A twisting radial readout module

The previous lesson wrote a Cartesian readout module with trapezoidal gradients. This lesson writes a non-Cartesian readout module: the trajectory is stated as a k-space path, solved into a gradient waveform under the gradient limits, and published as a module.

A radial spoke samples the centre of k-space far more densely than the periphery: at radius \(k\), adjacent spokes of an \(N\)-interleaf set are \(2\pi k / N\) apart, which exceeds the Nyquist spacing \(1/\mathrm{FOV}\) beyond a transition radius

\[k_t = \frac{N}{2\pi\,\mathrm{FOV}}.\]

A twisting radial line [JNM92] departs from the spoke beyond that radius and accumulates azimuth with radius, so that the perpendicular distance between neighbouring interleaves stays at the Nyquist spacing.

The module concept, and the events a module publishes, are described in Sequence modules.

Learning objectives

After this lesson, you should be able to:

• state a twisting radial interleaf as a polyline in k-space from the transition radius \(k_t\);

• solve the path into a time-optimal gradient waveform under the amplitude and slew limits;

• subclass NonCartesianReadout so that a module designs its own interleaf;

• compare the readout duration with that of a constant-density spiral of the same coverage;

• rotate one solved arm per shot with a rotation extension in a scan loop.

The path

The arm is a polyline in k-space: its samples set the geometry and nothing else. Arbitrary passes it to the time-optimal solver, which assigns the timing under the amplitude and slew limits and builds the gradient events, the acquisition window and the rewinder back to k = 0.

import numpy as np
from scipy.integrate import cumulative_trapezoid

import pypulseqpp as pp
import pypulseqpp.sequences as design


def twirl_path(fov: float, matrix: int, interleaves: int, samples: int = 2048):
    """Return the ``(samples, 2)`` k-space path of one twisting radial arm, in 1/m."""
    radius = np.linspace(0.0, matrix / (2 * fov), samples)
    transition = interleaves / (2 * np.pi * fov)
    twisting = radius > transition
    slope = np.zeros_like(radius)
    slope[twisting] = (
        np.sqrt((2 * np.pi * fov * radius[twisting] / interleaves) ** 2 - 1.0)
        / radius[twisting]
    )
    angle = cumulative_trapezoid(slope, radius, initial=0.0)
    return np.column_stack([radius * np.cos(angle), radius * np.sin(angle)])

The interleaf

Arbitrary stretches the readout to hold the samples it is given, so the sample count has to match the arm rather than the matrix. The time-optimal duration is not known until the arm is solved, so it is solved once at two samples to measure that duration and once more with the number of samples the duration holds at the requested rate.

def twirl_interleaf(
    system: pp.Opts,
    fov: float,
    matrix: int,
    interleaves: int,
    *,
    readout_bandwidth_hz: float = 250e3,
):
    """Solve one twisting radial arm and fill it with samples."""
    path = twirl_path(fov, matrix, interleaves)
    probe = design.Arbitrary(
        system, path, matrix=2, bandwidth_hz_px=readout_bandwidth_hz
    )
    samples = int(probe.read_duration * readout_bandwidth_hz)
    return design.Arbitrary(
        system, path, matrix=samples, bandwidth_hz_px=readout_bandwidth_hz
    )

The readout module

NonCartesianReadout plays any two-channel interleaf: it places the prewinder, the acquisition and the rewinder against the pulse it is given, solves the echo time and the repetition time, and adds the spoiler. A family that designs its own interleaf subclasses it, builds the trajectory in init_module and forwards the rest, which is how the shipped spiral and rosette readouts are written.

class TwirlReadout2D(design.NonCartesianReadout):
    """One twisting radial arm in a plane.

