FULL PAPER
Magnetic Resonance in Medicine 00:000–000 (2013)
Mitigate B11 Inhomogeneity Using Spatially Selective
Radiofrequency Excitation with Generalized Spatial
Encoding Magnetic Fields
Yi-Cheng Hsu,1,2 I-Liang Chern,1 Wei Zhao,3 Borjan Gagoski,4 Thomas Witzel,3 and FaHsuan Lin2,5*
High-field magnetic resonance imaging (MRI) shows a
great promise due to its high signal-to-noise ratio (1,2).
Yet one challenge of the high-field MRI is the inhomogeneous flip-angle distribution when a volume radiofrequency (RF) coil is used for RF excitation (3). This
artifact is due to the deleterious interaction of the dielectric properties of the sample with the shorter wavelength
of the RF fields in a strong main magnetic field B0 (>4T)
(4–6). Consequently, an imaging object with the size
approximating to a human head can have a spatially
varying flip-angle distribution with larger flip angles at
the center of the field-of-view (FOV) and smaller flip
angles at the periphery of the FOV (2). This causes
images with a spatially dependent T1 contrast, which is
difficult for clinical diagnosis.
Different methods for mitigating B1þ inhomogeneity
have been proposed. Dedicated volume RF transmit coils
have been designed (7–9). Spatially selective RF excitation (10) can be used to design RF and gradient waveforms in order to generate a homogeneous flip-angle
distribution after considering the inhomogeneous B1þ
generated by a volume coil (11). Under the practical
limit of relatively short RF excitation (in a few milliseconds), this approach can only mitigate a rather smooth
flip-angle distribution below 7T. Using simultaneous RF
excitation from multiple RF coils, it has been shown that
RF shimming (12–16) and transmit SENSE (17,18) methods can use a shorter RF pulse than the spatially
selective RF excitation method to achieve the desired
flip-angle distribution. Several parallel RF excitation
strategies have been demonstrated at 3T (19,20), 4.7T
(20), and 7T (21,22). However, the challenges include
the complexity of the RF electronics and coils to achieve
simultaneous excitation, the necessity of accurate estimates of phases and amplitudes of the B1þ maps for
each RF coil, and the specific absorption rate (SAR) management (23). Alternatively, a more homogeneous image
intensity can be obtained after appropriately combining
images of different modes of a volume coil (24). However, this approach is designed to improve the image
intensity inhomogeneity rather than to reduce the flipangle inhomogeneity.
Recently, it has been demonstrated that nonlinear spatial encoding magnetic fields (SEMs) can be used in MRI
spatial encoding in order to improve spatiotemporal
resolution (25,26). Preliminary studies using quadratic
SEMs for RF excitation (27–32) and small FOV imaging
(33) have also been reported. Under the small flip-angle
approximation, a formulation to describe the spatial
Purpose: High-field magnetic resonance imaging (MRI) has
the challenge of inhomogeneous B1þ, and consequently inhomogeneous flip angle distribution, which causes spatially dependent contrast and makes clinical diagnosis difficult.
Method: We propose a two-step pulse design procedure in
which (1) a combination of linear and nonlinear spatial encoding magnetic fields (SEMs) is used to remap the B1þ map in
order to reduce the dimensionality of the problem, (2) the locations, amplitudes, and phases of spoke pulses are estimated
in one dimension. The advantage of this B1þ remapping is that
when the isointensity contours of a linear combination of
SEMs are similar to the isointensity contours of B1þ, a simple
pulse sequence design using time-varying SEMs can achieve
a homogenous flip-angle distribution efficiently.
Results: We demonstrate that spatially selective radiofrequency (RF) excitation with generalized SEMs (SAGS) using
both linear and quadratic SEMs in a multi-spoke k-space trajectory can mitigate the B1þ inhomogeneity at 7T efficiently.
Numerical simulations based on experimental data suggest
that, compared with other methods, SAGS provide a formulation allowing multiple-pulse design, a similar average flip-angle
distribution with less RF power, and/or a more homogeneous
flip-angle distribution.
Conclusion: Without using multiple RF coils for parallel transmission, SAGS can be used to mitigate the B1þ inhomogeneity
in high-field MRI experiments. Magn Reson Med
C 2013 Wiley Periodicals, Inc.
000:000–000, 2013. V
Key words: 7T; RF inhomogeneity; nonlinear gradient; SAR;
fast imaging
1
Department of Mathematics, National Taiwan University, Taipei, Taiwan.
Department of Biomedical Engineering and Computational Science, Aalto
University School of Science, Espoo, Finland.
3
A. A. Martinos Center, Department of Radiology, Massachusetts General
Hospital, Charlestown, Massachusetts, USA.
4
Center for Fetal-Neonatal Neuroimaging and Developmental Science,
Boston Children’s Hospital, Boston, Massachusetts, USA.
5
Institute of Biomedical Engineering, National Taiwan University, Taipei,
Taiwan.
Grant sponsor: National Science Council (NSC), Taiwan; Grant number:
101-2628-B-002-005-MY3, 100-2325-B-002-046. Grant sponsor: Ministry
of Economic Affairs, Taiwan; Grant number: 100-EC-17-A-19-S1-175. Grant
sponsor: National Health Research Institute, Taiwan; Grant number: NHRIEX102-10247EI. Grant sponsor: Finland Distinguished Professor (FiDiPro)
programme.
*Correspondence to: Fa-Hsuan Lin, Ph.D., Institute of Biomedical Engineering, National Taiwan University, Taipei, Taiwan. E-mail: fhlin@ntu.edu.tw
Received 26 August 2012; revised 26 March 2013; accepted 17 April 2013
DOI 10.1002/mrm.24801
Published online in Wiley Online Library (wileyonlinelibrary.com).
2
C 2013 Wiley Periodicals, Inc.
