Improved gradient waveforms for small-tip 3D spatially tailored excitation using Iterated Local Search

Jon-Fredrik Nielsen^{1}, Hao Sun^{2}, Jeffrey A Fessler^{1,2}, and Douglas C Noll^{1}

The joint design of gradient and radiofrequency (RF) waveforms for small-tip 3D spatially tailored excitation poses a non-linear, non-convex, and constrained optimization problem that remains unsolved. Most existing approaches to tailored RF design either pre-define the gradients and only solve the least-squares RF design sub-problem, or restrict the gradients to certain classes such as echo-planar^{1} or concentric shells^{2} that permit low-order parametrization. Recently, a more general method for joint gradient and RF design was proposed^{3}, in which the excitation k-space trajectory is expressed using a 2nd-order B-spline basis leading to the following constrained optimization problem:

$$ \arg\min_{\mathbf{c},\mathbf{b}} ||\mathbf{A}(\mathbf{c}) - \mathbf{b}||_2^2 + R(\mathbf{b}) \textrm{, subject to gradient hardware constraints,} \qquad \qquad \qquad (1) $$

where $$$\mathbf{c}$$$ is the B-spline coefficients, $$$\mathbf{A}$$$ is the small-tip system matrixOur approach is based on the Iterated Local Search (ILS) method^{6}, an iterative optimization strategy that employs a local optimization routine at each iteration. Our ILS search begins by finding a local optimum $$$k^*_0$$$ starting from some initial k-space trajectory (e.g., stack-of-spirals or concentric shells). We then attempt to sample "nearby" local minima by "perturbing" $$$k^*_0$$$ multiple times to produce a set of $$$N_{test}$$$ candidate seed trajectories $$$k_{1,j}$$$, $$$j=1...N_{test}$$$, for the next iteration (Fig. 1). For each $$$k_{1,j}$$$, Eq. (1) is descended (in parallel on a 32-core desktop computer) using the method in [3] to yield a local optimum $$$k^*_{1,j}$$$, and among all $$$k^*_{1,j}$$$, $$$j=1...N_{test}$$$, the trajectory producing the miminum cost function value is chosen as the value $$$k^*_1$$$ for the next iteration. This procedure is then repeated until convergence. We generate the candidate trajectories $$$k_{i+1,j}$$$, $$$j=1...N_{test}$$$, at iteration $$$i$$$ by expressing the locally optimal trajectory $$$k^*_i$$$ using a reduced number $$$N_{basis,j}$$$ ($$$<N_{basis}$$$) of B-spline basis functions, i.e., by projecting $$$k^*_i$$$ onto a set of $$$N_{test}$$$ lower-dimensional B-spline bases. For a given $$$j$$$ we do this projection by first obtaining an unconstrained least-squares fit $$$\mathbf{c}_j^u$$$ to $$$k^*_i$$$ using $$$N_{basis,j}$$$ 2nd-order B-spline basis functions, and then minimizing the following constrained cost function using Matlab's quadprog function:

$$\arg\min_{\mathbf{c}_j} ||\mathbf{c}_j - \mathbf{c_j^u}||_2^2 \textrm{, subject to gradient hardware constraints.}$$

Here we use $$$N_{basis}=50$$$ and $$$N_{basis,j} = \{ 10,12,14,...,50 \}$$$ (i.e., $$$N_{test}=20$$$, with the last value of 50 corresponding to the unperturbed trajectory). In essence, the $$$k_{i+1,j}$$$, $$$j=1...N_{test}$$$, correspond to smoothed versions of the current iterate $$$k^*_i$$$ with varying degree of smoothness. We evaluated three different k-space initializations: stack-of-spirals, SPINS^{5}, and the extended-KT trajectory from [3]. The target pattern was a 3D cube. RF pulse duration was approximately equal (4ms) for all three k-space initializations. Simulations and experiments were done for single-coil transmission, however the principles introduced here would also apply to parallel transmission. We ran a total of five ILS iterations.

Figure 2 shows the design cost function value (Eq. (1)) versus computation time for both local and the proposed global (ILS) optimization, for the three different k-space initializations. Each curve shows the cost-vs-time for the best local minimum $$$k^*_i$$$ obtained at each iteration. A sharp spike along a curve indicates that the ILS algorithm identified a nearby local minimum with lower cost than the unperturbed trajectory. Stack-of-spirals produces significantly higher cost (poorer excitation accuracy) after local optimization compared to the other two initializations, however after the ILS search all three k-space intializations produce more similar final costs.

Figure 3 shows Bloch simulated excitation patterns obtained with both local and ILS optimization. All three k-space initializations produce similar excitation accuracy only after the proposed ILS optimization, as expected from Fig. 2. Figure 4 validates these excitation results experimentally, for the case of stack-of-spirals initialization [GE 3T scanner; uniform gel phantom; body coil RF transmission; 8-channel headcoil reception; matrix 240x240x48; FOV 24x24x24 cm^{3}; TR=100ms; flip angle 7.5^{o}].

1. Yip CY, Grissom WA, Fessler JA, Noll DC. Joint design of trajectory and RF pulses for parallel excitation. Magn Reson Med. 2007 Sep;58(3):598-604.

2. Davids M, Schad LR, Wald LL, Guérin B. Fast three-dimensional inner volume excitations using parallel transmission and optimized k-space trajectories. Magn Reson Med. 2015 Nov 3. doi: 10.1002/mrm.26021.

3. Sun H, Fessler J, Noll D, Nielsen JF. Joint design of excitation k-space trajectory and RF pulse for small-tip 3D tailored excitation in MRI. IEEE Trans Med Imaging. 2015 Sep 15.

4. Yip CY, Fessler JA, Noll DC. Iterative RF pulse design for multidimensional, small-tip-angle selective excitation. Magn Reson Med. 2005 Oct;54(4):908-17.

5. Malik SJ, Keihaninejad S, Hammers A, Hajnal JV. Tailored excitation in 3D with spiral nonselective (SPINS) RF pulses. Magn Reson Med. 2015 May;67(5):1303-1315

6. Lourenço HR, Martin O, Stützle T. Iterated Local Search: Framework and Applications. Handbook of Metaheuristics, 2nd Edition. Kluwer Academic Publishers, International Series in Operations Research & Management Science 146: 363–397. doi:10.1007/978-1-4419-1665-5_12

Iterated Local Search strategy. The local optimization (red paths) is done by descending Eq. (1) using the method in [3].

Cost function (Eq. (1)) vs. computation time for three different k-space initializations, for both local-only search (dashed curves) and the proposed ILS search (solid curves). When using the proposed ILS approach, all three k-space initializations converge to nearly the same final cost function value.

Simulated excitation patterns for the three k-space initializations (seeds) tested in Fig. 2, after either local search (left) or the proposed ILS search (right).

Experimental validation of the Bloch simulation results in Fig. 3, for stack-of-spirals k-space initialization. Simulated and observed excitation patterns are in excellent agreement.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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