum the transverse relaxation time,
T2 , that is itself limited by the longitudinal relaxation time, T1 .
known as self-refocusing pulses that have a transverse phase response that is independent of ∆Bz
2.3
Semi-Classical Approach, the Bloch Equations
24
Relaxation Mechanisms
What are the mechanisms of relaxation? Both through the quantum approach (Section 2.2.5) and the classical approach (Section 2.3.2), it was demonstrated that for
a spin to undergo a transition between two states the transition should be stimulated by a photon “or” a magnetic field varying with a frequency, ω0 . Through the
interactions of the ensemble there are a variety of ways that such a mechanism can
occur. There are two interactions which are most important in spin
1
2
samples:
• Nuclear dipole-dipole interactions of which some examples are dipole-dipole,
scalar of first kindand chemical shift anisotropy [25]. Most of these rely on
molecular motion, rotation or vibration to generate the needed time-varying
magnetic fields;
• Paramagnetic relaxation, in which a molecule with one unpaired electron which
has a large magnetic moment, ≈ 1000 times larger than the proton’s, that
through random tumbling generates a random magnetic field at the site of a
nucleus. This effect will propagate through the sample through a mechanism
called spin diffusion, that tries to compensate the magnetisation gradient that
exists in the radial direction starting at the impurity centre. The mechanism
of spin-diffusion is in most cases due to dipole-dipole interactions [25]. Paramagnetic materials are often used to reduce T1 and therefore allow higher SNR
when a sequence is repeatedly applied;
These mechanisms are Larmor frequency dependent. In the case of longitudinal
relaxation, they are stimulated emissions due to the transverse components of the
field, Bxy =
q
Bx2 + By2 , at the position of spin j generated by a neighbouring spin
k. For transverse relaxation the z component of the field, Bz , is also relevant. These
fields are given by
Bxy = µ0
3γ sin 2Θjk
3
2rjk
B z = µ0
γ(3 cos2 Θjk − 1)
3
2rjk
(2.25)
where the notation is consistent with Figure 2.3
Most dipole-dipole relaxation mechanisms (all but spin rotation) rely on molecular movement by translation or rotation to modulate the field changes in nearby
molecules. As a result of the fluctuating/random translation or rotation of a molecule,
a function h(t) can be associated with the fluctuation of the local field. The autocorrelation of such a function is given by G(t) =< h(t)h(t + τ ) >. If the motion
2.3
Semi-Classical Approach, the Bloch Equations
25
is truly random/Brownian, the autocorrelation function is expected to be proportional to e−t/τc , where τc is the correlation time. This result can be interpreted as a
decrease of the probability of finding a correlation between the local fields as time
increases (imagine a random motion where the steps are small compared with the
order of magnitude of the positions, after a small time it is likely that the positions
still have the same sign, while after a long time they may either have the same sign
or opposite sign, that will null each other as the integration is carried out). The
frequency spectrum or spectral density J(ω) of the autocorrelation function is simply the Fourier transform of G(t). The Fourier transform of an exponential decay is
known to be a Lorentzian given by
J(ω) =
τc
.
1 + τc2 ω 2
9
x 10
2
1.8
2
10
solids τc=10-8 s
viscous liquidsτc=10-9 s
nonviscous liquids τc=10-10s
T1
T2
T1
T2
1
10
1.6
at
at
at
at
100MHz
100MHz
400MHz
400MHz
0
10
1.4
Relaxation Times (s)
Spectral Density J(ω)
(2.26)
1.2
1
0.8
-1
10
-2
10
Liquids
Solids
-3
10
0.6
Viscous
Liquids
0.4
-4
10
0.2
0
-5
-8
10
-10
frequency, ω /2 π
10
10
-12
10
-10
10
-8
τc (s)
10
-6
10
Figure 2.6: (a)Typical spectral densities as a function of frequency for different states of matter.
(b) Relaxation Times as a function of τc . Figure based on data from [26]
The relationship between relaxation times and spectral densities was originally
derived through time-dependent perturbation theory by Bloembergen, Purcell and
Pound [6]. For a two spin rotating molecule the relaxation times are
1
= K[J(ω) + 4J(2ω)]
T1
1
3
5
= K[ J(0) + J(ω) + J(2ω)]
T2
2
2
(2.27)
2.4
Magnetic Resonance Imaging
³
´2
26
4 2
µ0
3γ h̄
. The spectral densities with arguments 0, ω, 2ω correspond
with K = 4π
10r6
to the effects of static field (affects T2 through a dephasing that needs no energy
transition, but not T1 ), one spin flip and two spin flips respectively. J(0) is the
dominant effect in solids where T2 ¿ T1 . The effect of temperature on the relaxation
times results from the dependence of the correlation time on temperature, which
generally tales the form τc ∝ eE/kb T , where E is an activation energy.
2.3.4
Diffusion, D
Diffusion is an intrinsic property of matter and hence has to be included in the
Bloch equations. The diffusion coefficient is defined as a proportionality constant
between average of the squared radial displacement of a particle and elapsed time,
< r2 >= 6Dt. Diffusion was first introduced into the Bloch Equations by Torrey [27],
who used a simple argument of magnetisation transport acting to compensate a
gradient of magnetisation. Neglecting all the other terms in the Bloch equations,
the dependence of the evolution of magnetisation on the diffusion constant is given
by,
~ (~r, t)
dM
~ (~r, t)
= D∇2 M
(2.28)
dt
Diffusion per se has no effect on the evolution unless there is some spatial de~ (~r, t) 6= 0). This spatial dependence of magpendence of the magnetisation (∇2 M
netisation can either be related to the structure of the sample, or to local field
inhomogeneities or can be externally induced by the application of gradients. The
tighter the modulation of the magnetisation, the faster is the diffusion related decay. Consider the case where all the magnetisation, M0 , is rotated parallel to
the x-axis, and then a gradient pulse G is applied along z during an interval δt
leading to a state where M + (δt) = M0 e−iγGδtz = M0 e−ikm z , refocusing is possible
by simply applying a pulse with the same duration, but the opposite amplitude,
M + (2δt) = M0 e−iγ(Gδtz−Gδtz) = M0 . If diffusion occurs between these two pulses,
2
separated by ∆T , the signal will be attenuated by e−km D∆T (the attenuation while
the gradients are applied is neglected here). Such an attenuation is not reversed in
a Hahn spin echo.
2.4
Magnetic Resonance Imaging
In conventional NMR, as the FID is acquired and subsequently Fourier transformed,
it is possible to distinguish protons in different chemical environments because of
2.4
Magnetic Resonance Imaging
27
their varying frequencies. In MRI, as was hinted in Section 2.3.1, gradients are used
before and during acquisition to localise spatially the precessing nuclei. The aim of
this section, is to present Magnetic Resonance Imaging, without going into historical
detail. An initial discussion centres on the mathematical formalism of MR imaging,
going into some details of the sequences used for imaging during this thesis, before
moving on to a consideration of some common artifacts and limitations.
2.4.1
The Signal Equation and k-space
The wavelengths associated with typical Larmor frequencies ω0 (> 10cm) make
imaging using NMR in a “conventional” way unappealing (“conventional” can be
understood as relating to optical imaging, where it is essential to have a wavelength
that is much smaller than the object being observed). Assuming that the sensitivity
of the RF Coil (see Section 2.7) is constant over the whole sample, neglecting any
field inhomogeneities or varying chemical environment, the signal after a 90o RF
pulse that would tip all the magnetisation in the sample can simplistically be written
as:
Z
sr (t) =
Z
+
vol
M (~r, t)dV =
M0 (~r)e−t/T2 e−iω0 t e−iγ
Rt
0
~ )·~
G(τ
rdτ
dV
(2.29)
To simplify the maths, relaxation will be considered not to affect the acquisition
period and, since in this thesis all the imaging sequences used are planar/2D, our
attention will focus on the 2D case. If a slice of width ∆z is initially excited centred
around the position z0 , the previous equation can be rewritten by defining m(x, y) =
R z+∆z /2
z−∆z/2
M0 (x, y, z)dz. Equation 2.29 can be reformulated to focus on the properties
of the envelope/baseband signal, s(t), by simply dividing sr (t)13 by e−iω0 t , giving
rise to the final expression known as the Signal Equation,
Z Z
s(t) = sr (t)eiω0 t =
x
y
m(x, y)e−i(kx (t)x+ky (t)y) dxdy
(2.30)
where
kx (t) = γ
ky (t) = γ
13
Z t
0
Z t
0
Gx (τ )dτ
Gy (τ )dτ
(2.31)
The process of extracting the carrier wave information is referred as demodulation and will be
described in Section 2.6.
2.4
Magnetic Resonance Imaging
28
are the time integrals of the gradient waveforms. If the signal equation is compared with a 2D Fourier Transform of m(x, y)
Z Z
M (kx , ky ) =
x
y
m(x, y)e−i(kx x+ky y) dxdy
(2.32)
the similarities are obvious and it can immediately be concluded that s(t) =
M (γ
Rt
0
Gx (τ )dτ, γ
Rt
0
Gy (τ )dτ ). The understanding that the signal obtained is re-
lated to the Fourier transform of the magnetisation distribution is essential for the
design and understanding of any imaging sequence. With some basic understanding
of Fourier Transforms, as described in Appendix A, the aim of any imaging sequence
is to sample k-space in such a way that, following the inverse Fourier transformation,
enough information is gathered to provide a good representation of the object in real
space (with the desired field of view and resolution).
So far the discussion of the signal equation has assumed that the signal is a
continuous function, in reality the envelope signal is digitised using an analog to
digital converter, ADC, leading to a discretised sampling of k-space. Here, the
spacing between adjacent points and the extent of k-space sampled in the frequency
and phase encoding directions are given respectively by ∆kx , ∆ky and Wkx = ∆kx Nx ,
Wky = ∆ky Ny where Nx and Ny are the number of points sampled in the x and ydirections. Using the relations introduced in Appendix A between the spacing of
sample points and the extent of sampling, the extent of the representation and
resolution of its Discrete Fourier Transform (DFT), it can be concluded that the
field of view, FOV, in each direction is given by,
2π
∆kx
2π
F OVy =
∆ky
F OVx =
(2.33)
while the spatial resolution is
2π
W kx
2π
δy =
W ky
δx =
(2.34)
2.4
Magnetic Resonance Imaging
ky
Wk x
(a)
29
kx
Wk y
∆k
2DFT
y
∆ kx
y
(b)
FOV
FOV
δx
FOV
FOV
δx
FOV
δy
δy
FOV
FOV
δx
FOV
δy
FOV
FOV
x
δx
FOV
δ
FOV
FOV
δy
FOV
δx
δy
FOV
δx
FOV
δy
δx
FOV
δy
δy
FOV
x
δx
FOV
δy
Figure 2.7: Representation of the effect of applying a 2 dimensional Discrete Fourier Transform
(2DFT) to a limited and digitised k-space region (a). The grey area of the real space (b) is the
image obtained, that will be periodically repeated. The notation used in the figures is defined in
the main text
2.4.2
Imaging Sequences
There are various ways of generating images in MRI: in multiple or single shots, in
Cartesian or in radial coordinate systems or in other more complex non-Cartesian
ways, such as spiral sampling of the k-space.
In the following sections, attention will be focused on one multi-shot and one
single-shot Cartesian technique. These being Spin Warp and Echo Planar Imaging,
respectively.
2.4
Magnetic Resonance Imaging
30
Spin Warp Imaging
Spin Warp imaging is essentially equivalent to a preceding technique called 2D
Fourier imaging (see Section 2.1), Spin Warp will be explored here as it is more
robust due to the form of the phase encoding gradients employed. The Spin Warp
imaging sequence is shown in Figure 2.8, and it consists of an initial frequency selective pulse (refer to Section 2.3.2) applied simultaneously with a gradient along the
z-direction. Once the pulse finishes, the magnitude of the gradient is inverted for
a time approximately equal to half the RF pulse length, in order to refocus all the
through-slice magnetisation along the same transverse direction. After an evolution
time, that will define the contrast weighting of the image, the gradients along x and
y are turned on with an amplitude −Gx and Gy for a time t2x (these are not necessarily simultaneous and for notational simplicity ty will be defined as the time when
Gy is on). The polarity of the gradient along x is changed and remains positive for a
time tx until it is switched off. While Gx is on, sampling is performed. The imaging
sequence is repeated NP Esteps times for various amplitudes of Gy .
The function of Gy is to phase encode (PE) the signal. At the end of the time
ty , spins will have varying phase shifts depending on their y position, such that
their phase will be encoded as ∆φ = γGy yty . Note that during one acquisition,
Rt
ky (t) will remain unchanged and will be given by ky (t) = γ 0 y Gy dτ = γGy ty . The
phase encoding is applied NP E times with the gradient Gy being incremented from
−Gy max to Gy max in steps of size Gyi
RF channel
K y
Gz
K x
ty
Gy
Gx
t x /2
tx
Figure 2.8: Diagram of the spin warp imaging sequence
The role of the gradient along x, that is applied during acquisition is to frequency
encode the signal. For this reason, this is often referred as the Read Out (RO)
2.4
Magnetic Resonance Imaging
31
gradient. Section 2.3.1 showed that the effect of a gradient was to make spins precess
at different frequencies depending on their spatial position, ∆ω = γGx x. Note that
R t /2
during the acquisition, kx (t) is varying and is given by kx (t) = −γ 0 x Gx dτ +
R
γ 0t Gx dτ with 0 ≤ t ≤ tx . The importance of the anteceding negative lobe is
to dephase the magnetisation, so that maximum signal and kx = 0 is reached at
t = tx /2.
If the expressions obtained earlier are rewritten as a function of the sequence
parameters, noting that ∆kx = γGx ∆tx and ∆ky = γGyi ty , then the field of view
will be given by:
2π
2π
=
F OVx =
∆kx
γGx ∆tx
2π
2π
=
(2.35)
F OVy =
∆ky
γGyi ty
while the spatial resolution is
2π
2π
'
W kx
γGx tx
2π
2π
δy =
'
Wky
γ2Gy max ty
δx =
(2.36)
Echo Planar Imaging
RF channel
K y
Gz
K x
Gy
Gx
Figure 2.9: Diagram of an ideal echo planar imaging sequence
Echo Planar Imaging, EPI, was first proposed in 1977 as a technique allowing the
entire k-space14 to be sampled after one single excitation. With time it has undergone
14
Only one half of the k-space was sampled in the sequence described in the original paper.
2.4
Magnetic Resonance Imaging
32
a large number of improvements. Figure 2.9 shows an ideal EPI sequence in which
the discrete k-space is sampled in a perfect rectilinear grid after the application
of a slice selective excitation. This is accomplished by rapidly switching the read
out gradient, to generate a forward and backward movement along the kx -direction
whilst blipping the phase encoding gradient in between the switches of the read out
gradient in order to produce an “instantaneous” jump in the ky -direction.15 Because
a gradient coil is effectively an inductor, the gradients can not be switched on and off
instantaneously and thus in a realistic EPI sequence, the gradient waveforms applied
in the read direction are effectively trapezoids or sine waves. The consequences and
implications of this are straightforward, if the signals are sampled with a constant
dwell time (time between consecutive samples), k-space will not be filled by points
positioned on a rectilinear grid. To deal with this there are three possibilities: use a
variable sampling rate; use the non-rectilinear k-space data to interpolate the signal
values at points located at the desired coordinates (regridding); simply ignore those
effects, as they will be more relevant in regions of high |~k| and consequently low
signal.
It is also important to remark that as a consequence of a long acquisition time
(acquisition of N read out lines each requiring a single echo is time consuming, as
the gradient strength can not be increased endlessly due to safety issues and hardware limitations), the assumption used at the start of this section that the intrinsic
T2 decay can be ignored (during signal acquisition) is not necessarily true, and EPI
sequences have an inherent Lorentzian blurring that affects the quality of the image.
To overcome this problem, apart from improving the gradient coils (increasing their
strength to minimise acquisition time and minimising their inductance so they can
be switched more quickly while trying to avoid peripheral nerve stimulation) some
alterations can be made to the sequence. One option is to use two excitations in
which only half the k-space is acquired in each [28]; in the first, odd lines of the
phase encoding are acquired, whilst in the second the even lines are acquired, this
technique is called interleaved EPI. Another option is to use parallel imaging [23],
where N RF receiver coils are used and at least 1/N of the k-space is acquired, the
information can be recombined either in real space or in k-space to obtain a complete image. When the limiting factor of the quality of the images is patient/organ
movement (eg: cardiac imaging) interleaved-EPI is not an option as, in the absence
of a reliable triggering system, it will introduce even more artifacts than conventional EPI. Parallel Imaging is again a good way to get around the problem due to
15
Note that using this methodology the even echoes are effectively acquired in an opposite kx
sense so, prior to 2D Fourier Transformation, the even lines therefore have to be reversed.
2.4
Magnetic Resonance Imaging
33
its short acquisition time. Another valid option is the use of spiral trajectories, that
often have a shorter acquisition time and have the advantage of over-sampling the
centre of the k-space (This technique is less sensitive to motion, although it may
have extra blurring as the errors in high |k| regions will be greater).
2.4.3
Image Artifacts
In this section attention will be centred on artifacts associated with image acquisition. Three of these are especially relevant and are common to any MR imaging
procedure: Aliasing, Truncation Artifacts and Distortions.
Aliasing Artifacts
Aliasing artifacts arise if the sampling rate is not sufficiently high. The minimal
sampling rate is dictated by the sampling theorem. For a sampling to be representative of a signal it has to be carried out at a rate at least two times greater than
the maximum frequency present. If a signal with frequency ωlarge is sampled at a
frequency ω (with ωlarge < ω < 2ωlarge ), then its Fourier transform will appear not
at ωlarge , but at a position ω − ωlarge . In the specific case of imaging, it is as if an
object that is larger than the field of view has the out of range points folded back
into the region of the field of view. Aliasing may happen both along the RO and
the PE directions, but due to the methodology used to acquire each direction, it is
a greater problem in the PE direction. This is because in the readout direction, the
sampling frequency may be either increased until the whole object is inside the field
of view 16 or the signal may be low-pass filtered before digitisation. There is a higher
price to pay for increasing the sampling frequency in the PE direction, since it will
imply either acquiring more PE steps (more excitations in the spin warp sequence
or more echoes in the EPI sequence), or using the same number of PE steps, but
using a lower amplitude PE gradient, thus reducing the image resolution.
Truncation Artifacts
Truncation artifacts are a consequence of the finite extent of the region of k-space
that is sampled. The low sampling extent not only affects the resolution of the
image obtained (see Eq. 2.34) but also generates an effect known as the Gibb’s
ringing artifact. This artifact is due to the multiplication of the infinite k-space
16
This approach is only valid if the object is spatially bounded in a reasonable area, which is
not the case for coronal imaging of the head, as it is preferable not to have to increase the field of
view until the feet lie inside the FOV.
2.4
Magnetic Resonance Imaging
34
representation by a “square” function. In the image domain, this truncation causes
a convolution of the object with a sinc function (see Appendix A). The effect is most
noticeable in regions near a transition from high to low signal. The Gibbs ringing
artifact can be attenuated by multiplying the raw data by an apodisation function
before applying the 2DFT to the sampled k-space (the effect will be a smoothing of
the final image).
Image Distortion
When discussing the signal equation at the start of this section, the effects of local
frequency inhomogeneities, ∆ω(~r), where neglected. If they are considered when
writing the signal equation, after application of a slice selective pulse along z to
excite a slice positioned at z1 , the signal would be given by,
Ã
Z Z Z
s(t) =
x
y
!
∆ω(~r)
m(x, y, z) δ z +
− z1 e−i(kx (t)x+ky (t)y) e−i∆ω(~r)t dxdydz (2.37)
γGz
z
From this equation it can be seen that the selected slice is not necessarily a plane
r)
defined by z = z1 , but a surface where z + ∆ω(~
= z1 . As well as this slice selection
γGz
anomaly, the image inside the plane will also be distorted. The distortion due to the
local field inhomogeneities is dependent on how k-space is sampled. In the case of
the Spin-Warp sequence where each PE line is acquired after a different excitation,
the phase, ∆ω(~r)t, along the PE direction will be constant. The same is not true
for the RO direction, where kx (t) = − 12 γGx tx + γGx t, and the signal equation for
spin warp imaging becomes:
Z Z Z
s(t) =
x
y
Ã
!
∆ω(~
r)
∆ω(~r)
m(x, y, z) δ z +
− z1 e−iky (t)y e−ikx (t)(x+ γGx )+iconst dxdydz
γGz
z
(2.38)
The spin warp image will therefore suffer a distortion characterised by shifts in
r)
r)
the x and z directions given by ∆ω(~
and ∆ω(~
respectively.
γGx
γGz
In the case of the EPI sequence, the gradient waveforms are more complex as they
are “more” time dependent and it is important to define an average PE gradient
strength Ĝy =
1
tacq
R tacq
0
Gy (τ )dτ , where tacq is the length of the EPI echo-train.
To first order, ky = γ Ĝy t and in addition to the shifts observed in the spin-warp
r)
sequence, there will also be one along y with a magnitude ∆ω(~
. The magnitude
γ Ĝ
y
of the distortion will be much greater in the PE direction than in the RO direction
as the ratio of Gx and Ĝy is approximately given by (assuming the same resolution
2.5
Basic Pulse Sequences in Quantitative NMR
35
in both directions) the number of phase encoding lines. Because distortion in the
PE direction in EPI is so significant, EPI images are often characterised by the
bandwidth per pixel or frequency per point, that is the number of Hertz needed to
create a one voxel displacement in the PE direction, which is equal to (1/tacq ).
The same arguments used to describe image distortion can also be used to explain
chemical shift artifacts in images. Suppose the water of the sample/subject, ρw (~r),
is on resonance leading to an image with no distortions. A second substance with a
distribution ρCS (~r) with a different chemical shift, ∆ωCS , will be represented with
no distortion (as there is no spatial dependency), but will be shifted from the first
image by a distance dependent on ∆ωCS and the image parameters, as described in
the previous paragraph. The most common chemical shift artifact in human imaging
is due to fat.
2.5
Basic Pulse Sequences in Quantitative NMR
In Section 2.3 many of the NMR parameters (T1 , T2 , T2∗ , D and M0 ) that characterise matter were described. Since in NMR spins can be tagged both by selective
RF pulses and gradients, it also has great potential for measuring physiological phenomena such as flow rate and perfusion, or at a more chemical level, magnetisation
transfer. Here, attention will be centred on the first class of parameters.
Measuring T1
The most common technique for measuring T1 is inversion recovery (see Fig. 2.10a).
This uses an inversion pulse followed by an inversion recovery delay, TI, after which
an RF excitation is performed just before the acquisition is started. The acquisition
can be a simple µ
FID acquisition
or an imaging sequence. The signal will be given by
¶
−T I
S(~r, t) ∝ M0 (~r) 1 − 2e T1 (~r) . If this procedure is repeated several times for various
values of IR (keeping the repetition time greater then 5T1
spatial location can be fitted to the function A(1 − 2e
max ),
−T I/T1
signal from each
), in order to find the
value of T1 .
Measuring T2
T2 is often measured using a spin echo sequence (see Fig. 2.10b). The spin echo
sequence consists of a 900 RF excitation pulse followed by a 1800 refocusing pulse
at a time T E/2 later and an acquisition after a further time T E/2 has elapsed.
−T E
The signal at a time T E will be given by S(~r, t) ∝ M0 (~r)e T2 (~r) . If the procedure is
2.5
Basic Pulse Sequences in Quantitative NMR
36
repeated several times, for varying values of T E, an exponential decay can be fitted
to the signal, allowing the measurement of T2 . The argument is that most dephasing
results from field inhomogeneities and therefore a simple 1800 pulse will have the
ability to refocus all the magnetisation.
When the spatial field inhomogeneities are of the same length scale as the average
displacement through diffusion during the evolution time, total refocusing is not
possible, and the measured T2 is significantly affected by field inhomogeneities and
diffusion. In this case it is necessary to use the CPMG (Carr-Purcell-MeibomGill) sequence that instead of employing one single refocusing pulse uses a train of
2N refocusing pulses (that should grouped in pairs with opposite phases to avoid
accumulation of errors) after the excitation pulse (see Fig. 2.10c) with a pulse
separation of 2τ , the final echo time is T E = 4N τ . The sequence is repeated with
various values of N and a fitting as suggested for the spin echo will reveal a value of
T2 where the contribution of diffusion is greatly reduced. As a reference, if the field
inhomogeneities are assumed to take the form of a linear gradient, the attenuation
2
for the spin echo and CPMG will be respectively A(T E) = e− 3 γ
− 4N
3
2 G2 D(T E/2)3
and
γ 2 G2 Dτ 3
A(T E) = e
. Knowing that T E = 4N τ it can be seen that the SE
attenuation is much greater because (2N )3 À 2N .
Measuring T2 ∗
T2∗ is measured using a gradient echo sequence (see Fig. 2.10d). The gradient echo
sequence consists of an excitation pulse directly followed by an acquisition after
a certain time, T E, has elapsed. Measurements at various echo times allow an
exponential fitting that will quantify T2∗ . The measured value can be extremely
dependent on the scanner settings such as the quality of the shimming (anything
that contributes to dephasing causes T2∗ decay).
Measuring δω
Estimating the values of the field shifts (that are simply related to frequency offsets
by δω = γδB)can be important as it allows images to be undistorted and facilitates
the choice of shim coil current settings that optimise the field homogeneity. To
measure δω, several images (at least two) are acquired using different dephasing
times (see Fig. 2.10d for a gradient echo based sequence and Fig. 2.10e for a spin
echo based sequence). As opposed to the other cases considered above, where the
important information was encoded in the amplitude/modulus image, in this case
the interest is centred on the phase image. The phase image, arctan
h
Imag(Image)
Real(Image
i
, will
2.5
Basic Pulse Sequences in Quantitative NMR
37
180
90
TI
(a)
ACQUISITION
ACQUISITION
180
TE/2
90
TE/2
(b)
180
180
a
180 a
a+180
180
ACQUISITION
a+180
90
(c)
τ
τ
τ
τ
τ
τ
τ
τ
ACQUISITION
h90
TE
dephasing time
(d)
ACQUISITION
180
90
TE/2
TE/2
dephasing time
(e)
90
(f)
180
TE/2
G
δ
∆
ACQUISITION
TE/2
G
δ
ACQUISITION
Figure 2.10: Diagram of sequences used to measure: (a) longitudinal relaxation time, (b-c)
transverse relaxation time, (d) apparent transverse relaxation times, (d-e) field maps, and (f) the
diffusion constant
be given by ei(δω(~r)t+Φ(~r)) , where Φ(~r) is an intrinsic phase change due to the imaging
settings (it has a strong dependence on whether the signal is properly centred in
k-space). With measurements at more than one dephasing time, the dependence on
Φ(~r) can be eliminated and δω(~r) can be calculated.
Measuring D
The apparent diffusion coefficient, D, can be measured using a spin echo sequence
with one gradient pulse applied on each side of the refocusing pulse (see Fig. 2.10f).
If the two gradient pulses are considered to be perfectly square with a duration δ and
2.6
The Basic Hardware
38
amplitude of G and they are distanced apart (from the start of the first to the start
of the second) by a time, ∆, it can be shown from the Bloch equations that there will
2 2 2
be an attenuation given by e−γ δ G D(∆−δ/3) . Repeating such a sequence several times
with different gradient amplitudes allows the calculation of the diffusion constant.
So far, it was assumed that the diffusion is isotropic , which is not necessarily the
case in the presence of structure. In the latter case, diffusion is better represented by
a tensor. The most direct consequence of this, is the dependence of the magnitude
of the attenuation on the direction of the applied gradients. The calculation of all
the elements of the diffusion tensor allows structures which are smaller then the
voxel size to be investigated. This information can be, for example, used to infer
the connectivity between different areas of the brain through following white matter
tracts [29].
2.6
The Basic Hardware
The four main units of a NMR instrument are the magnet (responsible for producing
the net magnetisation), the gradient system (which enables the spatial encoding,
vital for imaging), the radio frequency coils (responsible for both the transmission
and reception of RF) and the console (that drives the whole system). A schematic
representation of an MR scanner is shown in Figure 2.11. A brief overview of the
function of each part of the hardware is subsequently given.
Magnets
The magnet is obviously essential in NMR as it generates the B0 field responsible
for producing the net magnetisation. There are three types of magnet: permanent
(fields < 0.4T ); resistive - air-core or iron core (fields < 0.4T ); and superconductive
( 0.4T < fields < 21T ); The first two were the most common types in the early days
of NMR, but with the rise of superconducting technology the latter became more
and more popular due to the high field strength obtainable from them.
Permanent magnets are made of blocks of ferromagnetic material such as alloys of
iron and cobalt (ALNICO) or rare earth metals such as Neodymium Iron Boron (NdFe-B)and for even a modest field such as 0.2 T a huge mass is needed (a 0.2 T whole
body magnet requires ∼ 23 Tonnes of ALNICO or ∼ 4 Tonnes of Nd-Fe-B [30]).
Another reason why permanent magnets are not very competitive is their temporal
instability (mostly due to temperature variation) and poor field inhomogeneity. The
advantages of such magnets are their low operating cost (virtually none), lower fringe
fields and their price.
2.6
The Basic Hardware
39
RF shielding
Main Magnet
Gradient
gradient amplifiers
RF coil
duplexer
TX amplifier
X
Y
pre-amplifier
Z
waveform generator
waveform generator
x
x
low pass filter
low pass filter
ADC
ADC
RF synthesizer
COMPUTER
data storage
terminal
Figure 2.11: Scheme representing the various hardware components and how they relate to one
another.
Resistive magnets generate the field by passing current through an arrangement
of conductive coils, which means that they need continuous current requiring a significant power (a 0.15 T whole body magnet requires 50kW [30]) and thus high
energy dissipation as heat (requiring a cooling system, often based on water circulation). Resistive magnets may either be air-cored, in which case the windings are
arranged as a solenoid and generate a coaxial field, or iron-cored in which case the
wires are wound around a ferromagnetic material, e.g. a C-shaped Iron yoke (giving the same field geometry as a permanent magnet). The advantages of resistive
magnets relate to their low price and the possibility of easily turning them off.
Superconductive (SC) magnets are generally air-cored solenoids constructed from
windings that become superconductive at low temperatures. Consequently, they
will present zero resistance to the flow of current and once energised will remain on
indefinitely (once the magnet is energised it will remain energised as long as it is kept
under the superconductive conditions). The superconductor used in most modern
magnets is a Niobium-Titanium alloy embedded in a copper matrix. This has a
critical temperature (temperature at which it ceases behaving as a superconductor)
of 10K. To keep the coils at such low temperatures, liquid Helium (4K) is used.
2.6
The Basic Hardware
40
Superconductive magnets can generate an extremely wide range of fields (there are
NMR magnets up to 20 T) with very good homogeneity and high temporal stability.
One drawback is the extent of the fringe field, which can be tackled by the use of
either passive or active shielding.
Passive shielding often involves constructing a symmetric iron box surrounding
the magnet (it has to be symmetric for force cancellation in the magnet coils, and
conservation of homogeneity of the B0 field). The walls of this box are kept at
some “trade-off” distance from the magnet (the closer the box is placed around
the magnet, the thicker the steel has to be to avoid complete magnetic saturation,
the further the box is placed around the magnet, it will have a greater weight and
therefore higher cost). Active shielding involves using an extra solenoid winding
concentric with the main magnetic field solenoid winding, but of larger radius and
opposite current. The parameters can be chosen to optimise the field cancellation
outside the magnet, but this will reduce the field both inside and outside the inner
solenoid, so this method is only usable when there is field strength to spare.
To compensate for any inhomogeneities that may exist in the static magnetic
field (both induced by magnetic susceptibility variation in the sample, or magnet
imperfections present in the hardware) a set of shim coils wound on a cylinder inside
the superconducting solenoid, is employed.
Gradients
The gradient coil set is located inside the shim set and consists of three separate
sets of windings, responsible for creating the gradients of the z-component of the
magnetic field along the x-, y- and z-directions. Each of these are usually made
up of two layers, the inner layer creates most of the gradient whilst the outer is
responsible for cancelling fields outside the coil region (the effect of cancellation is
not so relevant on the coil inside due to the unequal radius). This arrangement
aims to avoid the the generation of induced currents in conductors surrounding the
coil set (eddy currents). Such eddy currents create uncontrolled gradients which,
because they can not be “disconnected”, can seriously modify the signal evolution
and hence create artifacts in the final image.
Each of the gradient coils is driven by a separate waveform generator whose
signal is amplified prior to delivery to the coil by the gradient amplifiers. These
amplifiers have to be able to create high currents (high gradients) and high voltages
in order to be able to switch the gradients as rapidly as possible.
2.6
The Basic Hardware
41
Radio Frequency chain
Radio Frequency Coils can be used both for transmission and reception of the RF
radiation. The RF coil itself can be represented as an LCR circuit. The simplest
case is that of a surface coil consisting of a single loop that produces a coaxial B1
field. This is connected in parallel to a capacitor, which is chosen to tune the circuit
so that it resonates at the desired frequency. Furthermore, this circuit is connected
in series with another capacitance (matching capacitor) which is chosen to match the
RF circuitry to the rest of the scanner so that the signal transmission is optimised
(this RF circuit typically has an impedance of 50Ω). Volume coils are arranged with
a more complex geometry and circuitry, in order to produce a more uniform RF
field. The RF coils are positioned closer to the sample/subject than the gradient
and shim coils in order to maximise the detected signal.
Although the same coil can be used for both transmission and reception, the
transmitter and receiver paths are inherently different. They are separated immediately after the coil by a switching system called a duplexer or T/R switch. The
transmitter path starts with an RF synthesizer that generates a sine wave at the
Larmor frequency. This goes to the waveform generator, where it is split into two
sine waves 900 out of phase with one another that are modulated so as to create the
RF pulse, B1 (t). This signal then passes to the transmitter amplifier where the amplitude of B1 (t) is set just before passing through the T/R switch and then arriving
at the coil. Once the RF pulse is applied, the duplexer is switched into the receiver
mode. The signal detected by the coil is pre-amplified to minimise further noise
contamination and is split into real and imaginary components, which are separately demodulated by analog mixers. The signals are then low-pass filtered, so that
the components with frequencies higher then half the sampling rate are eliminated,
avoiding aliasing at the stage of conversion of the analogue signal into a digital one
at the analog to digital converters (ADC). Once the signal is digitised, it is then
stored in the computer for further processing.17
Because the coils are sensitive to radio-frequency waves such as those broadcasted
by radio stations, it is important that the entire room housing the scanner is RFshielded. RF shielding is implemented by constructing a “box” around the scanner
made of a mesh of conducting material with spacing between wires expected to be
much smaller than the wavelength at the Larmor frequency.
17
In more recent scanners the digitisation is performed straight after the pre-amplifier.
2.7
Signal to Noise Ratio in NMR and MRI
42
Console
The setting and synchronisation of the gradients, RF pulses and sampling is controlled by the console, where the pulse and gradient programs are compiled and the
data are stored and processed.
2.7
Signal to Noise Ratio in NMR and MRI
In this section, the dependence of the Signal to Noise Ratio, SNR, upon the static
field magnitude, repetition time and resolution will be explored.
2.7.1
SNR as a function of B0 and B1
For the purpose of characterising SNR as a function of B0 some approximations, that
are not necessarily fair, will be made by considering that some parameters (such as
RF coil parameters18 and relaxation19 ) remain constant as the field increases.
Signal as a function of B0 and B1
So far, it has been described how the signal is simply proportional to the bulk magnetisation, which has a linear dependence on the static field (as described in Section
2.2.6) M0 =
ργ 2 h̄2 B0
.
4kb T
Furthermore, because of the signal detection methodology based on current
H
~ · d~l =
induced in a coil due to an oscillating magnetic field, Faraday’s law, E
− dtd
R
l
~ ~
s B · dS, states that the induced current is proportional to the rate of change of
the magnetic flux with time caused by the precessing transverse magnetisation. In
the laboratory frame the transverse magnetisation will be given by M + (0)e−t/T2 e−iω0 t
so that the main cause of time dependence is the precession around B0 at the Larmor
frequency (the effect of relaxation os negligible). In conjunction with the proportionality of the magnetisation and field strength, this yields, a quadratic dependence
of the signal on the static field strength.
The detected signal has further dependencies on the RF coil’s geometry and
efficiency. One is the coil’s ability to rotate the bulk magnetisation vector in a
18
As B0 increases, coil design becomes a more complex issue due to the reduction of the wavelength associated with the increased Larmor frequency.
19
In Section 2.3.3 it was shown that there is an obvious dependence of relaxation times on the
static field strength.
2.7
Signal to Noise Ratio in NMR and MRI
43
certain region into the transverse plane (whether an exact 900 is achieved). The
amount of transverse magnetisation is proportional to sin(γB1+ τ ) where τ is the
pulse length. The other term is related to the principle of reciprocity derived by
Hoult [31] and is an extension of Faraday’s Law of induction. It states that if
~ 1 (x, y, z) is the magnetic field produced at a position (x, y, z) by unit current in
B
the RF coil, the incremental EMF produced in the coil by precessing magnetisation
at the same location is given by
d² = −
∂ ~
~ (x, y, z, t)]dV
[B1 (x, y, z) · M
∂t
(2.39)
~ 1 (x, y, z) can also be seen as being indicative of the receiver coil’s sensitivity
and B
within the volume.
Finally it can therefore be written that the signal will be proportional to B02 B1 (x, y, z)
and the nutation angle at each position.
Noise as a function of B0
Noise is often defined as the random component of an image (random noise) or
“aspects” of the image that do not directly relate to the distribution of magnetisation
(e.g. imperfections in the image processing). It is the random noise that will be
quantified here. Its origin is the thermal motion of charges and dipoles in the coil
or in the sample. These can induce signal fluctuations whose standard deviation is
q
σVnoise = 4kB BW (Tcoil Rcoil + Tsample Rsample ), where Rcoil , Tcoil and Rsample , Tsample
are the equivalent resistances and temperatures of the coil and sample, and BW
is the bandwidth used in the signal acquisition. The assumption in the previous
expression is that the sample and coil are effectively in series. Alternating electrical
currents tend to flow on the outer surfaces of conductors, due to the skin effect. The
exact current distribution depends on the shape of the conductor, but for a round
wire with conductivity, σc , and permeability, µc , the cross-section through which the
current flows is 2πrδ = 2πr
q
2
,
ω0 µc σc
where r is the radius of the wire and δ is the skin
depth. The resistance of this wire will be inversely proportional to the cross section
1/2
of the conductor and thus proportional to B0 . Note that the noise originated by
the coil can be reduced by cooling it. On the sample side, if Faraday’s Law is written
H
~ · d~l = −iω R B
~ · d~s, and acknowledging that the power
in a time harmonic form, E
l
s
dissipated at each point is proportional to σt E 2 , where σt is the conductivity of the
tissue, will mean that the power dissipation in the tissue is proportional to ω 2 and
therefore B02 [32].
2.7
Signal to Noise Ratio in NMR and MRI
2.7.2
44
Signal to Noise Ratio as a function of B0 and B1
Combining the discussion of the last two sections it can be written that the SNR
will have the following dependence on B0 and B1 ,
B02 B1
SN R ∝ q
1/2
aB0
+ bB02
(2.40)
1/2
where the term aB0 in the denominator relates to RF coil power dissipation, whilst
bB02 relates to sample power dissipation. At low fields the SNR will therefore increase
7/4
at a rate of B0 while at high fields it will tend to increase linearly with B0 .
Although some very relevant parameters were not brought into this discussion,
mainly the homogeneity of the RF field and the fact that T1 increases with B0 (and
consequently more time is needed to recover the saturated signal) while T2 decreases
with B0 (note that noise is proportional to the square root of the bandwidth of the
acquisition, meaning that to minimise noise a small bandwidth should be used,
which implies a long acquisition time) it is fair to state that using a detection based
on induction makes the increase of the static magnetic field rewarding in terms
of increasing the SNR (although that increase might be sub-linear at high fields).
Furthermore, there are two relevant discussions relating to signal to noise ratio. How
much will the signal increase with averaging and how does SNR relate to the chosen
image parameters, such as resolution.
Signal Averaging and Sampling Efficiency
To increase SNR, one of the options is to average several signal acquisitions. As they
are combined, NMR signal is expected to add up coherently whereas the noise, that is
supposed to be uncorrelated between acquisitions, adds incoherently. Consequently
the signal increases linearly with the number of averages, N , while the noise will scale
√
√
as N leading to an increase of SNR with N . That is why to compare techniques
q
that require different times for acquisition (tacq ), an efficiency, η = SN R/ N tacq is
used. To optimise efficiency, many techniques try to improve the percentage of time
that k-space is being traversed during the image acquisition period (such as FLASH,
SSFP and EPI). FLASH (Fast Low Angle Shot) is a technique used for structural
imaging that uses a gradient echo, in which the excitation flip angle, α, is reduced
from 900 , thus allowing the use of short repetition times, T R, of a few miliseconds.
The signal in each acquisition in the steady state is given by:
M0 sin α(1 − e−T R/T1 ) −T E/T2∗
e
(2.41)
1 − e−T R/T1 cos α
It is possible to see that the signal in the steady state will have an extra dependence
on B0 because T1 itself is a function of B0 (see Eq. 2.27).
S(α, T E, T R) ∝
2.8
Safety
45
SNR in a voxel
It can also be shown that SNR has some further dependence on the imaging sequence. It depends on the voxel volume (∆x · ∆y · ∆z), the number of phase
encoding steps (NP E ), number of acquisitions in the frequency encoding direction (NRO ) and the
signal bandwidth. One way of writing such a dependence is
q
NP E NRO
SN R ∝ ∆x∆y∆z
[33]. Thus, to increase the SNR, the options are either
BW
to increase the voxel volume which will have resolution costs, or to decrease the
acquisition bandwidth (increasing the dwell time).
2.8
Safety
Magnetic Resonance Imaging is often said to be innocuous, this is to a large extent true, but it is important to know the limits of such a statement and why the
technique is so inoffensive. There are mainly three ways in which there can be interactions between NMR and the human body which have safety implications: through
the static field, the gradients, and the RF pulses. In this section, a coarse overview
of how these three essential components of MRI can affect a subject being scanned
will be given.
B0 effects
Static fields can interact with the subject mainly in two ways: either by generating
forces or torques on substances with magnetic susceptibility or by inducing electric
fields in moving charges/conducting materials.
The most serious safety concern in MR is the ferromagnetic missile effect. The
z-directed acceleration, a, experienced by material of density ρ and magnetic susceptibility χ will be given by:
a=
χ ∂B
B
,
µ0 ρ ∂z
(2.42)
where B is the main magnetic field and ∂B
is its spatial derivative. From this
∂z
expression it is possible to conclude that there will be no forces exerted in the centre
of the magnet where the field homogeneity is high. The force will be maximised at
the extremities of the bore, where the derivative is maximised and the field is still
high. For analogous bore geometries, it is possible to conclude that the forces in
ferromagnetic materials tend to grow quadratically with the field. Luckily the human
body contains no ferromagnetic materials and has a very low magnetic susceptibility
(human tissue is mainly composed of water that is diamagnetic and has 11 orders
2.8
Safety
46
of magnitude lower magnetic susceptibility than a ferromagnetic material such as
iron).
A conductive material moving in the static magnetic field will have currents
induced in it and therefore will experience a torque due to the interaction of the
static magnetic field with the induced currents.
Screening over and over again the materials that are brought into proximity with
the scanner and certifying that patients have no undesired implants is the only way
to tackle such problems.
Charge carriers such as flowing blood experience a Lorentzian force in the static
magnetic field and this can induce an electric field transverse to the magnetic field
and flow directions. This effect is similar to the Hall Effect. This is most significant
in the aorta and some studies have been carried out in order to estimate whether this
effect could lead to induction of electric fields above cardiac stimulation thresholds.
Some changes in the pattern of the human electrocardiogram have been observed:
mainly an enhancement of the T-wave (refractory period) [34]. No changes in diastolic blood pressure have been observed in fields up to 8T, whilst in the systolic
blood pressure the increase is approximately one-half of the increase seen when the
subject changes from a supine position to a sitting position. [35].
To date, the most obvious biological effect of B0 is vertigo, which is felt when a
subject moves around or towards the scanner, but ceases once motion in the gradient
magnetic field ceases. This is likely to be related to eddy currents generated in the
endolymph flowing in the three semicircular canals of each middle ear. The hair cells
sense these currents and confuse this stimulation with the normal flow stimulation
used to sense head motion and positioning. (The effect is not as dramatic as would
be for some birds, insects or other animals that use the earth’s magnetic field for
orientation [36])
At a more chemical level there are also some effects. Some reactions come to
a different equilibrium in the presence of magnetic fields due to the existence of
magnetic components in the reaction process. Reactions involving free radicals may
also be affected as the energies of spin states can be changed (vitamin B12 is one of
the few confirmed examples). [37]
Gradient effects
There are mainly two safety issues related to field gradients: peripheral nerve stimulation (PNS) and acoustic noise.
Peripheral nerve stimulation results from the induction of electric fields in the
tissue of the body by temporally varying magnetic fields, d|B|
. When these electric
dt
2.8
Safety
47
fields exceed the polarisation threshold of a nerve, nerve stimulation occurs. The
reason why it is peripheral, and not simply nerve stimulation, has to do with the
known fact that charges tend to accumulate in the surface of conductors which
will affect the electric potential and thus the induced electric field (It should also
be remarked that d|B|
for a gradient has a minimum at ~r = 0, and that also the
dt
gradients of the field components Bx and By , although not contributing to imaging,
contribute to stimulation). PNS poses a limit to fast imaging where it is extremely
relevant to have high amplitude gradients and to switch them as fast as possible to
enable a fast coverage of the entire k-space.
Work on nerve stimulation was done by Reilly [38], and resulted in the setting of
safety limits that represent a threshold for PNS stimulation in terms of dB/dt [39].
Ã
dB
dt
!
Ã
threshold
chronaxie
= rheobase 1 +
τduration
!
(2.43)
where rheobase is the mean dB/dt stimulation threshold for an infinite pulse, chronaxie is the duration at which the stimulation threshold is twice the rheobase and
τduration is the duration of the field change. 20
Acoustic noise results from forces experienced by wires of a gradient coil carrying
current in static magnetic fields. These forces induce pressures that are linearly
dependent in B0 leading to a sound intensity with a square dependence in B0 . As
acoustic noise is usually quoted in decibels, doubling the field strength increases the
noise by 6dB. The most common protection against scanner noise is ear protection
(ear-plugs or headphnes). Some effort has been directed towards designing coils that
are insensitive to certain vibration modes and to operating the gradient coils using
waveforms mainly composed of frequencies of those modes of vibration. The forces
on the gradient coils can also generate physical vibration of the subject that can be
uncomfortable (and cause image artifacts). This can be overcome by decoupling the
gradient system from the patient’s bed.
Radio Frequency effects
Radio frequency pulses have as a main physical effect, deposition of energy by magnetic induction in the body, leading to tissue heating. Some organs are particularly
delicate due to their composition or low vascularity, preventing efficient energy dissipation (eg: eyes, testicles). Although the heating potential depends not only on
the RF power deposition per unit of mass, but also on the relative humidity, airflow
20
This relation is currently used to establish the safety limits by the International Electrotechnology Commission, IEC.
2.8
Safety
48
rate, blood flow and patients’ insulation, only the first factor is used to calculate the
Specific Absorption Ratio, SAR, that is effectively a measure of heating potential.
The SAR at a point is
SAR =
σDC|E|2
σDC|ωA|2
'
2ρ
2ρ
(2.44)
where σ is the electric conductivity, DC is the duty cycle (defined as the ratio
of the period while the RF pulse is on and the repetition time), ρ is the tissue
density, |E| is the induced electric field, |A| is the magnetic potential vector, and
ω is the pulse frequency (the approximation E ' −∂A/∂t is made). For coils that
produce homogeneous B1 fields, the SAR can be calculated for specific geometries by
neglecting coupling between patients and the coil (special care has to be taken with
conductive implants, tatoos or patient-formed “closed loops”). A useful analytical
solution can be obtained for a tissue sphere of radius R [40]
SARave =
σDCω 2 B12 R2
= 0.4SARpeak
20ρ
(2.45)
Note that due to the distribution of A (and therefore E) regions further way from
the center of the coil are subjected to greater heating (which facilitates dissipation
to the environment). One simple way of reducing SAR is to use circularly polarised
RF pulses instead of a linear polarisation. Calculating realistic B1 field deposition
in more complex geometries by numerical simulations and developing techniques to
measure SAR experimentally both form current research areas.
Further references and guidelines can be found in references [40–43].
3
Long Range Dipolar Field
49
Chapter 3
Long Range Dipolar Field
The Dipolar Magnetic Field in a certain region of a sample is the sum of the magnetic
fields generated by all the magnetic dipoles present within it. In most liquid NMR
experiments these fields are small (and tend to cancel out as discussed in Section
2.2.7) and therefore are usually neglected. The first time it was realised that the
Long Range Dipolar Magnetic Field could have an important role in the evolution
of magnetisation was in the late 70’s [44] by G. Deville et al.. In an experiment
on solid He using two radio-frequency pulses applied at times 0 and τ , a train of
echoes occurring at times 2τ , 3τ , 4τ , . . ., nτ , . . . was observed instead of the single
echo expected at a time 2τ . These echoes were christened Multiple Spin Echoes
(MSE). Only in the early 90’s [45] was this effect observed in liquids (distilled water) at room temperature by Bowtell and coworkers (the echoes had already been
observed by Dürr et al. [46], but they were unable to find any convincing explanation of this phenomenon). The sequence used consisted of two rf-pulses separated
by a time τ with a constant gradient applied during the entire sequence. It was
also shown that the Dipolar Field and MSE’s could be used to couple different spin
species [47] and to make investigations of sample structures of size comparable to
that of the modulation of the magnetisation [48]. At the same time, some unexpected cross-peaks between independent molecules in solutions were observed in 2D
experiments [49,50]. In particular in the COSY-revamped by asymmetric z-gradient
echo detection (CRAZED) experiment, which consists (again) of two rf-pulses separated by a time, t1 , with the second pulse being sandwiched between two gradient
pulses, the first with area, Gτ , and the second with n times the first area, nGτ . To
explain the appearance of signal after such a sequence, a Quantum Mechanical approach was developed. It is now agreed that the quantum and classical approaches
to explain these effects are equivalent [51].
3.1
The Quantum Description
50
The two different approaches to describe the signal evolution will be presented in
the following two sections. To be consistent with the structure adopted in the previous chapter, the quantum approach will be discussed first, followed by the classical
approach. Although the classical formalism makes evaluation of the evolution of the
signal much easier, an analytical solution for the Bloch equations involving dipolar
fields, diffusion and relaxation is not achievable. Numerical methods for evaluating
the evolution of the signal will be presented. The accuracy of such numerical methods will be assessed by comparison with the results of some in vitro experiments at
different magnetic fields.
3.1
The Quantum Description
Quantum Mechanics predicts the correct evolution of any system as long as the
Schrödinger Equation can be solved, either by using the wave function formalism,
ih̄ dtd |ψspin (x, t)i = Ĥspin |ψ(x, t)i, or by using the density matrix formalism, ih̄ ∂∂tρ̂ =
[Ĥ, ρ̂] (see Section 2.2). The full Hamiltonian for N spins interacting via the dipoledipole interaction in the rotating frame can be written as
H = 2h̄
N X
N
X
Dij (3Iˆzi Iˆzj − Îi · Îj )
i=1 j>i
Dij =
µ0 γ 2 h̄ 1 − 3 cos2 (θij )
3
4π 4
rij
(3.1)
Îi and Îj represent the spin angular momentum operator for spins i and j, rij
is the distance that separates these spins, and θij is the angle between the internuclear vector and the applied field. In liquids, diffusion averages to zero the nearby
couplings since < 1 − 3 cos2 (θij ) >= 0. As the average displacement of a spin is
√
< r2 >1/2 = 6Dt (D is the diffusion constant, 2.3 × 10−9 m2 s−1 for water, and t the
time over which diffusion occurs, which can be tens to hundreds of milliseconds) the
dipolar term does not average to zero in a macroscopic sample (it will only average
to zero in the case of an isotropic liquid, in a spherical sample in the absence of some
externally imposed modulation). In fact, it is considered that in a liquid, every pair
of spins acts coherently only for a time t such that Dij t ¿ 1 [52].
3.1
The Quantum Description
51
In solids, the dipolar Hamiltonian generates splittings between spin states which
are typically only 10−4 of the Zeeman Hamiltonian. This means that they can be
ignored in the evaluation of the equilibrium density matrix.
e−H/kB T
ρeq =
=
T r (e−H/kB T )
QN
e−h̄ω0 Izi /kB T
T r (e−H/kB T )
i=1
(3.2)
If all spin couplings are ignored, the equilibrium density matrix can be expressed
as a ”sum” of the contributions of the equilibrium density matrixes of each individual
spin (σi,eq ), so that ρeq = σ1,eq ⊗σ2,eq ⊗. . .⊗σN,eq . It is therefore possible to construct
basis functions as direct products of the eigenfunctions of individual spins [52],
leading to the following expression
ρeq = 2
−N
N
Y
i=1
Ã
Ã
h̄ω0
1 − 2 tanh
2kB T
!
!
Izi
(3.3)
This is the exact expression for the density operator. Expanding it leads to the
expression
N
h̄ω0 X
1
ρeq = 1 −
Izi +
kB T i=1
2
Ã
h̄ω0
kB T
!2
N
X
i, j = 1
i 6= j
1
Izi Izj +
6
Ã
h̄ω0
kB T
!3
N
X
Izi Izj Izk ...
i, j, k = 1
i 6= j 6= k 6= i
(3.4)
This expansion is often only made to first order in
and is known as the high
temperature approximation, since it is based on the assumption that the maximum
energy difference is much smaller than the thermal energy [53], h̄ω0 ¿ kB T , which
is perfectly valid in the absence of collective effects. The high temperature approximation for the density matrix has been used for a long time by the Solid State
NMR community (it is possible to deal with line narrowing in solids using such
an approach). Starting from this initial state, it is possible to calculate the evolution of magnetisation throughout any sequence using the density matrix formalism
introduced in Section 2.2.
Although the terms of order greater than one in Iz are very small, their noninclusion in the high temperature approximation will mean the disappearance of the
signal in an n-quantum CRAZED experiment (these terms manifest themselves in
simple COSY experiments, as multiple ”satellites” in the ω1 direction, separated
by multiples of the offset frequency δω [54]). The usage of such an expansion of
the density operator to predict the appearance of signal in an n-quantum CRAZED
sequence was first carried out by Warren [49, 50, 52] and was said to show a ”spectacular breakdown of the high temperature approximation”. The evolution of the
h̄ω0
kB T
3.1
The Quantum Description
52
system during an n-quantum CRAZED sequence (such as that represented in Figure
3.1) can be ”simply” evaluated using the density matrix formalism:
−1
−1
−1
−1
ρ(t) = UDip (t)UnGτ (t)Uθ (t)UGτ (t)U90 (t)ρeq (0)U90
(t)UGτ
(t)Uθ−1 (t)UnGτ
(t)UDip
(t)
(3.5)
How this is actually done will be shown in the following section.
3.1.1
Hands on the Quantum Approach, the n-CRAZED
case study
90
τ
θ
τ
nG
G
t1
t2 =nt1
Figure 3.1: Standard n-CRAZED pulse sequence (”n” is an integer value).
Immediately after the first hard 900 pulse is applied, the equilibrium density
π
−1
matrix (see Eq. 3.4) is transformed as U90 ρeq (0)U90
, where U90 = ei 2 Ix (assuming
B1 is oriented along the x-axis in the rotating frame) leading to
ρ1
N
´
h̄ω0 X
−i ³ −
= 1−
Ii − Ii+
kB T i=1 2
1
+
2
1
+
6
Ã
Ã
h̄ω0
kB T
h̄ω0
kB T
!2
N
X
i, j = 1
i 6= j
!3
(3.6)
´
−1 ³ + +
−Ii Ij − Ii− Ij− + Ii− Ij+ + Ii+ Ij−
4
N
X
i, j, k = 1
i 6= j 6= k 6= i
−i
8
"
Ii+ Ij+ Ik+ − Ii− Ij− Ik−
Ii+ Ij− Ik− − Ii+ Ij+ Ik−
#
During the period t1 , the spins evolve under a pulsed gradient of amplitude G
and length τ applied in the z-direction and the spins at different positions may have
different frequency offsets, ∆ω. The Hamiltonian during this period will be given by
(∆ωt1 + γGzτ )Iz , making terms proportional to I + evolve as e−i(∆ωt1 +γGzτ ) whilst
terms proportional to I − will evolve as ei(∆ω+γGzτ ) . Therefore, the density matrix at
a time t1 will be given by,
3.1
The Quantum Description
ρ2 = 1 −
1
+
2
1
+
6
53
N
´
h̄ω0 X
−i ³ − i(∆ωt1 +γGzτ )
Ii e
− Ii+ e−i(∆ωt1 +γGzτ )
kB T i=1 2
Ã
Ã
h̄ω0
kB T
h̄ω0
kB T
!2
N
X
i, j = 1
i 6= j
!3
−1
4
N
X
i, j, k = 1
i 6= j 6= k 6= i
"
−Ii+ Ij+ e−i2(∆ωt1 +γGzτ ) + Ii+ Ij−
−Ii− Ij− ei2(∆ωt1 +γGzτ ) + Ii− Ij+
−i
8
"
(3.7)
#
Ii+ Ij+ Ik+ e−3i(∆ωt1 +γGzτ ) − Ii− Ij− Ik− ei3(∆ωt1 +γGzτ )
Ii+ Ij− Ik− ei(∆ωt1 +γGzτ ) − Ii+ Ij+ Ik− e−i(∆ωt1 +γGzτ )
#
After the time t1 , a second RF pulse is applied. For the sake of simplicity, it will
be assumed that this second RF pulse is also a 900 pulse applied along the x-axis
and attention will be concentrated upon the terms that may generate observable
signal after evolution under the action of the dipolar field (as discussed previously),
and only n-spin one quantum operators will be kept. The effect of the 900 pulse on
the single and double quantum operators is:
1 +
(I + I − ) + iIz
2
1 +
−1
(I + I − ) − iIz
U90 I − U90
=
2
i
1
−1
U90 Ii− Ij+ U90
=
(−Izi Ij− + Ii− Izj − Izi Ij+ + Ii+ Izj ) + (Ii+ + Ii− )(Ij+ + Ij− ) + Izi Izj
2
4
1
i
−1
(Izi Ij− − Ii− Izj + Izi Ij+ − Ii+ Izj ) + (Ii+ + Ii− )(Ij+ + Ij− ) + Izi Izj
U90 Ii+ Ij− U90
=
2
4
...
...
(3.8)
−1
U90 I + U90
=
As this is the last RF pulse, following only the in-phase vectors (proportional
to I − ) during the t2 period will be relevant in the discussion (as only such terms
produce an observable signal). A ”reduced” form of the the density matrix at the
start of the t2 period is,
ρ3 = 1 −
1
+
2
N
´
h̄ω0 X
−i ³ − i(∆ωt1 +γGzτ )
−Ii e
+ Ii− e−i(∆ω+γGzτ )
kB T i=1 4
Ã
h̄ω0
kB T
!2
N
X
i, j = 1
i 6= j
−i
8
"
(−Ii− Izj − Izi Ij− )e−i2(∆ωt1 +γGzτ )
+(Ii− Izj + Izi Ij− )ei2(∆ωt1 +γGzτ )

