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Target specific optimization of the transmit k-space trajectory stack-of-spirals and SPINS for pTx pulse design

Ole Geldschläger¹, Dario Bosch^1,2, and Anke Henning^1,3
¹High-field Magnetic Resonance, Max-Planck-Institute for biological Cybernetics, Tübingen, Germany, ²Biomedical Magnetic Resonance, University Hospital Tübingen, Tübingen, Germany, ³Advanced Imaging Research Center, University of Texas Southwestern Medical Center, Dallas, TX, United States

Synopsis

Four basis transmit k-space trajectories (a single variable density spiral-in, a two stack of variable density spiral-in, a three stack of variable density spiral-in and a SPINS trajectory) were optimized for pTx radiofrequency pulse design in order to match the excitation target pattern. The parameter to be optimized where the parameter of the analytical equations of the basis trajectories. The procedure was tested on local excitation and whole brain-like excitation target patterns. Optimized trajectories enabled considerably improved radiofrequency pulse performance, compared to radiofrequency pulses based on unsuited trajectories. The optimization code is available online as open source (https://github.com/ole1965/workflow_OTUP.git).

Introduction

The parallel transmission (pTx) technique^1,2 is the most promising approach to overcome the inhomogeneity issue of the radiofrequency (RF) field, typical for ultra high field.
Usually a pTx RF pulse is calculated based on a pre-defined transmit k-space trajectory. For a desired target excitation pattern ‘the choice of a suitable transmit k-space trajectory is crucial’³.
In this work, we investigated and optimized four basis transmit trajectories (a single variable density spiral-in^4,5,6 (1SOS), a two stack of variable density spiral-in^7,8 (2SOS), a three stack of variable density spiral-in (3SOS) and a SPINS⁹ trajectory) for pTx RF pulse design. The analytical equations of spirals⁴ and SPINS⁹ were considered in order to optimize the trajectories.
This approach is similar to the works from Davids et al.¹⁰ and Herrler et al.¹¹. However, the former optimized the k-space trajectories by setting control points (instead of the analytical equations), while the latter only optimized SPINS for whole brain excitation.

Methods

A single variable density spiral-in trajectory⁴ can be represented by:
\begin{equation}
k(t) = \lambda t^{\alpha}e^{i\omega t}
\end{equation}
with $$$t = 1,1-\frac{1}{T-1},1-\frac{2}{T-1},...,0$$$ is the time and additional parameters are $$$\lambda, \alpha$$$ and $$$\omega$$$.
For 1SOS the spiral equation was implemented in a cost-function having the equation parameter $$$T,\lambda,\alpha,\omega$$$ and the position $$$z$$$ of the spiral on the $$$k_z$$$-axis as input parameter. Within the cost-function, the spiral $$$k$$$ was created according to these input parameters.
For that trajectory $$$k$$$, a RF pulse was designed with the ‘Spatial domain method’¹². The output parameter of the cost-function was the root mean square error (RMSE) between the flip angle (FA) profile of the designed pulse (Bloch simulated) and the target pattern.
For 2SOS and 3SOS the procedure was analogous with two and three spirals, respectively.
A SPINS trajectory⁹ can be created using
\begin{equation}
k_r(t)=\frac{k_{max}}{1+e^{\alpha(\frac{t}{T}-\beta)}}\\
k_{\theta}(t)=ut\\
k_{\varphi}(t)=vt
\end{equation}
where $$$k_r$$$ is the radial coordinate, $$$k_{\theta}$$$ and $$$k_{\varphi}$$$ are the polar and azimuthal angles and $$$t = 1,...,T$$$ is the time. Additional parameters are $$$k_{max},\alpha,\beta,u$$$ and $$$v$$$.
Similar to 1SOS, 2SOS and 3SOS a cost-function with the SPINS parameter $$$k_{max},\alpha,\beta,u,v$$$ and $$$T$$$ as input parameter was implemented in order to create a SPINS trajectory for each set of parameters and to design a pulse based on this trajectory.
Each basis k-space trajectory was optimized separately with the ‘Particle swarm optimization’¹³, implemented in Matlabs particleswarm function. The particleswarm function was executed 20 times for each cost-function. From the 20 optimized parameter sets, the one that provided the lowest RMSE was set as the optimum. The maximum allowed trajectory duration was 10ms.
The RF pulse design was based on $$$B_0/B_1^+$$$ maps¹⁴ from one human subject measured in advance on a 9.4T whole-body MR scanner (‘Magnetom 9.4T’, Siemens Healthcare, Erlangen, Germany). The utilized in-house-built 16 channel coil¹⁵ consisted of eight transceiver loops.
The optimization procedure was tested on two local excitation and one whole brain-like excitation target pattern (Figure 1). The optimized trajectories were tested in simulations and in vivo.

