Gradient amplitude

Gradient amplitude#

TL;DR

  • check_max_grad() compares the largest per-axis amplitude, after each block’s rotation, with max_grad from the system limits. A nonpositive max_grad disables the comparison.

  • The report also states each axis peak with its 1-based block and the largest simultaneous vector magnitude. Only the per-axis quantity is compared with a limit, because max_grad in a Pulseq system description is a per-axis limit.

  • The axis peaks are in general attained at different times, so their root-sum-square is an upper bound on the vector magnitude, and usually a loose one.

  • A rotation preserves the vector magnitude and changes its per-axis components, so a sequence within max_grad unrotated can exceed it at an oblique prescription. A prescription is evaluated by applying it with TransformFOV and checking the result.

  • A readout traversing \(\Delta k = N/\mathrm{FOV}\) at constant amplitude in a window of duration \(T\) requires \(G = \Delta k/T\) in Hz/m. At fixed field of view and matrix size, halving \(T\) doubles both the receiver bandwidth and the required amplitude.

A gradient amplifier has a maximum output current, and therefore a maximum gradient amplitude on the axis it drives. check_max_grad() compares the largest per-axis amplitude of the sequence with max_grad from the system limits.

Quantity compared with the limit#

The check reconstructs the gradient waveforms after each block’s rotation and evaluates

\[ \max_{t}\;\max_{a \in \{x,y,z\}} |G_a(t)| \;\le\; \mathtt{max\_grad}. \]

The report also states the peak of each axis, with its 1-based block, and the largest simultaneous vector magnitude

\[ \max_{t}\;\sqrt{G_x(t)^2 + G_y(t)^2 + G_z(t)^2}. \]

Only the per-axis quantity is compared with a limit, because max_grad in a Pulseq system description is a per-axis limit. Whether a system also constrains the vector magnitude is a property of its gradient chain; the reported vector peak permits that comparison outside the check. A nonpositive max_grad disables the comparison, and the report is returned regardless.

Simultaneous vector magnitude and per-axis peaks#

The three axis peaks are in general attained at different times, so their root-sum-square is an upper bound on the vector magnitude, and usually a loose one. The reported vector peak is evaluated at each instant.

../../_images/axis_peaks_against_vector.png

A radial gradient echo. The slice-selection lobe reaches 40 mT/m on z while x and y are idle, and the rotated readout reaches 22 mT/m on each of x and y at angles the slice lobe is not played at. The root-sum-square of the three axis peaks is 50 mT/m; the largest simultaneous vector magnitude is 40 mT/m.#

Dependence on rotation#

A rotation preserves the vector magnitude at every instant and changes its per-axis components. In a plane, a vector of magnitude \(|G| \le\) max_grad is within the per-axis limit at every orientation; a longer one exceeds it at some orientations. A sequence within max_grad unrotated can therefore exceed it at an oblique prescription.

../../_images/rotation_against_per_axis_limit.png

The in-plane gradient vector of a Cartesian gradient echo at the instant its magnitude is largest, drawn at prescription rotations 15 degrees apart. Left, a design solved against max_grad: its prewinder and its rewinder-and-spoiler block play the readout and phase-encode axes together, and each reaches the limit, so the vector is \(\sqrt2\) times that amplitude. Only the orientations that leave it on a diagonal of the box keep both components inside. Right, the same prescription solved against max_grad divided by \(\sqrt2\), which fits the vector inside the circle. Below, the largest per-axis amplitude over the whole scan against the prescription angle.#

The check reads the rotations the sequence holds. A prescription is evaluated by applying it with TransformFOV and checking the result; apply_system_derates() returns reduced limits to design against.

Readout amplitude and resolution#

A readout traversing \(\Delta k = N/\mathrm{FOV}\) in an acquisition window of duration \(T\) at constant amplitude requires

\[ G = \frac{\Delta k}{T} \]

in Hz/m, or \(\Delta k/(\gamma T)\) in T/m. At fixed field of view and matrix size, halving \(T\) doubles both the receiver bandwidth and the required amplitude.

See also#