Synopsis

Keywords: System Imperfections: Measurement & Correction, Pulse Sequence Design

MRI systems are usually engineered to give 'ideal' performance, and acquisition methods are generally developed using this assumption. This work proposes an alternative sequence learning framework that includes a model of realistic scanner performance, allowing acquisition sequences to be designed to directly account for system imperfections. In this proof-of-concept demonstration we designed pulse sequences to account for eddy current perturbations with different realistic time constants, while also respecting hardware limits. The flexibility of this approach could be used to design new types of pulse sequence to operate on lower performance, lower cost hardware in future.

Introduction

MRI scanners are usually engineered to give close to ideal performance such that acquisition methods can operate under this assumption. Separate strategies are developed to overcome individual imperfections which can lead to an inefficient use of resources; a voltage overhead is required for pre-emphasis, for example. Here we propose a sequence design framework that explicitly accounts for system imperfections and can learn strategies to overcome them.

We have built on the recently proposed "MRzero" supervised self-learning pipeline¹ that can discover MRI pulse sequences via knowledge of the Bloch equations only. The effectiveness of our approach is shown by optimizing sequences in the presence of eddy currents, with or without applying hardware performance limits.

Theory

Sequences are usually designed by considering a gradient waveform and respecting fixed amplitude and slew rate limits on the gradients themselves. In most scanners, gradient systems are driven under voltage control, often with corrections (e.g. pre-emphasis) as a hidden layer. Thus the designer works with a specified gradient waveform $$$\mathrm{G}(t)$$$ but the gradient amplifier delivers a different hidden voltage waveform $$$\mathrm{V}(t)$$$. Instead we directly formulate the sequence design problem in terms of $$$\mathrm{V}(t)$$$ and connect this to $$$\mathrm{G}(t)$$$ via a physical model. The model could include many effects but here we focus on exponentially decaying eddy currents as in Equation 1:

$$[1]\;\mathrm{G}(t)=\beta\left\{\mathrm{V}(t)-\frac{Δ\mathrm{V}}{Δt}\otimes\mathrm{H}(t)\sum_{n}\alpha_n\exp\left(-\frac{t}{\tau_n}\right)\right\}$$

Here $$$\beta$$$ is the 'sensitivity' of the gradient system (mT/m/V), $$$\alpha_n$$$ and $$$\tau_n$$$ are amplitude and time constants for each eddy current term² and $$$\mathrm{H}(t)$$$ is a unit step function. Physical constraints actually apply to $$$\mathrm{V}(t)$$$ whereas $$$\mathrm{G}(t)$$$ defines the k-space trajectory for imaging, incorporated into forward MR signal simulation within the optimization. The pipeline compared to a more conventional approach is presented in Figure 1.

Methods

Starting from a base sequence with defined initial waveforms $$$\mathrm{V}(t)$$$, the sequence is first simulated assuming perfect system performance (i.e. $$$\mathrm{G}(t)=\beta\mathrm{V}(t)$$$) to determine a 'target' k-space $$$\mathrm{k_T}$$$ and via Bloch simulation, image $$$\mathrm{I_T}$$$. Subsequently, a true forward model is used to predict the actual k-space $$$\mathrm{k_C}$$$ and image $$$\mathrm{I_C}$$$; crucially this image is always reconstructed via iFFT assuming k-space locations $$$\mathrm{k_T}$$$. Optimization of $$$\mathrm{V}(t)$$$ then uses a composite loss function comprising: (i) an image loss term; (ii) a k-distance loss term; (iii) a voltage term; and (iv) a slew rate term. $$$\mathrm{V_{max}}$$$ and $$$\mathrm{S_{max}}$$$ are voltage and slew rate limits respectively and $$$w$$$ are tunable regularization weights.