+
1
6
Ã
h̄ω0
kB T
!3
N
X
i, j, k = 1
i 6= j 6= k 6= i
−i 


16 
(3.9)
#
+(Ii− Izj Izk + Izi Ij− Izk + Izi Izj Ik− )e−3i(∆ωt1 +γGzτ )
−(Ii− Izj Izk + Izi Ij− Izk + Izi Izj Ik− )ei3(∆ωt1 +γGzτ )
+(Ii− Izj Izk + Izi Ij− Izk − Izi Izj Ik− )ei(∆ωt1 +γGzτ )
+(Ii− Izj Izk + Izi Ij− Izk − Izi Izj Ik− )e−i(∆ωt1 +γGzτ )





3.1
The Quantum Description
54
As spins evolve during the time t2 , the effects of chemical shift and of a second
gradient pulse will be felt (the Hamiltonian is the same as that used during t1 ). As
only in-phase terms were kept, it is equivalent to simply multiply Eq. 3.9 by a term
ei(∆ωt2 +γGzτ ) . Because t1 and t2 are necessarily positive, only the terms in Eq. 3.9
that have an exponential whose argument is negative in i∆ωt1 will be observable.
The density matrix after a time t2 , neglecting the dipolar hamiltonian is then given
by,
ρ4 = 1 −
1
+
2
+
1
6
N
−i ³ − i(∆ω(t2 −t1 )+(n−1)γGzτ ) ´
h̄ω0 X
+Ii e
kB T i=1 4
Ã
Ã
h̄ω0
kB T
h̄ω0
kB T
!2
N
X
i, j = 1
i 6= j
!3
´
−i ³
(−Ii− Izj − Izi Ij− )ei(∆ω(t2 −2t1 )+(n−2)γGzτ )
8
 



−i 


16

i, j, k = 1


i 6= j 6= k 6= i
N
X
(3.10)
+Ii− Izj Izk

−
 +Izi Ij Izk
Izj Ik−
 +Izi
+Ii− Izj Izk

−
 +Izi Ij Izk
−Izi Izj Ik−


 i(∆ω(t2 −3t1 )+(n−3)γGzτ ) 

e






 i(∆ω(t2 −t1 )+(n−1)γGzτ ) 
e

All spin terms involving the product of two spin operators are not directly observable (only one spin operators are observable), but any n-spin one-quantum operator
can produce observable signals through commutation with the dipolar Hamiltonian
(two-spin zero-quantum operator)1 . An n-spin one-quantum operator can be converted into an observable operator by n − 1 successive dipolar interactions. Having
in mind the commutation rules for spins, [Ix , Iy, ] = iIz , [Iy , Iz ] = iIx , [Iz , Ix ] = iIy ,
it can be demonstrated that the evolution of the spin operators under the operator
UDipi,j = e−iDij Izi Izj t , will be given by
−1
UDipi,j Ii− UDip
= Ii− cos(Dij t/2) + i2Ii− Izj sin(Dij t/2)
i,j
1
−1
UDipi,j Ii− Izj UDip
= Ii− Izj cos(Dij t/2) + i Ii− sin(Dij t/2)
i,j
2
−
−1
−1
I
I
cos(D
t/2)
cos(D
=
I
U
UDipi,k UDipi,j Ii− Izj Izk UDip
ij
ik t/2)
i zj zk
Dipi,k
i,j
1
+i Ii− Izj cos(Dij t/2) sin(Dik t/2)
2
1 −
+i Ii Izk sin(Dij t/2) cos(Dik t/2)
2
1 −
(3.11)
− Ii sin(Dij t/2) sin(Dik t/2)
4
1
An m-quantum operator remains an m-quantum operator when commuting with a zero quantum operator.
3.2
The Classical Approach
55
It is now possible to conclude that all the terms in Eq. 3.10 can become observable after sufficient time, and for this to happen it is essential that the exponential
multipliers are real and time independent. One-quantum coherences can consequently be selected by setting n = 1 and t2 = t1 , there will also be some contribution
from higher order operators, but their magnitude will be smaller (in this example
it can be seen that the one-quantum three-spin operator that is a three-quantum
three-spin operator during the period t1 will still have some effect). Higher order
coherences such as 2 or 3-quantum coherences can be selected by making n = 2 and
t2 = 2t1 or n = 3 and t2 = 3t1 respectively.
Any attempt to quantify signals with this formalism has to deal with large products of quantum mechanical operators 2 . The inclusion of mechanisms such as relaxation, radiation damping or diffusion (which are ”straightforward” in the classical
interpretation) are not easily incorporated in the Quantum Picture, making quantification of the signal evolution difficult. Potentially such a quantum description
has the advantage of straightforwardly predicting the location of spectral peaks in
2D-spectroscopy. In this description (as in the classical one), the unexpected peaks
in COSY are interpreted as resulting from a very large number of very small longrange multiple-quantum coherences. The concept of coherences introduced here is
slightly different from the conventional coherences in NMR, where a coherence is a
signature of the interaction between the participating spins [53]. A coherence in this
description is no more than the alignment of two independent spins
3.2
The Classical Approach - Long Range Dipolar Demagnetising Field
In the classical approach, the evolution of magnetisation during any sequence can
be predicted using the Bloch equations. The dipolar field however is a complicated
integral of the field generated by the spins over the whole volume, and is consequently
not straightforward to evaluate. Deville et al [44] realised that the complicated nonlocal description of the dipolar field,