V
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2
distribution of the flip angle when RF pulse is transmitted with time-varying linear and nonlinear SEMs has
been described earlier (28,29,34). Furthermore, methods
of using linear and quadratic SEMs to compensate B1þ
inhomogeneity have been proposed. Specifically, parallel
RF transmission can be combined with higher spatial frequency encoding at the periphery of the FOV generated
by a quadratic SEM to reduce flip-angle inhomogeneity
(35,36). Alternatively, driving linear and quadratic SEMs
between two excitation pulses can generate spatially
dependent transverse magnetization phase, which can
counteract the inhomogeneous B1þ and consequently
leads to more homogeneous flip-angle distribution (37).
Here, under the small flip-angle approximation (10),
we propose the spatially selective RF excitation using
generalized SEMs (SAGS), a two-step pulse design procedure in which (1) a combination of linear and nonlinear
SEMs is used to remap the B1þ map, so as to reduce the
dimensionality of the problem, (2) the locations, amplitudes, and phases of spokes pulses are estimated in one
dimension. The advantage of this B1þ remapping is that
when the isointensity contours of a linear combination
of SEMs are similar to the isointensity contours of B1þ, a
simple pulse sequence design using time-varying SEMs
can be used to achieve a homogenous flip-angle distribution efficiently. Using simulations based on empirical
data at 7T, we compare our method with fast-kz (11)
and the method of tailored excitation using nonlinear
B0-shims (37). Results demonstrate that SAGS can
achieve homogeneous flip-angle distribution with a low
RF power without parallel RF transmission.
THEORY
Hsu et al.
over time are related to each other by an inverse Fourier
transform (28,29,34):
Z
Mxy ðrÞ ¼ W ðkÞBþ
1 ðrÞexp ½j2pf ðrÞ ksðkÞdk
K
W ðkÞ ¼ W ðkðtÞÞ ¼
jgBþ
1 ðtÞ
jk0 ðtÞj
kðtÞ ¼ ½k1 ðtÞ; ; kn ðtÞ ¼ g
[1]
Z
T
gðsÞds
t
Note that the notation k(t) in this study is different
from that in conventional MRI (10). We chose k(t) to
express the maximal phase difference of the transverse
magnetization precession within the imaging object at
time instant t. k(t) is the product between the gradient
moment and the gyromagnetic ratio g. Since a k-space
trajectory has the one-to-one correspondence between
k(t) and t, we omit the t argument in k(t) and use k in
the following. Additionally, we use a delta function s(k)
in k-space to describe the k-space trajectory.
To achieve a practical slice-selective RF excitation, we
propose to use a spoke k-space trajectory (11). Without
losing generality, we consider that the slice selection is
in the z-axis and that only the central slice (z ¼ 0) is
excited. Accordingly, Eq. 1 in a spoke k-space trajectory
becomes
Mxy ðx; yÞ ¼ Bþ
1 ðx; yÞ
X
WF ðkf Þexp ½2pjkf f ðx; yÞ; [2]
spokes
where Mxy(x,y) is the excited transverse magnetization at
the z ¼ 0 plane.
Small Flip-Angle Approximation Using Nonlinear SEMs
With Inhomogeneous B1þ
For an MRI system with n distinct configurations of
SEMs turning on during RF excitation, we use the
dimensionless variable f(r)¼[f1(r), . . ., fn(r)] to describe
the spatial distributions of the z-components of these
SEMs. To facilitate the description of the arbitrary spatial
distribution of f(r), including nonpolynomial distributions, we define that the maximal and the minimal
values among all components of f(r) within the imaging
object are 1 and 0, respectively. g(t)¼[g1(t). . .gn(t)]
describes the instantaneous magnetic field strength of
each individual SEM in a physical unit. Accordingly,
each component of g(t) clearly defines the instantaneous
difference between the minimal and maximal z-component of the magnetic field generated by each SEM within
the imaging object. The instantaneous additional z-component of the magnetic field at location r is thus the
inner product g(t)f(r). Here, we assume that the RF
transmit field B1þ(r,t) is spatiotemporally separable
B1þ(r,t) ¼ B1þ(r) 3 B1þ(t), where B1þ(t) is a waveform of
the RF transmit field, and B1þ(r) is a spatial distribution
of the ratio between B1þ(r,t) and B1þ(t).
Taking the small flip-angle approximation and assuming the initial magnetization [Mx(r,0), My(r,0), Mz(r,0)]T ¼
[0, 0, 1]T, the spatial distribution of the transverse magnetization Mxy(r), RF pulse waveform B1þ(t), and temporal integral of SEMs (linear/nonlinear MRI gradients)
The k-Space Dimension Reduction Using Nonlinear SEMs
When linear X and Y SEMs are used, Eq. 2 shows that
WF(kf) ¼ WF(kx, ky) is the 2D discrete Fourier transform
coefficient of Mxy(x,y)/B1þ(x,y). Suppose that we aim to
achieve a target transverse magnetization distribution
Mxy(x,y) subject to the B1þ field B1þ(r) ¼ B1þ(x,y) at z
¼0. If Mxy(x,y)/B1þ(x,y) can be parameterized by f(x,y),
then Eq. 2 becomes
Mxy ðx; yÞ=Bþ
1 ðx; yÞ
~ xy ðf ðx; yÞÞ
¼M
X
¼
WF ðkf Þexp ½2pjkf f ðx; yÞ
:
spokes
[3]
This shows that the WF(kf) is the discrete Fourier
~ xy ðf ðx; yÞÞ. The dimension of
transform coefficient of M
this discrete Fourier transform depends on the dimension of f(x,y). Using generalized SEM can be advantageous if there is an one-dimensional f(x,y) such that
~
Mxy ðx; yÞ=Bþ
1 ðx; yÞ M xy ðf ðx; yÞÞ:
Then, Eq. 3 becomes
[4]
Mitigate B1þ Inhomogeneity
~ xy ðf ðx; yÞÞ ¼
M
X
3
WF ðkf Þexp ½2p jkf f ðx; yÞ:
[5]
spokes
WF(kf) is the 1D discrete Fourier transform coefficient
~ xy ðf ðx; yÞÞ. Reducing the dimension of the discrete
of M
Fourier transform from 2D to 1D implies that a shorter
k-space trajectory can be used to achieve the similar
distribution of the transverse magnetization.
