.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "generated/autoexamples/03-sequence-optimization/03-rf-pulse-design.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 or to run this example in your browser via Binder. .. rst-class:: sphx-glr-example-title .. _sphx_glr_generated_autoexamples_03-sequence-optimization_03-rf-pulse-design.py: ======================================== Designing an RF pulse by optimal control ======================================== The scope of this notebook is to design the samples of an RF pulse directly, by gradient descent through a Bloch simulation of what they do [1]_. A slice-selective pulse is designed for one transmit field. Where B1 varies -- by a fifth either way across a body at 3 T -- the flip inside the slice varies with it. Here a 90 degree excitation is reshaped so that it stays as close to 90 degrees across that range as its samples allow. .. [1] Conolly S, Nishimura D, Macovski A. Optimal control solutions to the magnetic resonance selective excitation problem. IEEE Trans Med Imaging 1986;5(2):106-115. .. GENERATED FROM PYTHON SOURCE LINES 20-24 .. colab-link:: :needs_gpu: 0 !pip install torchsim .. GENERATED FROM PYTHON SOURCE LINES 24-57 .. code-block:: Python import math import torch from torchsim import SequenceDesign, compose_spinor .. GENERATED FROM PYTHON SOURCE LINES 58-70 Pulse and slice --------------- The pulse is ``SAMPLES`` samples long, and under its slice-select gradient each sample turns a spin at ``x`` slice thicknesses from the centre by ``2 pi TBW x / SAMPLES`` about z. Its drive is in radians per sample, so the design is stated without a raster or a gradient amplitude: any pulse of this time-bandwidth product plays it. The starting point is a Hamming-windowed sinc with the area of a 90 degree flip -- a small-tip design, played at a large tip. .. GENERATED FROM PYTHON SOURCE LINES 71-82 .. code-block:: Python SAMPLES, TBW = 128, 4.0 x = torch.linspace(-2.0, 2.0, 161, dtype=torch.float64) turn = 2 * math.pi * TBW / SAMPLES * x t = torch.arange(SAMPLES, dtype=torch.float64) - (SAMPLES - 1) / 2 sinc = torch.sinc(TBW * t / SAMPLES) * ( 0.54 + 0.46 * torch.cos(2 * math.pi * t / SAMPLES) ) start = sinc / sinc.sum() * (math.pi / 2) .. GENERATED FROM PYTHON SOURCE LINES 83-89 Transmit field -------------- Every spin is simulated at five transmit scalings, from 0.8 to 1.2 of nominal. The pulse is one; what it does at each scaling is not. .. GENERATED FROM PYTHON SOURCE LINES 90-101 .. code-block:: Python B1 = torch.tensor([0.8, 0.9, 1.0, 1.1, 1.2], dtype=torch.float64) def excited(real, imag): """``|Mxy|`` after the pulse, from ``+z``: ``(B1, x)``.""" drive = (real + 1j * imag)[:, None, None] * B1[None, :, None] a, b = compose_spinor(drive, turn.expand(len(B1), -1)) return (2 * a.conj() * b).abs() .. GENERATED FROM PYTHON SOURCE LINES 102-109 The cost -------- Inside the slice the magnetisation should be all transverse, outside it untouched; the transition band between is left free. A small penalty on the pulse's energy keeps it from buying flatness with power. .. GENERATED FROM PYTHON SOURCE LINES 110-122 .. code-block:: Python inside = (x.abs() < 0.4).double() outside = (x.abs() > 0.75).double() def cost(real, imag): transverse = excited(real, imag) miss = inside * (transverse - 1.0) ** 2 + outside * transverse**2 energy = (real**2 + imag**2).sum() / (start**2).sum() return miss.sum() / (inside.sum() + outside.sum()) / len(B1) + 1e-4 * energy .. GENERATED FROM PYTHON SOURCE LINES 123-130 Optimized pulse --------------- The real and imaginary parts of every sample are the designed parameters, free of limits: the scanner's peak B1 would be a :class:`~torchsim.Bounded` on them. .. GENERATED FROM PYTHON SOURCE LINES 131-159 .. code-block:: Python design = SequenceDesign( cost, real=start.clone(), imag=torch.zeros(SAMPLES, dtype=torch.float64) ) result = design.minimize(iterations=100, learning_rate=2e-4) real, imag = result.parameters["real"], result.parameters["imag"] with torch.no_grad(): before = excited(start, torch.zeros_like(start)) after = excited(real, imag) .. image-sg:: /generated/autoexamples/03-sequence-optimization/images/sphx_glr_03-rf-pulse-design_001.png :alt: Starting sinc, Optimized, Cost :srcset: /generated/autoexamples/03-sequence-optimization/images/sphx_glr_03-rf-pulse-design_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 160-163 Inside the slice the flip now varies less across the transmit range, most of all where B1 is low, while outside it the leakage stays at the sinc's level: .. GENERATED FROM PYTHON SOURCE LINES 164-171 .. code-block:: Python centre = inside.bool() for label, profile in (("starting sinc", before), ("optimized", after)): mean = profile[:, centre].mean(dim=1) print( f"{label:>14}: |Mxy| in the slice at each B1 {[round(float(v), 3) for v in mean]}" ) .. rst-class:: sphx-glr-script-out .. code-block:: none starting sinc: |Mxy| in the slice at each B1 [0.895, 0.94, 0.967, 0.976, 0.969] optimized: |Mxy| in the slice at each B1 [0.945, 0.981, 0.996, 0.989, 0.963] .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 6.639 seconds) .. _sphx_glr_download_generated_autoexamples_03-sequence-optimization_03-rf-pulse-design.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/firmlab-pisa/torchsim/gh-pages?urlpath=lab/tree/v0.0.8/examples/generated/autoexamples/03-sequence-optimization/03-rf-pulse-design.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: 03-rf-pulse-design.ipynb <03-rf-pulse-design.ipynb>` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: 03-rf-pulse-design.py <03-rf-pulse-design.py>` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: 03-rf-pulse-design.zip <03-rf-pulse-design.zip>` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_