.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "generated/gallery/02-tours/03_epi.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_03_epi.py: =================== Echo planar imaging =================== A gradient echo acquires one k-space line per excitation. Echo planar imaging (EPI) acquires the whole matrix after a single excitation. This Tour first follows the readout gradient with further readout gradients of alternating polarity, each forming an echo, and then adds a phase-encode blip between the echoes so that each echo acquires a different line. You then measure the two effects that limit such an acquisition: the signal decay over an echo train tens of milliseconds long, which blurs the image along the phase-encode direction, and the sensitivity of a train of alternating readouts to a delay between the gradient waveform and the acquisition, which produces the half-field-of-view ghost. **Prerequisites:** lessons 1 and 2 of the :doc:`course `. .. GENERATED FROM PYTHON SOURCE LINES 20-30 .. GENERATED FROM PYTHON SOURCE LINES 31-41 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 gradients are played back to back, so the echo spacing is the duration of one readout gradient, ramps included. Every second echo is acquired in the opposite direction, and its samples are in the reverse order of the odd echoes'. .. GENERATED FROM PYTHON SOURCE LINES 41-98 .. 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 = 64 THICKNESS = 5e-3 FLIP_ANGLE_DEG = 90.0 DWELL = 4e-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, ) 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) gx_pre = pp.make_trapezoid( channel="x", area=-(gx.amplitude * gx.rise_time / 2 + (MATRIX / 2 + 0.5) / FOV), duration=5e-4, system=system, ) ECHOES = 6 train = pp.Sequence(system=system) train.add_block(rf, gz) train.add_block(gx_pre, gz_reph) for echo in range(ECHOES): train.add_block(pp.scale_grad(gx, (-1.0) ** echo), adc) .. GENERATED FROM PYTHON SOURCE LINES 99-102 The analysis gives the k-space location of every sample of every acquisition window. Along the readout axis the train is a triangle wave between the two ends of the line, and an echo is where it crosses zero. .. GENERATED FROM PYTHON SOURCE LINES 102-139 .. code-block:: Python k_train, _, t_train_excitation, _, t_train = train.calculate_kspace() kx_train = k_train[0] * FOV / MATRIX * 2 echo_times = np.array( [ t_train[ echo * MATRIX + int(np.argmin(np.abs(kx_train[echo * MATRIX :][:MATRIX]))) ] - t_train_excitation[0] for echo in range(ECHOES) ] ) print( "echo times (ms): " + ", ".join(f"{1e3 * time:.2f}" for time in echo_times) + f"\nspacing {1e3 * np.diff(echo_times).mean():.3f} ms, " f"readout gradient {1e3 * pp.calc_duration(gx):.3f} ms" ) .. image-sg:: /generated/gallery/02-tours/images/sphx_glr_03_epi_001.png :alt: 03 epi :srcset: /generated/gallery/02-tours/images/sphx_glr_03_epi_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none echo times (ms): 1.99, 2.69, 3.39, 4.09, 4.79, 5.49 spacing 0.700 ms, readout gradient 0.700 ms .. GENERATED FROM PYTHON SOURCE LINES 140-145 The samples the analysis reports are not one line acquired six times: the even echoes run from :math:`+k_\mathrm{max}` to :math:`-k_\mathrm{max}`, so a reconstruction has to reverse them before they are lines of the same matrix. The last section measures what a delay between the gradient and the acquisition does to such a train. .. GENERATED FROM PYTHON SOURCE LINES 147-153 Single-shot acquisition ----------------------- A phase-encode blip between the echoes advances the acquisition by one line, and the train runs the length of the matrix. The prewinder on the phase-encode axis starts the train at one edge of k-space. .. GENERATED FROM PYTHON SOURCE LINES 153-180 .. code-block:: Python gy_pre = pp.make_trapezoid( channel="y", area=-MATRIX / (2 * FOV), duration=5e-4, system=system ) blip = pp.make_phase_blip(channel="y", fov=FOV, steps=1, system=system) seq = pp.Sequence(system=system) seq.add_block(rf, gz) seq.add_block(gx_pre, gy_pre, gz_reph) for echo in range(MATRIX): readout = pp.scale_grad(gx, (-1.0) ** echo) if echo == 0: seq.add_block(readout, adc) else: seq.add_block(readout, blip, adc) ok, errors = seq.check_timing() echo_spacing = pp.calc_duration(gx) print( f"timing {ok}, {seq.num_blocks} blocks, " f"echo spacing {1e3 * echo_spacing:.3f} ms, " f"train {1e3 * MATRIX * echo_spacing:.1f} ms, " f"blip {1e6 * pp.calc_duration(blip):.0f} us" ) seq.paper_plot() .. image-sg:: /generated/gallery/02-tours/images/sphx_glr_03_epi_002.png :alt: 03 epi :srcset: /generated/gallery/02-tours/images/sphx_glr_03_epi_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none timing True, 66 blocks, echo spacing 0.700 ms, train 44.8 ms, blip 80 us .. GENERATED FROM PYTHON SOURCE LINES 181-187 k-space trajectory ------------------ One shot, coloured by the rank of each echo in the train: the acquisition starts at one corner of k-space and works across it, reversing direction at every line. .. GENERATED FROM PYTHON SOURCE LINES 187-190 .. code-block:: Python pp.plot.plot_kspace(seq, color_by="echo", plane="xy") .. image-sg:: /generated/gallery/02-tours/images/sphx_glr_03_epi_003.png :alt: 03 epi :srcset: /generated/gallery/02-tours/images/sphx_glr_03_epi_003.