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IEEE Trans Med Imaging. Author manuscript; available in PMC 2009 April 06.
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Published in final edited form as:
IEEE Trans Med Imaging. 2008 September ; 27(9): 1213–1229. doi:10.1109/TMI.2008.920605.
Sparsity-Enforced Slice-Selective MRI RF Excitation Pulse
Design
Adam C. Zelinski, IEEE* [Student Member] and
Research Laboratory of Electronics, Department of Electrical Engineering and Computer Science,
Massachusetts Institute of Technology (MIT), Cambridge, MA 02139 USA.
Lawrence L. Wald
HST and the Department of Radiology, Harvard Medical School, Longwood, MA 02115 USA.
Kawin Setsompop and Vivek K Goyal, IEEE [Senior Member]
Research Laboratory of Electronics, Department of Electrical Engineering and Computer Science,
Massachusetts Institute of Technology (MIT), Cambridge, MA 02139 USA.
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Elfar Adalsteinsson
Research Laboratory of Electronics, Department of Electrical Engineering and Computer Science,
Massachusetts Institute of Technology (MIT), Cambridge, MA 02139 USA and also with the
Harvard-MIT Division of Health Sciences and Technology (HST), Cambridge, MA 02139 USA.
Abstract
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We introduce a novel algorithm for the design of fast slice-selective spatially-tailored magnetic
resonance imaging (MRI) excitation pulses. This method, based on sparse approximation theory,
uses a second-order cone optimization to place and modulate a small number of slice-selective
sinc-like radio-frequency (RF) pulse segments (“spokes”) in excitation k-space, enforcing sparsity
on the number of spokes allowed while simultaneously encouraging those that remain to be placed
and modulated in a way that best forms a user-defined in-plane target magnetization. Pulses are
designed to mitigate B1 inhomogeneity in a water phantom at 7 T and to produce highly-structured
excitations in an oil phantom on an eight-channel parallel excitation system at 3 T. In each
experiment, pulses generated by the sparsity-enforced method outperform those created via
conventional Fourier-based techniques, e.g., when attempting to produce a uniform magnetization
in the presence of severe B1 inhomogeneity, a 5.7-ms 15-spoke pulse generated by the sparsityenforced method produces an excitation with 1.28 times lower root mean square error than
conventionally-designed 15-spoke pulses. To achieve this same level of uniformity, the
conventional methods need to use 29-spoke pulses that are 7.8 ms long.
Keywords
B1inhomogeneity mitigation; high field strength; magnetic resonance imaging (MRI) radiofrequency (RF) pulse sequence design; parallel transmission; sparse approximation; threedimensional (3-D) RF excitation
© 2008 IEEE
*
(e-mail: zelinski@MIT.edu).
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I. Introduction
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SLICE-SELECTIVE radio-frequency (RF) excitation pulses are ubiquitous throughout
magnetic resonance imaging (MRI). They are widely used to excite only those magnetic
spins within a thin slice of tissue while leaving spins outside the slice in alignment with B0,
the system's magnetic main field. This type of excitation is extremely useful because it
simplifies the encoding stage of the MRI experiment by permitting the system to record and
reconstruct an image from only two dimensions of data. RF pulses that are not only able to
excite a thin slice, but able to spatially-tailor the in-plane excitation (i.e., produce a varying
in-plane flip angle) are also highly useful because they are able to reduce field of view
(FOV) requirements in dynamic imaging applications [1], decrease susceptibility artifacts in
functional MRI [2], and mitigate
transmit profile inhomogeneity occurring at high field
[3].
A. Spoke-Based Slice-Selective Pulses
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In this work we focus on a class of slice-selective pulses comprised of segments that
resemble sine cardinal (sinc) functions, which are applied in the presence of oscillating z
gradients. We refer to these pulse segments as “spokes” [3]-[5] because as each is played, its
trajectory in k-space is a straight line. These pulses rely on the small-tip angle
approximation, where a Fourier relationship (approximately) holds between the energy
deposited by the RF pulse and the resulting transverse magnetization, giving rise to
excitation k-space [6]. In this regime, a rectangle-like slice profile along a logical z axis is
achieved by placing a sinc-like RF pulse segment (a spoke) in the kz direction of excitation
k-space [7]. In practice, a true sinc function along kz is replaced by a finite-length waveform
[8], e.g., a Hanning-windowed sinc [5]. The time-bandwidth product of the pulse segment
and its extent in kz influence the thickness and transition edges of the resulting thin-slice
excitation.
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Playing a single sinc-like spoke at (kx, ky) = (0, 0) (dc) in k-space is one way to excite a thin
slice. This approach, where each of the system's excitation channels encodes a single
amplitude and phase along the spoke, is known as RF shimming [4], the goal of which,
typically, is to mitigate an inhomogeneous B1 field profile. A single frequency sample in
(kx, ky), however, does not provide any ability to tailor the resulting in-plane excitation
pattern when using a single-channel excitation system; even with a multichannel excitation
system, significant tailoring is not possible with this approach. To strongly influence the inplane excitation flip angle in each of these cases, one must place a number of spokes at
various locations in (kx, ky), modulating the amplitude and phase of each to form a desired
in-plane transverse magnetization [3]. Unlike RF shimming, this approach does not alter the
B1 field profile—rather, it is the gradient modulation of the excitation process that produces
the desired magnetization. In this multiple-spoke case, the complex weightings in (kx, ky)
form the in-plane excitation, while the sinc-encoded kz traversals provide slice selectivity in
z. Simultaneous slice selection and in-plane tailoring are possible due to the separability of
the 3-D Fourier transform that relates the spatial excitation to the deposition of energy in kspace. Unfortunately, using multiple spokes has the negative consequence of increasing
pulse duration. Thus, an ideal spoke-based pulse is one that not only achieves the userspecified in-plane excitation, but does so using as few spokes as possible.
B. Spoke-Based Pulse Applications
To date, spoke-based waveforms (also referred to as echo-volumnar trajectories) have been
used to mitigate B1 inhomogeneity on a single-channel excitation system at 3 T [3] and
perform spatially-tailored slice-selective excitations on multichannel excitation systems [5],
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[9]-[11]. In these cases, only small numbers of spokes were placed at low spatial frequencies
because only smoothly-varying in-plane excitations were necessary.
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There are many applications, however, where more than a few spokes are needed. For
example, 25-spoke pulses were needed for signal recovery in -weighted functional MRI
[12]. Another application of multispoke pulses is the mitigation of B1 inhomogeneity [13]
that occurs in the human body at 4 T [14] and in the head at 7 T due to wavelength
interference effects [15], [16] and tissue-conductive RF amplitude attenuation [17]. The
presence of B1 inhomogeneity causes a system's transmit and receive profiles to exhibit
high-frequency spatial variations. The nonuniformity of the transmit profile causes standard
slice-selective pulses to produce a nonuniform flip angle across space, one that varies widely
throughout the field of excitation (FOX), causing images to exhibit severe centerbrightening, spatial contrast variation, and SNR nonuniformity despite the use of
“homogeneous” excitation and reception coils [18]-[20]. In this situation, an ideal sliceselective mitigation pulse is one that produces a spatially-tailored excitation across the FOX
that is the exact multiplicative inverse of the inhomogeneity, strongly flipping those regions
where the B1-effect causes under-flipping of the magnetization and vice versa, such that
when the pulse is applied, a uniform flip angle arises across the FOX. In practice, forming
such a high-spatial-variation excitation requires spokes to be placed with care at both low
and high frequencies throughout (kx, ky).
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C. Conventional Spoke Placement
One way to achieve a desired high-variation in-plane excitation is to place a large number of
spokes throughout (kx, ky), but this leads to lengthy, impractical pulses. A more intuitive
approach is to compute the Fourier transform of the desired in-plane excitation and place
spokes at (kx, ky) frequencies where the Fourier coefficients are largest in magnitude [12],
[21]. We will show shortly, however, that this method yields suboptimal placements.
D. Sparsity-Enforced Spoke Placement
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In this paper we introduce a spoke-based pulse design framework based on sparsity and
simultaneous sparsity [22]-[27]. Our method is applicable to all of the problems mentioned
above, yet general enough to apply to both conventional single-channel and multichannel
parallel transmission systems [5], [9]-[11], [28], [29]. In our approach, we pose optimization
problems that determine the minimal number of spokes needed to produce a user-defined inplane excitation, the placement of these spokes in (kx, ky), and their proper amplitudes and
phase modulations. For computational tractability, we use a convex relaxation [25], [26],
[30]-[32], formulating a second-order cone (SOC) program [33] that promotes sparsity on a
(kx, ky) grid of candidate spoke locations. This sparsity-enforcement algorithm discourages
the use of many spokes while simultaneously encouraging those that remain to be placed
and modulated in a way that best achieves the desired in-plane excitation. Preliminary
results of this technique were presented in [34] and [35]. This design method automates
spoke-based RF pulse design and frees the designer from the task of spoke placement. The
method has several control parameters that allow designers to easily trade off pulse duration
(number of spokes) with excitation fidelity (squared error relative to a target). This
technique automatically generates echo-volumnar thin-slice excitation pulses, differing
greatly from other automated design algorithms that optimize 2-D trajectories and produce
pulses incapable of slice selection [36].
E. Overview of Experiments
The utility of the sparsity-enforced spoke placement method is demonstrated by conducting
a B1 mitigation experiment in a head-shaped water phantom on a human scanner at 7 T,
along with a spatially-selective excitation experiment on an eight-channel parallel
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transmission system at 3 T. In each experiment, the sparsity-enforced placement algorithm is
compared to the conventional Fourier-based placement method and to an extension of the
latter, which we call “inversion-based” placement. After demonstrating the superior
performance of the sparsity-enforced placement algorithm, we proceed to characterize it in
detail by exploring its sensitivity to its control parameters. Results for these various
experiments are first obtained via Bloch-equation simulations and then validated with trials
on the 7 T and 3 T systems; thus the sparsity-enforced method is investigated in both theory
and practice.
