Vincent Gras^{1}, Franck Mauconduit^{2}, Alexandre Vignaud^{1}, Alexis Amadon^{1}, Denis Le Bihan^{1}, Tony Stöcker^{3}, and Nicolas Boulant^{1}

The k_{T}-point technique exploits the
dynamic RF shimming capability of parallel transmission to uniformly excite the
spins. That technique allows homogenizing the flip angle in RF-spoiled sequences
but also the rotation angle in sequences involving non-selective refocusing
pulses. As far as the flip angle is concerned, it has been shown that so-called
universal k_{T}-point pulses can be designed to work robustly on any
subject without having to tailor the pulse to the subject. In this study we propose to extend
the universality concept to refocusing pulses, and to give an experimental
demonstration with 3D fast spin echo brain imaging.

The
SPACE refocusing pulse train (RPT) was built, as proposed in ref. 4, from a
unique scalable k_{T}-point
pulse (RF
coefficients $$$\mathbf{b} \in \mathbb{C}^{N_c\times N_{k_T}}$$$ and k_{T}
locations $$$\mathbf{k} \in \mathbb{R}^{3\times N_{k_T}}$$$, $$$N_c$$$ and $$$N_{k_T}$$$ representing the number of transmit channels
and k_{T} points respectively). The design of that pulse yet was
performed “offline” in two steps using a database of field maps acquired on ten
adults of a previous study^{7}.

1) An optimized symmetric parameterization $$$(\mathbf{k}_{\text{sym}},\mathbf{b}_{\text{sym}})$$$ was first obtained by solving numerically under
temporal symmetry constraints the usual spatial domain magnitude
least-squares problem^{7} $$$\min_{\mathbf{k},\mathbf{b}} C_{\text{STA}}(\mathbf{k},\mathbf{b})$$$, where $$$C_{\text{STA}}(\mathbf{k},\mathbf{b})$$$ is the root-mean-squares (RMS) distance
between the maximum RA of the RPT (the target) and the actual flip angle
profile calculated with the small tip angle (STA) approximation.

2) to better mitigate the effect of the static field heterogeneity which tends to degrade the performance of symmetric solutions especially at high flip angles, that solution was used as the initialization point of a second optimization which consisted in minimizing the scaling error (SE):

$$\text{SE}(\mathbf{k},\mathbf{b}) = \text{RMS} \left(\left|\mathbf{U}(\mathbf{k},\mathbf{b})\mathbf{U}_{\text{STA}}^{-1}(\mathbf{k},\mathbf{b})\right|_{\text{f}} - \textbf{Id} \right)$$

where $$$|\cdot|_{\text{f}}$$$ denotes
the Fröbenius norm and $$$\mathbf{U},\mathbf{U}_{\text{STA}} \in \mathbb{C}^{2\times 2}$$$ denote
the Bloch-simulated RF pulse SU(2) rotation matrix and the STA approximation
thereof. Importantly,
for this optimization, the time-symmetry constraint was disabled and replaced
by the constraint: $$$C_{\text{STA}}(\mathbf{k},\mathbf{b})\le C_{\text{STA}}(\mathbf{k}_{\text{sym}},\mathbf{b}_{\text{sym}})$$$. This
second optimization step aimed at preserving the correct rotation matrix when
the reference pulse is scaled, in order to faithfully implement the nominal RA
train and maintain the CPMG condition. The optimization problems
were solved using the Active-Set algorithm under explicit hardware (peak and
average power) and SAR constraints^{8}.

Using Average Hamiltonian Theory^{9}, it is possible to express the rotation $$$\mathbf{U}$$$ as $$$e^{-i(\mathbf{A}^{(0)}+\mathbf{A}^{(1)}+\cdots)}$$$. Interestingly, the expression for $$$\mathbf{A}^{(0)}$$$ is in perfect analogy with the STA while $$$\mathbf{A}^{(1)}$$$ appears to vanish at pure resonance when the RF pulse is symmetric. At off-resonance (as is the case when static field is not homogeneous), $$$\mathbf{A}^{(1)}$$$ does not vanish and is found to scale with the energy of the RF pulse. That AHT analysis thus explains why in off-resonant voxels $$$\mathbf{U}$$$ can deviate more rapidly from its STA approximation as the RF amplitude is scaled up.

