Tailored Excitation using Non-Linear B₀-Shims

Abstract

In high field MRI, RF flip angle inhomogeneity due to wavelength effects can lead to spatial variations in contrast and sensitivity. Improved flip angle homogeneity can be achieved through multi-dimensional excitation, but long RF pulse durations limit practical application. A recent approach to reduce RF pulse duration is based on parallel excitation through multiple RF channels. Here, an alternative approach to shorten multi-dimensional excitation is proposed that makes use of non-linear spatial variations in the stationary (B₀) magnetic field during a B₀-sensitive excitation pulse. As initial demonstration, the method was applied to 2D gradient echo (GE) MRI of human brain at 7T. Using B₀ shims with up to second order spatial dependence, it is demonstrated that root-mean-squared flip angle variation can be reduced from 20% to 11% with RF pulse lengths that are practical for general GE imaging applications without requiring parallel excitation. The method is expected to improve contrast and sensitivity in GE MRI of human brain at high field.

INTRODUCTION

In MRI, RF transmission field (B₁⁺) imperfections can lead to undesired spatial variations in image signal-to-noise ratio (SNR) and contrast. This potential problem becomes increasingly apparent at higher field strengths, where the shorter RF wavelengths lead to increasing spatial variations in the amplitude and phase of B₁⁺(1). B₁⁺ imperfections are significantly affecting MRI of human body at 3T and MRI of human head at 7T, for which the RF wavelength has approached or become smaller than the dimensions of the imaging target.

The mitigation of this problem is an active area of research (2–12), with proposed solutions falling into two categories: homogenization of the amplitude of B₁⁺, and homogenization of the RF flip angle in the presence of inhomogeneous RF fields.

To improve the uniformity of B₁⁺, a specialized RF transmit structure can be designed to which a single or multiple RF sources are connected. For example, multiple RF signals can be applied to amplify independent oscillatory modes (2) or independent current loops in the transmit structure (3,4,11) or coupled coil elements (12). This strategy, known as RF shimming (1) allows substantial but limited improvement of B₁⁺ uniformity, generally leaving an undesired level of inhomogeneity (5).

The second strategy is based on the notion that uniform flip angle for the RF pulses employed in a pulse sequence does not necessarily require uniform B₁⁺. One example of this is the use of adiabatic pulses that compensate for variations in B₁⁺ (6). Alternatively, RF pulses can be designed that have a spatial selectivity in multiple spatial dimensions, i.e. in one or more directions additional to the slice or slab direction. Such pulses can be designed to have an in-plane spatially varying effect that compensates for the inhomogeneities of B₁⁺. This in-plane spatial selectivity is generated in a manner analogous to one-dimensional slice selective excitation (7), by combining a frequency selective RF pulse with a spatially varying resonance frequency generated by the imaging gradients (8–10). This in-plane selectivity is generally achieved by using a sequence of (one-dimensional) slice selective pulses interspersed with gradient pulses (8–10). These pulses are generally referred to as “spokes” for the patterns they excite in k-space (13).

Despite the fact that multi-dimensional pulses allow excellent flip angle uniformity, their extended duration can result in significant sensitivity to off-resonance effects as well as restricting their practical application. One way to mitigate this problem and shorten RF pulse duration is the use of transmit-SENSE techniques (14). Although these techniques allow shortening of the RF pulse by several fold, they require multi-channel transmit capability which currently is not generally available.

In the following we propose a method to shorten the RF pulse duration for B₁ inhomogeneity mitigation based on the use of non-linear gradients. Previous work using radial or PatLoc gradients has demonstrated efficient multi-dimensional excitation for specific shapes of the desired volume (15–18). The additional flexibility available with the combination linear and non-linear gradients may facilitate the use of multi-dimensional excitation for B₁ mitigation.

To investigate the effectiveness of this approach for imaging the human brain at 7T, we used the linear gradients together with the non-linear (second order) field gradients available with the system’s resistive B₀ shim coils.