F. Organization of the Paper
Section II covers background material on parallel excitation systems, RF excitation pulse
design, transmit and receive profile estimation, and sparse approximation. Section III
presents the conventional Fourier-based spoke placement method, the inversion-based
technique, and the proposed sparsity-enforced algorithm. Section IV discusses how to design
a set of RF pulses and gradients based on a chosen spoke placement pattern. Section V
describes the 7 T and 3 T systems and their associated experiments, while Section VI
presents and discusses the experimental results. Concluding remarks appear in Section VII.
II. Background
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A. Parallel Excitation RF Pulse Design
1) System Overview—A parallel excitation system differs from a standard single-channel
system in that its RF excitation coil is comprised of multiple elements, each capable of
independent, simultaneous transmission. The presence of multiple elements allows one to
undersample a given excitation trajectory and yet in many cases still form a high-fidelity
version of the desired excitation; undersampling the trajectory is greatly beneficial because
it reduces the distance one travels in k-space, which in turn reduces the duration of the
corresponding pulse [5], [9]-[11], [28], [29]. This ability to “accelerate” in the Fourier
domain and reduce pulse duration arises due to the extra degrees of freedom provided by the
system's multiple transmit elements, in direct analogy with the concept of readout-side
acceleration where the use of multiple reception coils permits one to undersample readout kspace and substantially reduce readout time [37], [38]. We now describe one way to design a
set of P RF pulses to play through the elements of a P-channel parallel excitation system in
order to generate a user-defined target excitation. This formulation, largely due to Grissom
et al. [39], holds for P ≥ 1 and thus applies to both single- and multichannel systems.
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2) Linearization—To establish a design methodology for RF pulses, assume for now that
the set of gradient waveforms, G(t) = [Gx(t), Gy(t), Gz(t)]T, is fixed, i.e., the k-space
trajectory, k(t) = [kx(t), ky(t), kz(t)]T, is predetermined, and that the spatial transmit profile
(a
map) of each array element is known. Finally, in order to make the design process
tractable, we impose the small-tip angle approximation on the problem, enabling a Fourierbased design [6]. This results in the following system of linear equations:
(1)
where M0is the steady-state magnetization, r = [rx, ry, rz]T a spatial variable, t a temporal
variable (sec), m(r) the approximate transverse magnetization due to the transmission of the
RF pulses in the presence of the time-varying gradients (rad), the gyromagnetic ratio (rad/
transmit profile of the pth coil (T/V), b1,p(t) the RF pulse
T/s), Sp(r) the complex-valued
played along the pth coil (volts), ΔB0(r) a field map of B0 inhomogeneity (rad/s),
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eiΔB0(r)(t–L) the phase accrual due to this inhomogeneity, L the duration of each waveform
(sec), and
[39].
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3) Discretization—Discretizing (1) in space and time leads to
(2)
Here,
is a diagonal matrix comprised of Ns samples of the pth spatial profile
taken within a user-defined FOX, which in many cases is smaller than the FOV. The next
matrix,
, incorporates the effect of main field nonuniformity and brings energy
placed in k-space into the spatial domain at the Ns locations where each coil profile is
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are
sampled. Formally, F(m, n) = i M0ΔteiΔB0(rm)(tn–L)eirm·k(tn). Each of the
comprised of samples of the pth RF waveform, b1,p(t), spaced by Δt. Finally, Atot and btot
are implicitly defined in (2) [39]. Users are free to choose how finely to sample the spatial
and temporal variables in (1) and thus control whether m = Atotbtot is under- or overdetermined [40], [41]. Furthermore, by sampling r only within the FOX, (2) becomes blind
to the rest of the FOV and is freed from needless spatial constraints.
4) RF Waveform Generation—To excite a pattern d(r) using a P-channel system, P RF
pulses are needed. To determine these pulses, one may form Ns samples of the desired
pattern into the vector
. Then, by estimating a ΔB0 map and each spatial coil profile,
one may generate F, the Sps, and in turn Atot. Finally, a set of RF pulses that
(approximately) achieves the desired excitation may be generated by solving
(3)
for btot. Solving (3) via direct inversion (or pseudoinversion) of Atot tends to result in
poorly-conditioned btot vectors [42], so we instead solve the following Tikhonovregularized problem [43] that penalizes the energy of the solution:
(4)
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Setting to a small positive value and then solving (4) typically produces a btot with
reasonable and voltages without seriously impacting the residual error [39], [40],
[44]. In our work, we use the LSQR algorithm [45] to solve (4), which has been shown in
practice to yield favorable multichannel pulse designs [42].
5) Spoke-Based RF Pulse Generation—For most spoke-based pulse designs, one fixes
not only k(t) and d(r), but desired slice thickness and spoke type as well. In this work, slice
thickness is fixed at 10 mm and each spoke is a Hanning-windowed sinc. For all T-spoke
pulses, the spoke at dc in (kx, ky) has a time-bandwidth product of 4, while the T – 1 off-dc
spokes have kz-lengths half that of the former. (Using shorter off-dc spokes lets one reduce
the duration of a pulse without noticeably impacting slice selection performance [5].)
Overall, these constraints fix the shape of the P RF pulses and all properties of the slice
profile (e.g., sidelobes and sharpness) [3], [5], while still letting the user retain control over
the amplitude and phase each channel encodes along each spoke.
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For a T-spoke pulse on a P-channel system, these constraints cause major changes to (2) and
(3). First, all but T columns of F are discarded, since only T locations in (kx, ky)-space
(those where the spokes are located) are available for weighting. Further, each bp reduces
from Nt to T elements. Finally, the FOX reduces to a 2-D in-plane region, leading to a far
smaller Ns than in a 3-D case. The system now has only PT unknowns, giving the user
control over PT weights to influence the in-plane excitation pattern.
B. Transmit and Receive Profile Estimation
In order to generate RF pulses for use on a P-channel system, one sees from (1) and (2) that
estimates of the spatial transmit profiles,
, are first required. In our work, we only
need coil profile estimates in the 2-D plane where the thin-slice excitation occurs, so here r
indexes samples only within a 2-D FOX.
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1) Magnitude Estimation—Here we use a parametric fitting method to obtain |Sp(r)| in
T/V. This technique also estimates |R(r)|, the system's receive profile magnitude. For fixed
p, the following steps are performed: first, NV intensity images are collected. Each is
obtained by playing a standard slice-selective pulse with duration τ through channel p
(leaving all other P – 1 channels dormant) and reading out the result on the receive coil with
a gradient-recalled echo (GRE). This pulse's peak transmit voltage, V, is varied from one
image to another, and increased enough so that several high-flip images are obtained. Then,
for all coordinates r in the FOX, the intensities of these NV images are fit to
(5)
where SIp,V is the intensity image due to playing the pulse with peak voltage V through
channel p, |α(r)| = τ|Sp(r)|, and E1(r) = e–TR/T1(r) [46]-[48]. In this work, experiments are
conducted in uniform-T1 phantoms, so E1(r) is known.
2) Phase Estimation—To estimate the phase of each channel profile, we keep V constant
and collect one low-flip image per channel; each phase profile is then simply the phase of
each respective image. This yields the phase relative to the receive coil, which is sufficient
[5], [42].
C. Sparse Approximation
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,
1) Single-Vector Case—Consider a linear system of equations d = Fg, where
,
, and d and F, are known. It is common to use the pseudoinverse of F,
denoted F†, to determine ĝ = F†d as an (approximate) solution to the system of equations.
When d is in the range of F, ĝ is the solution that minimizes ∥ĝ∥2, the Euclidean or norm
of ĝ. When d is not in the range of F, no solution exists; ĝ minimizes ∥ĝ∥2 among the
vectors that minimize ∥d–Fĝ∥2. In the design problems in this paper, it is necessary for the
analogue to ĝ to have only a small fraction of its entries different from zero. We are faced
with a sparse approximation problem of the form
(6)
where ∥·∥0 equals the number of nonzero elements of a vector. The subset of {1,2,...,N}
where there are nonzero entries in g is called the sparsity profile. For general F, solving (6)
essentially requires a search over the 2N – 1 nonempty sparsity profiles. The problem is thus
computationally infeasible except for very small systems of equations (e.g., even for N = 30,
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one may need to search 1 073 741 823 subsets). Formally, the problem is NP-complete [22],
[23].
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For problems where (6) is intractable, a large body of research supports the relaxation of (6)
[25]
(7)
This is a convex optimization and thus may be solved efficiently [49]. The solution of (7)
does not always match the solution of (6)—if it did, the intractability of (6) would be
contradicted—but certain conditions on F guarantee a match [26], [31] and (7) is generally a
useful heuristic [50]. The optimization
(8)
has the same set of solutions as (7). The first term of (8) keeps residual error down, whereas
the second promotes sparsity of g [25], [51]. As the control parameter, λ, is increased from
zero to one, the algorithm yields sparser solutions and the residual error increases; sparsity is
traded off with residual error.
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2) Multiple-Vector Case (Simultaneous Sparsity)—Now suppose we have the
system of equations
(9)
and the
are known. In the design problems posed in this paper, it
where
is natural to constrain the number of positions in which any of the gps are nonzero. Thus we
seek approximate solutions in which the gps are not only sparse, but the union of their
sparsity patterns is small; this is called simultaneous sparsity [30], [32]. Unfortunately,
optimal approximation with a simultaneous sparsity constraint is even harder than (6). Thus,
we propose to use a relaxation similar to (8)
(10)
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where
and
, i.e., the norm of the
norms of the rows of G. This latter operator is a simultaneous sparsity norm: it penalizes the
program (produces large values) when the columns of G have dissimilar sparsity profiles
[30]. An equivalent way of writing (10), which will be useful later, is to replace ∥G∥s with
, where
element of this vector contains the
sees that (8) is a base case of (10).
, and
, i.e., the nth
norm of the nth row of G. Finally, by setting P = 1, one
To the best of our knowledge, (10) is a novel instance where simultaneous sparsity is
required. To date, others have sought simultaneously sparse gps such that [d1...dP] =
F[g1...gP] holds [30], [32], [52], whereas here we are seeking simultaneously sparse gps that
solve (9). In other words, in prior work, there are many different ds and a single F, whereas
here, we have a single d and many different Fs.
