Spiral K-Space Trajectory

Plan an Archimedean k-space spiral and its analytic gradient derivative for a supplied duration, radius and turn count; no gradient/slew constrained optimization is performed.

Description

An Archimedean spiral increases radial k-space position linearly while rotating at a fixed angular speed.

An Archimedean spiral increases radial k-space position linearly while rotating at a fixed angular speed. Differentiating the trajectory gives its ideal gradient waveform.

The tool returns kCoordinates (k coordinates); gradientMilliTeslaPerMeter (gradient milli tesla per meter). Input measurements and model parameters are supplied by the caller; the tool computes the stated summary rather than acquiring those measurements.

When to use Spiral K-Space Trajectory

Use Spiral K-Space Trajectory when you already have timesSeconds, turns, maximumRadiusCyclesPerMeter, gammaHzPerTesla and want kCoordinates, gradientMilliTeslaPerMeter. This is useful for checking a supplied calculation, documenting a measurement convention, or comparing observations prepared on the same basis.

How to use Spiral K-Space Trajectory and understand its results

Enter timesSeconds, turns, maximumRadiusCyclesPerMeter, gammaHzPerTesla using the field table below. Match the units, ordering and data boundaries described in the calculation.
Run the tool after checking the stated bounds and matching paired observations or categories. A malformed request gives a tool error rather than a usable measurement.
Read kCoordinates, gradientMilliTeslaPerMeter. Use the calculation and interpretation below to distinguish amounts, ratios, signs and decision flags where applicable.
Review the supplied parameters and assumptions before using the result in a report or comparing it with another observation.

Input field	Meaning	Value type and bounds	Requirement
timesSeconds	Times seconds	list of number	Required
turns	Turns	number	Required
maximumRadiusCyclesPerMeter	Maximum radius cycles per meter	number	Required
gammaHzPerTesla	Gamma hz per tesla	number	Required

Use 2–128 increasing times starting exactly at zero, positive turns, maximum radius and gamma. The final time defines duration.

k(t) = radialRate×t×[cos(omega t),sin(omega t)]; GmT/m = 1000×dk/dt/gammaHz/T

The symbols refer to the supplied quantities or generated values described above; unit conversions and comparison boundaries follow the stated implementation.

json

{
  "input": {
    "timesSeconds": [
      0,
      0.25,
      0.5
    ],
    "turns": 1,
    "maximumRadiusCyclesPerMeter": 1,
    "gammaHzPerTesla": 1
  },
  "output": {
    "kCoordinates": [
      [
        0,
        0
      ],
      [
        -0.5,
        6.123233995736766e-17
      ],
      [
        1,
        -2.4492935982947064e-16
      ]
    ],
    "gradientMilliTeslaPerMeter": [
      [
        2000,
        0
      ],
      [
        -2000.000000000001,
        -6283.185307179586
      ],
      [
        2000.0000000000032,
        12566.370614359172
      ]
    ]
  }
}

Limitations and assumptions

This analytic plan does not enforce amplitude, slew, PNS or density-compensation constraints; it is not a hardware-ready sequence.

Numerical inputs must be finite. The tool checks its declared input shape and calculation constraints; those checks do not verify the quality, calibration or representativeness of the supplied observations.

Alternative or Complementary methods

Cartesian K-Space Trajectory: Integrate two-axis piecewise-linear gradients by trapezoids into Cartesian k-space from a zero origin; refocusing sign changes must already be applied to supplied gradients. Gradient Slew Rate: Measure signed piecewise-linear gradient slew from timestamped mT/m samples; mT/m per ms equals T/m/s and between-sample excursions are not observable.

Methodological background: RAD229: MRI signal and sequence lecture notes.^[1]

References

RAD229: MRI signal and sequence lecture notes — Brian Hargreaves and colleagues, Stanford University

Similar or alternative tools

Cartesian K-Space Trajectory
Integrate two-axis piecewise-linear gradients by trapezoids into Cartesian k-space from a zero origin; refocusing sign changes must already be applied to supplied gradients.
Gradient Slew Rate
Measure signed piecewise-linear gradient slew from timestamped mT/m samples; mT/m per ms equals T/m/s and between-sample excursions are not observable.