Introduction

Imaging at ultra-high field (≥7T) can be challenging due to the increased inhomogeneity in the radiofrequency and static magnetic fields, $$$B_1^+$$$ and $$$B_0$$$, and restrictive SAR constraints. Parallel transmit (pTx)¹ provides flexibility in the pulse design to overcome these challenges, and can achieve more uniform flip angle distributions. Conventional pulse designs targeting flip angle² ($$$\alpha$$$) are not applicable for design of pulses intended to saturate semisolid magnetization, since the semisolid $$$T_2$$$ is too short for the magnetization to be rotated by normal RF pulses. The recently proposed PUSH^3,4 design framework instead designs pulses based on their root-mean-squared $$$B_1^+$$$ ($$$\beta$$$), which can then be used to directly control saturation of the semisolid. This has so far been applied to design of off-resonance saturation pulses that do not excite the water magnetization directly.
However, magnetization transfer imaging also requires excitation pulses to create images. These can interact with the semisolid magnetization in two ways: (1) these pulses may themselves have non uniform $$$\beta$$$; and (2) non-uniform $$$\alpha$$$ (i.e., rotation of the water magnetization) will transfer to the semisolid pool. These effects are illustrated in Figure 1 which shows simplified simulations of a MT-weighted (MT_w) gradient echo scan with saturation pre-pulse: part A shows the case for a non-uniform saturation pulse and part B shows the effect for an idealized perfectly uniform saturation pulse. Even assuming the latter case (as might be achieved by PUSH) both non-uniform $$$\alpha$$$ and non-uniform $$$\beta$$$ of the excitation pulse can lead to a non-uniform achieved MT ratio (MTR).
In this work we propose a pulse design method that considers both $$$\alpha$$$ and $$$\beta$$$ and apply it to design of excitation pulses for MT imaging at 7T.

Theory

Conventional pTx pulse design optimizes RF and gradient waveforms⁵, $$$\mathrm{b}$$$ and $$$\mathrm{k}$$$ respectively, to obtain a desired flip angle distribution $$$\alpha_{des}$$$. On the other hand, PUSH design optimises RF pulses to obtain a desired $$$\beta$$$ distribution $$$\beta_{des}$$$. In this work we propose a generalized pulse design that considers both flip angle $$$\alpha$$$ and $$$\beta$$$ properties:
$$\begin{equation}\begin{aligned}\mathrm{\left\{\mathbf{\hat{b}},\mathbf{\hat{k}}\right\}=\underset{\mathbf{b,k}}{\arg\min}\left\{\frac{1-\lambda}{{\|\alpha_{des}\|_2}}\|\alpha(\mathbf{b,k})-\alpha_{des}\|_2+\frac{\lambda}{\|\beta_{des}\|_2}\|\beta(\mathbf{b})-\beta_{des}\|_2\right\}}\quad\quad[1]\\\mathrm{s.t.\,\quad{}SAR\,\,constraints\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\,}\\\mathrm{Hardware\,\,constraints\,\,\,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad}\end{aligned}\end{equation}$$
where $$$\lambda{}\in[0,1]$$$ balances between error in $$$\alpha$$$ and $$$\beta$$$. If $$$\lambda{}=0$$$ then Eq.[1] reduces to the conventional pulse design that optimizes just for $$$\alpha$$$. If $$$\lambda{}=1$$$ Eq.[1] reduces to the PUSH design that optimizes just for $$$\beta$$$.
It is necessary to ensure the two targets $$$\alpha_{des}$$$ and $$$\beta_{des}$$$ are not incompatible. Considering a target $$$\alpha_{des}$$$ then the minimum $$$\beta$$$ required to achieve it is
$$\begin{equation}\mathrm{\beta_{min}=\frac{\alpha_{des}}{\gamma\sqrt{TR\,\tau\,p_1^2\,p_2}}}\quad\quad[2]\end{equation}$$
where $$$\mathrm{TR}$$$ is the repetition time, $$$\tau$$$ is the RF duration, and $$$\mathrm{p_1}$$$ and $$$\mathrm{p_2}$$$ are the integrals of the amplitude and squared amplitude of the normalized RF waveform ($$$\mathrm{p_1}=\mathrm{p_2}=1$$$ for a rectangular pulse).