$$[\mathrm{2a}]\;\mathrm{Total\;Loss}=L_{\mathrm{I}}+L_{\mathrm{k}}+L_{\mathrm{V}}+L_{\mathrm{S}}$$
$$[\mathrm{2b}]\;L_{\mathrm{I}}=w_{\mathrm{I}}\left(\sqrt{\sum\left|\left(\mathrm{I_T}-\mathrm{I_C}\right)\right|^2}\right)$$
$$[\mathrm{2c}]\;L_{\mathrm{k}}=w_{\mathrm{k}}\left(\sum\left(\mathrm{k_T}-\mathrm{k_C}\right)^2\right)$$
$$[\mathrm{2d}]\;L_{\mathrm{V}}=w_{\mathrm{V}}\sum\left(\left|\mathrm{V}(t)\right|-\mathrm{V_{max}}\right)\mathrm{H}\left(\left|\mathrm{V}(t)\right|-\mathrm{V_{max}}\right)$$
$$[\mathrm{2e}]\;L_{\mathrm{S}}=w_{\mathrm{S}}\sum\left(\left|\frac{Δ\mathrm{V}}{Δt}\right|-\mathrm{S_{max}}\right)\mathrm{H}\left(\left|\frac{Δ\mathrm{V}}{Δt}\right|-\mathrm{S_{max}}\right)$$

Optimizations used the Adam³ algorithm with an initial learning rate of 0.001.

Throughout, a spoiled gradient echo sequence with FA = 5° and TR = 10ms was used as the starting point. A numerical brain phantom with feasible B₀, B₁, T₁, T₂, T₂', PD and isotropic diffusion coefficient values was used for signal simulation using a novel phase graph approach⁴. Time was discretized into steps $$$Δt$$$ = 0.1ms, with the first 2ms of each TR reserved for RF pulses (no gradients) and an acquisition window of 3.2ms (0.5ms after the RF pulse). Images were simulated on a 32x32 grid, and all TR periods were simulated to allow for longer-term eddy currents whose effect persists over multiple periods.

An unconstrained scenario (no hardware limits; $$$w_{\mathrm{V}}=w_{\mathrm{S}}=0$$$) was considered with eddy currents having $$$\tau_n$$$ = 0.5ms or 50ms, and together as a bi-exponential perturbation; $$$\alpha_n$$$ = 0.5 was used for all cases. Constrained optimizations using $$$\beta\mathrm{V_{max}}$$$ = 2mT and $$$\beta\mathrm{S_{max}}$$$ = 20mT/m/ms were subsequently explored.

Results and Discussion

Figure 2 shows that for a short time constant eddy current (0.5ms), unconstrained optimization recovers the target image by distorting the input. The optimizer has learnt the expected solution of pre-emphasis. Figure 3 shows equivalent results for longer $$$\tau_n$$$ (50ms); the compensation is larger and not quickly time-varying, as expected. Interestingly, small negative gradients have appeared after the spoiler and spoiler amplitude has changed to mitigate this longer-term eddy current. Figure 4 represents a more realistic scenario of multiple $$$\tau_n$$$ and shows more significant gradient changes that are not simply a linear combination of the results in Figures 2 and 3.

Figure 5 compares results when activating strict hardware constraints. Under these scenarios, pre-emphasis is not possible, and the optimization returns more novel waveforms. When amplitude is limited, extra area is added after the pre-winder to compensate (blue arrow) whilst when only slew rate is limited, gradients ramp more slowly but to a higher amplitude (yellow arrow). For both amplitude constrained cases, the optimization produces a bipolar spoiler that destroys transverse magnetization yet counteracts prolonged eddy current effects (grey boxes). Other innovative strategies include dual-negative lobes pre-readout (red arrow) and post-readout (green box) that ensure that sufficient (though incomplete) pre-winding and spoiling respectively is attained without violating gradient constraints. Each scenario reaches a compromise with low image reconstruction error.

Conclusions

This work presents a framework to model hardware imperfections and overcome them with novel acquisitions, rather than develop separate correction strategies. Consideration of eddy current perturbations provides a simple demonstration of the effectiveness of our approach; optimizations adapt to imperfections with different temporal characteristics. Under strict gradient constraints, a conventional pre-emphasis solution would either force base sequence changes or fail altogether whereas here, acquisition-based solutions are learnt. Future scanner validation of these sequences will be performed.

Acknowledgements

The research was funded/supported by core funding from the Wellcome/EPSRC Centre for Medical Engineering [WT203148/Z/16/Z] and by the National Institute for Health Research (NIHR) Biomedical Research Centre based at Guy’s and St Thomas’ NHS Foundation Trust and King’s College London and/or the NIHR Clinical Research Facility. The views expressed are those of the author(s) and not necessarily those of the NHS, the NIHR or the Department of Health and Social Care.