Design the Flip-Angle Distribution Using Arbitrary SEMs
and a Spoke k-Space Trajectory
A target transverse magnetization distribution can be
achieved efficiently using arbitrary SEMs and a spoke kspace trajectory, if Eq. 4 is satisfied. If targeting on
achieving a spatially homogeneous flip-angle distribution
Mxy(x,y) ¼ mxy, this criterion is equivalent to finding one
configuration of SEMs such that the isointensity contours of f(x,y) is geometrically similar to the isointensity
contours of B11(x,y). Since in this situation the value of
Mxy(x,y)/B11(x,y) can only depend on which isointensity
curve of f(x,y) it lies on.
When Eq. 4 is satisfied, the next is to design the location kf and the strength WF(kf) as requested by Eq. 5. A
~ xy ðf ðx; yÞÞ can be generally approximated accurately by
M
a sufficient number of spokes satisfying the Nyquist
~ xy ðf ðx; yÞÞ is typically
sampling rate. However, since M
spatially smooth, it may be more efficient using irregularly spaced spokes to achieve the approximation of
Eq. 5 using fewer spokes. This idea is similar to the fastkz method, which uses spokes near the center of the kspace to achieve a quadratic transverse magnetization
distribution. In practice, we propose using irregularly
spaced spokes, whose strengths WF(kf) can be estimated
by subjecting to the least-square error criterion (error ¼
X
WF ðkf Þexp ½2pjkf f ðx; yÞ)
and
Mxy ðx; yÞ B1 ðx; yÞ
spokes
locations can be exhaustively sought over a set of candidate locations.
trajectory. Two spokes were at kf ¼ 0.077 and kf ¼
0.077 with 49.85% amplitude of the central spoke in
the fast-kz method. The locations, amplitudes, and
phases of two spokes were manually tuned to generate a
jMxyj distribution close to the jMxyj distribution generated by five spokes described above.
The RF power for the fast-kz method and SAGS was
separately calculated as the sum of the squares of the
amplitude of each spoke. We also plot the k-space trajectories and the corresponding pulse sequences for both
methods (Fig. 1). The pulse sequences were calculated
based on a 25 25 25 cm3 FOV and a 5-mm slice
using SEMs with a maximal slew rate of 800 T/s within
the FOV and with the maximal strength generating 160
mT difference in the z-component of the magnetic field
within the FOV. This slew rate and maximal strength
was equivalent to the 200 T/m/s slew rate and 40 mT/m
maximal strength in classical MRI using linear SEMs.
Note that a recent study using the Z2 SEM for spatial
encoding used a higher maximal strength than our simulation specification (38).
jB1þj Map of Saline Phantom and Human Head
The B1þ map of a saline phantom was acquired on a 7T
system (Siemens, Erlangen, Germany) with a 16-channel
receive array, a SC72 gradient (maximum amplitude 70
mT/m, maximum slew rate 200 T/m/s), and a 16-channel
transmit array. Two healthy subjects with the informed
consents were also scanned on the same 7T scanner with
a 16-channel transmit array driven by a Butler matrix in
the birdcage mode and a 16-channel wrap-around receive
coil array. We collected the B1þ map at the magnet isocenter with 10 mm thickness, 240 240 mm2 FOV, and
64 64 image matrix. The B1þ mapping method consisted of two parts: first, acquiring low flip-angle images
and then stepping through different flip angles of the
presaturation pulse (39). Imaging parameters were as follows: TR/TE ¼ 1000 ms/2 ms.
METHOD
SAGS Design Procedure to Mitigate jB1þj Inhomogeneity
Comparison on Relative RF Power and Pulse Duration
Between SAGS and Fast-k Methods
Find the SEM With Profile f(x,y) Satisfying Eq. 4
We used simulations to study if a quadratic flip-angle distribution can be more efficiently achieved by a configuration of SEM using a two-spoke trajectory than the fast-kz
method. The fast-kz method used five spokes: one central
spoke at the center of the kx–ky plane (kx, ky) ¼ (0, 0).
Four side spokes were at the (kx, ky) ¼ (0, 0.32), (0,
0.32), (0.32, 0), and (0.32, 0). The ratio of the amplitude between the central spoke and a side spoke was
1:0.175. The locations, amplitudes, and phases of each
spoke were chosen manually to generate a quadratic transverse magnetization distribution with the magnitude of
the transverse magnetization jMxyj ¼ 0.25 at center and
jMxyj ¼ 0.5 at the corner of the FOV without considering
the jB1þj inhomogeneity. This flip-angle distribution can
be used to mitigate the jB1þj inhomogeneity, where jB1þj
tends to be higher in the center of the FOV (11).
In this simulation, the SAGS method used the Z2 SEM
(fZ2(x,y,z) / z2 1/2(x2 þ y2)) and a two-spoke k-space
In this study, we only used the linear and quadratic
SEMs. At the z ¼ 0 plane, f(x,y) is a quadratic function
with unknown coefficients. Since we desire a homogeneous flip-angle distribution with a constant Mxy(x,y) ¼
~ ðf Þ such that
mxy, the goal is to find f(x,y) and M
þ
~
M xy ðf ðx; yÞÞ mxy ðx; yÞ=B1 ðx; yÞ. This equation suggests
that we need to find f(x,y) such that the isointensity contour of f(x,y) matches the isointensity contours of
B1þ(x,y). It is possible to determine f(x,y) based on observation. For example, the isointensity contours of the
B1þ(x,y) of a saline phantom are nearly concentric ellipses. Thus, we used isocontours of B1þ(x,y) to determine
the center of the ellipses. Then, we averaged ratios
between the long and short axes of each ellipse to estimate the shape of the ellipse.