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3) Solution via Second-Order Cone Programming—One may solve (10) using
second-order cone (SOC) programming, a method that encapsulates conic, convex-quadratic
[33], and linear constraints. Quadratic programming is not an option because the terms of
(10) are complex. Second-order conic constraints are of the form s : ∥[sl,...,sn–1]T∥2 ≤ sn, and
the generic format of an SOC program is
(11)
, H is the complex conjugate,
is the N-dimensional positive
where
orthant cone, and the Lns are second-order cones [33]. SOC programs are useful because
they can be solved via efficient interior point algorithms [53], [54]. To convert (10) into the
SOC format, we first write
(12)
This makes the objective function linear, as required. Now we change variables, rewriting
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∥G∥s ≤ q as
, for n = 1,...,N, and 1Hy ≤ q, where 1 is an N × 1 vector of
ones and y = [y1,...,yN]T. Letting z = dtot – Ftotgtot, we arrive at
(13)
With these steps, the problem is a fully-defined SOC program that may be implemented and
solved numerically. In our work, we implement (13) in SeDuMi (self-dual-minimization)
[53], and find that it is capable of solving problems in a reasonable amount of time (e.g.,
3−25 min) when the number of unknowns, PN, is in the thousands. It is possible that a more
efficient method for solving (10) can be found, but techniques for (10) lag behind those for
(7) and (8).
III. Spoke Placement and Weighting Methods
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We now describe three methods for determining the placement of spokes in (kx, ky) when
designing a spoke-based RF waveform whose goal is to produce a user-defined in-plane
target excitation, d(x, y), using a P-channel system whose coil profiles are known. The
second and third methods will suggest spoke weights in addition to placement. Assume that
the FOV is a rectangle, so when d(x, y) is sampled, its NFOV samples may be assembled into
. Furthermore, assume the FOX is smaller than the FOV and not necessarily
rectangular. It is discretized and comprised of NFOX < NFOV samples, which are a proper
subset of the FOV samples. (The NFOV – NFOX pixels in the FOV but not in the FOX
comprise a “don't care” region.) Finally, assume the system's P coil profiles are sampled
within the FOX and assembled into the diagonal matrices
.
The goal now is to place spokes at T locations in (kx, ky) such that the resulting in-plane
excitation is close (in the sense) to d(x, y) within the FOX (not the entire FOV).
A. Fourier-Based Spoke Placement
An intuitive way to determine where to place each spoke is to compute the discrete Fourier
transform (DFT) of D and from the resulting discrete grid of (kx, ky) frequencies, choose the
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T whose Fourier coefficients are largest in magnitude [12], [21]. If the complex weights at
these T locations are brought back into the spatial domain via an inverse transform,
Plancherel's theorem implies that this image, D̂T, is the best approximation of D within
the FOV [7], [55]. Unfortunately, there are problems with this method. 1) The FOX is
typically smaller than the FOV, and the RF profile in the “don't care” region is not
important. The Fourier method, however, has no concept of the FOX or the “don't care”
region. This FOX–FOV mismatch means that the T chosen locations do not necessarily
yield the best representation of d(x, y) within the FOX. 2) The influence of each spatial
profile, Sp(r), is not accounted for, even though each has a major impact on the in-plane
excitation. 3) There is no concept of transmit channels, so the method cannot suggest a
weight for each channel to place at each spoke location.
B. Inversion-Based Spoke Placement
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Single-Channel Derivation: This method retains the logic of the Fourier technique while
accounting for its shortcomings. For now, assume the system has a single channel and
imagine a grid of Nf arbitrarily-spaced points in (kx, ky) located at {k1,...,kNf}. Each is a
Dirac delta that produces a complex exponential in the spatial domain. An arbitrary choice
of complex weights at different points on this grid results in a nominal spatial domain
excitation, which is related to the weighted grid by a Fourier transform. The nominal
excitation is then multiplied (pointwise) by the coil profile to yield the actual excitation,
m(x, y). One may sample this excitation within the FOX at {r1,...,rNFOX} and arrange the
. Likewise, the weights on the grid may be formed into
. The
samples into
relation between m and g is represented by
, where F(m, n) = i M0Δteirm·kn,
where the nth column of F describes how the excitation at the rms is influenced by the
, this leads to
weight at kn. When samples of the coil profile are arranged into
(14)
This is analogous to the derivation in Section II-A, In short, (14) expresses the excitation
that forms at a set of points in the FOX when an arbitrary set of complex weights is placed
on an arbitrary k-space grid. For fixed kn, the column of F here is nearly identical to the
column of its counterpart in Section II-A, except now there is no notion of time. For the
most part, this is tolerable: kn is known rather than k(t) at time tn. The difference here is that
time-dependent ΔB0 phase accrual cannot be incorporated because k's dependence on t is
not yet known. Fortunately, if B0 is nearly homogeneous or pulse duration is short, then the
column of F here (nearly) equals the column of its counterpart in Section II-A.
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, and by solving
By sampling d(x, y) analogously to m(x, y), it is possible to form
d = SFg for g, the weighting of the (kx, ky) grid that best achieves the target excitation in the
sense within the FOX may be obtained. This solution is generated via regularization
rather than direct pseudoinversion
(15)
This yields a well-conditioned g that (approximately) solves the system and provides a set of
grid weights that form the desired excitation. At this point, two pitfalls of the Fourier
method have been mitigated: the coil profile has been explicitly incorporated and the
unnecessary constraints on the excitation outside the FOX have been removed. Now, using
the intuition that motivated the Fourier method, we decide to place spokes at the T points on
the grid that correspond to the T largest-magnitude elements of g.
1) Multichannel Derivation—For multichannel systems, (14) generalizes to
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(16)
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analogously to (2). A regularization analogous to (15) is then applied to solve d = Atotgtot,
yielding a well-conditioned solution, gtot, which is disassembled into
, where gp
contains the weights for channel p. When an element of gp is large in magnitude, channel p
is using the corresponding grid point to strongly influence the resulting excitation, so
placing a spoke here may be worthwhile. However, at this same location, imagine that the
other P – 1 channels have weights of zero; should a spoke still be placed here, if only one
channel suggests doing so, or should it be placed at an alternate location, one where each
channel contributes toward the formation of d(x, y)?
The presence of multiple channels has led to a quandary about how to choose the T best
spoke locations. A solution to this problem is to “compress” the energy of the resulting gpweighted grids along the channel dimension using the norm. This produces a single grid
that represents the overall energy placed at each (kx, ky) location by all P channels, denoted
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, where
. Using this norm to combine the energy deposited
by all P channels at the ith grid location has no sparsifying effects, so
will not
radically differ if one channel deposits a large amount of energy at the ith grid point, or if
this same amount of total energy is contributed by many channels. Forming
is sensible
because when performing excitations using a multichannel system, it does not matter if it is
one or many channels that place energy at a given point if such a deposition improves the
excitation. Based on this reasoning, we reapply the intuition that inspired the Fourier-based
method, and decide to place spokes at the T locations corresponding to the largestmagnitude elements of
.
Unfortunately, using Tikhonov regularization to find gtot yields a dense solution, leading to
, one where all grid locations experience moderate amounts of energy
a dense
deposition. Frequently, the energies at each of the T chosen locations are not much greater
than the energies present at most other points. Up until this point, the magnitude of energy at
a grid point has been used as proxy for whether that point is a good spoke location, but since
the energies being deposited across all candidate locations are now similar, it is no longer
clear if this intuition is useful. The ideal weights for each channel to place on the candidate
grid have indeed been determined, but this process has not restricted the number of points at
which a channel may deposit energy.
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This is a serious drawback, because when T spokes are chosen, only T locations in (kx, ky)
remain for the P channels to modulate and attempt to form d(x, y). Thus in practice, when all
weights are zeroed out except those PT weights associated with the chosen locations, the
resulting excitation does not closely resemble the desired one. The achieved excitation is of
low quality because the PT chosen weights relied on the weights placed at all other grid
locations to form an accurate excitation. To mitigate this problem, the PT weights need to be
retuned (see Section IV). In short, this method does not encourage the channels to form the
excitation by modulating only a small number of locations, despite the fact that only a small
subset of candidate grid points are able to be retained for the final pulse design.
C. Sparsity-Enforced Spoke Placement
This method—our main contribution—resolves the problem of the former by finding a
sparse
that reveals a small set of spoke locations (and weights) that alone form a highfidelity excitation. In Section VI, we demonstrate that this approach is superior to the
inversion method at revealing a small number of locations at which to place spokes.
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1) Single-Channel Derivation—Since P = 1, there is only one grid vector, g, and the
system of equations we desire to solve is d = SFg. Rather than using Tikhonov
regularization to reveal spoke locations, we instead promote the sparsity of g
(17)
Because of the relationship between sparsity and regularization [25], [51], this
optimization yields a sparse g that approximately solves the system.
2) Multichannel Derivation—With P > 1 channels, the resulting excitation may be
expressed as (16), so we again arrive at d = Atotgtot. To mitigate the inversion-based
method's dense
grid problem, we rely on sparse approximation and pose an optimization
that promotes the sparsity of
(18)
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This is equivalent to (10), the only difference being notational (
is used rather than
∥G∥s). With large enough λ, this method produces gps that approximately solve the system
. Typically, solving (18) yields a
with many elements
while also yielding a sparse
close to zero, with only several high-energy grid points. (The small grid weights are not
exactly zero; this is because an regularization is used to promote sparsity rather than an
ideal but computationally intractable penalty.)
In this way, the sparsity-enforced spoke placement reveals a small subset of the overall grid
that alone is able to form the excitation. At this stage, T spoke locations are selected by
choosing those points on the grid corresponding to the largest-magnitude elements of
. In
contrast with the inversion-based method, the PT weights associated with these T locations
alone are typically able to generate a reasonable version of the desired excitation when other
weights are zeroed out. This does not occur by chance. Rather, it is because the T columns
of F in Atot are nearly identical to the T columns of the similarly-named matrix in Section
II-A. (This claim holds if ΔB0 is small or the pulse will be of short duration, the latter of
which is always the case for small T.) Thus this method determines a small set of spoke
locations and weightings that form an acceptable-quality excitation. Like the inversionbased method, the weights undergo retuning (discussed next), but in many cases they do not
radically change.
