Parallel transmission technology (pTX) [1] helps to improve excitation uniformity and allows for excitation acceleration in ultra-high field MRI. However, the question how to design a 3D volumetric excitation pulse with short duration in order to avoid signal fading due to T2 relaxation remains unaddressed. A related method based on the concept of designing a k-space trajectory limited to within a “k-space trajectory container” prior to the design of a tailored RF (TRF) pulse was proposed by Shao T, et al [2]. However this method was only demonstrated by simulation results and it was impractically time-consuming to find the boundary of the “k-space trajectory container”. In this work, a simplified, direct and fast method of the calculation of the contained k-space trajectory is proposed, and the concept is verified with experimental results at a 9.4 Tesla whole-body human scanner equipped with 8 independent transmit channels.

B1 map and basic imaging settings The B₁+ field maps were acquired with a home-built 8-channel TxRx array head coil [3] and a spherical spectroscopy phantom filled with an aqueous solution of acetate and lactate on a 9.4T whole body human MRI scanner (SIEMENS Healthcare, Erlangen, Germany), using the 3D actual flip angle imaging (AFI) method [4]. The B₁+ field map acquired in CP mode is shown in Figure 1. The desired excitation patterns are a uniform cube (4cm*4cm*4cm), a uniform slab (thickness = 4cm) and a specified “MPI” pattern (thickness = 4cm), all blurred by convolution with a Gaussian kernel of FWHM = 1.2cm. The 3D parallel excitation experiments are based on a fast 3D gradient echo sequence (GRE) (TE 10ms / TR 100ms / resolution 1.72mm*1.72mm*5mm).

K-space trajectory determination Assuming that all the channels fully contribute to the RF modulation during the entire duration of the k-space trajectory, the Point-Spread-Function (PSF) induced by RF-modulation of all channels can be calculated by summing up the absolute values of the inverse FFT of the B₁+ maps from all individual channels. The de-convolution of the RF distribution of the spatial target profile (its inverse FFT) with the PSF leads to a weighting map that demonstrates the distribution of the required deposition of RF energy throughout the excitation k-space. By finding a certain weighting threshold based on the histogram of the weighting map, the boundary of the high weighting area can be used to define the “k-space trajectory container”. As seen in Fig. 2 that corresponds to the excitation patterns for the cube (c-d) and the slab (g-h) the resulting k-space trajectories are restricted to traverse through the pre-defined k-space container.

To maintain the continuity of the k-space sampling and to meet the Nyquist criterion, the k-space trajectory is designed to connect separated parts of the trajectory container with the constraint of being k-space central-point-symmetrically shaped. To efficiently cover the interior space of the “trajectory container”, an acceleration factor of 4 is chosen for the transverse direction and 2 is chosen for the longitudinal direction, according to the dimension of the estimated PSF. In this work we used a stack-spiral trajectory and tailored its shape according to these criteria, as can be seen in Fig 3. Stack-spiral trajectory is chosen because of its traversing efficiency and the accordance of its gradient amplitude and the weighting map distribution.

RF design method The parallel RF pulses are designed based on the iterative small-tip-angle (STA) approach [5] after trajectory optimization. A constraint on maximum RF amplitude is applied during the RF pulse calculation. The pulse duration was limited to a maximum of 16ms considering the ultra-short T2 relaxation time at 9.4T.

Fig 4 shows the experimental results of the aforementioned excitation cases as acquired with a GRE sequence including one pTX excitation pulse, among which (a, c, e) show the transversal views and (b, d, f) the longitudinal views of the excitation profiles: the cube (a,b), the slab (c,d) and the specified “MPI” pattern (e,f). Table I shows the general root-mean-square excitation errors (RMSE), pulse durations, maximum RF amplitudes and RF integrations. The latter two meet the concerns of RF safety. It can be seen that the excitation profiles matched the target profiles as previously described well, which verifies the feasibility of our proposed calculation scheme of the “trajectory container” and hence the determination of the contained k-space trajectory. The intensity drops in the center of the phantom are due to vendor implemented image reconstruction that does not consider receive sensitivities.