METHOD

Background

RF pulses of finite length and amplitude are sensitive to B₀ variations through differences between the NMR resonance frequency and the transmit frequency. The flip angle variations that result from this sensitivity to off-resonance RF have been exploited in spectroscopy to selectively excite spectral moieties (19,20). A popular implementation of this spectral selectivity is the 2-pulse “jump-return” sequence used for water suppression in NMR spectroscopy (21). In imaging, spectral selectivity can be used to perform slice selective excitation. However, with certain excitation pulses, B₀ variation across the object may cause inadvertent flip angle variations that may be detrimental to SNR and contrast. On the other hand, the B₀ sensitivity of the RF excitation be exploited to adjust the spatial excitation pattern and counter flip angle variations due to inhomogeneity of B₁⁺; for this purpose one would need to intentionally introduce a spatially varying B₀-field during the RF excitation, the precise shape of which is dictated by the desired flip angle correction. This, in effect, is what is done with approaches based on multi-dimensional excitation pulses (8,10,13,22,23); however, because these pulses attempt to compensate for the generally non-linear spatial variations in B₁⁺ with only linear B₀ gradients, they often require rather elaborate designs that incorporate a number of slice-selective sub-pulses. This results in long RF pulse length, and increased sensitivity to T₂* and background B₀ variations. Here we attempt to ameliorate this issue by using both linear and non-linear spatial variations in B₀.

Pulse Sequence

To demonstrate the principle of the method, we used a simple 2-pulse slice selective excitation, combined with a gradient echo (GE) readout (Fig.1a). The two RF pulses excite the same slice, inside which the spins experience a combined effective flip angle θ that is dependent on the individual flip angles α₁ and α₂, as well as the accrued phase φ between the pulses, similar to the classical jump-return pulse. An alternative implementation that allows reduced pulse spacing is shown in Fig.1b (inspired by pulses used in (24)). The introduction of an in-slice position dependent phase accumulation is indicated by the trapezoidal shim pulse S in Figs.1a and b, which represents a temporary spatially dependent frequency shift effectuated by a combination of transmitter frequency shift Δf₀, x-, y-, and z-gradients, and higher order shims. The effect of the two RF pulses and the shim pulse on the magnetization can be calculated by applying 3 rotation matrices to the magnetization vector (0,0,M_z). These matrices represent the successive rotations α₁, φ and α₂, around the x, z and x axis respectively (instead of x one could have chosen any axis in the transverse plane, leading to the same result). By neglecting relaxation effects and B₀ heterogeneity, for any spatial location r⃗, the effective flip angle θ can be derived by taking the arccosine of the ratio of the final longitudinal magnetization to the starting longitudinal magnetization, leading to:

$$\mathrm{\theta }(\overrightarrow{\mathbf{r}})=\text{arccos}(\text{cos}{\mathrm{\alpha }}_1(\overrightarrow{\mathbf{r}})\text{cos}{\mathrm{\alpha }}_2(\overrightarrow{\mathbf{r}})-\text{cos}\mathrm{\phi }(\overrightarrow{\mathbf{r}})\text{sin}{\mathrm{\alpha }}_1(\overrightarrow{\mathbf{r}})\text{sin}{\mathrm{\alpha }}_2(\overrightarrow{\mathbf{r}})).$$ (1)

Figure 1.

Shimming of B₁ excitation using B₀-sensitive RF. Slice selective pulses α₁ and α₂, are separated by a time ΔT, during which a pulse S is played out to generate a spatially dependent phase shift. (a) Basic version. (b) To minimize ΔT, the version used in this work had inverted polarity of the second slice select gradient (inspired by pulses used in (24)).

For the simplified situation α₁=α₂=α, we have:

$$\mathrm{\theta }(\overrightarrow{\mathbf{r}})=\text{arccos}(1-(\text{cos}\mathrm{\phi }(\overrightarrow{\mathbf{r}})+1){\text{sin}}^2\mathrm{\alpha }(\overrightarrow{\mathbf{r}})).$$ (2)

This relationship is presented graphically in Fig.2a. For φ=0, the effective flip angle of the excitation becomes 2α, equal to the sum of the individual flip angles whereas for nonzero φ, the effective flip angle is reduced. The fractional reduction in θ with increasing φ is similar for different α, except for α approaching 90°. In Eq.(2), both α and φ are dependent on r. In fact, angle α directly reflects the spatial variation of B₁⁺, as they are proportionally related according to: $\mathrm{\alpha }(\overrightarrow{\mathbf{r}})={\displaystyle \int }c(t)\mathrm{\gamma }{B}_1^{+}(\overrightarrow{\mathbf{r}})\mathit{\text{dt}}=C․B_1^{+}(\overrightarrow{\mathbf{r}})$, with the integral running over the (sub)pulse duration, c(t) representing the modulation amplitude (i.e. voltage) of the pulse, γ the gyromagnetic ratio, and B₁⁺ representing the amplitude of the transmit RF field generated per unit voltage. The goal is to generate a spatial distribution φ(r⃗) that counteracts the effects of α(r⃗) on θ(r⃗) such that the latter is constant over space. To achieve this we need to generate a φ_target(r⃗) according to:

$$\mathrm{\phi }(\overrightarrow{\mathbf{r}})=\text{arccos}\left(\frac{1+\text{cos}2\mathrm{\alpha }(\overrightarrow{\mathbf{r}})-2\text{cos}\mathrm{\theta }(\overrightarrow{\mathbf{r}})}{1-\text{cos}2\mathrm{\alpha }(\overrightarrow{\mathbf{r}})}\right).$$ (3)

Figure 2.

Behavior of the 2-pulse excitation of Fig.1 calculated from the Bloch Equations. Only positive values of φ are shown; behavior is symmetric around φ=0.

(a) Dependence of θ on φ at selected values of αas expressed by Eq.(2).

The effective flip angle of the 2-pulse excitation reduces with increasing φ. The relationship between θ and φ is dependent on α, and for α=90°, θ becomes inversely proportional to φ.

(b) Dependence of φ_target on α for selected target values of θ. θ uniformity optimization can be achieved around a range of (α,φ) combinations. For low φ and θ, relatively large changes in φ are needed to compensate for changes in α.

As seen in Fig.2b, optimization of θ can be performed around different (φ,α) points, with the lowest RF power required for optimizations around φ=0.

Shim Optimization

Generation of a φ_shim(r⃗) that approximates φ_target(r⃗) is achieved by applying current pulses S_i to the n individual shim coils indexed by i, resulting in:

$${\mathrm{\phi }}_{\mathit{\text{shim}}}(\overrightarrow{\mathbf{r}})={\displaystyle \sum _{i=1}^n}{\mathrm{\phi }}_i(\overrightarrow{\mathbf{r}})={\displaystyle \sum _{i=1}^n}{\displaystyle \int }\mathrm{\gamma }{c}_i(\overrightarrow{\mathbf{r}})S_i(t)\mathit{\text{dt}},$$ (4)

with the time integral running over the pulse interval ΔT. Shim currents S_i can be optimized by minimizing the root-mean-squares (r.m.s.) difference between φ_shim(r⃗) and φ_target(r⃗) from eq.(3), using multi-linear regression with the fields associated with each shim term as regressors. This requires knowledge of the magnetic field distribution c_i(r⃗) generated by each shim term, which can be obtained from calibration scans (25).

EXPERIMENTS

Experiments were performed on phantoms and on human brain using a Siemens Magnetom 7T (Erlangen, Germany) whole body scanner based on an Agilent 7T-830-AS (Oxford, UK) shielded magnet design. The system provided 5 second order shims producing z²(i.e. x²+y²), zx, zy, xy, x²−y² field dependencies with maximum strengths of 1.6kHz/cm². Together with a zero order term (effectuated through either the reference frequency or the RF pulse phase) and the linear gradients, in principle a total of 9 degrees of freedom were available (i.e. n=9 in eq.(4)) for flip angle optimization. However, for the single (axial) slice demonstration presented below, z, zx, and zy were ineffective and therefore not used. Also, due to the lack of rapid switching capability of the second order terms, all shims were kept on continuously. This was not expected to affect image quality substantially under the conditions of the experiments used in this study.

Our B₀-based flip angle optimization involved the following steps:

1. Shim calibration: Calibration of the fields c_i generated by the individual shim terms, involving the acquisition of B₀ field maps during application of known current amplitudes. This procedure needs to be performed only once, and can be done on a phantom. In this work, a silicon oil phantom was used to minimize B₁ inhomogeneity due to wavelength effects. Shim values of around 50µT/m for linear and 50µT/m² for second order terms were used for this purpose.