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3) Necessity of Simultaneous Sparsity—What if rather than promoting the sparsity of
, we simply promote that of gtot? This would produce a sparse gtot, and thus individuallysparse gp. But this does not encourage simultaneous sparsity among the gps, so their sparsity
profiles differ, and thus when
is formed, it is not necessarily sparse, contradicting our
goal. Further, ∥gtot∥1 discourages solutions where many channels deposit energy at a single
location, and thus penalizes solutions which are in fact useful.
4) Variants—One may modify (18) so that an rather than penalty is imposed on the
system. Furthermore, if the conditioning of the gps is a concern, constraints may be placed
on their and energies. These variants and their combinations are still SOC programs.
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IV. Generation of Gradients, Retuned Weights, and RF Pulses
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At this point, we have a P-channel system whose profiles are known and have used one of
the methods to determine T locations at which to place spokes. Assume that ΔB0(r) and the
gradient coil's amplitude and slew rate constraints are known, all spoke parameters are fixed,
and the assumptions from Section III are in effect. To begin, a set of gradients must be
designed to visit and traverse a spoke at each location in an efficient manner. To accomplish
this, a genetic algorithm is used to find a near-optimal minimal-length Euclidean path
through the T points. Knowing this short path allows us to design the gradients, G(t), to
drive to each (kx, ky) coordinate in a way that minimizes pulse duration. The time-dependent
k-space trajectory, k(t), is obtained in turn by integrating these gradients. Additionally,
because the spoke parameters have been fixed, the shape of each channel's RF pulse is
known.
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The next task is to obtain the weight each channel should encode along each spoke location,
i.e., to find PT weights that best generate d(x, y). If the Fourier method is used for
placement, this step is necessary because it does not generate weights during the placement
stage. If the inversion method is used, weight retuning is also a necessity (see Section III-B).
If the sparsity-enforced method is used, the weights it generates often do form a reasonablyaccurate excitation, but still benefit from retuning because they have a slight reliance on the
various small weights that get discarded when only T locations are retained. Further, if
ΔB0(r) ≠ 0, retuning helps the inversion and sparsity-enforced methods because the weights
these techniques initially determine cannot take B0 inhomogeneity into account (see Section
III-B).
We now apply the pulse design method of Section II-A. With k(t), the coil profiles, and
ΔB0, we update the T columns of F to account for ΔB0 phase accrual and then generate Atot
in (2). We then solve (4), generating the PT weights that the RF pulses should use to best
produce the excitation. This concludes the design process. (Note that the PT weights may be
obtained via an alternative approach, e.g., [9], [28], [56].)
V. System and Experiment Setup
A. B1 Mitigation in a Head-Shaped Water Phantom on a Single-Channel 7 T Scanner
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1) Overview—Each placement method will design pulses that mitigate B1 inhomogeneity
present in a head-shaped water phantom at 7 T using different numbers of spokes. Here, the
ideal mitigation pulse is one that excites the multiplicative inverse of the inhomogeneity [3]
and produces a uniform magnetization across the FOX. Each method will attempt to produce
this ideal excitation.
2) Hardware—Work is conducted on a 7 T whole-body human scanner (Siemens Medical
Solutions, Erlangen, Germany). The scanner is built around a 90-cm-diameter magnet
(Magnex Scientific, Oxford, U.K.). The amplitude and slew rate of the system's head
gradient coils are always constrained to 35 mT/m and 600 T/m/s, respectively. Potential
gradient imperfections are minimized by operating at amplitude and slew rates below the
hardware's maximum specifications and performing a one-time RF-gradient delay
calibration. A single-channel, 28-cm-diameter, quadrature bandpass birdcage coil is used for
transmission and reception [57], [58].
3) Spatial Profiles, Target Pattern, and Spoke Parameters—A GRE magnitude
image of the head-shaped water phantom is obtained using a conventional low-flip sliceselective pulse with a repetition time (TR) of 20 ms, an echo time (TE) of 5 ms, a bandwidth
(BW) of 390 Hz/pixel, and a FOV of 25.6 cm in each dimension. The dielectric properties of
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the water in the phantom cause its transmit and receive profiles to exhibit significant spatial
variation, causing the resulting image intensity to be highly nonuniform. One sees from (5)
that this image is linearly related to both the transmit profile and the receive profile when in
the small-tip angle regime, because sin(α) ≃ α for small α.
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We use this image—the combination of the transmit and receive profiles—as proxy for the
transmit profile. Naturally, when one implements a
mitigation pulse for in vivo use, one
should mitigate only the transmit profile, as this achieves optimal contrast and SNR, but
here we will mitigate the combination of profiles, for three reasons: 1) The combination
exhibits more spatial variation than the transmit profile alone, making the spoke selection
problem more challenging. 2) A pulse designed to mitigate this combination, rather than
simply the transmit profile, produces a result that is easy to understand and evaluate, since it
ideally will produce a uniformly image, whereas a pulse that mitigates only the transmit
profile still produces a nonuniform image because the receive profile is not mitigated. 3)
Evaluating the performance of a pulse that mitigates only the transmit profile is difficult
because to do so we must quantify the postmitigation flip angle. In order to obtain a flip
angle map via fitting, the pulse must be played at increasing voltages to obtain high-flip
images (cf. Section II-B), but the spoke-based pulse does not maintain its intended excitation
profile in this regime, because like all spoke-based designs, it relies on the small-tip angle
approximation. This is in contrast with conventional RF pulses, which may be used to
estimate the
maps for each coil, but not to validate the flip angles of spoke-based pulses.
The hypothetical transmit profile to be mitigated appears in Fig. 1 in nT/V, with a 25.6 cm ×
25.6 cm FOV and 4-mm resolution. The FOX is where the phantom is present. The in-plane
target magnetization magnitude is a uniform 10° flip and target phase is zero. (We have no
desire to shape the nearly-constant phase that occurs naturally across the target.) Spoke
parameters for all designs are given in Section II-B. Based on these parameters and gradient
constraints, the duration of a T-spoke pulse, L(T), is close to 0.26T+0.34 ms, i.e., each spoke
adds roughly 1/4 ms to pulse duration. The in-plane (kx, ky) traversals take negligible time
because most time is spent playing spokes in kz. Finally, ΔB0 is estimated from the phase of
two GRE images (TE1 = 5 ms, TE2 = 6 ms).
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4) Experiment Details: Simulations, Quality Metrics, and Validation Trial on
Real System—First, simulations are conducted as a function of T to demonstrate the
superior performance of the sparsity-enforced technique; here, we assume we know a good
value for the sparsity-enforced method's λ parameter. The quality of the in-plane excitation
produced by each method is evaluated by computing the within-FOX, in-plane root mean
square error (RMSE) and maximum error between each resulting excitation and the desired
magnetization. Because the desired pattern is in degrees, each of these error terms are in
degrees as well. Further, because the resulting pulses are in volts, we calculate the rms
(VRMS) and peak voltage (Vpeak) of each. After the simulations, a sparsity-enforced pulse is
played on the scanner to show that the experimental result closely matches the one predicted
via simulation. We then return to simulations and characterize how the sparsity-enforced
placement algorithm behaves when provided grids of oversampled candidate spoke
locations. Finally, we investigate how RMSE varies as a function of λ and T. Results appear
in Section VI.
B. Phase-Encoded, Spatially-Selective Excitation in an Oil Phantom on an Eight-Channel 3
T Scanner
1) Overview—Here we evaluate how well the spoke placement methods design pulses that
form a highly-structured excitation pattern in a 17-cm oil phantom, where the FOX is the
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phantom itself and transmission is conducted using an eight-channel parallel excitation
system.
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2) Hardware—The parallel system is built around a 3 T Magnetom Trio scanner (Siemens
Medical Solutions, Erlangen, Germany). The coil array is composed of eight circular surface
coils arranged on a 28-cm-diameter acrylic tube [59]. After a set of pulses is designed for
use on the scanner, the eight resulting waveforms are transmitted simultaneously through
their respective coils and modulated in magnitude and phase as dictated by the pulse design.
The system's body coil is used for reception. The amplitude and slew rate constraints of the
gradients are 35 mT/m and 150 T/m/s, respectively. Gradient imperfections are mitigated by
operating in a region well within the hardware's maximum specifications; RF-gradient
mismatch is prevented via a one-time delay calibration.
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3) Spatial Profiles, Target Pattern, and Spoke Parameters—Spatial profiles are
obtained by collecting ten images per channel and fitting the results to (5) using the Powell
method [60]. The images are obtained by sending ten pulses through each of the eight
transmit elements, one at a time, and receiving on the body coil using a GRE readout (TR =
20 ms, TE = 6 ms, BW = 390 Hz/pixel). This yields 25.6 cm × 25.6 cm FOV, 4-mm
resolution magnitude and phase maps, half of which are in the high-flip regime. The profile
magnitudes, in nT/V, are illustrated in Fig. 2. The smooth variations exhibited by each
profile occur without smoothing the fitted results, leading us to believe that the fitting
method is robust and accurate. Further, the method determines that the receive profile is
smooth, varying less than 5% within the FOX. Spoke type and thickness are the same as in
the single-channel case. The duration of a T-spoke pulse, L(T), fits well to 0.47T+0.65 ms;
ΔB0 is again obtained from two phase maps.
With the profiles of Fig. 2, it is possible to run the spoke placement algorithms and produce
pulses. For all experiments, d(x, y) is the phase-encoded bifurcation depicted in Fig. 3. This
target has a high degree of spatial selectivity, experiencing a 10° flip within only two thin
“veins” and no flip across the rest of the FOX. The left vein is 90° out of phase with the
right vein. Exciting this pattern is worthwhile because highly-structured excitations may
have applications to clinical MRI. Furthermore, the strength of multichannel excitation will
be demonstrated. Finally, the phase-encoded nature of this excitation may have applications
to phase-contrast magnetic resonance angiography (MRA) [61], [62], perhaps allowing
MRA concepts to be ported to general MRI.
