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

At ultra-high field, transmit RF field inhomogeneity mitigation methods exploiting parallel transmission and multi-spoke pulses are readily applicable for brain imaging with small flip angle (FA) protocols. The extension to large FAs yet appears more challenging due to the inherent computationally extensive calculations. Dealing now with multi-slice applications, where specific absorption rate (SAR) issues often come into play, a SAR-aware slice-specific pulse design algorithm can improve significantly the output. This work thus presents a large FA slice-specific multi-spoke pulse design approach enforcing explicitely SAR constraints and, to reduce computations, where the spin dynamics is approximated with Average Hamiltonian Theory.

At ultra-high field, transmit RF field
inhomogeneity mitigation methods exploiting parallel transmission (pTx) and
multi-spoke pulses are becoming mature
for in vivo brain imaging but to date only in the small flip angle (FA) regime^{1-4}.
To handle large FAs, pure numerical approaches consisting of discretizing RF
envelopes into piecewise constant intervals^{5} can be used but, doing
so, the computational complexity is increased dramatically. Additionally, when
dealing with large FA multi-slice 2D-sequences, the user is often confronted to
specific absorption rate (SAR) issues. Here, a SAR-aware slice-specific pulse
design can provide a quasi-optimal solution provided that the spokes parameters
(RF coefficients and spoke locations) of all
slices are optimized within a single
SAR-constrained optimization problem^{6}. The latter, may be written as:

$$\mathbf{JP}:\min_{\left\{{p}_j\right\}}\sum_{j=1}^{M}\left\|\mathbf{A}_j(\mathbf{p}_j)/\alpha_t-1\right\|_2^2$$

where $$$M$$$ denotes the number of slices, $$$\alpha_t$$$ the target FA, $$$\mathbf{p}_j$$$ the parameterization of the RF pulse exciting the $$$j$$$th slice and $$$\mathbf{A}_j$$$ the operator returning the FA profile for all voxels of interest. We thus present a pulse design framework where multiple slice-specific multi-spoke pulses are designed jointly while enforcing explicitly SAR and RF power constraints. In this approach, the action of the RF waveforms is approximated with Average Hamiltonian TheoryFor a constant RF envelope $$$s(t)$$$ (hard pulse) of duration $$$T$$$, the action of the RF pulse at a static field offset $$${\Delta}\!B_0$$$ is described by the rotation matrix $$$R$$$ of angle $$$\theta={\gamma}{T}\sqrt{\left|s(T/2)B_1^+\right|^2+{\Delta}\!B_0^2}$$$ and rotation axis $$$\left[\text{Re}(s(T/2)B_1^+),\text{Im}(s(T/2)B_1^+),{\Delta}\!B_0\right]/\sqrt{\left|s(T/2)B_1^+\right|^2+{\Delta}\!B_0^2}$$$. For a time-varying envelope, the first order AHT approximates $$$R$$$ as $$$e^{-i\gamma{\Delta}\!B_0\sigma_z{T}/4}e^{-i(A^{(0)}+A^{(1)})}e^{-i\gamma{\Delta}\!B_0\sigma_z{T}/4}$$$ where, for a time-symmetric RF envelope, the terms $$$A^{(0,1)}$$$ take the form:

$$A^{(0)}=\frac{\gamma}{2}\left(\sigma_x\text{Re}(s(T/2)B_1^+)+\sigma_y\text{Im}(s(T/2)B_1^+)\right)\times\int_{-T/2}^{T/2}s_0(t+T/2)\cos(\gamma\Delta\!B_0{t})dt,$$

and:

$$ A^{(1)}=\frac{\gamma^2}{4}\sigma_z\left|s(T/2)B_1^+\right|^2\times\int_{0}^{+\infty}g(t)\sin(\gamma \Delta\!B_0{t})dt,$$

where $$$B_1^+$$$ denotes the complex transmit sensitivity, $$$\sigma_{x,y,z}$$$ are the Pauli matrices, $$$s_0(t)=s(t)/s(T/2)$$$ and $$$g(t)=\int_0^T{s_0}(t+u)s_0^{\star}(u)du$$$. Approximating $$$\mathbf{A}_j(\mathbf{p}_j)$$$ by using AHT thus reduces considerably the complexity compared to a numerical Bloch integration because no discretization of $$$s(t)$$$ is required.

