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Volume RF coils based on metamaterials for ultra-high field magnetic resonance imaging

Abstract and Figures

This manuscript aims to create new kinds of volume radiofrequency coils for UHF MRI (B0 ≥ 7 T) derived from the well-known birdcage coil. Their designs are inspired from metamaterials. Based on transmission line analysis, the radiofrequency B1 field and coil’s geometry can be shaped almost as will in order to obtain coils with unique features. At first, we propose a volume coil that eases access to its inner volume. This “opencage” coil does not only improve the comfort during MRI examinations, especially for patients suffering from claustrophobia but it can also be useful for some specific applications of MRI such as motion correction. A small-animal and a head-size prototypes working at 7 T have been built and tested. Then we propose to apply this metamaterial approach to develop dualband birdcages that are sensitive simultaneously to two nuclei. Finally, we introduce a new method of passive B1 shimming based on phase conjugation technique. We theoretically investigate this method for developing a head coil for 7 T imaging, whereas later we build a prototype for imaging of rodents in a 17.2 T scanner.
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Préparée à ESPCI Paris, France
Bobines volumiques à base de métamatériaux pour
l’imagerie par résonance magnétique à très haut champ
Volume RF coils based on metamaterials for ultra-high field
magnetic resonance imaging
Soutenue par
Le 28 Septembre 2020
Ecole doctorale n° 564
Physique en Île-de-France
Composition du jury :
Université de Lille, France Président
Christophe, CRAEYE
Universit´e catholique de Louvain, Belgium Rapporteur
Université de Nice-Sophia Antipolis, France Rapporteur
ESPCI Paris, France Examinateur
MPI for Biological Cybernetics, Germany Invited
Institut Fresnel, France Invited
Julien, de ROSNY
ESPCI Paris, France Co-directeur de thèse
Abdelwaheb, OURIR
ESPCI Paris, France Directeur de thèse
First of all, I would like to express my genuine gratitude to my supervisors, Julien and Abdel.
I could not imagine accomplishing of most of our goals without your wonderful scientific and
life advising during my PhD. I had been so lucky to work with you!
At second, I would like to acknowledge the members of the defense committee for their work.
Further, I glad to write thank you to some colleagues who was willing to contribute to my
work, and whom I was fortunate to work with.
Foremost, I would like to thank my colleagues from CEA NeuroSin (Gif-sur-Yvette, France).
I am very thankful to Alexandre Vignaud, Benoit Larrat, and Luisa Ciobanu for their support
during our joint experiments conducted in MRI. Benoit and Alexandre, I am also glad of your
help with tremendous amount of administrative work we did together.
I cannot not mention all my colleagues from Marseille team (Institute Fresnel, CRMBM
and Multiwave, France). Especially, I would like to mention Redha Abdeddaim who strongly
involved me into the M-CUBE project and helped a lot during all the time I know this person.
Marc, Tania and Marine thank you for your help over my project and for the nice time we
spent during the conferences.
I also would like to express my regards to Nikolai Avdievitch with MPI (Tübingen, Germany)
for his scrupulous advises over the engineering aspects of RF coils and its electronics!
Of course, I cannot forget my Russian colleagues Stas Glybovski, and Alexey Slobozhanyuk
from ITMO University (St. Petersburg, Russia). Thank you for huge support before and during
my PhD. I also would like to express best regards to my first supervisors from ITMO, Alexander
Timofeev and Andrei Anisimov thank you for the effort you put in teaching me in the very
beginning of my professional career.
Now, I would like to mention all my friends I had great time with. Pasha, Kolian, Kammel
and Slava thank you for the entertaining moment we spent together in Paris! Dima, we had
best fishing experience on the “Oyat” and “Izhora” rivers with you! Thank you also for being
close at the difficult times. Nikita, I would say just one thing: “there was a guy, but now there
is no a guy” (local joke).
Polina, I am very happy of being with you!
Lastly, I would emphasis the sincere appreciation to my parents Vladimir, Tania, and to my
sister Ksenia for their support during my life.
Introduction 1
1 State-of-the-art and novel techniques in ultra-high field magnetic resonance
imaging 5
1.1 Historical overview and emergence of MRI . . . . . . . . . . . . . . . . . . . . . 6
1.2 PrincipleofNMR................................... 7
1.3 Ultra-higheldMRI ................................. 10
1.4 RF coil classification and parameters . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.1 Surfacecoils.................................. 15
1.4.2 Volumecoils.................................. 17
1.4.3 Phasedarraycoils............................... 25
1.4.4 Metamaterial-based coils . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.5 Conclusion....................................... 31
2 Metamaterials theory for volume RF coils design 33
2.1 Birdcage principle and design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 Electrical representation of a birdcage . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Simplified electrical representation of a birdcage . . . . . . . . . . . . . . . . . . 36
2.4 Transmission line-based approach . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.5 Comparison of the proposed theory to the reference approach . . . . . . . . . . . 42
3 Opencage RF coil 45
3.1 Opencage coil for preclinical imaging at 7 Tesla . . . . . . . . . . . . . . . . . . 46
3.1.1 Theoretical aspects of the preclinical opencage coil . . . . . . . . . . . . 47
3.1.2 Numerical investigations of the opencage coil . . . . . . . . . . . . . . . . 49
3.1.3 Experimental prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.4 MRI assessment and comparison to the birdcage coil . . . . . . . . . . . 54
3.2 Quadrature opencage coil for clinical brain imaging at 7 Tesla . . . . . . . . . . 55
3.2.1 Theory of a quadrature opencage coil . . . . . . . . . . . . . . . . . . . . 58
3.2.2 The design of the opencage coil and full-wave simulations . . . . . . . . . 61
3.2.3 SARassessment................................ 62
3.2.4 The prototype and Tx/Rxinterface..................... 64
3.2.5 Experimental assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2.6 Phantomimaging............................... 68
3.2.7 In-vivoimaging ................................ 70
3.3 Conclusion....................................... 70
4 Dual-band volume RF coils 73
iv Table of contents
4.1 Dual-band nested coil for 19F/1H preclinical imaging at 7 Tesla . . . . . . . . . . 74
4.1.1 Design of the proposed coil . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1.2 Numerical simulations of separated structures . . . . . . . . . . . . . . . 76
4.1.3 Investigation of the combined structure . . . . . . . . . . . . . . . . . . . 76
4.1.4 Experimental validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 Dual-bandopencagecoil ............................... 80
4.2.1 Theory of dual-band opencage coil . . . . . . . . . . . . . . . . . . . . . 81
4.2.2 Head opencage coil for 13C/1Hat7Tesla.................. 82
4.2.3 Numerical investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3 Conclusion....................................... 86
5 Phase conjugate inspired metacage coil for UHF MRI 89
5.1 Phase conjugated volume coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1.1 Phase conjugation principle . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1.2 ApplicationtoMRI.............................. 91
5.2 Conjugated birdcage coil for 17.2 Tesla preclinical imaging . . . . . . . . . . . . 95
5.2.1 Theory of conjugated birdcage coil for 17.2 Tesla . . . . . . . . . . . . . 96
5.2.2 Numerical optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2.3 Experimental assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.4 Imaging with the conjugated birdcage . . . . . . . . . . . . . . . . . . . . 101
5.3 Conclusion.......................................102
Conclusions and perspectives 105
Publications 109
A Résumé Français 111
A.1 Introduction......................................111
A.2 Chapitre 1: Techniques de pointe et nouvelles technologies d’ imagerie par réso-
nance magnétique à ultra-haut champ . . . . . . . . . . . . . . . . . . . . . . . . 112
A.3 Chapitre 2: Bobine volumique à base de métamatériaux . . . . . . . . . . . . . . 116
A.4 Chapitre 3: Bobine RF d’ouverture . . . . . . . . . . . . . . . . . . . . . . . . . 116
A.4.1 Bobine d’ouverture pour l’imagerie préclinique à 7 Tesla . . . . . . . . . 118
A.4.2 Bobine opencage en quadrature pour l’imagerie clinique du cerveau à 7
A.5 Chapitre 4: Bobines de volume à double bande . . . . . . . . . . . . . . . . . . . 120
A.5.1 Bobine de volume à double bande pour l’ imagerie préclinique 19F/1H à 7T121
A.5.2 Bobine d’ouverture bibande . . . . . . . . . . . . . . . . . . . . . . . . . 121
A.6 Chapitre 4: Bobine de métacage inspirée du conjugué de phase pour l’IRM UHF 122
References 125
List of abbreviations
AFI: Actual flip-angle
BP: Band-pass
BS: Band-stop
CP: circular polarized
ESR: Equivalent series resistance
FA: Flip angle
FEM: Finite element method
FOV: Field of view
GRAPPA: Generalized autocalibrat-
ing partial parallel acquisition
GRE: Gradient echo sequence
HMA: Hybridized meta-atom
HP: High-pass
LHTL: left-handed transmission line
LP: Low-pass
MRI: Magnetic Resonance Imaging-
Magnetic Resonance Imaging
MRS: Magnetic Resinance Spec-
MTL: Microstip transmission line
NRMSE: Normalized root-mean-
square error
PC: Phase conjugation
pTx: Prallel transmit
RHTL: Right-handed transmission
ROI: Region of interest
SD: Standard deviation
SE: Spin echo sequence
SENSE: Sensitivity encoding
SIV: Signal invensity variation
SLR: Split-loop resonator
SNR: Signal-to-Noise Ratio
SRR: split-ring resonator
TEM: transverse electromagnetic
TR: Time Reversal
UHF: Ultra-high field
XFL: Magnetization-prepared turbo-
Fast Low Angle Shot sequence
Magnetic Resonance Imaging (MRI) is a non-invasive apparatus using non-ionizing radiation
developed for in-vivo imaging. MRI is based on nuclei magnetic resonance (NMR) that appears
with some nuclei having angular momentum (spin) [1]. Thanks to the high concentration of
nuclei (1H), the main application of MRI scanners is to provide quantitative anatomical images
of tissues based on proton resonance. However, MRI scanners are also useful for other purposes.
For instance, being combined with magnetic resonance spectroscopy (MRS), it can be used for
studying cellular metabolic processes [2], [3]. Taking benefit of the paramagnetic property
of deoxygenated red cells, cerebral activity can be monitored (functional MRI - fMRI) [4].