2. Selecting α: Determination of subject-specific distribution of α(r⃗) based on B₁⁺ field mapping. The latter was done using the Bloch-Siegert method (26) using a GE sequence and an 8 ms, Fermi-apodized irradiation pulse. The pulse was applied at resonance frequency shifts of +4kHz and −4kHz respectively, after which $B_1^{+}(\overrightarrow{\mathbf{r}})$ was determined from the phase difference between the images. TE and TR were 12 and 900ms respectively, image matrix 64×64, FOV was 27cm, and slice thickness 3.5mm. This resulted in a subject-specific map $B_1^{+}(\overrightarrow{\mathbf{r}})$ per unit transmitter voltage; from this α(r⃗) follows by taking into account the duration, shape of the RF pulse and choosing its amplitude (i.e. transmitter voltage). Choice of the RF amplitude could be included in the optimization procedure; here we optimized at a small range of amplitudes that resulted in minimal RF power deposition (see following steps).

3. Calculating φ: Calculation of φ_target(r⃗) by substituting α(r⃗) into eq.(3). To minimize the effect of B₀ inhomogeneities caused by background susceptibility effects, an object specific phase φ_object(r⃗), was calculated from a separately acquired field map, and subtracted from φ_target(r⃗). Calculation of the required strengths of the available shim terms from multi-linear regression. This required choice of the sub-pulse amplitude as this determines the amount of phase accumulation φ that is needed to produce the target flip angle θ. Due to the non-linear character of eq.(3), larger values of α will require different shim distribution (Fig.2).

4. Shim optimization: It is possible to optimize the uniformity of θ over a range of sub-pulse amplitudes and choose the amplitude value that leads to the best θ uniformity or has other favorable attributes, for example minimal sensitivity to temporal variations in φ or minimal RF power. Here, the optimization was performed by selecting a nominal sub-pulse flip angle α of 10°, which was close to the expected minimum required for the selected target angle θ, which was set at 20°, and calculating the desired B₀ distribution from eq.(3), followed by a linear least squares optimization of the shim currents to create this distribution (eq.(4)). Note the solution of eq.(3) (φ_target(r⃗)) is symmetric for either a positive of negative offset frequency, that is the sign of φ_target(r⃗) can be chosen freely. Therefore, both positive and the negative solutions were considered in combination with the B₀ distribution remaining after the baseline shimming was performed on the slice of interest. The solution that resulted in the smallest deviation from the desired field was then chosen. Note that the available shim strengths were sufficiently large to not require constraints on the solution: the optimized shim strengths were always within the hardware limits.

5. Evaluation: Evaluation of the optimized B₀ shim distribution φ_shim(r⃗) was performed by combining the 2-pulse excitation (Fig.1b) with a GE readout using the following parameters: ΔT=1.35ms, TE=5ms, TR=600ms, 64×64 resolution FOV 27cm, slice thickness 3.5mm. The flip angle θ was measured by dividing the image from the 2-pulse excitation by the receive coil profile; the latter was derived from a low flip angle GE acquisition after division by the B₁⁺ profile. The effective flip angle was also predicted from eq.(1) using background and applied B₀ field, and the B₁⁺ map.

Simulations were performed to compare the optimized 2-pulse excitation with conventional 2- and 3-spoke pulses that did not apply higher order shims. The spoke pulses were optimized by minimizing the flip angle variance while varying the flip angles of the individual pulses as well as the phase shifts and linear gradients applied between the pulses. Likewise, the pulse with higher order shims was iteratively optimized for minimal effective flip angle variance, adjusting both α and the relevant shim currents. Note this is somewhat different than the procedure used for the actual measurements as outlined above, as here the sub-pulse flip angles were allowed to vary and the optimization minimized the spatial variation in the effective flip angle, rather than the deviations of the resulting B₀ distribution from the calculated φ_target(r⃗).

RESULTS AND DISCUSSION

Figure 3 shows the effect of applying a non-linear shim during selective excitation, demonstrating the flexibility of the proposed method in manipulating the effective flip angle θ within an axial slice selected in the oil phantom. Linear, circular, and ellipsoidal flip angle distributions were generated by applying the x gradient, the z² shim, and combinations of z²(i.e.x²+y²), x²−y², and xy shims respectively. Strong spatial variations can be generated with each of these φ_shim(r⃗) distributions. Note that the z² shim generates a circular (x²+y²) pattern within the axial slice, addition of x²−y² and xy terms allow ellipsoidal patterns, and x and y terms allow shifting these patterns across the slice.

Figure 3.