Z
´
~ ~0
~0 ³
1
~ d (~r) = µ0
~ (r~0 ) − 3 M (r ) · (~r − r ) ~r − r~0  d3 r~0
M
B
4π | ~r − r~0 |3
| ~r − r~0 |2
(3.12)
could be written as a local function in positional space, if the magnetisation is
spatially modulated and there are no significant inhomogeneities at the length scale
2
Assuming that Dij t ¿ 1, all the terms can be simplified as sin(Dij t/2) ≈ Dij t/2 and
cos(Dij t/2) ≈ 1 which can be useful for further calculations
3.2
The Classical Approach
56
of the modulation. Under these circumstances it can be written as (for further
details refer to Appendix B)
³
´
~ d (~r) = µ0 Λ(~s) 3Mz (~s)ẑ − M
~ (~s)
B
3
(3.13)
where ŝ is the direction along which the modulation is applied, k̂ is a unit vector
2
in the field direction and Λ(ŝ) = 3 cos2 θ−1 is the second order Legendre polynomial
(meaning that the dipolar field varies with the angle, θ, between ẑ and ŝ).
In a system where the effects of relaxation (both transverse and longitudinal)
and diffusion can be neglected, the magnetisation would then evolve obeying the
following reduced version of the Bloch Equation,
µ
¶
³
´
~ (~s, t)
dM
~ (~s, t) × µ0 Λ(~s) M
~ (~s, t) − 3Mz (~s, t)ẑ
(3.14)
= γM
dt
3
~ does not afNaturally, the term of the dipolar field which is parallel to M
~ ×M
~ = 0) and the term which is parallel to ẑ only affect the evolution (as M
fects the transverse magnetisation. So, Eq. 3.14 can be rewritten as
dM + (~s,t)
dt
=
−iγM + (~s, t)µ0 Λ(~s)Mz (~s, t) that has a solution given by,
M + (~s, t) = M + (~s, 0)e−iγµ0 Λ(~s)Mz (~s,t)t
(3.15)
To produce the previous equations, it was assumed that the magnetisation had
a well defined modulation that often takes the form Mz (~s) = mz cos (km s). In such
circumstances [55], it is useful to use the Bessel-function expansion
eiδ cos(φ) =
∞
X
in Jn (δ)einφ
(3.16)
n=−∞
and therefore the solution to the Bloch Equation can be written as
M + (~s, t) = M + (~s, 0)
∞
X
in Jn (γµ0 Λ(~s)mz t)einkm s
(3.17)
n=−∞
This means that an echo will only be formed if there is a term M + ∝ e−inkm s ,
for which the evolution of the magnetisation will be characterised by the Bessel
function, Jn (see Fig. 3.2).
3.2
The Classical Approach
57
1
J
(x)
0
J
(x)
1
J (x)
J
(x)
3
2
0.5
J
(x)
4
J
(x)
5
0
−0.5
0
2
4
6
8
10
x
Figure 3.2: Different order Bessel functions
3.2.1
Hands on the Analytical Classical Approach,
the n-CRAZED case study
In this section, attention will be focused on the expected characteristics of magnetisation evolution in a sequence in which the dipolar field effects are predominant,
and in which this dipolar field is due to modulated magnetisation so that it can
be described using Eq. 3.13. When a sequence, of the form represented in Fig.
~ = M0 k̂), the
3.1, is applied to a fully recovered medium of one spin species (M
magnetisation after the first 900 RFpulse applied along x will be given by
Mz1 = 0
(3.18)
+
M(1)
= −iM0
During the period t1 the magnetisation will evolve under the effect of a field gradient
of amplitude G and duration τ , and any local frequency offsets, ∆ω,
Mz(2) = 0
(3.19)
+
M(2)
= −iM0 e−i(γGzτ +∆ωt1 )
where it is assumed that the field gradient is applied along z.
Immediately after a second RF pulse of nutation angle θ is applied along x, the
magnetisation is transformed into,
Mz(3) = −M0 sin (θ) cos (γGzτ + ∆ωt1 )
+
= −M0 sin (γGzτ + ∆ωt1 ) − iM0 cos (θ) cos (γGzτ + ∆ωt1 )
M(3)
(3.20)
3.2
The Classical Approach
58
rearranging the previous equation and considering the effect of a gradient pulse of
amplitude nG and duration τ , and of a frequency offset during time t2 yields the
following state of magnetisation
Mz(4) (z) = −M0 sin (θ) cos (γGzτ )
³
i
+
M(4)
(t2 , z) =
M0 − (1 + cos (θ)) e−i((n+1)γGzτ +∆ω(t1 +t2 ))
2
´
+ (1 − cos (θ)) e−i((n−1)γGzτ +∆ω(−t1 +t2 ))
(3.21)
No signal is observable at this point as the magnetisation once integrated over
z will vanish. However, during the subsequent evolution, the components of the
magnetisation will be coupled due to the effect of the Dipolar Field, so that,
+
M(4)
(t2 , z) = M + (0, z)e−iγµ0 Λ(~z)Mz (~z)t2
(3.22)
which may be written in terms of Bessel functions using Eq. 3.16. Summing the
magnetisation over the varying z locations so that all the z dependencies disappear
gives,
µ
M + (t2 ) =
in
M0 (1 + cos (θ) )Jn+1 (M0 µ0 γ sin(θ)t2 ) e−i∆ω(t2 −nt1 )
2
¶
+(1 − cos (θ) )Jn−1 (M0 µ0 γ sin(θ)t2 ) e−i∆ω(t2 −nt1 )
(3.23)
As in the previous section it is possible to demonstrate that an echo will appear
centred at t2 = t1 if n = 1. If n = 2, the frequency offset cancellation occurs at
t2 = 2t1 and so on. In the above analysis, the effects of relaxation, diffusion and
radiation damping were ignored, if we also consider that there are no frequency
offsets, the observable magnetisation will simply follow a combination of two Bessel
functions of order n − 1 and n + 1. All these ignored effects will however cause an
attenuation of the dipolar field signal, making it hard to observe this behaviour.
When the effects of relaxation, diffusion and radiation damping are included in
the Bloch Equations, an analytical solution becomes more complicated. Relaxation
is relatively straightforward to introduce into the solution. Mz(4) decays as e−t2 /T1 ,
+
decays as e−t2 /T2 , implying that the longitudinal relaxation term will
whilst M(4)
modify the argument of the Bessel functions while the transverse relaxation term
will be a simple multiplying factor. Diffusion however is more complex; attempts
have been made to include the effect of diffusion for the case of a DQC sequence via
3.2
The Classical Approach
59
modified analytic expressions, but this is only possible under certain limiting conditions [56]. If diffusion is assumed to operate on a more rapid time scale (τm = k21D )
m
2
than the dipolar field (τd = γµ01M0 ), solutions of the form M + (t2 ) = M + (0)e−km Dt2 ,
2
Mz (t2 ) = Mz (0)e−km Dt2 can be inserted into the Bloch Equations. After integration
of the differential equation, the result obtained also including the effect of T1 and
T2 relaxation is
+
n
−
M (t2 ) = i M0 e
(t1 +t2 )
T2
Ã
à !
2
cos
à !
!
θ
θ
Jn−1 (ζ) + sin2
Jn+1 (ζ)
2
2
(3.24)
where
2
sin(θ) −t1 /T2 1 − e−(2km D+1/T1 )t2
ζ=
e
2 D + 1/T
τd
2km
1
(3.25)
It can be immediately shown that this solution is only an approximation, as it
2
assumes that diffusion will always tend to make the signal decay as e−km Dt2 which
is only true at the start of the evolution and only for the term varying as J1 . The
2
term varying as J3 will in fact initially decay as e−9km Dt2 ...
3.2.2
Hands on the Numerical Classical Approach, more
of the n-CRAZED case study
A full solution describing the signal evolution in the n-CRAZED experiment requires the use of numerical methods. In previous work [57], simulations for one
spin species were introduced. The method involved solving the Bloch equations
for a 1-D distribution of magnetisation. The dipolar field was evaluated at each
time step by Fourier transforming the 1-D magnetisation distribution and using the
expression [44],
2
³
´
~ (~k) − 3Mz (~k)ẑ
~ d (~k) = µ0 3cos(β) − 1 M
B
3
2
(3.26)
where ~k is the coordinate position in k-space and β is the angle between the main
magnetic field and ~k.
Following inverse Fourier transformation, the effect of the long range dipolar
field, diffusion, relaxation and radiation damping can then be introduced at each
time and spatial point.
An alternative approach is described here, involving solution of a set of coupled
differential equations obtained by expanding the magnetisation as a Fourier series,
based on a method described by Bedford et al [47]. The Fourier components are then
3.2
The Classical Approach
60
inserted into the Bloch equations [47]. Bedford’s implementation of this approach
was carried out for two different spin species whose magnetisation evolves under a
constant gradient applied along the modulation direction and is based on a technique
first suggested by Einzel et al. [58]. Here the approach is adapted to the use of pulsed
gradients of the form shown in Fig. 3.1. If the magnetisation of the two spin species
(1 and 2) at a time immediately after the second gradient pulse is expanded as
M1+ = M0,1
Mz,1 = M0,1
M2+ = M0,2
X
X
n
X
Mz,2 = M0,2
bn eiγGz(nτ )
(3.27)
n
iγGz(nτ ) −iδωt2
cn e
n
an eiγGz(nτ )
e
X
dn eiγGz(nτ )
(3.28)
n
where, to simplify the maths, it has been assumed that the modulation was applied
along z and the 1st spin species is on resonance with the rotating frame whilst
species 2 is off resonance by an amount δω (M0,1 and M0,2 represent the amount
of equilibrium magnetisation of the two spin species, T1,n and T2,n represent the
longitudinal and transverse relaxation times of the nth species). Once these terms
are inserted into the Bloch equations, they lead to an infinite series of coupled
equations
µ
X
dan
2
= −iγµ0
ap M0,1 bn−p + ap M0,2 dn−p + M0,2 cp bn−p e−iδωt1
dt
3
p
an
−
− Dγ 2 G2 n2 τ 2 an
T2,1
´
dbn
iγµ0 M0,2 X ³
= −
an+m c∗m eiδωt1 − a∗−n−m c−m e−iδωt1
dt
6
m
bn
1
−
+ δn0
− Dγ 2 G2 n2 τ 2 bn
T1,1
T1,1
Xµ
2
dcn
cp M0,2 dn−p + cp M0,1 bn−p + M0,1 ap dn−p eiδωt1
= −iγµ0
dt
3
p
cn
− Dγ 2 G2 n2 τ 2 cn
−
T2,2
´
ddn
iγµ0 M0,1 X ³
= −
cn+m a∗m e−iδωt1 − c∗−n−m a−m eiδωt1
dt
6
m
dn
1
−
+ δn0
− Dγ 2 G2 n2 τ 2 dn
T1,2
T1,2
¶
¶
(3.29)
(3.30)
3.3
High Field Signal Evolution
61
of which only the lower orders are relevant3 and the observable signal is described
by the terms a0 and c0 . The terms containing exponential factors relating to the
frequency shift between the two spin species can effectively be ignored as their
contribution averages to zero on the time-scale of detection.
3.3
High Field Signal Evolution
3.3.1
Methods
Most experiments on the long range dipolar field to date have been carried out in
the regime where it can be assumed that the signal grows linearly with time. This
assumption is valid when γµ0 M0 t =
t
τd
¿ 1. This section describes experiments
where the previous approach is not valid, since t reaches values significantly larger
than τd , leading to a signal that does not grow linearly with time. The experiments
were performed at three different magnetic fields strengths (corresponding to proton
resonant frequencies of 400, 600 and 750 MHz) and show for the first time that the
evolution of the signal can be described by a Bessel function series, leading to nulling
of the signal at predictable values of time. For this to be possible, care was taken
to minimise all the effects that contribute to signal attenuation and tend to mask
the effect of the dipolar field. The data were then successfully fitted to numerical
integrations of the Bloch Equations as described in Section 3.2.2. For generating
numerical data, the set of coupled differential equations was solved using the ode23tb
matlab solver, which uses an implicit Runge-Kutta formula with a first stage that is
a trapezoidal rule step and a second stage that is a backward differentiation formula
of order two.
h90
h180
τ/2
G
τ/2
G
t1
s180
h180
θ
t2
1.8τ
0.2τ
n/2 G
ACQUISITION
ACQUISITION
n/2 G
Figure 3.3: Modified sequence for high field n-quantum coherence detection, h90 and h180 stands
for a hard 900 and 1800 pulses, while s180 is a soft (frequency selective) refocusing pulse.
3
Most of the studies performed to date, show that the inclusion of terms with n > 10, does not
change the evolution of the signal, a0 (t).
3.3
High Field Signal Evolution
62
As the aim of this work was to follow the evolution of the signal, due to long
range dipolar fields produced by modulated magnetisation, over a long time, it was
essential to minimise the effects of diffusion, radiation damping, B0 homogeneity
and RF homogeneity. This was achieved by using the modified sequence shown in
Fig 3.3.
To minimise the effects of field inhomogeneities a 1800 RF pulse was inserted in
the t1 period to eliminate T2∗ decay. To minimise effects of diffusion and radiation
damping, the second gradient pulse was divided into two parts, one applied immediately after the second RF pulse (made smaller in area than the initial gradient
pulse to avoid complete signal refocusing, and thus radiation damping, and also to
make the transverse modulation as insensitive as possible to diffusion), and one just
before the acquisition (to refocus the signal).
As the evolution of the signal has a significant dependence on the flip angle used,
it is important to minimise RF inhomogeneity over the sample. The experiments
reported here were therefore performed using a 5 mm slice thickness and the tube
containing the sample was of 5 mm diameter although the RF coil had a diameter
of 1 cm. The slice selection was accomplished via the final soft 1800 degree pulse.
To avoid undesired evolution of the magnetisation at the edges of the slice where the
dipolar field will be perturbed (note that the modulation pattern of Mz is changed
as inside the slice the magnetisation is inverted whilst outside remains the same) the
slice selective 1800 rf pulse was applied just prior to detection. A hard 1800 pulse
is applied approximately half way through the evolution time and is responsible for
refocusing attenuation due to field inhomogeneities while the last 180 pulse has a
”very insignificant refocusing role” as its aim is simply to select a slice over which
all the hard pulses are accurately calibrated.
The substance chosen for the study was DMSO which, although having a smaller
equilibrium magnetisation, M0 , than water (M0 DM SO = 0.66M0 W ater ), is known
for having a diffusion constant that is approximately three times smaller than that
of water (DDM SO = 0.7 × 10−9 m2 s−1 ).
The experiments were performed on three different Bruker MR systems, operating at frequenciesd of 400 MHz (Nottingham University), 600 MHz (University of
Florida) and 750 MHz (University of Florida).
Relaxation times and the diffusion constant were measured in situ. Although
at high field T2 is usually measured using a CPMG sequence due to the large field
inhomogeneities that can be generated by magnetic susceptibility variations, such a
procedure was avoided as the final aim here is to use the T2 as parameter for the
simulations, which do not account for diffusion in inhomogeneous fields. Thus, it was
3.3
High Field Signal Evolution
63
important to have a T2 measurement in which diffusion in any local inhomogeneities
was equivalent to that experienced in the nQC experiments. The Spin Echo sequence
used for T2 measurement was made equivalent to the final part (starting from the θ
pulse) of the nQC sequence shown in Fig. 3.3.
The n-QC experiments on the 400 MHz, 600 MHz and 750 MHz systems, were
performed with both RF excitation pulse tipping angles (h90 and θ) equal to 900
degrees (these are not the flip angles that maximise the signal [59], but the main
aim was to avoid any factor that would introduce variability between all the experiments. Setting θ = 900 also ensures faster evolution of the Bessel functions). The
Zero QC experiment was an exception, since here the second pulse produced a 450
tipping angle (if the second pulse were a 900 pulse, no signal would appear as the
contributions from the two Bessel functions of Eq. 3.23 would cancel one another).
The data points presented in the next section correspond to the integral in the
region of the peak obtained after Fourier transforming each FID. The data obtained
were then fitted to the numerically generated values which had to be evaluated
independently for each different set of parameters (n, km , M0 , B0 , T1 , T2 and D).
The value of M0 water per Tesla for water at standard pressure and temperature of
0.0033Am−1 T −1 was used for reference. The value of M0 of DMSO was found to be
approximately 0.66M0
water
(this was calculated considering DMSO has a relative
density of 0.95 and that the relative weight of protons per mole of DMSO in respect
to that of water is 0.69). The fit was made by simply matching the maximum
amplitudes of the experimental and numerical data, and this is the only arbitrary
parameter used in the fitting process.
Experiments were also performed on samples containing two spin species. The
chosen species were the protons from water and from DMSO. The same initial
measurements were made, yielding values of all the basic parameters: longitudinal
and transverse relaxation times, diffusion constants and M0 ratio between water and
DMSO, this was obtained from the T2 measurements. Due to the application of an
1800 refocusing pulse in the t1 period. the initial values for the magnetisation when
solving Eq. 3.29, correspond to those obtained from Eq. 3.21 for t1 = 0.
3.3.2
Results
The values obtained for T1 , T2 and D for the samples used at 400, 650 and 700MHz
are presented in Table 3.1. The values obtained for T1 and T2 as a function of
magnetic field are in general agreement with the expected behaviour, showing an
increase of T1 and a decrease of T2 with increasing B0 .
3.3
High Field Signal Evolution
Field Strength \ Parameter
400 MHz
600 MHz
750 MHz
64
T1 (s)
2.3
2.8
2.7
T2 (s)
1.20
0.55
0.53
D (m2 s−1 )
0.53 × 10−9
0.66 × 10−9
0.64 × 10−9
Table 3.1: Relaxation times and diffusion constant of DMSO measured at different field strengths
Fig. 3.4 shows some of the data acquired on the 400 MHz system. It is possible to
compare the effect that diffusion has on the evolution of the signal when varying the
modulation length (λ = 2π/km ) from 403µm to 101µm. The tighter the modulation
is, the less visible is the Bessel function behaviour, and after the maximum growth
is achieved the signal essentially follows an exponential T2 decay. It is also possible
to observe, in the data acquired with λ = 403µm, the way in which the first zero
of the signal is moved to later times as the order of the CRAZED experiment is
increased. Note that the ZQC data do not obey this trend due to the fact that the
RF pulse used for the second excitation is not a 900 pulse.
Similar beviour can be seen in Figs. 3.5 and 3.6 which show data acquired
at 600 and 750 MHz with a modulation length of 357µm. Figures 3.5 and 3.6
demonstrate that increasing the B0 field leads to a faster evolution of magnetisation,
visible through a decrease of the time needed to reach the first zero of the signal.
Note that the y-scale is in arbitrary units in Figs. 3.4, 3.5 and 3.6 as different
receiver gains were used for each experiment, the elevated noise fluctuations in the
quintuple QC data shown in Fig. 3.4 are evidence of the reduced absolute signal
in the data corresponding to higher order coherences. Discrepancy between the
expected values of M0 and that found from the best fit to the data occurred for the
experiments carried out at 600 and 750 MHz (although they were consistent across
the two experiments, which used the same physical sample). M0 was found to have
decreased by 12%, possible explanations being: a lower accuracy in the setting of
the hard pulse flip angles (which would have the same apparent effect as reducing
M0 ); broader linewidth (backed up by the shorter values of T2∗ values found at higher
field) at high fields potentially invalidating the on-resonance assumption made when
applying Eq. 3.29; that these experiments were performed at a higher temperature
(backed up by the higher values of diffusion); some water contamination of the
sample.
The importance of the long range dipolar field is highly evident in the case of
the SQC experiment, where several zeroes of signal amplitude can be observed (see
Fig. 3.7 ). Ignoring the Long Range Dipolar Field, would lead to signal evolution characterised by simple exponential decay due to T2 relaxation, as opposed to
3.3
High Field Signal Evolution
65
evolution dominated by the variation of the Bessel function J0 shown in Fig. 3.7.
2
1
2
3
1
2
3
signal (a.u)
10
0x 10
1
2
3
n=-3 2
10
0x 10
1
2
3
2
0
0
3
1
2
3
1
2
3
1
2
3
0x 10
1
2
3
0
1
2
3
10
0x 10
10
0x 10
2
0
n=-4
0 10
0x 10
4
1
2
3
10
0x 10
2
1
0
10
n=-5 2
n=-5 2
0
2
n=-3 1
signal(a.u)
signal (a.u)
signal(a.u)
n=-4
1
n=-2 2
n=-2 1
0
signal (a.u)
signal (a.u)
10
0x 10
2
0
10
0x 10
n=-1 2
n=-1 2
0
0
signal (a.u)
10
0x 10
signal (a.u)
signal (a.u)
n= 0 2
1
0
10
x 10 mod length=101µm
signal (a.u)
10
x 10 mod length=403µm
signal (a.u)
n= 0
signal (a.u)
This observation means that care should be taken when measuring diffusion with
0
1
time (s)
2
3
0
time (s)
Figure 3.4: Plot of experimental measurements (crosses) and simulations (continuous lines) of
various n-QC signals from DMSO as a function of evolution time at 400 MHz for two different
modulation lengths.
3.3
High Field Signal Evolution
mod length=357µm
8
Signal(a.u)
66
x 10
4
n= 0 2
Signal(a.u)
0
0x 108
0.5
1
1.5
2
2.5
3
0x 108
0.5
1
1.5
2
2.5
3
0x 108
0.5
1
1.5
2
2.5
3
0x 108
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
n=-1 2
Signal(a.u)
0
n=-2 2
1
Signal(a.u)
0
n=-3 2
n=-4
Signal(a.u)
0
2
0
time (s)
Figure 3.5: Plot of experimental measurements (crosses) and simulations (continuous lines) of
various n-QC signals from DMSO as a function of evolution time at 600 MHz for a modulation
length of 357 µm.
stimulated echo techniques at high field.
In Fig. 3.8 it is possible to see that the methodology described in Section 3.2.2
allows correct estimation of the evolution of the signal under dipolar fields produced
by two different spin species. The relative amount of each spin species was calculated
from the relative amplitudes of signals in the spin echo data. In this case the
amounts of magnetisation in each chemical environment were approximately equal,
nevertheless it is possible to observe that the water that is present in a slightly
greater quantity shows an earlier zero. This observation suggests a new approach
for solvent suppression, in which the spectra/image would be acquired at times when
the solvent reaches its first zero while any dilute solutes are still producing strong
3.3
High Field Signal Evolution
8
n=-2
Signal(a.u)
n=-1
Signal(a.u)
n= 0
Signal(a.u)
x 10
0
Signal(a.u)
Signal(a.u)
0x 108
0.5
1
1.5
2
2.5
3
0x 108
0.5
1
1.5
2
2.5
3
0x 108
0.5
1
1.5
2
2.5
3
0x 108
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
5
0
4
2
2
0
n=-4
mod length=357µm
5
0
n=-3
67
5
0
time (s)
Figure 3.6: Plot of experimental measurements (crosses) and simulations (continuous lines) of
various n-QC signals from DMSO as a function of evolution time at 750 MHz for a modulation
length of 357 µm.
signals. Note that under such circumstances, the signal of the solvent evolves as in
Eq. 3.23, whilst the solute is described by a slightly different equation. If we refer
again to Eq. 3.23, in the case of a solute the magnetisation terms outside the Bessel
functions would all relate to the solute while those inside the Bessel function would
relate to the solvent, but would be multiplied by a factor, 2/3, meaning that the
Bessel-like evolution occurs more slowly (Fig. 3.9 shows such an effect).
An interesting result, which is not directly related to the aim of this work was
the reduction in the measured diffusion constants in the water/DMSO mixture.
The diffusion of DMSO was found to be reduced from ∼ 0.6 × 10−9 m2 s−1 to ∼
0.3 × 10−9 m2 s−1 while the diffusion constant of water which is known to be ∼
3.3
High Field Signal Evolution
68
9
10
4
x 10
2.5
(a)
3.5
2
3
1.5
(b)
1
Single QC (a.u)
2.5
Single QC (a.u)
x 10
2
1.5
0.5
0
-0.5
1
-1
0.5
-1.5
0
-2
-0.5
-2.5
0
2
time (s)
4
6
1
2
3
time (s)
4
5
Figure 3.7: Plot of experimental data for a single QC experiment as a function of evolution time
at 400 MHz. Three zeroes of the signal are visible on the enlarged region of the graph shown on
the right.
8
x 10
4
Water DQC signal(a.u)
3
2
1
0
1
2
time (s)
8
x 10
3
2
1
0
mod length=202 µm
1
2
time (s)
8
x 10
3
2.5
2
1.5
1
0.5
0
1
2
time (s)
3
mod length=604 µm
4
3
2
1
3
0
mod. length=403 µm
1
2
time (s)
8
x 10
3
mod length=604 µm
3.5
3.5
DMSO DQC signal(a.u)
DMSO DQC signal(a.u)
8
x 10
4
3
3.5
mod. length=403 µm
DMSO DQC signal(a.u)
Water DQC signal(a.u)
mod length=202 µm
Water DQC signal(a.u)
8
x 10
3
2.5
2
1.5
1
0.5
0
1
2
time (s)
3
3
2.5
2
1.5
1
0.5
0
1
2
time (s)
3
Figure 3.8: Plot of experimental measurements(cross) and simulations (continuous line) of DQC
signal in a 2 spin species environment as a function of evolution time at 400 MHz for three different
modulation lengths.
1.77 × 10−9 m2 s−1 was reduced to ∼ 0.6 × 10−9 m2 s−1 in the mixture. More extensive
discussion of this effect can be found in reference [60].
2D-Spectroscopy, the DQ-CRAZED experiment
Water DQC signal(a.u)
Water SQC signal(a.u)
12
10
8
6
4
2
7
Water TQC signal(a.u)
3.4
6
5
4
3
2
69
4
3
2
1
1
0
0.5
1
time (s)
1.5
0
-1
0.5
1
time (s)
0
1.5
-1
x 10
0
0.5
1
1.5
1
1.5
time (s)
-2
x 10
x 10
4
2
1.5
3
2
1
0.5
1
0
DMSO TQC signal(a.u)
DMSO DQC signal(a.u)
DMSO SQC signal(a.u)
15
5
0
0.5
1
time (s)
1.5
0
0
0.5
1
time (s)
1.5
10
5
0
0
0.5
time (s)
Figure 3.9: Plot of the simulated evolution of magnetisation following different nQC sequences
applied to a sample with 95% water and 5% DMSO at 400 MHz the grey rectangles show regions
where the DMSO signal is stronger than that of the water
3.4
2D-Spectroscopy, the DQ-CRAZED experiment
High resolution NMR spectroscpy is a powerful tool for studying molecular structure
and dynamics. Conventional spectroscopy depends crucially upon achieving narrow
linewidths through making the static magnetic field highly homogeneous. If such
shimming is possible, either 1D spectroscopy or COSY can provide a large amount
of useful information in a relatively short space of time. However, most biological
samples are heterogeneous and made of materials of different magnetic susceptibility
and when exposed to a strong magnetic field significant field inhomogeneity results
(for example they might contain small air pockets). Therefore 1D spectroscopy or
COSY may not provide the required information. In this situation, 2D sequences
based on Long Range Dipolar Field such as the zero-quantum CRAZED HOMOG-
3.4
2D-Spectroscopy, the DQ-CRAZED experiment
70
ENIZED [61] [62] or the double-quantum CRAZED IDEAL [63] sequences, which
can yield narrow lines in the presence of large field inhomogeneity may be employed.
The implementation of a 2D sequence of this form is briefly described here. One
aim of this work was to explore the conditions that are needed for the formation of
signals in the CRAZED experiments.
It is often described in the literature, when referring to 2D spectroscopy that
for the appearance of cross peaks in solution NMR one needs a definite period t1 in
which the coherences are formed. In COSY experiments (see Fig. 3.10) this is true,
as the different spins evolve during the whole t1 period under an Hamiltonian that
contains the various terms described in Section 2.2.7 such as dipole-dipole coupling
and J-coupling.
90
90
ACQUISITION
t1
t2
Figure 3.10: Pictorial representation of the COSY sequence
In fact the quantum picture of the CRAZED coherences doesn’t require the
molecules involved in coherences to be proximal during the t1 period, and only
expects them to be similarly prepared [53]. In CRAZED sequences, the spin Hamiltonian in the t1 period simply reflects the application of a gradient pulse, as long as
the same gradient pulse is experienced by spins that subsequently interact via the
dipole-dipole coupling, it is not relevant that the gradients are applied at the same
time or even at the same place. We consider the case where cross-peaks produced
by dipole-dipole coupling between two distinct species (1 and 2) are of interest. The
two different modulated spin populations, only need to exist simultaneously during
the evolution time. To prove this point, a 2-step 2D-DQ-CRAZED experiment (see
Fig. 3.12) was designed to generate a signal similar to a modified 2D-DQ-CRAZED
experiment (see Fig. 3.11) without ever having Double Quantum Coherences, I + S + ,
between the two species present during the preparation time, t1 . By using selective
pulses, only one of the spin species will be undergoing the gradient pulse, as if
the two spin species would be spatially (as well as temporally) separated when the
modulation is applied.
3.4
2D-Spectroscopy, the DQ-CRAZED experiment
3.4.1
71
Classical Approach to Peak Location in 2D Spectroscopy
Here we present the mathematical expressions that describe the appearance of the
peaks in the Double-Quantum Crazed sequence as shown in Fig. 3.11. This sequence
was applied to a mixture containing equal quantities of DMSO and water percent
volume, and their magnetisation will be referred as Ma and Mw respectively. We
consider the water protons to be on resonance, whilst the DMSO is off resonance due
to chemical shift by a frequency δω. The sample also naturally has some frequency
inhomogeneity that is characterised by ∆ω(~r) (for simplicity we will refer to this as
∆ω). In this discussion, diffusion and relaxation processes have been neglected. It
is straightforward to observe that the magnetisation after the first two RF pulses,
the application of a modulation (characterised by φ = γGzτ , assuming that the
gradient is applied along the z direction) and the application of the crusher gradient
pulses will all be stored along the z-axis and described by,
Mzw = −M0w cos (φ + ∆ωt1 )
Mza = −M0a cos (φ + ∆ωt1 + δω)
(3.31)
Once the 450 pulse has been applied followed by the ”double-quantum filter”,
the magnetisation can be described as
180
90
90
τ/2
G
45
CRUSHER
τ/2
G
2τ
ACQUISITION
G
t1
t2
Figure 3.11: Pictorial representation of the Double Quantum CRAZED pulse sequence used for
2D spectroscopy
τ/2
G
G
t1
45
90
90
CRUSHER
τ/2
τ/2
τ/2
G
G
t1
CRUSHER
90
90
2τ
G
ACQUISITION
t2
Figure 3.12: Pictorial representation of the Double Quantum CRAZED pulse sequence used for
2D spectroscopy in which the two substances’ magnetisation are modulated at different stages.
Frequency selective pulses in black excite one of the substances, while substance 2 is excited by
pulses in gray
3.4
2D-Spectroscopy, the DQ-CRAZED experiment
π
4
π
= −M0a cos (φ + ∆ωt1 + δω) cos
4
π i(2φ+∆ωt)
= −M0w cos (φ + ∆ωt1 ) sin e
4
π
= −M0a cos (φ + ∆ωt1 + δω) sin ei(2φ+(∆ω+δω)t)
4
72
Mzw = −M0w cos (φ + ∆ωt1 ) cos
Mza
Mw+
Ma+
(3.32)
(3.33)
So far the effect of long range dipolar fields has been neglected, but in the period
following the double quantum filter the signal effectively evolves under the effect
of this field, not as primarily introduced in Eq. 3.15, but rather described by an
equation, which has the following appearance for the term affecting the evolution of
the transverse water magnetisation,
³
´
dMw+ (~s, t)
µ0
= γ Λ(~s) Mw+ (~s, t) (−3Mzw (~s, t)ẑ − 2Mza (~s, t)ẑ) + Mzw (~s, t)Ma+
dt
3
(3.34)
A similar equation can be written for the evolution of the transverse magnetisation of DMSO on its rotating frame 4 The last term of Eq. 3.34 will average to
zero, as a consequence of its variation occurring on a time scale that is so much
shorter then any other effect (in the water rotating frame the phase of transverse
magnetisation of the second spin will be e−iδωt ). Attention is first focused on the
evolution of the transverse water magnetisation, Mw+ (~s, t), which can be written as,
Mw+ (~s, t) = M + (~s, t)e−iγµ0 Λ(~s)(Mzw (~s,t)+2/3Mza (~s,t))t
(3.35)
then using the Bessel expansion introduced earlier in this chapter,
π
Mw+ (~s, t) = −M0w cos (φ + ∆ωt1 ) sin ei(−2φ+∆ωt)
4
Ã
!
X
π in(φ+∆ωt1 )
n
i Jn (γµ0 M0w cos )e
4
n
Ã
!
X
2
π im(φ+∆ωt1 +δωt1 )
m
i Jm ( γµ0 M0a cos )e
3
4
m
(3.36)
This equation can be rearranged so that the dependence on the parameters φ,
δω and ∆ω can be understood. This leads to the expression,
4
In this case the the chemical shift term would have an opposite sign.
3.4
2D-Spectroscopy, the DQ-CRAZED experiment
73
1
π³
Mw+ (~s, t) = − M0w sin
(3.37)
2X
4
in+m ei(−1+m+n)φ ei∆ω(t1 (n+m+1)+t) eimδωt1 Jn (Aw )Jm (Aa )
n,m
+
X
´
in+m ei(−3+m+n)φ ei∆ω(t1 (n+m−1)+t) eimδωt1 Jn (Aw )Jm (Aa )
n,m
√
√
where Aw = γµ0 M0w / 2 and Aa = γµ0 M0a 2/3.
There are two combinations of the indices, n and m, which will yield a signal at
the water frequency (in the direct detection direction, F2 , of the two dimensional
spectrum) that is independent of the modulation imposed by the gradient pulses:
m + n = 3 or m + n = 1. They both imply that the refocusing of field inhomogeneity occurs when t = −2t1 . Therefore, to be able to refocus the effect of field
inhomogeneity a refocusing 1800 RF pulse has to be used and the signal will be only
refocused at a time 2t1 after the expected spin-echo5 . The two n, m combinations
each give an infinite number of possible solutions. This implies in the F1 direction
(indirect detection direction, F1 ) there are an infinite number of peaks6 located at
frequencies mδω. This can be further simplified to yield,
i
π³
Mw+ (~s, t) = − M0w sin
(3.38)
2
4
´
X
eimδωt1 ei∆ω(2t1 +t) ( − J3−m (Aw ) + J1−m (Aw ))Jm (Aa )
m
If the same analysis is carried out for the DMSO protons, a similar expression
is obtained yielding a similar pattern of peaks, However it is found to be inverted
in the indirect detection (because the chemical shift of the water with respect to
the DMSO is −δω) so that the spectra are, in the frequency space, centered at
(ω1 = δω, ω2 = δω).
The spectra produced using this double quantum sequence show the same important differences from those generated using the zero quantum sequences. A zero
quantum CRAZED experiment produces spectra that are independent of ∆ω in the
F1 dimension, all the dependence on δω occuring in the F2 direction. As a consequence, the peaks will be narrower in the F1 direction [61] [62]. On the other
5
For simplicity the role of the 1800 pulse has been neglected, it is straighforward though to
conclude that it will not affect significantly the results discussed so far. The time constant appearing in the argument of the Bessel functions remain unchanged, whilst the term t multiplying
∆ω should be replaced by 2t180 − t, where t180 is the time after the 450 RF pulse, at which the
refocusing pulse is applied.
6
Naturally not all the peaks will be observable due to the reduced amplitude of the higher order
Bessel functions.
3.4
2D-Spectroscopy, the DQ-CRAZED experiment
74
hand, double quantum coherences, because of the term ei∆ω(2t1 +t) produce narrower
peaks that are narrower along the direction perpendicular to the direction (2ω1 , ω2 ).
This was first experimentally shown by Zhong et al [64], where the spectra had to
be projected into a direction perpendicular to that in which line broadening takes
place, in order to produce a maximum spectral resolution unaffected by the field
inhomogeneities.
The sequences shown in Figs. 3.11 and 3.12 are not exactly equivalent, in the
two-step sequence (Fig. 3.12) the component whose magnetisation is modulated
first will suffer from two attenuation effects before the common evolution time: T1
relaxation and attenuation due to diffusion in the modulation direction. The time
during which such effects take place, the t1 period of the second substance to be
modulated, varies as the 2D spectra is acquired due to variation in the duration of
t1 . To minimise these undesirable effects, the protons to be first modulated were
those of the DMSO in order to take advantage of its longer T1 and smaller diffusion
constant.
To maximise the signal in the 2D spectra it is important to make sure that the
signal is refocused at the time detection starts. In obtaining 2D spectra, the t1 period
is varied in order to encode the information along the indirect detection direction.
To make sure that the detection always starts at the maximum signal amplitude, the
1800 refocusing pulse had to be moved accordingly. This also means the spectrum
peaks are narrower along F1 (unlike the experiments by Zhong et at [64]).The 2D
Spectra acquired , were made up of 2560×128 points and had a frequency resolution
of 47Hz in the F1 direction and 1Hz in the F2 direction. 2D spectra were acquired
at echo times of 0.13, 0.3, 0.6, 0.9 and 1.5 seconds.
3.4.2
Experiments
In the linear regime (t < τD ) four peaks are expected, two arising from the direct
couplings (Water-Water and DMSO-DMSO) and two from cross couplings (DMSOWater and Water-DMSO). Figure 3.13 shows that the two dimensional spectra produced using two different sequences (Fig. 3.11 and 3.12) are to a large extent equivalent and all peaks appear in the expected positions. The fact that the two sequences
produce similar patterns implies that the only time when coupling is important is
during the evolution period.
In both DQ-CRAZED sequences more than the 4 peaks usually described in
simplified models were found (see Fig. 3.13). In Fig. 3.14 special attention is
focused in the peaks occurring at the water frequency in the F2 direction, when the
3.5
Conclusions
75
spectra is acquired at echo times of 0.13, 0.3, 0.6, 0.9 and 1.5 seconds. As expected,
the pattern becomes more complex as longer evolution times are used and higher
order Bessel functions become more important (Eq. 3.38).
These observations are in agreement with what has been also suggested by Jeener
in a review article on collective effects in Liquid NMR [54] and experimentally observed for triple-quantum coherences [65].
3.5
Conclusions
For the first time experiments carried out at various field strengths have shown
long range dipolar effects in the evolution of magnetisation at times larger than τd ,
leading to a signal evolution with a Bessel function form with evident zeros and sign
changes. A numerical method for calculating the evolution of the magnetisation
that may be described by the Bloch equations (containing relaxation, diffusion and
dipolar field effects) was developed and tested by comparison with experimental
data acquired at different magnetic fields for varying modulation lengths. A new
3000
3000
+
M Mz
W
DMSO
2000
+
M Mz
W
DMSO
0
+
M Mz
W
1000
F1 (Hz)
F1 (Hz)
1000
2000
0
W
-1000
-1000
+
M Mz
-2000
DMSO
-3000
-1000
-2000
DMSO
0
-3000
1000
F2 (Hz)
-1000
0
1000
F2 (Hz)
Figure 3.13: 2D spectra obtained using a (a) 2 Step COSY experiment and (b) 1 step COSY
experiment, with an echo time of 300ms in both cases.
F1 (Hz)
echo time=0.15s
2000
echo time=0.3s
2000
0
echo time=0.6s
2000
0
-2000
0
F2 (Hz)
50
2000
0
-2000
-50
echo time=0.9s
0
F2 (Hz)
50
2000
0
-2000
-50
echo time=1.5s
0
-2000
-50
0
F2 (Hz)
50
J1-J-1
J2-J0
J3-J1
J4-J2
-2000
-50
0
F2 (Hz)
50
-50
0
F2 (Hz)
50
Figure 3.14: Zoom into the peaks obtained at the water frequency at 5 different echo times.
3.5
Conclusions
76
method for solvent suppression, that as the advantage of being neither chemical shift
selective nor T1 sensitive, has also been proposed.
4
Optimising the Sequence Parameters for DQC Imaging
77
Chapter 4
Optimising the Sequence
Parameters for DQC Imaging
The evolution of magnetisation during repeated application of the double-quantum(DQ)-CRAZED (COSY revamped with asymmetric z-gradients) sequence is analysed, with the aim of identifying the sequence parameters that maximise sensitivity