To solve **JP** under constraints, the following procedure was used: 1) Choose randomly $$$N_{\text{seed}}$$$ k-space trajectory seeds^{9}, 2) For each trajectory seed and for each
slice separately, solve the slice-specific
problem using the active-set algorithm (A-S) while multiplying the RF pulse duty-cycle by $$$M$$$ for
evaluating the constraints ($$$M\times N_{\text{seed}}$$$ *independent* optimizations), 3)
Select for each slice the best solution out of the $$$N_{\text{seed}}$$$ trials and “merge”
the solutions (slice-by-slice design), 4) Starting from this feasible but likely suboptimal solution,
solve **JP** using A-S.

The proposed pulse design
approach was applied to the Siemens 2D interleaved multi-slice GRE sequence (0.5×0.5 mm^{2} voxels, 25 slices, 2 mm slice thickness, 3 mm slice gap,
TR=2 s, bandwidth=50 Hz/pixel, TE=16 ms, nominal FA=90°, GRAPPA 2, TA~8 min).
That protocol was first applied in the usual circularly polarized (CP)
transmission mode and subsequently with optimized 3-spoke RF pulses. Here the
optimization of the pulses was based upon patient specific field maps which were
measured beforehand^{10}.

A
comparison of the CP- and 3-spoke-GRE images is provided in Figure 1. The
relative signal-to-noise ratio (SNR) comparison obtained by computing the ratio *R* of both images, is shown in Figure 2 together with a simulation of $$$\sin(\alpha_{\text{CP}})/\sin(\alpha_{\text{3-spoke}})$$$, $$$\alpha_{\text{CP}}$$$ and $$$\alpha_{\text{3-spoke}}$$$ denoting the simulated FA profile of the CP
and 3-spoke pulses. Despite the presence of incomplete T_{1}
which can bias the comparison by up to 20%, both images reveal qualitatively similar patterns. The FA simulations of
the optimized 3-spoke pulses by numerical Bloch integration indicate that the
overall FA normalized root mean square error (NRMSE) is below 3%, while none of the 25 slices exceeds 5%
NRMSE. With the proposed method, the computation time required to obtain the 25
pulses was 542 s on a Xeon E5-2690 bi-processor and could be decreased down to
228 s by porting the AHT calculation on a K40 Nvidia (Santa Clara, CA, USA) GPU
card. The precision of the first and second order AHT, reported in Figure 3, indicate that the first order AHT is perfectly adequate for the purpose of designing multi-spoke pulses. Figure 4 compares in simulation the FA profiles obtained with the slice-by-slice approach and the joint design approach. Figure 5 reports, slice-wise, the respective NRMSE and local SAR distributions and describes how performances can be increased with the proposed joint design.

This work demonstrates that the B1 inhomogeneity problem can be mitigated efficiently in a multi-slice near whole-brain coverage UHF protocol involving large FA pulses using a rapid pulse design active-set based algorithm exploiting AHT.

Comparison of the 90° 2D-GRE images obtained in
the usual CP-mode with single-spoke RF pulse (top) and with optimized pTx
3-spoke RF pulses. The 3-spoke excitation allows restoring a 90° excitation
especially at lower slices where the transmit sensitivity of the CP-mode is
several times smaller than at the center of the brain. Note that the image
shown are native images, i.e. are not corrected for the receive profile.

Experimental R-map (left) representative of the
relative SNR between the CP- and 3-spoke-GRE images and simulated sin(α_{CP})/sin(α_{3-spoke}) map. Qualitatively, both images are fairly
similar. Quantitatively, a bias of up to 20% between both ratio maps may appear
due to an incomplete T_{1} relaxation (TR = 2 s ~ T_{1} of gray
matter at 7T).

Relative difference between the numerical
Bloch integration and the zeroth (red), the first (green) and the small tip angle approximation (STA) (blue) in the FA
evaluation plotted against the frequency offset. While the STA and the
zeroth order approximation appear too crude to return a good approximation of **A**_{j}(**p**_{j}), the first order AHT seems
already sufficient for the pulse design intended here.

FA profile Bloch simulation based on the measured B_{1}^{+} and ΔB_{0} profiles for a) 90° 3-spoke RF pulses designed slice-by-slice b) 90° 3-spokes RF pulses designed according to the 4-step procedure described in the Method section (joint design). Several slices display a better excitation uniformity with the proposed joint design.

Slice-specific a) FA NRMSE and b) maximum local SAR profiles obtained with the slice-by-slice (blue curve) and the joint design approaches (red curve). For the slice-by-slice approach, the sum of the maximum local SAR values by construction does not exceed the prescribed limit of 10 W/kg. For the joint design, the maximum local SAR values being not additive (the position of the SAR hotspots indeed is pulse dependent), the latter could be increased at several slices without violating the 10 W/kg limit.