.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "generated/gallery/02-tours/04_multi_echo.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_generated_gallery_02-tours_04_multi_echo.py: ========================= 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 :math:`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 :doc:`course `. .. GENERATED FROM PYTHON SOURCE LINES 21-31 .. GENERATED FROM PYTHON SOURCE LINES 32-41 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 :math:`1/\mathrm{FOV}`. .. GENERATED FROM PYTHON SOURCE LINES 41-119 .. code-block:: Python 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" ) .. rst-class:: sphx-glr-script-out .. code-block:: none dwell 4.0 us, readout 0.960 ms, amplitude 26.69 mT/m, slew 121 T/m/s .. GENERATED FROM PYTHON SOURCE LINES 120-126 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. .. GENERATED FROM PYTHON SOURCE LINES 126-165 .. code-block:: Python 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) .. image-sg:: /generated/gallery/02-tours/images/sphx_glr_04_multi_echo_001.png :alt: 04 multi echo :srcset: /generated/gallery/02-tours/images/sphx_glr_04_multi_echo_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none timing True, 1280 blocks, 5.120 s for 6 echoes on each of 128 lines .. GENERATED FROM PYTHON SOURCE LINES 166-174 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. .. GENERATED FROM PYTHON SOURCE LINES 174-255 .. code-block:: Python # 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"]] .. image-sg:: /generated/gallery/02-tours/images/sphx_glr_04_multi_echo_002.png :alt: 04 multi echo :srcset: /generated/gallery/02-tours/images/sphx_glr_04_multi_echo_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none 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 .. GENERATED FROM PYTHON SOURCE LINES 256-286 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: .. code-block:: text 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. .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.226 seconds) .. _sphx_glr_download_generated_gallery_02-tours_04_multi_echo.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: 04_multi_echo.ipynb <04_multi_echo.ipynb>` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: 04_multi_echo.py <04_multi_echo.py>` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: 04_multi_echo.zip <04_multi_echo.zip>` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_