The k_T-points parametrization¹ is a powerful technique to mitigate the RF inhomogeneity at ultra-high field (UHF) (≥7T). This method uses a sparse k-space trajectory, thus making the calculations robust, fast and tractable with second-order optimization algorithms². This technique can also be used to achieve homogeneous non-selective inversion profiles with less SAR than adiabatic RF pulses³, thus making the large flip angle (FA) regime an interesting domain of application of the k_T-points technique. However, to fully exploit this technique, it is necessary to optimize the placement of the k_T-points in k-space^4-7, which has been shown in 2D to depend on the imposed constraints⁸. The simultaneous optimization of the RF ($$$b$$$) and k-space trajectory coefficients ($$$k$$$) (rad/m) here is proposed and validated, whereby SAR and power constraints are handled explicitly.

The optimization problem involves the magnitude least squares⁹ cost function:

$$f(b,k)=\|\text{FA}(b,k)-\pi\|_2,$$

and includes 4 sets of constraints: the local and global SAR constraints defined by a set of virtual observation points¹⁰ and the average and peak power constraints, which account for hardware limits. The optimization problem was solved numerically using the Active-Set (A-S) algorithm. The objective function and its gradient were computed with full Bloch simulations by using CUDA on a K40 Nvidia (Santa Clara, CA, USA) graphics processing unit card. The proposed approach was first studied in simulations, based on B₁ maps calculated on a numerical head model and a home-made pTx coil model using HFSS (Ansys, Canonsburg, PA, USA) and on a calculated field offset (ΔB₀) map¹¹. Because the optimization problem is non-convex, the A-S solution can depend on the initialization. The simultaneous optimization of $$$b$$$ and $$$k$$$ was thus run on 10⁴ random initial k-space trajectories ($$$k^{(0)}$$$) contained in a hypersphere of radius 8 m^-1, while the initial RF coefficients ($$$b^{(0)}$$$) were calculated from the initial trajectory using the variable-exchange method^2,9. For comparison, the optimization of the RF coefficients was rerun for the same initial trajectories while keeping $$$k=k^{(0)}$$$ fixed. The two approaches were finally tested experimentally on a spherical water phantom on a Magnetom 7T scanner (Siemens Healthcare, Erlangen, Germany) equipped with an 8 channel pTx system. In this experiment, two RF pulses composed of 7 kT-points were designed, in one case with and in the other case without trajectory optimization, and their respective performance were assessed by using the Quantum Process Tomography (QPT) technique¹².

The histograms of the FA normalized root-mean-square error (NRMSE) are displayed in Fig. 1 for 5, 7 and 9 k_T-points with the optimization of $$$b$$$ only (dashed) and $$$(b, k)$$$ (solid lines). Without optimization of the trajectories, average NRMSEs are 11.4%, 7.8% and 5.9% for 5, 7 and 9 k_T-points respectively, the respective computation times being 9, 23 and 45 s. With simultaneous optimization of $$$b$$$ and $$$k$$$, the average NRMSEs reduce to 6.8%, 4.6% and 3.6% while the computation times reached in this case 26, 76 and 88 s. As expected, increasing the number of k_T-points improves pulse performance, but, more interestingly, 5 k_T points with optimization of the trajectory on average performs better than 7 k_T-points without trajectory optimization. Furthermore, by analyzing separately for 5, 7 and 9 k_T-points the distribution of the NRMSE ratios, it appeared that optimizing the trajectory simultaneously with the RF coefficients reduces on average by 40% the NRMSE. For a robust implementation of trajectory optimization in practice, one unique random initialization is unfortunately not sufficient due to the large NRMSE distribution spread. However, we found that running 4 times the optimization, each time starting with a new random trajectory and taking the best result, has 95% chance to yield a better NRMSE than the distribution’s average NRMSE. Starting from initial k-space trajectories generated by the Fourier or sparsity-enforced methods [6] returned, after further trajectory optimization, NRMSEs significantly better than the average NRMSEs, but still substantially worse than the best NRMSEs. Alternatively, the greedy method⁸ could also be attempted as another initialization technique. The FA map, measured experimentally with QPT, and the corresponding simulations (based on the acquired B₁ and ΔB₀ maps) are displayed in Fig. 2. Experimentally, the inversion pulses reached 11.4% with and 18.7% without optimization of the trajectory, in fair agreement with the FA profile simulations (10.1% and 15.6%).The present study demonstrates, for large FA pulse design, possible substantial improvement in pulse performance with a simultaneous optimization of the k-space trajectory and RF coefficients under explicit constraints, and describes a tractable method to approach nearly optimal solutions in practice.