Direct design of 2D RF pulses using matrix inversion

Journal of Magnetic Resonance 235 (2013) 115–120

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Communication

Direct design of 2D RF pulses using matrix inversion Rolf F. Schulte ⇑, Florian Wiesinger GE Global Research, Freisinger Landstr. 50, 85748 Garching bei München, Germany

a r t i c l e

i n f o

Article history: Received 10 April 2013 Revised 17 July 2013 Available online 31 July 2013 Keywords: Radio-frequency pulses Spectral–spatial excitation 2D RF pulses Spatio-temporal encoding SPEN Metabolic imaging 13 C

a b s t r a c t Multi-dimensional pulses are frequently used in MRI for applications such as targeted excitation, fat–water separation or metabolic imaging with hyperpolarised 13C compounds. For the design, the problem is typically separated into the different dimensions. In this work, a method to directly design two-dimensional pulses using the small-tip angle approximation is introduced based on a direct matrix representation. The numerical problem is solved in a single step directly in two dimensions by matrix inversion. Exemplary spectral–spatial excitation and spatio-temporal encoding (SPEN) pulses are designed and validated. The main benefits of the direct design approach include a reduction of artefacts in case of spectral–spatial pulses, a simple and straightforward computer implementation and high flexibility in the pulse design. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction Multi-dimensional radio-frequency (RF) pulses are used in special applications such as those involving targeted or spectral– spatial (SPSP) excitation [1–4]. A recent application of SPSP pulses is metabolic imaging of hyperpolarised 13C compounds, which demands highly efficient sampling of the decaying and non-replenishing magnetisation. A single metabolite is excited spectrally selectively in a single slice, hence not requiring spectral encoding during the acquisition. The image can be encoded by sampling the metabolite’s spatial dimensions for example with a single-shot imaging readout such as EPI or spirals [5–8]. Another class of pulses are pulses with a quadratic phase. Overlaying a quadratic phase onto the profile leads to an approximately quadratic phase in the other domain, the pulse coefficients, as well [9]. The introduction of the quadratic phase can be used to spread out the central main lobe, thereby reducing the peak B1 amplitude of that pulse. The quadratic phase is approximately the same as a linear gradient sweep and can be used for spatio-temporal encoding (SPEN) [10,11]. The SPEN concept works by exciting spins sequentially in time with a linear frequency sweep. Together with a linear gradient, this sweep translates into a spatially sequential Abbreviations: FOV, field of view; MRS, magnetic resonance spectroscopy; RF, radio-frequency; ROI, region of interest; SNR, signal-to-noise ratio; SPSP, spectral– spatial; SPEN, spatio-temporal encoding; EPI, echo-planar imaging; CSI, chemicalshift imaging. ⇑ Corresponding author at: GE Global Research, Freisinger Landstr. 50, 85748 Garching bei München, Germany. Fax: +49 89 5528 3180. E-mail address: [email protected] (R.F. Schulte). 1090-7807/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmr.2013.07.014

excitation, which can be read out directly with a gradient of opposite polarity. The use of SPEN does not require Fourier transformation normally used for image reconstruction. The spatial information is extracted from the magnitude of the acquired data, while the spectral information can be reconstructed from the phase of the acquired data. Two-dimensional (2D) pulses are advantageous for slice-selective imaging [11]. Commonly, 2D pulses are designed using the so-called separable design, by first choosing a suitable gradient trajectory, and subsequently designing a 1D spectral (or quadratic phase) and a 1D spatial filter function and finally combining this into the actual 2D pulse, possibly using a correction function [5]. In this work, we introduce a simple 2D pulse design approach using direct matrix inversion, which reduces sideband artefacts. Furthermore, exemplary SPSP and SPEN pulses are generated to demonstrate the benefits of the design. Three different kinds of ‘‘all purpose’’ SPSP pulses are designed for metabolic imaging with hyperpolarised pyruvate: (1) pulse with bidirectional gradient modulation and its excitation shifted to the first sidelobes; (2) pulse with bidirectional gradient modulation and its excitation centred; and (3) pulse with flyback gradient modulation and its excitation centred. These three pulses are implemented on the MR scanner and the resulting profile and performance are validated using phantoms. These results are compared to simulations obtained by solving the Bloch equations. Two different kind of SPEN pulses are designed: one with a Cartesian gradient modulation and a quadratic phase along the slower, phase-encoded direction; and one with a spiral trajectory and a quadratic phase in both directions.

