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Magn Reson Med. Author manuscript; available in PMC 2009 August 10.

Published in final edited form as:

PMCID: PMC2723802

NIHMSID: NIHMS66977

Adam C. Zelinski,^{1,}^{*} Lawrence L. Wald,^{2,}^{3} Kawin Setsompop,^{1} Vijayanand Alagappan,^{2} Borjan A. Gagoski,^{1} Vivek K. Goyal,^{1} and Elfar Adalsteinsson^{1,}^{3}

The publisher's final edited version of this article is available at Magn Reson Med

See other articles in PMC that cite the published article.

A novel radio-frequency (RF) pulse design algorithm is presented that generates fast slice-selective excitation pulses that mitigate inhomogeneity present in the human brain at high field. The method is provided an estimate of the field in an axial slice of the brain and then optimizes the placement of sinc-like “spokes” in *k _{z}* via an

High-field MRI systems significantly increase signal-to-noise ratio (SNR) (1), but in vivo imaging at high field is impeded by the presence of severe inhomogeneity (2) arising due to wavelength interference effects (3,4) and tissue-conductive radio-frequency (RF) amplitude attenuation (5). Inhomogeneity is also a concern at low field when structures such as the spine (6) and body (7) are imaged. When standard slice-selective RF excitation wave-forms are used for imaging, inhomogeneity causes images to exhibit center brightening, spatial contrast variation, and SNR nonuniformity, despite the use of homogeneous volume RF excitation coils (3,4,8 –10).

The three-dimensional (3D) RF pulse designs proposed in (11–13) describe a class of slice-selective pulses capable of mitigating inhomogeneity that offer improvements over high specific absorption rate (SAR) adiabatic pulses (14) and image postprocessing methods (15). These pulses are played in the presence of echo-volumnar-like 3D gradients. They consist of modulated sinc-like pulse segments (“spokes”) in the *k _{z}* direction of excitation

In prior work, relatively few spokes have been used for inhomogeneity mitigation on single-channel (11) and multichannel parallel transmission systems (12,13,17–20). In all cases, work is performed at field strengths below 7T, where inhomogeneity in the brain is less severe, resembling a quadratic function in space (11). In contrast, inhomogeneity at 7T exhibits significant spatial variation and is not quadratic (3,4). This means that spoke designs that utilize single-channel transmit systems and rely on quadratic assumptions about (11) are unlikely to mitigate brain inhomogeneity at 7T. Parallel excitation systems, on the other hand, are indeed useful for mitigation at high field, but are expensive in terms of hardware and complexity: each transmission channel requires an RF power amplifier as well as a SAR monitor. Based on the above, it is evident that a method is needed to design fast, slice-selective, mitigation pulses for use on high-field single-channel systems.

Since is highly nonuniform at 7T (3,4), one approach to mitigating it would be to extend prior spoke-based designs by placing a large number of modulated spokes throughout (*k _{x}*,

At high field, the in-plane transmit and receive profiles of a system, and, exhibit significant variation across space, indexed by **r**. When a low-flip-angle pulse is transmitted, its nominal excitation, *p*(**r**), is multiplied (point-wise) by to yield the actual magnetization to within a multiplicative constant. Applying a standard slice-selective excitation, |*p*(**r**)| = 1, thus results in a nonuniform magnetization, proportional to. In contrast, an ideal mitigation pulse produces *p*(**r**) such that is constant for all **r** in the FOX, i.e., the ideal |*p*(**r**)| equals to within a multiplicative constant. This pulse is ideal in the sense that it mitigates the magnitude of the inhomogeneity; it does not impose phase uniformity because the latter is not stringently required in most clinical imaging applications. Clearly then, to design a mitigation pulse we must first estimate the magnitude of the transmit profile. There are several ways to accomplish this (26–31). Here we fit a set of intensity images to a signal intensity equation.

