Simple analytic spiral K-space algorithm

  • Glover G
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The exact, hardware-constrained design of a spiral k-space trajectory requires the solution of a differential equation, thereby making real-time prescription difficult on scanners with limited computational power. This study describes a closed-form ap-proximate solution for interleaved Archimedian spiral trajecto-ries that closely matches the exact design. Both slew rate-limited and amplitude-limited regimes are incorporated. Magn Reson Med 42:412–415, 1999. Spiral magnetic resonance imaging (MRI) has been found to be effective for dynamic studies, with applications that include cardiovascular (1–3), renal (4), breast (5), spectros-copy (6), and functional (7,8) neuroimaging. The spiral trajectory excels in such applications because of its intrin-sic moment-nulling motion compensation (1,9) and effi-cient use of gradient power (10). Despite these advantages, spiral methods are not avail-able routinely on clinical scanners, partly because compu-tation of the gradient waveforms can be cumbersome. King et al. (11) have formulated the design of an Archimedian spiral trajectory as a differential equation in which the constraints are established by the physical characteristics of the scanner hardware. Solution by this approach can be lengthy and can require the calculation to be performed off-line, so that only precomputed waveforms are used at scan time. This approach restricts the choices of scan parameters and limits flexibility in optimizing protocols. Recognizing these difficulties, Duyn and Yang (12) de-rived an approximate closed-form algorithm for the slew rate-limited case. This formulation has the advantage of simplicity, but it is valid only for k-space points away from the origin. In fact, this algorithm has a singularity at k ϭ 0, where the approximation breaks down and the slew rate approaches infinity. Amann et al. (13) modified Duyn and Yang's formulation near the origin by modeling the behav-ior of the exact solution. Hardy and Cline (14) utilized a resampling algorithm to incorporate amplitude and slew-rate limitations. Here, we propose a simpler approach that is well behaved at the origin and approaches the exact slew-rate and amplitude-limited solutions in their respec-tive regimes. THEORY

Author-supplied keywords

  • Rapid magnetic resonance imaging
  • Spiral k-space trajectory design
  • Spiral magnetic resonance Imaging

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  • Gary H. Glover

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