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2. Theory and methods 2.1. Spectral–spatial pulses Spectral–spatial (SPSP) pulses are 2D radio-frequency (RF) pulses that are used for combining frequency- and slice-selective excitation, as shown in Fig. 1. Typically, a large bandwidth in the spatial dimension and a narrow bandwidth in the spectral dimension is required. This naturally leads to using the sublobes for the spatial dimension and the overall pulse envelope for the spectral dimension Fig. 1. Hence, the spatial dimension is quasi-continuously sampled, while sidelobe artefacts will occur along the spectral dimension in the excitation profile due to its coarse sampling [12]. Sidelobe artefacts in the spatial dimension will be outside the region of interest if the discretisation frequency is high enough (e.g., on the used MR scanner 500 MHz). The SPSP pulse design starts by defining the number n and duration of the pulse’s sublobes s, and by choosing the kind of gradient modulation, bidirectional or flyback, for applying RF during both or only one gradient polarity, respectively. With a maximum gradient strength gmax and slew rate smax given by the scanner hardware, the gradient trajectory is determined, as the gradients are typically driven at their limits. The minimum slice thickness z determines the spatial bandwidth

c BW ¼  z  g max ; 2p

profile). The overall pulse duration is given by the product of duration and number of sublobes plus the spatial rewinder T = ns + trewinder. The overall duration is the main constraint for the spectral time-bandwidth product. Choosing these parameters involves a delicate trade-off; the duration of sublobes s should be sufficiently long in order to obtain a suitable spatial time-bandwidth product. At the same time, s determines the position of the spectral sidelobe artefacts stemming from the coarse sampling. These artefacts should not overlap with other peaks, ideally lying outside the spectral region of interest, hence requiring a short duration of the sublobes s. The overall pulse duration T should be sufficiently long to obtain a suitable spectral profile, but sufficiently short for reducing artefacts such as sensitivity to flow or decay due to T 2 . The overall quality of the pulse is determined by the time-bandwidth products; however, error ripples can be traded off against selectivity. A more selective pulse with a narrow transition band will have larger error ripples and vice versa. The 2D excitation profile bm needs to be chosen in alignment with the constraints of the overall parameter selection mentioned above. In other words, the profile must be physically realisable, as otherwise the fitting errors will be too large. The desired excitation profile is given by

bm ¼ ð1Þ

which together with the duration of the sublobes determines the spatial time-bandwidth product (measure of the quality of the pulse



1

for jf1;m j < fP1 and jf2;m j < fP2 ;

0

for jf1;m j > fS1 or jf2;m j > fS2 ;

with the passband and stopband frequencies being defined by fP ¼ 12 BWð1  FTWÞ and fS ¼ 12 BWð1 þ FTWÞ, respectively. The fP parameter BW = FP + Fs denotes the bandwidth and FTW ¼ fSBW denotes the fractional transition width. The transition band between fP and fS is undefined and no sampling points are included in this band for the fit. For the ‘‘shifted’’ pulse, this profile is shifted in the spectral dimension to the position of the first sidelobe. The time reference of all designed pulses in the spectral profile is shifted towards the end of the pulse, thereby leading to some self-refocusing in the spectral domain. This leads to the formation of a main lobe towards the end of the pulse, which reduces relaxation, flow and motion effects. The excitation matrix A describes the forward model of the excitation process in the small tip-angle approximation and is composed of k-space locations and the discrete spatial and spectral frequencies f1,m and f2,m, respectively. This matrix is given by

Am;n ¼ exp ð2piðf1;m k1;n þ f2;m k2;n ÞÞ:

Fig. 1. Principles of spectral–spatial (SPSP) excitation pulses. The pulse coefficients are shown on top, while the corresponding excitation profile is shown below. The overall envelope of the pulse determines the spectral domain, while the individual sublobe determine the spatial dimension. Different metabolites and slices can be acquired in subsequent excitations by modulating the frequency accordingly.