*When a standard slice-selective pulse is played with transmit voltage V* and an intensity image SI is generated via a gradient-recalled echo (GRE), the following holds:

(1)

where *c* is a catch-all gain constant, ρ(**r**) is proton density, *E*_{1}(**r**, TR) = exp(−TR/T_{1}(**r**)), and α(**r**) is the flip angle achieved in radians/volt. For a standard pulse, the latter term equals, where γ is the gyromagnetic ratio, τ is pulse duration, and is in Tesla/volt (31,32). Estimating from Eq. [1] is nontrivial because either fully-relaxed images are needed (where TR *T*_{1}) or an accurate *T*_{1} map must be available. Fortunately, one may eliminate flip angle dependence on *T*_{1} by playing a magnetization reset pulse (27) after the standard pulse, yielding

(2)

where is the proton-density-weighted receive profile and *q*(**r**, TR) is implicitly defined. Equation [2] holds even for TR < T_{1} (27). Finally, consider a case where *V* is small enough such that *V*α(**r**) is small every-where and a reset pulse is not used; here, cos(*V*α(**r**)) ≈ 1 and sin(*V*α(**r**)) ≈ *V*α(**r**), causing the (1 – *E*_{1}(**r**, TR)) terms of Eq. [1] to cancel, yielding an intensity image *L* where

(3)

Equation [3] holds for any excitation pulse, not just a standard slice-selective one.

If inhomogeneity is not severe, one may exploit Eq. [2] to obtain α(**r**) (and subsequently) simply by collecting two short-TR images with voltages *V*_{1} and *V*_{2}, where *V*_{2} = 2*V*_{1} (using the reset pulse each time), dividing the magnitude of the second image by the first (point-wise), and taking the inverse cosine (27,28). This method relies on the voltages being large enough such that the flip angle across the FOX is no longer in the linear regime (i.e., such that sin(*V*α(**r**)) ≠ *V*α(**r**)). Unfortunately, when inhomogeneity is severe, the voltages *V*_{1} and *V*_{2} fail to produce flip angles that fall outside of the linear regime across the entire FOX, and as a result we find that at 7T, the double-angle procedure consistently fails to produce stable estimates at all spatial locations of interest. Therefore, we adopt a different approach: using the reset pulse each time, we vary *V* over a wide enough range to ensure that both low-flip and high-flip angles are achieved at each spatial location **r** and collect *N* short-TR images. For each **r**, we then fit the *N* corresponding intensity samples to Eq. [2] in the least-squares sense using the Powell (33) method; this obtains in Tesla/volt as well as *q*(**r**, TR). The *V*s are chosen such that, for each **r**, at least several of the *N* samples are in the large-tip-angle regime (29,31).

Fitting the transmit profile yields α(**r**) and *q*(**r**, TR) as a byproduct, but obtaining the desired *R*(**r**) from *q*(**r**, TR) is nontrivial because the latter depends on *T*_{1}. Instead, we collect a low-flip image without using the reset pulse, averaging multiple times, such that Eq. [3] holds and SNR is large. We divide this image (point-wise) by α(**r**) to obtain *R*(**r**) (to within a multiplicative constant).

The weighted receive profile, *R*(**r**), does not depend on the excitation pulse. Exploiting this, we may estimate the flip angle map achieved by any pulse, even a nonstandard one such as a spoke-based mitigation waveform. First, we collect a low-flip image using a mitigation pulse (without using the reset pulse); the intensity of the resulting image thus obeys Eq. [3]. We then divide this image by the *R*(**r**) estimate to obtain a “postmitigation” flip angle map, α_{m}(**r**), giving us an estimate of the actual magnetization that arises when the mitigation pulse is played on the scanner (to within a multiplicative constant). We then judge mitigation performance by studying the uniformity of α_{m}(**r**).

Once an estimate of the transmit profile is known, we may pursue our goal of exciting a thin uniform slice in the presence of the inhomogeneity. To achieve slice-selectivity, we will place sinc-like spokes in *k _{z}*. To ensure excitation uniformity, we will place and modulate the spokes such that the magnitude of the resulting in-plane nominal excitation closely resembles the point-wise inverse of the transmit profile. Finally, to minimize pulse duration, we will use as few spokes as possible.

Assume we modulate the entire **k** = [*k _{x}*,

(4)

where *m* is the (approximate) transverse magnetization in radians and *M*_{0} is the steady-state magnetization. We now discard the leading constants, discretize space at locations **r**_{i}, *i* = 1,…, *N*_{s} within a chosen FOX, and discretize *k*-space at locations **k**_{i}, *i* = 1,…, *N*_{f}. Applying the formalism of (34) to Eq. [4] yields

(5)

where **m** is an *N*_{s}-element vector of samples of *m*(**r**), **S** an *N*_{s} × *N*_{s} diagonal matrix containing samples of |*S*(**r**)| (we ignore profile phase), **F** an *N*_{s} × *N*_{f} matrix where **F**(*m, n*) = exp(^{ir}*m* · **k**_{n}), and an *N*_{f}-element weight vector. The *i*th element of is the weight placed at **k**_{i}. Equation [5] thus computes the excitation *m*(**r**) at a set of spatial locations that is produced by a complex-weighted grid of *k*-space points while accounting for *S*(**r**).