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4) Experiment Details: Simulations, Quality Metrics and Validation Trial on
Real System—Analogously to the single-channel system experiments, simulations are
first conducted and RMSE and metrics are used to evaluate the performance of the
different methods as a function of T. Voltages of the resulting pulses are also analyzed. Each
design consists of eight RF pulses, so to succinctly present the data, the maximum peak and
maximum RMS voltages observed across each set of eight waveforms are recorded. After
the Bloch-simulated trials, a sparsity-enforced pulse is played on the scanner and the
resulting excitation is compared to the one predicted via simulation. We then return to
simulations, analyzing how the sparsity-enforced algorithm behaves when provided grids of
candidate spoke locations with different extents in k-space, along with the method's
sensitivity to its control parameter, λ.
VI. Results and Discussion
1) Single-Channel System: Bloch-Simulated Spoke Placement Analysis
Here, each method is used to place T = 1,...,40 spokes. Based on Section V, we are able to
generate all matrices and vectors needed to run the placement and pulse design routines,
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e.g., 1138 within-FOX samples of the transmit profile, S(x, y), are used to form the spatial
profile matrix, S, all elements of d are set to 10° based on the desired excitation d(x, y), etc.
The pointwise inverse of the inhomogeneity, S−1(x, y), is provided as the desired excitation
to the Fourier placement method (see Section III-A). The inversion-based method's
Tikhonov parameter in (15) is set to 0.1, and the pulse design Tikhonov parameter in (4) is
set to 0.001.
The frequency grid of candidate spoke locations provided to the inversion-based and
sparsity-enforced methods is centered at dc, Nyquist-spaced at (1/25.6) cm−1, extends
.
outward equally in both kx and ky, and is comprised of 172 points, leading to
Providing this same grid to the inversion and sparsity-enforced methods promotes fair
comparisons between the algorithms. Ideally, this grid would extend further out in k-space
to reach the maximum and minimum frequencies suggested by the transmit profile's 4-mm
sample spacing, but using such a large grid increases the sparsity-enforced method's runtime
to 25 min (versus 3 min for the 172-point grid). The smaller grid is acceptable, however,
because the Fourier-based method—which selects spokes on the large, ideal grid—never
chooses to place a spoke outside of this 172 region when T ≤ 40, i.e., the majority of k-space
energy lies on the small grid. (Later we will show how grid extent affects the sparsityenforced method's performance.)
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The final step is to run the sparsity-enforced routine for R = 40 values of λ, solving (17)
each time and storing each resulting weighted grid, g(r). When the sparsity-enforced method
is used to place T spokes, R possible placements are evaluated—spoke locations being
chosen based on the T maximum values of each g(r)—and the placement yielding the
smallest residual error,
, is retained. Thus when the sparsity-enforced
method's results are presented, they implicitly assume that a good value for λ is known.
(Later on, we will analyze sensitivity to λ.) We do not find it necessary to loop over the
inversion-based method's Tikhonov term because we observe that once it is tuned past a
certain threshold, all resulting solutions suggest essentially the same set of T spoke
locations.
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Each method designs pulses with T = 1,..., 40 spokes, which are then simulated. The RMSE
and maximum error of each resulting excitation pattern with respect to the uniform target are
computed, along with the voltage characteristics of each corresponding pulse. Fig. 4 depicts
these results, illustrating how each metric varies with T. For each method, RMSE decreases
with T, because using more spokes allows for more spatial tailoring and inhomogeneity
mitigation. For fixed T, the Fourier method is outperformed by the inversion-based method,
which in turn is outperformed by the sparsity-enforced technique. For all T, the sparsityenforced placement method produces the lowest RMSE.
When T < 15, the RMSE for each method is large, and the resulting magnetization still
highly nonuniform. When T ∈ [15, 30], the sparsity-enforced method, on average, produces
an excitation with 1.32 times lower RMSE relative to the inversion method. The inversionbased method's excitations, in turn, have on average 1.06 times lower RMSE than those due
to the Fourier method. For T > 30, the RMSE of the sparsity-enforced and inversion
methods converges, but the resulting pulses are over 8-ms long and thus impractical.
Overall, it is clear that the sparsity-enforced method produces higher-quality excitations than
the other methods for all practical pulse durations. To see the strength of the sparsityenforced method, consider its 21-spoke 5.7-ms pulse with 0.68 RMSE. In order for the
inversion method to produce the same quality excitation, a 29-spoke, 7.8-ms pulse is
required. In other words, the sparsity-enforced pulse achieves the same quality excitation yet
is 1.37 times shorter.
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Unlike RMSE, maximum error does not vary smoothly with T, because the algorithms do
not explicitly penalize error. In general, the maximum error curves of the inversion and
sparsity-enforced methods are similar. The RMS voltages of pulses generated by each
method exhibit a downward trend as T increases, because with more spokes, each individual
spoke does not need to be modulated as intensely to form the excitation. Further, the peak
voltages behave more erratically than the RMS voltages, analogously to how the error
fluctuates more than RMSE. Finally, for T ∈ [10, 20], the sparsity-enforced method often
has the lowest voltages among all methods, whereas for T > 20, its voltages are higher.
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Fig. 5 depicts the results of each method when T = 21. The left, center, and right columns
show the Fourier, inversion, and sparsity-enforced method's results, respectively. The top,
middle, and bottom rows depict the simulated excitation before accounting for transmit
profile nonuniformity, the resulting magnetization after accounting for inhomogeneity, and a
2-D view of k-space showing where each method places its spokes and how they are
traversed in-plane. Metrics are also given in correspondence with those in Fig. 4. From the
top row, one sees that each method produces an excitation that approximates the inverse of
the nonuniform profile in Fig. 1. Looking at the middle row, one appreciates the ability of
each method to produce a relatively flat magnetization. The bottom row shows that the
Fourier and inversion methods cluster spokes around dc. The sparsity-enforced method,
however, places its spokes further out on the grid, leading to an excitation with 1.28 times
lower RMSE. This improvement is achieved with no increase in pulse duration and only a
moderate voltage increase. The spoke placement determined by the sparsity-enforced
algorithm is not obvious, but by placing spokes at slightly higher spatial frequencies, the
sparsity-enforced method produces a nominal excitation that is less symmetric, better
resembling the pointwise inverse of the transmit profile than the excitations produced by the
other methods.
2) Single-Channel System Validation
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To validate our simulations, we play the 21-spoke sparsity-enforced pulse from Fig. 5 on the
scanner and perform a GRE readout. Since the pulse is designed to mitigate both the
transmit and receive profiles, we validate our simulation by analyzing the magnitude of the
readout image. Fig. 6 shows the RF magnitude and gradients of the 21-spoke pulse, along
with the in-plane and through-plane result from the real system. The through-plane image
demonstrates excellent slice selection. Further, there is a striking resemblance between the
in-plane result and simulation (middle row, right column of Fig. 5). Note how Fig. 5 Fourier
and inversion-based simulated patterns have two bright spots slightly north-west and northeast of center, whereas the simulated pattern and the scanner result in Fig. 6 have only a
single bright spot northeast of center. This close match between the experimental and
simulated result lends strong support to the simulation method and results.
3) Multichannel System: Bloch-Simulated Spoke Placement Analysis
Here, each method places 1−40 spokes, and RMSE, error, VRMS, and Vpeak are
calculated for each of the resulting 120 pulses and excitations. The goal here is to produce
the dual-vein target in Fig. 3. Since there are 8 channels, the matrices are 8 times larger, e.g.,
Atot in (18) is now 1416 × (289 × 8). For all trials, the inversion method's Tikhonov value in
(15) is 0.01 and the pulse design Tikhonov term in (4) is 0.01. The frequency grid is the
same as in the single-channel experiment, so each gp vector (see Section III-B) is comprised
of 289 elements. With these variables, it takes about 25 min to solve (18) for one value of λ.
Analogously to the single-channel experiment, the sparsity-enforced routine is run for R =
32 different values of λ, so when the method is requested to place T spokes, it evaluates R
possible placements and chooses the one yielding the smallest error. Here, in contrast with
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the single-channel experiment, λ ∈ (10−9, 10−1). Small values of λ are necessary because
the application of ∥·∥s to the gps results in grids with barely any energy when λ > 10−1.
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After these steps, the algorithms place 1−40 spokes and design pulses for each. Fig. 7 shows
the error and voltage metrics of each method as a function of T. The RMSE plot shows that
the sparsity-enforced routine outperforms the other methods. For T ∈ [10, 40], the sparsityenforced technique, on average, has 1.18 times and 1.31 times lower RMSE than the Fourier
and the inversion methods, respectively. Consider the sparsity-enforced method's 15-spoke,
7.5-ms pulse with 2.01 RMSE. For the Fourier method to attain this RMSE, a 26-spoke, 13ms pulse is required, i.e., the sparsity-enforced pulse generates the same quality excitation
yet is 1.73 times shorter in duration. In terms of voltages, all methods exhibit similar trends;
this differs from the single-channel case where the sparsity-enforced method at times
produced higher-voltage pulses. This means that the sparsity-enforced method produces
pulses that yield higher-quality excitations without significantly increasing pulse duration,
VRMS, or Vpeak.
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These error plots also reveal a surprising result: for all T, the inversion method performs
worse than the Fourier one, even though the former was designed to account for pitfalls of
the latter. This differs from the single-channel case of Fig. 4, where the inversion method
outperformed the Fourier one. The inversion method fails here because of the “grid
compression” step explained in Section III-B. This problem does not occur in the singlechannel experiment because in the latter case there is only a single gp vector and the
never occurs.
compacting of various grids to form a dense, problematic
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Fig. 8 shows each method's excitation and spoke placement for T = 15. Its left, middle, and
right columns show the results of the Fourier, inversion, and sparsity-enforced techniques.
The rows, from top to bottom, depict the simulated excitation magnitude, excitation phase,
and spoke placement in (kx, ky). Each method succeeds, to some extent, in producing the
desired excitation, but the sparsity-enforced method produces one that best resembles the
target. Its excitation is better because it is not only less blurry than the others, its lower vein
exhibits a degree of curvature not present in the veins of the other methods. This leads to the
sparsity-enforced method's 2.01 RMSE, which is 1.18 and 1.24 times lower than those of the
Fourier and inversion techniques. The sparsity-enforced method's spoke placement exhibits
the same trend that it did in the single-channel case: it places spokes at slightly higher spatial
frequencies than the Fourier method. The discussion about the poor performance of the
inversion method (see Section III-B) is bolstered by its placement shown here: the dense
grid causes the inversion-based technique to tightly cluster its spokes around dc, and
because of this, its resulting excitation completely lacks distinct edges and is only a
“lowpass” version of the target in Fig. 3.