ð2Þ

ð3Þ

The spatial k-space locations are obtained by integrating the Rt 0 zig-zag gradient modulation k1;n ¼ 2cp T Gðt0 Þdt . The spectral k-space locations for a pulse of duration T are given by k2,n = T  t. The kspace modulation follows the convention in [12] and is shown in Fig. 2 for both bidirectional and flyback gradient modulation. The sampling points along the spatial and spectral frequencies f1,m and f2,m have to approximately fulfil the Nyquist criterion. Because the k-space is not equidistant and because of the undefined transition region, oversampling is required. The sampling density used in this work is slightly non-uniform, placing more sampling points f1,m and f2,m within the central part of the excitation profile and near the transition region in order to move some of the fitting errors to the sides. Three general-purpose SPSP pulses were designed for metabolic imaging of [1-13C]pyruvate and its downstream metabolites: (1) a spectrally shifted and (2) centred profile both with bidirectional gradient modulations, and (3) a spectrally centred profile with flyback gradient modulation. The requirements for this metabolic imaging application are that the spectral profile is suitable for exciting a single resonance of any of the five present metabolites (pyruvate, lactate, alanine, bi-carbonate and pyruvate-hydrate)

Communication / Journal of Magnetic Resonance 235 (2013) 115–120

spatial

bidirectional k

k−space

117

trajectory and a quadratic phase along both dimensions, which in combination with a spiral gradient trajectory could enable even faster chemical-shift imaging (CSI). The amount of quadratic phase chosen according to Eq. 19 in [9] is given by

kK

3:6 ð2p  BWÞ2  FTW

:

ð4Þ

flyback kspatial

The quadratic phase is multiplied onto the desired excitation profile bm. In case of the Cartesian SPEN pulse it is applied only along the phase-encoded dimension, while in case of the spiral SPEN pulse it is applied along both dimensions with the same limits as given in Eq. 4. 2.3. Pulse design by direct matrix inversion

kspectral Fig. 2. Different k-space trajectories for SPSP pulses with bidirectional (top) and flyback (bottom) gradient modulations. Bidirectional gradients are more samplingefficient but they are also more sensitive to gradient delays as compared to flyback gradient modulation. Furthermore, bidirectional gradients lead to a non-equidistant k-space distribution, and hence, the first sidelobe artefact at 21p cannot be completely eliminated as is the case for flyback gradients.

with minimal contamination from the other resonances. Furthermore, it is required to maintain a sufficiently large spectral passband against typical B0 offsets. The most closely spaced resonances are those corresponding to alanine and pyruvate-hydrate with a 90-Hz shift difference (B0 = 3 T). Typical B0 variations in the abdominal regions of rats at 3 T on proton frequency (128 MHz) are up to 120 Hz, which translates to 30 Hz for 13C spins due to the smaller gyromagnetic ratio. Therefore, a suitable compromise for the SPSP pulse design was to use a spectral bandwidth of 85 Hz and fractional transition width of FTW = 0.7. Sideband artefacts were chosen to alias into spectral areas with minimal contamination of other metabolites. The sampling frequency for fitting is given by one over the respective sampling times plus some oversampling in the spectral dimension to compensate for the non-equidistant kspace. For the spatial dimension, a sampling time of 16 ls (62.5 kHz) is used to reduce matrix size. It is later interpolated to the actual scanner sampling time of 2 ls. For the spectral dimension, the sampling times (i.e., duration of each sublobe) are s = 1.12, 1.12 and 1.216 ms for the shifted, centred and flyback pulses, respectively. In combination with an oversampling factor of 1.25, this leads to sampling bandwidths of 1116, 1116 and 1028 Hz, respectively. The number of discrete spatial and spectral frequencies (1, m) and (2, m) is given by the number of points for a sublobe and the number of sublobes times an oversampling factor of 8 and 20, respectively. For a well-posed, physically realisable problem, the linear least-squares fit is stable and yields the optimal solution. The profile beyond the sampling frequency is simply a repetition of the sampling field of view. If oversampling factors for the sampling frequency or the number of frequency points are too low, the fit becomes instable, leading to large and fluctuating RF coefficients and large errors in the profile.

