This Matlab program spiral_vdsn.m generates a constant density spiral starting from the center of k-space out to a selected portion of k-space, and then switches over to a variable density spiral. In this scheme, only the outer regions of k-space are under-sampled. The program uses a Matlab GUI to generate a user-friendly interface, which is brought up by typing vdsn at the Matlab command prompt. The program is generally based on considerations discussed in Kim et al. (1) and includes a scheme analogous to that of Zhao et al. (2) to prevent exceeding the gradient slew rate while near the center of k-space. Inputs to the program include the image matrix size, the field of view, values of alpha (1) for the outer regions of k-space, and the transition point to switch over to the value of alpha for the outer region of k-space. As discussed by Chang and Glover (3), this approach is more efficient for under-sampling the outer regions of k-space than a variable density spiral, which losses efficiency by unnecessarily over-sampling the center portion of k-space.
Addition inputs to the program include the gradient raster time, the gradient maximum, the gradient maximum slew rate, the number of interleaved spirals, and the acquisition dwell time to be used (which may be different from the gradient raster time). Various graphical outputs, including waveforms and the point density function, are provided to show the results. Additional outputs include the maximum dwell time for Nyquist sampling at the center of k-space, an estimate of the improvement in time resulting from under-sampling the outer region of k-space, and the total number of k-space points. The program generates a number of global variables to make outputs available for use in subsequent calculations. While the program has been tested for reasonable input values, the program can fail if extreme input values are used. In addition, the program expects the gradient maximum to be reached prior to switching from constant density to variable density.
SPIRAL K-SPACE POINTS OUTPUTS
For spiral out, variable density spiral function: spiral_vdsn.m
Global Outputs
| G_K_SPACE | Spiral k-space points at gradient raster times (gradient table) |
| S_KSPACE | Spiral k_space pts (x + i*y) |
| S_ARMS | Number of spiral k_space arms |
| S_DWELL | Spiral dwell time in us |
| S_RESOL | Nominal resolution (mm) at unit point density |
| S_FOV | Spiral fov in mm |
| C_PTS | Size of Cartesian matrix (one dimension) |
The Matlab program will run on Windows, Linux and other systems with a Matlab license.
References
- Kim DH, Adalsteinsson E, Spielman DM. Simple analytic variable density spiral design. Magn Reson Med 2003;50(1):214-219.
- Zhao T, Qian Y, Hue Y-K, Ibrahim TS, Boada F. An Improved Analytical Solution for Variable Density Spiral Desgn. 2008. p 1342.
- Chang C, Glover GH. Variable-density spiral-in/out functional magnetic resonance imaging. Magn Reson Med 2011;65(5):1287-1296.