to signal produced by the distant (or long range) dipole field (DDF). Phase cycling
schemes that allow cancellation of signals following undesired coherence pathways
are also described. Simulations and imaging experiments carried out at 3T on phantoms and the human head have been used to verify the analysis. The results indicate
that in the absence of phase cycling, the DDF-related signal-to-noise ratio (SNR) per
unit time is maximised using flip angles α = 90o and β = 60o , and delays T E = T2
and T R = 2.05T1 , but that at this value of TR there can also be a significant signal
contribution due to stimulated echo effects (up to 60% of the DDF-related signal for
water at 3T and T E = 70ms). Using the two-step phase cycle, the stimulated echo
signal is eliminated and the maximum SNR per unit time occurs for T R = 2.76T1 .
It is also demonstrated that sensitivity to signal changes caused by small variations
in T2 is maximised by setting T E = 2T2 .
4.1
Introduction
The long-range field generated by the nuclear magnetisation itself has been shown
to produce anomalous effects in liquid state NMR. These include the formation of
multiple spin echoes in simple sequences employing just two RF pulses [44, 45], and
the production of unexpected multiple quantum coherences between spins located
on different molecules [49, 50]. Such effects result from the coherent action of the
4.1
Introduction
78
distant dipolar field (DDF) on the evolution of transverse magnetisation and are
most strongly manifested when the magnetisation is spatially modulated by the application of magnetic field gradients. An important feature of the DDF in such a
situation is that it acts over a distance that is set by the length scale of the imposed spatial modulation [48, 66–69]. This feature imparts an interesting structural
sensitivity to the signal formed by the action of the DDF, which has been exploited
in bulk measurements of structure in simple model systems [48, 66, 67, 70], emulsions [68] and trabecular bone [69]. In addition, it has been suggested that this
length-scale dependence extends to the sensitivity to magnetic field inhomogeneity
of signals formed using the DDF [71]. The idea being that significant magnetic field
variation on the length-scale of the modulation causes local attenuation of the DDF
and thus the signal that it generates. By varying the strength and/or duration of the
magnetic field gradients used to modulate the magnetisation spatially it is possible
to change the length scale over which significant field variation must occur to cause
signal loss. This potentially, length-scale tuneable contrast has engendered a considerable amount of interest, which has motivated some initial experiments exploring
the use of DDF effects in fMRI [72–74]. The non-standard dependence of the NMR
signal generated by the DDF has also opened up other possibilities for exploiting
DDF effects in biomedical MRI. In particular, the dependence of the signal strength
on the square or higher powers of the equilibrium magnetisation, M0 , facilitates
mapping of the absolute value of M0 using the effects of the DDF [75]. The enhanced attenuation of the signal due to DDF effects by molecular diffusion may also
allow the implementation of new approaches to diffusion-weighted imaging [76, 77].
Unfortunately, however, the signal generated by the DDF is generally much
smaller than that which can be produced by conventional means. It is therefore
particularly important to optimise the parameters of any sequences employing DDF
effects so as to yield the maximum signal or contrast to noise per unit time. The
reduced signal also makes such sequences very sensitive to contamination by signals generated due to the evolution of magnetisation following undesired coherence
pathways [78]. It is also necessary therefore to adopt experimental sequence modifications, such as phase cycling [79,80], so as to eliminate undesired coherences whilst
preserving the signal generated via the effects of the DDF.
The work presented here, is therefore devoted to the analysis of the evolution of
magnetisation in the double-quantum (DQ) CRAZED (COSY revamped with asymmetric z-gradients) sequence that is most commonly used to generate DDF-related
signals [50, 81]. The results of this analysis have been used to identify the sequence
parameters (specifically the RF pulse tipping angles, the repetition time, TR, and
4.2
Theory
79
the echo time, TE) that provide the optimum signal to noise ratio (SNR) per unit
time, and also to devise phase-cycling schemes that can be used to eliminate undesired signals. Simulations and experimental results obtained at 3 T on phantoms
and the human head have been used to verify the analysis.
4.2
Theory
Figure 4.1 shows the DQ-CRAZED sequence [50], which forms the basis of the measurements described here. On first inspection, no signal is expected to be formed
by this sequence because of the unequal areas of the gradient pulses, however the
~ d (~r), during the period after the second RF pulse refocuses
action of the DDF, B
a fraction of the transverse magnetisation. The DQ-CRAZED signal results from
~ ×B
~ d introduced into the Bloch equations by
the non-linear term of the form γ M
~ d (~r) is a complicated function of the distribution
the DDF. In the general case, B
of magnetisation throughout the sample [44]. However when the magnetisation is
spatially modulated by the application of magnetic field gradients it can be shown
that the DDF at a particular position is largely sensitive to the magnetisation found
within a surrounding region whose extent is equal to the length scale of the modulation [48,66–68]. For the sequence shown in Fig. 4.1, this length scale is given by k2πm ,
where km = γGτ . In the case where the sample is homogeneous on the length scale
of modulation, the DDF becomes a simple function of the local magnetisation [44]
such that
µ
¶
~ d (~s) = µ0 ∆ 1 M
~ (~s) − Mz (~s)ŝ
B
3
where ẑ is a unit vector in the direction of the main magnetic field, B0 ,
∆=
´
1³
3(ŝ · ẑ)2 − 1
2
(4.1)
(4.2)
and ŝ is a unit vector in the s-direction along which the magnetisation is modulated. In the work described here, it is assumed that Eq. 4.1 is made approximately
valid via the use of sufficiently strong spatial modulation of the magnetisation. The
~ d (~r) is largest when the magnetisation is modulated
form of Eq. 4.2 means that B
along the z-direction, takes a value that is opposite in sign and half in magnitude
when the modulation direction is orthogonal to z and is zero when the modulation
is at the magic angle (54.7o ) with respect to the field direction. The term in Eq.
4.1 that is co-linear with the magnetisation vector has no effect on the evolution of
4.2
Theory
80
magnetisation so that the effects of the DDF are only manifested through the term
proportional to Mz , with the Bloch equation taking the form
Ã
!
~ (~s)
dM
~ (~s) × B~d + ∆ω ẑ + D∇2 M
~ + M0 − Mz ẑ − Mx x̂ − My ŷ
= γM
dt
γ
T1
T2
T2
(4.3)
where D is the self-diffusion coefficient and ∆ω is the frequency offset due to any
magnetic field inhomogeneities.
t1
α
τ
G
TE/2+t1
TE/2-t1
β
180
EPI
module
2τ
G
t
Maximum signal
Figure 4.1: Schematic representation of the Double Quantum CRAZED pulse sequence, with
echo planar imaging module.
Considering a single application of the sequence shown in Fig. 4.1, with the
magnetisation initially in the state, Mx = My = 0, Mz = m, then including the
effect of the dipolar field, but neglecting relaxation and diffusion, the magnetisation
at time, t, after the application of the 180o pulse is described by
Mz = −m (cos(α) cos(β) − sin(α) sin(β) cos(a − b − km s − ∆ωt1 ))
(4.4)
and
µ
M
+
³
´
β
β
= m sin(α) cos2 ( )ei(a−km s−∆ωt1 ) + sin2 ( )ei(a−2b−km s−∆ωt1 )
2
2
¶
+ cos(α) sin(β)eib e−i(∆ω(t−
TE
+t1 ))
2
e−iµ0 γ∆Mz (t+
TE
−t1 )
2
(4.5)
Here M + = Mx +iMy , while a and b describe the phases of the α and β RF pulses
(as the angle between the B1 -direction and the x−axis in the rotating frame). In
most experiments carried out on biological samples at currently accessible magnetic
fields, the argument of the last exponential term in Eq. 4.5, which represents the
effect of the dipolar field, is small enough for the approximation
e−iµ0 γ∆Mz (t+
TE
−t1 )
2
≈ 1 − iµ0 γ∆Mz (t +
TE
− t1 )
2
(4.6)
4.2
Theory
81
+
to hold. In this case, we can simply divide M + into a contribution ( Mdip
) that
+
is produced by the DDF, and a contribution ( Mnon−dip
) that is independent of DDF
effects, stemming from the second and first terms in Eq. 4.6 respectively. These are
given by
TE
− t1 )
2
µ
TE
β
sin(α) cos2 ( )e−i(a+km s) e−i∆ω(t− 2 ) +
2
TE
β
− sin(α) sin2 ( )ei(a−2b−3km s) e−i∆ω(t− 2 +2t1 ) +
2
¶
+
= −mµ0 γ∆Mz (t +
Mdip
+ cos(α) sin(β)e−i(b+2km s) e−i∆ω(t−
TE
+t1 )
2
(4.7)
µ
TE
β
+
Mnon−dip
= −im sin(α) cos2 ( )e−i(a+km s) e−i∆ω(t− 2 ) +
2
TE
β
− sin(α) sin2 ( )ei(a−2b−3km s) e−i∆ω(t− 2 +2t1 ) +
2
¶
+ cos(α) sin(β)e−i(b+2km s) e−i∆ω(t−
TE
+t1 )
2
(4.8)
In order to calculate the amount of signal generated by each of these terms it is
necessary to average Eqs. 4.7 and 4.8 over the extent of the voxel or sample, ∆s,
in the s−direction. The result of this averaging process will clearly depend on the
s−dependence of m. When the sequence of Fig. 4.1 is repeatedly applied, m is in
fact the steady-state longitudinal magnetisation, but before considering this case,
we briefly evaluate the signal produced when the sequence is applied:
- only once (with m = M0 );
- twice using a repetition time, T R (which helps in understanding the
origin of the signal produced in the steady-state case);
- many times, so the steady state is reached (with and without phase
cycling).
4.2.1
Single Experiment
+
If m = M0 , most of the terms in the expression for Mdip
(Eq. 4.7) have a phase which
depends on s−position, so that averaging over the voxel leads to signal cancellation.
However this is not the case for the first term in the square brackets in Eq. 4.7,
which yields a contribution whose phase is independent of s and given by
4.2
Theory
82
cos(β) + 1 i(b−2a) −i∆ω(t− T E +2t1 )
TE
2
− t1 ) sin2 (α) sin(β)
e
e
2
4
(4.9)
TE
This produces an echo when t = 2 +t1 , whose phase and magnitude is described
+
Mdip
= −µ0 γ∆M02 (t +
by
1
cos(β) + 1 i(b−2a) − T E+2t
e
e T2
(4.10)
4
Here we have introduced the effect of T2 -relaxation, but have assumed that
2
km DT2 ¿ 1 so that the effect of diffusion on the transverse magnetisation can be
neglected. As has been previously described, this signal is maximised when α = 90o
and β = 60o [59].
+
With m = M0 , all of the terms in the expression for Mnon−dip
have an s−dependent
phase, so that as expected the unbalanced gradient pulses mean there is no strong
signal contribution in the absence of the DDF. Analysis of Eq. 4.8 indicates that
there are terms varying as e−ipkm s , with p = 1, 2 and 3. Complete cancellation of the
signal due to these terms on integrating across the voxel only occurs if pkm ∆s = n2π
or pkm ∆s À 1. When these conditions are not fulfilled, or when the modulation
is disturbed by the presence of large local field inhomogeneity, there may be a significant non-dipolar signal contribution after each execution of the DQC sequence.
The largest contribution is likely to be from the e−ikm s term, which has the weakest
s-dependence [82]. This takes the form
+
Mdip
= −µ0 γ∆M02 T E sin2 (α) sin(β)
+
Mnon−dip
= −iM0 sin(α)
at t =
4.2.2
TE
2
1)
cos(β) + 1 −i(a+km s+∆ωT E) − (T E+t
T2
e
e
2
(4.11)
+ t1 .
Two Experiments
We now consider the case where the sequence is applied twice with a repetition time,
T R, using RF pulse phases a1 , b1 in the first execution and a2 , b2 in the second.
Assuming that T R À T2 , so that no transverse magnetisation survives at the end of
the T R period, the signal produced by the second application of the sequence can be
calculated from Eqs. 4.7 and 4.8, using the following expression for the longitudinal
magnetisation present the second time the sequence is applied,
³
´
t
m
−TR
−TR
− 1
= (1−e T1 )−e T1 cos(α) cos(β)−e T2 sin(α) sin(β) cos(−km s−∆ωt1 +a1 −b1 )
M0
(4.12)
4.2
Theory
83
where we have assumed that T E and t1 ¿ T1 . In this case, the dominant
+
contribution to Mdip
, resulting from the terms in Eq. 4.12 which are independent of
³
´
−TR 2
s, is reduced in magnitude by a factor of 1 − (1 + cos(α) cos(β))e T1
compared
with Eq. 4.10 due to incomplete recovery of the longitudinal magnetisation. Some
new contributions to the signal due to the spatially modulated magnetisation in
Eq. 4.12 are also introduced. Their amplitudes are reduced by a factor of at least
−2 T R
e T1 relative to those due to the spatially uniform part of m.
Substituting Eq. 4.12 into the expression for (Eq. 4.8) indicates that when the
sequence of Fig. 1 is applied twice, a strong signal is formed even in the absence of the
DDF - this is the stimulated echo formed from spatially modulated magnetisation
stored along z during the T R period. This magnetisation also forms an echo at
t = T2E + t1 , whose phase and magnitude, including the effects of relaxation and
diffusion during the T R period, are described by
+
Mnon
dip
= iM0 sin2 (α) sin(β)
1 −T R(k 2 D+ 1 )
cos(β) + 1 i(b1 −a1 −a2 ) − T E+2t
m
T1
e
e T2 e
4
(4.13)
This has identical flip angle and T2 -dependence to Eq. 4.10, which makes it
difficult to distinguish this stimulated echo signal from that produced by the DDF
[78]. However an important difference is that the phase of the stimulated echo
depends on the phases of the RF pulses used in both the first and second execution
2 D+ 1 )
−T R(km
T1
of the sequence. In addition this signal is attenuated by a factor of e
,
which characterises the decay of longitudinal magnetisation during the TR period.
4.2.3
Steady State
When the sequence of Fig. 1 is repeatedly applied with a repetition time, T R, we
can calculate the steady-state value of m in the usual manner by equating the values
of m before and after application of one cycle of the sequence [83]. In doing this, the
assumption that T R À T2 has been made (so that there is no steady-state transverse
magnetisation), and the effect of diffusion during T R has also been neglected. When
the RF pulse phases, a and b, are the same in all repetitions, we find
³
´
1 − e−T R/T1
m
=
M0
1 + (cos(α) cos(β) − sin(α) sin(β) cos(−km s − ∆ωt1 + a − b)) e−T R/T1
(4.14)
More complicated expressions for m can be straightforwardly derived for the
situation where RF phase cycling is employed. In addition, the case where the
condition T E ¿ T1 does not hold, so that the refocusing 1800 RF pulse in the
sequence of Fig. 1 has some effect on the recovery of longitudinal magnetisation can
4.2
Theory
84
−TR
−
T R−T E/2
T1
be accommodated by changing the numerator of Eq. 4.14 to 1 + e T1 − 2e
.
The s−dependence of the denominator of Eq. 4.14 makes calculation of the nonzero signal contributions following averaging of Eqs. 4.7 and 4.8 over the voxel
more complicated. The simplest approach is to expand Eq. 4.14 as a power series
−TR
in e T1 . For a more exact solution, we can substitute for m in Eqs. 4.7 and 4.8
and then evaluate the integral of the transverse magnetisation over one cycle of the
R π
modulation (i.e. −kmπ M + ds). The latter approach was found to only be necessary
km
when T R ¿ T1 , since using the series expansion up to 4th order leads to only a 10%
underestimation of the dipolar signal at T R = 0.37T1 , which is reduced to 0.4% at
−TR
T R = T1 . Expanding Eq. 4.14 to first order in e T1 and re-introducing the effect
of relaxation in the t1 period yields Eq. 4.12.
4.2.4
Phase Cycling
The above analysis indicates that when repeatedly executing a DQ-CRAZED sequence there will be a stimulated echo contribution to the signal, which is indepen−TR
−TR
dent of the DDF. To first order in e T1 (i.e. in the regime where e T1 ¿ 1), the
ratio of signal amplitudes is
|
+
2 D+ 1 )
Mdip
T R(km
T1
+ | = µ0 γ∆M0 T Ee
Mdip
(4.15)
The relative stimulated echo contribution can thus be minimised by using a long
T R or a large value of km . However, the generally small value of µ0 γM0 T E(∼ 0.33
for water at 3 T with T E = 100 ms) means that the exponential factor has to
be large to ensure that the signal due to the DDF dominates the stimulated echo.
Unfortunately, the optimum SNR per unit time, can not always be achieved using
1
values of T R that are significantly larger than k2 D+
, and utilisation of large
1
m
T1
values of km may also not always be possible. In such circumstances other methods
must be used in order to measure a signal purely due to the DDF. In previous
work, a common approach has been to vary the direction [72, 74] or magnitude [78]
of the applied field gradient across cycles of the sequence. For example, Zhong et
al. [74] used a gradient mainly directed along the z-axis, but with an additional weak
component along the y-direction (approximately 10% of the z-gradient amplitude).
The polarity of this weak component was alternated so as to prevent the formation
of stimulated echoes. For imaging with the ”zero-quantum” CRAZED sequence,
there is a significant problem with FID signal produced by the β RF-pulse and a
more radical scheme in which the signal generated with the gradient along the xdirection (∆ = −0.5) is subtracted from that produced when the gradient is applied
4.2
Theory
85
Phase Cycle I
dipolar signal
stimulated echo
spin echo
αx βx
-x
x
-y
αy βx
x
x
-x
Table 4.1: Phase of dipolar signal, stimulated echo and spin echo signals during the Phase Cycle
I (−αx βx , αy βx ).
Phase Cycle II αx βx
dipolar signal
-x
stimulated echo
-y
spin echo
-y
α−x βx
-x
y
y
α−y βy
y
-x
x
αy βy
y
x
-x
Table 4.2: Phase of dipolar signal, stimulated echo and spin echo signals during the Phase Cycle
II (−αx βx , −α−x βx , −iα−y βy , −iαy βy ).
along z (∆ = 1.0) has been employed [71]. This prevents stimulated echo formation
and more importantly, largely eliminates the FID signal while preserving the signal
produced by the DDF. Varying the gradient direction or strength, does however have
some disadvantages including the introduction of varying signal sensitivity to field
inhomogeneity, potentially different structural sensitivity of the signal produced by
the DDF, and varying eddy current effects. We have consequently devised phase
cycling schemes that can also be used to eliminate signal that is not generated by
the DDF.
The similar dependence of the stimulated echo and DDF signal contributions on
RF pulse phase makes the generation of a phase cycle which distinguishes between
these two signals harder. Analysis of Eqs. 4.10 and 4.13 indicates that a twostep cycle of the form (−αx βx , αy βx ), which we call Phase-Cycle I, has the required
effect, when terms up to only first order in are considered. Table 4.1 depicts the
phase dependence of the various signals through the phase cycle, and shows that
this cycle will not eliminate the signal of the form described in Eq. 4.11 (although
√
its contribution is reduced by a factor of 2). To cancel this signal as well, it
is necessary to adopt a four-step phase cycle (−αx βx , −α−x βx , −iα−y βy , −iαy βy )
(Phase-Cycle II). Table 4.2 shows the phase dependence of the various signals for
− TTR
this phase cycle. Again this only holds strictly for small e
−ikm s
that the signal varying as e
cycle.
1
, but it is worth noting
in Eq. 4.8 can also be eliminated using this phase
4.3
Methods
4.2.5
86
Echo-Time Dependence
Equation 4.10 indicates that the strength of the signal produced by the DDF depends
−TE
on the product of T E and e T2 , which is maximised when T E = T2 . In fMRI
experiments based on using the DDF [72–74], the goal is to maximise sensitivity
to BOLD-related signal changes. When the signal is sampled at the peak of the
echo [73], the signal change due to a small reduction, −∆R2 , in the T2 -relaxation
rate (R2 ), is proportional to
µ
T E e−
R2 −∆R2
TE
R2
− e− T E
¶
≈ ∆R2 T E 2 eR2 T E
(4.16)
where we have assumed that t1 ¿ T2 . This change is maximised when T E = 2T2 .
4.3
Methods
The values of α, β and
TR
T1
that give the largest SNR per unit time were found
M+
by evaluating the steady-state values of qdip
for the sequences with and without
TR
T1
phase-cycling. This was accomplished via the use of Eqs. 4.7, 4.8 and 4.14 and
their variants including the effect of phase-cycling. In each case an expansion up
−TR
to fourth order in e T1 was employed. The truncation of the series at this order is
likely to lead to some errors in the calculations at small values of TTR1 .
Experiments were performed at 3T using four, 2.2 cm diameter, sample tubes
(A-D), filled with Agar gel, doped with Gd-DTPA (T1 =3.22(A), 1.07(B), 1.14(C),
1.18(D) s; T2 =0.178(A), 0.147(B), 0.108(C), 0.081(D) s). 64 × 64 DQC images
(FOV= 19×19 cm2 , slice thickness= 10 mm) were acquired via echo-planar imaging
with different values of T R and T E, using Phase-Cycle I. Modulation was applied
both along z and at the magic angle. The same procedure was also used to image
the human brain. Imaging of the Agar phantom was additionally performed using
Phase-Cycle II. For this experiment, Sample B was replaced by a sample tube filled
with water, into which a smaller air-filled 5 mm diameter glass tube was partially
inserted (Sample E). This generated significant local magnetic field inhomogeneity
in the imaging plane, located near the end of the air-filled tube. In all experiments,
km was set to 6300 mm−1 corresponding to a spatial period of modulation 1 mm.
4.4
Results
+
+
Figure 4.2 shows the variation of Mdip
and Mnon−dip
with TTR1 for the sequence of
+
Fig. 4.1 applied with α = 90o and β = 60o and no phase-cycling. Mdip
is shown in
4.4
Results
87
0.32
0.3
0.08
(a)
(b)
0.28
0.26
0.24
0.06
Mnon-dip
0.22
+
+
Mdip
0.2
0.18
0.16
0.04
0.14
0.12
0.1
0.08
0.02
0.06
0.04
0.02
0
1
2
3
4
5
0
1
2
TR/T1
3
4
5
TR/T1
+
+
Figure 4.2: Variation of (a) Mdip
and (b) Mnon−dip
with T R/T1 for repeated application of the
+
DQ-CRAZED sequence without phase-cycling. Mdip
is shown in units of µ0 γ∆M02 T Ee−
while units of M0 e
−
−
T E+2t1
T2
are used for
T E+2t1
T2
,
+
Mnon−dip
.
T E+2t1
−
T E+2t1
+
units of µ0 γ∆M02 T Ee T2 , whilst Mnon−dip
is shown in units of M0 e T2 . It
2
has been assumed here that km
DT1 ¿ 1 so that the effect of diffusion during the
+
T R period is negligible. Mdip
rapidly decreases at small values of TTR1 because the
dependence of this term on the square of the available magnetisation makes it very
+
sensitive to saturation [84]. Mnon−dip
shows behaviour characteristic of a stimulated
echo, formed by magnetisation that is stored along the z-direction for at least one
T R period. For long repetition times, this signal decays exponentially with TTR1 , as
predicted by Eq. 4.13. For small values of TTR1 the effect of saturation dominates.
+
Mdip
Figure 4.3 shows contour plots depicting the variation of √
T R/T1
(which is pro-
portional to the available signal to noise ratio per unit time) with the flip angles
α and β, for four different values of TTR1 . For TTR1 = 5 (Fig. 4.3a) the magnetisaM+
tion fully recovers during the T R period, and the flip-angle dependence of qdip
is
TR
T1
o
well characterised by Eq. 4.10, with the maximum value occurring at α = 90 and
β = 60o . At TTR1 = 2.5 (Fig. 4.3b) there is little change in the contour pattern
with the maximum still occurring approximately at α = 90o and β = 60o , but the
reduction in T R/T1 has produced a significant increase in the maximum value of
M+
√ dip . As TTR is further reduced, the location of the maximum shifts to a larger
T R/T1
1
value of α and a slightly smaller value of β. These changes in the flip-angle values
act to reduce the constant term in the denominator of Eq. 4.14 thus increasing the
+
part of m which most significantly contributes to Mdip
. The maximum attainable
+
Mdip
value of √
T R/T1
is however reduced on decreasing
TR
T1
from 2.5 to 1. This is because,
4.4
Results
88
180
160
180
(a)
160
140
140
120
120
100
100
β
β
80
80
-4
x 1.29*10
60
40
20
20
20
40
60
80
α
100
120
140
x 1.58*10
60
40
160
180
20
180
180
160
(b)
(c)
160
140
140
120
120
100
40
60
80
α
100
120
-4
140
160
180
160
180
(d)
100
β
β
80
80
-4
x 1.43*10
60
40
40
20
20
20
40
60
80
α
100
120
140
x 1.02*10-4
60
160
180
20
40
60
80
α
100
120
140
√
+
Figure 4.3: Contour plots showing the variation of Mdip
/ T R with the flip-angles, α and β (in
√
+
degrees), for T R/T1 = (a)5, (b)2.5, (c)1 and (d)0.5. The site of Mdip
/ T R maximum magnitude of
is marked on each plot, along with the value for M0 = 7.3 × 10 − 3Am−1 ,T E = T2 and T1 = 1.2s.
+
as indicated by Fig. 4.2a, in this range of TTR1 values, Mdip
decreases rapidly as T R
TR
is reduced. When T1 is further reduced to 0.2, there is a further decrease in the
+
Mdip
maximum value of √
T R/T1
and the contour pattern becomes more complicated as
higher order terms in the series expansion of Eq. 4.14 become more important. A
+
full exploration of the α, β , TTR1 space indicates that the maximum Mdip
signal to
TR
o
o
noise ratio per unit time occurs at α ≈ 90 , β ≈ 60 and T1 = 2.05.
The optimal values of the flip-angles are not significantly changed when phase
cycling is employed. The same is not true for the optimum value of TTR1 , which
increases to 2.76 when Phase Cycle I is employed. This increase results from the
+
cancellation of contributions to Mdip
from modulated magnetisation that is stored
longitudinally for at least one T R period.
In order to explore the behaviour of the mixed signal formed from the sum of
+
+
Mnon−dip and Mdip
it is necessary to consider a specific value of M0 because the ratio
4.4
Results
89
(a)
(b)
0.0004
-1
M (Am )
1/2
0.0002
0.0002
+
+
0.0003
-1 -1/2
M /TR (Am s )
0.0003
0.0001
0.0001
0
1
2
TR
3
0
5
4
1
2
TR
3
5
4
(c)
(d)
0.0007
0.00025
0.0006
-1 -1/2
M /TR (Am s )
0.0004
0.00015
1/2
1/2
-1 -1/2
M /TR (Am s )
0.0002
0.0005
+
+
0.0003
0.0001
0.0002
5e–05
0.0001
0
0
1
2
TR
3
4
5
1
2
TR
3
4
5
Figure 4.4: (a) Variation of M + of grey matter at 3 T with T R, for the first√(continuous line)
and second (dashed grey line) steps of Phase Cycle I. (b) Variation of M + / T R for the first
+
(continuous line) and second (dashed grey line) steps of Phase Cycle I. (c) Mdip
(continuous line)
+
and Mnon−dip (dashed grey line) produced by summing signals from the two steps of Phase Cycle
√
+
+
I and scaling by T R. (d) Mdip
(continuous line) and Mnon−dip
(dashed grey line) produced by
√
subtracting signals from the two steps of Phase Cycle I and scaling by T R.
of the two signals scales as M0 . Here we used the M0 value for grey matter at 3
T (7.3 × 10−3 Am−1 ) [75], and also employed grey matter relaxation times for this
field strength (T1 = 1.2s and T2 = 0.08s). Figure 4.4a shows the TR dependence
of the absolute value of M + produced in consecutive applications of the sequence
of Fig. 4.1 using Phase Cycle I and α = 90o , β = 60o , T E = 80ms and with the
modulation gradient applied in the z-direction. It is clear from these plots that the
+
+
signal is made up of a mixture of contributions from Mnon−dip
and Mdip
, which have
different relative phases in consecutive applications of the pulse sequence. Figure
√
4.4b, shows the resulting variation of the signal scaled by T R, which dictates the
SNR per unit time. Table 4.1 indicates that considering the signal contributions
−TR
to first order in e T1 for experiments employing Phase Cycle I, the signal due
4.4
Results
90
to DDF effects changes sign in consecutive experiments while the stimulated echo
signal stays the same. Subtraction of signals from consecutive experiments therefore
preserves the signal that originates from the DDF, while addition yields a signal
mainly resulting from stimulated echo effects. This is indicated in Figs. 4.4c and
√
4.4d, which respectively show the signal scaled by T R produced by adding or
subtracting data from alternate experiments. These curves indicate that the phasecycle can be used to separate the signal due to the DDF and stimulated echo signal
effectively, when T R > T1 . For smaller values of T R, signal contributions varying
as higher powers of T R/t1 become significant and subtraction of the signal from
alternate experiments does not entirely eliminate the stimulated echo contribution.
(a)
(b)
D
B
A
C
(c)
(d)
Figure 4.5: Modulus echo planar images of Agar samples A-D produced using Phase Cycle I. (a)
αx βx image; (b) αy βx image; (c) sum of αx βx and αy βx images; (d) difference of αx βx and αy βx
images.
Figure 4.5 shows echo planar images of the Agar-filled samples, which were produced using Phase Cycle I with T R = 2.7s, T E = 110ms and the modulation
gradient applied in the z-direction. Figures 4.5a and 4.5b show the images produced
in consecutive applications of the sequence, showing the difference in signal intensity
+
+
when the contributions due to Mdip
and Mnon−dip
have the same (Fig. 4.5a) or the
opposite sign (Fig. 4.5b). Figure 4.5c shows the result of adding the images of Figs.
4.5a and 4.5b, while Fig. 4.5d, shows the difference image. Figure 4.5c displays the
contrast expected from a signal with stimulated echo characteristics; Tube A, which
has the longest T1 , appears with highest intensity. Tube B, which has a relatively
short T1 (and long T2 ) yields the strongest signal in the image formed from the difference of the two signals, as would be expected for a signal generated by the DDF.
Figure 4.6 shows the result of imaging a similar set of tubes using Phase Cycle
II with T R = 2.5s and T E = 51ms. The data have been combined in the manner
indicated by Table 4.2, so as to produce images displaying the signal due to the
4.4
Results
(b)
(a)
A
91
(c)
(d)
E
C
D
Figure 4.6: Modulus echo planar images of Agar samples A-D produced using Phase Cycle II.
(a) image formed from−(αx βx ) − (α−x βx ) − i(α−y βy ) − i(αy βy ) combination of signals; (b); image
formed from i(αx βx ) − i(α−x βx ) + (α−y βy ) − (αy βy ) combination of signals; (c) image formed from
i(αx βx ) − i(α−x βx ) − (α−y βy ) + (αy βy ) combination of signals; (d) image from αx βx step of the
phase cycle.
DDF (Fig. 4.6a), the incompletely dephased conventional signal (Fig. 4.6b) and
the stimulated echo contribution (Fig. 4.6c). Figure 4.6d shows the image produced
using the αx βx step of the phase cycle. Once again the signal due to the DDF is
strongest in the samples with the lower T1 relaxation times, while the stimulated
echo image shows the reverse contrast. The major contribution to Fig. 4.6b comes
from Sample E, which contains a small air-filled tube. This generates significant
local field variation that in some locations will cancel the effect of the modulation
of magnetisation due to the applied field gradients, leading to significant non-DDF
signal contributions from the terms like those described in Eq. 4.11.
Figure 4.7 depicts the variation with T R of the SNR per unit time of the DDFbased signal from the Agar phantom formed from the difference of consecutive echo
planar images acquired using Phase Cycle I with T E = 110ms and the modulating
gradient applied in the z-direction. The data values (points) were produced by
sampling the signal in regions of interest (ROI) encompassing each tube (A-D)
√
and scaling by T R. The continuous lines show data calculated using the known
values of T1 and fitted to the experimental data only by varying the average signal
amplitude. The experimental and calculated data are in good agreement, showing
a definite maximum in the obtainable SNR per unit time for T R ≈ 2.8T1 .
Figure 4.8a shows the variation with T E of the DDF-related signal from the four
Agar samples. Data were taken from ROI in the difference images produced from
consecutive echo planar image acquisitions using Phase Cycle I with T R = 2.5s and
the gradient applied in the z-direction. As expected from previous work, the signal
is largest when T E ≈ T2 . The variation of the difference in the measured signal
from Samples C and D (dotted line) with T E is shown in Fig. 4.8b, along with
the difference in the fitted curves (continuous line). The difference is largest when
4.4
Results
92
4
12
x 10
10
Signal/ TR1/2(a.u.)
B
8
A
C
6
D
4
2
0
0
1
2
3
TR(s)
4
5
6
7
√
Figure 4.7: Variation with T R of the DDF-based signal scaled by T R, produced using Phase
Cycle I. Experimental measurements from Samples A (crosses), B (points), C (filled circles) and D
(open circles), along with calculated fits A (dot-dashed line), B (continuous line), C (dashed line)
and D (dotted line).
T E is approximately equal to twice the average value of the T2 -relaxation times
of the two samples. Figure 4.8c shows the variation of the signal from Samples
A and D in the image formed by adding signals from consecutive images acquired
using Phase Cycle I with the gradient applied in the z-direction. This figure also
shows the signal measured for the same samples when the gradient is applied at
the magic angle and no phase-cycling is employed. In this situation the only signal
contribution is due to the stimulated echo, since no DDF is present (the Legendre
polynomial in Eq. 4.2 will be equal to zero). The similarity of the two curves and the
simple exponential decays that they follow, indicate that Phase Cycle I successfully
separates the stimulated echo and DDF-based signal.
Figure 4.9a shows the variation of the SNR per unit time of the DDF-based signal
from the human brain (grey matter, white matter and CSF) with T R, while Fig.
4.9b depicts the variation of the signal with T E. Experimental data were generated
√
from the average signal (scaled by T R in the case of Fig. 4.9a) in ROI of difference
images formed from the combination of data from 16 acquisitions. The behaviour is
similar to that seen in Figs. 4.7 and 4.8a, with maximum SNR per unit time being
generated when T R ≈ 2.8T1 and T E ≈ T2 .
4.5
Discussion
93
5
1.8
4
x 10
4
1.6
3.5
B
Signal Difference (a.u.)
1.4
Signal (a.u.)
1.2
C
1
0.8
0.6
D
A
3
2.5
2
1.5
1
0.4
0.2
0.05
x 10
0.1
0.15
0.2
TE (s)
0.25
0.3
0.5
0.05
0.1
0.15
0.2
0.25
0.3
TE (s)
4
14
x 10
12
Signal (a.u.)
10
8
6
4
2
0
0.05
0.1
0.15
0.2
0.25
0.3
TE (s)
Figure 4.8: (a) Variation of the DDF-based signal produced using Phase Cycle I with T E.
Experimental measurements from Samples A (crosses), B (points), C (filled circles) and D (open
circles), along with calculated fits A (dot-dashed line), B (continuous line), C (dashed line) and D
(dotted line). (b) Variation of the difference of signals from Samples C and D with TE. Measured
data are marked with circles, calculated data are shown with a continuous line. (c) Variation of
the summed signal from Sample A (open circles) and Sample D (crosses) with TE, produced using
Phase Cycle I, with the gradient applied in the z-direction. The continuous lines show the variation
of signal from the same samples produced with the gradient applied at the magic-angle and with
no phase cycling.
4.5
Discussion
In the absence of phase cycling, the SNR per unit time of the DDF-related signal
produced by the DQ-CRAZED sequence is maximised when T R = 2.05T1 . This
is significantly less than the optimal value of T R estimated by just considering the
magnetisation that recovers between experiments [84]. The signal from this magneti-
4.5
Discussion
94
4
2.4
14000
(a)
x 10
2.2
(b)
12000
2
10000
Signal (a.u.)
Signal/ (TR)
CSF
1.8
1/2
(a.u.)
CSF
1.6
8000
1.4
GM
6000
WM
1.2
WM
4000
GM
1
2000
0
1
2
3
4
5
TR (s)
6
7
8
0.8
0.05
0.06
0.07
0.08
0.09
TE (s)
0.1
0.11
0.12
√
Figure 4.9: (a) Variation with T R of the DDF-based signal scaled by T R, produced using Phase
Cycle I applied to the human brain. Experimental measurements/fitted curves from Cerebrospinal
fluid (CSF) [diamonds/dashed line], Grey Matter (GM) [points/dot-dashed line] and White Matter
(WM) [crosses/continuous lines]. (b) Variation with T E of the DDF-based signal produced from
CSF, along with grey and white matter, using Phase Cycle I.
− TTR 2
1
sation alone varies as (1 − e
) so that the SNR per unit time is maximised when
T R = 2.34T1 . The discrepancy results from the signal due to modulated magnetisation that is stored longitudinally during the T R period. This signal contribution,
−n T R
which is composed of terms varying as e T1 where n ≥ 2, has the same sign as
the signal produced by magnetisation that recovers during T R. Consequently the
maximum SNR per unit time occurs at a reduced value of
√
TR
,
T1
where this con-
tribution is larger. The results of the simulations and experiments also show that
when the DQ-CRAZED sequence is used without phase-cycling a significant amount
of stimulated echo signal can be produced and that this signal is maximised when
T R = 0.6T1 . In addition, the ratio of
+
Mdip
+
Mnon−dip
is approximately 7µ0 γ∆M0 T E at
T R = 2.05T1 , where the SNR per unit time of the DDF-related signal is maximised.
For grey matter at 3 T with T E = 70ms and the modulation gradient applied along
z, this ratio is approximately equal to 1.63, so that nearly 40 % of the total signal
is due to stimulated echo. It is clearly therefore necessary to employ phase cycling
or variation of the gradient direction [71, 72, 74, 76] across consecutive repetitions of
the sequence if a pure DDF related signal is to be measured.
When Phase Cycle I is employed, the DDF-related signal resulting from magnetisation that is stored longitudinally during TR has the opposite sign to that
produced by magnetisation that recovers during T R. Consequently the maximum
4.5
Discussion
95
SNR per unit time occurs at a larger value of T R (2.76T1 ) at which the first contribution is relatively reduced in magnitude. The experimental results and simulations
show that subtraction of consecutive signals acquired using this Phase Cycle allows
the stimulated
echo signal to be largely eliminated. With T R = 2.76T1 , the ratio
+
Mdip
of M +
in the subtracted signal is increased to 1400µ0 γ∆M0 T E , which takes
non−dip
a value of 3260 for grey matter at 3 T (T E = 70ms). The results indicate that
Phase Cycle I can be used to separate the signal resulting from the DDF form that
due to stimulated echo. It is interesting to note that when the value of T1 is known
the absolute value of M0 can be calculated directly from the ratio of these two
contributions.
The analysis used to identify the optimum values of TTR1 assumed that the decay
of modulated magnetisation in the TR period due to T1 relaxation is significantly
2
faster than that due to diffusion. This will be the case provided that km
DT1 ¿ 1.
Considering grey matter at 3 T (D = 10−9 m2 s−1 ) the rate of decay due to T1 is five
times faster than that due to diffusion when the pitch of the modulation is 500µm or
greater. For tighter modulation, the effect of diffusion will reduce the strength of the
stimulated echo signal relative to that due to the DDF, and shift the optimal value
of TR towards the value of 2.34T1 [84]. When the pitch of the modulation is less
