Mechanical resonance

TL;DR

• check_mech_resonance() compares a windowed amplitude spectrum of the physical-axis gradient waveforms, after each block’s rotation and then the rotation argument, with the forbidden bands of a supplied table.

• Each window of window_width (40 ms by default) is mean-subtracted, multiplied by a Hann taper, zero-padded and transformed with a real FFT. The amplitude is scaled so that a sustained sinusoid of amplitude \(A\) at a bin frequency reads \(A\), in mT/m.

• A train of trapezoids of alternating polarity at echo spacing \(\Delta t\) has its fundamental at \(f = 1/(2\,\Delta t)\), with an amplitude between \(8/\pi^2\) and \(4/\pi\) of the plateau amplitude. Echo spacing is therefore the parameter a forbidden band constrains for an echo-planar readout.

• A band is violated by each window whose largest amplitude on a bin inside the band exceeds the threshold on any axis the band applies to. The threshold is the band’s tolerance where it is positive and min_threshold (10 mT/m by default) otherwise.

• The criterion is spectral and is applied to the commanded waveform. It does not model the coil’s transfer function or the acoustic output of the assembly, and a passing result does not establish that a sequence is quiet or within any acoustic-noise regulation.

The Lorentz force on a gradient coil in the static field is proportional to its current, and the gradient assembly has narrowly resonant mechanical modes. Sustained gradient drive at a mode’s frequency produces acoustic output and mechanical stress well above those at other frequencies.[1] Vendors state the frequency ranges that must not be driven as forbidden bands. check_mech_resonance() compares a windowed amplitude spectrum of the physical-axis gradient waveforms with those bands.

Windowed amplitude spectrum

Each physical axis is sampled at the centres of the sequence’s own gradient raster, after each block’s rotation and then the rotation argument. A window of window_width (40 ms by default) starts every stride (half the window by default); the last window is zero-filled to the end of the sequence. Each window is mean-subtracted, multiplied by a Hann taper \(w\), zero-padded to frequency_oversampling times its length and transformed with a real FFT. The amplitude of bin \(k\) is

\[ A_k = \frac{2\,|X_k|}{\sum_n w_n}, \]

so a sustained sinusoid of amplitude \(A\) at a bin frequency reads \(A\), in mT/m. The window length sets the time scale over which drive counts as sustained; a single transform of the whole sequence would not distinguish one brief crossing of a band from drive held inside it. Mean subtraction removes the constant component, the taper limits leakage of low-frequency content into distant bins, and zero-padding interpolates the spectrum without adding information. mech_resonance_spectrum() returns the spectrum of one window from the same pass.

Spectra of echo-planar and spiral readouts

A train of trapezoids of alternating polarity at echo spacing \(\Delta t\) is periodic, with fundamental

\[ f = \frac{1}{2\,\Delta t} \]

and harmonics at odd multiples. The fundamental’s amplitude lies between \(8/\pi^2\) (triangular) and \(4/\pi\) (square) of the plateau amplitude; a 500 µs echo spacing places it at 1 kHz for the length of the train.

The readout-axis spectrum of a 40 ms window at the middle of two sequences, both 64 x 64. The spoiled gradient echo spreads its power below a few hundred hertz; the single-shot echo-planar train concentrates it in one line at the reciprocal of twice its echo spacing, several times higher than anything the gradient echo reaches.

Echo spacing is therefore the parameter a forbidden band constrains for an echo-planar readout. A spiral readout sweeps its instantaneous frequency and spreads its power over a range; the window length determines whether the interval the sweep spends inside a band is resolved.

Bands and thresholds

A ForbiddenBand is an axis (None for all three), an inclusive frequency range in Hz and a tolerance in mT/m. The threshold is the tolerance where it is positive and min_threshold (10 mT/m by default) otherwise. A band is violated by each window whose largest amplitude on a bin inside the band exceeds the threshold on any axis the band guards; a band narrower than one bin is read at the bin nearest its centre. The report states each band’s worst window whether or not it violates.

Source read by read_forbidden_bands()	Band	Axis	Tolerance
Siemens .asc acoustic resonances	centre ± bandwidth / 2	all	0, so min_threshold applies
GE epiesp.dat forbidden echo-spacing ranges	\(1/(2\,\mathrm{ESP})\) over the range	x, y or z, by table section	plateau amplitude, G/cm converted to mT/m

The epiesp.dat tolerance is a plateau amplitude and is compared unscaled with the spectral amplitude, which for an alternating train is between \(8/\pi^2\) and \(4/\pi\) of the plateau.

Limitations

The criterion is spectral and is applied to the commanded waveform. It does not model the coil’s transfer function, the acoustic output of the assembly or the scanner’s own predownload assessment, and a passing result does not establish that a sequence is quiet or within any acoustic-noise regulation. It states that no window’s gradient amplitude spectrum exceeds a threshold inside a band of the supplied table.

See also

• check_mech_resonance(), mech_resonance_spectrum(), read_forbidden_bands() and ForbiddenBand — the calls.

• Constraint checks — running the check over a sequence and reading its report.

References

[1]

Hedeen RA, Edelstein WA. Characterization and prediction of gradient acoustic noise in MR imagers. Magnetic Resonance in Medicine. 1997;37(1):7–10. doi:10.1002/mrm.1910370103.