    Parameters
    ----------
    fov : float
        Isotropic in-plane field of view (m).
    matrix : int
        In-plane matrix size.
    interleaves : int
        Number of arms used to set the pitch and transition
        radius. The loop may acquire any number of rotated copies.
    readout_bandwidth_hz : float, optional
        Requested ADC sampling rate (Hz).
    """

    def init_module(
        self,
        system: pp.Opts,
        rf,
        gz=None,
        gz_reph=None,
        *,
        fov: float,
        matrix: int,
        interleaves: int,
        readout_bandwidth_hz: float = 250e3,
        **kwargs,
    ) -> None:
        trajectory = twirl_interleaf(
            system,
            fov,
            matrix,
            interleaves,
            readout_bandwidth_hz=readout_bandwidth_hz,
        )
        super().init_module(system, rf, gz, gz_reph, trajectory=trajectory, **kwargs)

Readout duration against a spiral

A constant-density spiral designed for the same interleaf count samples the same field of view at the same resolution with a longer readout: the twisting arm crosses the centre of k-space radially, where the spiral has to wind through it at the Nyquist pitch.

system = pp.Opts(
    max_grad=40.0,
    grad_unit="mT/m",
    max_slew=150.0,
    slew_unit="T/m/s",
    rf_dead_time=100e-6,
    rf_ringdown_time=30e-6,
    adc_dead_time=10e-6,
)

FOV = 220e-3
MATRIX = 128
INTERLEAVES = 16

arm = twirl_interleaf(system, FOV, MATRIX, INTERLEAVES)
spiral = design.Spiral(system, FOV, MATRIX, design_interleaves=INTERLEAVES)
for name, interleaf in (("twisting radial", arm), ("spiral", spiral)):
    print(
        f"{name:16} {interleaf.n_samples:5d} samples, readout "
        f"{interleaf.read_duration * 1e3:5.2f} ms, interleaf "
        f"{interleaf.duration * 1e3:5.2f} ms"
    )
print(
    f"transition radius {INTERLEAVES / (2 * np.pi * FOV):.1f} of "
    f"{MATRIX / (2 * FOV):.1f} 1/m"
)

twisting radial    968 samples, readout  3.88 ms, interleaf  4.60 ms
spiral            1100 samples, readout  4.42 ms, interleaf  5.04 ms
transition radius 11.6 of 290.9 1/m

One repetition

excitation = design.SpatialSelectiveExcitation(
    system, flip_angle_deg=15.0, thickness_m=5e-3, duration_s=3e-3
)
readout = TwirlReadout2D(
    system,
    excitation.rf,
    excitation.gz,
    excitation.gz_reph,
    fov=FOV,
    matrix=MATRIX,
    interleaves=INTERLEAVES,
    te=None,
    tr=None,
    spoiling_cycles=4.0,
)
print("events:", ", ".join(sorted(vars(readout.events))))
print(
    f"TE {readout.echo_time * 1e3:.3f} ms over a {readout.duration * 1e3:.3f} ms "
    "repetition"
)

seq = pp.Sequence(system=system)
for block in readout.blocks:
    seq.add_block(*block)
seq.paper_plot(tr=1)

/home/runner/work/pypulseqpp/pypulseqpp/docs/build/site/pypulseqpp/_events.py:273: UserWarning: Specified RF delay 0.00 us is less than the dead time 100 us. Delay was increased to the dead time.
  made = factory(*args, **kwargs)
events: adc, gx, gx_rew, gy, gy_rew, gz, gz_reph, gz_spoil, rf, wait_pre, wait_rew
TE 2.080 ms over a 9.240 ms repetition

Scan loop

One solved arm is turned per shot by a rotation extension, which the loop adds to every block that drives an in-plane gradient.

angles = 2 * np.pi * np.arange(INTERLEAVES) / INTERLEAVES
rotations = [pp.make_rotation(float(angle)) for angle in angles]

scan = pp.Sequence(system=system)
for shot, rotation in enumerate(rotations):
    scan.add_block(excitation.rf, excitation.gz, pp.make_label("LIN", "SET", shot))
    for block in readout.blocks[1:]:
        events = list(block)
        if any(getattr(event, "channel", None) in ("x", "y") for event in events):
            events.append(rotation)
        scan.add_block(*events)

print(
    f"{scan.num_blocks} blocks, {scan.duration()[0] * 1e3:.1f} ms, "
    f"timing {scan.check_timing()[0]}"
)

64 blocks, 147.8 ms, timing True

The arms the loop acquired.

pp.plot.plot_kspace(scan, plane="xy")

References

[JNM92]

Jackson JI, Nishimura DG, Macovski A. Twisting radial lines with application to robust magnetic resonance imaging of irregular flow. Magnetic Resonance in Medicine. 1992;28(2):251-263. https://doi.org/10.1002/mrm.1910280209