The isointensity contours of the B1þ(x,y) in the human
head can be too complicated to be determined by observation. Under the assumption that the B1þ map is spatially smooth, we first chose to approximate the desired
4
Hsu et al.
FIG. 1. The k-space trajectories (a, c) and the pulse sequence diagrams (b, d) of SAGS using two spokes and the fast-kz.
mxy/B1þ(x,y) by a distribution consisting of a linear
~ ðf ðx; yÞÞ ¼
combination
of
spatial
harmonics:
M
L
X
al sin ðlf ðx; yÞÞ þ bl cos ðlf ðx; yÞÞ, where al and bl are
l¼0
the coefficients for the l-th harmonic used in the
approximation. Realistically al and bl were estimated
using a gradient decent method with the initial guess
~ 0 ðf ðx; yÞÞ estimated by using only the first harmonic:
M
~ 0 ðf ðx; yÞÞ ¼ b1 cos ðf ðx; yÞÞ. b1 is the smallest number
M
such that mxy/b1 < 1, and f(x,y) ¼ ax2 þ by2 þ cxy þ dx
þ ey þ g can be uniquely estimated by the least square fitting of arccos (mxy/b1 B1þ(x,y)) using a quadratic function
2
ða; b; c; d; e; gÞ ¼ argminjjarccosðmxy =b1 Bþ
1 ðx; yÞÞ a x
2
þ b y þ c xy þ d x þ e y þ gjj2 ;
[6]
Overall, the cost function of this procedure is
min
a;b;c;d;e;g;al ;bl
L
X
jjBþ
ðal sin ðlf ðx; yÞÞ þ bl cos ðlf ðx; yÞÞÞ
1 ðx; yÞ
l¼1
mxy jj2
[7]
We used the gradient descent method to optimize coefficients in Eq. 7. Afterward, f(x,y) was linearly scaled
such that its minimum and the maximum are 0 and 1
within the imaging object, respectively.
To evaluate if Eq. 4 is satisfied, we plot the distribution between f(x,y) and normalized mxy/B1þ(x,y), which
was linearly scaled mxy/B1þ(x,y) such that the maximum
¼ 1. Ideally, the distribution of all data points should be
on a curve, which was estimated by fitting all points
using a 10th-order polynomial. We calculated the normalized standard deviation, which was the standard
deviation of the difference between the fitted polynomial
and each data point divided by the mean of the normalized mxy/B1þ(x,y) to quantify the fitting error in Eq. 4.
Find Spoke Locations and Compute the Amplitudes and
Phases for Each Spoke
In the second step, the locations, amplitudes, and phases
of spokes were estimated. First, design the location of
~ xy ðf ðx; yÞÞ is a
spokes. Physically, we know that M
smooth function and nearly monotonically increasing
from the center to the periphery of the FOV. Hence, a
spoke at kf 僆 [0.5, 0.5] may be sufficient to approximate this distribution, because a spoke at kf ¼ þ1 or -1
represents one full cycle oscillation within the imaging
Mitigate B1þ Inhomogeneity
5
object. To find the locations of the n spokes, we also
assumed that spokes are symmetrically located at
Kf¼{6ki}, where k0 < 0.5 and ki < ki-1 þ 1, i ¼ 1 . . . n21.
We then estimated the amplitude and phase of each
spoke. For any Kf, we estimated Wf({6ki}) by the leastsquares solution of Eq. 2. Since this least-squares solution can be calculated rather fast, we sought all possible
ki in steps of 0.05. We calculated results of two and four
spokes in this study.
Comparison on Relative RF Power and jMxyj
Homogeneity Between One-Pulse, SAGS, Fast-kz, and
Two-Pulse “Tailored-Excitation” Methods
For comparison, we also estimated the jMxyj distributions generated by (1) one spoke using only the linear z
SEM, (2) a fast-kz five-spoke trajectory using only linear
SEMs (11), and (3) a two-spoke “tailored-excitation” trajectory using both linear (X and Y) and quadratic (Z2,
X2–Y2, XY) SEMs (37). Note that the two-spoke tailored
excitation has a similar k-space trajectory as SAGS
method. With a predetermined spoke amplitude and
constrained by the desired flip-angle distribution, the
two-spoke tailored excitation first estimates the phase
distribution accrued between two spokes. Given the latency between two spokes, SEMs are then chosen independently to approximate the desired spatially
dependent phase shift to approximate this phase distribution, which is subsequently used to compensate the
jMxyj inhomogeneity. Based on the chosen spoke amplitudes a priori, there are either the most homogeneous
jMxyj (the most homogeneous mode) or the least RF
power dissipation (the minimal SAR mode). For the
fast-kz method, the locations of side spokes were symmetrically located on the jkxj < 0.5 and jkyj < 0.5. At
each possible spoke location, we estimated spoke amplitude and phase using the least-squares solution of
Eq. 2 without using any regularization. The locations of
spokes in the fast-kz trajectory were sought exhaustively
at a step of 0.05. The five spokes with the least-squares
error were chosen as the fast-kz five-spoke trajectory.
All jMxyj distributions were calculated by solving the
Bloch equation numerically with the provided trajectory
and SEMs. In addition, we chose to achieve 20 flipangle distribution when we designed parameters for different methods.