Measurements were performed on five healthy adults on a 7T scanner equipped with the Nova 8TX-32RX head coil. All scans were run under local SAR supervision mode. A T2-weighted SPACE sequence was first run in circularly-polarized (CP) mode with 1-ms square pulses, then with the scaling-optimized k_{T}-point pulse obtained from the above optimization (9 k_{T}-points, total k_{T}-point pulse duration 1.1 ms). Sequence parameters were: TR 3 s, ES 8.6 ms, BW 290Hz/pixel, ETL 117, 0.8×0.8×0.8 mm^{3} voxels, Flip Angle (FA) and nominal signal evolution displayed in Figure 1 and TA 7:47min.

In Fig. 2a, the SE of the square pulse as well as of the symmetric and scaling-optimized solutions are displayed as functions of the nominal RA. As expected, when isolating highly off-resonant voxels, the SE remains small for the scaling optimized solution (Fig. 2b).

In the SPACE images (Fig. 3 and 4), the CP mode exhibits severe signal dropouts in the temporal lobes and the cerebellum. The use of UPs allows excellent signal recovery to make the UHF SPACE sequence“truly” whole brain.

1. Cloos MA, Boulant N, Luong M, Ferrand G, Giacomini E, Le Bihan D, Amadon A. kT-points: Short three-dimensional tailored RF pulses for flip-angle homogenization over an extended volume. Magn Reson Med 2012;67:72–80.

2. Cloos MA, Boulant N, Luong M, Ferrand G, Giacomini E, Hang M-F, Wiggins CJ, Bihan DL, Amadon A. Parallel-transmission-enabled magnetization-prepared rapid gradient-echo T1-weighted imaging of the human brain at 7T. NeuroImage 2012;62:2140–2150.

3. Mugler JP. Optimized three-dimensional fast-spin-echo MRI. J. Magn. Reson. Imaging 2014;39:745–767.

4. Massire A, Vignaud A, Robert B, Le Bihan D, Boulant N, Amadon A. Parallel-transmission-enabled three-dimensional T2-weighted imaging of the human brain at 7 Tesla. MRM 2015;73:2195–2203.

5. Eggenschwiler F, O’Brien KR, Gruetter R, Marques JP. Improving T2-weighted imaging at high field through the use of kT-points. Magnetic Resonance in Medicine 2014;71:1478–1488.

6. Gras V, Vignaud A, Amadon A, Le Bihan D, Boulant N. Universal pulses: A new concept for calibration-free parallel transmission. Magnetic Resonance in Medicine 2017;77:635–643. doi: 10.1002/mrm.26148.

7. Gras V, Boland M, Vignaud A, Ferrand G, Amadon A, Mauconduit F, Bihan DL, Stöcker T, Boulant N. Homogeneous non-selective and slice-selective parallel-transmit excitations at 7 Tesla with universal pulses: A validation study on two commercial RF coils. PLOS ONE 2017;12:e0183562.

8. Grissom W, Yip C, Zhang Z, Stenger VA, Fessler JA, Noll DC. Spatial domain method for the design of RF pulses in multicoil parallel excitation. Magn Reson Med 2006;56:620–629.

9. Warren WS. Effects of arbitrary laser or NMR pulse shapes on population inversion and coherence. The Journal of Chemical Physics 1984;81:5437–5448. doi: http://dx.doi.org/10.1063/1.447644.

Figure 1. Rotation angle evolution of the refocusing pulse train used for the SPACE (solid red) protocol and generated for T1 = 1400 ms, T2 = 40 ms. The overlaid dashed curve display the respective nominal signal evolutions during the application of the refocusing pulse train, after a 90° excitation with π/2 phase offset.

Figure 2. Scaling
error simulations for the 1 ms-long CP pulse (gray), symmetric (red) and scaling-optimized
(blue) 1.1 ms-long 9-k_{T}-point universal pulses. The average error on
all voxels pooled together over the five subjects is shown in a) while b)
returns the same error this time on average over the voxels satisfying |∆B_{0}| > 200 Hz. In a), the CP curve is almost
indistinguishable from the scaling optimized one.

Figure 3. Comparison of the SPACE images
acquired in CP mode (top row) and with pTx UP pulses (bottom row) in all five
subjects in representative sagittal slices. The reception profile of the coil
was not removed. In all subjects, successful signal recovery in the cerebellum
and the occipital cortex is obtained, and in the mid-brain in subject 2.

Figure 4. Coronal SPACE images acquired in
CP mode (top row) and with pTx UP pulses (bottom row). A clear signal
enhancement is observed in the cerebellum and temporal lobes.