Towards a geometry factor for projection imaging with non-linear gradient fields

INTRODUCTION: O-Space parallel imaging combines a Z2 gradient field with conventional radial k-space trajectories to perform highly-efficient parallel imaging [1]. Because the Z2 field varies as r2 = x2+y2 in the axial plane, it provides spatial encoding that is complementary to that obtained from a circumferential surface coil array. However, non-linear projection encoding invites the exploration of a great multitude of trajectories, including differently-weighted combinations of X, Y, Z2 producing projections with different center placements (CPs). Time-varying readout pulses such as those used in spiral acquisitions present yet another degree of freedom for non-linear encoding. At present it is unclear how to best traverse this large space of potential encoding functions. A good first step is to develop a metric for evaluating the performance of arbitrary sets of surface coils, gradients, and pulse shapes. This work presents a protocol for evaluating the performance of these trajectories with a given surface coil array via and attempts to formulate an O-Space equivalent to the conventional g-factor. In conventional parallel imaging, the spatially-varying noise amplification of SENSE [2] and GRAPPA [3] reconstructions is described by the geometry factor, or g-factor. Maps of the g-factor illustrate the degree of noise amplification throughout the image while assuming that resolution remains invariant. A direct extension of Cartesian SENSE into the domain of non-linear encoding gradients has been proposed in the form PatLoc imaging [4]. PatLoc uses pairs of orthogonal, multipolar gradients, one of which is used for readout and the other for phase encoding. Because the shapes of the gradients do not change in between readouts, the Jacobian of the linear-to-multipolar transformation can be used to calculate voxel size throughout the FOV. The PatLoc g-factor is thus simply the SENSE g-factor weighted by the Jacobian. By contrast, in O-Space imaging, the spatially-varying resolution is a complicated function of the CPs selected for each projection.