The performance of jB1þj mitigation was evaluated by
the relative standard deviation r (37):
¼ stdðjMxy jÞ=meanðjMxy jÞ;
[8]
where std(•) and mean(•) denote taking the standard
deviation and the mean, respectively. Note that r
changes moderately when the target flip angle is small,
because the standard deviation and mean of transverse
magnetization are linearly proportional to each other.
To explore the limit of SAGS and to justify the number
of spokes is sufficient, we parametrically increased the
number of equally spaced spoke from 1 to 99 and calculated the relative standard deviation r in phantom and
head imaging experiments. We also defined the relative
RF power g:
FIG. 2. The distribution of jMxyj by fast-kz using five spokes and
SAGS using two spokes. SAGS used 44% of the RF power of
fast-kz. Results were generated using the pulse sequence and the
k-space trajectory in Figure 1.
h¼
X
jWðkf Þj2 =jWsingle ðkf ¼ 0Þj2 ;
[9]
kf
where jWsingle ðkf ¼ 0Þj is the amplitude of the single
spoke in the least-square solution to the target transverse
magnetization distribution.
RESULTS
Comparison on Relative RF Power and Pulse Duration
Between SAGS and Fast-kz Methods
Figure 1 shows the k-space trajectories and the pulse
sequence diagrams of SAGS with two spokes and the
fast-kz method. Note that with the simulated SEM, the
duration of blips for the Z2 SEM was 9 ms. The simulated jMxyj distributions without considering physical
B1þ were shown in Figure 2. Both show similar lower
jMxyj at the FOV center and higher jMxyj at the FOV periphery. However, compared with the fast-kz method
using five spokes, SAGS used only two spokes to achieve
this jMxyj distribution. Importantly, SAGS used only
44% of the RF power of fast-kz.
jB1þj Map of Saline Phantom and Human Head
The estimated jB1þj maps of a saline phantom and two
different brain image slices from two different subjects
are shown in Figure 3. In the saline phantom, the ratio
between the highest and the lowest jB1þj was 3.1. This
ratio in head slice 1 and 2 was 2.65 and 3.39,
respectively. We found that the jB1þj map of the saline
phantom has elliptical isointensity contours. These isointensity contours for the human head jB1þj maps were
more irregular.
Mitigate jB1þj Inhomogeneity Using SAGS
Figure 4 shows the spatial distribution of the z component of the SEM f(x,y) used for jB1þj inhomogeneity mitigation in a saline phantom and two different head
imaging slices. Visually, the isointensity contours in Figure 3 are similar to those in Figure 4. The coefficients to
generate these SEMs were (X, Y, Z2, X2-Y2, XY) ¼
(0.04, 0.03, 1.98, 0.05, 0.17), (0.21, 0.22, 2.87,
1.19, 0.13), and (0.14, 0.24, 0.16, 0.80, 0.64) for
6
Hsu et al.
FIG. 3. The estimated jB1þj from
a saline phantom and a human
head.
phantom, head slice 1, and head slice 2, respectively.
These coefficients were calculated based on two specifications: (1) FOV ranged between 0.5 and þ0.5, and (2)
the minimal and maximal values of the SEM in the imaging object were 0 and 1, respectively. To quantify the
similarity, Figure 5 shows the distribution of pairs of
mxy/B1þ(x,y) and f(x,y) for all image pixels within the
imaging object. We found that for the saline phantom,
the locations of pairs of mxy/B1þ(x,y) and f(x,y) are
almost on a curve when f(x,y) < 0.2. This suggests that
the approximation required by Eq. 4 was accurate
around the center of the imaging object (region corresponding to f(x,y) < 0.2). However, at locations further
away from the center of the imaging object, the approximation became poor as the plot showing a scattered distribution. The normalized standard deviation for the
saline phantom is 0.0828. For human head imaging, the
approximation required by Eq. 4 was reasonably
achieved, since the data points were distributed over a
narrow curved band. The normalized standard deviation
for the head slice 1 and 2 are 0.1015 and 0.1304,
respectively.
Based on the estimated experimental jB1þj distribu
tions and an arbitrarily chosen 20
flip angle
(jMxyj¼0.342), Figure 6 shows simulated jMxyj in the saline phantom using a single spoke, fast-kz, two-spoke tailored excitation, and SAGS with two or four spokes. The
spatial distribution of jMxyj in single-spoke excitation
was similar to the spatial distribution of jB1þj with r ¼
28.3% (relative RF power h ¼ 1; Fig. 6a). The two-spoke
tailored excitation method with the minimally required
RF power can generate the distribution of jMxyj with r ¼
21.8% and g ¼ 0.937 (minimal SAR mode; Fig. 6d) or
the most homogeneous distribution of jMxyj (most
homogeneous mode) with r ¼ 7.5% using a much larger
RF power (g ¼ 19.16; Fig. 6e). The fast-kz with five
spokes had r ¼ 15.5% and with g ¼ 0.837 (Fig. 6f).
SAGS using two spokes had r ¼ 10.2% and g ¼ 0.64
(Fig. 6b). SAGS using four spokes had r ¼ 7.4% and g ¼
0.646 (Fig. 6c). We found that r did not further decrease
as the number of spoke increased from 4 to 6 (data not
shown). Table 1 lists the spoke amplitudes and phases
as well as the associated average and standard deviation
of jMxyj.
For the human head slice 1, the spatial distribution of
jMxyj in the single-spoke excitation had r ¼ 18.6% (g ¼
1; Fig. 7a). The two-spoke tailored excitation method
with the minimally required power (minimal SAR mode)
can generate the distribution of jMxyj with r ¼ 9.9%
with g ¼ 1.73 (Fig. 7c) or the “most homogeneous” distribution of jMxyj with r ¼ 3.9% using much larger RF
power g ¼ 34.8 (most homogeneous mode; Fig. 7d). The
fast-kz with five spokes had r ¼ 11.26% and with g ¼
1.14 (Fig. 7e). SAGS using two spokes had r ¼ 9.9% and
g ¼ 0.92 (Fig. 7b). SAGS using four spokes had similar r
and g with that using two spokes (data not shown). Table
2 lists the spoke amplitudes and phases as well as the
associated average and standard deviation of jMxyj.