than 100µm, diffusion largely eliminates any modulated longitudinal magnetisation
during the inter-experimental delay, when T R = 2.34T1 . In this case there is no
stimulated echo and analysis of Eqs. 4.10 and 4.13 indicates that a simple two step
phase-cycle (αx βx , α−x βx ) can be used to eliminate the dominant residual non-DDF
signal [72]. In fact this cycle also eliminates the term varying as e−i3km z in Eq. 4.8.
A 4-step phase cycle (αx βx , −αy βx , α−x βx , −α−y βx ) must be used if the term varying
as e−i2km z is to be eliminated as well. The SNR per unit time is also maximised for
T R = 2.34T1 when gradient cycling schemes [71, 72, 74, 78] which prevent refocusing
of magnetisation stored longitudinally during T R are employed. As noted above,
such sequences have some disadvantages in yielding ill-defined sensitivity to structure
and field inhomogeneity.
To account properly for the effect of diffusion on the steady state longitudinal magnetisation it is necessary to make a Fourier expansion of the steady-state
P
inkm s
(with cn = c∗n ), and then to solve the set of
magnetisation, m = n=N
n=−N cn e
2 2
coupled equations which separately include the effect of diffusion e−n km DT R on each
harmonic during the T R-period
´
1
2 2
−TR
sin(α) sin(β) (cn+1 + cn−1 ) e−n km DT R e T1
2
³
´
− TTR
(4.17)
+δn,0 M0 1 − e 1
³
cn = − cn cos(α) cos(β) −
4.5
Discussion
96
Although this is possible, we found that the complexity of the solutions meant
that this approach was not very practical.
Our analysis indicates that with and without phase-cycling the DQ-CRAZED
sequence yields the maximum SNR per unit time when the flip angles, α and β are
approximately 90o and 60o respectively - values which also maximise the signal for
long T R. This is a consequence of the relatively weak dependence of the steady-state
longitudinal magnetisation (Eq. 4.14) on flip angle for the high values of
TR
T1
that
maximise the SNR.
The experimental results confirm the previously reported finding that the DDFrelated signal is largest when the echo time, T E = T2 . They also show for the
first time that the sensitivity to small changes in T2 is greatest when T E = 2T2 .
Simple analysis indicates that for a small change ∆R2 in the relaxation rate, R2 ,
such that ∆R2 T E ¿ 1, the signal change is made 4/e ≈ 1.5 times larger by setting
T E = 2T2 rather than equal to T2 . This and the fact that T2 > T2∗ implies that fMRI
experiments based on using the fully refocused DQ-CRAZED sequence [72,73] should
employ echo times that are more than twice as long as those used in conventional
gradient echo experiments. In fact, recent work has shown that as might be expected,
the BOLD contrast in DQ-CRAZED experiments is increased by acquiring the signal
when its evolution under the action of field inhomogeneities is not fully refocused
[74, 85]. This can be achieved for example by employing equal evolution times on
either side of the 180o refocusing pulse of Fig. 4.1 [74]. In this case, the optimum
T E is a more complicated function which depends on the values of t1 , ∆R2 , T2 , ∆R2∗
and T2∗ [85, 86], and a topic which is explored in Chapter 5.
5
Simulation of the DQC in inhomogeneous media
97
Chapter 5
Simulation of the DQC in
inhomogeneous media
In this work a numerical method has been used to simulate the evolution of the
signal arising from a Double Quantum CRAZED (DQC) sequence applied to a
magnetically inhomogeneous sample. The aim was to explore the sensitivity of the
DQC sequence to the BOLD (Blood Oxygenation Level Dependent) effect. The
results indicate that: BOLD contrast is maximised when a 180o refocusing pulse
is employed, with a total evolution time of T2 and the signal allowed to dephase
nT ∗
2
where n = TT2∗ ; the contrast does not depend strongly on the
over a time of (n−1)
2
modulation length; the DQC produces a larger percentage, but smaller absolute,
signal change in activation, than a GE sequence.
5.1
Introduction
There has been much recent interest in exploiting the distant dipolar field to yield
novel forms of contrast in biomedical applications [50, 69, 72, 74]; in particular it has
been suggested that the dipolar field could provide contrast which is sensitive to
magnetic field inhomogeneities occurring on a length-scale set by the experimenter.
This would provide tuneable contrast which could be especially useful in fMRI,
where the sequence could be tuned to be sensitive only to susceptibility changes
in capillaries. Such changes should be better localised to the region of neuronal
activity than those due to larger vessels. Although a lot of emphasis has been
placed on this idea, it has never been clearly proven that long range dipolar fields
provide an effective mechanism for generating length scale contrast [87] [88].
5.2
Numerical Simulations in Inhomogeneous Media
98
The parameters that maximise the DQC signal have been extensively discussed
in the literature (and in Chapter 4) including: optimum T R [89], optimum T E [71]
and optimum flip angles [59]. Recently, due to interest in the application of this
technique to fMRI, parameters for maximal T2 contrast [89] (or see Chapter 4) and
optimum BOLD sensitivity [85] have also been discussed.
In the work described in this chapter we have used numerical simulations to
explore how the physiological/haemodynamic changes during brain activation may
cause a change in the signal detected using the Double Quantum CRAZED (DQC)
sequence (see Fig 5.1) and to identify which parameters maximise this signal change.
We have also tested whether the contrast can be tuned to be sensitive to the effect
of vessels of a particular size.
t1
90
τ
G
TE/2+t1
TE/2-t1
θ
180
Maximum
signal
2τ
G
t
Figure 5.1: Pictorial representation of the DQC sequence
Particular questions that need clarification include: what is the effect of varying
the length of the t1 period and the imposed modulation length on the values of
∆R2
DQC
and ∆R2+ DQC . Ultimately the question is how important is the effect of
the inhomogeneity gradients on the dipolar field coupling. Only using numerical
simulations can these effects be straightforwardly probed.
5.2
Numerical Simulations in Inhomogeneous Media
The evolution of the magnetization during the DQC sequence in inhomogeneous
media, consisting of isotropically oriented blood vessels embedded in grey matter,
was calculated by numerically applying the Bloch equations to an N x N x N matrix
representing the magnetisation of the sample. The Bloch equations included the
effects of relaxation, frequency shifts due to field inhomogeneities, the long range
dipolar field and diffusion. The first two terms are easy to calculate in real space,
5.2
Numerical Simulations in Inhomogeneous Media
99
the same is however not true for the dipolar field and diffusion. In real space, the
dipolar field, due to an inhomogeneous sample is a complex function of the spatial
distribution of magnetisation, M (~r) within the sample, but can be written as a local
function in k−space,
2
³
´
~ d (~k) = − µ0 3cos(β) − 1 3Mz (~k) − M
~ (~k)ẑ
B
(5.1)
3
2
where ẑ is a unit vector in the field direction, ~k describes the co-ordinates in
k−space and β is the angle between ẑ and ~k. This field introduces a non-linear
term into the Bloch equations, which is responsible for the formation of signal in
the DQC experiment. The effect of diffusion during time dt can also be calculated
in k−space [90] as
dM (~k)
= −Dk 2 M (~k)
dt
or in real space using a finite difference solution [91]
M (~ri,j,k , t + δt) − M (~ri,j,k , t)
=
δt
D
(M (~ri−1,j,k , t) − 2M (~ri,j,k , t) + M (~ri+1,j,k , t))
(δx)2
D
+
(M (~ri,j−1,k , t) − 2M (~ri,j,k , t) + M (~ri,j+1,k , t))
(δy)2
D
+
(M (~ri,j,k−1 , t) − 2M (~ri,j,k , t) + M (~ri,j,k+1 , t))
(δz)2
(5.2)
(5.3)
The real space method proved to be less time consuming to compute and far more
effective when restricted diffusion close to boundaries is considered (the k−space
method allows exchange of magnetization between the two environments, it is therefore not suitable for simulating diffusion in structured samples). To make the
problem tractable, the vessel walls were assumed to be impermeable. This assumption was implemented by imposing some restrictions on the simple finite difference solution. This involved simply inserting a delta function multiplier term
δSubst(ri,j,k ),Subst(ri0 ,j0 ,k0 ) that reduces diffusion attenuation when the ”substance” of a
neighbouring pixel is different (by imposing a reflection). Since substance changes
only occur at blood/tissue boundaries, this modification makes the vessel walls effectively impermeable. Its implementation is decribed here for a 1D case where
diffusion is considered to occur only in the x−direction,
5.2
Numerical Simulations in Inhomogeneous Media
100
Main Program
START
Parameter
Definition
Bo
2r
θ
Medium
Characterisation
χ
Evolution
M(r,t)
Sequence
Diffusion
Output data
Field Shift
Evolution
Inversion
Relaxation
3DFT
Bd(k)
3DIFT
Bd(r)
END
M(r,t+δt)=M(r,t)+(dM(r,t)/dt)δt
Figure 5.2: Flow chart representing the numerical simulation procedure
M (~ri,j,k , t + δt) − M (~ri,j,k , t)
=
δt
D
(M (~ri−1,j,k , t) − M (~ri,j,k , t)) δSubst(ri,j,k ),Subst(ri−1,j,k )
(δx)2
D
+
(−M (~ri,j,k , t) + M (~ri+1,j,k , t)) δSubst(ri,j,k ),Subst(ri+1,j,k )
(δx)2
(5.4)
The simulation process, which is described in the flow chart shown in Fig. 5.2,
therefore involved defining N x N x N matrices describing the spatial distribution
of magnetisation, field inhomogeneity and relaxation times. At each time step, the
incremental effects of field inhomogeneity and relaxation on the magnetization were
5.2
Numerical Simulations in Inhomogeneous Media
101
calculated from the Bloch equations and Eq. 5.4 was used to apply the effect of
~ (~r), and the resulting value of M
~ (~k)
diffusion. A 3DFFT was then applied to M
~ d (~k). An inverse 3D-Fourier
was used to evaluate the dipolar field in k−space, B
~ (~r) in real
transformation allowed the effects of the dipolar field to be applied to M
space. To simulate the effect of susceptibility differences between blood and tissue
on the DQC-signal, blood vessels were approximated as infinite cylinders. The
frequency shift due to such a vessel,
µ
¶
1
1
∆ω(~r) = − γB0 ∆χ cos2 (θ) −
2
3
¶
µ
1
Rcylinder 2 2
∆ω(~r) = − γB0 ∆χ
sin (θ) cos(2φ)
2
r
inside
(5.5)
outside
depends on the intra/extra vascular susceptibility difference (∆χ), the angle between
the vessel and the magnetic field, θ, the radius of the cylinder, Rcylinder , the angle
between the projection of the position vector and magnetic field vector in the plane
transverse to the vessel, φ, and the radial distance r from the centre of the cylinder.
Simulations were carried out using 32 x 32 x 32 matrices, with an effective blood
volume fraction of 1.3, 2.8 and 4.9 %. To simulate the BOLD effect, calculations
were carried out for vessels at 5 different orientations with respect to the applied
magnetic field (θ =
π
n
10
with n=1,..,5) and different values of blood/tissue suscepti-
bility difference (∆χ), corresponding to the typical values occurring in the absence
of contrast agents. Values of ∆χ in SI units of 0.45 ppm (oxygenation fraction,
Y =0.5), 0.36 ppm (resting state, Y =0.6), 0.27 ppm (intermediate state, Y =0.7)
and 0.18 ppm (activated state, Y =0.8) were used to simulate BOLD susceptibility
changes in the absence of a contrast agent (these values were calculated using the
relationship ∆χ = HCT (1 − Y )4πχhaemoglobin ).
The values used for the relaxation times were T2GM = 80ms and T2B = 100ms
while for longitudinal relaxation the values used were of T1GM = 1.2s and T1B = 2s
(the subscripts GM and B refer to grey matter and blood respectively). In order to
measure R2∗ DQC the sequence represented in Fig. 5.1 was simulated without inclusion of the 1800 refocusing pulse. In order to measure R2 DQC the 1800 refocusing
pulse was positioned at a time T E − t1 after the second RF pulse (T E was chosen
to be 2T2
GM ). Diffusion
−9 2 −1
0.7 × 10 m s
coefficient values of 1.45 × 10−9 m2 s−1 inside the vessel and
outside the vessel were employed, and the equilibrium magnetisa-
tion was set to 0.01Am−1 in both compartments. In order to simulate the various
cylinder orientations, while avoiding problems arising from variation of the partial
volume of the cylinders inside the simulated matrix, the option chosen was to rotate
5.2
Numerical Simulations in Inhomogeneous Media
102
the static magnetic field B0 (and, as a consequence, also the direction in which the
modulation was applied) with respect to the ẑ direction of the matrix.
The results of simulations employing different vessel orientations were then integrated with the appropriate weighting (sin θ) in order to yield the signal of a
voxel containing vessels with an isotropic distribution of orientations. Chapter 4
described the risk of generating signal that is not due to the long-range dipolar field
effect after the application of the second gradient pulse, due to the effect of field
inhomogeneities. Such signal can be cancelled by the application of a phase cycle of
the first RF pulse (x,-x) (the results shown in the next section all refer to the summation of the signal from successive simulations with different phases of the first RF
pulse). Such a phase cycle leaves the dipolar signal unchanged, whilst the unwanted
rephased signal cancels out. This phase cycle had to be applied in the simulations
due to the reduced size of the matrix (32 voxels) with respect to the modulation
length (32.0, 16.0, 10.7, 8.0, 6.4 and 5.3 voxels) and also the fact that with the
rotation of the direction at which the modulation was applied with respect to the
z-axis of the cube, a perfect cancellation of the signal was not achieved through
summation over the voxel dimensions.
5.2.1
Results
As a preliminary test of the simulation procedure, it was applied to the previously
studied case of a single vessel aligned along the y−direction, with the B0 field applied
along z, and the modulation applied along y. The simulated image shows the 6-fold
symmetric pattern expected [92] (see Fig. 5.3), in contrast to the 4-fold symmetric
pattern of the frequency shift [93].
Figure 5.3: Simulated (a) and experimental [92](b) images showing the effect of the presence of
a tube of 6 mm diameter with a susceptibility difference with respect to the surrounding medium
of 1 ppm
5.2
Numerical Simulations in Inhomogeneous Media
70
No Diffusion
60
signal amplitude (a.u.)
signal amplitude (a.u.)
70
50
40
30
20
10
0
0
40
30
20
10
0
0
8
No Diffusion
4
2
0
50
0.2
6
-1
k = 19 mm
m
-1
km= 32 mm
-1
km= 58 mm
-1
k = 78 mm
m
-1
km= 98 mm
-1
km=117 mm
60
contrast (a.u.)
contrast (a.u.)
8
0.05
0.1
0.15
evolution time (s)
With Diffusion
0.05
0.1
0.15
evolution time (s)
0.2
With Diffusion
-1
km= 19 mm
-1
k = 32 mm
m
-1
k = 58 mm
m
-1
k = 78 mm
m
-1
k = 98 mm
m
-1
km=117 mm
6
4
2
0
0.05
0.1
0.15
evolution time (s)
0.2
0
103
0
0.05
0.1
0.15
evolution time (s)
0.2
Figure 5.4: Signal evolution without (a) and with (b) diffusion for different values of km , with a
susceptibility difference of 0.18ppm (continuous lines) and without any perturber (dashed lines).
Signal difference without (c) and with (d) diffusion for different values of km , with a susceptibility
variation from 0.18ppm to 0.36ppm
To test for any length-scale dependence of the DQC signal, modulation lengths
were varied from sizes smaller than the diameter of the vessel to significantly larger.
Figure 5.4 shows the effect on the signal evolution in a GE-DQC sequence of
varying the modulation length from 320 µm to 6.5 µm (corresponding to km =
2π/(modulation length) values from 19 to 137 mm−1 ) in an environment of isotropically oriented cylinders of diameter 9 µm without diffusion (a) and with diffusion
(b), with (continuous lines) and without (dashed lines) the effect of field inhomogeneity. The results showed no significant dependence on modulation length, apart
from the extra signal attenuation due to diffusion, as the applied magnetic gradient
strength is increased (see Fig. 5.4). Figures 5.4(c) and 5.4(d) show the variation
of the difference in the signal produced with intra/extra vascular susceptibility differences of 0.18 and 0.36 ppm (corresponding to BOLD susceptibility changes) as
a function of echo time. The lack of any strong dependence on modulation length
in the absence of diffusion implies that the DQC contrast cannot be tuned to be
sensitive to susceptibility changes in vessels of a particular size.
The importance of the preparation time t1 was also evaluated, the results shown
in Fig. 5.5 indicate that increasing t1 leads to a loss of signal as a result of T2 decay
5.2
Numerical Simulations in Inhomogeneous Media
104
during the t1 period (Fig. 5.5a). Diffusion along the modulation direction (Fig.
5.5b) will also reduces the contrast (Fig. 5.5c and d) as t1 increases.
This evaluation of the signal dependence on field inhomogeneities was qualitative;
to quantify the effect of the field inhomogeneities it is important to parameterise the
dependence of the relaxation rates ∆R2 DQC and ∆R2∗ DQC on the volume fraction
of cylinders and their susceptibility, for the different values of km and t1 . This
evaluation allows a quantitative comparison with the relaxation processes occurring
due to field inhomogeneities in spin echo and gradient echo sequences.
Since there is no analytical expression for the signal evolution following a DQC
preparation when diffusion is present, relaxation rates were measured by comparing
the signal simulated in the absence of the any susceptibility difference between blood
and tissue, to that occurring when the field perturbation was present. The effective
relaxation rates were then given by
!
Ã
∆R2 (V, χ, t1 , km )DQC
70
No Diffusion
60
signal amplitude (a.u.)
signal amplitude (a.u.)
70
SignalSEDQC (V, χ, t1 , km , t)
= ln
/(2t1 + t)
SignalSEDQC (V, 0, t1 , km , t)
50
40
30
20
10
0
0
40
30
20
10
0
0.2
0
8
No Diffusion
6
4
2
0
50
contrast (a.u.)
contrast (a.u.)
8
0.05
0.1
0.15
evolution time (s)
t1=00ms
t1=11ms
t1=23ms
t1=34ms
t1=46ms
t1=57ms
t1=69ms
t1=80ms
With Diffusion
60
0.05
0.1
0.15
evolution time (s)
0.2
With Diffusion
t1=00ms
t1=11ms
t1=23ms
t1=34ms
t1=46ms
t1=57ms
t1=69ms
t1=80ms
6
4
2
0
0.05
0.1
0.15
evolution time (s)
0.2
0
0
0.05
0.1
0.15
evolution time (s)
0.2
Figure 5.5: Signal evolution without (a) and with (b) diffusion for different values of t1 , with a
susceptibility difference of 0.18ppm (continuous lines) and without any perturber (dashed lines).
Signal contrast without (c) and with (d) diffusion for different values of t1 , with a susceptibility
variation from 0.18ppm to 0.36ppm (continuous lines)
5.2
Numerical Simulations in Inhomogeneous Media
0.6
∆R
105
1.3
=α V χ
2 DQC
2.5
No Diffusion
2
2
1.5
1.5
α
α
2.5
1
1
0.5
0.5
0
0
0.02
0.04
t (s)
0.06
0
0.08
0
0.02
0.04
t (s)
1
0.08
2.5
No Diffusion
t =00 ms
1
t1=11 ms
t =23 ms
1
t1=34 ms
t1=46 ms
t =57 ms
1
t 1=69 ms
With Diffusion
2
1.5
1.5
α
2
α
0.06
1
2.5
1
0.5
0
-1
km= 19 mm
-1
k = 32 mm
m
-1
km= 58 mm
-1
k = 78 mm
m
-1
km= 98 mm
-1
km=117 mm
-1
km=137 mm
With Diffusion
1
0.5
0
50
100
k (mm-1 )
0
150
0
m
50
100
k (mm-1 )
150
m
Figure 5.6: Plots showing the proportionality constant, α, using the relationship ∆R2 DQC =
αV 0.6 χ1.3 as a function of t1 and km with (right) and without (left) diffusion. The grey line
represents the value found when considering a conventional spin echo sequence (note that in the
absence of diffusion R2 SE = 0).
Ã
∆R2∗ (V, χ, t1 , km )DQC = ln
!
(5.6)
SignalGEDQC (V, χ, t1 , km , t)
/(2t1 + t)
SignalGEDQC (V, 0, t1 , km , t)
where it is assumed that the relaxation processes occur over a time, 2t1 + t. By
calculating relaxation in this manner, the effect of diffusion attenuation due to the
externally imposed modulation is eliminated.
The values of ∆R2
DQC
and ∆R2∗
DQC
were then fitted by minimization of the
mean square deviation from a function of the form αV β χγ (which is a simple model
characterising the relaxation dependence on the volume fraction, V , and susceptibility difference, χ [94]). Because of the long times needed to perform the simulations,
it was only possible to calculate relaxation rates for 12 different combinations of V
and χ, which makes a 3-parameter model seem excessive. Because the final aim is
to be able to compare the relaxation rates with those found for gradient echo and
spin echo experiments, the values of β and γ were set to those found for a simiπ
lar distribution of infinite cylinders (θ = n 10
, with n=1,...,5). Using the simulated
5.2
Numerical Simulations in Inhomogeneous Media
∆ R∗
1.1
1.1
=α V χ
2 DQC
3.2
3.2
No Diffusion
km= 19 mm
-1
km= 32 mm
-1
=
58
mm
km
-1
k = 78 mm
m
-1
km= 98 mm
-1
km=117 mm
-1
km=137 mm
3
α
α
With Diffusion
-1
3
2.8
2.6
2.4
2.8
2.6
0
0.02
0.04
t (s)
0.06
2.4
0.08
0
0.02
0.04
t (s)
1
3.2
3.2
No Diffusion
0.08
With Diffusion
t1=00 ms
t =11 ms
1
t1=23 ms
t1=34 ms
t1=46 ms
t1=57 ms
t 1=69 ms
3
α
α
0.06
1
3
2.8
2.6
2.4
106
2.8
2.6
0
50
100
-1
k (mm )
150
2.4
0
m
50
100
-1
k (mm )
150
m
Figure 5.7: Plots showing the proportionality constant, α, using the relationship ∆R2∗ DQC =
αV 1.1 χ1.1 as a function of t1 and km with (right) and without (left) diffusion. The continuous
lines represent the values obtained by the fitting, whilst the dashed lines represent the standard
deviation of those values. The grey line represents the value found when considering a standard
gradient echo sequence.
data described in Chapter 7 the dependence found for spin echo and gradient echo
sequences were ∆R2
SE
= 0.7V 0.6 χ1.3 and ∆R2∗
GE
= 3.1V 1.1 χ1.1 respectively. The
values for α then obtained for different preparation times, t1 , and modulation, km ,
are shown in Figs. 5.6 and 5.7.
Figure 5.6 reveals a clear trend, that the relaxation time ∆R2
DQC
increases
relative to the value obtained in a SE sequence as both the preparation time, t1 , and
the modulation length increase. The dependence on the modulation length is not
surprising, since the tighter the modulation, the less inhomogeneities are present
at the length scale over which the dipolar field ”acts”, making the signal more
insensitive to dephasing and diffusion attenuation. Although the results apparently
imply that the use of a long t1 would be a sensible approach to maximising relaxation
rate sensitivity, it should be remembered that the increase in sensitivity to relaxation
will not necessarily translate into a greater signal change due to the enhanced effects
5.3
Sequence Parameter Optimisation for Maximum Contrast
107
of diffusion and relaxation during the preparation time (see Fig. 5.5), which will
reduce the absolute signal.
In contrast, the ∆R2∗ DQC results shown in Fig. 5.7 do not show such a clear
dependence on either km or t1 , and although the value of ∆R2∗ DQC is consistently
less than that found from simulating conventional gradient echo experiments, the
difference is only approximately 10%.
5.3
Sequence Parameter Optimisation
for Maximum Contrast
Once the relaxivity has been characterised, we can then consider how the DQC
sequence may be optimised to give the largest changes in the signal strength due to
BOLD effects.
To first order, considering that the gradient induced modulation dominates the
effect of field inhomogeneities, the dipolar field can be written as a simple function
of the local magnetisation [44] (see Appendix B) such that
¶
µ
~ (~s)
~ d (~s) = µ0 ∆ Mz (~s)ŝ − 1 M
B
3
where ẑ is a unit vector in the direction of the main magnetic field, B0 ,
(5.7)
´
1³
3(ŝ · ẑ)2 − 1
(5.8)
2
and ŝ is a unit vector in the s-direction along which the magnetisation is modulated. In the linear regime, in which the effect of the dipolar field, is small enough
for the approximation e−iγµ0 Mz t ≈ 1 − iγµ0 ∆Mz t to be valid, the signal arising from
the DQC sequence shown in Fig. 5.1 depends upon,
∆=
Signal(t) ∝ te−R2
DQC (t+2t1 )
+
e−R2
DQC |2t180 −t+2t1 |
(5.9)
where t1 is the period separating the first two RF pulses, t is the total evolution
time starting at the second RF pulse and t180 is the time between the second RF
pulse and the 1800 refocusing pulse. Finally, the conventional notation is used for
relaxivity, where R2 = T12 and R2∗ = R2+ +R2 = T1∗ . (2t180 −t+2t1 ) can be understood
2
as the effective dephasing time. BOLD contrast arises when a certain region of the
brain undergoes a small change in relaxation rate. The attention will be focused
in the variation of R2+ as it is likely to have the greatest contribution. The change
in signal arising from an infinitesimal change in R2+ is the partial derivative of the
signal,
5.3
Sequence Parameter Optimisation for Maximum Contrast
contrast(t, t180 , t1 , R2
+
DQC , R2 DQC )
∂Signal(t)
∂R2+
∝ |2t180 − t + 2t1 |te−R2
∝
+
e−R2
108
(5.10)
DQC (t+2t1 )
DQC |2t180 −t+2t1 |
By solving the following system of equations,
∂contrast(t, t180 , t1 , R2
∂t180
∂contrast(t, t180 , t1 , R2
∂t
∂contrast(t, t180 , t1 , R2
∂t1
+
DQC , R2 DQC )
+
DQC , R2 DQC )
+
DQC , R2 DQC )
=0
=0
(5.11)
=0
A global maximum is not achieved, because of a monotonic dependence on t1 .
If t1 is considered to be zero (a reasonable assumption when the magnetisation
is rapidly modulated so as to minimise signal loss), a simple analytical approach
suggests that sensitivity to variations in R2+ is optimised at an evolution time t = 2T2∗
T2
if T2 < 2T2∗ or at an evolution time t = T2 , with an effective dephasing time of (n−1)
,
where n = TT2∗ when T2 > 2T2∗ . In the latter case the conditions mean that, although
2
we may have a longer effective evolution time, the refocusing pulse yields a smaller
effective dephasing time.
OF F −SignalON
The consequences in terms of the maximum fractional change, ( SignalSignal
),
OF F
in a BOLD experiment can then be analytically evaluated. For a refocused DQC experiment with the echo time optimised for sensitivity to changes in R2 DQC (optimum
echo time is 2T2 ) the fractional change in the signal will be given by 2∆R2 DQC T2 .
If the DQC sequence is optimised for a variation of R2+ (as discussed in the previous
paragraph) the fractional contrast will be given by
2∆R2
DQC T2 DQC
+ ∆R2+ DQC
n
T2∗ DQC
n−1
(5.12)
This values can be compared to those obtained with standard spin or a gradient
+
∗
T2∗ = n1 ∆R2GE T2 + ∆R2GE
echo experiments which are ∆R2SE T2 and ∆R2GE
T2∗
respectively1 . In both cases the fractional change is expected (assuming there are
no major changes in relationship of ∆R2∗ and ∆R2 measured using the different
techniques) to be greater with the DQC based sequences.
1
The reason for the appearance of the subscript SE and GE is because these are the values as
measured using a spin echo and a gradient echo respectively
5.4
Discussions
109
In a set of simulations, based on the methodology described in Section 5.2, the
effect of varying the time of application of the 1800 pulse was evaluated, and an
additional fixed R2+ effect was included in order to simulate an environment in
which T1 = 1.2s, T2 = 0.13s [95, 96] and T2∗ = 0.035s, corresponding to grey matter
at 3 T. The time of application of the 1800 refocusing pulse, t180 , was extensively
varied in order to choose the value that maximises the BOLD contrast.
5.3.1
Results
The results of this set of experiments are shown in Fig. 5.8. Figure 5.8a shows how
the integrated signal evolves with time for the resting and active values of ∆χ when
the 180o refocusing pulse is applied at timesof 30, 65 and 110 ms after the second
RF pulse, with t1 = 0. As expected (see Chapter 4), both signals were maximised
when the signal was refocused after an evolution time, approximately equal to T2
(130 ms), by applying the 180o RF pulse after a time t180 = 65ms.
Signal Evolution
160
120
Signal (a.u.)
t180
t180
t180
7
= 110ms
= 80ms
= 60ms
Signal Change(a.u.)
140
6
100
5
80
4
60
3
40
2
20
1
0
Difference
8
∆χ=0.36ppm
∆χ=0.18ppm
0
0.1
0.2
time (s)
0.3
0.4
0
0
0.1
0.2
time (s)
0.3
0.4
Figure 5.8: Simulations for varying times of application of the refocusing pulse. (a) It is possible
to see the evolution of the signal and the effect of a refocusing pulse. (b) The difference between
activated and resting state signal is plotted, confirming the expected position for the maximum T2∗
contrast. Arrows indicate the position of the 1800 refocusing RF pulses.
The optimum sensitivity to T2∗ change was confirmed in the simulations, Fig. 5.8,
where the maximum contrast for T2 = 130ms and T2∗ = 35ms such that n = TT2∗ =
2
3.71 was obtained at an evolution time of 130ms with a refocusing pulse applied at
80ms, giving an effective dephasing time of 50 ms which is approximately equal to
T2
.
n−1
5.4
Discussions
The idea that the long range dipolar field could allow the sensitivity to inhomogeneities generated by susceptibility perturbations occurring on different length
5.4
Discussions
110
scales to be tuned by variation of the modulation length seems not to be feasible.
Nevertheless it was possible to optimise some of the parameters so as to provide
maximum signal contrast in functional experiments. In addition it was demonstrated that, not only can DQC sequences be made more sensitive to the effect of
field inhomogeneities in refocused sequences than in conventional spin echo experiments, due to the different temporal dependence (te−tR2 vs e−tR2 ), the contrast will
also be increased due to the greater sensitivity to field inhomogeneities of ∆R2
compared with ∆R2
SE .
DQC
6
Calculations of Field Inhomogeneity
111
Chapter 6
Calculations of Field
Inhomogeneity due to Spatial
Variation of Magnetic
Susceptibility
6.1
Introduction
Magnetic susceptibility is an important source of contrast in MRI, as in the case
of functional magnetic resonance imaging (fMRI), where the variation of magnetic
susceptibility inside blood vessels due to changes in oxygen concentration is responsible for the BOLD (Blood Oxygenation Level Dependent) effect [97]. This effect
can be magnified by increasing the applied static magnetic field, the drawback being
that as well as acting as a useful source of contrast, magnetic susceptibility differences on a coarser length-scale are also a very powerful source of image artifacts.
Such artifacts include image distortion, that is a particular problem in echo planar
imaging (EPI) [98], and signal drop-out due to intra-voxel dephasing that occurs in
all gradient echo images [99]. The increasingly large magnetic fields used in MRI
exacerbate the magnetic field inhomogeneities induced by magnetic susceptibility
variation within the body and make it increasingly important to be able to quantify
the induced field variation, and how it changes as a result of motion and physiological
effects.
To indicate the importance of such effects, we briefly consider a 64 × 64 echo
planar image acquired in 40 ms, via a gradient echo EPI sequence with a 30 ms
echo time, T E, and a slice thickness, ∆s, of 6 mm. With these parameters, the
6.1
Introduction
112
presence of a local resonant frequency offset of 25 Hz would be enough to distort
the image by one pixel in the phase-encode direction. At 3 T, this frequency offset
is much smaller than that which is found next to air/tissue boundaries, such as
those occurring in the sphenoid sinus, making the task of accurately locating the
site of brain activation in nearby regions difficult in fMRI experiments. In addition,
a susceptibility-induced field gradient, g, assumed to occur in the slice direction,
will cause a modulation of the signal intensity by a factor of sinc(γgT E ∆s
). For the
2
parameters listed above, the signal is nulled (corresponding to the first zero of the
sinc function) for a gradient strength of just 55 Hzcm−1 , whereas it is found that
induced gradients at 3 T can be as large as 270 Hzcm−1 in brain regions close to
the sphenoid sinus. Although it is possible to optimize the shim settings so as to
eliminate field variation in such regions, movement may lead to re-introduction of
significant field inhomogeneity. During an fMRI experiment, such movement causes
an unwanted increase in the variance of the signal [100] resulting from changes in
the distortion and signal attenuation. It is consequently important to be able to
characterize both the field variation generated by a particular spatial distribution
of magnetic susceptibility in the human body and the way in which the induced
magnetic field changes with body movement during an MRI experiment.
The most extensive air/tissue boundary inside the human body is found in the
lungs, making these organs extremely difficult to image using magnetic resonance.
Not only do susceptibility effects create significant image artefacts near to and within
the lungs, they also lead to the generation of respiration induced resonance offsets
(RIRO) which occur as far away from lungs as the head [101, 102]. Field variation
due to RIRO adds extra variance to the MR signals in fMRI, which may result from
both the effect of lung volume changes and that of variation in oxygen concentration
in the lungs and airways. Understanding such effects is therefore of some importance.
Most techniques which have previously been used to calculate susceptibilityinduced magnetic field variation address the problem in real-space using iterative
methods [103–107]. These methods generally employ a finite-difference appproach
in which an initial solution corresponding to a homogeneous magnetic field, B0 , is
assumed, and the following differential equation is then solved iteratively
∂Φ
= ∇ · (µr ∇Φ)
∂T
(6.1)
~ = −µ0 µr ∇Φ), T is a pseudowhere Φ is the magnetic scalar potential (so that B
time and µr is the spatially varying, relative permeability, which is related to the
volume magnetic susceptibility, χ via µr = 1 + χ. Iteration is stopped when ∂Φ
∼ 0.
∂T
6.1
Introduction
113
Once this convergence has been achieved, the effect of the sphere of Lorentz [108]
is introduced at each point. The sphere of Lorentz is a local susceptibility construct which was introduced in electrostatics by Lorentz [108] and was first applied
to magnetic resonance by Dickinson [109]. It is used to account for the fact that
the large susceptibility shift effects due to the host molecule and the nearest solvent shells effectively cancel or are taken into account in including the effect of the
chemical shift. In calculating the susceptibility-induced fields via Maxwell’s equations, the resonant nucleus is therefore considered to be sited inside an imaginary
sphere (sphere of Lorentz) of zero magnetic susceptibility [110], which should be
large enough so that the effects of any external molecules can be viewed as that of
a homogeneous medium [111] .
Recently, Jenkinson et al. [112] [113] introduced a convolution-based method
of calculating susceptibility-induced magnetic field inhomogeneity, where an approximate analytical solution is calculated for a cube of material of unit magnetic
susceptibility. The resulting solution is then convolved with the spatial pattern of
susceptibility variation throughout the sample. This can be accomplished by numerically Fourier transforming the analytical solution for the field generated by the cube
shape and multiplying it by the Fourier transform of the magnetization distribution,
and then applying an inverse Fourier transform to produce the real-space field variation. The effect of the sphere of Lorentz must then be added to the resulting field
at each spatial position.
Recently, two groups have introduced an alternative method of calculating magnetic field perturbations via Fourier analysis of the heterogeneous magnetic susceptibility distribution [114, 115]. Salomir et al. [115] derived this approach from
analysis of Maxwell’s equations, leading to the generation of a differential equation
for the contribution to the magnetic scalar potential resulting from the magnetic
susceptibility of the object. An alternative derivation based on analysis of the
three-dimensional Fourier transform of the susceptibility distribution and an analytical solution of the Fourier transform of the point-dipole field was introduced
by Marques and Bowtell [114] and is further described here. Generally, the Fourier
method is straightforward to implement, allows rapid calculation of the field variation in heterogeneous samples, where a three-dimensional map of the susceptibility
variation is available and naturally includes the effect of the sphere of Lorentz. The
method was initially tested on simple arrangements (cylinders and sphere) for which
the analytical form of the induced magnetic field variation is known. Subsequently
it has been employed in evaluating the spatially varying fields generated in a human
head model due to the intrinsic variation of magnetic susceptibility with tissue type
6.2
Theory
114
and to evaluate the changes in these fields due to small head rotations. The method
was also used to model RIRO effects occurring in the head, allowing a separate
evaluation of the effects of lung expansion and changes in oxygen concentration in
the lungs.
6.2
Theory
Many molecules possess no intrinsic magnetic moment because of the net cancellation of both the orbital and spin angular momentum of the electrons. However,
in the presence of an applied magnetic field, precession of the orbital moments is
induced, leading to the generation of an extra magnetic moment that is oppositely
aligned to the applied field. This is known as a diamagnetic effect. Some other
molecules in which the net angular momentum does not cancel, possess an intrinsic
dipole strength that is larger than the induced diamagnetic moment. In the presence
of an external field, these molecules on average adopt the least-energy state, with the
dipole oriented along the applied field. Such materials are said to be paramagnetic.
To quantify these effects it is useful to use the volume magnetic susceptibility, χ,
~ = χH,
~ where M
~ is the induced magnetization and H
~ is the applied
such that M
magnetic field in Am −1 . χ of diamagnetic materials is negative while paramagnetic
materials have positive values. Biological tissues typically have χ values similar to
that of water which is approximately equal to −9 × 10−6 .
If a sample described by a magnetic susceptibility distribution, χ(~r), is exposed
to a strong external magnetic field, B0 , applied along the z -direction, only the z component of the induced magnetization is significant and can be written as
Mz (~r) ≈ χ(~r)
B0
B0
B0
= χ(~r)
≈ χ(~r)
µ0 µr (~r)
µ0 (1 + χ(~r))
µ0
(6.2)
where the approximations are based on the assumption that χ << 1.
The magnetic field generated at a position, ~r, by the resulting magnetization distribution is a non-local function, representing the sum of the dipolar fields generated
~ (r~0 ) , and is given by:
by each element of the magnetization distribution, M


