Ashley G Anderson III^{1}, Dinghui Wang^{2}, and James G Pipe^{2}

A flexible 3D “Spiral Staircase” (SSC) trajectory is introduced that reduces *g*-factor losses from through-plane parallel imaging acceleration, regardless of coil geometry. Results demonstrate up to a 5x *g*-factor improvement over Cartesian SENSE for through-plane acceleration in axial brain acquisition with R = 3.

**Introduction**

**Methods**

We propose a novel 3D trajectory (Figure 1) comprising spiral arms in parallel k_{x}-k_{y} planes, each with a unique position along the k_{z} axis. We refer to this trajectory as a “spiral staircase” (SSC). This trajectory is similar to previously reported distributed spiral^{3} and rotated stack-of-spirals^{4} (SOS) trajectories, but with periodic arm-rotation (rather than incremented by the golden angle) to maintain uniform sampling in k_{z}, and flexibility for sub-Nyquist distances between arms along k_{z}. While we believe this trajectory has many benefits, this work focuses on its potential for improved geometry factors for through-plane parallel imaging acceleration.

Reconstruction of fully-sampled SSC data (Figure 1) was performed in three-steps: 1) a 1D inverse FFT along the Cartesian kz dimension to transform the data into a hybrid k_{x}-k_{y}-Z space, 2) a Z-dependent phase applied to the data to compensate for the shifts in k_{z}, collapsing sets of spiral arms into slices, and 3) 2D gridding and FFT to reconstruct each slice.

In the case of through-plane aliasing, the phase correction in the second step is different for the nominal and any aliased slices. The result is reduced aliasing coherence and ultimately improved *g*-factor as SENSE encoding matrix becomes easier to invert. Figure 2 shows how SSC phase correction was incorporated in an iterative SENSE (conjugate gradient) technique to reconstruct data with through-plane undersampling.

A volunteer was scanned on a 3T system (Ingenia, Philips, The Netherlands) in under an IRB approved protocol. Data were acquired using a 15-channel head coil using an FFE sequence with 𝜏 (readout duration) = 9.76 ms, flip angle = 36°, T_{E1}/T_{E2}/T_{R} = 1.36/2.51/30 ms, voxel size = 1.0 x 1.0 x 2.0 mm^{3}, axial imaging volume = 240 x 240 x 130 mm^{3} (65 prescribed slices plus 20% oversampling). Each k_{z} segment contained 30 spiral arms with 'mixed'^{5} arm-ordering (also controlling the sub-k_{z} shifts). Total scan times were 2:32 (fully sampled), 1:26 (R = 2), and 1:02 (R = 3). All parameters were the same for SOS and SSC acquisitions.

SSC *g*-factor maps were approximated numerically^{6} using 25 noise realizations, followed by median filtering with a 5x5 kernel. SOS *g*-factor maps were calculated analytically. All *g*-factor maps were calculated from T_{E1} images before deblurring.

**Results**

Figure 3 shows central sagittal slices from *g*-factor maps comparing SSC and SOS for R = 2 and R = 3. Mean *g*-factor improved from 1.45 to 0.96 for R = 2 and 2.88 to 1.24 for R = 3. Figure 4 shows sagittal reformats from reconstructed images for R = 2 SSC and SOS data after joint fat-water separation and off-resonance deblurring^{7} (water-only images are shown).

Inclusion of a regularization factor (tissue presence map) resulted in *g*-factors < 1.0 outside the brain, but were masked for the mean *g*-factor calculations. Mean *g*-factor < 1.0 in the R = 2 SSC data is likely the result of too few iterations in the numerical calculations.

Discussion and Conclusions

Long-𝜏 3D spiral acquisitions are well suited for parallel imaging acceleration due to high SNR efficiency. These results demonstrate that the spiral staircase trajectory reduces *g*-factor to near optimal (1.0), expanding the utility of parallel imaging even with non-ideal coil geometries. Note as well as reduced overall *g*-factor there is reduced structure in the *g*-factor map, which further improves image appearance.

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The SSC trajectory and reconstruction scheme for fully-sampled data. Here the SSC trajectory is shown with linear arm ordering for illustrative purposes. In reality, arm ordering (and sub-∆k_{z} shifts) are adjusted to control the aliasing point spread function.

The iterative SENSE algorithm including phase correction terms for SSC aliasing.

Central sagittal slices of *g*-factor maps for SSC (A, B) and SOS (C, D) data for R = 2 (A, C) and 3 (B, D). Maps are scaled from 0 (black) to 5 (white).

Representative axial slices and sagittal reformats of deblurred water-only images for SSC (A, B, C) and SSC (D, E, F) data with through plane acceleration R = 1, 2, and 3.