For head slice 2, the spatial distribution of jMxyj using
single-spoke excitation had r ¼ 22.9% (g ¼ 1; Fig. 8a).
The two-spoke tailored excitation method with the minimally required RF power can generate the distribution of
jMxyj with r ¼ 14.1% with g ¼ 1.61 (minimal SAR mode;
Fig. 8d) or the most homogeneous distribution of jMxyj
with r ¼ 7.7% using much larger RF power (most homogeneous mode; g ¼ 26.9; Fig. 8e). The fast-kz method
using five spokes had r ¼ 17.0% and g ¼ 0.92 (Fig. 8f).
SAGS using two spokes had r ¼ 13.4% and g ¼ 1.13
FIG. 4. The spatial distributions of
the SEMs used to mitigate the
transverse magnetization Mxy
inhomogeneity in saline phantom
(left), human head slice 1 (center),
and human head slice 2 (right)
experiments.
Mitigate B1þ Inhomogeneity
7
FIG. 5. The mxy/B1þ(x,y) f(x,y) plot for saline phantom (left), the human head slice 1 (center), and the human head slice 2 (right). Blue
dots represent the normalized empirical mxyn/B1þ(x,y) and calculated f(x,y). The red solid curve represents the least-squares error fitting
to blue dots using up to the 10th order polynomials.
(Fig. 8b). SAGS using four spokes had r ¼ 12.6% and g
¼ 1.51 (Fig. 8c). To facilitate the comparison, we plot
the profiles of jMxyj through the center of the image in
Figure 9. Table 3 lists the spoke amplitudes and phases
as well as the associated average and standard deviation
of jMxyj.
Figure 10 shows the relative standard deviation r
when the number of equally spaced spokes increased
from 1 to 99 in phantom and head imaging experiments.
We observe that r decreases very quickly when only
nine spokes were used. There was only marginal
improvement when more spokes were used. Using 99
spokes had r ¼ 7.3%, 9.9%, and 12.5% for phantom,
head slice 1, and head slice 2, respectively.
DISCUSSION
Under the small flip-angle approximation (10), we propose a method to remap the B1þ map into a lower
FIG. 6. The simulated jMxyj using
measured saline phantom B1þ
generated by a: one spoke, b:
SAGS using two spokes, c: SAGS
using four spokes, d: two-spoke
tailored excitation with the minimally required power, e: twospoke tailored excitation with the
most homogeneous result, and
f: the fast-kz method. r is the
relative standard deviation of jMxyj
and g is the relative RF power.
dimension coordinate system. The advantage of this
remapping is that when the isointensity contours of either linear or nonlinear SEMs are similar to the isointensity contours of B1þ, a simple pulse sequence design
using time-varying linear and nonlinear SEMs can be
used to achieve a homogenous flip-angle distribution
efficiently. Specifically, we demonstrate the benefit of
using linear and quadratic SEMs to reduce the dimension of the k-space (and thus faster and less RF power)
to improve flip-angle homogeneity, when the isointensity
contours of the chosen SEM are similar to the isointensity contours of the desired transverse magnetization divided by jB1þj. This advantage was demonstrated in
simulations using linear z SEM and quadratic SEMs and
a two-spoke SAGS trajectory to generate a transverse
magnetization distribution similar to that generated by
the fast-kz method with only 44% of the RF power
(Fig. 2). Empirical results in saline phantom (Fig. 6) and
human head (Figs. 7 and 8) further demonstrate that the
8
Hsu et al.
Table 1
Locations, Amplitudes, and Phases of Spokes and the Associated Average and Standard Deviation of the Amplitude of the Transverse
Magnetization in the Saline Experiment at 7T
Method
Spoke
Amplitude (a.u.)
Phase (rad.)
Avg. (jMxyj)
Std. (jMxyj)
Single spoke
Fast-kz
1
Central
Side 1
Side 2
Side 3
Side 4
1
2
1
2
1
2
1
2
3
4
1
0.8956
0.1141
0.0769
0.0769
0.1141
0.7000
0.7000
3.0952
3.0952
0.5657
0.5657
0.0816
0.5642
0.5642
0.0816
0
0
p
p
p
p
0
0
0
0
1.2293
1.2293
2.6496
1.0441
1.0441
2.6496
0.3186
0.3316
0.0903
0.0514
0.3602
0.0722
0.3439
0.0255
0.3304
0.0338
0.3318
0.0245
Tailored excitation (minimal power)
Tailored excitation (most homogeneous)
SAGS (2-spoke)
SAGS (4-spoke)
two-spoke SAGS method can generate more homogeneous flip-angle distribution (compared with single-spoke
k-space trajectory and fast-kz method using only linear
SEMs) or a similar flip-angle homogeneity with much
lower RF power (compared with the two-spoke tailored
excitation method).
While our method and fast-kz both analyze the spatial
distribution of Mxy using the small flip-angle approximation, fast-kz only uses the linear SEMs, and SAGS uses
both linear and nonlinear ones. Nonlinear SEMs are not
commonly available in most MRI scanners. Thus, the
applicability of the proposed method may be limited.
However, SAGS can achieve a more homogeneous distribution of jMxyj than the fast-kz method with a shorter
pulse duration (Fig. 1) and a lower RF power (Figs. 6
and 7).