~ (r~0 ) · (~r − r~0 )
M
µ0 Z
1
~ (r~0 ) d3 r~0
~
3
(~r − r~0 ) − M
Bd (~r) =
0
3
0
2
~
~
4π | ~r − r |
| ~r − r |
(6.3)
Deville et al. [44] showed that this complex, non-local expression becomes simple
and local when calculated in the Fourier domain (k -space), using the rotating frame
of reference:
6.2
Theory
115
´
µ0 (3 cos2 β − 1) ³ ~ ~
~
~
Bd (k) =
M (k) − 3Mz (~k)ẑ
(6.4)
3
2
~ (~k) and Mz (~k) are the three dimenwhere ~k is the coordinate position in k -space, M
~ (~r) and Mz (~r), ẑ is a unit vector in the z -direction
sional Fourier transforms of M
and β is the angle between the direction of the main magnetic field and ~k, such that
3 cos2 β − 1 = 3
kz2
− 1.
kx2 + ky2 + kz2
(6.5)
The derivation of Eq. 6.4, which has not to our knowledge been previously published,
is detailed in the Appendix B. It should be noted that the transverse magnetization
referred to in Eq. 6.4 is measured in the rotating reference frame, and is of no
relevance to the problem under consideration here, where only z -magnetization is
present, and only the z -component of the dipolar magnetic field, Bdz , is important.
Under these circumstances Eq. 6.4 reduces to:
~
~ d (~k).ẑ = − µ0 Mz (k) (3 cos2 β − 1).
Bdz (~k) = B
(6.6)
3
Substitution of the Fourier transform of Eq. 6.2 into Eq. 6.6 followed by inverse
Fourier transformation yields the field perturbation produced by χ(~r).
Geometry/Compartment
sphere
internal
χe +χe χi
B0 3+2χ
e +χi
external
B0 χ3e
infinite cylinder (⊥B0 )
e −χi +2χe χi
B0 3χ6+3χ
e +3χi
B0 χ3e
infinite cylinder (kB0 )
B0 χ3i
B0 χ3e
Table 6.1: Uniform field shifts in internal and external compartments, including the effect of the
sphere of Lorentz, for different geometries.
An interesting property of this method is that it directly includes the effect of
the sphere of Lorentz in Eq. 6.4. This results from the fact that the integrand of the
first part of Eq. B.5 does not converge as ρ~ → 0, and consequently a Cauchy limit
has to be considered. Such a limit, is equivalent to the use of the sphere of Lorentz,
that is an empty spherical region of infinitesimal radius centred at each point.
In evaluating Eq. 6.6 the value of the Fourier transform of the field at ~k = 0,
which effectively describes the average value of the field [116], is not well defined.
This results from the fact that cos β is undefined at ~k=0. Here, we assume that the
object of interest is surrounded by a region of infinite extent whose susceptibility is
6.3
Method
116
equal to χe , and then set the value of Bdz (~k = 0) equal to − B03χe . This makes the
average field offset equal to that found in a homogeneous medium of susceptibility,
χe , including the effect of the sphere of Lorentz.
6.3
Method
The first step in implementing this method is to generate a three dimensional matrix over which the spatial distribution, χ(~r), of the magnetic susceptibility is represented. The induced magnetization distribution, Mz (~r), can then be calculated
at each point via Eq. 6.2. Subsequently, by applying a three-dimensional, discrete,
fast Fourier transform (3DFFT), Mz (~k) is obtained and Eq. 6.6 can then be used
to calculate Bdz (~k). Finally, the magnetic field perturbation Bdz (~r) due to the distribution of spatially varying magnetic susceptibility, χ(~r), can be calculated via an
inverse 3DFFT of Bdz (~k).
In carrying out the 3DFFT’s, care must be taken to avoid problems due to the
finite extent of the real and k -space domains considered. In particular, this means
that the three dimensional matrix over which the values of χ(~r) are defined should
be of significantly larger spatial extent than the actual region of interest.
As an initial evaluation of the method, the results obtained for the cases of
a sphere of different magnetic susceptibility from its surroundings and of infinite
cylinders (both parallel and perpendicular to the magnetic field) with a similar
difference of internal and external magnetic susceptibilities were compared to the
known analytic forms of the field. Once the method had been shown to work for
these simple geometries, susceptibility effects in the human body were studied using
the digitised HUGO body model (Medical VR Studio GmbH, Lorrach). Values of
the magnetic susceptibility of different tissues in the head previously described in
the literature [106] were used to calculate the susceptibility-induced field variation
in a typical human head. For these studies a spatial resolution of 2 × 2 × 2 mm3
was used and a portion of the HUGO model spanning the head and neck (26 cm
axial extent) only was considered. This model was sited at the centre of a 256 ×
256 × 256 matrix which spanned 51.2 cm in each direction.
The effects of small rotations of the head were also evaluated. Rotation of a
rigid body relative to the applied field can be straightforwardly incorporated into
the simulation via: (i) rotation of the object in real space; or (ii) rotation of the
direction of the main magnetic field. The second method was found to be the
most practical, as changes in partial volume effects are introduced by rotation of
6.3
Method
117
50
50
45
45
40
40
35
35
30
30
25
25
20
20
15
15
10
10
5
5
10
20
30
40
50
5
10
15
20
25
30
Figure 6.1: (a)Coronal and (b) axial slices of the body model used to simulate RIRO effects are
shown. Varying red hues are used to represent different tissue types which were associated with
different values of magnetic susceptibility. Field plots shown in Fig. 6.8 were calculated along the
vertical black line. Colored lines indicate positions where duplicate planes of susceptibility values
were inserted to simulate lung expansion: yellow = abdominal; green = thoracic.
the digitised object and in the case of field rotation, Mz (~k) does not need to be
recalculated. A clockwise rotation of the magnetic field by an angle θ about the x axis can be introduced when calculating the susceptibility-induced field in k -space
(Eq. 6.6), by using the substitution
Ã
!
3 cos2 β − 1
1
(kz cos θ − ky sin θ)2
=
3
−1
2
2
kx2 + ky2 + kz2
(6.7)
This has the advantage of keeping all calculations in the co-ordinate system defined
relative to the object, rather than fixing the co-ordinate system with respect to the
applied field. Here, we considered rotation about medio-lateral (pitch) and anteriorposterior (roll) axes. Rotation about the superior-inferior axis (yaw), assumed to
be parallel to the applied field, produces no change in the induced field [100] as
can be readily deduced from Eq. 6.6. In both cases, we evaluated the incremental
distortion introduced by changes in the magnetic field and the change in attenuation
resulting from variation of the local, through-slice, gradients.
6.4
Results
118
The field change in the head due to the movement of the lungs and chest cavity
during respiration (RIRO) was also investigated using the HUGO body model. Abdominal breathing due to movement of the diaphragm was simulated by shifting the
bottom of the lungs down by 1.5 cm [117], while thoracic breathing was simulated
by expanding the lungs by 0.76 cm anteriorly and laterally (see Fig. 6.1). These
alterations corresponded to changes in lung volume of 420 and 630 ml, respectively.
Expansion of the lung-space in the model was accomplished by inserting portions of
one or two duplicate slice(s) of susceptibility values at appropriate positions. The
effect of a 3% change in the oxygen concentration (typical oxygen variation for an
adult during tidal breathing) in the main respiratory airways, trachea and nasal
sinus, corresponding to a volume of 150 ml [118] was also considered, as was the
effect of assuming a quadratic variation in oxygen concentration with axial position
in the lungs (varying from 140 mmHg at the apex to 80 mmHg [118]) at the base.
These studies were performed on two different models of the susceptibility distribution: (i) only two magnetic susceptibility compartments, consisting of the air-filled
lungs and a surrounding region of water of infinite extent, were considered; (ii) all
tissues were given a susceptibility value (as found in the literature [106]), such that
the external surface of the body is properly represented. For these simulations, a
spatial resolution of 7.6 × 7.6 × 7.6 mm3 was used, along with a 2563 matrix size.
In the second model, the body was defined over an axial extent of 58 cm. Previous
experimental studies have shown that the pattern of field variation induced in the
human brain by RIRO is well modelled in terms of the field produced by a single
magnetic dipole located within the lungs [101, 102]. To evaluate the results of the
numerical analysis we consequently fitted the pattern of RIRO induced in the brain
to the field variation due to a single dipole. The anterior-posterior and superiorinferior position of the dipole along with its strength were variable parameters in the
fitting process (it was assumed that the apparent dipole position would be located
at the mid-point of the body along the medio-lateral direction).
Calculations were implented on a Sun Workstation (E280R with a 900 MHz
UltraSPARC-III+ processor and 4096MB memory). The programming language
used was Fortran 77.
6.4
Results
Field plots produced from both analytical and numerical solutions for the case of
infinite cylinders (oriented parallel and perpendicular to the applied field) are shown
in Fig. 6.2. Figure 6.3 shows similar plots for the case of a sphere. In all cases the
6.4
Results
0.1
119
a
b
analytical solution
numerical solution
0.12
field change (ppm)
0.14
field change (ppm)
analytical solution
numerical solution
0.05
0.16
0.1
0.15
0.18
0.2
0.2
0.25
0.22
0.3
0.24
20
40
60
80
grid location
100
20
40
60
grid location
80
Figure 6.2: Field Shift in ppm, generated by infinite cylinders of radius 12 units and an internal
magnetic susceptibility of χi = −0.72 ppm, centred within a cubic region of 2563 points with a
suceptibility value of χe = −0.36 ppm, exposed to a uniform, z -directed, magnetic field. The
variation of the z-component of the field perturbation with y-position along a line passing through
the centre of the cylinder is shown for the cases where the axis of the cylinder is: (a) parallel (along
z) and (b) perpendicular (along x)to B0 .
analytical solution
numerical solution
0.02
field change (ppm)
0.04
0.06
0.08
0.1
0.12
10
20
30
40
50
60
grid location
70
80
90
Figure 6.3: Field Shift in ppm, generated by a sphere of radius 12 units and an internal magnetic
susceptibility of χi = −0.72 ppm , centred within a cubic region of 2563 points with a suceptibility
value of χe = −0.36 ppm, exposed to a uniform, z-directed, magnetic field. The plot shows the
variation of the z-component of the field perturbation with z-position along a line passing through
the centre of the sphere.
6.4
Results
120
analytical solution
numerical solution
0
field change (ppm)
0.02
0.04
0.06
0.08
0.1
0.12
0
50
100
150
grid location
200
250
Figure 6.4: Field Shift in ppm, generated by a sphere of radius 86 units and an internal magnetic
susceptibility of χi = −0.72 ppm, centred within a cubic region of 2563 points with a suceptibility
value of χe = −0.36 ppm, exposed to a uniform, z -directed, magnetic field. The plot shows the
variation of the z -component of the field perturbation with y-position along a line passing through
the centre of the sphere.
susceptibilities of the internal, χi , and external, χe , compartments were set to values
of -0.72 and -0.36 ppm, respectively, and the diameters of the sphere and cylinders
span 24 volume elements of the matrix. As expected, the results for the sphere, and
cylinder arranged perpendicular to the field, show that a spatially varying field is
induced in the external compartment, along with a perturbation of the strength of
the uniform field present in both compartments. In the case of the cylinder arranged
parallel to the field, only the latter effect occurs. For comparison, Table 6.1 details
the expected average perturbations of the uniform field for these model systems,
including the effect of the sphere of Lorentz [110].
In each case, the maximum absolute error and average relative error in the numerically calculated field was evaluated over the whole matrix, and the results are
presented in Table 6.2. The average relative error is always less than 0.7 %, with
significant deviations from the analytic solution only occurring close to the boundary of the sphere or cylinder. These deviations, which result from truncation and
discretization effects in the discrete 3DFFT were found only to occur within approximately two voxel spacings of the boundary. They are unlikely to cause a significant
problem when dealing with models of biological tissue whose segmentation will introduce inaccuracy on a similar length scale. The average value of the field has been
6.4
Results
121
correctly assessed for these sample geometries using the approach for evaluating the
value of Bdz (~k) at ~k = 0 , outlined in the Method section (Sec. 6.3).
sphere
infinite
cylinder (⊥B0 )
infinite
cylinder (kB0 )
maximum
deviation (ppm)
0.09
percent
deviation (%)
0.09
0.15
0.46
8.1 × 10−4
0.67
Table 6.2: Table showing the maximum absolute deviation in ppm and the average relative error
for the field perturbations calculated using the numerical approach.
Figure 6.4 shows field plots for a larger sphere diameter (172 volume elements)
indicating the limitations of the approach when the spatial extent of the matrix
is not sufficiently large relative to the object size. The artifacts introduced in the
calculated frequency shift result from the effective repetition of the χ-distribution
pattern outside the defined matrix, which is a consequence of the use of the discrete
Fourier transform. This means that the calculated field is actually that which would
be generated by an infinite cubic grid of spheres, where the sphere spacing is equal
to the spatial extent of the matrix. The resulting artifacts are most evident at the
edge of the matrix, where the symmetry of the cubic array means that the spatial
derivative of the field in the normal direction must be zero.
6.4.1
Field Inhomogeneity in the Human Head
Figure 6.5 shows field offsets in ppm resulting from susceptibility variation in the
human head, which were calculated using the HUGO body model, with the main
field oriented in the superior-inferior direction. Results from one sagittal plane,
and three axial planes in which significant field inhomogeneity occurs as a result of
proximity to the sinuses. These results are in qualitative agreement with previously
published data [106, 107] produced from head and neck models. In particular, they
show that: the highest spatial gradients of the field occur adjacent to the highly
curved air/tissue interfaces of the sinuses; there is a significant axial field gradient
in the neck; the field outside the upper part of the head shows a dipolar pattern,
with a reduced field just above the head and a generally elevated field adjacent to the
head in mid-axial planes. Figures 6.6 and 6.7 show the effect of a rotation of 1o about
the medio-lateral and posterior-anterior axes, respectively. The induced change in
the z -component of the magnetic field and in the axial magnetic field gradient (with
6.4
Results
122
a
b
4
2
1.5
2
b
c
1
0
d
0.5
0
-2
-0.5
-4
-1
-1.5
-6
c
a
d
4
2
3
1
2
0
1
-1
0
-2
-1
-2
-3
-4
Figure 6.5: Maps of the susceptibility-induced field variation (ppm) generated by a realistic
model of susceptibility distribution in a human head. (a) mid-sagittal slice; (b-d) axial slices
whose locations are indicated in (a). The contour spacing is 1 ppm in all cases.
the axial direction defined relative to the body model) is shown for both rotations,
assuming a 3 T static magnetic field is applied in the superior-inferior direction. In
both cases, the region where changes are largest is adjacent to the sphenoid sinus.
Significant effects due to rotation about the lateral axis also occur at the back of
the head, whilst for the rotation about the posterior-anterior axis changes in the
orientation of the flat, lateral surface of the head relative to the applied field have a
noticeable effect in adjacent tissue.
In order to assess the significance of these rotation-induced changes in the field
more quantitatively, further analysis was carried out on the boxed region located
above the sphenoid sinus (of size 6×3×1.4 cm3 ), which is shown in Figs. 6.6 and
6.7. Tables 6.3 and 6.5 detail the fraction of voxels within this region that experience
a frequency change greater than 7.2, 12.9 or 25.9 Hz for a range of rotation angles.
These indicate that rotation about a medio-lateral axis (pitch) causes larger changes
in the field, with more than half the voxels in the boxed region experiencing a large
6.4
Results
Frequency \ Rotation
Change
7.2 Hz
12.9 Hz
25.9 Hz
123
10
30
8.5% 91.4%
0% 46.2%
0%
3.5%
50
70
97.7%
89.3%
28%
97.7%
97.7%
64%
Table 6.3: Evaluation of the percentage of pixels in the defined region of interest that experience
a frequency shift larger than a certain value following rotation by varying angles about the lateral
axis.
Gradient(z) of \ Rotation
Frequency Change
5.8 Hz.cm−1
9.6 Hz.cm−1
15.7 Hz.cm−1
19.6 Hz.cm−1
10
30
50
70
6.8%
2.0%
0%
0%
40.0%
18.4%
7.9%
5.8%
61.7%
39.9%
19.8%
13.8%
72.8%
54.8%
33.4%
23.7%
Table 6.4: Evaluation of the percentage of pixels in the defined region of interest that experience
a change in spatial z -gradient (in Hz.cm−1 ) that is greater than a certain value following rotation
by varying angles about the lateral axis.
Frequency \ Rotation
Change
7.2 Hz
12.9 Hz
25.9 Hz
10
30
1.3% 37.1%
0% 11.5%
0%
0.7%
50
70
60.8%
35.2%
7.5%
72.4%
52.2%
19.5%
Table 6.5: Evaluation of the percentage of pixels in the defined region of interest that experience a
frequency shift larger than a certain value following rotation by varying angles about the anteriorposterior axis.
Gradient(z) of \ Rotation
Frequency Change
5.8 Hz.cm−1
9.6 Hz.cm−1
15.7 Hz.cm−1
19.6 Hz.cm−1
10
30
50
70
3.8%
1.7%
0%
0%
23.8%
10.9%
5.2%
3.7%
42.9%
25.7%
12.2%
8.7%
55.5%
38.3%
22.1%
15.2%
Table 6.6: Evaluation of the percentage of pixels in the defined region of interest that experience
a change in spatial z -gradient (in Hz.cm−1 ) that is greater than a certain value following rotation
by varying angles about the anterior-posterior axis.
6.4
Results
124
enough field change for a rotation of 7o to cause a change in distortion of one pixel
magnitude in an echo planar image with an acquisition window of 40 ms duration.
a
b
c
10
5
Hz
0
d
e
f
-5
10
5
Hz cm
0
-1
-5
-10
Figure 6.6: Maps of the change in magnetic field in Hz (a-c) and the change in the z -gradient
in Hz.cm−1 (d-f) in representative sagittal, axial and coronal slices resulting from a rotation of 1o
about the lateral axis.
a
b
c
10
5
Hz
0
-5
d
e
f
-10
10
5
0
Hz cm
-1
-5
-10
Figure 6.7: Maps of the change in magnetic field in Hz (a-c) and the change in the z -gradient
in Hz.cm−1 (d-f) in representative sagittal, axial and coronal slices resulting from a rotation of 1o
about the anterior-posterior axis.
Tables 6.4 and 6.6 list the fraction of voxels that experience a change in the
axial gradient of the field in the same boxed region that is greater than 5.8, 9.6 15.7
or 19.6 Hz cm−1 . The relation between the rotation-induced change of the spatial
gradient of the field (Grot ) and signal attenuation due to through-slice dephasing is
not direct, as it depends on the quality of the original shim settings via any residual
through-slice gradient (Ginit ), the slice thickness (∆s), echo time (T E) and is given
by sinc(γ(Ginit + Grot )T E ∆s
).
2
6.4
Results
125
To evaluate the significance of these results, we again consider a gradient echo
acquisition using a slice thickness of 6 mm and an echo time of 40 ms, which will
mean that a total frequency gradient γ(Ginit + Grot )/2π of 9.6, 15.8 or 19.6 Hzcm−1
causes an attenuation of the signal approximately by factors of 1.1, 1.3 or 1.5. For
example, if a region experiences a residual gradient of 9.6 Hz.cm−1 , almost 7 % of
the pixels in the region above the sinus would in the worst case show a signal reduction of about 10 % following a 1o rotation about a medio-lateral axis.
6.4.2
Respiration Induced Resonance Offsets
The effect of each independent contribution to RIRO at 3 T is shown in Fig. 6.8,
for the two different models. In each case the induced field variation is plotted along
an axial line passing through the centre of the brain (as indicated in Fig. 6.1). The
figure also shows the combined effect of all contributions to RIRO for the two models,
along with the field variation generated by a single equivalent dipole. In both cases
the combined field variation on axis is very similar to that produced by a single
dipole, as has been found experimentally [101, 102]. Theoretically this is explained
by the fact that at a significant distance from any arrangement of magnetic dipoles,
the combined field will always be predominantly dipolar in character because of the
more rapid decay with distance of the higher order terms in the multipole expansion
of the magnetic field [119]. In the case of the simple two-compartment model in
which air-filled lungs are surrounded by a homogeneous region of water, effects of
similar magnitude are generated by expanding the bottom, sides and front of the
lungs. The effect of these changes on the simulated magnetic field perturbation
at 3 T is however much larger than than would be predicted from experimental
measurements of RIRO at 7 T [102] (which showed a variation across the brain of
about 6 Hz at 7 T). In addition the location of the effective magnetic dipole is found
to be significantly shifted in the axial direction away from the head compared with
the position found experimentally [102]: the combined effect of the thoracic and
abdominal breathing considered here could be represented by a dipole located at an
axial distance of about 40 cm from the top of the head, whereas experimental data
implied a ∼ 29 cm axial distance [102]. Including a quadratic variation of oxygen
concentration with axial position in the lungs [118], marginally moves the centre of
the equivalent dipole towards the apex of the lungs in the simulated data.
However, when the second model, in which the tissues surrounding the lungs
are properly represented is employed, the results are found to be in much better
6.4
Results
126
70
40
b
a
35
60
30
50
25
40
Hz
Hz
30
20
15
10
20
5
10
0
0
-10
-5
Head region
35
40
45
-10
Head region
35
40
45
Distance from bottom of the lungs (cm)
Figure 6.8: Plots showing the field variation induced by different changes in lung volume and
susceptibility. (a) Shows the results for a model in which the air-filled lungs are surrounded by an
infinite region containing water; (b) shows results for a more realistic model in which all tissues are
properly represented (see Fig. 6.1). The dots represent the effect of frontal thoracic respiration,
the squares represents lateral thoracic respiration, the crosses represent abdominal respiration,
whilst the diamonds represent the effect of variation of oxygen concentration in the main airways.
Finally, the combined effects of all changes are represented by open circles. The overlapping black
line shows the field variation produced by a single equivalent dipole located at (a) z = 15 cm (b)
z = 21.2 cm from the bottom of the lungs (using the co-ordinate system shown in the figure and
in Fig. 6.1).
agreement with experimental measurements. Simulated thoracic and abdominal
respiration both yield equivalent axial dipole locations at ∼ 33 cm from the top of
the head, with a further superior shift when changes in oxygen concentration in the
large airways are considered, leading to a ∼ 31 cm dipole position with respect to
the top of the head. The magnitude of the field shifts found with this model are
also closer to values that would be predicted to occur at 3 T from experimental
measurements [102]. This is a consequence of the changes in the position of the
tissue/air interface at the body surface which result from chest expansion. Such
6.5
Discussion
127
changes generally produce a field change in the head which is opposite in sign to
that generated by the expansion of the lung space. The larger transverse spatial
offset of this boundary from the axis linking the centre of the lungs to the centre of
the head (along which field plots are shown) gives a different spatial characteristic
to the field variation with axial position as can particularly be seen in the case of
the expansion of the lungs in the anterior-posterior direction.
6.5
Discussion
A novel Fourier-based method for calculating field variation induced by spatially
varying magnetic susceptibility has been described. This method is straightforward
to implement and computationally time efficient. Each of the studies described
here involved calculation of the field over a 2563 matrix, which was accomplished
in one minute on a Sun Workstation (E280R with a 900 MHz UltraSPARC-III+
processor). This time compares very favourably with that needed for real-space calculations using iterative methods carried out on similar matrix sizes [106, 107]. The
most time-consuming part of the computation is the calculation of the forward and
inverse 3DFFT’s, consequently the computational time is expected to scale approximately as 3N 3 log2 (N ) for matrix size N 3 . A further advantage of this method is the
ease with which it is possible to calculate the effect of rotations of the susceptibility distribution, which can be evaluated by effectively rotating the direction of the
magnetic field rather than the object under consideration. It is consequently unnecessary to recalculate the 3DFFT of the susceptibility distribution for each rotation,
thus speeding up calculations further. It is interesting to note that the field change,
δB(~r), produced by a small rotation, δθ (again assumed to be about the x-axis) can
be calculated directly. The k-space form, δB(~k), is given by the difference between
the values of Eq. 6.6 obtained using Eq. 6.5 and Eq. 6.7, which using the small
angle approximation takes the form
~ ~k) = 2µ0 δθ
δB(
kx2
kz ky
Mz (~k)ẑ
+ ky2 + kz2
(6.8)
Using the Fourier-based method described here, care has to be taken to limit
effects of the intrinsic repetition of the susceptibility distribution resulting from
the use of a discrete 3DFFT. This can be achieved by surrounding the object with
a ‘buffer’ region of uniform magnetic susceptibility, whose presence moves the repeated versions of the susceptibility distribution away from the central region of
6.5
Discussion
128
interest. The spatial extent that this buffer region needs to have depends
on the
h 2 i3/2
<r >
accuracy desired. The error can be estimated by considering the ratio <r02 >
,
where < r2 >1/2 is an estimate of the average distance between the position where
the field is calculated and the regions of the model that generate the dominant field
perturbation, and < r02 >1/2 is an estimate of the average distance between the
field point and the field generating region in the nearest neighboring repetition of
the susceptibility distribution. When the object is centred within the 3-D matrix,
it is straightforward to see that at the edge of the matrix this ratio will be equal to
one and will tend to zero close to the centre of the matrix. For this reason, in the
simulations of field perturbation in the head, where the dominant source of field perturbations is located near to the regions where the field is evaluated, the surrounding
buffer does not need to be very large (the model occupies a central region of extent
207 × 134 × 134 voxels within a 2563 matrix), whilst in the RIRO simulations the
buffer had to be made larger (the model occupies a central region of 76 × 38 × 69
extent within a 2563 matrix) to ensure that the field generated by changes in the
lung volume in the central model are much larger than those resulting from changes
in the neighboring repetitions of the lung model. The presence of a ‘buffer’ region
between the object of interest and the edge of the simulation matrix is also required
in the real-space methods [103, 106, 107]. In these methods the buffer region should
be large enough that the magnetic scalar potential at the extremes of the matrix is
not affected by the presence of the object [107].
Equation 6.6 forms an invertible relationship between the field perturbation and
induced magnetization in k -space, which can be converted to a direct relationship
between the Fourier transforms of χ and Bdz via Eq. 6.2:
3Bdz (~k)
1
χ(~k) = −
(6.9)
2
B0 3 cos β − 1
This offers the possibility of using an experimentally measured pattern of magnetic field variation to calculate the magnetic susceptibility distribution within an
object. Unfortunately there are a number of difficulties with this approach. First,
χ(~k) can not be evaluated for ~k values where β is equal to the magic angle (54.7o ),
because the denominator (and numerator) of Eq. 6.9 are both zero on the resulting
conical surface in k -space. This problem could be overcome by measuring the field
shift for two or more different orientations of the object relative to the field. This
would provide complementary data in which the conical surface over which χ(~k) is
undefined corrupts different regions of k -space, such that an uncorrupted set of χ(~k)
data could be formed. A second problem is that the field can obviously only be measured via MRI in regions which generate an 1 H NMR signal and have a sufficiently
6.5
Discussion
129
long T2 relaxation time. Consequently it is generally not possible to make a complete
measurement of Bdz (~r) because there will be regions of interest that do not generate
sufficient MR signal (e.g. bone, air-filled sinuses or air surrounding the body). It
may however be possible to use an iterative approach, in conjunction with a priori
information about the susceptibility values of different MR-silent tissues to produce
useful information. In addition it would be necessary to consider the effects of externally imposed inhomogeneities in the field, e.g. due to main magnet imperfections,
that do not result from susceptibility variation within the region over which the field
has been mapped. Availability of such an approach would also allow the prediction
of field changes expected to occur due to rotation of an object using only a previously measured field map, via insertion of the calculated value of Mz (~k) into Eq. 6.8.
The speed of the proposed method has recently been exploited in the evaluation
of MR signal variation resulting from changes in blood oxygenation and volume [120]
in models of the rat cerebral vasculature obtained from electron microscopy [121] as
described in Chapter 7. This evaluation required the calculation of field perturbations occurring in multiple versions of the model exposed to a strong uniform field
applied in different directions, which could be straightforwardly implemented using
the approach described in this chapter.
7
BOLD Simulations
130
Chapter 7
BOLD Simulations
7.1
Introduction
Although functional magnetic resonance imaging based on blood oxygenation level
dependent (BOLD) contrast has become a widely used tool in neuroscience, the
physiological changes that underpin the BOLD effect are still not completely understood. In particular the relationship between the changes in blood oxygenation
(∆Y ) and cerebral blood volume (CBV), occurring in active brain tissue, and the
variation in signal intensity of T2∗ , or T2 -weighted images is not fully characterised.
In probing this relationship, it has generally been assumed that the brain vasculature can be represented as an arrangement of randomly oriented, infinitely long
cylinders of varying size [122, 123]. Here, we probe the effect of this assumption by
using a 3D finite element method to simulate BOLD signal changes produced by infinite cylinders and a more realistic model of the human cortical vasculature, based
upon scanning electron microscopy measurements of the terminal vascular bed in
the superficial cortex of the rat [121].
7.2
Realistic Vasculature Model vs
Infinite Cylinders Model
When simulating BOLD effects in MRI, most researchers have centred their attention on the change of hemodynamics in the brain vasculature when a certain region
of the brain undergoes activation. Two approaches have been employed:
7.2
Realistic Vasculature vs Infinite Cylinders
131
-2D Finite element numerical simulations of the magnetisation, considering an environment where the field inhomogeneity is characterised by
infinite cylinders, whose internal magnetic susceptibility varies as the
state changes between resting and activated (Boxerman et al [123], Bandettini et al [124]);
-Analytical parameterisation of the problem, also assuming independent
infinite cylinders, the BOLD response is calculated as a function of parameters such as blood volume, HCT (haematocrit), Oxygen concentration, etc. (Buxton et al [125]);
Parameter variations during activation can then be calculated (Buxton et al
[125]) by fitting the simulated Hemodynamic Response Function (HRF) to a real
BOLD response. Although the methods described above are very different, they
employ a common assumption: the brain vasculature can be represented as an
arrangement of independent, randomly oriented, infinite cylinders. Furthermore,
intravascular and extravascular contributions to the contrast are largely discussed as
independent mechanisms and the literature for each of these mechanisms of contrast
is vast, but not always coincident.
In this chapter, the validity of the ”infinite cylinders” approach is assessed via
a 3D finite element simulation. Although an exact description of the vasculature
for the whole human brain will never be available, the terminal vasculature bed in
the superficial cortex of the rat, whose morphology has been measured by Scanning Electron Microscopy [121] (see Fig. 7.1) is a good starting point to address
the validity of the infinite cylinder assumption. The aim is to find differences of
signal behavior predicted by (i) the infinite cylinders model and (ii) a more realistic
vascular distribution where, for example, the loss of extravascular and intravascular magnetisation coherence results not only from the isotropic distribution of the
angular orientation of the infinite cylinders, but is also due to diffusion in the field
inhomogeneities generated by the ensemble of neighboring vessels. The effect of
blood movement within vessels in which frequency shifts vary spatially will also be
addressed.
Once the dependence of R2 and R2∗ have been well characterised for both models,
the consequences of draining vein effects in the correct localization of activated
regions will be discussed. This is done using prior knowledge of the variation of
intravascular ∆Y with distance from the activated region, which results from dilution
effects, as described by Turner [126].
7.2
Realistic Vasculature vs Infinite Cylinders
132
Figure 7.1: Rat Brain Micro Vasculature model obtained by Scanning Electron Microscopy [121].
20
40
60
80
100
120
20
40
60
80
100
120
Figure 7.2: Magnetic field variation in ppm in a 2D section calculated using the Rat Brain Micro
Vasculature model obtained by Scanning Electron Microscopy.
7.3
7.3
Methods
133
Methods
The infinite cylinder model (ICM) and the realistic vascular model (RVM) were
compared by simulating the evolution of magnetisation in these two environments
of frequency shift:
- ICM assumes that blood vessels are approximated by infinite cylinders,
allowing a straightforward analytical solution of the problem, giving spatially varying frequency shifts for a single vessel:
µ
¶
1
1
∆ω(~r) = − γB0 ∆χ cos2 (θ) −
2
3
µ
¶2
1
Rcylinder
∆ω(~r) = − γB0 ∆χ
sin2 (θ) cos(2φ)
2
r
inside (7.1)
outside
Thus, the frequency shift due to such a vessel, depends on the ratio of
the cylinder radius, Rcylinder , to the radial distance from the centre of
the cylinder, r, the intra/extra vascular susceptibility difference, ∆χ,
the angle between the vessel and the magnetic field, θ, and the angle
between the projection of the position vector and magnetic field vector in the plane transverse to the vessel, φ. The susceptibility of blood
plasma is similar to that of surrounding tissue, so that the dominant
susceptibility difference is between red blood cells and plasma/tissue.
The assumption made for the intravascular susceptibility is that after time averaging the blood has an equivalent susceptibility given by
∆χ = HCT (1 − Y )4πχdHb , where HCT is the hematocrit partial volume, Y is the partial oxygenation and χdHb is the susceptibility of deoxyhaemoglobin (that has a value of 0.18 ppm in cgs units [127]). The
results of simulations employing different vessel orientations were integrated in order to yield the signal of a voxel containing vessels with an
isotropic distribution of orientations. Several length scales were also considered so that a wide range of vessel diameters was simulated.
- The RVM uses the distribution of magnetic susceptibility given by the
rat brain microcirculation model [121] as a starting point. The model
gives the coordinates of the different nodes of the vasculature, and the
radius of the cylinder connecting these nodes, along with the speed of
blood flow in each of the vessels. This model spans a cube of the cortical
bed with 150 µm side incorporating 50 vessel segments with diameters
7.3
Methods
134
that vary from 4 to 9 µm. The model was converted into an integer
N × N × N cube matrix, Subst(ri,j,k ) , with each entry being labelled as
”1” or ”0” depending whether it corresponds to blood or grey matter.
The frequency shift map was calculated (see Fig. 7.2) using the method
introduced in Chapter 6, in which the Fourier Transform of the frequency
offset is given by
γ
∆ω(~k) = (3cosβ 2 − 1)∆χ(~k)B0 ẑ
3
(7.2)
where ~k is the coordinate position in k-space and β is the angle between
the main magnetic field and ~k. Applying an inverse Fourier transform
yields the frequency shift variation in the real space. Several orientations
of the model with respect to the static magnetic field and various length
scales were again simulated.
Once the environment had been defined, the evolution of the magnetization during a gradient echo (GE) or spin echo (SE) sequence was calculated by numerically
applying the Bloch Equations, taking into account the effects of relaxation, frequency shifts due to field inhomogeneities and diffusion. At each time step, the
integrated effects (as opposed to the incremental effects which yield a more stringent convergence criterion) of field inhomogeneity and relaxation on the transverse
magnetization, M + (~r, t), were evaluated using the Bloch equations,
³
´
M + (~r, t + δt) = M + (~r, t)eiγB(~r)δt + D(~r)∇2 M + (~r, t) e−δt/T2
(7.3)
where D(~r) is the spatially varying diffusion coefficient.
Only the term reflecting the effect of diffusion has to be calculated at each time
step, while the others may be calculated only once at the start of the simulation.
The diffusion term was calculated using a finite difference solution in real space [91]
M (~ri,j,k , t + δt) − M (~ri,j,k , t)
=
δt
D
(M (~ri−1,j,k , t) − 2M (~ri,j,k , t) + M (~ri+1,j,k , t))
(δx)2
D
+
(M (~ri,j−1,k , t) − 2M (~ri,j,k , t) + M (~ri,j+1,k , t))
(δy)2
D
+
(M (~ri,j,k−1 , t) − 2M (~ri,j,k , t) + M (~ri,j,k+1 , t))
(δz)2
(7.4)
To make the problem tractable, the vessel walls were assumed to be impermeable.
This assumption was implemented by imposing some restrictions on the simple finite
7.3
Methods
135
difference solution. This involved simply inserting a delta function multiplier term
δSubst(ri,j,k ),Subst(ri0 ,j0 ,k0 ) that reduces diffusion attenuation when the ”substance” of the
neighbouring pixel is different. Since substance changes only occur at blood/tissue
boundaries, this modification makes the vessel walls effectively impermeable. Its
implementation is described here for a 1D case where diffusion is considered to only
occur in the x−direction,
M (~ri,j,k , t + δt) − M (~ri,j,k , t)
=
δt
D
(M (~ri−1,j,k , t) − M (~ri,j,k , t)) δSubst(ri,j,k ),Subst(ri−1,j,k )
(δx)2
D
+
(−M (~ri,j,k , t) + M (~ri+1,j,k , t)) δSubst(ri,j,k ),Subst(ri+1,j,k )
(δx)2
(7.5)
The maximum allowed time step, δt, is set by the largest value of the diffusion
coefficient, Dmax , and the distance between neighbouring pixels (δx, δy, δz), via
δt < (δx2 + δy 2 + δz 2 )/(6Dmax ) or by the desired temporal resolution.
Calculations were carried out using two different regimes of blood/tissue susceptibility difference (∆χ), corresponding to the typical values occurring in the absence
or presence of exogeneous contrast agents. Values in SI units of 0.45 ppm (oxygenation fraction, Y =0.5), 0.36 ppm (resting state, Y =0.6), 0.27ppm (intermediate
state, Y =0.7) and 0.18 ppm (activated state, Y =0.8) were used to simulate BOLD
susceptibility changes in the absence of a contrast agent (these values were calculated
using the relationship ∆χ = HCT (1 − Y )4πχdHb , χdHb = 0.18 ppm in cgs units).
For the case of exogeneous contrast agents, such as MION or gadolinium, susceptibility differences of 1.41 ppm, 2.82 ppm, 4.23 ppm and 4.64 ppm (corresponding
to MION concentrations in the blood of 1, 2, 3 and 4 mM) were considered. These
values are based on susceptometry studies using MRI described by Weisskoff [127].
Blood volume fractions of 1.29%, 2.56%, 3.76% and 4.90% (the value set by the
form of the RVM was 1.29% and all the other values were obtained by multiplying
√ √
√
all the vessel radii given in the original model by 2, 3 and 4) were considered
in the simulations.
Diffusion coefficients of DB = 1.45 × 10−9 m2 s−1 inside and DGM = 0.7 ×
10−9 m2 s−1 outside vessels were assumed. For the SE simulation the echo time was
set to the T2 of grey matter (the values used for relaxation time were T2GM = 80ms
and T2B = 100ms that are similar to values described in literature [95,96] ). Several
length scales were considered (by simply rescaling the distance between neighbouring volume elements in the RVM or ICM matrix), so that a wide range of vessel
diameters from capillaries (4µm) to large venules (300µm) could be simulated.
7.3
Methods
136
Two different matrix sizes (323 and 643 ) were used in the ICM to test whether
reduction of pixel resolution would be an issue when simulating the RVM ( an
1283 matrix was the largest that could be simulated in an acceptable time of 30
hours). Simulations of the ICM in a 323 matrix were also performed using field
maps calculated via the FFT-based approach (Chapter 6) rather than the analytical
method, to test whether the FFT method introduced any intrinsic differences. No
significant differences in the ICM contrast were observed when reducing the matrix
size or calculating the field shift with the FFT method, leading us to believe that
any differences in the results between ICM and RVM models are not due to the
methodology used, but to the model under study.
The simulations spanned a range of susceptibility values, blood volume and vessel
sizes, potentially allowing the overall behaviour of the relaxation processes to be
parameterised. Such parameterisation is valuable since it potentially allows the
evaluation of BOLD and contrast agent effects over a wide range of field strengths.
7.3.1
Extravascular Contrast
Intra and extravascular compartments were studied separately. The characterisation
of extravascular signal was carried out by considering two different models [128].
(i) Considering that the extravascular distribution of field values (Bint )
σB
can be represented by a Cauchy distribution, p(Bint ) = √1π σ2 +B
2 , where
int
B
σB characterises the width of the distribution. The signal decay in
the presence of such a field distribution will be given by, G(T E) ∝
exp(−T EγσB ), leading to a relaxation rate in the static regime R2∗ =
γσB .
2
2
(ii) Considering a Gaussian field distribution p(Bint ) = √ 1 2 e−Bint /(2σB ) ,
2πσB
it can be shown that the signal decay is characterised by [128],
µ
µ
TE
G(T E) = exp − < ∆ω 2 > τc2 e− τc − 1 +
TE
τc
¶¶
(7.6)
where < ∆ω 2 >= γ 2 σB2 is the mean square frequency deviation of the
Gaussian field distribution and τc is the corresponding correlation time
for field fluctuations produced by diffusion in spatially varying fields.
When τc is very large (no diffusion approximation) the signal decay naturally becomes Gaussian.
Fitting the simulated gradient echo data to these two different models was accomplished by using least square minimization on the entire signal decay from t = 0
7.3
Methods
137
to t = T2GM ). In simulating spin echo signals, time constraints meant that it was
only possible to consider the application of a single 1800 RF pulse. It therefore had
to be assumed that the spin-echo was described by an exponential decay with a time
constant R2 (calculated using only two data points at times t = 0 and t = T2GM ).
The dependence of all the parameters (< ∆ω 2 >, τc , R2 and R2∗ ) on the total
blood volume fraction, V , and the magnetic susceptibility, χ, for different length
scales were calculated by fitting the parameters to an expression of the form αV β χγ .
Such an expression for characterising relaxivity was initialy proposed by Fisel et
al. [94] (Note that effectively the power of χ also characterises the dependence of
these parameters on the B0 field).
In the case of exponential signal decay (i), it is possible to consider the total
effect of a distribution of vessel diameters as being simply the sum of the relaxation
P
rate contributions due to vessels of different sizes, R2comb = R2GM + i αi V βi χγi .
The validity of this approach is based on the observation that the number of vessels,
N (d), with a specific diameter, d, in the brain scales as 1/d3 [126], while the volume
corresponding to the RVM scales with d3 . The value of V to be used is that of the
blood fraction of each specific vascular length scale. In this approach, the effect
of capillaries is essentially considered to have an additive background effect on the
relaxation rate due to venules and so on (the same argument is valid for venules and
pial veins).
7.3.2
Intravascular Contrast
There are various contributions to the intravascular signal decay: (a) phase variation
due to the distribution of cylinder orientations; (b) dephasing resulting from field
inhomogeneities generated by neighbouring vessels; (c) and finally a contribution
due to diffusion and flow between connected vessels having different orientations
(this term is not calculable with the simple infinite cylinder model):
(a) The intravascular FID due to a large number of isotropically-oriented
infinite cylinders containing a substance with susceptibility χ is given by,
I(t) =
S0 Z π
2
sin θeik(3 cos θ−1)t dθ
2 0
(7.7)
where k = 2π
γB0 χ [122]. This expression can be simplified by using the
3
relationship 3 cos2 θ−1 = 23 cos 2θ+ 32 and then expanding the exponential
P
n
of the integrand as eix cos φ = J0 (x) + ∞
n=1 2i Jn (x) cos nφ, leading to a
decay of the form
7.3
Methods
138
Ã
I(t) = S0 ei3kt/2
∞
X
!
1
Jn (3kt/2)
J0 (3kt/2) − 2
in 2
4n − 1
n=1
(7.8)
(b) The decay resulting from dephasing due to fields generated by neighbouring vessels, can be analysed using the histogram of field values in
the intravascular compartment.
(c) The third term can be assessed by considering the effect of diffusion
in the gradients present inside the vessels as an exponential perturbation
to Eq. 7.8 (this has to be done for all different length scales).
7.3.3
Draining Veins
Figure 7.3: Typical venous drainage structures, with overlaying definition of some of the relevant
parameters (reproduced from reference [126]).
Once the basic work focusing on the variation of intra- and extra-vascular contrast as a function of the blood volume fraction and the susceptibility difference
between grey matter and blood had been completed, the effect of draining veins was
analysed by considering that in the activated region there is a fractional variation
of the deoxyhaemoglobin concentration and volume that is constant over the entire
range of vessel diameters (as discussed earlier in this section), whilst the draining
vein is modelled as an infinite cylinder with a well defined direction relative to B0 .
7.4
Results
139
In Turner’s work [126] it is stated that, on average, a vessel that drains an area of
cortex, D, will have a diameter
D1/3
mm
(7.9)
8
(naturally there is a significant uncertainty in this estimate as it makes assumpdv =
tions regarding average length and diameter of capillaries and small venules, their
volume fraction and the thickness of the cortex). The maximum distance this vein
can be from the region of activation without containing blood draining from nonactivated regions of the cortex is calculated to be 0.42D0.5 [126]. At greater distances
from the region of activation, the vein will contain blood draining from other inactive
cortical regions that do not undergo a variation of deoxyhaemoglobin concentration.
The fractional change in deoxyhaemoglobin concentration in the draining vein, γ,
was calculated in reference [126] to be
γ=
1+δ
(L0 +(4D/π)0.5 )2
2.25D
+δ
(7.10)
where D is the cortical area of activation (that is assumed to be circular), L0
is the shortest distance from the vein to the edge of the activated region and δ is
defined to be ∆CBF/CBF , the fractional change in blood flow in the active region.
Which is directly related to the decrease of concentration of deoxyhaemoglobin.
Elevation of intravascular pressure following CBF change, means that there is also
an increase of the partial volume occupied by the draining vein, this will scale with
distance from the activated region, varying as 1/γ (with δ = 0). The variation of
volume will be spread over vessels of different sizes along the blood flow stream from
the arterioles where the increase of blood flow initially occurs, and it is sensible to
expect it will not to be confined to the region of activation, but will also occur in a
vein that drains the active region of cortex.
7.4
7.4.1
Results
Extravascular Contrast
To help in understanding the extra-vascular relaxation processes it is useful first to
visualize the frequency shift distribution produced using the RVM model and ICM
model. Figure 7.4 shows the field histograms for varying values of partial blood
volume using the two different models. The field distributions shown in Figs. 7.4a
and 7.4b are very similar in form and range of values, indicating that the ICM
7.4
Results
140
is generally a good model to characterize extravascular field distributions. The
best fit to the RVM data was achieved using a product of a Gaussian and a Cauchy
7
V=1.29%
V=2.56%
V=1.25%
V=2.50%
V=3.75%
V=5.00%
5
10
(a) V=3.76%
V=4.90%
Number of Voxels
Number of Voxels
10
4
10
6
10
3
10
5
10
2
10
4
10
1
3
10
10
2
10
-80
-60
-40
-20
0
20
40
60
80
80
100
60
40
Frequency (rad/s)
20
0
20
40
60
80
100
Frequency (rad/s)
Figure 7.4: Histograms of the frequency shift in gray matter for the (a) RVM and (b) ICM model
considering a susceptibility difference of 0.18ppm between intra and extravascular compartments
for partial volumes of 1.29% (blue), 2.56% (red), 3.76% (green) and 4.90% (yellow) at a B0 field
of 3 Tesla. The histograms were fitted to a product of Cauchy and Gaussian distributions (black
lines)
χ = 0.18 ppm
χ = 0.45 ppm
V= 1.29%
7
10
7
10
V= 1.29%
V= 2.5%
a.u.
a.u.
V= 2.5%
V= 3.7%
V= 3.7%
V= 4.9%
V= 4.9%
0
0.02
0.04
0.06
echo time (s)
0.08
0.1
0
0.02
χ = 1.4 ppm
0.04
0.06
echo time (s)
0.08
0.1
χ = 5.6 ppm
7
10
7
10
V= 1.29%
V= 3.7%
V= 2.5%
V= 1.29%
a.u.
a.u.
V= 2.5%
6
10
V= 4.9%
6
10
V= 4.9%
0
0.02
0.04
0.06
echo time (s)
0.08
0.1
0
0.02
0.04
0.06
echo time (s)
V= 3.7%
0.08
0.1
Figure 7.5: Simulated FID for the RVM using varying partial volumes at a B0 field of 3 Tesla,
considering four different susceptibility differences between blood and grey matter.
7.4
Results
141
distribution1 . The low frequencies are best characterised by the Cauchy distribution,
while the Gaussian characterises the high frequencies. It is interesting to note that
the width parameter of the Gaussian remains constant as the blood volume fraction
varies, whilst the width of the Cauchy distribution increases linearly with volume
fraction. This is natural since the Gaussian characterizes the highest frequency
variations and the maximum frequency shift generated by a cylinder is independent
of its radius (naturally both widths increase linearly with χ). The shape of the
histogram of extravascular frequency distribution will have a direct relationship to
the form of the extravascular FID. In the absence of diffusion, this relationship is
the Fourier transform. FID’s generated by Fourier transform the field distributions
are shown in Fig. 7.5 for various volume fractions and susceptibility differences at
3T. It is possible to see that the curves have different forms in the different regimes:
(a) for low intra-extravascular susceptibility differences as found in the activated
brain, the decay is non-exponential at short echo times, but such non-exponential
behavior does not occur at high concentrations of deoxy-haemoglobin (b) or low
concentration of paramagnetic contrast agents (such as MION) (c) for which the
data are well characterised by a single exponential decay time constant. At very
high susceptibility differences (corresponding to 4M of MION at 3T)(d) the signal
decay shows a bi-exponential behaviour at high volume fractions (as if there are two
different evironments, one close to the vessel shows rapid signal decay, and a second
in which the signal decays more slowly).
Figure 7.6 shows the dependence of the rate changes due to BOLD effects in spin
echo and gradient echo data as a function of the average vessel diameter. It can be
seen that, although the values for the two different models are not coincident, the
dependence on vessel diameter is very similar. This dependence means that in spinecho BOLD experiments the capillaries and small venules are the dominant source
of contrast, whereas the main contribution to contrast in gradient echo experiments
comes from venules and veins, as has previously been reported [123].
Figures 7.7 and 7.8 show the parameters that best fit the R2∗ and R2 values of
the extravascular signal as a function of the vessel diameter (in the case of the RVM
the average diameter present in the model is used) and it is possible to observe the
similarity of the dependence of the parameters obtained for the ICM and RVM. The
parameter that most significantly varies between the two models (both for GE and
SE data) is the proportionality constant α, that is approximately 10% higher for
1
In the non-diffusive regime, a Gaussian distribution causes a Gaussian decay, whilst a Cauchy
distribution generates an exponential decay.
7.4
Results
142
the RVM.
There will generally be vessels with a large range of diameters in a voxel. Considering a range from 4 µm to 110 µm where all the vessels have a number density
∝ 1/d3 , the combined dependence on the percentage blood volume fraction and
blood susceptibility was found to be
R2RV M = R2GM + 1.23(±0.04)V 0.87(±0.01) ∆χ1.74(±0.02)
(7.11)
1.13(±0.01)
∗
∆χ1.29(±0.01)
R2RV
M = R2GM + 2.73(±0.07)V
For the infinite cylinders model, the aggregate effect was found to be
R2ICM = R2GM + 0.88(±0.04)V 0.98(±0.02) ∆χ1.79(±0.02)
(7.12)
∗
R2ICM
= R2GM + 2.49(±0.09)V 1.15(±0.01) ∆χ1.38(±0.02)
Figure 7.9, shows the results for an alternative fitting (as suggested by Kennan et al [128]). Using the function given in Eq. 7.6. This reduces the sum of
squares deviation between simulated and fitted signal decay curve by two orders of
magnitude. In particular, this model gives a better description of the initial decay
of the extravascular signal, that as shown in Fig. 7.5 is not expected to decrease
∆R*2
∆R 2
0.45
1.2
0.4
0.35
∆χ=0.27ppm
1
0.3
0.8
0.25
0.2
∆χ=0.18ppm
0.6
0.15
0.4
0.1
∆χ=0.09ppm
0.2
0.05
0
5
-5
10
diameter (m)
4
-4
10
5
-5
10
diameter (m)
4
-4
10
Figure 7.6: Variation of relaxation rate changes (a) ∆R2 and (b) ∆R2∗ for changes in bood
susceptibility ∆χ of 0.09, 0.18 and 0.27 ppm as a function of the vessel diameter, a blood volume
fraction of 1.29% was used. Solid lines show the values obtained using the RVM whilst the dashed
lines represent values obtained using the ICM
7.4
Results
143
*
R2= R
3.8
1.7
3.6
+ α V β χγ
2GM
1.9
1.8
1.6
3.4
1.7
1.5
3.2
1.6
1.4
1.5
2.8
γ
β
α
3
1.3
1.4
2.6
1.2
1.3
2.4
1.1
2.2
2
-5
10
diameter (m)
1
-4
10
1.2
-5
1.1
-4
10
diameter (m)
10
-5
10
diameter (m)
-4
10
Figure 7.7: The variation of the parameters α, β and γ with the average vessel diameter (continuous lines) obtained for the ICM (black) and RVM (grey). In each case R2∗ values obtained for
varying values of V and χ were fit to the equation R2∗ = R2GM + αV β χγ . The dashed lines indicate
standard deviations.
R =R
2
+ α V β χγ
2GM
1.9
1.3
1.6
1.2
1.4
1.85
1.1
1
1.2
1.8
0.9
γ
β
α
1
0.8
0.8
1.75
0.7
0.6
0.6
1.7
0.5
0.4
0.4
1.65
0.3
0.2
-5
10
diameter (m)
-4
10
-5
10
diameter (m)
-4
10
-5
10
diameter (m)
-4
10
Figure 7.8: The variation of the parameters α, β and γ with the average vessel diameter (continuous lines) obtained for the ICM (black) and RVM (grey). In each case R2 values obtained for
varying values of V and χ were fit to the equation R2 = R2GM + αV β χγ . The dashed lines indicate
standard deviations.
exponentially at very short echo times. Although the fits are more accurate (even
considering that the degrees of freedom are reduced due to the existence of one extra
fitting parameter), it should be noted that in Fig. 7.9, the proportionality constant,
α, for τc becomes smaller as the length scale (diameter of vessels) increases when
it would be expected to increase. As a consequence, these parameters may be used
7.4
Results
144
β γ
τc= α V χ
3
12
x 10
0.3
0
0.2
10
-0.5
0.1
0
-0.1
6
-0.2
4
-5
10
diameter (m)
-4
10
-0.4
1.5
diameter (m)
β γ
=
1.4
1000
1.3
800
1.2
600
1.1
400
1
-5
10
diameter (m)
-4
10
0.9
-2
-4
10
-5
10
diameter (m)
-4
10
2.5
2.4
2.3
β
1200
α
-5
10
2
<∆ ω > α V χ
1400
200
-1.5
-0.3
γ
2
-1
γ
β
α
8
2.2
2.1
-5
10
diameter (m)
-4
10
2
5
10
-4
diameter (m) 10
Figure 7.9: The variation of the parameters α, β and γ with the average vessel diameter (continuous lines) obtained for the ICM (black) and RVM (grey) that characterise τc (top) and < ∆ω 2 >
(bottom). In each case the τc and ∆ω 2 values obtained for varying values of V and χ were fit to
the function αV β χγ . The dashed lines indicate standard deviations.
to predict the signal decay at different fields, length scales, susceptibility differences
and partial volume but it might be misleading to give them too much physical relevance. Nevertheless, it is tempting to remark that the dependence of the parameter
< ∆ω 2 > on the partial volume (approximately linear) and susceptibility (quadratic
dependence) could be predicted from analysing the histograms of Fig. 7.4, as could
the insensitivity of τc to variations in the volume fraction and its dependence on the
susceptibility (the negative power means that τc decreases with the increasing χ).
Figure 7.10 shows the R2∗ parameters relating to the extravascular signal behavior
for values of susceptibility difference as found when using intravascular contrast
agents. The exponential fitting used to identify R2∗ was carried out in the early
