A performance measure for MRI with nonlinear encoding fields

th coil and is the accumulated phase for the k th echo and i th time sample. For simplicity, susceptibility and relaxation effects are ignored. In the case of conventional Fourier imaging the frame matrix is diagonal, whereas for both SENSE and PatLoc, a pixel basis can be chosen such that the matrix is block diagonal and the invertibility analysis is simplified to the analysis of individual blocks. No such basis can be chosen for other techniques such as O-space imaging. Instead, we define the simple yet powerful metric in Eq.(3), comparable to the width of the point-spread-function. Applied to any encoding scheme, the proposed metric in Eq.(3) highlights regions that exhibit poor resolvability. This is analogous to the covariance matrix in SENSE encoding which identifies areas in the image with large reconstruction variance. The metric is also related to the Gerschgorin circle theorem concerning the eigenvalues of the frame matrix, from which the matrix condition number can be defined. METHODS The universal applicability of the proposed performance metric to PatLoc, O-Space and SENSE techniques is demonstrated through simulated data, as follows. Coil sensitivity profiles were generated according to the Biot-Savart equation for eight receive coils circumferentially distributed around the field-of-view. To validate the qualitative behaviour of the performance map, we compared our map to a 64×64 reconstruction variance map computed by full inversion of the frame matrix, with elements as per Eq.(2). PatLoc, SENSE and O-space define their own as follows. PatLoc: A radially quadratic field was used for the readout field and a quadrapolar field, with sinusoidal variation in azimuth was used for phase encoding, as in [1]. O-Space: A quadratic field with 64 different centre points arranged in concentric rings was simulated, with 256 time samples, as in [3]. SENSE: The frame operator for SENSE was constructed using 4-fold acceleration in the phase direction. In all cases, FOV=10cm × 10cm, and a total of 64 echoes were simulated. The proposed performance metric, Eq.(3), was applied with the same imaging parameters to generate maps of each imaging scheme’s performance. In order to demonstrate the efficacy of the proposed performance measure, an O-Space construction, similar to [6], was formulated in which ten different centre placement schemes were evaluated, using the metric C = ∑ν mν. This performance analysis is computationally tractable, avoiding the need to compute the reconstruction. RESULTS The proposed performance metric, Eq.(3), captures the qualitative behaviour of reconstruction performance for each of the PatLoc, SENSE and O-Space imaging techniques (Fig.1). It is evident that the predicted reconstruction error for O-Space is more uniform than PatLoc, which exhibits the characteristic “hole” in the centre. Fig. 2 displays the performance of the ten centre placement schemes, and the performance map for each of the best and worst schemes, confirming the results of [6] that concentric rings is an optimised coil arrangement. Importantly, this result was achieved without performing the cumbersome inversion associated with reconstruction or variance analysis.