Note
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Multi-echo gradient echo#
A multi-echo gradient echo follows the first readout gradient with further readout gradients of alternating polarity, each with its own ADC event. Each gradient traverses the same k-space line, in the opposite direction to the one before it, so the train forms one echo per gradient and needs no rewinder between them. The echoes sample the line at increasing echo times, which is what lets you map \(T_2^*\) from a single repetition.
In this Tour you build the repetition and then ask how many echoes fit in it. The echo spacing is the duration of one readout gradient, and it depends on the receiver bandwidth through the ramps. You sweep the bandwidth, read the echo spacing and the number of echoes that fit in the repetition time, and see where the gradient amplitude limit stops the sweep.
Prerequisites: lessons 1 and 2 of the course.
Readout train#
A readout gradient traverses one k-space line from one end to the other. A second gradient of the opposite polarity traverses it back, so a train of them needs no rewinder between the echoes and forms one echo per gradient. The even echoes are therefore acquired in reverse order along the line. The readout is designed as in lesson 1, with the amplitude set so that one sample advances k-space by \(1/\mathrm{FOV}\).
import numpy as np
import pypulseqpp as pp
system = pp.Opts(
max_grad=32.0,
grad_unit="mT/m",
max_slew=130.0,
slew_unit="T/m/s",
rf_dead_time=100e-6,
rf_ringdown_time=20e-6,
adc_dead_time=10e-6,
)
FOV = 220e-3
MATRIX = 128
THICKNESS = 5e-3
FLIP_ANGLE_DEG = 20.0
REPETITION_TIME = 40e-3
BANDWIDTH_HZ = 250e3
ECHOES = 6
rf, gz, gz_reph = pp.make_sinc_pulse(
flip_angle=np.deg2rad(FLIP_ANGLE_DEG),
duration=2e-3,
slice_thickness=THICKNESS,
apodization=0.5,
time_bw_product=4.0,
delay=system.rf_dead_time,
system=system,
use="excitation",
return_gz=True,
)
def readout(bandwidth_hz):
"""The readout gradient, its acquisition window and its prewinder.
The dwell time is put on the ADC raster and the flat top on the gradient
raster, which are different rasters, so the flat top is the acquisition
window rounded up rather than equal to it. The amplitude is set so that a
sample advances k-space by ``1 / FOV`` however long the window is.
"""
dwell = pp.round_to_raster(1.0 / bandwidth_hz, system.adc_raster_time)
acquisition = MATRIX * dwell
raster = system.grad_raster_time
gx = pp.make_trapezoid(
channel="x",
amplitude=MATRIX / FOV / acquisition,
flat_time=raster * np.ceil(acquisition / raster),
system=system,
)
adc = pp.make_adc(
num_samples=MATRIX, dwell=dwell, delay=gx.rise_time, system=system
)
# The prewinder cancels the ramp, one step per sample before the echo and
# the half step to the centre of the first sample.
gx_pre = pp.make_trapezoid(
channel="x",
area=-(gx.amplitude * gx.rise_time / 2 + (MATRIX / 2 + 0.5) / FOV),
duration=1e-3,
system=system,
)
return gx, adc, gx_pre
gx, adc, gx_pre = readout(BANDWIDTH_HZ)
gy_pre = pp.make_trapezoid(
channel="y", area=MATRIX / (2 * FOV), duration=1e-3, system=system
)
print(
f"dwell {1e6 * adc.dwell:.1f} us, "
f"readout {1e3 * pp.calc_duration(gx):.3f} ms, "
f"amplitude {1e3 * gx.amplitude / 42.576e6:.2f} mT/m, "
f"slew {gx.amplitude / gx.rise_time / 42.576e6:.0f} T/m/s"
)
dwell 4.0 us, readout 0.960 ms, amplitude 26.69 mT/m, slew 121 T/m/s
Repetition#
The train replaces the single readout block. The gradients of the train are played back to back, so the echo spacing is the duration of one readout gradient, ramps included.
def multi_echo(echoes, bandwidth_hz=BANDWIDTH_HZ, lines=MATRIX):
"""A multi-echo gradient echo with the given train length."""