4) Multichannel System Validation
Simulation results are validated by playing the 15-spoke sparsity-enforced pulse on the
system. The magnitude and phase of the center slice are shown in Fig. 9, corresponding with
the simulated images in the right column of Fig. 8. The experimental and simulated
magnitude images are quite similar: there is a dark ridge in both images where the two veins
intersect, the left vein in both images has barely any curvature, and the right vein of each
image exhibits slight curvature. The lower-left of the experimental image contains ghosts of
the left vein, an artifact not present in the simulation. We believe this occurs because
gradient infidelities cause a spoke to be slightly misplaced in (kx, ky); this misplaced spoke
ends up modulating spatial frequencies different from those intended, and because the
system's coil profiles are being driven and superposed to cancel each other out in the
majority of the FOX, this slight deviation perturbs the intended cancellation of the profiles
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and creates noticeable artifacts. In other words, commanding no flip in most of the FOX
while asking for a 10° flip within two thin veins is an ill-conditioned problem.
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The phase of the experimental image in Fig. 9 resembles the phase map predicted by the
simulation in Fig. 8. On average, the phase in the left vein differs from that in the right vein
by 81.6°, 8.4° off from the ideal 90° separation. The majority of this 8.4° error occurs where
the two veins intersect. In the ideal target in Fig. 3, we see there is a sharp 90° phase cutoff
between the left and right veins, whereas in the experimental map, there is a gradual change
in phase in this region. This behavior is reasonable: it is unrealistic to think that the real
system is capable of exciting a target with a discontinuity. Besides this difference, the
observed magnitude and phase maps match closely with the simulated ones, lending
credence to the simulation results and sparsity-enforced pulse design.
5) Sparsity-Enforced Placement: Grid Oversampling Analysis
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We have seen that the sparsity-enforced placement algorithm is superior to the other
methods, yielding improvements in excitation quality with negligible changes to pulse
duration and, at most, moderate increases in voltage. We now investigate how oversampling
the grid of candidate spoke locations affects the sparsity-enforced algorithm's RMSE
performance. This is done via simulations in the context of the single-channel system's
inhomogeneity mitigation scenario. Here, for T = 1,3,...,29, we run the sparsity-enforced
method using the 17 × 17 Nyquist grid discussed earlier, along with a 33 × 33 2×oversampled grid and a 49 × 49 3×-oversampled grid. These oversampled grids extend out
to nearly identical maximum and minimum spatial frequencies in k-space as does the
Nyquist grid, so the only difference among the grids is their oversampling factor. With other
parameters held constant, we run the sparsity-enforced method with various λs for each
grid, design T-spoke pulses, and compute RMSE. The sparsity-enforced method's runtime
increases when provided the oversampled grids, because they have roughly 4× and 8× as
many candidate locations than does the Nyquist grid. Fig. 10(A) shows the results of this
Bloch-simulated experiment. Here, RMSE as a function of oversampling factor is relatively
constant, e.g., for T < 7, T > 17, oversampling the grid has no RMSE benefit. For fixed T ∈
[7, 15], RMSE decreases with increasing oversampling factor, but only slightly.
6) Sparsity-Enforced Placement: Grid Extent Analysis
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We now investigate whether increasing (reducing) the extent of the grid out to higher
frequencies is of any benefit (detriment). Specifically, we alter the extent of the grid used in
the multichannel experiment and measure RMSE. We run the sparsity-enforced algorithm
using the original Nyquist-spaced 172 grid, along with Nyquist-spaced grids that are
72,92,...,152, and 192 in size. In terms of runtime relative to the 172 grid, the sparsityenforced algorithm runs 5.9, 3.6, 2.4, 1.7, and 1.3 times faster when the 72 through 152 grids
are used, and 1.2 times slower when the 192 grid is used. For each grid, we sweep over λ
(like in the earlier experiments) and then compute the RMSE of various T-spoke pulses for
T = 1,3,...,29. Fig. 10(B) shows RMSE as a function of grid extent and T. Grids of size 112
and up yield relatively the same RMSE, which means that runtime may be reduced without a
loss of performance by using smaller grids.
Based on the grid-extent results in Fig. 10(B) and the over-sampling results in Fig. 10(A), it
is sufficient for our applications to simply pick a Nyquist-sampled grid that extends out a
moderate rather than far distance into k-space. There is no need to extend the grid to high
spatial frequencies or oversample it by even a factor of two. The strength of the sparsityenforced algorithm does not come from placing spokes at high or finely-sampled
frequencies, but in simply tuning placements outward from low frequencies and making
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Zelinski et al.
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slight—but certainly not obvious—alterations to the placements suggested by the Fourier
method.
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7) Sparsity-Enforced Placement: λ Sensitivity Analysis
The single-channel simulation results presented in Fig. 4 assumed that a good choice of λ
was known. We now do away with this assumption, presenting RMSE as a function of λ
and T ∈ {11, 15, 21, 25} in Fig. 11. For small T, the choice of λ is crucial in order to best
reduce RMSE, but as T increases, the algorithm's sensitivity to λ decreases significantly.
Further, for fixed λ, RMSE does not exhibit a consistent trend across T.
Fig. 12 depicts the multichannel experiment's sensitivity results, showing RMSE as a
function of λ and T ∈ {11, 15, 21}. As T increases, the RMSE versus λ curve moves
smoothly downward. We are unsure why more erratic behavior occurs in the single-channel
context (see Fig. 11).
A fast technique for finding ideal values of λ is an open problem. We are investigating
several approaches to automated selection: the “L-curve” method [63], universal parameter
selection [64], and min–max parameter selection [65].
8) Runtime
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In the single-channel experiments where a 172 grid is used, it takes approximately 3 min to
solve (18) using our MATLAB SeDuMi implementation on a Linux-based computer with a
3.0-GHz Pentium IV processor. In the eight-channel case with the 172 grid, runtime
increases nearly linearly to 25 min. In general, the random-access memory footprint of the
SOC program ranges from 200−700 MB.
Runtime may be reduced via a multiresolution approach (as in [30]) by first running the
algorithm with a coarse grid, noting which spoke locations are revealed, and then running
the algorithm with a grid that is finely sampled around the locations suggested by the coarse
result. This is faster than providing the algorithm a large, finely-sampled grid and attempting
to solve the problem in one step. Further, because the desired solution to the spoke
placement problem is sparse and the matrices involved are dense, solving this problem using
iterative shrinkage techniques [66], [67], greedy-pursuit algorithms [25], [52], or specialpurpose solvers (e.g., [68], [69]) may lead to major runtime improvements.
VII. Conclusion
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We have introduced a novel algorithm for the design of fast, slice-selective MRI excitation
pulses. The algorithm uses sparse approximation theory and a second-order cone
optimization to place and modulate a small number of spokes in k-space to produce
excitations that are capable of both exciting a thin slice and tailoring the in-plane
magnetization.
The strength of sparsity-enforced spoke placement was demonstrated by designing fast,
slice-selective RF pulses that mitigated B1 inhomogeneity present in a head-shaped water
phantom on a 7 T single-channel system and that achieved a complex-valued target pattern
using an eight-channel 3 T parallel excitation system. In both cases, the sparsity-enforced
method outperformed conventional methods, producing excitations with lower RMSE when
pulse duration across the methods was fixed, and producing pulses with significantly shorter
durations when excitation quality across the methods was fixed. The simulation results
presented throughout this paper were validated by experiments on both the single-channel 7
T system and the eight-channel 3 T parallel excitation system and showed that sparsityenforced pulses are applicable in real scenarios. Throughout both experiments, the sparsityIEEE Trans Med Imaging. Author manuscript; available in PMC 2009 April 06.
Zelinski et al.
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enforced algorithm automated the task of spoke placement and the design of the
corresponding gradients and RF waveforms, yielding placement patterns that were not
obvious and would be difficult or impossible to design by hand, freeing the designer from
the task of slice-selective pulse design. The algorithm was shown to be highly robust to the
choice of many of its input parameters, such as the extent and sample spacing of the grid of
candidate spoke locations, and relatively robust to the choice of its sparsity-enforcing
control parameter, λ, when larger numbers of spokes were placed during the single-channel
experiments and in general during the multichannel experiments.
With further investigation into automated λ selection and fast sparsity-enforcement
algorithms, the sparsity-enforced pulse design method will have clinical impact in the areas
of B1 inhomogeneity mitigation, multichannel parallel transmission, high-field imaging, and
spatially-selective excitation, all of which comprise the vanguard of emerging MRI
technology.
Acknowledgments
The authors would like to thank the reviewers for providing very useful advice.
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This work was supported in part by the National Institutes of Health under Grant 1P41RR14075, Grant
1R01EB000790, Grant 1R01EB006847, and 1R01EB007942, in part by the National Science Foundation under
Grant CCF-643836, in part by the United States Department of Defense National Defense Science and Engineering
Graduate Fellowship F49620-02-C-0041, in part the MIND Institute, in part the A. A. Martinos Center for
Biomedical Imaging, and in part by the R. J. Shillman Career Development Award.
References
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1. Hu X, Parrish T. Reduction of field of view for dynamic imaging. Magn. Reson. Med. 1994;
31:691–694. [PubMed: 8057824]
2. Glover, G.; Lai, S. Proc. Int. Soc. Magn. Reson. Med. (ISMRM). 1998. Reduction of susceptibility
artifacts in BOLD fMRI using tailored RF pulses; p. 298
3. Saekho S, Yip CY, Noll DC, Boada FE, Stenger VA. Fast-kz three-dimensional tailored
radiofrequency pulse for reduced B1 inhomogeneity. Magn. Reson. Med. 2006; 55(4):719–724.