However, for some applications the signal-to-noise (SNR) ratio provided by current clinical
MRI scanners of field strength 1.5 Tesla and 3 Tesla is not sufficient [5], [6]. Because SNR
is directly related to field strength, ultra-high field (UHF) MRI scanners such as at 7 T are
promising and becoming more and more efficient tools [7]. They also open the door to new
applications such as X-nuclei imaging (exploiting of other nuclei than 1H) [3].
However, UHF MRI has certain unsolved issues [8], [9]. One of them is the decrease of
wavelength due to the increase of Larmor frequency. For instance, the wavelength in tissues for
proton at 7 T (298 MHz) is decimetric. Therefore, RF coils cannot be anymore considered as
magnetostatic probes but more as radio frequency antennas. It makes the development of such
antennas more tedious.
In this manuscript, we propose a method based on metamaterial analysis to develop volume
coils for UHF MRI. More precisely, we develop a new family of metasurfaces to build volume
coils with innovative features. The metasurface is made of a set of periodic resonators where the
last resonators are connected to the first one. The metamaterial approach allows to precisely
adjust the amplitude and the phase of the field that propagates along the ladder-like looped
metamaterial and therefore can be used to shape the magnetic field almost as will. Actually,
the birdcage coil that is one of the most efficient volume coils developed since the 80’s for MRI
belongs to this metasurface family.
The first application, to which we apply this approach, is for head imaging at 7 T. Typically,
in that case, a phased array is used in reception and a birdcage in the transmission [10]. Despite
having satisfactory performance, the combination of birdcage coil and phased array makes a
bulky setup. Thereby, it reduces patients’ comfort. In addition, this combination may not be
used for some applications, such as motion correction [11].
To that aim, here, we propose to use this metasurface approach to develop a birdcage-like coil
wherein the distance between the elements is not constant. We have called this optimized coil
- opencage [12]. This opencage coil facilitates access to a patient or a sample to be scanned by
the enlarged distance between certain elements of the coil. Two prototypes have been developed
at 7 T. The first one is dedicated to preclinical imaging of small rodents and the second one
to clinical imaging of human heads. Later, because X-nuclei becomes a common tool in UHF
MRI, this concept of opencage coil is expanded for dual nuclei applications.
The second application is related to the lack of B1field homogeneity in UHF MRI. It implies
SNR reduction in some regions wherein B1field is weak. This issue is especially associated
with brain imaging or body imaging because the wavelength (λ) in tissues is smaller than the
size of the volume under study. Parallel acquisition or transmission with phased coil arrays is
one solution to mitigate this issue. Actually, parallel acquisition is already in use for clinical
scanners working at 1.5 and 3 T [13], [14] as well as for 7 T [10]. As for transmission, the
technique is called passive [9] or active B1shimming [15]. The combination of transmit array
and receive array together provides outstanding performance, however, the cost and complexity
of the coils increases, while its reliability decreases.
A simpler solution is passive shimming exploiting dielectric non-resonant pads [16] or hy-
bridized resonator [17]. Here, we propose an intermediate alternative between full parallel
transmission and passive B1shimming with pads or resonators. This alternative is called con-
jugated birdcage coil or metacage. It allows focusing the B1field on another location than the
coil center. At first, we study this concept for 7 T brain imaging, then we show an experimental
demonstration at 17.2 T for preclinical imaging of small animals.
This manuscript is divided into one review and four original chapters:
Chapter 1 provides a historical overview of MRI emergence. The fundamental principles
of MRI are also described. Then, the pros and cons of UHF MRI are discussed. Subse-
quently, we explain the main properties and requirements for RF coils. We start from
the simplest example of a loop surface coil and then proceed with aspects of designing
conventional volume coils, and finally explain the interest of phased array coils for UHF
brain imaging. In the last section one provides short overview of metamaterials involved
in RF coil design for UHF imaging.
In Chapter 2 we describe the proposed principle of the metamaterial coils. Because it
can be interpreted as a generalization of birdcage coils, in the first part we remind the
theory of a conventional birdcage coil. In a second part, we provide an alternative to
the conventional theory that is based on the transmission matrix approach. The unit
cell is described in terms of phase shift and Bloch impedance. These 2 parameters, used
commonly in metamaterial theory, play a key role to develop metamaterial coils with
other geometries or functionalities than the birdcage ones.
Chapter 3 presents the new type of coil - opencage coil. First, we show that due to the
optimization, the conventional birdcage coil can be properly modified to have different
spacings between the unit cells (rungs). In the first part, the design is done for 1H
preclinical imaging of small animals at 7 T. In this case the coil is used as a transceive
Introduction 3
coil. The numerical simulations and experimental validations as well as in-vivo part are
presented. The second part of the chapter shows the result of transceive head opencage
coil for brain imaging at 7 T. Enlarged size of the coil and quadrature operation bring
new challenges in the coil design. In the end, the coil is tested in MRI with a phantom.
Chapter 4 shows two different dual-band coils. In the first part of this chapter, we develop
a nested dual-band coil based on a birdcage coil combined with a coil made with split-loop
resonators. This coil is optimized for 19F/1H imaging. The numerical analysis of the two
structures and numerical investigation are provided. In the second chapter we show that
the approach used previously for optimization of single frequency opencage coil can be
expanded for the dual nuclei operations wherein two resonant frequencies are far enough,
for example 13C/1H. In this chapter, we mainly focus on the theoretical and numerical
parts of the dual band opencage coil.
Chapter 5 raises some aspects of the design of a conjugated birdcage coil. This coil may be
an effective replacement of passive B1shimming that can be done using dielectric pads or
hybridized resonators. In this chapter, we provide a theoretical investigation of the setup
for brain imaging at 7 T. Then the proposed approach is applied for 17.2 T preclinical
imaging of small rodents. To that end, we perform the numerical and experimental
Eventually, the conclusion of the manuscript is summarizing achievements of the manuscript
and giving a glance of the future of RF coils in UHF MRI.
State-of-the-art and novel techniques in ultra-high field magnetic
resonance imaging
Table of contents
1.1 Historical overview and emergence of MRI . . . . . . . . . . . . . . . . . . 6
1.2 PrincipleofNMR ................................ 7
1.3 Ultra-higheldMRI............................... 10
1.4 RF coil classification and parameters . . . . . . . . . . . . . . . . . . . . . 14
1.4.1 Surfacecoils................................ 15
1.4.2 Volumecoils................................ 17
1.4.3 Phasedarraycoils............................. 25
1.4.4 Metamaterial-based coils . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.5 Conclusion .................................... 31
Chapter 1. State-of-the-art and novel techniques in ultra-high field magnetic resonance
In this chapter we describe the most important aspects of magnetic resonance imaging
(MRI). A short history of the nuclear magnetic resonance (NMR) applications is reviewed in
section 1.1. The physical principle of MRI is given in section 1.2. The next section reveals the
pros and cons of the recent introduction of ultra-high field (UHF) magnetic resonance imaging.
As this work is devoted to the development of new RF coils for UHF MRI, section 1.4 presents
the basic principles and parameters of RF coils. Subsequently, in sections 1.4.1-1.4.3, we show
currently existing solutions of surface, volume, and phased arrays coils. Eventually, in 1.4.4 we
focus on metamaterial-based coils for MRI.
1.1 Historical overview and emergence of MRI
The story of NMR originated in 1938, when Isidor Rabi discovered and measured NMR effect in
molecular beams [18]. The NMR implies resonant absorption and emission of electromagnetic
waves at the so-called Larmor frequency of ω0by the matter containing nuclei with nonzero
spins (angular momentum) and being placed inside a static magnetic field B0. For this discovery,
Isidor Rabi was honored by the Nobel Prize in 1944. Subsequently, his discovery was improved
in 1946 by Felix Bloch and Edward Purcell, who observed NMR in liquids [19] and solids [20].
However, at that moment these discoveries did not turn the NMR into the imaging applications.
The application of NMR for imaging, initially called NMR zeugmatography, started with the
work of Lauterbur in 1973 [1]. In his work, he showed that planar images based on NMR effect
can be obtained when the object under interest is placed in a spatially varying static magnetic
field. Indeed, in such a case, because the resonance frequency of a nucleus is dependent on
the magnetic field strength, the position can be recovered from a spectral analysis of the RF
signal generated by the nucleus spins. Later, in the late 1970s the fast post-processing method
was developed by mathematician Peter Mansfield [21]. With his development, and NMR as a
multidimensional medical imaging technique became possible. For these investigations, Paul
Lauterbur and Peter Mansfield shared the Nobel Prize in medicine won in 2003.
In 1980s these methods as well as general improvements of semiconductors-based electronics
made possible industrial production of NMR scanners, which was renamed MRI because the
fear of nuclear danger, despite non-ionizing radiation used in NMR. At that time, most of the
MRI scanners were low-field with field strength (B0) below 0.5 T [22]. These scanners had a
lot of restrictions and limitations, for instance the low SNR of acquired images.
To overcome these limitations, GE Research Center developed a full-body MRI scanner in
1980 [23]. This strength of static magnetic field of this scanner was 1.5T. Such increase in
strength allowed to solve many issues attributed to low-field full-body MRI scanners such
as low SNR and engineering difficulties of designing relatively small RF coils compared to
wavelength [23]. During the 1980s, 1.5 T MRI scanners became essential and flexible clinical and
preclinical non-invasive tools with shipment of over 20000 units at these days [22]. Nowadays,
the vast majority (70%) of the MRI scanners is still working at 1.5 T [22]. More recently 3 T
1.2. Principle of NMR 7
scanners also became largely distributed in hospitals [24], [25]. Later, more complicated systems
working at 4 T [26], 4.7 T, 7 T [8] and 8 T [27] have been introduced for research purposes.
However, most of them are not considered as a common system for daily operations except 7
T systems. Eventually, one concludes that currently promising system for clinical routine is 7
T [25].
These 7 T systems provide ultimately high SNR compared to lower field systems such as 1.5
T and 3 T [5]. However, the technology becomes more expensive and difficult due to several
reasons. For, instance the higher resonance frequency at 7 T leads to RF field inhomogeneities
because of the smaller wavelength (λ1m in free space and λ12 cm in tissues). Increase
of magnetic field strength requires also more care about magnetic compatibility of the used
components and the environment around the scanner. Phased array coils are preferred for
UHF imaging [13], [10], because they allow to get rid of most flaws of the conventional coils,
such as birdcage coil [9] or TEM coil [28]. The additional drawback is that the magnet employed
in 7 T MRI is heavy, which requires especially adapted large rooms for carrying the system.
