Synopsis

Keywords: RF Pulse Design & Fields, RF Pulse Design & Fields

This study proposed a novel algorithm for designing multidimensional RF in spin-domain using auto-differentiation in PyTorch. Our algorithm uses spin-domain description of rotations to formulate the RF pulse design into optimization problems. We designed 3D refocusing pulses and verified the performance using spin-echo based sequences in phantom at 1.5T and in vivo in the brain at 3T.

Introduction

The goal of this study is to develop an RF pulse design algorithm in spin-domain for a 3D refocusing pulse used in reduced field-of-view (rFOV) imaging. rFOV imaging techniques excite inner volume to reduce the encoding burden for higher spatial resolution or faster acquisition, compared to the imaging technique for the full FOV. While spin-echo based rFOV imaging can be achieved by using orthogonal RF excitation¹ or 2D spatially-selective RF pulse^2,3, a 3D spatially-selective refocusing pulse could potentially further reduce the encoding burden. However, to design a 3D tailored refocusing pulse is challenging due to the large flip angle when most RF pulse design algorithms use small-tip approximation⁴. While recent work⁵ using an auto-differentiable Bloch simulation demonstrated its potential to design pulses with large flip angles, the objective function using the Bloch equation in designing a refocusing pulse is complicated to formulate by specifying the initial and desired magnetization pattern.
In this study, we propose an algorithm to design multidimensional RF pulses in the spin-domain using the auto-differentiable simulation. We exploit the spin-domain description of rotations to efficiently formulate the refocusing pulse design problem without implicitly specifying the initial magnetization.

Methods

In the spin-domain, the rotation of the magnetization is described by Cayley-Klein parameters $$$\alpha$$$ and $$$\beta$$$, which are complex parameters and can be used to describe the functions of different pulses. The refocusing pulse can be designed by requiring: 1) $$$|\beta^2| = 1$$$ for the refocusing region; 2) $$$\beta^2$$$ have the same phase within the refocusing region.
In this study, We designed RF pulses over a 30x30x30cm³ using a matrix with size of 40x40x40 to target certain ROIs for different experiments. We formulated our RF design as an optimization problem in terms of $$$\beta$$$ using l1 norm: $$\begin{aligned}\mathop{\arg\min}_{\boldsymbol{b},\boldsymbol{g}} \mathcal{L} = \sum_{i=1}^{N} w_i |\beta_{S,i}^2(\boldsymbol{b},\boldsymbol{g}) - \beta_{D,i}^2|, \\ \text{s.t.} \quad \text{hardware limits}\end{aligned}$$ where $$$\boldsymbol{b}$$$ is a matrix of size $$$2\times N_t$$$ representing the RF waveform, $$$\boldsymbol{g}$$$ is a matrix of size $$$3\times N_t$$$ representing the gradient waveforms, and $$$N_t$$$ is the number of timepoints used. $$$N$$$ is the number of spins of different locations, $$$\beta_{S,i}^2$$$ is the simulation result for spin $$$i$$$, $$$\beta_{D,i}^2$$$ is the desired pattern for spin $$$i$$$, and $$$w_i$$$ is the weights assigned to that spin. Our algorithm was constrained by the hardware limitations of the MRI scanner, such as peak B1 of 25 uT, gradient strength of 50 mT/m, and slew rate of 120mT/m/s.
Our simulation for $$$\alpha$$$ and $$$\beta$$$ used the hard pulse approximation with dwell time of 4us. We implemented the optimization using auto-differentiation in PyTorch, which provides the derivative for optimization and makes the computation fast especially when the number of spins are relatively large in our design. We treated $$$\alpha$$$ and $$$\beta$$$ with their real and imaginary components to avoid directly computing the derivative of complex numbers.
Our design started by initializing the RF and gradient using the joint design method for small-tip 3D tailored excitation⁶. Then we jointly optimized the RF and gradient using an alternating scheme: 1) update RF using Frank-Wolfe method; 2) update gradient waveform based on gradient descent with constrained step size. During the update of RF and gradient waveform, we re-estimate desired pattern $$$\beta_{D,i}^2$$$ in the objective by computing the average of simulated $$$\beta_{S,i}^2$$$ within the ROI then scaling the magnitude to 1. This strategy leads to a faster decrease of the objective function.
To evaluate the performance of our algorithm, we replaced SINC refocusing pulse by our 3D refocusing pulse in both 2D and 3D spin-echo sequences implemented in Pulseq⁷. The phantom experiment was performed at Siemens 1.5T with an RF pulse designed to refocus the region of 10x10x5cm³, and in vivo brain imaging was performed at Siemens 3T with a refocusing region of 7x7x6cm³.

Results

Figure 1 shows a comparison of initial input excitation pulse and optimized output refocusing pulse. A visible change on the RF waveform is produced by our algorithm, and the change of gradient waveform can be noticed from their k-space trajectories. Figure 2 shows a comparison of $$$\beta^2$$$ pattern for initial and optimized refocusing pulse based on our simulation. It can be seen that optimized $$$\beta^2$$$ within the ROI have magnitudes close to 1 and have the same phase. Figure 3 shows 2D images acquired in phantom. We compared images acquired with 300x300mm³ with and without 3D refocusing RF pulse, and small FOV=150x150mm² using 3D refocusing RF pulse. Figure 4 shows 3D spin-echo images in phantom at 1.5T with reduced FOV=150x150x70mm³. We also performed experiments in vivo in the brain at 3T. Figure 5 shows these acquired 2D and 3D SE images. This demonstrates our 3D refocusing pulse can allow acquisition of reduced FOV.

Conclusion and Discussion

In this study, we developed a novel algorithm for designing multidimensional RF pulses in spin-domain using auto-differentiation in PyTorch. Our results show the rFOV images can be acquired using our designed refocusing pulse, demonstrating the proposed algorithm provides an efficient way to design 3D refocusing RF pulse. While focusing on the design of the refocusing pulse in this study, our algorithm can also be applied in designing various RF pulses, such as excitation, saturation, and inversion pulses.

Acknowledgements

We receive research support from Siemens Healthineers, NIH grant R01CA208236 and R37CA263583.