2.2. SPEN pulses Two different kinds of quadratic-phase RF pulses were designed for SPEN Fig. 5. A more conventional pulse with a Cartesian grid is shown on the left in Fig. 5. The quadratic phase is applied along the slower, phase-encoded dimension, also referred to as the SPEN direction [11]. The sublobes are again used for spatial selection. Shown on the right side in Fig. 5 is a pulse with a spiral gradient

The small tip angle approximation is used to design pulses in this work. The exemplary pulses are used with various flip angles, typically with u  10°  20°, but always with u 6 90°. Therefore, the error in the profile due to violation of the small tip-angle approximation is negligible, particularly considering the low time-bandwidth products and hence resulting in poor profiles. With the excitation matrix Am,n (Eq. 3) and the excitation profile bm (Eq. 2) given, the RF pulse design problem amounts to solving the inverse problem

Aq ¼ b:

ð5Þ

This system is overdetermined (m > n) and can be solved by various approaches, for instance by calculating the Moore–Penrose pseudo-inverse A . In this work, the equation was solved in a linear least-squares sense by computing 1

q ¼ ðA> AÞ ðA> bÞ:

ð6Þ

This is numerically most efficient and because it is well-conditioned and there is no noise, numerical differences are negligible. A regularisation is typically not necessary because Eq. 5 is overdetermined and noiseless. The condition number of matrix A is given by

condðAÞ ¼

rmax rmin

ð7Þ

where rmax and rmin are the maximum and minimum singular values, respectively. A weighting function can be optionally included with the diagonal weighting matrix W



q ¼ A> WA

1 

 A> Wb :

ð8Þ

For the exemplary pulses designed in this work, no weighting was applied, i.e., all diagonal element of W were equal to one. Typical computation times solving the problem using Eq. 6 are a few seconds on standard computers. 2.4. Experimental All designed pulses were verified by solving the Bloch equations in an efficient manner using spinor notation with SU (2) rotation matrices [13] for a flip angle of u = 90°. The three SPSP pulses were implemented on a 3 T GE HDx whole-body scanner (GE Healthcare, Milwaukee, WI, USA) with gmax = 40 mT/m and smax = 150 T/m/s. For multi-slice excitation, the gradient trajectory was used to modulate the frequency board as well. The profiles were measured in an NMR test tube filled with water doped with 2:100 parts of Dotarem (Guerbet, Villepinte, France) with the test tube aligned along the z direction. Multi-slice excitation and the readout gradient were both set along the same spatial direction as the phantom (z), leading to a direct measurement of the profile. The scanner excitation frequencies were stepped through from 256 to

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+256 Hz in steps of 4 Hz in different excitations. The pulse design and Bloch simulation code was implemented in Matlab (MathWorks, Natick, MA, USA).

The k-space trajectories for both bidirectional or flyback gradient modulations are depicted in Fig. 2. Each gradient modulation has different properties with specific advantages and disadvantages. Bidirectional gradients are more sampling-efficient; however, they are more sensitive to gradient-RF delay times. This delay time has to be accurately calibrated to within 2 ls for bidirectional gradient modulations, as otherwise sidelobe artefacts are exacerbated. Fortunately, this delay time remained constant over time and different orientations on the used MRI scanner. The flyback gradient modulation is more robust against errors in gradient delay times, but it is less sampling efficient requiring longer pulses and/or larger minimum slice thicknesses. The maximum error (ratio stopband maximum to passband maximum) as simulated with the Bloch equations is 5.6%, 30.3% and 5.5% for shifted, centred and flyback pulse, respectively. For the measurements the respective values are 8.0%, 26.5% and 8.6%, showing good agreement with the simulations, although the sidelobe artefacts start to disappear in the noise for the smaller values. The main advantage of the direct design approach is the better suppression of the sidelobe artefacts as compared to the separable design approach. This is due to the fact that the separable design

3. Results and discussion 3.1. SPSP The three exemplary SPSP pulses are shown in Fig. 3. The simulations with the Bloch equations closely match the experimental measurements, also in the off-isocentre slices (data not shown). The spectrally shifted and centred pulses exhibit a minimum slice thickness of 8 mm, while the flyback pulse has a minimum thickness of 14 mm (for 13C nuclei with gmax = 40 mT/m, smax = 150 T/ m/s). This translates to slices that are approximately four times thinner for proton than for carbon spins. The pulse durations are 17.5, 24.2 and 22.4 ms for the shifted, centred and flyback pulses, respectively. A short pulse duration is important to minimise relaxation, flow and motion effects. The condition numbers of matrix A are 7.1, 4.5 and 7.2 for the shifted, centred and flyback pulses, respectively.

Centred

Flyback

grad rf

Shifted

5

10

15

5

10

15

20

5

10

15

20

Measurem.