[PubMed: 16526012]
4. Ulloa, J.; Hajnal, JV. Proc. Int. Soc. Magn. Reson. Med. (ISMRM). Miami Beach, FL: 2005.
Exploring 3D RF shimming for slice selective imaging; p. 21
5. Setsompop K, Wald LL, Alagappan V, Gagoski BA, Hebrank F, Fontius U, Schmitt F,
Adalsteinsson E. Parallel RF transmission with 8 channels at 3 Tesla. Magn. Reson. Med. 2006;
56(5):1163–1171. [PubMed: 17036289]
6. Pauly JM, Nishimura D, Macovski A. A k-space analysis of small-tip angle excitation. J. Magn.
Reson. 1989; 81:43–56.
7. Oppenheim, AV.; Schafer, R. Discrete-Time Signal Processing. Prentice-Hall; Englewood Cliffs,
NJ: 1989.
8. Bernstein, MA.; King, KF.; Zhou, XJ. Handbook of MRI Pulse Sequences. Elsevier; New York:
2004.
9. Zhu Y. Parallel excitation with an array of transmit coils. Magn. Reson. Med. 2004; 51(4):775–784.
[PubMed: 15065251]
10. Ullmann P, Junge S, Wick M, Seifert F, Ruhm W, Hennig J. Experimental analysis of parallel
excitation using dedicated coil setups and simultaneous RF transmission on multiple channels.
Magn. Reson. Med. 2005; 54(4):994–1001. [PubMed: 16155886]
11. Zhu, Y.; Watkins, R.; Gianquinto, R.; Hardy, C.; Kenwood, G.; Mathias, S.; Valent, T.; Denzin,
M.; Hopkins, J.; Peterson, W.; Mock, B. Proc. Int. Soc. Magn. Reson. Medicine (ISMRM). Miami
Beach, FL: 2005. Parallel excitation on an eight transmit channel MRI system; p. 14
IEEE Trans Med Imaging. Author manuscript; available in PMC 2009 April 06.
Zelinski et al.
Page 21
NIH-PA Author Manuscript
NIH-PA Author Manuscript
NIH-PA Author Manuscript
12. Yip CY, Fessler JA, Noll DC. Advanced three-dimensional tailored RF pulse for signal recovery in
T2*-weighted functional magnetic resonance imaging. Magn. Reson. Med. 2006; 56(5):1050–
1059. [PubMed: 17041911]
13. Bottomley PA, Andrew ER. RF magnetic field penetration, phase shift and power dissipation in
biological tissue: Implications for NMR imaging. Phys. Med. Biol. 1978; 23:630–643. [PubMed:
704667]
14. Bomsdorf H, Helzel T, Kunz D, Roschmann P, Tschendel O, Wieland J. Spectroscopy and imaging
with a 4 Tesla whole-body MR system. NMR Biomed. 1988; 1(3):151–158. [PubMed: 3275125]
15. Vaughan JT, Garwood M, Collins CM, Liu W, DelaBarre L, Adriany G, Anderson P, Merkle H,
Goebel R, Smith MB, Ugurbil K. 7T vs. 4T: RF power, homogeneity, and signal-to-noise
comparison in head images. Magn. Reson. Med. 2001; 46(1):24–30. [PubMed: 11443707]
16. van de Moortele PF, Akgun C, Adriany G, Moeller S, Ritter J, Collins CM, Smith MB, Vaughan
JT, Ugurbil K. B1 destructive interference and spatial phase patterns at 7T with a head transceiver
array coil. Magn. Reson. Med. 2005; 54:1503–1518. [PubMed: 16270333]
17. Collins CM, Wanzhan L, Schreiber W, Yang QX, Smith MB. Central brightening due to
constructive interference with, without, and despite dielectric resonance. J. Magn. Reson. Imag.
2005; 21(2):192–196.
18. Jin J, Chen J. On the SAR and field inhomogeneity of birdcage coils loaded with the human head.
Magn. Reson. Med. 1997; 21:192–196.
19. Collins CM, Li S, Smith MB. SAR and B1 field distributions in a heterogeneous human head
model within a birdcage coil. Magn. Res. Med. 1998; 40:847–856.
20. Ibrahim TS, Lee R, Baertlein BA, Robitaille P-ML. B1 field homogeneity and SAR calculations in
the high pass birdcage coil. Phys. Med. Biol. 2001; 46:609–619. [PubMed: 11229737]
21. Yip, CY.; Fessler, JA.; Noll, DC. Proc. Int. Soc. Magn. Reson. Med. (ISMRM). Miami, FL: 2005.
A novel, fast and adaptive trajectory in three-dimensional excitation k-space; p. 2350
22. Davis, G. Ph.D. dissertation. New York Univ.; Sep.. 1994 Adaptive nonlinear approximations.
23. Natarajan BK. Sparse approximate solutions to linear systems. SIAM J. Comput. 1995; 24(2):227–
234.
24. Gorodnitsky IF, Rao BD. Sparse signal reconstruction from limited data using FOCUSS: A reweighted minimum norm algorithm. IEEE Trans Signal Process. Mar.1997 45(3):600–616.
25. Chen SS, Donoho DL, Saunders MA. Atomic decomposition by basis pursuit. SIAM J. Sci.
Comput. 1999; 20(1):33–61.
26. Donoho DL, Huo X. Uncertainty principles and ideal atomic decomposition. IEEE Trans. Inf.
Theory. Nov.2001 47(7):2845–2862.
27. Tropp, JA. Ph.D. dissertation. Univ. Texas; Austin: 2004. Topics in sparse approximation.
28. Katscher U, Bornert P, Leussler C, van den Brink JS. Transmit SENSE. Magn. Reson. Med. Jan.
2003 49(1):144–150. [PubMed: 12509830]
29. Graesslin, I.; Vernickel, P.; Schmidt, J.; Findeklee, C.; Roschmann, P.; Leussler, C.; Haaker, P.;
Laudan, H.; Luedeke, KM.; Scholz, J.; Buller, S.; Keupp, J.; Bornert, P.; Dingemans, H.; Mens,
G.; Vissers, G.; Blom, K.; Swennen, N.; vd Heijden, J.; Mollevanger, L.; Harvey, P.; Katscher, U.
Proc. Int. Soc. Magn. Reson. Med. (ISMRM). Seattle, WA: 2006. Whole body 3T MRI system
with eight parallel RF transmission channels; p. 129
30. Malioutov DM, Cetin M, Willsky AS. A sparse signal reconstruction perspective for source
localization with sensor arrays. IEEE Trans. Signal Process. Aug.2005 53(8):3010–3022.
31. Tropp JA. Just relax: Convex programming methods for identifying sparse signals. IEEE Trans.
Inf. Theory. Mar.2006 52(3):1030–1051.
32. Tropp JA. Algorithms for simultaneous sparse approximation: Part II: Convex relaxation. Signal
Process. 2006; 86(3):589–602.
33. Nemirovski, A.; Ben-Tal, A. Lectures on Modern Convex Optimization. Analysis, Algorithms and
Engineering Applications. SIAM; Philadelphia, PA: 2001.
34. Zelinski, AC.; Setsompop, K.; Goyal, VK.; Alagappan, V.; Fontius, U.; Schmitt, F.; Wald, LL.;
Adalsteinsson, E. Proc. Int. Soc. Magn. Reson. Med. (ISMRM). Berlin, Germany: 2007.
IEEE Trans Med Imaging. Author manuscript; available in PMC 2009 April 06.
Zelinski et al.
Page 22
NIH-PA Author Manuscript
NIH-PA Author Manuscript
NIH-PA Author Manuscript
Designing fast 3D RF excitations by optimizing the number, placement, and weighting of spokes
in k-space via a sparsity-enforcement algorithm; p. 1691
35. Setsompop, K.; Zelinski, AC.; Goyal, VK.; Wald, LL.; Adalsteinsson, E. Proc. Int. Soc. Magn.
Reson. Med. (ISMRM). Berlin, Germany: 2007. Sparse spokes slice-selective design for B1inhomogeneity correction at 7T; p. 356
36. Yip CY, Grissom WA, Fessler JA, Noll DC. Joint design of trajectory and RF pulses for parallel
excitation. Magn. Reson. Med. 2007; 58(3):598–604. [PubMed: 17763362]
37. Sodickson DK, Manning WJ. Simultaneous acquisition of spatial harmonics (SMASH): Fast
imaging with radio frequency coil arrays. Magn. Reson. Med. Oct.1997 38(4):591–603. [PubMed:
9324327]
38. Pruessmann KP, Weiger M, Scheidegger MB, Boesiger P. SENSE: Sensitivity encoding for fast
MRI. Magn. Reson. Med. Nov.1999 42:952–962. [PubMed: 10542355]
39. Grissom WA, Yip CY, Zhang Z, Stenger VA, Fessler JA, Noll DC. Spatial domain method for the
design of RF pulses in multicoil parallel excitation. Magn. Reson. Med. 2006; 56(3):620–629.
[PubMed: 16894579]
40. Yip CY, Fessler JA, Noll DC. Iterative RF pulse design for multidimensional, small-tip-angle
selective excitation. Magn. Reson. Med. 2005; 54(4):908–917. [PubMed: 16155881]
41. Katscher, U.; Graesslin, I.; Dingemans, H.; Mens, G. Proc. Int. Soc. Magn. Reson. Med. (ISMRM).
2006. Experimental verification of over- and underdetermined transmit SENSE; p. 600
42. Zelinski AC, Wald LL, Setsompop K, Alagappan V, Gagoski BA, Goyal VK, Hebrank F, Fontius
U, Schmitt F, Adalsteinsson E. Comparison of three algorithms for solving linearized systems of
parallel excitation RF waveform design equations: Experiments on an eight-channel system at 3
Tesla. Concepts Magn. Reson. Part B: Magn. Reson. Eng. Aug.2007 31B(3):176–190.
43. Tikhonov AN. On the solution of incorrectly stated problems and a method of regularization. Dokl.
Acad. Nauk SSSR. 1963; 151:501–504.
44. Katscher U, Bornert P, van den Brink JS. Theoretical and numerical aspects of transmit SENSE.
IEEE Trans. Med. Imag. Apr.2004 23(4):520–525.
45. Paige CC, Saunders MA. LSQR: An algorithm for sparse linear equations and sparse least squares.
ACM Trans. Math. Software. Mar.1982 8(1):43–71.
