Magnetic resonance imaging (MRI) is limited in many cases by long scan times and low spatial resolution. Recent advances in gradient systems hardware allow very rapid imaging sequences, such as steady-state free precession (SSFP), which has repetition times (TRs) of 2-5 ms. The design of these rapid sequences demands time-optimal preparatory gradient waveforms to provide maximum readout duty-cycle, and preserve spatial resolution and SNR while keeping TRs low. Time-optimal gradient waveforms can be synthesized analytically for certain simple cases. However, certain problems, such as timeoptimal 2D and 3D gradient design with moment constraints, either may not have a solution or must be solved numerically. We show that time-optimal gradient design is a convex-optimization problem, for which very efficient solution methods exist. These methods can be applied to solve gradient design problems for oblique gradient design, spiral imaging, and flow-encoding using either a constant slew rate or the more exact voltage-limited gradient models. Ultimately, these methods provide a time-optimal solution to many 2D and 3D gradient design problems in a sufficiently short time for interactive imaging. Scan times in magnetic resonance imaging (MRI) are generally limited by either the signal-to-noise ratio (SNR) or the gradient amplitude and speed. SNR limitations are being met by advances in higher-field systems, as well as improved receive coils. Additionally, many advances are being made in gradient systems to allow rapid imaging with high spatial resolution. These advances include increased switching rates and gradient amplitudes. Also, as the demand for continuous rapid imaging increases, gradient amplifiers are being made more resilient to heating limitations. Acquisition and reconstruction hardware has also improved in response to the development of parallel and real-time imaging techniques.These advances in MR system hardware have enabled the use of new rapid pulse sequences. Typical rapid sequences include rapid gradient-echo (fast low-angle shot (FLASH), gradient-recalled acquisition in steady state (GRASS)) (1), and multi-echo spin-echo (e.g., turbo spinecho (TSE), fast spin-echo (FSE), and rapid acquisition with relaxation enhancement (RARE)) sequences (2). Refocused steady-state free precession (SSFP), fast imaging with steady precession (True-FISP), fast imaging employing steady-state acquisition (FIESTA), and balanced-FFE sequences (3,4), which produce high signal and contrast, are becoming common as improved gradients allow imaging with minimal artifacts from off-resonance. All of these sequences demand efficient gradient waveform design. Efficient acquisition methods include echo-planar imaging (EPI) (5) and spiral imaging (6). Aside from imaging trajectories, gradient waveform design includes preparatory waveforms such as phase-encoding, prewinder, and rewinder gradients. In rapid sequences with short repetition times (TRs), the design of these latter gradients is an important consideration for improving imagin...
