SPIRiT¹ is a self-calibrated parallel imaging reconstruction method that is flexible on sampling patterns. The number of elements in the reconstruction kernel (the SPIRiT kernel) increases rapidly as the kernel becomes larger, especially for 3D imaging – a kernel of 5X5X5 for a 32-channel head coil contains about 4000 elements. A large kernel generally prompts two problems for self-calibrated parallel imaging. First, a sizable calibration region/volume is required for kernel calibration, which ultimately limits the achievable acceleration. Second, calibration of a large kernel requires solving a large set of linear equations and can be computationally time consuming. A complete set of the 5X5X5X32(channel) kernel takes minutes to compute using a modern desktop computer. Therefore it is desirable to reduce the number of elements in a SPIRiT kernel without compromising its performance. In this work we sought such possibility by exploring the geometry of the SPIRiT kernels. We refer to this technique as SPOT (SPIRiT with custOm kernel geomeTry).It was known that for a 2D SPIRiT¹ or GRAPPA^2,3 kernel the elements that are closer to the missing sampling point are generally more important in computing the missing point^4,5. Therefore it may be feasible to remove the corner elements of a kernel without affecting the kernel performance. By doing so a 2D kernel is similar to a disk rather than a square, which we call a SPOT, as illustrated in Fig. 1. Likewise a 3D SPOT kernel is ball-shaped. The 2D and 3D SPOT kernels constructed this way contain approximately 79% (π/4) and 52% (π/6) of the elements of the original kernel, respectively. Since the computation complexity is roughly proportional to N³, where N is the number of unknowns⁶, such reduction in the number of elements may significantly shorten the calibration time.

We tested SPOT in both 2D and 3D imaging on a 3T whole-body scanner (Siemens Tim Trio) with a 32-channel coil. For 2D, a RARE image was acquired with a FOV of 256x256 mm² and matrix size of 448X448 on a healthy volunteer. A random-sampling mask was applied to the k-space data to obtain a 3X under-sampled data. The images were reconstructed using both the SPIRT and SPOT 7X7 kernels shown in Fig. 1.

For 3D, stack-of-spirals⁷ were used for a 3D pCASL arterial spin labeling (ASL) acquisition of 2 mm isotropic resolution, collected in 4 segments (4-shot) in 35 s on a healthy volunteer. The sequence was highly accelerated and self-calibrated: the effective acceleration factor is 2.4 in the partition direction and is about 3 within each 2D axial partition. The variable-density spiral interleaves for each partition acquisition is shown in Fig. 2. The FOV is 216X216X144 mm³ (108X108X72 matrix). To maintain the high effective acceleration a moderate volume not large enough for the 7X7X7 SPIRiT kernel calibration was fully sampled and images were reconstructed using the 5X5X5 SPIRiT kernel and a 187-element 7X7X7 SPOT kernel.

Figure 3 summarizes the results of 2D SPOT validation, showing that the 7X7 SPIRiT and SPOT kernels have very similar performance – their 10X magnified difference maps from the original image showed no visible difference. However, the SPIRiT calibration took 85 s while the SPOT only took 38 s. Figure 4 compares the control-label different maps of the 3D ASL image in the sagittal view. Arrows indicate improvement of image quality by using the 7X7X7 SPOT kernel.Our results in this study showed that for the same kernel size, a SPOT kernel a) permits a smaller calibration region (3D example) and b) significantly speeds up calibration (2D example). Conversely, for the same calibration data, SPOT may allow for a larger kernel to be used (3D example). There are potentially more benefits of SPOT that are yet to be explored. For example, a larger kernel is possibly more effective at computing missing data in each iteration and therefore fewer iterations may be required to reach the desired image quality¹. The kernel geometry in SPOT is also not limited to disks or balls. In a sense, GRAPPA is a special case of SPOT where the kernel geometry is custom to the sampling pattern.We showed that SPIRiT image reconstruction can be accelerated with little loss of image quality by adjusting the SPIRiT kernel geometry.