One way to find weights to place in *k*-space to form a desired magnetization is as follows: find that minimizes ||**d** – **m**||_{2}, where **d** contains samples of the desired magnetization, *d*(**r**). In our work, all elements of **d** are equal to a fixed nonzero value because we want a uniform magnetization. One may use the pseudoinverse of **SF**, denoted (**SF**)^{†}, to compute = (**SF**)^{†}**d** as an approximate solution, but this yields a dense (every element of is typically nonzero), implying that spokes should be placed at all *N*_{f} candidate locations. In contrast, our goal is to find a sparse (one with many zeros) that still produces a uniform magnetization; this will reveal a small set of good locations at which to place spokes. One may consider searching over all possible spoke placements to find a sparse , but this is computationally infeasible even for small grids—it may be necessary to search up to 2^{Nf} – 1 subsets of candidate *k*-space locations to find the sparsest that attains a desired residual error (35). Clearly, a tractable approach is needed.

Fortunately, there is compelling evidence that requiring the *L*_{1}-norm of to be small encourages to have many zero elements (24). We apply this concept by regularizing the original problem and placing an *L*_{1} penalty on . Specifically, we formulate a convex optimization that seeks out a sparse capable of producing a uniform magnetization:

(6)

The first term of Eq. [6] keeps the residual error down and ensures the magnetization is close to uniform; the second encourages to be sparse. The parameter λ trades off residual error with sparsity, or in other words, mitigation with the number of spokes (and hence pulse duration). To solve Eq. [6], we first formulate it into a Second-Order Cone program (25) and implement the latter in Self-Dual Minimization (SeDuMi; http://sedumi.mcmaster.ca), a MATLAB toolbox.

Note that Eq. [5] and Eq. [6] are unconstrained outside of the FOX because **S** and **F** are constructed from only those samples within the FOX (34). Thus as Eq. [6] searches for good spoke locations (i.e., a sparse ), it incurs no penalty for introducing aberrations outside of the FOX. This freedom from unnecessary spatial constraints is why Eq. [6] is able to find a spoke placement pattern that is sparse yet still capable of exciting the desired pattern within the FOX.

With the proper choice of λ, Eq. [6] finds a sparse whose majority of elements are zero (or close to zero) that produces a relatively-flat magnetization: the few elements of that are large in magnitude indicate good spoke locations, revealing a small set of points capable of producing the needed excitation. We now place *T* spokes at the **k**_{i}s corresponding to the *T* largest-magnitude elements of ; *T* is thus another parameter trading off pulse duration with mitigation.

We now know *T* locations in (*k _{x}*,

(7)

where *b*(*t*) is the RF waveform (volts), *e*^{jΔB0(r)(t-L)} the phase accrual due to main field inhomogeneity as defined by the field map Δ*B*_{0}(**r**) (radians/s), *L* the pulse duration (s), and Discretizing Eq. [7] using the formalism of (34) yields:

(8)

where **m** and **S** are as in Eq. [5], **b** is an *N*_{t}-element vector of samples of *b*(*t*) taken at times *t*_{1},…, *t _{Nt}* (spaced by Δ

(9)

Fixing slice thickness and spoke shape ends up fixing the pulse shape. All that remains is to calculate the complex weight to encode along each spoke. This means that Eq. [8] reduces to one where **A** is *N*_{s} × *T* in size and **b** has *T* elements (12). Spoke weights are computed via **b** = (**SA**)^{†}**d** (again, all elements of **d** are equal to promote uniform magnetization). The *T* retuned weights are then extracted from **b**. At this point, the gradients and pulse have been calculated. Note that sparsity-enforced spoke placement and pulse design may be extended to parallel transmission systems (22).

For all *T*-spoke mitigation pulses presented here, λ = 0.35, slice thickness = 20 mm, spokes are Hanning-windowed sincs, the DC spoke’s time-bandwidth-product equals 4, the *k _{z}*-lengths of the other

Choosing the number of spokes, *T*, is an essential part of the design process and is accomplished by solving Eq. [6], designing a series of candidate pulses with increasing numbers of spokes, simulating the magnetization that arises due to each pulse, and recording the within-FOX standard deviation of each magnetization; *T* is then the smallest number of spokes needed to drive the standard deviation (SD) below some chosen value. This automated process takes several seconds.