The reasons that the tailored excitation method gives
less homogeneous flip-angle distribution than the SAGS
method in the “minimum SAR” mode (Figs. 6–8) are as
follows: (1) in the minimum SAR mode, the RF power
has been chosen and it is no longer a free parameter.
Thus, the degree of freedom is identical between SAGS
and the tailored excitation methods. (2) The tailored excitation method attempts to use both linear and quadratic SEMs to approximate the required distribution of
the phase accrued between two RF pulses. Under the
small flip-angle approximation (as required to satisfy the
minimum SAR mode), this accrued phase distribution is
FIG. 7. The simulated jMxyj using measured human head slice 1 B1þ generated by a: one spoke, b: SAGS using two spokes, c: twospoke tailored excitation with the minimally required power, d: two-spoke tailored excitation with the most homogeneous result, and e:
the fast-kz method. r is the relative standard deviation and g is the relative RF power.
Mitigate B1þ Inhomogeneity
9
Table 2
Locations, Amplitudes, and Phases of Spokes and the Associated Average and Standard Deviation of the Amplitude of the Transverse
Magnetization in the Head Slice 1 Experiment at 7T
Method
Spoke
Amplitude (a.u.)
Single spoke
Fast-kz
1
Central
Side 1
Side 2
Side 3
Side 4
1
2
1
2
1
2
1
1.1606
0.1611
0.0232
0.0232
0.1611
0.9301
0.9301
4.1725
4.1725
0.6782
0.6782
Tailored excitation (minimal power)
Tailored excitation (most homogeneous)
SAGS (2-spoke)
arccos(mxy/bB1þ(x,y)). However, it is important to note
that the homogeneous distribution of Mxy(x,y) ¼ mxy is
what we aim to achieve ultimately, rather than the distribution of the accrued phase arccos(mxy/bB1þ(x,y)). The
nonlinear arccos function and the denominator term
B1þ(x,y) add undesired weightings on Mxy(x,y). In other
words, the spatial distribution of the accrued phase is
optimized with the least-square error, but not Mxy(x,y).
This may explain why there was only marginal improvement in the absolute value of transverse magnetization
homogeneity by the two-spoke tailored excitation using
the similar RF power (Fig. 6).
One limit of the SAGS is that the two-spoke tailored
excitation method can correctly describe the dynamics of
magnetization excited by two high-amplitude spokes,
while SAGS needs to use the small flip-angle approximation. Thus, the two-spoke tailored excitation method
may achieve better results in cases, where a higher RF
power is used to excite magnetization. This is demonstrated in the case, where a higher RF power can
improve the jMxyj homogeneity (Figs. 6d, 6e, 7c, 7d, 8d
and 8e). One challenge of the two-spoke tailored
FIG. 8. The simulated jMxyj using
measured human head slice 2 B1þ
generated by a: one spoke, b:
SAGS using two spokes, c: SAGS
using four spokes, d: two-spoke
tailored excitation with the minimally required power, e: two-spoke
tailored excitation with the most
homogeneous result, and f: the
fast-kz method. r is the relative
standard deviation and g is the relative RF power.
Phase (rad.)
0
0
p
p
p
p
0
0
0
0
0.9759
0.9759
Avg. (jMxyj)
Std. (jMxyj)
0.3307
0.3375
0.0613
0.0393
0.3468
0.0344
0.3425
0.0132
0.3303
0.0327
excitation method is that it is difficult to further generalize to design SEMs when there are more than two pulses.
By contrast, our method did not have this limitation.
To our knowledge, there is no method of identifying
the globally optimized k-space coordinates in n-dimension, where n is the number of SEMs. In some cases, it
might be easy to identify the appropriate combination of
the SEMs such that its isointesnity contours were similar
to those of the B11 map (like the saline phantom experiment; Fig. 3). There are other cases that the B11 map is
rather complicated and therefore identifying the combination of the SEMs becomes tedious. If such a combination is identified, optimizing spoke locations can become
rather straightforward. In fact, directly fitting SEMs to
the target excitation pattern in Step 1 as described in
this study is just one of many potential approaches to
registering their isocontours; different methods may
result in solutions that require fewer spokes than the one
described here.
SAGS differs from the method proposed by Grissom
et al. (35) by (1) they used parallel RF excitation, but our
method uses only one single RF transmitter, and (2) our
10
Hsu et al.
FIG. 9. Left: the horizontal profiles of the simulated jMxyj at the center of the image using measured saline phantom B1þ generated by A:
one spoke, B: SAGS using two spokes, C: SAGS using four spokes, D: two-spoke tailored excitation with the minimally required power, E:
two-spoke tailored excitation with the most homogeneous result, and F: the fast-kz method. Center: the horizontal profiles of the simulated
jMxyj at the center of the image using measured human head slice 1 B1þ generated by A: one spoke, B: SAGS using two spokes, C: twospoke tailored excitation with the minimally required power, D: two-spoke tailored excitation with the most homogeneous result, and E: the
fast-kz method. Right: the horizontal profiles of the simulated jMxyj at the center of the image using measured human head slice 2 B1þ generated by A: one spoke, B: SAGS using two spokes, C: SAGS using four spokes, D: two-spoke tailored excitation with the minimally
required power, E: two-spoke tailored excitation with the most homogeneous result, and F: the fast-kz method.
method provides a clear description of designing RF
pulses, including k-space dimension reduction (i.e.,
remapping for B1þ maps) and optimizing the location of
RF spokes.
While aiming at achieving a homogeneous flip-angle
distribution, SAGS requires no multiple RF amplifiers
and RF coils as required by the parallel transmit
(17,18,21,40) and RF shimming techniques (12–16).
However, SAGS require quadratic SEMs. Although quadratic SEMs are not commonly available in MRI scanner,
it has been demonstrated that the requirement of quadratic SEMs is not high: even quadratic shimming coils
can generate very satisfying results (37). Our calculations
were based on the second-order nonlinear gradients,
whose spatial magnetic field distributions were similar
to those of the realistic second-order shimming coils in
the most modern MRI, Patloc imaging (28,29), and Ospace imaging (36). Additionally, while parallel transmit
(17,18,21,40) and RF shimming techniques (12–16)
require accurate B1þ maps for each RF coils, SAGS only
require one accurate jB1þj map.
SAGS has two limitations: first, remapping B1þ into
the coordinates defined by the selected SEM can be
imperfect. The second limitation is the number of spokes
to approximate the desired Mxy distribution. The first limitation was studied in Figure 5, where we show the
error of this remapping. It shows that the more accurately we can remap B1þ, the more homogeneous flip
angle we can achieve. The second limitation was studied
in Figure 10, where we parametrically increased the
number of the spoke from 1 to 99. We found that using
only a few spokes can achieve a similar result as using
99 spokes.
In spoke pulse design, the orthogonal matching pursuit
method can efficiently determine suitable spokes location on a multidimensional k-space (41). Typically, the
orthogonal matching pursuit method only checks possible spoke locations separated by integer multiples of the
Table 3
Locations, Amplitudes, and Phases of Spokes and the Associated Average and Standard Deviation of the Amplitude of the Transverse
Magnetization in the Head Slice 2 Experiment at 7T
Method
Spoke
Amplitude (a.u.)
Single spoke
Fast-kz
1
Central
Side 1
Side 2
Side 3
Side 4
1
2
1
2
1
2
1
2
3
4
1
0.8896
0.1405
0.2112
0.2112
0.1405
0.8967
0.8967
3.6684
3.6684
0.7608
0.7608
0.1420
0.8565
0.8565
0.1420
Tailored excitation (minimal power)
Tailored excitation (most homogeneous)
SAGS (2-spoke)
SAGS (4-spoke)
Phase (rad.)
0
0
p
p
p
p
0
0
0
0
0.5972
0.5972
0.5298
1.7102
1.7102
0.5298
Avg. (jMxyj)
Std. (jMxyj)
0.3516
0.3279
0.0806
0.0557
0.3492
0.0493
0.3442
0.0264
0.3259
0.0437
0.3249
0.0408
Mitigate B1þ Inhomogeneity
FIG. 10. The normalized standard deviation r of jMxyj using SAGS
with 1 to 99 equally spaced spokes. The dashed line shows the r
generated by the SAGS method with two or four spokes (Figs. 6–8).
k-space distance determined by the inverse of the FOV
and the Nyquist sampling theorem. Grissom et al. (36)
generalizes this method by a local optimization method
such that the spoke location around these integer modes
can also be the candidate of the optimized solution. The
motivation of using all integer modes as candidate spoke
locations is to approximate any arbitrary Mxy distribution
by providing complete bases. However, when targeting at
reducing Mxy inhomogeneity with spatially smooth variation, searching spoke locations around lower-order
modes should generate satisfactory results. In fact, exhaustive searching around lower-order mode may generate more efficient excitation. This is demonstrated in our
study of using 99 equally spaced spokes (Fig. 10): SAGS
with as few as two or four spokes can generate a transverse magnetization distribution similar to the transverse
magnetization distribution generated by 99 equally
spaced spokes.
The spoke locations, amplitudes, and phases may be
estimated by taking the Fourier transform of the 1D target excitation pattern as a function of f. However, there
are two concerns. Using the Fourier transform to find
spoke locations/amplitudes/phases implies that (1) all
data points in the remapped domain f are equally important and (2) the optimized spoke locations are separated
by integer multiples of 1/FOV. However, the data points
in the remapped domain f are not uniformly distributed
(Fig. 5). Additionally, spokes separate by integer multiples of 1/FOV does not guarantee the most efficient solution (Fig. 10).
While we only demonstrated SAGS at the central (z ¼
0) slice, it is possible to use SAGS to excite other slice
by shifting the frequency of the excitation pulse to excite
a different slice and shifting the center of Z2 gradient by
adding a z SEM.
We only used the Gaussian RF pulse to achieve slice
selection in the z direction in our study. It is possible to
use other waveforms to achieve the similar slice selection. While it may be possible to simultaneously drive Z
and other quadratic SEMs to achieve the desired spatial
11
distribution of jMxyj, our study only demonstrates how to
design multiple-spoke excitations to drive the linear Z
and quadratic SEMs sequentially. Exploration of driving
multiple SEMs simultaneously is certainly interesting
yet it is outside the scope of this work.
At high-field MRI, the SAR management becomes an
important issue as the electromagnetic wave during RF
excitation can cause deleterious interaction to endanger
the subject (23). This concern is especially critical in
parallel transmission, where RF pulses from multiple
coils can potentially combine constructively to elevate
local electric fields and eddy currents (23). For SAGS,
the SAR management may be qualitatively evaluated by
comparing with the SAR of traditional slice-selective excitation using the g metric in this study. This is because
that under the assumption that the strength of the electric field inside the object is linearly proportional to the
square root of the transmitted RF power and that the
conductivity does not change with the transmitted RF
power, the local SAR defined as the product of conductivity and the square of electric field is proportional to
the transmitted power. Importantly, because SAGS does
not use multiple coils for parallel RF transmission, the
potential RF power hazard from the interaction between
RF coils can be avoided.
In conclusion, we proposed the SAGS as a method of
design spoke RF pulses using linear and nonlinear SEMs
with potential advantage of approximating the desired
flip-angle distribution in a lower dimension k-space.
Without using multiple RF coils for parallel transmission, SAGS can be one method to mitigate the B1þ inhomogeneity in high-field MRI experiments.
ACKNOWLEGMENT
The authors thank Dr. Lawrence L Wald for his help on
the 7T data.
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