Moreover, magnetic field penetrates thought walls, closets, and other environment, therefore,
the room the scanner must be well shielded.
Eventually, even higher field strength full-body systems of 9.4 T [29] and 10.5 T [30] were
invented and used purely for research purposes of imaging and spectroscopy. The cutting-edge
full-body system, being deployed at CEA NeuroSpin (France) has recently reached 11.7 T [31].
In contrast to full-body scanners, there is another branch of scanners, called preclinical
scanners for small animals or for humans. These scanners are mainly used for research purposes.
The field strength in this scanner is typically equal to 4.7 T or 7 T. However, the cutting-edge
MRI preclinical studies can be performed with magnets up to 21.1 T [32], 23.5 T [33], and
even very recently 28 T working at 1.2 GHz [33]. To conclude, we have quickly reviewed the
history of MRI, subsequently in thbee next chapter we reveal some fundamental aspects of that
1.2 Principle of NMR
NMR is based on the Larmor precession of some nuclei in a strong static magnetic field B0[34].
Indeed, a magnetic field B0exerts on a particle showing a magnetic moment µ, a torque:
Γ = µ×B0.(1.1)
The magnetic moment µof the particle is proportional to its angular momentum L, i.e., µ=γL
where γis gyromagnetic ratio. The angular momentum is solution of the fundamental relation
of dynamic, i.e.,
Γ = dL
dt .(1.2)
Consequently, the angular momentum is precessing with a frequency (Larmor frequency) given
Chapter 1. State-of-the-art and novel techniques in ultra-high field magnetic resonance
However, if this classical view provides an intuitive interpretation of the Larmor frequency, only
quantum mechanics can correctly describe the phenomenon. First, this theory is required to
justify the intrinsic spin of protons and neutrons and therefore the spin of nucleus. Moreover,
within this framework, the Larmor frequency is more related to the energy level difference
between two consecutive nucleus spin states. For instance, for the hydrogen 1H, the spin can
be either in two spin states. Either the component of spin parallel to magnetic field is in the
same direction as this last (state 1
/2or up) or in opposite direction (1
/2or down). Because of
the interaction of the magnetic moment with the magnetic field, these 2 states show different
Thereby, when a material is being placed into a strong static magnetic field B0, at thermal
equilibrium, the lowest state of energy of the spin is more populated than the higher one(s)
and the magnetization Mbecomes aligned to that B0field (Fig. 1.1).
Figure 1.1 Principle of NMR magnetization flip.
However, the population can be modified by applying an additional RF magnetic wave packet
1) oscillating at the Larmor frequency with a circular polarization that is transverse to
the static B0field. The rotation direction should be in the same as the one of the Larmor
precession [35]. Due to this stimulated excitation of the spin state, the magnetization M
becomes progressively parallel to B1field. Hence, the maximum flip angle (FA) is 90.
When the excitation of B+
1field is interrupted, the nuclear spins progressively stop being
aligned, and as a result, the magnetization Mgoes back to its initial state. During this process,
an echo, i.e., a RF field, B
1is generated by the spins at the Larmor frequency [35].
Two relaxation processes occur: the longitudinal relaxation (along axis parallel to B0) and
the transverse relaxation (in the plane perpendicular to B0). Two characteristic times are
associated with these processes, T1and T2respectively.
1.2. Principle of NMR 9
This echo can be collected by the same coil as the transmit one or another receive coil [36].
The resonant frequency of the particular nuclei and therefore, the working frequency of the
transmit/receive coil as well is defined by the Larmor equation stated before. Because, the
collected signal is very weak, the receive coil should be well tuned, and impedance matched at
the working frequency in order to minimize the noise. The concrete properties of receive and
transmit coils are presented in Chapter 1.4.
Spatial encoding of signal is based on an additional magnetic field Gcreated by a gradient
system that adds to the static field B0. Note that the gradient can be changed during the
acquisition. As a result, spatial encoding can be done by receiving the signal at different
frequencies according to:
ω0=γ[B0+G(x, y, z, t)].(1.5)
The raw data are the transversal components of the magnetization in the imaging object gath-
ered from the receive coil and stored as a function of time during the acquisition process. One
can show that in a transverse slice and with linear gradients, the raw data are directly linked to
the k-space Fourier transformation of the magnetization. By changing the gradient over time,
the complex k-space data can be sampled. Every point in the raw data matrix contains certain
information related to the complete image, however this point does not correspond to a point
in the image matrix. The “image” is then recovered by an inverse Fourier transform (see Fig.
Figure 1.2 On the left: raw k-space data and on the right the inverse Fourier transform corre-
sponding to an image [37].
Because of the gradients, the RF excitation pulse should be provided not at the single fre-
quency but in a narrow frequency band, which typically equals few a kHz [38].
The process of applying radiofrequency pulses, acquiring their echoes and tuning simulta-
neously the gradients is called a pulse sequence [39]. Generally, there are two basic types of
sequences, spin echo (SE) applying 90pulse called flip angle (FA) [40] and gradient echo se-
quence, with usually smaller FAs αfrom 10to 80[41]. Another type of sequence such as
AFI [42] or XFL [43] can be used to map the RF field. In addition, depending on the appli-
cations of MRI, magnetic resonance spectroscopy (MRS) and functional MRI (fMRI) different
specific sequences should be employed. However, developing pulse sequences is not a target
Chapter 1. State-of-the-art and novel techniques in ultra-high field magnetic resonance
of the current PhD thesis, which focuses on designing of new RF coils using metamaterials
As it was mentioned before, the frequency of NMR is defined by Larmor equation, wherein
the gyromagnetic ratio depends on the nuclei. This ratio for some compatible nuclei is shown
in Tab. 1.1 [35], [44].
Table 1.1 The gyromagnetic ratio of some nuclei commonly used in MRI, MRS and fMRI.
Nucleus γ/2π(MHz/T)
Proton 1H 42.6
Phosphorus 31P 17.2
Fluorine 19F 40.1
Sodium 23Na 11.3
Lithium 7Li 16.5
Oxygen 17O -5.7
Carbon 13C 10.7
Amid this plurality of nuclei, the appropriate one for anatomical image is proton 1H. There-
fore, it is exploited in most clinical and preclinical imaging applications. The proton 1H has
almost the highest frequency in the NMR spectra. Moreover, there are also a bunch of nuclei,
called X-nuclei (see Tab. 1.1), appropriate for imaging and spectroscopy used for the different
medical or research purposes [45]. The application of MRI and MRS is very broad, it can be
used for many different applications. For example, tracking of injected cells into a body or de-
tection and quantification of immune cell therapy for cancer using fluorine 19F [46], [47]; study-
ing tissue metabolism using phosphorus 31P associated with many disorders and diseases [48].
Sodium 23Na imaging can be used for elucidating the effects of sodium disequilibrium in mul-
tiple sclerosis pathology using [7]; Study of Alzheimer’s disease, Huntington’s disease, epilepsy
and other [3]. Carbon 13C can be used imaging breast cancer using hyperpolarized carbon [49],
Mapping of metabolism is possible with deuterium 2H [50]. Some examples of MRI images are
presented in Fig. 1.3.
1.3 Ultra-high field MRI
Since the clinical emergence of MRI in the early 80s, the strength of magnetic field in MRI
increased from bellow 0.5 T to 1.5 T and 3.0 T in clinical daily routine [6]. These values
1.3. Ultra-high field MRI 11
Figure 1.3 (a) Anatomical (1H) images of human brain [51]; (b) merged images (1H and 19F)
of mouse thorax [52]; (c) 23Na images of human brain at multiple field strength (1.5
T, 3 T and 7 T) [7].
correspond to moderate and high field strength, respectively. As explained before, the strength
of the next generation of scanners is going to Ultra High Field (UHF). Even though MRI
examinations can be conducted on scanners working between 9.4 T and 11.7 T, the most
promising field strength for clinical routine is 7 T field [6].
The main advantage of UHF scanners, such as 7 T, is to provide a greater SNR compared
to 1.5 T and 3 T systems [5], [7]. Higher SNR allows detecting earlier stage of diseases and
in general allows improving quality of acquired images. Moreover, for the most spread MRI
systems of 1.5 T the SNR for X-nuclei is dramatically small [see Fig. 1.3(c) for an example
of 23Na imaging] and its resonant frequency may be very low. Such a low SNR makes MRI
at 1.5 T using X-nuclei less informative, whereas at 7 T the SNR and resonant frequency is
acceptable [3].
Basically, the fundamental increase of SNR comes from spin magnetization M. Indeed this
last is expressed by the formula [22]:
4kT B0,(1.6)
where ρ0is the proton spin density, γis the gyromagnetic ratio, }is the reduced Planck
constant, kis the Boltzmann constant, Tis the absolute temperature (in Kelvin) and B0is the
static magnetic field. We observe that the magnetization Mscales linearly with field strength
B0. However, the SNR does not depend linearly on B0field. In practice, the SNR depends
on many factors, for example, the spatial resolutions (voxel size), thermal noise of the sample,
noise of the electronics, quality factor of the coil as well as the static B0field.
Chapter 1. State-of-the-art and novel techniques in ultra-high field magnetic resonance
This non-linear improvement of SNR with field strength can be observed in Fig. 1.4(a).
Increase of temporal SNR is especially important for functional MRI (fMRI) [see Fig. 1.4(b)]
where the duration of a sequence and therefore SNR is limited by the characteristic evolution
time of the function under study.
Figure 1.4 (a) The SNR of images obtained in different parts of brain using 3 T, 7 T and 9.4 T
MRI [5]. (b) The tSNR maps in the brain at 3 T and 7 T [31].
However, one of the major drawbacks of UHF MRI is the B1field homogeneity because of
the wavelength decrease (Fig. 1.5). For instance, at 7 T, the wavelength is as small as λ12
cm in tissues [8]. As a result, a conventional coil cannot generate a uniform field on some
large organs such as head. Dark spot occurs where the RF field is not high enough to properly
flip the magnetization (flip angle) leading to SNR drop. One solution to mitigate this issue is
transmit phased arrays wherein the amplitudes and phases of each element can be passively
tuned [9]. This technique to homogenize the field is called passive B1shimming. There is also
active B1shimming or parallel transmission (pTx) that allows to modify phases, amplitudes as
well as pulse shape during running the sequence [15], [53], [54]. However, pTxhas certain flaws
related to the fact that the B1field pattern from each independent element of the phased array
should be analyzed independently for each subject. Consequently, pTxis a time-consuming
procedure, which requires measuring subject-specific field maps and does not fit into a routine
clinical scanning. Furthermore, stated adjustment leads to fine adjustment of the transmit
power allowing to finely tune the SAR [9]. In contrast to phased arrays, conventional coils are
usually much closer to the SAR limits because of the higher power needed to converge the FA
on dark spot [22].
Figure 1.5 The simulated images of the human brain at different magnetic field strength using
a birdcage coil [9]. Inhomogeneity is clearly seen after 6T.
Of course, the same issue of homogeneity occurs in reception. Again it can be overcome using
phased arrays (parallel imaging) as shown for example in references [10] and [51]. A significant
1.3. Ultra-high field MRI 13
increase in SNR with 32 channels received phased array coil compared to birdcage coil can be
seen in Fig. 1.6(a). This effect is less explicit when a 8 channel Txarray replaces a birdcage
[Fig. 1.6(b)] where an overall increase in the mean SNR is about 10%[51]. However, use of Tx
arrays allows to homogenize B1field map according to RF shimming.
Figure 1.6 The SNR maps in the brain: (a) maps obtained with 32 channels receive coil (Tx
birdcage) and Tx/Rxbirdcage coil [10]; (b) maps obtained with 8Tx/32Rxarrays
and commercial 1Tx/32Rxarrays [51].
In addition, at UHF field, for instance 7 T, the examination takes more time than at 1.5 T
and 3 T. The cause is the increase of T1relaxation time. Moreover, it is also known that a
strong static magnetic field can introduce physiological effects such as dizziness, nausea, mag-
netophosphenes, or metallic taste [3], [55]. Therefore, it is important to decrease the scanning
time for the sake of patients’ comfort. One solution is to perform accelerations available in
parallel imaging. These acceleration methods are SENSE [56] (Sensitivity Encoding for Fast
MR) and GRAPPA [57] (Generalized Autocalibrating Partially Parallel Acquisition). Basically,
they are based on under sampling in phase-encoding steps. In fact, both technologies are not
especially done for UHF, and can be exploited with any scanner supporting multi-channel re-
ceive coils. However, acceleration techniques sacrifice the SNR of the images [56]. Indeed, in
case of SENSE the SNR can be estimated by the following formula:
In this formula gis a spatially dependent term called geometrical factor, or g-factor, Ris a
parallel imaging acceleration factor.
At last, one would like to briefly note that the increase of magnetic field strength requires
more care about magnetic compatibility of the electrical components and the environment
around the scanner, such as extinguishers, stretchers, furniture, and others. For example, one
situation appeared in 2001, when a boy was killed, when magnetized extinguisher started flying
in the MRI room [58].
Eventually, all stated above challenges make UHF MRI, and RF coils quite expensive com-
pared to 1.5 T and 3 T. To conclude, we have reviewed the advantages and challenges of UHF
MRI, and now we would like to describe the framework devoted to resolve some fundamental
issues of UHF MRI. The name of the project is M-CUBE.
Chapter 1. State-of-the-art and novel techniques in ultra-high field magnetic resonance
M-CUBE (MetaMaterials for MRI) project. Our investigation has been done in the
framework of European M-CUBE project [59]. This project gathers 8 universities and 2 com-
mercial partners located all around the globe. The project is aimed to go beyond the limits
of clinical UHF MRI. For example, UHF MRI scanners are constrained by the lack of homo-
geneity of the acquired images and by limitations related to the specific absorption rate (SAR)
associated with the transmit RF coil. By gathering MRI experts and physicist working in the
cutting-edge domain of electrodynamic-metamaterials, it is expected to solve some fundamental
restrictions in UHF MRI due to RF coils. Before considering metamaterial inspired coils, we
review conventional coils and phased array coils.
1.4 RF coil classification and parameters
Here we discuss the classification of RF coils employed for UHF MRI. Design and evaluation
of properly selected RF coils are critical for a safe and successful MRI imaging. To remind,
RF coils are required for exciting the nuclei’ spins (flipping their magnetization) with B1field
(transmit coil) and probing nuclei echo (receive coil). These two operations can be done by a
single coil (transceive), or by two distinct coils (transmit only receive only - ToRo) [36].
Transmit coils Contrary to receive coils (see below) the dimensions of transmit coils are
usually larger. The parameters that are important to assess performances of transmit coils are:
Transmit efficiency ( B+
P) related to the input power [60];
Homogeneity of magnetic field B+
1, which can be accessed with normalized root mean
squared error (NRMSE) or normalized standard deviation;
Specific absorption rate (SAR) measured in W/kg, or what is more relevant - SAR ef-
ficiency, defined as ( B+
). Thus, the input power has certain limits related to a
SAR limit [61]. Moreover, the inhomogeneity in electric field can produce some local hot
spot, which is undesirable. This problem can also be overcome with Txphased arrays by
adjusting amplitudes and phases in each element [3].
Receive coils In contrast to transmit coils, the ideal receive coils should be as small as
possible to tight fit and be the closest to the scanned sample. The Rx coil should be sufficiently
sensitive to provide high SNR. The average SNR of the image can be defined as mean signal
over the standard deviation of noise. Receive coils are mainly characterized by
Homogeneity of magnetic field B
1, which can be assessed with normalized root mean
squared error (NRMSE) or normalized standard deviation; Note that contrary to trans-
mission, in reception, the generated signal is due to the counter-clockwise circularly po-
larized magnetic field.
SNR map that is related to the received efficiency
1.4. RF coil classification and parameters 15
Generally, the plurality of RF coils can be classified into two groups depending on the appli-
cation: surface coils and volume coils
1.4.1 Surface coils
The loop coil is one of the basic coils for MRI imaging. This coil is useful for many applications,
wherein it is less important to obtain a signal with high SNR from the whole sample than to
acquire as much signal as possible from a small region of interest (ROI) [62]. The surface loop
coil originates in 1970, when it was used to measure the blood velocity using a magnet of 0.36
T [63]. These days, loop coils are widely used for both transmit and receive regimes [64], albeit
they can be used as receive probe only [65]. It is widely available commercially for any field
strength [66], [67]. For example, preclinical imaging of small animals’ organs; clinical imaging
at 1.5 T and 3 T of wrist, joints of toes and fingers, skin, temporo mandibular joints imaging,
knees. Several preclinical and clinical commercial examples related to surface loop coils are
depicted in Fig. 1.7. In practice, the surface coil may have different shapes [68] or it may be
flexible as well [69] in order to tightly fit the area under examination.
Figure 1.7 (a) A 20 mm balanced receive only surface loop coil [64]; (b) earlier surface loop
coil design performed for study of a rat brain [70]; (c) tuning, matching balancing
and detuning circuit for receive only surface loop coil [70]; (d) commercially available
clinical receive loop coils [66].
First of all, we would like to consider the physical principle of a surface loop coil. The sketch
of a typical surface loop coil is shown in Fig. 1.8(a, b). In such a loop, the current I flowing
around the loop produces a magnetic flux B that is coaxial to the ring axis.
The produced B field of this setup schematically depicted in Fig. 1.8(a) can be calculated
using Biot-Savart law [70], [72]. According to this law the expression of the axial component
(By) can be expressed as [71]:
(a2+y2)(3/2) .(1.8)
Chapter 1. State-of-the-art and novel techniques in ultra-high field magnetic resonance
Figure 1.8 (a) Schematic view of the surface loop coil [70], (b) B1field distribution of surface
loop coil, (c) magnetic field strength of surface loop coil [71] over the loop axis.
Where µ0is the magnetic constant, Iis the currents flowing around the loop, ais the loop’s
radius, yis the distance from the loop’s surface. As it can be seen and according to the formula
above, the field quickly decays along the axis of the loop [Fig. 1.8(c)], and therefore the efficiency
of the coil also decreases toward the axis of the loop. Thus, to increase the penetration depth,
the loop’s diameter can be increased. However, this diameter increase sacrifices the SNR [70].
In Fig. 1.9, we show the spatial dependence of SNR for several loop sizes.
Figure 1.9 Dependence of SNR as a function of distance for loops of different sizes and for the
volume coil [73].
Usually, the perimeter of the surface loop is much smaller than wavelength. From an electrical
point of view, this wire can be represented by an inductance Lthat can be easily calculated
according to [71], [74]. To be resonant at the desired frequency of ω, capacitances of value C
have to be inserted in the loop [Fig. 1.7(c)]. To resonate at ω, the capacitance value should be
given by
1.4. RF coil classification and parameters 17
where nis the number of loop segments. In the loop design, it is important to meet the rule
wherein the diameter of the coil should be smaller than λ/20 of wavelengths to stay much below
its geometrical resonance that occurs when the perimeter is of the same order as the wavelength
. In this case, the loop is considered as electrically small and it leads to homogeneous current
distribution on it. As a result, the magnetic field distribution is symmetrical. At moderate
field, for instance 1.5 T, λin free space equals 4.7 m, while at 7 T it becomes only 1 m.
Therefore, the maximal available diameter is 5 cm. In order to increase this geometrical size,
the loop must be electrically shortened with capacitors distributed around the circumference.
This segmentation also induces electrical field reduction, which is important to stay below the
SAR limit in the transmit regime.
The surface coils are also appropriate for multi nuclei imaging and spectroscopy. After small
modifications, the conventional loop may be suitable to resonate at two-frequency bands. It
can be achieved, for example, by switching the the matching/tuning circuit as it is explained
in [71]. Another method that can be used is multipole insertion method, wherein an additional
resonant LC circuit inserted to the coil makes that last resonate at 2 frequencies [70].
1.4.2 Volume coils
There are two kinds of cylindrical volume coils that can either produce axial [Fig. 1.10(a)] or
transverse [Fig. 1.10(b)] magnetic field with respect to their axis within their volume. In this
work we do not consider axial resonators, such as a solenoid [75] that is a very common coil
for example at 17.2 T imaging [76]. Transverse resonators are well adapted for head or body
imaging because they generate magnetic field perpendicular to the bore axis. The basic principle
of transverse volume coil shown on Fig. 1.10(b) is based on a cylinder with surface current
J(ϕ)flowing parallel to the cylinder axis(ϕis the azimuthal angle). When the phase of current
equals the angle ϕ, a homogeneous magnetic field B1is generated inside that cylinder [77] at
least for scanners up to 4 T when considering brain imaging for example.
There are many different ways known in the literature to realize the principle shown in
Fig. 10 (b) : cosine coil [78], saddle coil [79], birdcage coil [80], transmission line or TEM
coils [81], [82], [83], [84], ... All these realizations are based on the same approximation: the
continuous conductive cylinder is replaced by a set of conductive wires (legs) parallel to the
cylinder axis. The greater the number of legs and the more homogeneous the magnetic field.
Birdcage coils A birdcage coil is a volume resonator invented in 1985 by C. Hayes [80]. The
advantage of the birdcage coil is to generate a highly homogeneous magnetic field inside the
resonator’s volume. However, this general increase of homogeneity is at the cost of an increase
of complexity. In the beginning, the birdcage coil was proposed for full-body imaging [80], while
later it was used in many applications at various field strengths [85], [86], [87]. A birdcage coil
is composed of Nparallel conductors (legs or rungs). Each of them is at the edge of a regular
polygon as it is shown in Fig. 1.11(a). Leg extremities are interconnected by way of end rings.
Depending on the application; the number of legs Ntypically ranges between 4 and 32. The
currents along the longitudinal legs generate the transverse magnetic field results from . These
currents result from the electrical ladder network composed of Nunit cells. Each element
consists of one leg inductance and two end-ring inductances. In addition, each unit cell has
Chapter 1. State-of-the-art and novel techniques in ultra-high field magnetic resonance
Figure 1.10 (a) Surface current [77] of axial resonators; (b) Surface currentJshowing cosine-like
amplitude with azimuthal angles around the infinite cylinder produces homogeneous
B1field inside that cylinder.
capacitors placed in the legs in the end rings or in both segments [Fig. 1.11(b)]. These 3 kinds
of unit cell correspond to a low-pass, a high-pass and a band-pass filter, respectively
Figure 1.11 (a) Schematic view of the conventional birdcage coil composed of eight rungs; (b)
unit cell of the conventional birdcage coil; (c) low-pass birdcage coil for preclinical
imaging at 4.7 T [77]; (d) commercial head birdcage coil for clinical imaging from
GE healthcare [88].
By using an analysis based on the birdcage unit cell, it is possible to define the capacitance
needed to tune the fundamental mode of the birdcage coil to a desired Larmor frequency.
The detailed analytical analysis of the birdcage coil is provided in Chapter 2. Eventually, the
different examples of preclinical and clinical birdcages are presented in Fig. 1.11(c, d).
A birdcage is a resonant system that exhibits resonant modes. For instance, the reflection
parameter of a low-pass and a high-pass birdcage are shown in Fig. 1.12(a) and 1.12(b),
respectively. The total number of modes kdepends on the number of rungs N. Note that N
1.4. RF coil classification and parameters 19
positively acts on homogeneity of B1field, whereas negligibly affects the field amplitude in the
center [55]. The current distribution on the leg and the magnetic field of 6 modes of a low pass
birdcage are shown in Fig. 1.12(c).
The fundamental mode (k= 1) is the mode that generates a homogenous field inside the
birdcage. We observe in [Fig. 1.12(a)] (respect. [Fig. 1.12(b)]), that this mode occurs at the
lowest (respect. highest) resonant frequency for low-pass (respect. high-pass) birdcage.
Figure 1.12 S-parameter spectrum of the low-pass [77] (a) and high-pass (b) birdcages [71]; (c)
Top: current distribution associated to mode k. Bottom: Magnetic field distribution
associated to mode k.
These different topologies have different properties and features. For instance, high-pass is
preferable for the coils at UHF MRI, whereas low-pass is better for lower field MRI because
the fundamental (k=1) mode of the LP birdcage is lower in terms of frequency compared to
the HP birdcage coil of the same size [88].
In addition, a birdcage coil supports quadrature operations (circular polarized mode), which
can increase the SNR by q(2) [89], [90]. Circular polarization results from two orthogonal
linear modes that can be excited in a birdcage coil at the same frequency (degenerated modes).
These modes are well decoupled due to their orthogonality. They are excited by 2 ports that
are 90distant. The isolation level (|S12|) for the properly assembled coil is typically equal
to -20dB. At the same time, because the modes are well decoupled it is possible to tune and
match them independently. The frequency tuning is done by a capacitor that is 90apart if
the feeding ports are placed in ring (parallel to the capacitor) and 180apart if the feeding
ports are placed between ring and shield. The impedance matching can be done by placing a
capacitor in serial or parallel to the feeding port. The development of a quadrature birdcage
coil is slightly more complicated than a linear polarized birdcage coil because it also requires
a power splitter providing 90phase shift between two feeding ports and 2 Tx/Rxswitches for
each receiving channel.
Chapter 1. State-of-the-art and novel techniques in ultra-high field magnetic resonance
The head birdcage coils are convenient to handle at magnetic fields up to 7T for proton imag-
ing and even higher field when exploiting X-nuclei because they usually have lower frequency.
Talking about UHF fields such as 7 T, the birdcage coil plays the role of a transmit head
coil, while another phased array coil is typically employed for receive. In this case, the transmit
birdcage coil should be detuned out of the resonance frequency in the receive regime to not
perturb the receive array coils. One example of this detuned coil is provided in [36]. One of the
best ways to provide detuning is inserting at each leg or end-ring segment an active controlled
switch based on PIN-diodes creating open circuit, i.e., blocking currents flow. At the same
time additional current path should be provided for controlling the PIN diodes by DC current.
Moreover DC lines must be separated from RF signal using chokes [36].
A birdcage coil can also be designed for dual-nuclei operations. Many different designs are
known in the literature. For instance, four ring birdcage [91], dual-band, band-pass or band-
stop birdcage [92], nested birdcage for dual-nuclei operations [93], birdcage with even or odd
capacitors [94]. Some of them are presented in Fig. 1.13. Depending on the chosen applications,
the proper design must be selected. These approaches can be used when Larmor frequencies of
both nuclei are far enough.
Figure 1.13 Design of the several dual-tuned birdcages: (a) band-pass; (b) band-stop [92]; (c)
nested birdcage coil [93]; (d) four ring birdcages [91].
TEM coils TEM coils or, in other words, coupled microstrip line (MTL) coils [81], [82], [84]
can advantageously replace birdcage coil for head from 4.7 T up to 9.4 T. The typical examples
of TEM resonators are presented in Figure 1.14. Contrary to the birdcage coil wherein the
1.4. RF coil classification and parameters 21
adjacent legs are capacitively coupled through electric field, in the TEM resonators the adjacent
elements are mainly mutually coupled through the magnetic field (inductive coupling).
Figure 1.14 Geometry of different TEM coils: (a) the first prototype of TEM coil based on
tunable coaxial resonators [81]; (b, c) Coupled MTL [82], [84]; (d) feeding and
matching of coupled TEM resonator [84].
As in the birdcage coil, in a TEM resonator, a total number of modes depends on the number
of legs. However, among these modes, the mode “k= 1” has a homogeneous magnetic field
distribution inside the resonator. Despite, this mode locates at the lowest frequency in the
spectra, it can be tuned over a broad frequency range. The TEM coil supports this mode even
at 9.4 T for full-body imaging at UHF field [95]. The S-parameters and the modes of the TEM
coil are shown in Fig. 1.15.
Figure 1.15 (a) S-parameters of the coaxial tubes TEM resonator; (b) modal structure of the
TEM resonator (magnetic field) [81].
As it can be seen the separation between the modes is only 12 MHz that may lead to some
issues in experiments especially at UHF whereas for the HP birdcage coil this separation is
about 50 MHz.
The TEM coils are compliant with quadrature regime [84] and dual nuclei operations [96].
Being compared to the birdcage, the TEM coil may show greater performance, especially at
Chapter 1. State-of-the-art and novel techniques in ultra-high field magnetic resonance
higher frequency. This can be explained by the relatively high losses in the small capacitances
needed to tune head birdcage coil [Fig. 1.16(a)]. For a TEM coil of the same size as a birdcage
coil the self-resonance is higher in frequency compared to LP or HP birdcages [Fig. 1.16(a)].
Furthermore, the radiation losses are lower in a TEM coil compared to a shielded high-pass
birdcage [95]. The graph of radiation resistance for several volume coils with respect to the
frequency is shown in Fig. 1.16(a). Because TEM coils are less radiative and inductive than
birdcages, full-body TEM coil can be used up to 8 T (350 MHz), whereas birdcages are limited
to 3 T (128 MHz). In addition, here the authors observed that the extend of the magnetic field
B1generated by a TEM body coil is slightly larger than the one obtained with a shielded HP
body birdcage coils at 3 T [see Fig. 1.16(c)].
In the work [84] authors compared quadrature TEM head coil to the quadrature HP head
birdcage coil optimized for 4 T (170 MHz) brain imaging. The in-vivo images of these coils are
shown in Figure 1.16. The overall SNR increase with the TEM coil is about 53%in the human
head over that of the RF shielded HP birdcage coil [Fig. 1.16(d)]. In addition, TEM coil shows
a signal intensity variation (SIV) 43%lower than the shielded HP birdcage coil. In general,
the reported coil does not need an additional RF shield whereas it is strongly required by the
birdcage coil to avoid Q factor degradation.
Figure 1.16 (a) Equivalent capacitors with respect to the frequency plotted for several volume
coil [95]; (b) radiation resistance with respect to the frequency; (c) field distribution
along the axis of the several resonators [95], (d) images of a brain acquired with
two different coils [84].
Half-volume coils Another sort of volume coils that we would like to mention is half-volume
coils, which designs perfectly matches with one of our goal. The simplest way to design an open
1.4. RF coil classification and parameters 23
resonator is to remove the top half of a birdcage coil [97], [98] or half-volume TEM coil [99].
One example of half-birdcage coil devoted to shoulder imaging at 1.5 T has been proposed
in [97]. The sketch of that coil made of 9 rungs is shown in Fig. 1.17(a). The comparable
quadrature TEM coil and its interfacing circuit are depicted in Fig. 1.17(b, c). However, in
this work we are more focused on the birdcage-like coils rather than on TEM coils, therefore
further explanation is mainly devoted to the half birdcages.
Figure 1.17 (a) Sketch of the half-birdcage coil of 20 cm diameter [97]; (b) Sketch of the half-
TEM coil of 20 cm diameter [99]. (c) Interfacing of the coil and current distribution
on the rungs for two orthogonal modes.
Among the plurality of modes in such a resonator, there is the so-called volume mode, having
proper magnetic field distribution polarized in the transverse plane. The SNR of a half-volume
coil is greater over that one of a volume coil of comparable size [99]. However, this increase
is only valid close to the coil. The SNR profiles of a 16-rung birdcage coil and 9 rungs half-
birdcage coil are presented in Fig. 1.18(a), and the same kind of profiles for volume TEM coil
versus half-volume TEM coil are shown Fig 1.18(b). As it can be seen, the half-volume coil
cannot provide homogeneous signal as the one of fully closed volume coils [Fig. 1.18(b, d)]. In
conclusion, half-volume coil can facilitate access to the patient under examination, and, they
can improve the SNR in the vicinity of the coil. However, they cannot provide the homogeneous
SNR distribution, compared to the full-volume coils.
Optimized birdcages The last branch of the RF coils that we would like to review is exotic
birdcages with cross-sections that are not circular in order to better fit anatomical regions
and therefore reduce the coil volume. Indeed, as it is known, the smaller the volume of a coil
and the larger B1amplitude and consequently the larger the SNR. A few papers have been
published on this topic: elliptical birdcage coil [100] and its theory [101], analytical approach
to noncircular section birdcage coil design [102], asymmetric birdcage coil [103]. In this work
we mainly describe the elliptical birdcage coil reported in [100]. This coil was proposed for
head or body imaging at 3 T. The sketch of that coil is shown in Fig. 1.19(a).
Chapter 1. State-of-the-art and novel techniques in ultra-high field magnetic resonance
Figure 1.18 (a) The SNR profiles of the half-birdcage coil compared to the full-volume coil
[97]. (b) spin-echo images of the phantom obtained with (b) 16 rungs full-body
quadrature birdcage coil; (a) The SNR profiles of the half-volume TEM coil (solid)
compared to the volume TEM coil (dashed) and phased arrays (doted) [99]; (d)
spin-echo image of the phantom obtained with 9 rungs half-birdcage coil [97].
According to their method, capacitances and inductances in each element of the coil should
be calculated to satisfy current distribution on the coil’s rungs. It is important to note that to
calculate the mutual inductances between the elements, the same approach as in the conven-
tional coil is used [104]. However, that approach is slightly different according to the elliptical
cross-section of the coil.
Figure 1.19 Elliptical birdcage coil: (a) sketch of the coil; B1field profiles along: (b) long semi-
axis; (c) short semi-axis of ellipse. Spin-echo phantom images obtained using: (d)
non-optimized coil; (e) optimized coil [100].
1.4. RF coil classification and parameters 25
When the elliptical birdcage coil is being compared to a conventional circular birdcage, it
shows a 55%increase of SNR. In addition, the two types of coils are compared in that work:
non-optimized (when all capacitors are identical) and optimized. The B1profiles are presented
in Fig. 1.19(b, c). Image with much better homogeneity can be observed with the optimized
coil [see Fig. 1.19(d, e)].
1.4.3 Phased array coils
In the 90s, phased arrays already widely used for RADARS or in acoustics [13], [105] were
introduced for MRI. They are now widely available commercially. It is important to note that
any possible geometries of loops are available, circles and squares [13], triangle elements [106].
The basic principle of parallel MRI is based on the simultaneous or parallel acquisition the signal
from multiple independent local elements [107]. The coil composed of these small independent
elements provides higher SNR compared to the conventional coils sensitive to the same FOV.
An addition, transmit phased arrays allows to use passive or active B1shimming due to pTx
that can homogenize B1field or FA over the transverse plane [15]. Passive 3d shimming is also
possible by multi-row arrays reported for example in [108].
The first NMR phased array was composed of Nsurface loop coils [13]. At first, we remind
that conventional single electrically small loop coil provides high SNR in the small FOV seen
by a coil. Then there are two options for increasing that FOV. The first one implies increasing
of the coils size. Unfortunately, in this scenario the SNR is degraded [13], [70]. In another
scenario, the number of independent elements of the coil has to be increased.
The main issue with array is to mitigate the mutual coupling between the elements. Indeed,
let consider a single loop coil resonating at frequency f0. In order to understand the functioning
of coupled loops in a phased array, we consider two associated loops placed at a subwavelength
distance. Because of the coupling, two eigenmodes can be obtained (the two loops in phase or
out of phase) at 2 frequencies different from of their initial frequency f0(Fig. 1.20).
Figure 1.20 Two coupled loop coils supporting two modes [13].
Consequently, the aim is to bring these two modes at the same frequency f0. This can be
done by several ways. However, the most intuitive way is to reduce the coupling between the
Chapter 1. State-of-the-art and novel techniques in ultra-high field magnetic resonance
coils using different overlapping between the adjacent elements [13]. As it can be seen in Fig.
1.21(a), the ratio between distance and diameter of 0.9 leads to the zero of magnetic coupling
(km) between two loops. However, electric coupling (ke) equals 0.4 [Fig. 1.21(b)]. Practically
this corresponds to -14 dB of |S12| coefficient between two adjacent elements [Fig. 1.21(b)].
We can also cite another techniques as resonant inductive decoupling (RID) [109], capacitive
decoupling [110], [111].
Figure 1.21 (a) Decoupling of the elements can be simply done by the geometrical decoupling
between the elements; (b) electric (ke) and magnetic (km) coupling coefficients
depending on mutual geometry of adjacent loops [13].
Figure 1.22 The calculated SNR at the depth of 8 cm from the surface of three different coils:
(a) four element phased arrays made of square loops of 8 cm; (b) single 8 cm loop
coil; (c) a large 30 ×15 cm loop [13].
Thanks to these decoupled elements, a phased array can dramatically increase its FOV with-
out degrading the SNR as it is shown in Fig. 1.22. The presented graph considers several
coils: a phased array of four elements, one element of the array and a large loop having the
same FOV as the entire phased array. It is obvious that with the phased array, the SNR of the
entire image can be almost 4 times higher with the same FOV as for the large loop coil. As a
drawback, the system is significantly more complicated because array element should have its
1.4. RF coil classification and parameters 27
own matching and balancing circuits, its own a transmit/receive interface with a preamplifier
and detuning circuit of the coil when it is a ToRo coil [36].
In addition, the phased arrays cannot improve the SNR in the center of the image [70]. For
instance, we would like to consider a receive array composed of 12, 32, 96 channels for brain
imaging at 3 T. The setup of 96 channels coil is shown in Fig. 1.23(a). In Fig. 1.23(b) it can
be seen that increase in the number of channels does not increase the SNR in the center, while
it does in the distal cortex (periphery).
Figure 1.23 (a) Photograph of 96ch coil and sketch of its elements; (b) SNR maps depending
of number of channels [112].
As it was mentioned before, phased array channels have been initially used for reception,
however it is also can be used as a transmit coil. Below 4 T, transmit phased array are not
common, especially because the homogeneous magnetic field can be provided by conventional
Another existing branch of coil is transceiver phased arrays. These arrays are more compact
than ToRo phased arrays because the same elements are used for transmission and reception.
They are also simplier to build than ToRo array because they are usually composed of a small
number of channels (usually 8). However, because of this limitation, they do not provide as
high SNR as Rxphased arrays exploiting for example 32 small elements do.
Phased arrays for UHF imaging As we considered the main pros and cons of phased
arrays, now we would like to consider some features related to UHF brain imaging, which
opens borders for new types of elements in phased arrays [113] such as λ/2 dipoles [114] or
MTL lines (so-called TEM arrays) [115], [116]. These elements are shown in Fig. 1.24.
For example, at 7 T the length of λ/2 dipoles is 0.5 m. It is still relatively long compared
to the anatomical size of a head. Therefore it can be geometrically shortened with lumped or
meandered inductances [117] with folding [118], or it can be and bent [51] in order to better
circumscribing a head.
Being compared to a Tx/Rxbirdcage coil or a TEM coil, the phased arrays can also show
superiority in their performance. For instance, in the work [10] a receive only phased array
Chapter 1. State-of-the-art and novel techniques in ultra-high field magnetic resonance
Figure 1.24 Elements of the passed arrays: (a) loop; (b) radiating dipole; (c) microstrip trans-
mission line [114].
is proposed for brain imaging at 7 T. The array has 32 loop-elements arranged on a close-
fitting fiberglass helmet modeled [Fig. 1.25(a)]. This phased array can be embedded inside a
detunable 16 rung head birdcage coil of diameter 28 cm and 20 cm of length. When the array
is being compared to the same birdcage coil but used for both transmit and receive, it shows
5-fold greater SNR in the cortex and twice higher in the center [Fig. 1.25(b)]. The brain image
obtained with this setup of a birdcage coil and receive only phased arrays is presented in Fig.
1.25(c). In such a case the quality of the image demonstrates enough spatial resolutions to
distinguish small details inside the brain.
Figure 1.25 (a) A Detunable transmit only birdcage coil combined with 32-channel receive only
array; (b) the SNR maps of the Tx/Rxbirdcage coil compared to transmit bird-
cage combined with 32ch receive only array; (c) brain images obtained with that
configuration [10].
In the next work, the transceiver arrays composed of eight loops is reported [108] [see Fig.
1.26(a)]. This array is made elliptical to better fit a human head. We observe on Fig. 1.26(b),
the phased array provides much greater performance in transmit regime compared to the con-
ventional TEM coil of comparable size even though the input power is lower. Due to passive
B1shimming when the amplitudes and phases are adjusted the field map can be homogenized.
1.4. RF coil classification and parameters 29
Moreover, it is also reported in this work that by adding another row of elements it is possible
to perform passive 3d shimming, naturally impossible in single row coil.
Figure 1.26 (a) Eight channel transceiver array [108]; (b) B1field comparison between TEM
coil and 8-channel phased array [108].
d, phased arrays are very flexible tool especially for UHF MRI. It allows creating almost any
kind of geometries. For example, there is a coil proposed in [119] made of two open phased
arrays: one for transmit with 4 channels loop layout and other one with two rows with 8 loops
for receive. Although they are based on completely different principle, this coil is close in terms
of applications to the opencage coil proposed later in Chapter 3.
However, these coils are complicated to manufacture because of numerous elements leading to
a lot of electrical components. As a result, phased arrays are more expensive than conventional
coils. are more complex to use. Some problems of the conventional coils can be solved using
1.4.4 Metamaterial-based coils
Another class of RF coil is the metamaterials-based coils. Metamaterials are artificial periodical
materials that are not available in nature. The history of metamaterials originates in 1967 when
Victor Veselago [120] introduced the concept of negative index material showing simultaneously
negative permittivity and negative permeability. Subsequently, in 2000, John Pendry predicted
that these negative index materials acts as superlens because they can focus a wave on a smaller
spot than the one predicted by the classical diffraction limit, i.e., with a size smaller than half
a wavelength [121]. Such an artificial material with effective parameters is made of a sub-
wavelength (typically less than λ/10) periodic arrangement of meta-atoms, i.e., resonators. In
2001, the concept negative index material was confirmed experimentally in radiofrequency band
(X-band) by David Smith et al. [122]. They created a lens based on split ring resonators (SRR)
and dipoles having together both left-handed and right-handed behavior. After publishing of
this work, many scientific groups in the world started studying metamaterials.
Chapter 1. State-of-the-art and novel techniques in ultra-high field magnetic resonance
One of the first applications of metamaterial lens composed of SRR in MRI originates in
2008 [123], [124]. In this work authors showed that metamaterial structure with negative
permeability can redistribute the magnetic field created by a surface loop coil at 1.5T [Fig.
Figure 1.27 (a) Photograp of the metamaterials lens [124], (b) sketch of the setup on the left
side and the MRI images on the right side, (c) sketch of the setup with reflectors
in the left side and the MRI images in the right side.
In this work authors show that their lens acts basically as a secondary source of RF magnetic
field created by the loop coil. In this case, being installed between two legs, the metamaterial
lens can expand magnetic field to the second ankle [Fig. 1.27(b, c)]. Here, the lens increases the
penetration depth of RF magnetic field. Despite the fact it is a good example of metamaterials’
applications, increasing penetration depth can also be done by using a vast loop coil [76] or by
placing a secondary loop coil nearby the second ankle, or like it is done with wireless implants
in [125].
Another example of application of metamaterial is wire medium [126]. This media also found
use in MRI applications. First application of wire medium was pretty interesting but pretty
far from real clinical or preclinical routine [127]. In this work the metamaterial wire medium
was used in order take out the surface loop coil outside of the bore of 3T MRI scanner [Fig.
Figure 1.28 (a) Photograph of the metamaterials lens [127], (b) sketch of the setup, (c) several
phantom images obtained with and without wire media.
Different configurations of metalens (straight, convergent, divergent and bend) devoted to
magnetic field transmission from a loop coil placed outside the MRI bore [Fig. 1.28(b)]. Mean-
time, the quality of the acquired images with the metalens are comparable to the one obtained
1.5. Conclusion 31
with a single loop placed around the scanned object [Fig. 1.28(c)]. The real application of this
work can be found for using coils that are not MRI compatible. Indeed, some components, for
instance capacitors and pin diodes become more expensive due to their magnetic compatibility.
Many interesting applications were achieved by using metasurfaces from microwaves to optical
range [128]. For example, sub wavelength periodical wire structures that can be considered as
metasurface supports hybridization of modes [129] [Fig. 1.29(a)]. Using the different modes of
the wire medium, it is possible to provide better penetration depth into the sample. It is also
possible to reduce scanning time by enhancing the sensitivity of the local coil in the presence of
wire metasurface [130]. Meta solenoid can be designed and used as a wireless alternative to the
local coil [131] [Fig. 1.29(b)]. Orthogonal eigenmodes can be excited in the wire metasurface
for dual nuclei applications [132] (Fig. 1.29(c)).
Another type of metasurface is electrical band gap structures. These structures can also
be used in MRI for decoupling of the arrays’ elements [133]. In addition metamaterials and
metasurfaces can be used for other applications: magnetic shielding [134], dual-nuclei operations
[135], [136], locally homogenies magnetic RF field at 7 T for brain imaging using Kerker effect
[137]. This hybridized meta-atom (HMA) [137] also matches with our goal of passive RF
shimming. This HMA slows slightly improved transmit B1field, however, precise positioning
inside a birdcage coil may be difficult. In addition, it may degrade comfort of patients under
1.5 Conclusion
In this chapter we have first presented the emergence of MRI as a useful medical apparatus.
The fundamental principles and abilities of MRI have been reviewed. Subsequently, the pros
and cons of UHF MRI have been given. Then, we have defined the key parameters of RF coils
for MRI, such as B1field, transmit efficiency, SAR and SNR and their role and importance.
The several branches of coils, such as surface, volume and phased arrays have been reviewed.
The last section provided the overview of metamaterials in MRI.
To make an overall conclusion, one would say that conventional volume coil such as birdcage
or TEM remain one of the best transmit coil for moderate and high field MRI, respectively.
However, the entire domain is going toward UHF, wherein conventional coils are inefficient in
terms of field homogeneity. The issues of homogeneity may be mitigated with B1shimming
provided by pTxleading to more complexity in the coil design. Thereafter, this problem of
inhomogeneity can also be partially solved by passive shimming provided by the dielectric pads
or by hybridized meta atoms placed inside a transceiver conventional coil. Unfortunately, there
are no mechanical stability and reliability of these structures. Therefore, they did not yet
become routine medical tool. In addition, they cannot be combined with receive phased array.
In this manuscript, we propose a general framework inspired by the works on metasurface
to develop volume coils for UHF scanners. The details of the framework and the link with
birdcage are exposed in Chapter 2. The first application of this approach is the introduction
of a new kind of volume coils with a large aperture operating at 7 T. This coil does not only
improve the comfort of patients, but it can also facilitate access to the area under study to
perform other tasks. Chapter 3 and 4 are dedicated to the development of a preclinical and a
Chapter 1. State-of-the-art and novel techniques in ultra-high field magnetic resonance
Figure 1.29 (a) Hybridized resonator improving the sensitivity of surface loop coil [129], (b)
metasolenoid playing the role of local wireless coil [131], (c) hybridized resonator
for dual-nuclei applications [132].
clinical prototype of such coils, respectively. In chapter 5, we propose an elegant solution to
build a dual-band birdcage-like coil. Finally, in the last chapter, a coil that performs passive
B1shimming is introduced.
Metamaterials theory for volume RF coils design
Table of contents
2.1 Birdcage principle and design . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 Electrical representation of a birdcage . . . . . . . . . . . . . . . . . . . . . 35
2.3 Simplified electrical representation of a birdcage . . . . . . . . . . . . . . . 36
2.4 Transmission line-based approach . . . . . . . . . . . . . . . . . . . . . . . 39
2.5 Comparison of the proposed theory to the reference approach . . . . . . . . 42
34 Chapter 2. Metamaterials theory for volume RF coils design
This chapter reveals the main theoretical aspects of novel volume RF coil design for pre-
clinical and clinical ultra-high field (UHF) MRI. The coils, proposed in this work, are inspired
by birdcage coils [80], [138]. Since their introduction in 1985, birdcage coils have been playing
one of the key roles in clinical and preclinical MR imaging at various field strength [71], [70].
Several theoretical approaches to describe birdcage coils are already known in the literature, for
example in [139], [140], [141]. In addition, there are a few approaches, which are very convenient
from a practical point of view [104], [142]. All these approaches stated above can be used to
design the birdcage geometry and the values of lumped capacitors needed to tune a birdcage
coil at the desired frequency.
In this investigation we propose to apply transmission line theory to straightforwardly de-
scribe a birdcage coil [71]. Indeed, a birdcage coil can be considered as a periodical chain
network composed of unit cells with LC elements where a single unit cell or a rung can be
completely described by a transfer matrix or ABCD matrix [71]. This proposed approach is
commonly used in metamaterials theory [12].
From this analysis, each unit cell of a birdcage coil can be completely described in terms of
Bloch or characteristic impedance and phase shift. This approach allows to tune individually
each unit cell. Eventually, this work is the corner stone of this manuscript to design metacages,
i.e., birdcage-like coils showing other field distribution or other geometries.
2.1 Birdcage principle and design
As explained in Chapter 1, a birdcage coil is a resonator invented in 1985 by C. Hayes [80]. In the
beginning, the birdcage coil was proposed for full-body imaging [80], while later it was adapted
for many different applications at various field strength [85], [86], [87]. The advantage of the
birdcage coil is to generate a highly homogeneous magnetic field inside the resonator’s volume
at its fundamental mode (k= 1). The birdcage coil is composed of Nparallel conductors (legs
or rungs) distributed around the circumference of radius rwith a constant angle ϕ(ϕ= 2π/N)
forming a cylinder as it is shown in Fig. 2.1(a). At both ends, the legs are connected to the
end-rings. The field B1in an empty birdcage coil can be worked out in the quasi-static limit
from the Biot-Savart law [70], [72]. The transverse magnetic field B1principally results from
currents flowing inside the longitudinal legs. The currents are driven by the resonance of the
ladder network consisting of Nleg inductances, 2Nend-ring inductances, and 2Ncapacitors
placed in the legs (low-pass), in the end-rings (high-pass) or in both segments (band-pass) [Fig.
From analyzing a particular unit cell, it is possible to define the capacitance needed to tune
the fundamental mode of the birdcage coil to a Larmor frequency. The detailed analytical
analysis of the birdcage coil is provided in the next section
2.2. Electrical representation of a birdcage 35
Figure 2.1 (a) Sketch of the conventional birdcage coil composed of eight rungs; (b) different
unit cells of the conventional birdcage coil.
