Improved gradient waveforms for small-tip 3D spatially tailored excitation using Iterated Local Search
Jon-Fredrik Nielsen1, Hao Sun2, Jeffrey A Fessler1,2, and Douglas C Noll1

1Biomedical Engineering, University of Michigan, Ann Arbor, MI, United States, 2Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, United States


We propose a strategy for the joint design of gradient and radiofrequency waveforms for small-tip 3D spatially tailored excitation, that may lead to more globally optimal excitation k-space trajectories. Currently, gradients are either pre-defined or restricted to certain classes such as echo-planar or concentric shells. Our method makes use of a recently proposed optimization method that expresses k-space with a 2nd-order B-spline basis permitting arbitrary k-space trajectories. We employ this method in an Iterated Local Search strategy, and show that this approach reduces the sensitivity of the excited pattern to the choice of initial k-space trajectory that "seeds" the optimization.


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-planar1 or concentric shells2 that permit low-order parametrization. Recently, a more general method for joint gradient and RF design was proposed3, 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 matrix4, and $$$\mathbf{b}$$$ is the RF waveform. The method in [3] alternates between updating $$$\mathbf{c}$$$ and $$$\mathbf{b}$$$, resulting in locally optimal gradients that are not restricted to the heuristic trajectories of previous approaches. However, the gradients obtained depend on the initial k-space trajectory that "seeds" the algorithm. For example, in [3] it was observed that a stack-of-spirals initial trajectory produced a relatively poor excitation (after local optimization) compared to the other initial trajectories tested. Here we propose an extension of [3] that reduces the dependence of the final excitation patterns on the initial (seed) k-space trajectory, possibly leading to designs that are closer to being globally optimal.


Our approach is based on the Iterated Local Search (ILS) method6, 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, SPINS5, 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 cm3; TR=100ms; flip angle 7.5o].


We have proposed an Iterated Local Search strategy for the joint design of gradients and RF waveforms for small-tip 3D tailored excitation, that reduces the influence of the choice of initial (seed) k-space trajectory on excitation accuracy.


No acknowledgement found.


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)