Christopher Mirfin^{1}, Paul Glover^{1}, and Richard Bowtell^{1}

The Genetic Algorithm (GA) is motivated by the process of natural selection, allowing mutliple initialisations. Due to the stochastic nature of genetic algorithms they are beneficial in avoiding local minima, although they can require significantly more function evaluations to run than a traditional solver. In this work, motivated by the field of shape optimisation, an approach is taken to perform the joint design of RF and gradient waveforms using a GA with a GPU-accelerated iterative solver.

In certain applications, full three-dimensional (3D) spatial excitation is useful. The joint optimisation of both RF and gradient waveforms has been shown to reduce the excitation error in comparison to optimising RF pulses alone$$$^1$$$. Previous joint-methods$$$^{2,3}$$$ typically use local solvers that are susceptible to the initial conditions. The Genetic Algorithm (GA) is motivated by the process of natural selection, allowing mutliple initialisations. Due to the stochastic nature of genetic algorithms they are beneficial in avoiding local minima, although they can require significantly more function evaluations to run than a traditional solver. In this work, an optimisation approach is taken to perform the joint design of RF and gradient waveforms using a GA with a GPU-accelerated iterative solver.

The joint optimisation of RF$$$(\mathbf{b})$$$ and Gradient waveforms (represented by parameters $$$\rho$$$) over $$$N_t$$$ time points is formalised as:

$$\begin{split}\min_{\mathbf{b}\in\mathbb{C}^{N_tN_c},\rho^{3N_{\rho}}}\big\{||A(\rho)\mathbf{b}- &\mathbf{m}_{\text{target}}||^2+\lambda||\mathbf{b}||^2\big\}\\\text{s.t.}|\mathbf{G}_i[j]|&<\mathbf{G}_{\text{max}}\\|\Delta\mathbf{G}_i[j]|&< \mathbf{S}_{\text{max}}\Delta t\\\forall i\in\{x,y,z\},&j\in\{1,...,N_t\}\\\end{split}$$

where $$$A(\rho)$$$ represents the system encoding matrix including off-resonance effects, and $$$N_c$$$ coil-sensitivities. The 3D magnetisation target $$$\mathbf{m}_{\text{target}}$$$ consists of $$$N_s$$$ spatial voxels. The maximum gradient amplitude and slew rates are denoted by $$$\mathbf{G}_\text{max}$$$ and $$$\mathbf{S}_\text{max}$$$ respectively.

We parameterise the gradient waveforms as a linear transformation of $$$N_{\rho}$$$ parameters per axis, where $$$N_{\rho} \ll N_t$$$. As the gradients played out by the scanner hardware can be significantly distorted compared to the input waveforms; we convolve each basis function $$$g_k(t)$$$ with the associated Gradient Impulse Response Function$$$^5$$$ (GIRF) , $$$H(t)$$$ for each axis.

$$G_i(t)=\sum_{k=1}^{N_{\rho}}\rho_k^{(i)}\Big(H^{(i)}(t)*g_k(t)\Big)=\sum_{k=1}^{N_{\rho}}\rho_k^{(i)} \tilde{g}_k^{(i)}(t)$$

We can conveniently precompute the convoluted basis functions and rapidly calculate a new distorted gradient waveform by matrix multiplication.

Equation 1 is cast as an alternating minimisation consisting of a GA outer loop for gradient parameters and a GPU-accelerated conjugate gradient least squares (CGLS) inner loop for RF pulses. 50 parameters per axis were used to define the gradient waveform shapes.To minimise data transfer across the CPU-GPU bridge, all $$$B_1^+$$$, $$$B_0$$$ maps, target profiles, and transformation matrices reside on the GPU in device memory. After each GA update, the new gradient parameters are transferred to the GPU and uncompressed (we used both linear piecewise and discrete cosine transformations (DCT)). The system matrix $$$A$$$ is calculated directly, including any off-resonance terms. We implemented a CGLS solver using high level MATLAB constructs directly on the GPU. CGLS is dominated by two matrix multiplications $$$m=Ab$$$ and $$$b=A^Hp$$$ which are well suited for parallelisation. Only the residual error is returned to the CPU to be used in the next genetic update. An NVIDIA Titan X GPU was used for all work (28 streaming-multiprocessors; 12Gb memory).

The GA was seeded with an initial population of 100 randomly drawn parameter sets and 100 scaled versions of several "classic" trajectories including: Shells$$$^6$$$; extended SPINS$$$^7$$$; EVI$$$^8$$$; stack-of-spirals$$$^9$$$ extended kT-points$$$^{6,10}$$$; cross$$$^{3}$$$; "petal" - a rosette like trajectory$$$^{11}$$$ (see Figure 2). Gradient waveforms for each of the trajectories were generated using the time-optimal method$$$^{12}$$$ and accordingly adjusted to a duration of 6.4ms at a resolution of 6.4$$$\mu$$$s. The maximum allowed gradient amplitude tolerated was limited to 40mTm$$$^{-1}$$$, and maximum allowed slew rate was 150Tm$$$^{-1}$$$s$$$^{-1}$$$.

Bloch simulations were used to validate the performance of the designed waveforms using experimentally-acquired field maps (Nova Medical 8Tx/32Rx head-only RF coil). Two seperate 3D transverse magnetisation targets were defined: a cube of dimension (10cm$$$^3$$$), and a cuboid (5,20,7.5)cm. A local solver was also evaluated using the interior-point algorithm along with the best initial starting point in each case.

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Algorithmic workflow for hybrid genetic algorithm (solving for Gradient waveform parameters and executing on a CPU), and a concurrent Conjugate Gradient Least Squares solver (solving for parallel transmit RF waveforms, executing on a GPU). Data transfer across the slow GPU-CPU bridge is minimised by allocating permanent data arrays on the GPU. The initial gradient parameter population consists of several "classic" k-space trajectories. A local solver is used to refine the end solution from the genetic algorithm.

Examples of "classic" k-space trajectories used to seed initial population: (a) Cross, (b) "petal" a rossete like trajectory, (c) concentric shells, (d) Extended -SPINS, (e) Echo Volumar, (f) Stack-of-spirals. Multiple stretched and contorted versions of each trajectory were also included in the initial population. In each case, the trajectory duration was limited to 6.4 ms.

Magnitude excitation profiles (axial centre slice shown, FOV = 20 cm) for cubic target for active-set solver (a), and genetic algorithm (b). Corresponding magnitude excitation profiles shown for range [0,0.1] (c) and (d).

Example optimised excitation k-space trajectories for cubical target (a) and cuboidal target (d), associated gradient waveforms (b),(e) and parallel transmit RF pulses (c),(f). Corresponds to magnetisation profiles as shown in Figure 3 respectively.