Two dimensional Spatial Selective Shinnar Le Roux pulse design for arbitrary k-space trajectory

Introduction: The Shinnar Le Roux (SLR) method [1,2,3] has been widely used to design slice selection pulse and two-dimensional (2D) RF pulse on EPI trajectory [4] due to the advantage of allowing tradeoffs among the in-slice ripple, out-of-slice ripple and transition. In this work, the SLR method is extended to design 2D pulse on arbitrary k-space trajectory. Firstly, 2D filter coefficients are designed using McClellan transformation; then, the inverse gridding algorithm is applied to resample the 2D filter coefficients in Cartesian trajectory to arbitrary k-space trajectory needed. Finally, the SLR inverse transform is applied to convert these filter coefficients to a 2D RF pulse. A 90 excitation pulse is designed and the excitation profile is simulated to verify our proposed method. Theory and method: As in the traditional SLR method, the 2D RF pulse is assumed to be piecewise pulse, which is commonly used in commercial MR scanner. Thus, the rotation caused by each RF sample and the corresponding gradients can be treated as spinors, and the state of spin after each rotation can be expressed by a 2×1 complex vector: