Download Tailored RF Magnetic Field Distribution along the Bore of a 7

ICEAA 2013 / Torino / Centro Congressi
Session «Bioeffects and Medical Applications»
Paper #279 / September 11, 2013.
Tailored RF Magnetic Field Distribution along the Bore of a
7-Tesla Traveling-Wave Magnetic Resonance Imaging System
H. Yang1, T. Liebig1, A. Rennings1, J. Froehlich2, and D. Erni1
Abstract − This paper provides a highly efficient method for tailoring the RF magnetic field (B1) distribution along the cylindrical
bore of a high-field (7T) magnetic resonance imaging (MRI)
scanner operating in the advanced traveling-wave scheme. Here
the B1 wave field propagates as a circularly polarized TE11 waveguide mode and is excited, molded and dumped by a lengthwise
equidistant array of thin quadrature-fed (metamaterial) ring
antennas where each of them perfectly conforms to the inner
surface of the cylindrical MRI bore. All individual antenna excitations associated to the desired longitudinal field profile are
retrieved from an inverse problem that is efficiently solved in a
(weighted) least-squares sense. The electromagnetic modeling is
carried out with our equivalent-circuit (EC) FDTD simulation
platform openEMS, and a convincing showcase involving a narrow illumination window for larynx diagnostics is presented.
1
2
THE INVERSE PROBLEM
2.1 General setting
In our present research we are now tackling the inverse
problem, where both quadrature excitations, namely
the current amplitudes and phases of all periodically
arranged CRLH metamaterial ring antennas [cf. Figure 2 (b)] are subject to optimization in order to achieve
a customized magnetic RF field distribution along the
loaded bore (i.e. loaded with an adult human phantom
from the Virtual FamilyTM [11]).
INTRODUCTION
The distribution of radio frequency (RF) transversal
B1-fields within a high-field (7T) MRI scanner using
traveling waves – namely the propagating (circularly
polarized) fundamental waveguide mode (TE11) along
the cylindrical bore – have been pioneered by Brunner
et al. in their seminal publication [1]. Most of the actual
excitation schemes in traveling-wave MRI [2-6] rely on
closed-end antenna systems at the bore ends, which
pose a considerable challenge in terms of patient’s
comfort. In order to support either uniformity or selectivity in the longitudinal magnetic field profile we have
proposed an adaptive RF antenna system (cf. Figure 2)
consisting of multiple stripline-like composite right/left-handed (CRLH) metamaterial ring antennas that
perfectly conforms to the inner surface of the MRI bore.
[7-9]. Owing to their dispersion engineering capabilities, such CRLH metamaterial ring antennas are best
suited for providing a full-wave resonance [10] (at the
Larmor frequency of 298 MHz) along the circumference that closely mimics the surface current distribution
(and hence the associated azimuthal distribution of the
magnetic field’s longitudinal z-component) of the TE11
mode for any given bore diameter. While applying a
quadrature current feed (i) and (q) as sketched in
Figure 2 (b), any of the ring antennas is perfectly apt to
contribute to the excitation of propagating circularly
polarized B1 mode fields. The described setting (that
we may call «MetaBore») is now further analyzed as to
provide tailored B1 fields for specific MRI diagnostics.
Figure 1: Assessing the MetaBore’s «field responses»
for all N ring antennas with unit excitations. The resulting (computed) response is either stored as comprehensive electromagnetic field data or sampled within
M ≥ N longitudinally distributed cross-sectional planes.
Every design starts with the retrieval of the electromagnetic response of the loaded bore (cf. Figure 1),
where each ring antenna undergoes a unit current excitation [for both feeds (i) and (q)] while either storing or
sampling the resulting fields to setup a corresponding
data basis for further optimization. It’s worth mentioning
that a proper superposition of these «field responses»
can virtually yield any desired unidirectional propagating elliptically polarized or locally resonant field state
along the MRI bore. This includes e.g. local amplification, attenuation or even annihilation of the TE11-like
traveling wave in order to compensate reflections and
absorption entailed by the human phantom. The overall
electromagnetic analysis is carried out using a powerful
forward solver based on our 3D equivalent-circuit (EC)
FDTD simulation platform openEMS [12] with compu-
1
General and Theoretical Electrical Engineering (ATE), Faculty of Engineering, University of Duisburg-Essen,
and CENIDE – Center for Nanointegration Duisburg-Essen, D-47048, Duisburg, Germany,
e-mail: [email protected], tel.: +49-203-3794212, fax: +49-203-3793499.
2
Laboratory for Electromagnetic Fields and Microwave Electronics, ETH Zurich, CH-8092 Zurich, Switzerland,
e-mail: [email protected], tel.: +41-1-6324385, fax: +41-44-6321198.
978-1-4673-5707-4/13/$31.00 ©2013 IEEE
468
Figure 2: (a) Quadrature-fed CRLH ring antenna supporting a λ-resonance along the circumference for the
excitation of a circularly polarized propagating TE11 mode in the bare MRI bore; (b) EC-FDTD model of the
holistic excitation concept «MetaBore» [7-9] consisting of 15 ring antennas each having 24 up to 32 unit cells,
where (i) and (q) represents the in-phase respective quadrature excitation port (of each ring antenna).
tation times in the order of 1 h per single (ring antenna)
response on a state-of-the art CPU (Core-i7) [7, 8].
First design scenarios using iterative local search
heuristics based on e.g. the Levenberg-Marquardt algorithm to track down the current excitations for a confined
50 cm wide uniform B1 illumination profile (with preferably B1+ circular polarization) of the abdomen have
turned out very successful [7, 8], proving our «MetaBore»
concept a holistic approach to future multi-functional
traveling-wave MRI.
2.2 The forward solver
To keep up with a most efficient full-wave solution of
the forward problem we have developed a cylindrical
EC-FDTD implementation with sub-gridding capabilities
[12-14] that fully adapts to the specific bore geometry.
The EC-FDTD engine uses state variables such as
voltages and currents (instead of field quantities) in the
framework of a passive equivalent circuit as the corresponding representation of the standard Yee cell [15, 16].
Figure 3: Screenshot of the openEMS graphical user
interface (GUI) displaying the discretized geometry of a
specific set of CRLH ring antennas (with ground plane
sections). Local refinements of the cylindrical mesh
due to corresponding mesh grading are clearly visible.
The underlying voltages respective currents are
retrieved from the product of the electric respective
magnetic field times the length of the associated Yee
cell’s edge. Inherent to this very specific state variable
representation a reduced numerical effort is achieved
inside the iteration loop (due to the reduced number of
multiplications), which pays off with respect to a speed
and memory improvement of around 33 % [13]. Even if
there is a close formal interrelationship between the ECFDTD formulation and the conventional FDTD scheme,
the former has – besides the aforementioned reduced
numerical costs – clear benefits, such as e.g. the very
straightforward implementation of highly dispersive material properties with the introduction of a small corresponding filter section into the EC representation of the
Yee cell [15, 16]. The EC allows in addition the deduction
of an alternative energy-based stability criterion [16]
that is more relaxed than the standard Courant-FriedrichLevy (CFL) condition aiming at potentially larger time
steps, and hence shorter computing time.
Our EC-FDTD code is provided within the free and
open source simulation platform openEMS [12]. The
underlying engine is based on an extensive C++ class
concept that allows the modular extension e.g. to different
mesh types (Cartesian, cylindrical) and dispersion models
(multipolar Drude, Lorentz, Debye, and dispersive sheet
impedance) keeping the core engine thus as simple as
possible to maximize the numerical efficiency. The platform includes graded meshes and (sub-gridding capabilities), near-to-far-field transform, Mur and UPML boundary conditions. Numerical efficiency is achieved while
using multithreading, SSE processor instruction sets,
and/or MPI distributions for cluster computing where
an aggregate simulation speed > 2 Gcells/s has been
demonstrated on a Linux cluster with 15 off-the-shelf
PCs (Intel Core-i7 920 processor) [13]. The platform
relies on a user-friendly Matlab/Octave interface for
scripting purposes together with a corresponding GUI as
structural 2D/3D viewer (shown e.g. in Figure 3).
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Figure 4: Profiling scenario for larynx illumination: (a) amplitudes of the in-phase (i) and quadrature (q) current
excitations; (b) phases of the in-phase (i) and quadrature (q) current excitations; and (c) the resulting magnitude of
the RF transversal magnetic field. The gray-shaded region [in (a)] stands for the body’s cross-section area at the
corresponding position. To speed-up analysis a simplified excitation model was used with 18 «continuous» circular
current strips as ring antennas each having a strip width of 1 cm, a pitch of 15 cm, and a ring diameter of 64 cm.
2.3 Simplified excitation model
As already discussed in Section 1 the numerical acquisition of the MetaBore’s complete «field response» is a
laborious task mainly due to the structural complexity
inherent to the multiple CRLH ring antennas [cf.
Figure 2(b) and Figure 3 for the underlying highly graded
mesh] but also because of the distinct resonant nature
of the CRLH ring antennas. In order to speed up the
underlying field computation a simplified excitation
model has been defined. Instead of fully detailed CRLH
ring antennas (with a strip width of e.g. 3,7 cm) «contin-
uous» circular current strips are applied with a reduced
width of 1 cm [cf. Figure 4(c)] where each is mimicking
the desired ideal sinusoidal circumferential current distribution. These continuous current strips are excited in quadrature and act as (non-resonant) soft sources for the
longitudinal H1z components of the circularly polarized
TE11-like traveling waves. With this simplification (and
in comparison to the full-blown simulation model) speedup factors of three up to values way beyond one order
of magnitude are possible, depending on the specific
MetaBore scenario.
2.4 Direct least-squares solution of inverse problem
Figure 5: The larynx profiling scenario according to
Figure 3 displaying the spatial average (over the body’s
cross-sectional area) of the magnetic field amplitude
B1+ respective B1– of the right-hand/clockwise respective
left-hand/counter-clockwise circularly polarized field
along the scanner bore (i.e. along the z-axis). The black
dashed line stands for the target window (Gaussian with
FWHM = 15 cm) of the B1+ component’s illumination
profile, whereas the other target is B1– = 0 along the bore.
Both, the comprehensive exploration of the profiling
scheme’s potentiality and its clinical applicability
strongly hinges on the ultra-fast solution of the underlying inverse problem. As the main result of this paper
we propose a direct least-squares solutions using the
Moore-Penrose pseudo-inverse in conjunction with the
singular value decomposition [17]. The underlying
2M × 2N-matrix A is set up by the M averaged (over the
body’s cross-section) spatial samples of the circularly
polarized B1+ and B1– field components stemming from
the N unit quadrature excitations, and relates the unknown excitation currents i to the corresponding spatial
samples of the desired target field profile btarget.
A ⋅i = b target
(1)
The factor 2 emerges from the twofold nature of both
the quadrature current excitations (i) and (q) as well as
of the polarization directions (+) and (–). In the framework of the least-squares solution the Euclidian norm
⎥⎜η ⎥⎜ = √(η Η ⋅η ) of the residual η (i ) = A⋅ i – btarget is mini-
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mized leading to the following normal equation
( A A) ⋅i = A
H
H
⋅b target
(2)
that is inverted using Moore-Penrose pseudo-inverse in
conjunction with the singular value decomposition [17].
The matrix AH denotes the self-adjoint of A. Local
control of the spatial error distribution (residual) is
achieved when introducing a weight in the form of the
positive-definite diagonal matrix W, yielding
(A
H
) (
)
WA ⋅i = A W ⋅b
H
target
(3).
A further refinement of the (weighted) least-squares
solutions is achieved when the phases of btarget are
defined as additional degrees-of-freedom being used
for further error minimization within a corresponding
iterative procedure [17].
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Acknowledgments
We kindly acknowledge fruitful discussions with Klaas
Pruessmann (ETH Zurich) on the potential of conformal
and ergonomic excitation schemes in traveling-wave
MRI, as well as the valuable input from the University of
Magdeburg group (Johannes Bernarding) with respect to
bore constraints for the proper antenna design.
References
TEST CASE
A convincing showcase for the described least-squares
solution is depicted in Figure 4, where a 15 cm wide
illumination window was achieved for diagnostic MRI
of the larynx. Please compare this dimension to e.g.
the wavelength of the underlying propagating modes
of around 2.6 m … 3.5 m ! Here N = 18 ring excitations
are used and the field values are sampled at M = 137
equidistant cross-sections along the bore (i.e. typically 8
field samples between two neighboring rings). The
suppression of the unwanted B1–component (compared
to B1+) can be read-off e.g. from the field profiles in
Figure 5, which amounts to 16 dB and is considerably
improved to 21 dB when relying on the iterative error
minimization scheme using variable phases for the target
fields btarget. Highly selective illumination profiles as
shown here are strongly desired especially for MRI diagnostics of the neck region, where large values of the
local specific absorption rate (SAR) are usually expected
in the neck-shoulder transition.
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the same set of N ring antennas are therefore used for
both the excitation and the sampling of the loaded bore’s
field response yielding M := N. It is still open to future
inquiry whether M := N field samples are sufficient for
a successful solution of the underlying inverse problem,
or whether one has to rely on an ultra-fast numerical
field estimate that is e.g. based on a simplified water-fat
separated model [18] of the patient’s body.
CONCLUSION AND OUTLOOK
In the present numerical study we have demonstrated
the versatility of a «longitudinal shimming scheme»
within our multiple antenna system «MetaBore» that
relies on a highly efficient least-squares method for
tailoring the distribution of the transversal RF magnetic
field (B1) along the MRI scanner bore. The challenging
test case involving a highly selective illumination
window for larynx diagnostics renders our profiling
approach to become promising to future multi-functional
traveling-wave MRI. At this stage we are setting up the
MetaBore hardware to achieve first MRI images of a
successful longitudinal field profiling. Further research
has to handle the issue of applicability to clinical MRI
diagnostics, meaning that the desired field profiles
should be computed and generated in real time. Hence,
[1] D. O. Brunner, et al., Nature, 457 (7232), 994, 2009.
[2] D. O. Brunner, et al., Magn. Reson. Med., 66 (1), 290,
2011.
[3] Y. Pang, et al., Magn. Reson. Med., 67 (4), 965,
2012.
[4] J. Paška, et al., ISMRM 2013, April 20-26, Salt Lake
City, UT, USA, pp. 391, 2013.
[5] H. Shang, et al., ISMRM 2013, April 20-26, Salt Lake
City, UT, USA, pp. 2799, 2013.
[6] J. Mallow, et al., Magn. Reson. Mater Phy., (online first)
DOI: 10.1007/s10334-012-0358-z, Dec. 2012.
[7] D. Erni, et al., IEEE EMBS 2011, Aug. 30 – Sept. 3,
Boston, MA, USA, pp. 554, 2011.
[8] T. Liebig, et al., MAGMA, 24 (suppl. 1), 37, 2011.
[9] D. Erni, et al., German patent, reference no. 10 2011
111 996, 2011.
[10] A Rennings, et al., EuCAP 2006, Nov. 6-10, Nice,
France, pp. 1-6, 2006.
[11] A. Christ, et al., Phys. Med. Biol., 55 (2), N23, Jan. 2010.
[12] Full-wave 3D EC-FDTD simulation platform openEMS:
http://openems.de
[13] T. Liebig, et al. Int. J. Numer. Model., online, DOI:
10.1002/jnm.1875, 2012.
[14] T. Liebig, et al., MAGMA, 25 (suppl. 1), 627, 2012.
[15] A. Rennings, et al., J. Comput. Theor. Nanosci., 5 (4),
690, 2008.
[16] Andreas Rennings, Elektromagnetische Zeitbereichssimulationen innovativer Antennen auf Basis von Metamaterialien. PhD Thesis, University of Duisburg-Essen,
Sept. 17, 2008.
[17] Hongyi Yang, Design and optimization of longitudinal
B1-field distribution in the context of traveling-wave
magnetic resonance imaging. Master thesis, Laboratory
for General and Theoretical Electrical Engineering (ATE),
University of Duisburg-Essen, Nov. 7, 2012.
[18] H. Homann, et al., ISMRM 2011, May 7-13, Montréal,
Canada, pp. 489, 2011.
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-1/16ICEAA 2013 / Torino / Centro Congressi
Session «Bioeffects and Medical Applications»
Paper #279 / September 11, 2013.
Tailored RF magnetic field distribution
along the bore of a 7-Tesla
traveling-wave MRI system
H. Yang, T. Liebig, A. Rennings, and D. Erni
General and Theoretical Electrical Engineering (ATE)
Faculty of Engineering, and CeNIDE
University of Duisburg-Essen, D-47048 Duisburg
J. Fröhlich
Laboratory of Electromagnetic Fields and Microwave
Electronics, ETH Zurich, CH-8092 Zurich.
-2/16-
Agenda
! Concept of traveling-wave MRI.
! 1D electromagnetic metamaterials.
! Conformal metamaterial ring antennas.
! Unidirectional wave excitation.
! The «MetaBore» as a holistic excitation scheme.
! The openEMS simulation platform (EC-FDTD).
! Test case: Narrow B1-field profile for larynx MRI.
! Concluding remarks.
1
-3/16-
Traveling-Wave MRI
David O. Brunner, Nicola De Zanche, Jürg Fröhlich, Jan Paska, and
Klaas P. Pruessmann, Nature, vol. 457, pp. 994-998, Feb. 2009.
Underlying Concept
(1)! Standing waves:
! i.e. in birdcage coils
! resonant B1 field
constant
phase
amplitude
variations
!
B0
I
MR
7 Tesla
!
B1
© D. Brunner, ETH Zürich.
(2) Travelling waves:
! MRI bore as waveguide
! travelling B1 field along the bore
phase
variations
constant
amplitude
⇒
Uniform B1 field illumination along the MRI bore.
1D Electromagnetic Metamaterials
Tailoring transmission lines
Equivalent circuit of the CRLH*) unit cell:
-4/16-
dispersion
diagram
unit cell
synthesis and
periodic continuation
transmission line
*) CRLH: composite right-/left-handed
⇒
MTM
Transmission line with tailored wavelength at e.g. 297 MHz.
2
-5/16-
Metamaterial Ring Antennas
Compact CRLH ring antenna
! The quadrature excitation of a
TE11 mode yields a circularly
polarized traveling wave.
Excitation of circular waveguide modes:
! Interesting application in novel
traveling-wave MRI schemes.
excitation phase !-90°
conformal
CRLH ring
antenna
excitation of a fullwave resonance
on a CRLH ring antenna
! Ring antenna: bending around a
metamaterial transmission line.
excitation
phase !
!
Hz ! J
! The current density distribution is apt to
excite the circular waveguide TE11 mode.
-6/16-
Traveling-Wave MRI I
Excitation Concepts
(1) Excitation concept (298 MHz):
! uniform B1 field along z
! unidirectional TE11 wave
! circularly polarized
!
B0
(2) Excitation antennas:
(simplified for linear polarization)
! conventional
approach
7 Tesla
!
B1
MRI
© D. Brunner, ETH Zürich.
! ergonomic
approach
3
-7/16-
Traveling-Wave MRI II
Unidirectional wave excitation
!
J1
(1) Single current excitation:
B1 ( z,t )
(TE11
mode)
! !
J1 J2
(2) Dual current excitation:
B1 ( z,t )
(TE11
mode)
lambda /4
Traveling-Wave MRI III
New traveling-wave MRI concept
(exaggerated
frequency
for visibility
reasons)
-8/16D. Erni, T. Liebig, N. H. Koster, A. Rennings, «Meta-MRTAntennenvorrichtungen für die Wanderwellen-Magnetresonanztomographie», Dec. 13, 2012, German patent,
reference number 10 2011 111 996.
B1 ( z,t )
!
J1
!
J2
!
J3
!
J4
4
-9/16-
EM simulator «openEMS» I
(1) Equivalent-circuit (EC) FDTD:
! voltage and currents instead of E and H
! equivalent circuit representation of Yee cell
! intuitive inclusion of dispersion (filter section)
! energy-based stability criterion (relaxes CFL)
(2) Some features:
! Carthesian and cylindrical mesh
! sub-gridding capabilities
! MATLAB/Octave scripting and GUI
! > 2 Gcell/s (MPI, 15 Core-i7 cluster)
unidirectional
excitation
EM simulator «openEMS» II
-10/16T. Liebig, A. Rennings, S. Held, and D. Erni,
Int. J. Numer. Model., (online first), 2013,
DIO: 10.1002/jnm.1875.
Look at: www.openems.de
5
-11/16-
The «MetaBore» Concept I
«Holistic» excitation concept
(1) Acquisition of the «bore response»:
(2) Tailoring longitudinal field profiles:
! yielding a weighted superposition of the field basis to
achieve a desired longitudinal
B1(+) field profile.
! defining longitudinal profiles
for circularly polarized B1(+)and B1(–) fields (B1(–) ! 0).
! excitation of single ring antennas: n = 1…N.
! each excitation includes ( in(i) := 1, in(q) := 1).
! dumping of all resulting fields (i.e. loaded bore
responses) " n = 1…N in the frequency domain.
! loaded bore response represents a field basis.
! inverse problem for N times
( in(i)!!n(i), in(q) !!n(q) ) using the
full set of the field basis.
! The inverse problem is solved
using a weighted direct leastsquares solution (using the
Moore-Penrose pseudo-inverse in conjunction with SVD).
-12/16-
The «MetaBore» Concept II
Direct least-squares solution
(2) Field averaging:
(1) Longitudinal field sampling:
2M # 2N 2N # 1 2M # 1
(3) Analysis: 2N unit quadrature excitations [< 0,5 h]
! 2M body averaged field samples [B1(+), B1(–)]
(+),
! 2M target profile samples [b1
(–)]
b1
! 2N unknown complex current excitations [ in(i), in(q)]
A!i = b target
(A
weighting
) ( )
" ( i ) = ( A!i # b ) $ min
H
WA !i = AH W ! b target
2
target 2
6
The «MetaBore» Concept III
-13/16-
Test case: «Larynx illumination»
(1) Profiling scenario for confined illumination:
(c) total B1 field
N = 18 / M = 137
! 18 continuous circular current strips
(width: 1 cm / pitch: 15 cm / !: 64 cm).
! Gaussian profile: FWHM = 15 cm
!n(q) + 90°
body’s cross-sectional area
!n(i)
(a) current excitation amplitudes
(b) current excitation phases
The «MetaBore» Concept IV
-14/16-
Test case: «Larynx illumination»
(2) Field amplitudes constituting the illumination profile :
! Virtually purely circularly
polarized B-fields (B1(+)).
! Undesired component B1(–)
is suppressed by 21dB.
! Conforms perfectly to the
target profile.
! Achieved field confinement
(FWHM = 15 cm) is much
below the wavelength of
the TE11 waveguide mode
(2,6 m … 3,5 m) !
z [m]
! No hotspots in the neckshoulder region.
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-15/16-
Conclusion
! Proposal of a holistic scheme for molding
traveling-waves in MRI: The «MetaBore».
! The «MetaBore» relies on «flat» resonant
electromagnetic metamaterial ring antennas.
! Efficient full-wave modeling with the free and
open EC-FDTD solver openEMS.
! Longitudinal field profiling is based on a fast
least-squares solution to the inverse problem.
! Highly confined B1 profile for larynx MRI.
! Outlook: ! Real-time feasibility of the profiling
scheme (e.g. with M = N = 18).
! Setup of a demonstrator.
-16/16-
Thanks.
Further Information:
Check the site
on «Publications»
www.ate.uni-due.de
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