Demonstration of RF flip angle shimming on silicon oil phantom using 2-pulse excitation and (continuous) application of various shim terms. Signal amplitude variations reflect the effective θ, as both RF transmit and receive fields were uniform over space. Shim strengths: x = 82 Hz/cm (row 2) and 3.8 Hz/cm (row 4); y = 3.8 Hz/cm; z² = 7.5 Hz/cm²; x²−y² = 1.3 Hz/cm²; xy = 1.2 Hz/cm².

In human brain, substantially improved θ uniformity was achieved using B₀ shimming. Figure 4 shows an example of this for θ=20°; despite the 20% r.m.s. variation in B₁⁺ over the slice, the variation in estimated θ was within 11%. The optimization was performed around φ=0, with the average α around 10°.

Figure 4.

Demonstration of RF flip angle shimming on human brain using 2-pulse excitation and (continuous) application of optimized shim terms. (a) sagittal localizer indicating location of axial slice; (b) B₁⁺ amplitude estimated from Bloch-Siegert B₁-mapping; (c) original B₀, scale −200 to 100Hz; (d) measured θ using 2-pulse sequence with original B₀; (e) adjusted B₀ using shim terms, scale −200 to 100Hz; (f) measured θ using 2-pulse sequence with adjusted B₀. Shim strengths were: Δf₀ = 300 Hz, x = 2.2 Hz/cm, y = 9.1 Hz/cm, Z² = 2.0 Hz/cm², x²−y² = 0.3 Hz/cm², xy = 0.03 Hz/cm².

Results of the comparison with conventional 2-dimensional selective pulse that do not apply non-linear shims are summarized in Figure 5 and Table 1. Shown in columns from left to right (in both Table and Figure) are the effect of various pulses of 2 and 3 sub-pulses, with and without the use of non-linear shims. The simulation shows that for the excitations with 2 sub-pulses, a substantial improvement in flip angle uniformity can be achieved when using non-linear shims, whereas similar uniformity can only be achieved when one adds an additional sub-pulse (spoke) (column 3) if only linear gradients are used.

Figure 5.

Effect of the addition of non-linear shims to spatial uniformity of 2-dimensional excitation (see also Table 1). Top row shows calculated flip angle distribution over an axial slice, plots in row 2 and 3 show profiles along lines in indicated in MRI slices shown on left.

Single pulse excitation (first full column) shows substantial B₁ variation in both x (left to right) and y (top to bottom) directions. Two pulse excitations with only linear gradients can improve uniformity with limited extent (column 2). Addition of an additional RF sub-pulse (column 3) allows improvement over both image dimensions, resulting in a uniformity the approaches that of the 2-pulse excitation with non-linear shims included (column 4).

Table 1.

Effect of the addition of non-linear shims to spatial uniformity of 2-dimensional excitation (see also Fig.5). Uniformity was calculated for optimized excitations with 2 and 3 sub-pulses (columns 2 and 3 respectively) and compared with a single pulse (1-dimensional) excitation (data column 1). The pulse with non-linear shims (column 4) used two sub-pulses of equal flip angle. Target flip angle θ was 20°. The two bottom rows give calculated SAR values, relative to the single pulse excitation for two scenarios: constant RF duration for sum of all sub-pulses (SAR1) and for individual sub-pulse (SAR2).

	single pulse	2-pulse x,y-gradient	3-pulse x,y-gradient	2-pulse grads+shims
Pulse angle(s)	20	22.9, 7.46	24.1, 8.7, 3.5	12.9, 12.9
Rel. SD [%]	19.4	12.4	6.6	6.1
Rel. SAR1	1	2.89	5.03	1.66
Rel. SAR2	1	1.44	1.68	0.83

As can be seen from Table 1, SAR levels (i.e. specific absorption ratio, a measure for RF power deposition in tissue) vary substantially between the various pulses and are also dependent on how the pulse is executed. When total RF duration (length of concatenated sub-pulses, excluding periods when RF is off) is kept constant, the SAR of the 2-pulse excitation employing non-linear shims is 66% higher than that of a single pulse; however, it remains below that of the spoke pulses.

Using conventional spokes pulses to match θ uniformity of the proposed method requires increasing the overall pulse length and consequently increase sensitivity to T₂* decay and susceptibility-induced contributions to φ. On the other hand, it is possible to limit the number of spokes by transmitting through multiple RF channels. Two pulse designs have been shown to provide good θ uniformity in human brain at 7T (23). The results shown above suggest that this performance may be achieved without the need for independent RF channels, and that performance may be further increased by combining the two methods. Alternatively, the proposed method may relax the requirements for transmit coil uniformity and therefore lessen the burden on coil design.