2.2 Electrical representation of a birdcage
A birdcage of Nlegs can be modeled by an equivalent electric system made of a periodic
arrangement of lumped LC elements. Such a system is shown in Fig. 2.2. In this system the
Nth rung is connected to the first one because a birdcage is a close electrical system.
Figure 2.2 Full electrical representation of a birdcage coil composed of Nrungs.
On Fig. 2.2 the elements Lrand Llrepresent the self-inductances of the ring and leg conduc-
tive wires. According to the references [74], [104], [143], the self-inductance of a flat conductor
considered in this work can be expressed as:
Ls= 2l·ln 2·l
where lis its length, wis its width, tis its thickness.
36 Chapter 2. Metamaterials theory for volume RF coils design
It also exists strong inductive coupling between the parallel legs (not represented on Fig.
2.2). The mutual inductance Mij of two thin and parallel wires is given by:
Mij = 2l·ln
d+s1 + l2
d2s1 + d2
Where lis the length of the parallel wires and dthe distance between them. When a shield
is present, we use the image theorem to consider its effect. This mutual inductance also exists
between end-ring segments. Because the elements are not parallel, the expression is more
complex and can be found in [86]. Finally, the capacitances Crand Clare provided by lumped
capacitors that should be inserted in the rung and leg conductive lines to get a resonance
We only handle birdcages that are symmetric with respect to a plane that is normal to
the birdcage axis and intersects this last in the middle. Moreover, we only consider modes
wherein the current directions on the top and the bottom rungs are opposite. In such a case,
the aforementioned mid-plane plays the role of a virtual ground plane and the schematic of a
birdcage can be considerably simplified (see Fig. 2.3). Note that half of the mutual inductance
between the legs has to be then taken into account.
Figure 2.3 Simplified electrical representation of a birdcage coil’s unit cell
2.3 Simplified electrical representation of a birdcage
Before describing the metamaterial approach, we present the conventional way to calculate the
capacitance of low-pass (LP) and high-pass (HP) birdcage coils in order to tune the fundamental
mode (k= 1) at the operating frequency. The starting point is to apply the Kirchhoff’s voltage
law to one loop of a birdcage represented in Fig. 2.4. It comes:
n+1 = 0.(2.3)
2.3. Simplified electrical representation of a birdcage 37
Figure 2.4 To the calculation of the birdcage coil.
Where the rung and leg voltage drops are given by the following expressions:
n=· Lr
2· Ll
Due to the Kirchhoff’s current law, it comes:
n+1 Ir
Note that because the last rung is connected to the first one Ur,l
0and Ir,l
We are looking for the system’s resonant frequency, at which the current distribution on the
rings is given by the k-th CP mode (2.5), i.e.:
where I0is the maximum current intensity, kis the mode number and Nis the total number
of unit cell. The current flowing in the legs is then deduced from Eq. 2.5, i.e.,
Moreover, the introduction of effective inductances Lrand Llsimplifies equations (2.4) to the
38 Chapter 2. Metamaterials theory for volume RF coils design
The effective inductance of the ring or leg number mthat takes into account the mutual coupling
between conductors is given by:
nme2iπ k nm
where ˆ
mis the self-impedance of ring or leg m. Because all the legs and rungs are identical
and equally spaced, Mr,l
nm =Mr,l
nm. Therefore, the previous expression becomes:
nm cos 2πk nm
Combining equations (2.5) and (2.8), it comes.
n+1 2Ir
Eventually, replacing the current by its expression from 2.6, we obtain the following relation
between the capacitances, the frequency, and the order of the mode:
=2 sin2 πk
When two capacitors Crand Clare employed simultaneously, a band-pass (BP) or hybrid
birdcage coil [71] is obtained. It is important to note that such a birdcage is different from a
dual-band BP birdcage coil [92] as we are going to see in Chapter 4.
Now, we consider the case of low-pass (LP) and high-pass (HP) birdcages. In a LP birdcage,
only the capacitances on the legs are present while in a HP birdcage, the capacitances are
inserted only on the rings.
From Eq. 2.12, we can deduce the capacitance to obtain the k= 1 mode at the angular
frequency ω. For the HP birdcage, CHP is given by:
CHP =1
ω2Lr+ 2 sin2π
While the capacitance CLP for k= 1 of a conventional LP birdcage becomes:
CLP =2 sin2π
ω2Lr+ 2 sin2π
To prove the proposed theory and compare it to the conventional approach, we are going
to consider three different types of LP and HP birdcages, composed of 4, 8 and 16 rungs,
respectively. The schematically represented birdcage coil composed of 8 rungs is shown in Fig.
2.1. Despite, these considered birdcages are composed of a different number of rungs, their
sizes, such as, radii, length, and widths, are chosen to be the same. The effective inductances
are different due to different mutual coupling between the conductors. The following geometry
2.4. Transmission line-based approach 39
with length of 240 mm, width of rings and legs of 10 mm, radius of 130 mm and radius of shield
of 155 mm is chosen for head imaging at 7 T. Thereby, the working frequency is approximately
300 MHz. Using these dimensions, one can calculate the self and mutual inductances for 4, 8
and 16 rungs birdcages. The results are presented in Table 2.1.
Table 2.1 Effective inductances of 3 different birdcages at 300MHz.
Inductance type (nH) 4 rungs 8 rungs 16 rungs
Legs (Ll) 132.60 146.59 173.97
Rings (Lr) 149 86.05 42.09
In this table, Lris the effective inductance of an end-ring segment, Llis the effective induc-
tance of a leg, ωis an angular frequency at 300 MHz and Nis a legs number. Knowing the
effective inductances, the capacitances can be deduced from Eqs (2.13) and (2.14). The values
of capacitance to tune the k= 1 mode at the operating frequency of 300 MHz are shown on
Tab. 2.2.
Table 2.2 Ring and leg capacitances of the 3 different types of birdcages.
Capacitance type (pF) 4 rungs 8 rungs 16 rungs
CHP or (Cr) 1 2.18 5.09
CLP or (Cl) 1 0.64 0.39
As it can be seen in Table 2.2, LP birdcage calculation process leads to not feasible values of
capacitance in order to tune the k= 1 mode at the frequency of 300 MHz. Indeed these values
are below the self-parasitic capacitance [71]. Hence, at such a high frequency, the HP topology
is preferable.
2.4 Transmission line-based approach
In this section we analyze the birdcage in terms of the transmission line. This formalism is the
cornerstone of the metamaterial approach of this work. Indeed, a birdcage coil [Fig. 2.5(a)]
can be seen as a transmission line (TL) composed of a periodic arrangement of unit cells.
Fig. 2.5(b) shows three-unit cells (i1, i, i + 1), where Yrepresents leg admittance and Z
represents end-ring segment impedance. Because of the symmetry, as explained in section 2.2,
the analysis of one cell (rung in other words) can be reduced to an equivalent circuit composed
only of three elements [caption in Fig. 2.5(b)]: a lumped parallel admittance 2Yion the leg
section, and two equal lumped serial impedances Zi/2for the ring.
40 Chapter 2. Metamaterials theory for volume RF coils design
Figure 2.5 (a) General view of a birdcage coil composed of eight rungs. (b) Simplified equivalent
circuit of a birdcage coil segment. (c) The HP or LP unit cells in terms of inductances
and capacitances.
To describe such a periodic network we have to refer to the transfer or ABCD matrix formal-
ism [12], [71], which is very convenient to handle. This approach can be used even though the
unit cells are not identical. This formalism allows to deduce the 2 main intrinsic parameters of
a unit cell: the Bloch impedance and the phase shift [12]. The transfer matrix Tiof a single
unit cell is a 2 by 2 complex matrix that links the input voltage (Un) and current (In) to the
output (Un+1 and In+1) ones:
In+1 !=Ti· Un
From the schematic represented in the Fig. 2.5(b), it can be shown that the transfer matrix
(2.15) of the ith cell is given by:
1 + ZiYiZ2
iYi+ 2Zi
2Yi1 + ZiYi
In the case of a HP unit cell, Ziand Yiare given by:
In case of LP unit cell, the expressions become:
iωLl,i +1
2.4. Transmission line-based approach 41
Here Cr,i,Cl,i, are the ring and leg capacitance for HP and LP respectively. Ll,i, and Lr,i are
the leg effective inductance, and the end-ring effective inductance of the ith unit cell (rung)
respectively, ωis the angular frequency.
To evaluate the effective inductance, we use the same expression as the one presented in the
previous section. The underlying approximation is that the current distribution on the ring
and the legs are given by Eqs. (2.6) and (2.7) and may not exactly correspond to the ones
provided by the transmission line model.
Finally, the phase shift δϕiinduced by a unit cell can be determined by computing the
eigenvalues of matrix Ti[12]. Eventually, from matrix (2.16) one can deduce the following
formula for the phase shift:
δϕi= cos1(1 + ZiYi).(2.19)
The phase shift δϕicalculated in (2.19) provides the phase of the nodes of the unit cells.
However, the magnetic field is mainly generated by the current flowing on the legs. Therefore,
the important parameters are the amplitude and the phase on these lasts. According to Fig.
2.5(b), the current on the legs Il
ncan be derived from the current on rings Ir
nand Ir
n+1 as
n+1 Ir
Assuming that only the phase varies from one rung to another one, the current becomes:
n=en+1 en= 2ieiϕn+1 +ϕn
2sin ϕn+1 +ϕn
This expression shows that the current phase on a leg is the average of the current phases
between the two adjacent ring nodes plus a phase shift of 90. Note that in case of varying
phase shifts between legs (see opencage or metacage chapters), an amplitude modulation factor
is involved.
The second parameter characterizing the unit cell is the Bloch impedance, which is the ratio
between voltage and current at the nodes of unit cells [Fig. 2.5(b)]. The Bloch impedance can
be determined from eigenvector analysis [12] of the transfer matrix (2.16) as:
For birdcages, because all the cells are identical, the Bloch impedance is set the same from one
cell to another one by nature. This condition is important to avoid reflections of the progressive
mode between the unit cells. For this reason, when the unit cells are not identical because the
required phase shift between rings varying, one has to adjust the cell parameters to keep the
same Bloch impedance.
For instance, in case of HP cell, based on equations (2.19) and (2.22), one can find explicit
solutions of leg inductance and ring capacitance as the functions of Bloch impedance and the