To develop an ESPIRiT-based reconstruction method for simultaneous multi-slice imagingFor simultaneous multi-slice (SMS) acquisitions^[1] with SENSE^[2] reconstruction, accurate coil sensitivity maps estimation is essential to obtain high quality images. Traditionally, coil sensitivity maps are estimated with the ratio of individual-coil images over a uniform image, followed with a polynomial fitting process. The uniform image can be either acquired from an additional body coil scan or derived by taking the square root of the sum-of-square (SOS) of the multi-coil image^[3]. However, there are two problems associated with those existing techniques. First, the assumption that the SOS image from multi-coil is uniform is not exactly accurate and thus brings in-homogeneity in coil sensitivity maps estimated. Second, the estimated sensitivity maps depend on how many polynomial orders are chosen during the fitting process, which cause residual spatial oscillations or influence from imaging object, both as modulations to the estimated coil sensitivity maps. ESPIRiT (An Eigenvalue approach to auto-calibrating parallel MRI)^[4] is a new framework for calibration of coil sensitivity map and reconstruction in parallel MRI. Basically, ESPIRIT is a sub-space based method that is highly robust to many types of errors because the estimated subspace automatically adapts to inconsistencies in the data.

Coil sensitivity map estimation with ESPIRiT: Data used in this study came from the ISMRM2015 simultaneous Multi-Slice imaging workshop and was acquired at 3.0T Siemens Skyra using a T1 weighted FLASH sequence. ESPIRiT framework ^[1] is used to calculate the coil sensitivity map with the central 24*24 full sampled raw data. ESPIRiT determines the signal space spanned by local patches in k-space using singular value decomposition(SVD) of a calibration matrix constructed from calibration data. Because there are local correlation in k-space due to field-of-view limitation and correlations induced by the receiver coils, this signal space is a small subspace of the space from all possible patches. Specifically, ESPIRiT recovers the sensitivity maps of the receive coils as eigenvectors to eigenvalue 1 of a reconstruction operator, which is derived from signal space during calibration process. Calibration process with SVD follows this equation $$$A=UΣV^{H}$$$, where A is the calibration matrix constructed by sliding a window through calibration data. The column of the V matrix in the SVD are basis for the rows of A, and therefore are basis for all the overlapping blocks in the calibration data.Explicit sensitivity maps for each point follows the equation $$$G_{q}\overrightarrow{s_{q}}=\overrightarrow{s_{q}}$$$,where $$$G_{q}$$$ is the a reconstruction operator for each position in image space; and $$$\overrightarrow{s_{q}}$$$ is the sensitivities at spatial position q. The explicit sensitivity maps can be found by an eigenvalue decomposition of all $$$G_{q}$$$ chosing only the eigenvectors corresponding to eigenvalue”=1”. Reconstruction of SMS with extended SENSE:After computation of coil sensitivities, a standard SENSE reconstruction can be performed to unfold all 3 slices excited simultaneously. In the ideal case, ESPIRiT yields a single set of sensitivity maps. But for corrupted data, ESPIRiT often yields multiple sets of maps in an attempt to fit the data into a subspace. We use two sets of coil sensitivity maps weighted by S-curve transition function for an extended SENSE reconstruction^[4].

Figure1 demonstrates the SOS coil sensitivity maps for each of the 3 slices. It can be seen that the method based on traditional polynomial fitting of SENSE introduce imaging content leakage (see the arrow in slice 1) or contain sharp spatial variation at the edge of the object (see arrow in slice 2), while the ESPIRiT algorithm guarantees a very spatial smooth coil sensitivity maps. The advantage of good coil sensitivity maps can be seen more clearly in figure 2, which shows the images reconstructed with ESPIRiT and traditional SENSE, with full sampled image as reference. Imperfect coil sensitivity maps cause abnormal signal level for object out of the brain boundary in slice 1 for SENSE reconstruction and lead to bright point at the image edge and some wrapped content at the center of image in slice 2. In comparison, the reconstruction from ESPIRiT method preserves image quality quite well within same imaging areas. Figure 3 shows that the G-factor of ESPIRiT is not as concentrated as those from the traditional SENSE reconstruction^[2].In this work, we apply ESPIRiT reconstruction framework to SMS imaging. The ESPIRIT reconstruction framework helps to produce coil sensitivity maps without polynomial fitting process and thus obviates the dependency of sensitivities on the polynomial order. Multiple sets of sensitivity maps can be taken into account during the reconstruction process for enhanced robustness.