gx, adc, gx_pre = readout(bandwidth_hz)
spoiler = pp.make_crusher(4.0, FOV / MATRIX, channel="z", system=system)[0]
played = (
pp.calc_duration(rf, gz)
+ pp.calc_duration(gx_pre, gy_pre, gz_reph)
+ echoes * pp.calc_duration(gx)
+ pp.calc_duration(spoiler)
)
seq = pp.Sequence(system=system)
for step in np.linspace(-1.0, 1.0, lines, endpoint=False):
seq.add_block(rf, gz)
seq.add_block(gx_pre, pp.scale_grad(gy_pre, step), gz_reph)
for echo in range(echoes):
seq.add_block(pp.scale_grad(gx, (-1.0) ** echo), adc)
seq.add_block(spoiler)
seq.add_block(
pp.make_delay(
pp.round_to_raster(
REPETITION_TIME - played, system.block_duration_raster
)
)
)
return seq
seq = multi_echo(ECHOES)
ok, errors = seq.check_timing()
print(
f"timing {ok}, {seq.num_blocks} blocks, "
f"{seq.duration()[0]:.3f} s for {ECHOES} echoes on each of {MATRIX} lines"
)
seq.paper_plot(tr=1)
timing True, 1280 blocks, 5.120 s for 6 echoes on each of 128 lines
Echo spacing against receiver bandwidth#
The readout gradient has to cover the same area whatever the bandwidth, so a shorter flat top is a proportionally stronger gradient, until the amplitude limit is reached and the flat top can shorten no further. The echo spacing follows the flat top, and the train that fits in a repetition follows the echo spacing.
# The dwell time is the quantity the ADC raster quantizes, so the sweep is over
# dwell times and the bandwidth is read from them.
DWELLS = np.array([16e-6, 12e-6, 10e-6, 8e-6, 6e-6, 4e-6, 2e-6])
BANDWIDTHS = 1.0 / DWELLS
overhead = (
pp.calc_duration(rf, gz)
+ pp.calc_duration(gx_pre, gy_pre, gz_reph)
+ pp.calc_duration(
pp.make_crusher(4.0, FOV / MATRIX, channel="z", system=system)[0]
)
)
spacing = []
for bandwidth in BANDWIDTHS:
try:
gradient = readout(bandwidth)[0]
except ValueError:
spacing.append({"bandwidth": bandwidth, "realizable": False})
continue
spacing.append(
{
"bandwidth": bandwidth,
"realizable": True,
"spacing": pp.calc_duration(gradient),
"ramps": gradient.rise_time + gradient.fall_time,
"amplitude": gradient.amplitude / 42.576e6,
"echoes": int((REPETITION_TIME - overhead) // pp.calc_duration(gradient)),
}
)
realizable = [row for row in spacing if row["realizable"]]
bandwidth amplitude spacing ramps echoes
62 kHz 6.67 mT/m 2.180 ms 6% 15
83 kHz 8.90 mT/m 1.700 ms 9% 20
100 kHz 10.68 mT/m 1.480 ms 14% 23
125 kHz 13.35 mT/m 1.280 ms 19% 27
167 kHz 17.79 mT/m 1.060 ms 26% 32
250 kHz 26.69 mT/m 0.960 ms 46% 36
500 kHz beyond the amplitude limit
The acquisition window falls as the reciprocal of the bandwidth; the echo spacing does not. A shorter window at the same k-space extent is a stronger gradient, and a stronger gradient takes longer to ramp, at both ends of every echo. Over the sweep the window shortens fourfold and the spacing by a factor of little more than two, with the ramps growing from a sixteenth of the echo spacing to nearly half of it. Beyond the last point the amplitude the readout would need is above the limit, and the factory raises an error rather than lengthening the flat top.
The shorter spacing reduces the signal-to-noise ratio. The noise in a sample grows as the square root of the bandwidth, so the fourfold bandwidth of the sweep halves the signal-to-noise ratio of each echo, in exchange for the number of echoes and a shorter shortest echo time.
Specification#
What this Tour built, stated the way you would ask an agent for it:
Write a 2D multi-echo gradient echo with pypulseqpp: 220 mm FOV,
128 matrix, 5 mm slice, 20 degree sinc excitation, TR 40 ms. After
the excitation and the readout and phase prewinders, play a train of
6 readout gradients of alternating polarity at 250 kHz receiver
bandwidth, each with its own ADC event and no rewinder between them,
then a z spoiler. Check the timing. Sweep the dwell time from 16 us
to 2 us and report the echo spacing, the ramp fraction and the
number of echoes that fit in TR, stopping where the readout
amplitude exceeds the system limit.
Total running time of the script: (0 minutes 0.226 seconds)