To improve the fidelity of gradient waveforms using a case-by-case optimization approach, considering both the bandwidth of the gradient system and time-domain limitations.

Accurate gradient time-courses are crucial for MRI, especially in the context of fast imaging acquisitions. However, due to e.g. eddy curents, the gradient system acts as a filter on the input waveforms, thereby causing actual gradient time-courses to deviate from ideal. It is standard practice to pass the gradient input waveforms through preemphasis filters to improve the output waveform fidelity¹. Common to all preemphasis approaches is that they strive to equalize the system response within a given bandwidth. However, as any real-world system, the gradients have a limited range of operation, manifested as amplitude and slew rate limitations. These limits operate on the waveform in the time-domain and must be kept throughout. For any given preemphasis filter serving to boost certain frequencies there will exist valid input sequences that violate the time-domain limits after preemphasis. At the same time, the filter will not be able to deliver the optimal gradient output that the system would be capable of producing for all target gradient waveforms.

We here propose to treat the task of finding an ideal gradient input to achieve a targeted output gradient waveform as a case-by-case optimization problem. This allows for taking the inherent frequency response of the system into account while still ensuring time-domain system limits to be kept. We demonstrate the feasibility of the method by comparing measured gradient output using optimized and non-optimized input waveforms on a commercial MR system.

The proposed gradient input optimization is based on a model of the gradient system as linear and time-invariant, implying that the system can be fully described by its gradient impulse response function (GIRF). For MR imaging applications the time-course of the integral of the gradient waveform, i.e. $$$k(t)$$$, is commonly the crucial parameter. We therefore defined the objective function of the optimization to minimize the L2-norm of the difference between the actual $$$k(t)$$$ and the targeted k-coefficient, $$$k_T(t)$$$: $$min\| \gamma\int_{0}^{t}\int_{-\infty}^{\infty}i(\tau)girf(t'-\tau)d\tau dt'-k_T(t)\|_2$$ subject to the constraints: $$i(t)\leq G_{max}, \frac{\text{d}i(t)}{\text{d}t} \leq S_{max} \quad \forall t$$ where $$$i(t)$$$ is the input waveform, and $$$G_{max},S_{max}$$$ are the system amplitude and slew rate limits, respectively. The objective function describes a constrained quadratic problem, which we chose to solve with an active set algorithm. Optimization was performed based on the measured GIRF of a 3T Philips Achieva system². The measured GIRF was fitted to a rational system transfer function in order to eliminate measurement noise (Fig. 1).

Target k-space sampling patterns were defined as an EPI and an Archimedean spiral with 220x220mm² FOV and 3x3mm² resolution. From the selected k-space traversals, time-optimal target gradient waveforms were obtained using the algorithm described by Lustig et al³, with amplitude and slew rate settings of 25mT/m and 160mT/m/ms, respectively. The integral of the target gradient waveforms yielded the target k(t) for the objective function of the gradient input optimization. The input optimization was performed for each imaging axis (X and Y) separately, setting G_max and S_max to 31mT/m and 200mT/m/ms, respectively. To evaluate the resulting optimized input waveforms, the actual field output to the non-optimized and the optimized gradient waveforms were measured on the MR system using a dynamic field camera⁴.

The input optimization achieved GIRF-predicted gradient output waveforms that were visibly closer to the target compared to using non-optimized inputs (Fig. 2). Sharp corners of trapezoidal gradient lobes were better approximated in the EPI, and the sinusoidal target gradient of the spiral was better matched in amplitude and phase. The error to the target gradient waveform was reduced by about half (Fig. 3). As expected, the error reduction was even more pronounced in k(t), which defined the cost function. The measured k-space trajectories similarly showed that the optimization yielded output that more closely followed the target (Fig. 4). The improvement was especially noticable in bandwidth-limited features, such as the turns of an EPI and in the center of a spiral trajectory. We have here presented a method for achieving optimal gradient waveform fidelity that a given gradient system is capable of producing, considering both bandwidth and amplitude/slew rate limitations. The approach is not limited to gradient systems, but may e.g. be advantageous for driving higher-order shims dynamically, where the trade-off between bandwidth and time-domain capabilites is especially problematic. The method may further find uses in the design of gradient sequences, which currently typically only consider time-domain amplitude/slew rate limitations.