As mentioned above, there is some flexibility in optimization of θ with respect to the choice of α. This choice affects both the amount of RF power required for the excitation, as well as the sensitivity to temporal variations in resonance frequency or spatial variations related to magnetic susceptibility effects. Large α may lead to improved θ uniformity, however at the penalty of higher RF power deposition and increased sensitivity to temporal variations in resonance frequency. These penalties ideally should be accounted for in the optimization process. To minimize RF power deposition, an iterative optimization could be performed, varying both the shim terms and α. The optimization criterion can incorporate both the deviation of the flip angle from the desired target and a term reflecting SAR level. It is expected that such an optimization may become prohibitively time consuming for excitations with more than a few sub-pulses (or multiple excitation coils): these may require more sophisticated methods like those used for spoke pulses (i.e. without higher order shims) (27–29).

Another consideration for the optimization is the choice of inter-pulse spacing ΔT. Choosing ΔT too large may result in significant susceptibility-induced phase accumulation and T₂* decay between the RF sub-pulses, both of which can adversely affect θ uniformity. Large phase accumulation may become difficult to compensate for with the available shim terms. Shorter pulse spacing reduces this problem, but increases the required shim strength, gradient switching rates, and may require shorter RF pulse duration. The latter increases RF peak power and overall power deposition. The 1.35 ms spacing used above was limited by gradient slew rate.

The inter-pulse spacing ΔT also determines the range of chemical shifts the excitation will work over, although the precise range is dependent on α and φ In spectral domain, ignoring slice shifts, the 2-pulse version with ΔT=1.35ms has cyclical excitation bands separated by 1/ΔT=740Hz. As can be seen from Fig.2a, the full-width at half maximum of each band, for small flip angle, is around 250 degrees, which corresponds to about 500Hz. Therefore, the pulse will excite chemical species over roughly this range, provided the shim optimization is performed around φ=0.

Although the experiments described here were performed on a single axial slice, the method is extendable to multi-slice with oblique orientations. With the availability of a full set of second order shims, and the even larger range of shim terms available with most modern shim designs, extension to oblique slices appears trivial. One issue that requires attention with oblique slices, and also with off-center axial slices, is the minimization of through-plane dephasing while optimizing the in-plane field. For example, for the axial optimization described above, the in-plane x²+y² term was generated by the z². However, for off center slices, the use of a z² shim creates substantial through plane gradients. Fortunately, such gradients can be compensated by adjusting the linear gradients, in this case the z-gradient. In the optimization described above, this was not needed as the slices were selected at iso-center of the shim system.

Extension to multi-slice 2D will require rapid switching of the higher order shims, ideally according to pulses S in Fig.1, or alternatively with pulse S extending over the RF sub-pulses. Although not available on our scanner, shim hardware capable of millisecond-scale switching time has recently become available on human scanners and demonstrated for dynamic shimming applications (25,30,31). Extension to 3D scanning will likely require a complete set of third order shims and possibly even higher orders. There is ongoing development in this area, and recent work has demonstrated the application of up to fourth order shim sets for brain imaging (32).

Although the use of non-linear gradients for B₁ mitigation presented in this work is novel, the use of such gradients for spatial excitation has been presented before and the underlying mathematics is similar. It may therefore be possible to adopt theoretical methods described previously for optimization of multi-dimensional excitation (18,29) to the problem of B₁ mitigation presented here. This was not attempted here. In addition, it may be advantageous to design non-linear gradients specifically for the problem of B₁ mitigation. This may improve switching capability and pulse performance; however it would requires specialized hardware that is currently not available on MRI systems, in contrast with the gradients available with B₀ shim coils.

During submission of this manuscript, we found another report of the use non-linear gradients for B₁ inhomogeneity mitigation (33). This report, which also investigated the combination with transmit SENSE, found a substantial pulse reduction available with non-linear gradients, especially for lower numbers of transmit channels.

CONCLUSION

We propose the extension of multi-dimensional excitation with non-linear B₀ shims to improve excitation flip angle uniformity in high field MRI. The method is demonstrated for GE MRI for human brain at 7T. The method can be combined with existing multi-dimension excitation methods, including those that use parallel excitation, and is expected to lead to improved contrast and sensitivity in GE MRI at high field.