portion of the signal variation with TE that correspond to the mono-exponential
7.4
Results
145
R*
= R
2
+ α V β χγ
2GM
3.15
1.09
1.09
3.1
1.08
1.08
1.07
1.07
1.06
1.06
1.05
1.05
3.05
3
2.9
γ
β
α
2.95
1.04
1.04
1.03
1.03
1.02
1.02
1.01
1.01
2.85
2.8
2.75
2.7
2.65
-5
10
diameter (m)
-4
10
1
-5
10
diameter (m)
-4
10
1
-5
10
diameter (m)
-4
10
Figure 7.10: The variation of the parameters α, β and γ with the average vessel diameter
(continuous lines) obtained for the ICM (black) and RVM (grey). In each case R2 values obtained
for varying values of V and χ (in a range corresponding to exogenous contrast agent) were fit to
the equation R2 = R2GM + αV β χγ . The dashed lines indicate standard deviations.
decay shown Fig 7.5. The parameter values are significantly different from those
found for the typical deoxyhaemoglobin susceptibility differences at 3T (Fig. 7.7).
In particular, the dependence on the partial volume and susceptibility difference
are linear. The different dependences are partially related to the more adequate
fitting of the exponential decay for higher susceptibility differences since the initial
non-exponential decay is absent.
Figure 7.11 shows the effect that the change in the susceptibility and blood
volume fraction on activation has on the change in extravascular relaxation rate.
The haemodynamic response can be represented as a path on this surface. It is
possible to see how sensitive the haemodynamic response is to the initial parameters.
Negative values of ∆R2∗ cause a positive BOLD response, whilst a negative BOLD
response is caused by a positive change in relaxivity. The generation of a negative
BOLD response using the model R2∗ = αV β χγ implies that changes in volume, ∆V ,
+ γ ∆χ
> 0. Such a negative BOLD
and susceptibility, ∆χ, obey the inequality β ∆V
V
χ
response, or generally occurring as an initial dip, has mainly been reported at high
fields [129] and is elusive, but not undetectable at lower fields [130]. The initial
dip is expected to be mainly localized in the capillaries, at 3 Tesla the condition
for negative BOLD Response to occur will be 1.4 ∆V
+ 1.7 ∆χ
> 0 whilst at higher
V
χ
∆χ
∆V
2
fields the negative BOLD condition is 1.03 V + 1.02 χ > 0 (see Figures 7.7 and
2
Simulations of the effect of exogenous contrast agents are equivalent to those of endogenous
contrast agents at higher field. The relationship found for exogenous contrast at 3T should hold
for endogenous contrast at fields greater than 9T.
7.4
Results
146
radius 9µ m
radius30µ m
0
radius100µ m
0
0
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.6
0.8
0.8
0.8
χ
=0.36 ppm
OFF
0.2
VOFF=1.25
3.5
3
1
∆ V (%)
∆ V (%)
(a)
∆ V (%)
2.5
1
2
∆R *
2
1
1.2
1.2
1.2
1.4
1.4
1.4
1.6
1.6
1.6
1.8
1.8
1.8
2
2
2
1.5
1
0.5
-0.2
-0.1
0
0.1
-0.2
∆ χ (ppm)
-0.1
0
0.1
-0.2
∆ χ (ppm)
radius 9µ m
-0.1
0
0.1
0.5
∆ χ (ppm)
radius30µ m
0
radius100µ m
0
0
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.6
0.8
0.8
χ
=0.36 ppm
OFF
VOFF=3.75 0.2
0
5
4
1
∆ V (%)
∆ V (%)
(b)
∆ V (%)
3
2
0.8
1
1
1.2
1.2
1.4
1.4
1.4
1.6
1.6
1.6
1.8
1.8
1.8
∆R *
2
1
1.2
0
1
2
2
-0.2
2
-0.1
0
∆ χ (ppm)
0.1
-0.2
2
-0.1
0
∆ χ (ppm)
0.1
-0.2
-0.1
0
∆ χ (ppm)
0.1
Figure 7.11: ∆R2∗ = R2∗ (ON ) − R2∗ (OF F ) as a function of the variation of volume, ∆V , and
susceptibility ∆χ for vessels with different radii. Considering an OFF state with: (a) V = 1.25%
and χ = 0.36ppm and (b) V = 3.75% and χ = 0.36ppm.
7.10). The condition for the initial dip to occur with an increase of blood volume
fraction is less restrictive at high field, ∆χ > −1.01 Vχ ∆V than it is at low fields,
∆χ > −0.82 Vχ ∆V .
7.4.2
Intravascular Contrast
The main difference between the ICM and RVM models seems to be in the intravascular contribution to contrast. Figure 7.12 shows intravascular frequency distributions generated using the ICM (considering 10 different directions with an isotropic
weighting, sin θ) and RVM, (considering 5 different orientations of the model with
respect to B0 ). Differences between the two models (see the discussion in Section
7.3.2) definitely results from fields perturbations at junctions between vessels in the
RVM(the arrow in Fig. 7.2 indicates the location of such a junction). The difference in the distribution can also be partially explained by the fact that vessels in
the RVM experience fields generated by neighbouring vessels as well as the varying
field shift due to the range of vessel orientations Eq. 7.1). To assess the magnitude
of this effect we can imagine inserting an (infinite) cylinder randomly into the extravascular space of the RVM many times, and then assume that the distribution
of values inside the cylinder averaged over the different orientations and locations
provide a good representation of the distribution of the intravascular frequency shift.
As this infinite cylinder can be sited at any position and orientation, the frequency
7.4
Results
147
5
x 10
RVM
ICM
CONV
6
5
4
3
2
1
0
100
80
60
40
20
0
20
40
60
Intravascular Frequency Shift (rad.s -1)
80
100
Figure 7.12: Distribution of intravascular frequency shifts at 3 Tesla for a susceptibility difference of 0.18 ppm between intra and extravascular compartments. The frequency distribution of
isotropically oriented infinite cylinders is shown in black, while the histogram obtained for the realistic vasculature model is shown in grey. The dashed black line represents the expected frequency
distribution produced by convolving the extravascular field distribution with the intravascular
distribution expected for the ICM model.
distribution obtained after averaging over the ensemble of locations and orientations
can be understood as a convolution of the extravascular frequency distribution from
the vessel model with the intravascular frequency distribution due to an isotropic
distribution of infinite cylinders. The dashed line in Fig. 7.12 is the result of such
a convolution, which shows a good qualitative agreement with the grey line that
represents the intravascular field distribution calculated using the RVM.
The simulated intravascular signal produced using the infinite cylinder model
was naturally in very good agreement with Eq 7.8, once the the effect of the R2 of
blood was removed (as can be seen in Fig. 7.13a).
Various attempts were made to fit functions to the variation of the intravascular
signal of the RVM with T E. The approach that proved to be most robust was based
on using intravascular decay curve obtained from the larger length scale (average
vessel diameter of 100 µm) for each χ and V value, which we write as IV,χ (t). These
curves were subsequently used as models for the data obtained at smaller average
vessel diameters, because larger scales are expected to be independent of diffusion.
Using such a model, instead of simply using Eq. 7.8, has the advantage of eliminating
geometry dependencies of the vascular model that might be introduced by the lack
angular averaging, allowing just the effect of blood diffusion within the vascular
network to be evaluated. The effect of diffusion was modelled as an exponential
multiplier, so that the fitting function became IV,χ (t)e−R2 t . Some representative fits
7.4
Results
148
RVM
1
0.8
absolute intravascular signal (a.u.)
absolute intravascular signal (a.u.)
ICM
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.01
0.02
0.03
0.04 0.05
TE(s)
0.06
0.07
0
0.01
0.02
0.03
0.04 0.05
TE(s)
0.06
0.07
Figure 7.13: Plots showing the intravascular signal evolution: (a) shows the evolution of the
modulus of the intravascular signal for (a) the ICM and (b) the RVM as predicted by the simulations
(dashed lines) and by the respective fittings (continuous lines) for average diameters of the vessels
of ∼ 5µm (grey) and ∼ 17µm(balck).
R* = R
2
0
2B
+ α V β χγ
-0.35
1.35
-5
1.3
-0.4
-10
1.25
-15
-0.45
1.2
γ
β
α
-20
-25
1.15
-0.5
-30
1.1
-35
-0.55
1.05
-40
-45
-5
10
diameter (m)
-4
10
-0.6
-5
10
diameter (m)
-4
10
1
-5
10
diameter (m)
-4
10
Figure 7.14: The variation of the parameters α, β and γ with the average vessel diameter
(continuous lines) obtained for the RVM. The R2 values due to diffusion in the gradients present
in the intravascular compartment for varying values of V and χ were fit to the equation R2 =
R2B + αV β χγ superimposed on the decay in the static regime. The dashed lines indicate standard
deviations.
of the RVM intravascular signal are shown in Fig. 7.13b. Figure 7.14 shows the
parameters that characterise the effect of diffusion inside the vessels. The values of
”R2 ” are one order of magnitude greater then those observed in gray matter. Note
also that the R2 values shown are negative, meaning that diffusion tends to reduce
7.4
Results
149
the signal decay due to dephasing of intravascular signal. It can also be anticipated
that flow in the vessels will have a similar effect, but potentially larger, acting to
reduce the attenuation of intravascular signal.
7.4.3
Combining the Different Contributions to Contrast
1
450
0.9
400
extravascular
0.8
350
signal difference (au)
contrast fraction
0.7
0.6
_ . _100 µm
___ 5.9 µ m
0.5
0.4
0.3
0.2
intravascular
0.1
0
0
0.02
0.04
0.06
TE(s)
0.08
300
250
200
150
100
50
0.1
0
0
0.02
0.04
0.06
0.08
0.1
TE(s)
Figure 7.15: Dependence of the GE BOLD contrast in intra and extravascular compartments as
a function of echo time, T E. (a) The relative contribution to the total contrast in a voxel with
CBV=1.29% for a change in χ on activation from 0.36ppm to 0.18ppm is shown as a function of
echo time for extravascular (blue) and intra vascular (red) contributions. (b) The variation of the
total contrast (black) extra-vascular (blue) and intra vascular (red) is shown for the large length
scale regime where the average vessel diameter is 100 µm (dashed lines) and short length scale
regime where the average vessel diameter is 5.9 µm (continuous line)
As both contributions (intra- and extra-vascular) to contrast have now been
discussed independently, the next step is to analyse their combined effect. Figure
7.15 shows the different contributions to contrast for the specific case of a blood
volume fraction of 1.29% that does not vary with activation and a typical change
of susceptibility on activation of 0.18ppm (corresponding to a change of 20% in
the blood oxygenation). Figure 7.15(a) shows that at different echo times different
relative intra/extra-vascular contributions to the BOLD signal arise. At echo times
varying from 0 ms to 40 ms, the intravascular contribution to contrast is overrepresented (over 10 times greater than its volume fraction in the voxel). The
relative intravascular contribution increases for smaller vessels where diffusion plays
a greater role. The data shown correspond to a field strength of 3T. The range of
echo times for which the relative contribution of the intravascular compartment is
7.4
Results
150
significant will decrease with increasing magnetic field strength for two main reasons:
B0 appears in the argument of the Bessel function series of Eq. 7.8; R2∗ of blood
also decreases with field. But as the extravascular R2∗ will also tend to decrease,
the ratio of intra/extra-vascular contrast at the optimum contrast echo time should
remain relatively unchanged.
R*
= R
2
+ α V β χγ
2GM
2
5
1.9
1.04
1.8
4
1.03
1.7
1.02
1.6
γ
β
α
3
1.5
1.01
2
1.4
1
1.3
1
0.99
0
0
0.5
1
angle (rad)
1.5
1.2
0.5
1
angle (rad)
1.5
1.1
0.5
1
angle (rad)
1.5
Figure 7.16: Dependence of the parameters found for the R2∗ fit due to a cylinder as a function
of the angle in respect to the B0 field (black - cylinder centred in the pixel; grey - only half the
cylinder is located inside the pixel) .
7.4.4
Draining Vein Effect
To discuss the effect of draining veins upon the localization of activation in GE
experiments (from Fig. 7.8 it is clear that large vessels make a negligible contribution
to the contrast obtained in spin echo experiments) it is important to consider the
dependence of R2∗ upon the vessel orientation with respect to the applied magnetic
field. Figure 7.16 shows the parameters obtained by fitting the R2∗ dependence on
blood volume and susceptibility for the situation of a single cylindrical vessel located
within a voxel vary with the angle between the cylinder and the static magnetic field.
The two different curves show the variation of the R2∗ parameters when the cylinder
is centred in the voxel and when it is located at one of the sides of the voxel. From
Fig. 7.16 it is possible to conclude that only veins draining the cortex in a direction
almost perpendicular to the field might have a significant role in mislocalization
of the activated region and the effect of a large draining vein is only significant in
pixels in which the vessel is passing through the centre of the voxel (α is reduced by
a factor of two when only half the cylinder is inside the voxel and becomes negligible
7.4
Results
151
80
0.8
Bo
60
0.6
0.4
40
0.2
60
0
0
Angle between draining
vein and static
magnetic field
20
0
20
40
60
-0.2
scale
Region with active tissue
with 10mm radius
Region with a draining vein perpendicular to Bo
65 mm away from the active region
∆R 2
Distance from the centre
of the activated region in mm
when it is completely outside the voxel). The intravascular effect will appear simply
as a change of phase of the blood magnetisation.
Once this is known, it is possible to attempt to characterise the contrast in the
neighbourhood of the activated region. Since it is unlikely that the contrast can be
fully described, the aim is to impose some bounds upon what may happen. The
analysis relies on Eqs. 7.9 and 7.10. For example, the resolution is an important
parameter as it controls the partial volume occupied by the vessel (the finer the
resolution, the larger will be the effect of a draining vein due to the increase of the
partial volume occupied by the vessel, Vc ).
Figure 7.18 shows some examples of draining vein effects for different regimes.
Figure 7.17 provides a guide to the interpretation of Fig. 7.18. The graphs represents
the variation in R2∗ , ∆R2∗ = R2∗ (ON ) − R2∗ (OF F ): in a voxel in a region where all
the vessels show the same variation of susceptibility and fractional volume, which is
the effectively the active region; in a voxel located in the regions surrounding such a
patch of active cortex that are at certain distance from the active region and contain
a draining vein that is a well defined cylinder. The effect of varying the resolution
of the image is shown in the sub-figures along each column. A vein draining an
activated region can cause either a positive BOLD response (negative ∆R2∗ ) in the
case that it occupies a large percent of the voxel, or a negative BOLD response
when it occupies a smaller fraction of the volume and the variation of volume rather
than susceptibility has a dominant effect on the contrast. The four different rows
highlight the effect of pixel resolution, showing that increasing the pixel resolution
-0.4
80
Region with a draining vein parallel to Bo
30 mm away frome the active region
Figure 7.17: Visual aid to Figure 7.18: representation of ∆R2∗ variation with respect to the angle
between the draining vein and the static field and its distance from the active region.
7.4
Results
152
Active radius=10mm
χ (OFF)=0.36 ppm ∆χ =-0.09 ppm vol(OFF)=4% ∆ v=1.2%
∆vc =0 ∆v
volC=1.28%
Pixel res=2mm
volC=5.10%
Pixel res=1mm
80
-1
-2
20
0
-2.5
0
20
40
60
80
-3
-0.5 60
-0.4
-1
-0.6
-1.5
20
0
0
20
40
60
-1
-1.2
0
20
40
60
80
0.8
0
0.6
60
-0.1
-0.3
-0.4 40
-0.2
0.4
40
0.2
-0.5
-0.8
0
20
40
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80
0
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80
80
60
40
0.4 60
-0.2
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40
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80
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0
20
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80
20
40
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0
0
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40
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80
1.5
60
0.2 40
1
0.5
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20
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20
0.6
0.4
-0.35
0
0.5
-0.2 40
-0.25
-0.4
0
1
-0.15
-0.3
20
1.5
0.8 80
-0.1 60
40
80
-0.2 20
-0.05
60
60
40
80
80
40
0
-0.35 20
20
20
80
-0.1 60
-0.15
-0.3
0
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0
0.6
-0.4
0
0
-0.2
-0.05
-0.25
20
0
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-0.6 20
-0.7
20
-1.4
80
-0.2 60
40
-0.8
20
0
80
-0.1
60
40
-2
80
80
80
volC=0.32%
Pixel res=4mm
60
-1.5 40
40
∆v =1 ∆v
c
80
-0.2
-0.5
60
0
volC=0.08%
Pixel res=8mm
∆vc =0.5 ∆ v
80
-0.2
-0.4
0
20
40
60
80
20
0
0
0
20
40
60
80
Figure 7.18: ∆R2∗ (s−1 ) = R2∗ (ON ) − R2∗ (OF F ) as a function of the distance from the activated region and angle between the draining vein and the main magnetic field for different volume fractions of the voxel occupied by the draining vein close to the active region,
Vc = 0.08%, 0.32%, 1.28%, 5.01%. Changes in this volume fraction given by ∆Vc = 0, 0.5∆V, 1∆V
are considered and the susceptibility change is given by ∆χc = ∆χγ, where γ is described in Eq.
7.10. ∆V and ∆χ characterise the variation the variation of volume and susceptibility in the active
tissue, that was considered to have a radius of 10 mm.
7.5
Conclusion
153
will inherently create some problems in the localization as it increases the variations
on the relaxivity variation around the draining vein leading, ultimately generating
a contrast greater than that found on the active tissue. The difference between the
three columns is that they describe the expected change in ∆R2∗ in the case where
only a certain portion of the blood volume change in the active region affects the
subsequent draining vein, the fractions used in generating these images are 1, 0.5
and 0 respectively. Most of the dilation of vessels is expected to occur close to
the synapses of the active cortex where there is the release of various agents such
as nitric oxide, potassium, adenosine and neurotransmitters (such as glutamate)
that can behave as vasodilators [131]. The change of the volume at the onset of
activation occupied by the draining vein is therefore expected to be close to zero.
Due to the passive role of vessels in circulation it is expected that the variation of
blood volume in the region of activation will spread to vessels down stream from
the area of activation, ultimately making ∆Vc 6= 0. It is interesting to notice that in
some regimes, the contrast outside the region of interest can be effectively greater
than in the activated region. For example, if in the active region the variation of
blood susceptibility is given by ∆χ = −χ/2 and the variation in volume is given by
∆V = V /3 and in a draining vein perpendicular to B0 , ∆Vc = 0 then, the condition
for the positive BOLD response to be greater in the draining vein is simply that the
blood volume fraction of the draining vein is 1/3 of that of the active region.
7.5
Conclusion
To account totally for the intravascular contrast it would be also necessary to simulate the effect of diffusion of water around the red blood cells. Such a procedure
would be feasible, using the same approach as used for the realistic vasculature
model. However there are two problems that prevent the implementation of such
an approach: the time steps required to simulate the effect of diffusion would be
extremely small, making the simulation even more time consuming; the red blood
cell volume fraction in blood is approximately 40% (much larger than the partial
volume of blood in grey matter, of approximately 4%) and the field shifts for such a
high packing fraction are extremely dependent on the arrangement of the blood cells
(in preliminary simulations, arrangements such as cubic, cubic centred and hexagonal configurations showed very different histograms of frequency shift). Another
reason for not simulating the intravascular contrast dependence on red blood cell
concentration and oxygenation is that such dependence can be measured in vitro
with much more reliability [96].
7.5
Conclusion
154
Most of the results presented in this chapter imply that the infinite cylinder
model provides a good description of the extra vascular contrast, as is shown by
the close agreement between the parameters obtained using the RVM and ICM, this
is an important conclusion in itself. Nevertheless, using a realistic model of the
vasculature to simulate the BOLD effect brings a new insight into the importance of
different contributions to the overall BOLD contrast, mainly relating to the greater
role of intravascular contrast in small vessels. The effect of flow should now be
evaluated. Preliminary studies indicate that flow causes a line narrowing of the
intravascular signal that can be understood as a reduction of the phase accumulation
experienced by each individual spin due to flowing through the network of vessels
in which there is a significant distribution of field shifts.
The parametrisation of the extravascular relaxivity showed that the assumption
of linearity in blood volume fraction of the contrast of functional studies using
contrast agents is a good approximation, but is not necessarily valid for endogenous
contrast agents.
The effect of draining veins was assessed and the values presented show that
there is an inherent risk of the region of activation appearing blurred in the direction
perpendicular to the magnetic field in the situation where there is a draining vein
flowing in such a direction.
8
Field Map Estimation
155
Chapter 8
Field Map Estimation
8.1
Introduction
Inhomogeneous B0 -fields generate distortion in MR images, particularly those acquired using EPI [132]. This distortion can be corrected using information provided
by a field map [98, 100, 133] that is often generated from phase images acquired at
two (or more) echo times (TE) [98, 100], or by using alternative k-space trajectories [134]. The dominant source of field inhomogeneity in many MRI experiments is
the variation of magnetic susceptibility across different tissues of the human body.
In fMRI, head movement changes the susceptibility-induced field distribution leading to changes in distortion, which can induce signal variation and a consequent
reduction in BOLD contrast-to-noise ratio [100]. In the presence of significant motion (or other sources of variability such as respiration induced resonant offsets [102]
that may be similar in magnitude to the frequency changes due to head rotation),
the resulting field changes mean that image distortion-correction based on a field
map acquired prior or subsequent to an fMRI experiment is sub-optimal. In previous work, this problem was addressed by acquiring data at two echo times during
the fMRI experiment, thus providing a contemporaneous measurement of field and
BOLD signal variation [100]. Here we describe a method for measuring the field
variation in fMRI experiments that is based on monitoring phase variation in conventional EPI data acquired at a single echo time, T E.
8.2
Theory
EPI is one of the most sensitive imaging techniques to distortion due to field inhomogeneity as it has an intrinsically long phase encoding time [133]. An EPI acquisition
8.2
Theory
156
of 40 ms duration can lead to two neighbouring pixels along the phase encoding
direction with a 25 Hz frequency shift between them appearing superimposed in the
distorted image domain. Methods to perform the undistortion of EP images have
been extensively discussed in the literature [98, 133] and generally rely on the use of
a measured frequency map. This frequency should reflect the local magnetic field
that a region experiences in a period of time between excitation and signal acquisition. However a phase map obtained using GE-EPI, which may be used to form a
frequency map, has many other contributions other than this local magnetic field.
By considering the signal evolution following a slice-selective 900 pulse,
Z
sr (t) =
M0 (~r)e−i∆ωt e−iγ
Rt
0
G(τ )·~
rdτ −t/T2 −iΦRF (~
r) −iγ
e
e
e
Rt
0
Beddy (τ,r)dτ
dV
(8.1)
it is possible to conclude that the phase image obtained using a GE-EPI sequence
does not depend only on the evolution under the action of the local magnetic field
shifts(∆ω/γ) during, T E. Additional contributions to the phase may arise due to the
maximum signal not being centred in k−space (linear phase shift), intrinsic T2∗ decay
(in both these cases the hermitian condition Signal(k) = Signal(−k ∗ ) necessary to
obtain a purely real image is violated), on the excitation and acquisition phases (the
spatial variations on these quantities that arise at high field due to both dielectric
effects and reduced wavelength (ΦRF (~r))) and on fields generated by eddy currents
resulting from the process of gradient switching.
The local magnetic fields and the T2∗ decay are best viewed in the ”object space”
while the other additional phase contributions are best described in the ”image
space” (in the sense that they will appear in the same slices and pixels independently
of the position of the object). Another characteristic is that all the contributions in
the image space and the T2∗ effects are echo time independent. This is why methods
for estimating frequency shifts generally rely on the subtraction of two different
images acquired using the same excitation and acquisition procedure but different
echo times so as to cancel out the intrinsic additional phase.1 The method proposed
here relies on separating the two contributions: the frequency offset and the intrinsic
phase. The method assumes that the intrinsic phase accumulation is constant during
the fMRI time series (the assumption made is that the T2∗ contributions are small
compared with all the others, and there are no significant changes in the image
1
The term related with eddy currents will not fully disappear from the frequency map obtained
using this approach. Eddy currents effects generated by read and phase encoding gradients will
be the same in both acquisitions, whilst those generated by the slice selection procedure that have
not fully decayed by the time signal acquisition commences will still be present in the frequency
map.
8.2
Theory
157
acquisition timings during the experiment). The local frequency offset can be written
as
φ0 (~r, t2 ) − φ0 (~r, t1 )
(8.2)
t2 − t1
where φ0 (~r, t2 ) and φ0 (~r, t1 ) are the phase images acquired at echo times t2 and
t1 . On the other hand, the intrinsic phase will be given by
ω(~r) =
φACQ (~r) = φ0 (~r, t2 ) − ω(~r) ∗ t2 = φ0 (~r, t2 )
−t2
−t1
+ φ0 (~r, t1 )
t2 − t1
t2 − t1
(8.3)
This separation relies on the assumption that no movement occurs between the
two acquisitions. Such an assumption is ”most valid” at the start of the experiment
while no task execution has occurred and subject discomfort should not be an issue.
The initial field maps should be generated from multiple signal averages in order
to minimise noise in φACQ (~r) and ω(~r). As φACQ (~r) should be independent of the
object, and therefore not have a strong spatial dependence, it can be fitted using
a second order polynomial expansion, φACQf it (~r). The advantage of using a fit to
characterize the intrinsic phase as opposed to simply using the average as an estimate
is that this approach allows extrapolation of the phase estimate to regions of low,
or no signal. Regions of low signal may carry important information regarding the
field map, and as motion occurs regions that had generated no signal at the start
of the experiment, may generate signal, and an incorrect estimate of the intrinsic
phase would lead to a large error in the field map estimate.
If t2 is chosen to be the echo time used in the fMRI experiment, we can now
introduce φi (~r, t2 ) to represent the phase image acquired at a point i of the time
course. This phase can be corrected giving rise to φcorr,i (~r, t2 ) = φi (~r, t2 )−φACQf it (~r)
which is directly related to the frequency shift of the spins at time point i,
ωi (~r) =
φcorr,i (~r, t2 )
φi (~r, t2 ) − φACQf it (~r)
=
t2
t2
(8.4)
.
The noise in phase data in the regime of high SNR is given by δφ = 1/SN R,
∗
∗
where SN R = A0 e−tR2 /σ = SN R0 e−tR2 and σ is the noise in the image signal.
Assuming that data from a large number of acquisitions are averaged to form the
initial phase map, so that the noise in φACQ (~r) is negligible, the noise associated
with the frequency shift estimate, δ(ωi (~r)), is given by
√
∗
e2t2 R2
δ(ωi (~r)) =
SN R0 t2
(8.5)
8.3
Methods
158
Previous methods to estimate frequency map variation during an experiment
have relied on subtraction of phase images acquired with different echo times acquired during the fMRI experiment. In order to monitor the spatial pattern of
frequency variation throughout an fMRI experiment, the sequence should employ
at least two different echo times [100], allowing the frequency map at each pixel to
be evaluated throughout the entire time course. This relies on the assumption that
no movement occurs between pairs of acquisitions (at different echo times) over the
entire time course. In this situation, the noise in the frequency map will be given
by
√
δ(∆ωi (~r)) =
∗
∗
e2t2 R2 + e2t1 R2
SN R0 (t2 − t1 )
(8.6)
The parameters that minimise the noise are, to a first order, a maximum (t2 −t1 ),
but the maximum value of t2 is limited by the optimum fMRI echo time (usually
chosen to be the T2∗ of grey matter) and the minimum value of t1 is restricted by the
time it takes to acquire data in half of k-space (4). Comparing Eqs. 8.5 and 8.6, it
is straightforward to conclude that there is a gain in the frequency sensitivity with
the method proposed here involving comparison of consecutively acquired images
√
∗
when 1 − tt21 ≤ 1 ≤ 1 + e2(t1 −t2 )R2 . Furthermore, other than simple noise analysis,
the proposed method does not rely on subjects remaining still between the two
measurements during the entire time course.
8.3
Methods
To compare the result of image distortion correction using single (Eq. 8.4) and
double echo (Eq. 8.2) data, experiments were performed on a 3 T scanner. Phantom
studies were also carried out to test whether φACQ (~r) remained invariant with motion
of the phantom and to compare the signal-to-noise ratio (SNR) of frequency offsets
measured using the two methods. In each phantom position, 10 repetitions were
made so that the variability of field maps could be evaluated. The phantom consisted
of a plastic sphere filled with agar gel containing an off-centre, glass sphere filled
with agar gel, doped with Gd-DTPA. In further experiments on human subjects,
the first aim was to observe the change in field offset associated with head rotations
about different axes. Subjects were asked to make head rotations about the lateral
and anterior-posterior axes. Between movements they remained still while two image
volumes (64x64x52 matrix) with a resolution of 4x4x4 mm3 were acquired with echo
times of 25 and 37 ms. The EPI acquisition employed a bandwidth of 60 Hz per
point in the phase encoding direction.
8.3
Methods
159
In each case, the first position was used to be the reference image as it is supposed
to be less affected by movement. Other then the slices that were reduced to 26,
the image and acquisition parameters were the same as described for the previous
experiments.Using the two volumes acquired at different echo times, φACQ (~r) was
calculated as described in the theory section. The value obtained for φACQ (~r) is
not an absolute value (due to the unwrapping procedure), the constant offset value
however only causes a simple translation mainly along the phase encoding direction
when the undistortion is carried out. The modulus images were motion corrected,
and the transforms applied to the field maps calculated using either Eq. 8.2 or Eq.
8.4. Finally the frequency shift change between consecutive positions was calculated.
yplane27
xplane32
zplane15
4
x 10
3
2
1
0
Figure 8.1: Modulus images of the phantom in three different slice orientations, close to the
region of high field inhomogeneity
In order to evaluate the possibility of using the proposed approach in fMRI,
double-echo fMRI time series (100 volumes) were processed using two different approaches: (i) data from the two echoes were used to produce a field map via Eq.
8.2; (ii) φACQ (~r) was estimated from an average over the double-echo data and field
maps were then calculated from data acquired at the longer T E via Eq. 8.4.
In both cases it is important to perform an unwrapping of the field maps along
the 4th dimension (time course) using a reference point located in the middle of
the head, since the frequency offset in this region is postulated to be close to zero
(as referred to above, there is a constant ambiguity due to the wrapping process,
but this is not important, as it will simply introduce an extra constant shift to all
the image pixels). Once each image of the time course had been undistorted using
the corresponding field map, the motion correction and normalisation were applied.
Finally as a means to investigate potential advantages of the usage of this method,
8.4
Results
160
the standard deviation of each pixel intensity over time was evaluated. This evaluation was carried out for images obtained using the post-processing methodologies
described earlier and also for data to which no undistortion had been applied.
Phase unwrapping, motion correction and undistortion were implemented using
the FSL software while all other calculations were implemented using Matlab.
8.4
Results
2
0
40
2
4
z=19
30 40
50
0
40
x=27
10
20
30
10
40
0
y=22 60
10
20 30
0
x=27 y=22 z=1 : 30
20
10
10
15
20
25
30
0
2
x=1 : 64 y=22 z=19
4
20
20
20
20
5
20
20
60
60
Intrinsic phase (rad)
10 20
10
Intrinsic phase (rad)
60
Intrinsic phase (rad)
20
40
20
30
40
50
60
0.5
0
0.5
x=27 y=1 : 64 z=17
1
0
10
20
30
40
50
60
Figure 8.2: On the right the value of the intrinsic acquisition phase as calculated from acquisitions
with object rotated to different orientations (different colors stand for different orientations of the
phantom). On the left an image of the average intrinsic phase after 10 excitations with the phantom
in the same position
It is possible to draw two important conclusions from the experiments performed
with the spherical phantom, whose results are shown in Fig. 8.1 (in which the
phantom was rotated, allowing the intrinsic phase and the field map to be calculated
at several positions): the intrinsic phase is independent of the phantom position and
does not have a significant dependence in the object position. The support for these
conclusions is evident in Fig. 8.2, the left side images show the independence of the
intrinsic phase from the object (the intrinsic phase has no pattern resulting from
8.4
Results
161
the inner sphere), whilst on the right side various positions represented in different
colors show the independence of the phase estimation from the phantom position.
It was also found that in the region of high SNR a second order polynomial could
be well fitted to the phase variation.
In Fig. 8.3 the plots show that the values measured for the variation of the field
map from one position to the next between the two methods (using the intrinsic
phase, or using the the subtraction of data acquired at the two echo times) are in
good agreement. However the standard deviation of the measurements made using
the proposed method is significantly reduced with respect to the obtained with the
conventional method (on average, the standard deviation was reduced by 48%) as
frequency shift (rad/s)
was expected from the noise discussion in the theory section.
20
0
-20
frequency shift (rad/s)
15
25
30
x axis at (y,z)=(27,15)
35
40
0
-20
-40
15
frequency shift (rad/s)
20
20
25
30
35
y axis at (x,z)=(32,15)
14
16
18
20
z axis at (x,y)=(32,27)
40
45
50
20
0
-20
10
12
22
24
26
Figure 8.3: Calculated field variation along one line of the image (standard deviation shown using
dashed lines) (grey) using proposed method (black) using conventional double-echo method.
15
10
5
Hz
-----
0
-5
Figure 8.4: Change in frequency map (Hz) due to a rotation of 1.7 degrees about the lateral axis.
8.4
Results
162
Fig. 8.4 shows the pattern of field change induced by rotation of subject’s head
about the lateral axis, which is in good qualitative agreement with the results of
previous experimental [132] and numerical studies [114], showing that regions of
greatest field change are effectively located over the ethmoid and sphenoid sinuses.
The range of values obtained are in good agreement with our previous numerical
Intrinsic Phase (rad)
studies [114].
0
40
-1
20
-2
0
-3
-20
-4
-40
200
400
200
0
-200
-400
-600
-800
Field Map (rad/s)
0
-200
-400
-600
Figure 8.5: Top row shows images of the intrinsic phase calculated using Eq. 8.3 for one volume at
one single time point of the functional data, while the bottom row shows an image of the frequency
map for the same time point calculated using Eq. 8.2
zplane=15
yplane=25
xplane=22
1.5
1
0.5
σS
/<S
>
(ii)
Figure 8.6: Image representing the ratio σS(ii)
, the average and standard deviation are
(i) /<S(i) >
calculated in the time-course, and the index (i) stands for the conventional method, while (ii)
denotes the new method presented in this chapter
8.4
Results
163
4
Translation (mm)
3
x direction
y direction
z direction
2
1
0
-1
-2
-3
0
10
20
30
40
50
60
70
80
90
100
0.01
x rotation
z rotation
0.005
Rotation (rad)
0
-0.005
-0.01
-0.015
-0.02
-0.025
-0.03
0
10
20
30
40
50
60
70
80
90
100
Figure 8.7: Translation and Rotation motion correction parameters for 100 volumes, the color
coding (green, cyan and yellow) represents the different subsets with different movement characteristics used in the the analysis
Figure 8.5 shows the intrinsic phase and the field map calculated for one point
in the functional-time course. The relevant feature shown is the lack of object
dependence present in the intrinsic phase, when compared to the field map which
shows a localized area of inhomogeneity in regions around the sinuses.
The comparison of the weighted standard deviation, σS/ < S >, of images
acquired in a double echo experiment and then corrected using the two different
methods. This is depicted in Fig. 8.6 shows that the new method (ii) yields a 10%
reduction of the weighted standard deviation in regions just above the sphenoid
sinus, whilst in the middle of the head the improvement is approximately 3%. Other
regions visible in Fig. 8.6 where better performances is found for the new technique
are the inferior portions of the temporal lobes and regions such as the cerebellum.
The data shown are taken from a taste fMRI experiment. In this experiment subjects
are, as part of the task, expected to swallow, an activity which will naturally affect
the frequency shift of regions close to the mouth and palate.
When comparing the proposed methodology with the result of post-processing
using only motion correction, the standard deviation of the signal just above the
sphenoid sinus is reduced by 14%.
8.5
Conclusions
164
The consequences of breaking one of the 100 volume samples into three groups
of 30 volumes, corresponding to different range of movements (as shown in Fig. 8.7)
can also be interesting. These show a period of lower amplitude movement (green),
followed by periods of increased movement later in the time course (cyan and yellow).
The evaluation of the ratio of the weighted standard deviations
σS(ii) /<S(ii) >
σS(i) /<S(i) >
was
calculated in a region over the nasal sinus, and found to take values of 0.95 (green),
0.93 (cyan) and 0.92 (yellow) in the different periods. These results show a tendency
for the performance of the proposed method when compared to the conventional
method to be improved as the amount of subject movement increases. One of the
reasons for these findings might be related to the fact that the conventional method
relies on the subject remaining still for every double echo acquisition, whilst the
new method proposed only relies on such an assumption for the volumes used to
calculate the intrinsic phase.
8.5
Conclusions
In the functional studies, the reduction in the noise was not as significant as in
the phantom experiments (if it had found to be of the same order of magnitude,
this would imply that all the noise was motion related which is obviously not true).
Comparison of the standard deviations of the intensity of pixels calculated with and
without distortion correction can be misleading because the interpolation involved
in the undistortion process is equivalent to convolving the data using a triangular
function in the image space, which as an inherent spatial smoothing effect [100]. But,
as we compare two different methodologies with the same inherent smoothing the
proposed method proves to be more reliable in regions where time varying distortions
are relevant than the conventional method.2
As scanner field strengths increase, distortion becomes an increasingly serious
problem, while the associated reduction in T2∗ makes the double echo EPI acquisition
(required to calculate an instantaneous field map using the conventional approach)
less practical. At high fields our methodology can be applied with single echoes,
with data being acquired at a second echo time at the start of the experiment and
perhaps also sparsely along the time course. Such an approach would enable firstly
a robust estimation of the intrinsic phase at a time when movement is not significant
and allow any drift of the intrinsic phase that could occur with heating of gradients
to be monitored.
2
The ultimate test will be the usage of the method in the processing of the functional data,
which is presently being done.
8.5
Conclusions
165
As stated before, the most significant advantages of the new method are that:
it only relies on a lack of subject movement when data at multiple echo times are
acquired at the start of the experiment, to allow calculation of φACQ (r), whilst the
double echo approach assumes that the subject does not undergo significant movement between acquisition of echo-pairs during the whole study; it does not require
any complex methodological change of the sequences used for the data acquisition
during the study; the ability to estimate the field map with a higher SNR than using
the other methods found in the literature.
9
Conclusions
166
Chapter 9
Conclusions
The initial aim of this thesis was to look at the characteristics of Long Range Dipolar
Fields and some of the possible applications of this phenomenon. Some further understanding of the dynamics was achieved with work performed at high field, where
features predicted by theory were observed for the first time. The way different spins
interact via the long range dipolar fields was also confirmed by looking at the evolution of the amplitude of the different peaks in a 1D spectra of the signal originated
by two distinct proton species and by designing alternative 2D-spectroscopy schemes
showing the ”irrelevance” of the simultaneity of preparation of the magnetisation of
the two different species.
One of the biggest problems of sequences based on Long Range Dipolar fields
is the intrinsically low amplitude of the signal that is produced. In Chapter 4 the
DQC-CRAZED sequence was successfully optimized for imaging applications by
maximising either the Signal to Noise Ratio or Contrast to Noise Ratio. Phase cycles
allowing DQC signal to be distinguished from stimulated echo contributions were
also developed. Making sure stimulated echoes are properly eliminated is extremely
important if misleading interpretation of the DQC signal characteristics are to be
avoided [88] such effects might have been the origin of some reported structural
dependencies [69].
The theory and application of dipolar fields has generated a great deal of interest,
and debate, in the imaging community, but this interest has been fading away in the
past couple of years: For the theoreticians the reason for this reduction of interest
arises from the consensus achieved that the classical and quantum pictures of the
phenomenon are equivalent; For the clinical experimentalists loss of interest has
occurred because the role of dipolar fields in clinical imaging seems to be extremely
9
Conclusions
167
limited, due to the low signal to noise ratio, difficulty of implementation and the
complexity of the results obtained as a result of the role of diffusion, modulation
length, preparation time and relaxation mechanisms in dictating the signal strength.
There are still however some potential applications that might prove usefull at
high field:
• The dipolar field can be used as an amplifier of the signal of diluted species.
• Short-T2 imaging. It was discussed in Section 3.4 that when a DQC sequence
is applied to a sample containing two different spin species in the linear regime
where τ ¿ τd , four different signals will be generated (as were evident in the
2D CRAZED experiments): signal due to component ”a” due to longitudinal
magnetization of the same species, Ma+ Mza ; signal from ”a” due to the longitudinal magnetization of spin species ”b”, Ma+ Mzb ; signal from ”b” due to the
longitudinal magnetization of spin species ”a”, Mb+ Mza ; and signal ”b” due to
longitudinal magnetization of the same species, Mb+ Mzb . If component ”a” has
a long T2 relaxation time and ”b” a short T2 relaxation time, and recall that in
the linear regime the maximum signal after a DQC preparation is achieved at
time T2 , it is obvious the terms proportional to Mb+ are not good candidates
for detection when compared to the terms proportional to Ma+ . Therefore, the
aim, as in magnetisation transfer [135] [136], is to use long T2 signal to detect
the presence of the short T2 components.
90
90
90
h 180
RF
Long T2
selective
Gradient
G
t1
CRUSHER
τ
τ
2G
imaging
sequence
echo time
Figure 9.1: Pictorial representation of the proposed method for short T2 imaging, that is more
resistent to long T2 contamination. The duration of t1 and the echo time are not drawn to scale.
The most promising sequence designed for achieving this indirect detection
is shown in Fig. 9.1. In this sequence both T2 components are modulated
and brought back into the longitudinal direction after a very short t1 period.
Any remaining magnetization in the transverse plane will be crushed. Using
9
Conclusions
168
a 900 pulse of long duration allows the long T2 component’s magnetisation
to be rotated into the transverse plane, whilst the magnetisation of short
T2 components should remain in the longitudinal plane (to avoid excitation of
only a narrow bandwidth Sussman et al. [137] have suggested that the long 900
pulse should broken up into smaller sections interleaved with refocusing pulses.
These pulses were called TELEX pulses). Subsequently, during the evolution
time the transverse magnetisation of the long T2 component is dephased using
a gradient pulse, and then evolves under the action of the dipolar field due
to the longitudinal magnetisation of the short T2 component. This yields a
signal that allows indirect detection of short T2 components. The condition
for the resulting signal to be dominated by the short T2 component is that
M0a e−t1 /T2a fz (τ, T2a ) < 32 M0b e−t1 /T2b fz (τ, T2b ), where fz (τ, T2 ) is the amount of
magnetisation that remains in the longitudinal direction after the application
of an RF excitation of length τ to a component with transverse relaxation
time T2 . If the experiment is repeated with the duration of the t1 period
increased to a value t1l , which is much larger than T2b , and the new data
subtracted from that produced in the first experiment. The condition for short
T2 dominance, after the signal from the sequences with different values of t1
are subtracted, becomes to a first order M0a fz (τ, T2a )(e−t1 /T2a − e−t1l /T2a ) <
2
M0b e−t1 /T2b fz (τ, T2b ). Wood and high concentration BSA solutions (because
3
they are known to have different T2 pools) were used to test this hypothesis
in experiments at 9.4T. The experiments were not successful, one of the main
reasons being that the properties of the pulses described in literature were not
observable when carrying out simple CPMG sequences where TELEX pulses
were used for RF excitation.
• The greatest specificity of the long range dipolar fields is the ability of generating signal from coherences between spins modulated in different places/times
(as demonstrated in Section 3.4). If two different spin species have their magnetisation modulated in two different regions of space, the magnetisation of
one being kept along the longitudinal axis, whilst the other is in the transverse
plane, the existence of signal is a proof of spatial proximity during the evolution time. But it is not obvious in which environment this property could
have any application as the movement of both spin species would have to be
”slow” or coherent enough not to distort the modulation.
With or without practical applicability of the long range dipolar field effects, it
is a beautiful technique for its somehow counter intuitive behaviour.
9
Conclusions
169
The methods used for numerical calculations of the long range dipolar fields in the
presence of inhomogeneity were adapted to calculated magnetic fields due to susceptibility induced magnetisation. This method has the advantage, compared with others described in teh literature, of being faster (because it is a Fourier based method)
and automatically including the effect of the Sphere of Lorentz. The method was
applied to study field variations due to head movement and respiration induce resonant offsets. Such effects increase in proportionality to the static magnetic field,
and therefore will become of greater importance as fields increase and the results
obtained can be used as a reference guideline to evaluate when methodologies to
counteract these field shifts will need to be used. For example, the calculations
made using the body model are currently being used to decide new strategies for
dynamic shimming [138]. The method is also being applied to venograms in order
to see what regions might be more affected by draining vein effects. One of the interesting future developments is to try to invert this methodology, although there is
an obvious lack of information due to the lack of a field map in regions producing no
NMR signal, iterative methods might be able to yield a satisfactory answer regarding the susceptibility distribution. At a smaller length scale, in Chapter 7, BOLD
simulations were done using, for the first time, the field distributions generated by
a realistic model of the vasculature.
The simulations of the BOLD effect, apart from generally validating the infinite
cylinder model, have helped to provide a more unified understanding of the coupling
between intra and extravascular contrast, the increased role of intravascular contrast
at fields strengths of 3 Tesla (30%) and the increased role of intravascular contrast
in the microvasculature compared to large venules. R2∗ effects of draining veins were
for the first time quantified as a function of the angle of the vein with respect to the
static field and the distance from the active region.
It could be of interest to further develop this approach to study blood relaxation mechanisms. There is some dispute in the study of the blood relaxation as it
is not clear if the data obtained in vitro are better described by exchange mechanisms between red blood cells and plasma or simple dynamical averaging [139]. The
models used to analytically describe the dependence of relaxation on diffusion, oxygenation and spacing of the 1800 pulses of CPMG sequences are obtained using the
assumption that red blood cells can be represented as sphere, which is not the most
accurate approximation. Furthermore, considering that the volume fraction of RBC
is approximately 40%, the field distribution will be very dependent on the possible
packing arrangement, which will be very different depending on the chosen model.
To avoid extremely large computation times due to the small time step necessary at
9
Conclusions
170
xplane
xplane
20
0
20
40
60
rad/s
rad/s
40
20
0
20
40
60
80
80
yplane
yplane
40
20
40
60
rad/s
20
0
rad/s
40
20
0
20
40
60
80
80
zplane
zplane
20
30
0
0
rad/s
10
rad/s
20
20
40
60
Figure 9.2: Frequency shift for a sphere and for a RBC across free different plans
this small length scale (µm) and because various random distributions of red blood
cells should be evaluated, Monte Carlo methods [123] should be used.
A method to estimate field variations due to head movement, respiration and
field variations during fMRI experiments was proposed and the initial analysis indicates it has advantages when compared to other methods that have previously been
described in the literature. Whether it is a useful tool is something that only will
become clear once data from fMRI studies have been systematically processed using this approach. Such methods will have increasing importance as field strengths
increase and therefore distortions due to induced susceptibility increase. On the
other hand, with the advent of parallel imaging, this technique may become less
significant, as the distortions are already significantly reduced. For this method to
be applicable with parallel imaging the reconstruction of the images should, in the
case of SENSE, be done separately for the real and the imaginary images, allowing
therefore the calculation of the phase image which is essential to allow the field map
to be produced.
A
Fast Fourier Transforms
171
Appendix A
Fast Fourier Transforms
The Fourier Transform is an essential tool in NMR and MRI. Although much greater
detail can be found in various textbooks, some basic principles and theorems of the
Fourier Transform are introduced in this appendix to facilitate the reading of the
thesis.
A.1
Fourier Transform
The one dimensional Fourier Transform of a function f (x) is
F (kx ) = F T 1D {f (x)} =
Z +∞
−∞
f (x)e−ikx x dx
(A.1)
where F (kx ) is the frequency spectrum of f (x) and the units of x and kx are the
reciprocal of one another (their product is dimensionless).
The relationship between f (x) and F (kx ) can be inverted leading to the inverse
Fourier Transform,
1 Z +∞
F (kx )eikx x dkx .
(A.2)
f (x) =
=
2π −∞
Some properties relating f (x) and F (kx ) can be easily concluded: if f (x) is a
real valued function, F (kx ) will have Hermitian symmetry, F (−kx ) = F ∗ (kx ); if f (x)
is strictly an imaginary valued function, F (kx ) will have anti-Hermitian symmetry,
F (−kx ) = −F ∗ (kx );
Other properties are not so obvious but are also extremely useful. If f (x) is
N
a convolution of g(x) with h(x), f (x) = g(x) h(x), then its Fourier transform,
F (kx ), will simply be the product of the Fourier transforms of g(x) and h(x) that
are G(kx ) and H(kx ) respectively.
F T −1
1D {F (kx )}
A.2
Discrete Fourier Transform
172
f (x)
1
δ(x − x0 )
u(x) = 1 if |x| ≤ 1/2, u(x) = 0 otherwise
2
e−ax
e−xa
P∞
a i=−∞ δ(x − ia)
N
g(x) h(x)
F (kx )
δ(kx )
e−ikx x0
e
P∞
sin(kx /2)
kx /2
−kx2 /(a4π 2 )
a(2π)2
(2π)2 a2 +kx2
δ(kx /(2π) − i/a)
G(kx )H(kx )
i=−∞
Table A.1: Some examples of relevant analytical 1D Fourier transforms and properties
The N-Dimensional Fourier Transform can be formulated as,
F (k1 , ..., kN ) = F T N D {f (x1 , ..., xn )} =
Z +∞
−∞
...
Z +∞
−∞
f (x1 , ..., xn )e−i
PN
j=1
kj x j
dx1 ...dxN
(A.3)
If f (x1 , ..., xn ) can be written as f1 (x1 )...fN (xn ), then
F T N D {f (x1 , ..., xn )} = F T 1D {f1 (x1 )}...F T 1D {fN (xn )}
A.2
(A.4)
Discrete Fourier Transform
In real life, signals are not acquired in continuous analytical form but in a digitized form due to sampling. If the spacing between each point of digitisation is
∆kx 1 , instead of acquiring F (kx ), the sampled signal will be given by F̂ (kx ) =
F (kx )
P∞
its inverse Fourier transform, instead of being f (x),
n=−∞ δ(kx − n∆kx ) and
³
´
2π P∞
2πn
ˆ
will be f (x) = ∆k
f
x
−
. Which will be a repetition of the function
n=−∞
∆k
x
f (x) in steps of
2π
.
∆kx
x
The minimum sampling rate at which aliasing is avoided is called the Nyquist
rate. The Nyquist rate should be greater than the maximum extent of f (x) which
is equivalent to saying the frequency of sampling should be greater or equal to twice
the maximum frequency of F (kx ).
Another limitation in sampling is the finite extent over which it is carried out.
Assuming a sampling window of width Wkx = 2kxmax + ∆kx = ∆kx Nsamples , where
kxmax is the maximum sampling frequency and Nsamples is the number of samples.
1
The examples given in this section refer to inverse fourier transform due to our main interest
being the sampling of k-space for imaging (as has been seen previously, Fourier and Inverse Fourier
transforms are essentially equivalent).
A.2
Discrete Fourier Transform
173
³
´
The sampled signal will be given by F̂W (kx ) = F̂ (kx ) u Wkkx , leading to a Fourier
x
N
transform that becomes fˆW (x) = fˆ(x) Wkx sinc(Wkx x). The effect of the limited
sampling region in the kx space will be a ”sinc” blurring in the x space.
Furthermore, as the result fˆW (x) will also be given in a discretised fashion,
ˆ
fˆ (x), with a separation of data points given by 1 , it is straightforward to conW
Wkx
clude that the Fourier transform reflects a periodic repetition of F̂W (kx ) with a
periodicity of Wkx .
Recapitulating, the function F (kx ) is sampled at a rate ∆kx for a period of
Nsamples ∆kx leading to a DFT representation of f (x) that is repeated every
with spacing between adjacent points of
2π
Nsamples ∆kx
2π
∆kx
and
.
The relationship between such discretised functions in each domain, F̂W (kx ) and
fˆW (x), is given by the Discrete Fourier Transform which can be computable using
a methodology known as Fast Fourier Transform [140].
B
Dipolar Field. . . making it easy to use
174
Appendix B
Dipolar Field. . .
making it easy to use
The appendix shows how is it possible to transform the non-local relationship between the magnetization and dipolar field in real-space into a local relationship in
k -space, which was first described by Deville et al. [44].
Equation 3.12 is initially re-written as
Z
µ
³
1
~ (r~0 ) − 3 Mx (r~0 ) sin(θr0 r ) cos(φr0 r )
dr
M
| ~r − r~0 |3
¶
´ (~
r − r~0 )
0
0
~
~
+ My (r ) sin(θr0 r ) sin(φr0 r ) + Mz (r ) cos(θr0 r )
| ~r − r~0 |
~ d (~r) = − µ0
B
4π
3 ~0
(B.1)
The meaning of the angles θr0 r and φr0 r is shown in Fig. B.1. As we are interested
in the effects of induced magnetic fields in the rotating frame, the above expression
can be averaged with respect to φr0 r , the azimuthal angle in the spherical polar
co-ordinate representation of the vector ~r − ~r0 , giving
~ d (~r) = − µ0
B
4π
Z
µ
µ
¶
³
´
3
1
− Mx (r~0 )x̂ + My (r~0 )ŷ 1 − sin2 (θr0 r ) . . . (B.2)
0
3
~
2
| ~r − r |
³
´¶
. . . + Mz (r~0 )ẑ 1 − 3 cos2 (θr0 r )
d3 r~0 .
This can be rearranged so as to separate magnetization-related and geometrical
terms, yielding
~ d (~r) = − µ0
B
4π
Z
´
3 cos2 (θr0 r ) − 1 ³ ~ ~0
M (r ) − 3Mz (r~0 )ẑ d3 r~0 .
2 | ~r − r~0 |3
(B.3)
B
Dipolar Field. . . making it easy to use
175
z
M
θr-r'
z
r
o
r-r '
y
r'
φr-r'
x
~ at position
Figure B.1: Geometry used to calculate the field generated at ~r by magnetization M
0
~
r.
The next step is to Fourier transform the magnetic field in order to find its form
in k -space
~ d (~k) =
B
Z
~ d (~r)ei~k·~r d3 r~0
B
(B.4)
By introducing a new vector quantity, ρ~ = ~r − r~0 , it is possible to factorize the
integral into two independent parts,