Experiments are conducted on a 7T Siemens scanner (Siemens Medical, Erlangen, Germany) with standard body gradients (40 mT/m maximum amplitude, 180 T/m/s maximum slew rate). A quadrature bandpass birdcage coil is used for transmission and reception.

When collecting intensity images to estimate we use a standard slice-selective pulse followed by a 200-volt 16-ms B_{1}-insensitive rotation, type 4 (BIR4) adiabatic reset, collecting 128 × 128 GRE images with 25.6-cm FOV, 5-mm slice thickness, 2-mm in-plane resolution, 380 Hz/pixel bandwidth, 1-s TR, and 8-ms TE. To obtain a low-flip reference image, *L*_{0}(**r**), we apply a standard pulse without the reset and average eight times; parameters are the same as above except TR = 100 ms and TE = 8 ms. Finally, when applying a mitigation pulse, we perform 3D GRE readouts (without the reset) and collect 16 contiguous 5-mm slices, using the parameters above, except here TR = 100 ms and TE = 8 ms. In-plane mitigation performance is judged by analyzing the magnitude of the center slice of the volume; slice-selectivity is judged by analyzing the through-plane intensity profile.

To demonstrate the utility of sparsity-enforced spoke placement we first compare it to Fourier-based spoke placement (21). The latter computes the Fourier transform of the ideal in-plane excitation, , and places *T* spokes in (*k _{x}*,

Here we design a pulse to mitigate the inhomogeneity presented by the combination of transmit and receive profiles in a uniform-*T*_{1} head-shaped water phantom. The motivation is as follows: because this pulse seeks to mitigate the combination of profiles rather than simply the transmit profile, it will produce a result that is easy to understand and evaluate, since it ideally will produce a uniform image. In contrast, a pulse that successfully mitigates only the transmit profile will produce an image that is still highly nonuniform, because in this latter case the nonuniform receive profile is not mitigated (Note: a successful mitigation pulse produces a nonuniform magnetization and is not practical for clinical scenarios; in clinical practice we want to mitigate only.)

To begin, we collect a low-flip image *L*_{0}(**r**) using a standard pulse; Eq. [3] implies that. We then design a 23-spoke mitigation pulse by setting *S*(**r**) in Eq. [4] equal to *L*_{0}(**r**) and running the sparsity-enforced design algorithm. A stack of images is acquired using the pulse in conjunction with the 3D read-out. To quantify the degree to which the inhomogeneity is mitigated, we compare the SD, σ, and worst-case variation (WV) of the original image *L*_{0}(**r**) with those of the center slice of the postmitigation readout volume and also observe five 1D profiles. The WV of an image is the ratio of its brightest to its darkest pixel within the FOX. Unlike σ, WV is sensitive to the change of even a single pixel and thus reveals if the mitigation pulse causes the image to contain undesirable spikes or black holes.

We now transition to a practical scenario and design a pulse to mitigate solely the inhomogeneous transmit pro-file. We first estimate α_{0}(**r**) (and subsequently by collecting 10 images with transmit voltages *V* (20, 60, 100,…, 380) volts and performing the Powell fit; collecting the images takes 17 min, while fitting takes under a minute. A low-flip image, *L*_{0}(**r**), is also collected and *R*(**r**) is then estimated. We then design a 19-spoke pulse by setting *S*(**r**) in Eq. [4] equal to and running the sparsity enforcement algorithm. The desired magnetization is a 10° uniform flip across the FOX. After playing the pulse with the 3D readout, we extract the postmitigation center slice, *L*_{m}(**r**); taking *L*_{m}(**r**)/*R*(**r**) yields α_{m}(**r**), the postmitigation flip angle map. To calculate the performance of the pulse, five 1D profiles of and α_{m}(**r**) are considered along with the σ and WV of each.

Finally, we design a 19-spoke pulse to mitigate non-uniformity in an axial slice of the human brain at 7T. This experiment is conducted exactly like the water phantom mitigation trial. Experiments are conducted at the A.A. Martinos Center for Biomedical Imaging (Charles-town, MA, USA) and obey all safety and Institutional Review Board (IRB) requirements.