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 191-201 Signal decay along the train ---------------------------- The last line is acquired tens of milliseconds after the first, and the transverse signal has decayed over that interval. Along the phase-encode direction the acquired data are therefore the true k-space multiplied by a decaying envelope, and the image is convolved with that envelope's transform. The echo times come from the analysis, and the line each echo lands on from the k-space it reports, so the envelope follows the sequence's own ordering. .. GENERATED FROM PYTHON SOURCE LINES 201-260 .. code-block:: Python k_adc, _, t_excitation, _, t_adc = seq.calculate_kspace() echo_time = t_adc.reshape(MATRIX, MATRIX).mean(axis=1) - t_excitation[0] line = np.round(k_adc[1].reshape(MATRIX, MATRIX)[:, 0] * FOV).astype(int) order = np.argsort(line) T2_STARS = (20e-3, 40e-3, 60e-3, 100e-3) def point_spread(t2_star): """The phase-encode point-spread function the decay produces.""" envelope = np.exp(-echo_time[order] / t2_star) spread = np.abs(np.fft.fftshift(np.fft.fft(np.fft.ifftshift(envelope)))) return spread / spread.max() def width(spread): """Full width at half maximum, in pixels, by linear interpolation.""" above = np.flatnonzero(spread >= 0.5) first, last = above[0], above[-1] left = np.interp(0.5, spread[first - 1 : first + 1], [first - 1, first]) right = np.interp(0.5, spread[last : last + 2][::-1], [last + 1, last]) return right - left widths = {t2_star: width(point_spread(t2_star)) for t2_star in T2_STARS} .. image-sg:: /generated/gallery/02-tours/images/sphx_glr_03_epi_004.png :alt: 03 epi :srcset: /generated/gallery/02-tours/images/sphx_glr_03_epi_004.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none T2* decay over the train point spread 20 ms 0.100 1.51 px 40 ms 0.316 1.21 px 60 ms 0.464 1.13 px 100 ms 0.631 1.08 px .. GENERATED FROM PYTHON SOURCE LINES 261-273 The envelope is not centred on the middle of k-space: the train runs from one edge to the other, so the decay is monotonic across the lines rather than symmetric about the line the echo is on. The result is a point-spread function that is both wider than one pixel and asymmetric: the blurring along the phase-encode direction of an echo planar image. At the shortest :math:`T_2^*` here the signal at the last line is a tenth of the first and the point spread is half again as wide as a pixel; at the longest it is within a tenth of a pixel of the unblurred width. The remedies are a shorter echo spacing, fewer lines per shot (a segmented acquisition), and acquiring fewer lines outright, by partial Fourier or by parallel imaging. .. GENERATED FROM PYTHON SOURCE LINES 275-287 Gradient delay and odd-echo shift --------------------------------- A delay between the gradient waveform and the acquisition displaces every sample along the readout direction by the distance k-space travels in that delay. The readouts alternate in polarity, so the displacement alternates in sign: the odd and the even lines of the matrix are shifted in opposite directions. :meth:`~pypulseqpp.Sequence.calculate_kspace` takes the delay as a parameter, so the displacement is measured from the trajectory the analysis reports rather than computed beside it. .. GENERATED FROM PYTHON SOURCE LINES 287-322 .. code-block:: Python DELAYS = np.array([0.0, 1e-6, 2e-6, 4e-6, 8e-6]) displacement = [] for delay in DELAYS: delayed = seq.calculate_kspace(trajectory_delay=delay)[0] kx = delayed[0].reshape(MATRIX, MATRIX) # The echo of each line, where the ideal trajectory crosses zero. centre = kx[:, MATRIX // 2] * FOV displacement.append( { "delay": delay, "samples": float(np.abs(centre[::2].mean() - centre[1::2].mean()) / 2), } ) .. image-sg:: /generated/gallery/02-tours/images/sphx_glr_03_epi_005.png :alt: 03 epi :srcset: /generated/gallery/02-tours/images/sphx_glr_03_epi_005.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none delay odd-even displacement 0.0 us 0.000 samples 1.0 us 0.250 samples 2.0 us 0.500 samples 4.0 us 1.000 samples 8.0 us 2.000 samples .. GENERATED FROM PYTHON SOURCE LINES 323-348 The displacement is the delay divided by the dwell time, and it is the same for every line, so the odd and the even lines differ only in its sign. A quantity that alternates from one line to the next along the phase-encode direction is, after the transform, an image displaced by half the field of view, which is the ghost an uncorrected echo planar acquisition shows. Measuring the delay and correcting for it are reconstruction steps, outside the scope of this package. Specification ------------- What this Tour built, stated the way you would ask an agent for it: .. code-block:: text Write a pypulseqpp single-shot EPI for a scanner with 32 mT/m and 130 T/m/s, 100 us RF dead time, 20 us ringdown and 10 us ADC dead time. Protocol: 220 mm FOV, 64x64 matrix, 5 mm slice, 90 degree sinc excitation (2 ms, time-bandwidth product 4), 4 us dwell. First play a train of 6 readouts of alternating polarity and locate the echoes with calculate_kspace. Then play 64 readouts with a one-line phase-encode blip between them, check the timing, and plot the k-space trajectory coloured by echo. Compute the phase-encode point-spread function for T2* of 20, 40, 60 and 100 ms from the echo times, and the odd-even k-space displacement for gradient delays of 0 to 8 us with calculate_kspace(trajectory_delay=...). .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.723 seconds) .. _sphx_glr_download_generated_gallery_02-tours_03_epi.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: 03_epi.ipynb <03_epi.ipynb>` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: 03_epi.py <03_epi.py>` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: 03_epi.zip <03_epi.zip>` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_