Methods

The proposed pulse design (Eq.[1]) was tested in simulations using k_T-points⁶ pulses with $$$\mathrm{\tau=1ms}$$$, $$$\mathrm{\alpha_{des}=16^\circ}$$$ and for a range of $$$\beta_{des}$$$ and $$$\lambda$$$.
Experiments were conducted on a 7T scanner (MAGNETOM Terra, Siemens Healthcare, Erlangen, Germany) in prototype research configuration, with an 8-TX-channel head coil (Nova Medical, Wilmington MA, USA) using spoiled gradient-recalled-echo sequences (SPGR; Figure 2). A phantom (thermally denatured egg white) was scanned using MT_w and non-MT_w SPGR with $$$\alpha=\{4^\circ,16^\circ\}$$$ and $$$\mathrm{TR=20ms}$$$ to calculate MTR maps. The excitation pulse was designed using (i) circular polarized (CP) mode; (ii) 5 k_T-points $$$\lambda{}=0$$$ (conventional pulse design); or (iii) 5 k_T-points $$$\lambda=0.5$$$. The saturation pulse for the MT_w scans was designed using (i) CP mode or (ii) PUSH applied at an offset frequency $$$\mathrm{\Delta=-2kHz}$$$ and $$$\beta_{des}=1.2\mu{}T$$$ optimized only over a mid-axial slice. $$$B_1^+$$$ and $$$B_0$$$ mapping were done using AFI^7,8 and low flip angle SPGR sequences, prior to pulse optimization which was fully scanner-integrated within Matlab (MathWorks, Natick, MA, USA).

Results & Discussion

Figure 3 shows the errors in $$$\alpha$$$ and $$$\beta$$$ for different k_T-points designs as a function of $$$\lambda$$$. For $$$\beta_{des}<\beta_{min}$$$ the error in $$$\alpha$$$ and $$$\beta$$$ increases rapidly when moving away from $$$\lambda=0$$$ and $$$\lambda=1$$$, respectively. For $$$\beta_{des}>\beta_{min}$$$ it is possible to obtain a good solution for both quantities ($$$\lambda=0.5$$$ – Figure 3D).
Figure 4 shows acquired MTR maps. Use of PUSH (homogenized saturation pulse) led to improved MTR, but the effect of the excitation pulse was still apparent particularly for $$$\alpha=16^\circ$$$ (part B). K_T-points improves the MTR homogeneity, however when $$$\lambda=0$$$ this is compromised by the excitation pulse producing non-uniform $$$\beta$$$ (also illustrated in Figure 3C). Design with $$$\lambda=0.5$$$ corrects this.

Conclusion

This work proposed a pTx pulse design that jointly homogenizes $$$\alpha$$$ and $$$\beta$$$; the latter is not directly influenced by applied gradients while the former is. This approach is necessary for proper correction of $$$B_1^+$$$ inhomogeneity effects in MT_w sequences even if saturation pulses are designed independently with uniform properties. In addition to MT_w imaging, these methods would also apply to quantitative methods such as MPM^9,10 which uses essentially the same sequence structure. Phantom imaging at 7T showed improvement in MT contrast homogeneity from the proposed design compared to the conventional design that only homogenizes the flip angle.

Acknowledgements

The authors wish to thank Alexis Amadon and Franck Mauconduit (NeuroSpin, CEA Paris-Saclay, Gif-sur-Yvette, France) for providing the AFI sequence and reconstruction. This work was supported by the King’s College London & Imperial College London EPSRC Centre for Doctoral Training in Medical Imaging [EP/L015226/1] and by core funding from the Wellcome/EPSRC Centre for Medical Engineering [WT203148/Z/16/Z] and by the National Institute for Health Research (NIHR) Biomedical Research Centre based at Guy’s and St Thomas’ NHS Foundation Trust and King’s College London and/or the NIHR Clinical Research Facility. The views expressed are those of the author(s) and not necessarily those of the NHS, the NIHR or the Department of Health and Social Care.

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