Figure 1 depicts the map provided to both the Fourier method and sparsity-enforced method (Fig. 1a). The FOX is where is nonzero; pixels within it are assembled to form **S**. The (*k _{x}*,

Fourier-based vs. sparsity-enforced spoke placement. **a:** Each method attempts to mitigate |*B*_{1}^{+}|. **b,c:** Excitations due to Fourier-based and sparsity-enforced placement. **d,e:** Magnetizations due to 2.93-ms Fourier-based and 2.96-ms sparsity-enforced pulses. **...**

The original image, *L*_{0}(**r**), along with the mitigated image due to a 200-volt, 8.5-ms, 23-spoke pulse, *L*m(**r**), are presented in Fig. 2a and b, respectively. A through-plane profile of the mitigated readout volume (Fig. 2c) proves the pulse achieves slice selection. Recall that here the in-plane goal of the pulse is to mitigate the combined transmit and receive profiles, so ideally *L*_{m}(**r**) will be constant everywhere. From Fig. 2b and the associated 1D profiles, we see that the pulse has produced a more uniform image. Based on standard deviation, *L*_{m}(**r**) (Fig. 2b) is 2.6 times smoother than *L*_{0}(**r**) (Fig. 2a). Furthermore, WV has been reduced by a factor of 1.7. It seems that to some degree, the pulse mitigates the inhomogeneity presented by the combined profiles. Note that each image is scaled to display its entire dynamic range within the grayscale spectrum.

Water phantom: *R*(**r**)|*B*_{1}^{+}(**r**)| mitigation due to an 8.5-ms 23-spoke pulse. **a:** original image, *L*_{0}(**r**), collected using standard pulse; highly nonuniform. **b:** In-plane mitigated image, *L*_{m}(**r**); standard deviation, σ, and worst-case variation (WV) reduced **...**

Figure 3 shows pulse design details; sparsity-enforced spoke locations are shown in (*k _{x}*,

Here a 19-spoke pulse attempts to produce a uniform magnetization. Figure 4a, b, and c depict the low-flip image *L*_{0}(**r**), receive profile estimate *R*(**r**), and transmit profile estimate, respectively; the latter is highly nonuniform with σ = 0.16 and WV = 2.5. The transmit and receive profiles are not equal; in fact, each seems to be the mirror image of the other (consider a reflection across the *y* axis). Note the smoothness and lack of noise in the transmit profile estimate (Fig. 4c). This map is not smoothed; rather, it simply comes directly out of the fitting algorithm whose inputs are non-smoothed raw images. Overall, this suggests that the fitted transmit profile is a realistic estimate.

Water phantom: |*B*_{1}^{+} (**r**)| mitigation due to a 7-ms 19-spoke pulse. **a:** Original image, *L*_{0}(**r**), collected using standard pulse. **b:** Receive profile estimate *R*(**r**). **c:** Transmit profile estimate |*B*_{1}^{+} (**r**)| in nT/volt [proportional to unmitigated flip angle map **...**

The Bloch simulation of the in-plane excitation created by the 19-spoke pulse is given in Fig. 4d; it closely resembles, as intended. The similarity of the mitigated in-plane image, *L*_{m}(**r**) (Fig. 4e), to the receive profile, *R*(**r**) (Fig. 4b), suggests that the postmitigation flip angle map may be fairly uniform, while the through-plane profile of the mitigated volume (Fig. 4f) indicates the pulse succeeds at slice selection. The postmitigation flip angle map, α_{m}(**r**) (Fig. 4g), does indeed confirm that in-plane flip angle is fairly uniform across space. Quantitatively, σ and WV have been reduced by factors of 4 and 1.6, respectively.

The pulse itself is 7.5 ms long and transmitted at a peak value of 243 volts. Pulse design details appear in Fig. 5. Note here that the spoke locations chosen by the sparsity-enforced method differ from those chosen in the earlier experiment (see Fig. 3) because the spoke patterns and pulses generated by the sparsity-enforced method depend on both the desired excitation and map.

In this clinical scenario a 7.5-ms 19-spoke pulse attempts to produce a uniform magnetization in an axial slice of a healthy volunteer’s brain. Figure 6 depicts the low-flip angle image, receive profile, transmit profile, and other images; formatting here is identical to Fig. 4. The through-plane profile of the mitigated volume (Fig. 6f) confirms that the pulse excites only the intended region. The post-mitigation flip angle map, α_{m}(**r**) (Fig. 6g), is more uniform that the original transmit profile (Fig. 6c); this is apparent from the 1D profiles as well as the fact α_{m}(**r**) has three times and 1.7 times lower σ and WV, respectively, than the original profile and flip angle map α_{0}(**r**).