46. Ernst RR, Anderson WA. Applications of Fourier transform spectroscopy to magnetic resonance.
Rev. Sci. Instrum. 1966; 37:93–102.
47. Wang J, Qiu M, Yang QX, Smith MB, Constable RT. Measurement and correction of transmitter
and receivers induced nonuniformities in vivo. Magn. Reson. Med. 2005; 53:408–417. [PubMed:
15678526]
48. Kerr, AB.; Cunningham, CH.; Pauly, JM.; Gianquinto, RO.; Watkins, RD.; Zhu, Y. Proc. Int. Soc.
Magn. Reson. Med. (ISMRM). Seattle, WA: 2006. Self-calibrated transmit SENSE; p. 14
49. Boyd, S.; Vandenberghe, L. Convex Optimization. Cambridge Univ. Press; Cambridge, U.K.:
Mar.. 2004
50. Donoho DL. For most large underdetermined systems of linear equations, the minimal l1-norm
near-solution approximates the sparsest near-solution. Commun. Pure Appl. Math. Jul.2006 59(7):
907–934.
51. Tibshirani R. Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Nov.1996 B
58(1):267–288.ser. B
52. Tropp JA, Gilbert AC, Strauss MJ. Algorithms for simultaneous sparse approximation: Part I:
Greedy pursuit. Signal Process. 2006; 86(3):572–588.
53. Sturm J. Using sedumi 1.02, a MATLAB toolbox for optimization over symmetric cones.
Optimizat. Methods Software. 1999; 11−12:625–653.
54. Toh KC, Todd MJ, Tutuncu RH. SDPT3—A matlab software package for semidefinite
programming,” Optimizat. Methods Software. 1999; 11(12):545–581.
55. Luenberger, DG. Optimization by Vector Space Methods. Wiley; New York: 1969.
56. Ullmann, P.; Junge, S.; Seifert, F.; Ruhm, W.; Hennig, J. Proc. Int. Soc. Magn. Reson. Med.
(ISMRM). Berlin, Germany: 2007. Parallel excitation experiments using a novel direct calibration
technique for RF-pulse determination; p. 672
IEEE Trans Med Imaging. Author manuscript; available in PMC 2009 April 06.
Zelinski et al.
Page 23
NIH-PA Author Manuscript
NIH-PA Author Manuscript
57. Tropp, J. Proc. Soc. Magn. Reson. Med. (SMRM). Berlin, Germany: 1992. The hybrid bird cage
resonator; p. 4009
58. Barberi EA, Gati JS, Rutt BK, Menon RS. A transmit-only/receive-only (TORO) RF system for
high-field MRI/MRS applications. Magn Reson Med. 2000; 43:284–289. [PubMed: 10680693]
59. Alagappan, V.; Wiggins, GC.; Potthast, A.; Setsompop, K.; Adalsteinsson, E.; Wald, LL. Proc. Int.
Soc. Magn. Reson. Med. (ISMRM). Seattle, WA: 2006. An 8 channel transmit coil for transmit
sense at 3T; p. 121
60. Powell MJD. A fast algorithm for nonlinearly constrained optimization calculations. Lecture Notes
Math. 1978; 630:144–157.
61. Dumoulin CL, Hart HR. Magnetic resonance angiography. Radiology. 1986; 161:717–720.
[PubMed: 3786721]
62. Souza SP, Dumoulin CL. Phase-contrast magnetic resonance angiography. Proc. 11th Annu. Int.
Conf. Eng. Med. Biol. Soc. 1989; 2:514–515.
63. Hansen PC. Regularization tools: A matlab package for analysis and solution of discrete ill-posed
problems. Numer. Algor. 1994; 6:1–35.
64. Donoho DL, Johnstone IM. Ideal spatial adaptation by wavelet shrinkage. Biometrica. Aug.1994
81(3):425–455.
65. Johnstone IM. On minimax estimation of a sparse normal mean vector. Ann. Stat. Mar.1994 22(1):
271–289.
66. Daubechies I, Defrise M, Mol CD. An iterative thresholding algorithm for linear inverse problems
with a sparsity constraint. Comm. Pure Appl. Math. 2004; 57(11):1413–1457.
67. Elad M, Matalon B, Zibulevsky M. Image denoising with shrinkage and redundant representations.
Proc. IEEE Comput. Vis. Pattern Recognit. 2006; 2:1924–1931.
68. Saunders MA, Kim B. PDCO: Primal-dual interior method for convex objectives. [Online].
Available: http://www.stanford.edu/group/SOL/software/pdco.html
69. Kim, S-J.; Koh, K.; Lustig, M.; Boyd, S.; Gorinevsky, D. A method for large-scale-regularized
least squares problems with applications in signal processing and statistics. Stanford Univ.;
Stanford, CA: 2007. [Online]. Available: http://www.stanford.edu/~boyd/reports/l1_ls.pdf
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Fig. 1.
Quantitative
map of the head-shaped water phantom at 7 T (nT/V). This hypothetical
transmit profile is generated by collecting a GRE magnitude image and scaling the resulting
pixels.
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Fig. 2.
Quantitative
transmit profile magnitudes of the 3 T, eight-channel parallel excitation
system (nT/V) obtained by collecting ten intensity images per channel and fitting to (5).
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Fig. 3.
Dual-phase bifurcation target excitation used for all 3 T eight-channel system experiments.
Within the two veins the target commands a 10° flip and zero flip elsewhere. Each branch is
phase encoded: the left vein is 90° out of phase with the right. The ability to achieve this
structured excitation pattern will show the strength of both multichannel excitation and
sparsity-enforced spoke placement.
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Fig. 4.
Bloch-simulated spoke placement algorithm comparisons for mitigating B1 inhomogeneity
in the head-shaped water phantom at 7 T. Error and voltage statistics versus number of
spokes used (T) are shown for the Fourier, inversion, and sparsity-enforced spoke placement
methods. Upper-left: RMSE versus T. Lower-left: maximum error versus T. Upper-right:
VRMS versus T. Lower-right: Vmax versus T. For all three algorithms, the duration of a Tspoke pulse is close to 0.26T + 0.34 ms.
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Fig. 5.
Bloch-simulated 21-spoke pulses designed by the Fourier, inversion, and sparsity-enforced
spoke placement algorithms for mitigating B1 inhomogeneity in the head-shaped water
phantom at 7 T. Columns, from left to right: results of the Fourier, inversion, and sparsityenforced methods. Top row: excitations produced by each algorithm. Middle row:
magnetizations after accounting for the inhomogeneity. Bottom row: 2-D view of k-space
illustrating each spoke placement; each trajectory ends at the center of k-space. The sparsityenforced pulse produces the lowest-RMSE excitation.
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Fig. 6.
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Experimental result in the head-shaped water phantom on the single-channel 7 T scanner.
Here, the 21-spoke sparsity-enforced RF pulse whose simulations appear in Fig. 5 is played
on the actual system. Left: RF waveform magnitude and gradients. Upper-right: in-plane
scanner result. Lower-right: through-plane scanner result. This experimental result closely
resembles the Bloch-simulated image in Fig. 5, validating the simulation methodology and
proving that the pulse design process is feasible for use on real systems.
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Fig. 7.
Bloch-simulated spoke placement comparisons for forming the dual-vein target on the eightchannel parallel excitation system at 3 T. Error and voltage statistics versus number of
spokes (T) are shown for the Fourier, inversion, and sparsity-enforced placement methods.
Upper-left: RMSE versus T. Lower-left: maximum error versus T. Upper-right: max VRMS
across all eight channels versus T. Lower-right: max Vpeak across all eight channels versus
T. For all algorithms, the duration of a T-spoke pulse is close to 0.47T + 0.65 ms.
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Fig. 8.
Bloch-simulated 15-spoke pulses designed by the Fourier, inversion, and sparsity-enforced
spoke placement algorithms for forming the dual-vein target on the eight-channel system at
3 T. Columns, from left to right: results of the Fourier, inversion, and sparsity-enforced
spoke placement methods. Top row: excitation magnitudes. Middle row: excitation phases.
Bottom row: 2-D view of k-space showing each method's spoke placement; each trajectory
ends at the center of k-space. The sparsity-enforced excitation has the lowest RMSE,
error, VRMS, and Vpeak.
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Fig. 9.
Experimental result in an oil phantom on the eight-channel 3 T system. Here, the 15-spoke
sparsity-enforced RF pulse and gradients whose simulation results appear in Fig. 8 are
played on the actual system followed by a GRE readout. This experimental result closely
resembles the Bloch-simulated magnitude and phase of this same waveform in Fig. 8,
validating the simulation methodology and showing that the proposed pulse design process
is applicable to real multichannel systems.
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Fig. 10.
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A: Single-channel B1 inhomogeneity mitigation experiment: RMSE versus number of
spokes (T) when providing the sparsity-enforced method Nyquist-sampled, 2×-oversampled,
and 3×-oversampled frequency grids of candidate locations. Grid oversampling seems to
provide little RMSE benefit. B: Eight-channel system, dual-vein target: RMSE versus T for
Nyquist-sampled frequency grids of candidate spoke locations that are 7 × 7,9 × 9,...,19 × 19
in size. Each grid is centered at DC and candidate points extent outward in kx and ky. The 11
× 11 grid yields reasonable RMSE results, suggesting that smaller grids may be used to
decrease runtime. A: Effect of oversampling on RMSE. B: Effect of grid extent on RMSE.
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Fig. 11.
Single-channel B1 inhomogeneity mitigation experiment: λ sensitivity analysis. The
sparsity-enforced spoke placement algorithm's sensitivity to λ is analyzed when designing
pulses comprised of 11, 15, 21, and 25 spokes. As T increases, sensitivity to λ decreases.
Further, for fixed λ, RMSE does not exhibit a consistent trend across T.
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Fig. 12.
Eight-channel system, dual-vein target: λ sensitivity analysis. The sparsity-enforced spoke
placement algorithm's sensitivity to λ is analyzed when designing pulses comprised of 11,
15, and 21 spokes. For fixed λ, RMSE decreases smoothly with T, in contrast with the
behavior in Fig. 11.
NIH-PA Author Manuscript
NIH-PA Author Manuscript
IEEE Trans Med Imaging. Author manuscript; available in PMC 2009 April 06.