Results

Figure 2 shows the optimized transmit k-space trajectories.
For targetNuclei all four optimized trajectories provided RF pulses with similar RMSEs, however, the 2SOS result offered the shortest duration (6.50ms). The trajectories for targetM extended further into the outer k-space compared to the targetNuclei trajectories. All targetM results provided similar pulse durations and RMSEs. For targetWB all trajectories exhibited a drastically lower extend in k-space compared to the targetNuclei and targetM trajectories. The 3SOS result constituted the best compromise between duration (3.96ms) and RMSE (0.36) for targetWB.
As Figure 3 shows, for each target the associated optimal transmit trajectory enabled the best performing RF pulse. If the underlying trajectory was not optimized for the respective target, the RF pulse performance diminished.
For each target the optimized trajectory produced (together with the RF pulses) excellent FA profiles (Figure 4). The in vivo results for targetNuclei showed local excitation of the central region of the human brain at 9.4T (Figure 5).

Discussion

Figure 3 verified that the utilization of an optimized trajectory was crucial for the RF pulse performance. If the RF pulse design for a certain target was based on a corresponding optimized transmit k-space trajectory the pulse performance was better compared to a non-corresponding underlying trajectory. Unsuited trajectories diminished the pulse performance.
The optimized k-space trajectories revealed that with increasing complexity of the target excitation pattern the extent of the optimized trajectories in $$$k_x$$$ and $$$k_y$$$ direction in k-space increased. For the least complex targetWB the optimized trajectory ran close to the k-space center, covering only low frequencies. For that reason, the targetWB trajectory is unsuitable for excitation of targetM and vice versa (Figure 3).
The proposed transmit trajectory optimization method can be applied on the $$$B_0/B_1^+$$$ maps of one subject. The optimized trajectory can then be used for RF pulse design on any other subject (Figure 4 and 5). By means of an optimized transmit trajectory the RF pulse performances of universal pulses^16,17 can be improved most likely.

Conclusion

We introduced and validated a new pTx pulse design method that optimizes the transmit k-space trajectory (stack of spiral and SPINS) to best match the excitation target of interest. The workflow code is available online as open source (https://github.com/ole1965/workflow_OTUP.git).

Acknowledgements

Funding by the ERC Starting Grant (SYNAPLAST MR, Grant Number: 679927) of the European Union and the Cancer Prevention and Research Institute of Texas (CPRIT, Grant Number: RR180056) is gratefully acknowledged.

References

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Figures

Figure 1: The three test targets: ‘targetNuclei’ (a central region encompassing the red nuclei should be excited locally with a flip angle (FA) of 7° in eight consecutive transversal slices), ‘targetM’ (the letter ‘M’ should be excited locally with a FA of 7° in one central transversal slice) and ‘targetWB’ (all brain voxels located in 16 consecutive transversal slices should be excited with a homogenous FA of 7° (i.e. there are no voxels which were not allowed to experience excitation)).

Figure 2: Optimized transmit trajectories. In each column, the same basis trajectory (1SOS, 2SOS, 3SOS or SPINS) was utilized, respectively. Each row presents the results for one of the three test target patterns, respectively. For each optimized trajectory the duration of the trajectory (in ms) and the RMSE (in °) that the trajectories enables, is displayed. All plots have the same kx and ky axes limits. The kz axes of the SPINS plots differs from the limits of the kz axes of the SOS plots. The green boxes present the trajectory chosen as the optimum for each target, respectively.

Figure 3: Bloch simulated FA profiles of RF pulses designed for different target patterns, based on different transmit trajectories. In the first row pulses aiming at excitation of targetNuclei were applied. In the first column, the underlying trajectory for pulse design was the optimized trajectory for targetNuclei. In the second (third) column, the underlying trajectory was the optimized trajectory for targetM (targetWB). Analog for row two and three. Below each profile, the corresponding RMSE and the maximum amplitude of each TP is displayed.

Figure 4: The images present simulated FA profiles based on the B₀/B₁⁺ maps from seven different subjects measured at 9.4T. For each target the corresponding optimized trajectory and subject specific tailored RF pulses were applied.

Figure 5: The images from two different subjects are visible. The blue-yellow images in the respective left column show the pulse design masks with the target excitation area (yellow) and non-excitation areas (blue) with a representative transversal, sagittal and coronal slice for targetNuclei. The black-white images are GRE acquisitions (voxel size: 0.8x0.8x0.8mm³) at 9.4T utilizing the optimized trajectory and subject specific tailored pulses for targetNuclei. Only the voxels between the two red horizontal were taken into account for pulse design.

Proc. Intl. Soc. Mag. Reson. Med. 30 (2022)

3318

DOI: https://doi.org/10.58530/2022/3318