In vivo |*B*_{1}^{+} (**r**)| mitigation due to a 7-ms 19-spoke pulse. **a:** Original image, *L*_{0}(**r**), collected using standard pulse. **b:** Receive profile estimate *R*(**r**). **c:** Transmit profile estimate |*B*_{1}^{+} (**r**)| in nT/volt [proportional to unmitigated flip angle map α **...**

Figure 7 shows the design of the 19-spoke pulse. It is 7.5-ms long and transmitted at 203 volts. We see from the (*k _{x}*,

Empirically, we find that pulse designs are robust to the choice of λ in Eq. [6]. That is, for various λs and fixed *T*, the algorithm often finds similar sets of spoke locations and produces magnetizations with similar degrees of uniformity.

The spoke-based mitigation pulse design method (11) requires users to visually inspect and tune a control parameter while working on the scanner in order to produce a mitigated image, whereas the sparsity-enforced placement method is automated and seems robust to its λ parameter. Furthermore, prior work does not provide a means to estimate the postmitigation flip angle map in the presence of a nonuniform and possibly proton-weighted receive profile and is thus not able to truly characterize the extent to which inhomogeneity is mitigated by a pulse designed for that purpose.

The sparsity-enforced algorithm needs a map to design a pulse. This requirement poses a challenge because varies per slice and per subject and estimating this map for a given slice and subject takes 17 min (collecting 10 images at 1.7 min/image and then fitting). We are currently pursuing several ways to reduce this mapping time. First, it may not be necessary to collect 10 high-resolution images for mapping; it seems that five to six lower-resolution images may be sufficient, but at most this reduces mapping time to 4–5 min. Instead, or additionally, it may be possible to rapidly map in under a minute by exploiting some empirical trends we have observed: for example, varies slowly with *z*, so a map estimate obtained at *z* = *z*_{0} may be accurate within some range *z*_{0} ± δ, allowing to be mapped once per slab rather than once per slice. It also seems that does not differ radically across subjects for a fixed axial slice. Thus it may be possible to develop a prototypical slice-by-slice model of the average brain and retune the slice maps of this model for a given subject by simply collecting a small set of rapidly-acquired calibration scans to account for individual differences from the atlas.

The second limitation of this work is that the sparsity-enforced algorithm needs 3–5 min to design a pulse when given . The vast majority of time is spent solving Eq. [6]. We are working on overcoming this computational problem by implementing Eq. [6] using a multiresolution approach (25) combined with fast iterative shrinkage (36).

The final limitation of our work involves the slice thickness and duration of mitigation waveforms. In order to play 19 spokes in a feasible period of time given our gradient constraints, we chose to excite 20-mm slabs, but in many practical cases 5-mm slices are desired. Fortunately, this problem may be minimized by using commercially available fast insert head gradients that are already in use at a number of sites. These gradients have amplitude and slew rate limits of 80 mT/m and 800 T/m/s, respectively; if these limits are conservatively constrained to 35 mT/m and 600 T/m/s, the 19-spoke patterns discussed earlier are able to be implemented to excite only 5 mm (10 mm) of tissue in 8 ms (5.5 ms).

We have presented a novel sparsity-enforced RF pulse design algorithm that produces short slice-selective excitation pulses that mitigate inhomogeneity at high field. The method provides two control parameters that let pulse designers trade off mitigation with pulse duration. Imaging experiments at 7T showed that the sparsity-enforced spoke placement and pulse design method was capable of mitigating inhomogeneity in both a head-shaped water phantom and the human brain, producing fairly uniform transverse magnetizations in each case.

To the best of our knowledge, the algorithm’s *L*_{1}-penalty on the traversal of *k*-space, the genetic algorithm used to connect spoke locations using nearly the shortest path possible, and the optimized nature of its pulse designs are novel contributions to high-field MRI RF excitation pulse design, inhomogeneity mitigation, and in vivo brain imaging at 7T. We conclude by noting that sparsity-enforced pulse design is applicable to lower field systems, nonbrain applications, and parallel transmission arrays.

We thank the reviewers for their insightful comments and P-F Van de Moortele for suggesting that we include quantitative assurance that the mitigation process creates no extraordinary flip-angle map extrema. This work was supported by the MIND Institute (LLW) and an R. J. Shillman Career Development Award (EA).

Grant sponsor: National Institutes of Health (NIH), National Center for Research Resources (NCRR); Grant number: P41RR14075; Grant sponsor: National Institute of Biomedical Imaging and Bioengineering (NIBIB); Grant numbers: 1R01EB007942, 1R01EB006847, 1RO1EB000790; Grant sponsor: U.S. Department of Defense, National Defense Science and Engineering Graduate Fellowship; Grant number: F49620-02-C-0041.

Published online in Wiley InterScience (www.interscience.wiley.com).

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