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.
Description
Gradient area determines k-space displacement.
Gradient area determines k-space displacement. Using gamma in Hz/T yields cycles per metre and avoids an extra 2π factor.
The tool returns kSpaceCyclesPerMeter (k space cycles 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 Cartesian K-Space Trajectory
Use Cartesian K-Space Trajectory when you already have timesMs, gradientsMilliTeslaPerMeter, gammaHzPerTesla and want kSpaceCyclesPerMeter. This is useful for checking a supplied calculation, documenting a measurement convention, or comparing observations prepared on the same basis.
How to use Cartesian K-Space Trajectory and understand its results
- Enter timesMs, gradientsMilliTeslaPerMeter, 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 kSpaceCyclesPerMeter. 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 |
|---|---|---|---|
| timesMs | Times ms | list of number | Required |
| gradientsMilliTeslaPerMeter | Gradients milli tesla per meter | list of list of number | Required |
| gammaHzPerTesla | Gamma hz per tesla | number | Required |
Use at most 128 increasing times with aligned two-component gradient rows. Start at [0,0] and accumulate trapezoidal areas.
k[i+1] = k[i]+gammaHz/T×averageGradientT/m×deltaTimeSeconds The symbols refer to the supplied quantities or generated values described above; unit conversions and comparison boundaries follow the stated implementation.
{
"input": {
"timesMs": [
0,
1,
2
],
"gradientsMilliTeslaPerMeter": [
[
10,
0
],
[
10,
0
],
[
10,
0
]
],
"gammaHzPerTesla": 1000000
},
"output": {
"kSpaceCyclesPerMeter": [
[
0,
0
],
[
10,
0
],
[
20,
0
]
]
}
}Limitations and assumptions
The initial k-space coordinate is fixed to zero. Hardware response, delays and gradient nonlinearity are not corrected.
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
Gradient Amplitude Required: Derive constant readout gradient from desired k-space traversal span and duration; prephasers, ramps and gradient hardware feasibility are excluded. 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.
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
- Gradient Amplitude Required
Derive constant readout gradient from desired k-space traversal span and duration; prephasers, ramps and gradient hardware feasibility are excluded.
- 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.