Z
2
ik·~
ρ
~ d (~k) = − µ0  d3 ρ~ 3 cos (θρ ) − 1 · e 
B
4π
2
ρ3
~
µZ
³
´
~ (r~0 ) − 3Mz (r~0 )ẑ ei~k·r~0
dr M
3 ~0
¶
(B.5)
~ (~r) and
The second integral involves three-dimensional Fourier transforms of M
~ (~k) and Mz (~k). The first integral of Eq. B.5 is not
Mz (~r), which can written as M
as trivial. Considering the relations between the different vectors shown in Fig. B.2,
it can be shown that
cos θρ = cos α cos β − sin α sin β cos(φz − φρ )
(B.6)
Consequently averaging over φρ yields
3 cos2 θρ − 1
3 cos2 αρ − 1 3 cos2 β − 1
=(
)(
)
(B.7)
2
2
2
so that integration with respect to φρ produces the following simplification
B
Dipolar Field. . . making it easy to use
k
176
ρ
β
α θρ
φρ
z
φz
Figure B.2: Relationships between the vectors ρ
~, ~z and ~k.
Z
3 cos2 β − 1 Z ∞ Z 1
3 cos2 α − 1 eikρ cos α
3 cos2 θρ − 1 eik·~ρ 3
d
ρ
~
=
2π
dρ
d(cos
α)
·
2
ρ3
2
2
ρ
0
−1
Ã
!
Z ∞
2
sin(kρ) 3 cos(kρ) 3 sin(kρ)
3 cos β − 1
dρ
= 2π
2
+
−
(B.8)
2
ρ
kρ
(kρ)2
(kρ)3
0
~
Although this function is not defined as ρ → 0, its Cauchy limit can be obtained
leading to the following overall result:
2
³
´
~ d (~k) = µ0 3 cos β − 1 M
~ (~k) − 3Mz (~k)ẑ
B
(B.9)
3
2
Which is the expression obtained by Deville et al. [44]. It is interesting to note
that it was not necessary to consider the way in which the modulation was generated
to produce this result.
In the case of uniform magnetisation which is modulated with a characteristic
spatial frequency km due to a gradient applied along an arbitrary direction ~s, it
would be as if we would have the dipolar field multiplied by a delta function with
a value different from zero at ~k = km ŝ. It is then possible to Fourier transform the
previous result back into real space. The result is the value of the dipolar field in
real space at the position ~s
2
³
´
~ (~s) − 3Mz (~s)ŝ
~ d (~s) = µ0 3cos(θs ) − 1 M
B
3
2
(B.10)
References
177
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