Bloch Sim: z [mm]

time [ms] −20

0

20

−20

0

20

Profile/spectra

1

0.5

0 −1000

0

1000

0

1000

0

1000

freq [Hz] Fig. 3. Different SPSP excitation pulses and their profiles. The pulse shapes with RF and gradient modulations are shown in the top, the simulated and measured profiles in the middle and the spectral profiles in the bottom rows. The SPSP pulse is an all-purpose pulse to selectively excite a single resonance from the [1-13C]pyruvate spectrum with its downstream metabolites lactate, alanine and bicarbonate. It is all-purpose in the sense, that it can be used to excite any of the metabolites with a good suppression of the other resonances. The shifted spectra are shown in green, the cross-section through slice centre of the profile in blue, and the maximum along the spatial dimension in red. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Separable design

approach does not take the actual k-space trajectory into account, but approximates it with perpendicular trajectories, thereby neglecting the fact k-space lines are tilted due to temporal evolution Fig. 2. This deviation leads to additional errors in the

−20 −10 0 10 20

Matrix inversion

−30 −20

excitation profile, particularly near the positions of the sidelobes. For the regular centred pulse Fig. 3 middle the first sidelobe artefact (at 21s) cannot be completely removed due to the non-equidistant k-space sampling Fig. 2. However, the direct design enables a greater degree of artefact suppression, as shown in the direct comparison to a pulse generated with the separable design in Fig. 4. One approach to achieve a wider stopband in the separable design is to invert every other RF lobe [14]. In the direct design this is achieved by shifting the target profile to the location of the first sidelobe artefact Fig. 3 left, however achieving higher suppression levels at zero frequency due to full consideration of the non-equidistant k-space Fig. 2, hence obviating the need for applying a separate correction function [7]. For the pulse with the flyback gradient trajectory, it is possible to completely eliminate the sidelobe artefact with the direct design approach due to equidistant k-space sampling Fig. 2.

−10

3.2. SPEN

0 10

The two exemplary SPEN pulses are shown in Fig. 5. On the left is the 2D spatial pulse with Cartesian k-space sampling and a quadratic phase overlaid to the horizontal direction. On the right is a 2D spatial pulse designed with a spiral gradient trajectory and a quadratic phase along both spatial dimensions. For this pulse, the pulse coefficients are not smooth mainly because the target profile bm is rectangular. This is in contrast to designing a pulse with a circular response function, which exhibits smooth pulse coefficients. When the fit problem is not sufficiently well posed, errors are

20

−1000

−500

0

500

frequency [Hz] Fig. 4. Comparison of profiles obtained using the separable design (top) and the   direct 2D design (bottom). The first sidelobe 21s can be reduced but not completely eliminated because the bidirectional trajectory leads to a non-equidistant k-space (see Fig. 2).

1

|B | [μT]

20 1 10 0.5

∠B1 [rad]

0

0

2

2 1

0

0 −2 −1

G [mT/m]

40

20

20 0

0

−20 −20

−40 0

5

10

time [ms]

15

20

0

5

10

time [ms]

Fig. 5. 2D spatial pulses with an overlaid quadratic phase, left with a Cartesian and right with a spiral gradient trajectory.

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appearing in the form of a large spike at the end of the pulse (centre of k-space) and large and fluctuating pulse coefficients. The condition numbers for matrix A were 13.2 and 3.2 for Cartesian and spiral SPEN pulses, respectively. As shown again with these two exemplary quadratic-phase pulses, the direct 2D design approach enables easy pulse tailoring and adds flexibility to the RF pulse design. 4. Conclusion A direct design of 2D RF pulses was introduced in this work. The design is based on an explicit matrix representation of the entire pulse design problem and the determination of a direct solution through matrix inversion. This approach is flexible, simple to implement and can decrease artefacts. The method is demonstrated by designing exemplary SPSP and SPEN pulses. Acknowledgments This work was partly funded by Bundesministerium für Bildung und Forschung (BMBF) Grant Nos. 01EZ0827, 01EZ1114 and 13EZ1114. The authors are responsible for the contents of this publication. The authors would like to kindly acknowledge William Grissom, Adam Kerr, Marion Menzel, Martin Janich and Jonathan Sperl for fruitful discussions, and Guido Kudielka for technical support. References [1] J.M. Pauly, D.G. Nishimura, A. Macovski, A k-space analysis of small-tip-angle excitation, J. Magn. Reson. 81 (1) (1989) 43–56.

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