Nikolai J Mickevicius^{1}, L. Tugan Muftuler^{2}, Andrew S Nencka^{3}, and Eric S Paulson^{1}

Projection imaging has many advantages over Cartesian sampling. The unique point spread function makes it particularly useful for highly accelerated parallel imaging and compressed sensing reconstructions[1]. In this study, a projection-domain sensitivity encoding algorithm is developed for highly accelerated simultaneous multislice radial imaging. Since it operates in the projection-domain, no time expensive gridding, de-gridding, and FFT operations are required within each iteration of the solving algorithm. From an in vivo experiment, two slices were reconstructed from only 34 radial spokes.

Methods to accelerate MR data acquisition are constantly being developed. Two such methods include the use of non-Cartesian k-space trajectories and simultaneous multislice (SMS) imaging. When used as the only form of acceleration, SMS proves advantageous compared with in-plane acceleration techniques since there is no $$$\sqrt{N}$$$ SNR penalty.

A method known as highly accelerated projection imaging (HAPI) uses high resolution (HR) radial k-space measurements to reconstruct lower resolution images[2]. The HR radial data are first brought into the projection domain by performing an inverse FT along the readout dimension. Each point in the sampled projection is then traced back through all intersecting voxels in the reconstructed image grid. The fraction of the voxel covered by the width of the projection point is multiplied by the respective complex-valued coil sensitivities. The measured complex-valued signal intensity in the measured high-resolution projections is parameterized by a sum along the projection angle of the lower resolution reconstructed image times the respective coil sensitivity profile and times the voxel fractions. The present work extends the HAPI method to reconstruct highly accelerated radial SMS images.

The
goal of SMS HAPI is to obtain separated images from slice-aliased projections.
Since more than one slice is to be reconstructed, the coil sensitivity matrix
must now contain information from every simultaneously excited slice.
Additionally, phase modulations using RF or gradient encoding are typically employed
in modern SMS acquisitions[3,4].
These modulations, referred to as CAIPIRINHA (CAIPI) phase modulations, allow
supplemental control over the aliasing patterns seen in the data. These
modulations are stored in a phase modulation matrix, . See Yutzy et. al. (2011)
for more information[5].
The SMS HAPI problem is posed in Eq. 1. The desired reconstructed image, **x**,
contains image data for all SMS slices. A circular field-of-view is
reconstructed. **F** represents the fractional area of reconstructed voxels
covered by a point in a measured projection, **p** (see Figure 1). **C**
represents the coil sensitivity profiles, and **T** represents a
transformation of **x** to a sparse domain. The reconstruction pipeline for
SMS HAPI is shown in Figure 2. No spatial regularization was utilized in this
study.

$$x = \underset{x}{\operatorname{argmin}} \lVert \sum_{n=1}^{SMS}(\Phi F C_n x_n) - p \rVert _2^2 + \lambda \lVert Tx \rVert_1$$

A SMS=3 simulation was performed in a numerical brain phantom. Three 128x128 images were reconstructed from 16 spokes of simulated 16-channel projection data with a base resolution of 512x512.

An SMS=2 experiment was simulated from acquired abdominal 3D stack-of-stars radial data on a 1.5T Elekta MR-Linac from a consenting volunteer. The base resolution of the acquired data was 512x512. The desired reconstructed matrix size is 128x128. Coil maps for the 8-channel array were calculated from the full dataset. SMS HAPI images were reconstructed using 34 k-space spokes from two slices summed together. Comparisons were made to the SMS CG-SENSE algorithm[5] as well as to the single-pass version of the same algorithm (equivalent to phase demodulation and NUFFT). For NUFFT and CG-SENSE, the images were reconstructed at 512x512 resolution then resampled to 128x128. The same circular FOV as the HAPI reconstruction is shown.

The SMS=3 simulation results are shown in Figure 3. Considering only 16 spokes were used to generate 128x128 images for three slices, the reconstructed images resemble the ground truth images remarkably well. Some residual aliasing artifacts are still present in the highly accelerated simulated images, however.

The retrospectively generated in vivo SMS=2 results are shown in Figure 4. The CG-SENSE algorithm failed to converge to the global minimum for such highly accelerated images. The SMS HAPI algorithm, however, was able to largely remove streaks from the images at the expense of enhanced noise in the images relative to the fully sampled images.

[1] Feng L, Grimm R, Block KT, Chandarana H, Kim S, Xu J, et al. Golden-angle radial sparse parallel MRI: Combination of compressed sensing, parallel imaging, and golden-angle radial sampling for fast and flexible dynamic volumetric MRI. Magn Reson Med 2014;72:707–17. doi:10.1002/mrm.24980.

[2] Ersoz A, Arpinar VE, Muftuler LT. Highly accelerated projection imaging with coil sensitivity encoding for rapid MRI. Med Phys 2013;40:022305. doi:10.1118/1.4789488.

[3] Breuer FA, Blaimer M, Heidemann RM, Mueller MF, Griswold MA, Jakob PM. Controlled aliasing in parallel imaging results in higher acceleration (CAIPIRINHA) for multi-slice imaging. Magn Reson Med 2005;53:684–91. doi:10.1002/mrm.20401.

[4] Setsompop K, Gagoski BA, Polimeni JR, Witzel T, Wedeen VJ, Wald LL. Blipped-controlled aliasing in parallel imaging for simultaneous multislice echo planar imaging with reduced g-factor penalty. Magn Reson Med 2012;67:1210–24. doi:10.1002/mrm.23097.

[5] Yutzy SR, Seiberlich N, Duerk JL, Griswold MA. Improvements in multislice parallel imaging using radial CAIPIRINHA. Magn Reson Med 2011;65:1630–7. doi:10.1002/mrm.22752.

Figure 1. Calculating voxel
fractions, F, for HAPI data. (a) The
magnitude of a measured projection, which is the Fourier transform of an
acquired k-space spoke. Each point in the projection (red dashed line) is equal
to the sum of the intensity of intersecting voxels at the acquired projection
angle weighted by the fraction of each voxel covered by the projection. (b) The
fraction of voxels covered by a point in a projection acquired at a higher
resolution than the voxels to be reconstructed.

Figure 2. The
coil sensitivity profiles, C, are shown overlayed with the fractional area, F,
of each reconstructed voxel covered by a finite-width line passing through at
the projection angle. The coil sensitivity maps are weighted by the overlapping
fractions and are then multiplied by the applied CAIPIRINHA phase, Φ.
Projection data are synthesized from the current guess of the reconstructed SMS
images, x. The sum of the synthesized projections is calculated and compared
with the measured projections, p, for data consistency enforcement. The
reconstructed images are then updated using a conjugate gradient algorithm.

SMS=3
simulation results. These data were reconstructed from only 16 spokes of
simulated 16-channel data.

Figure 4.
In vivo results SMS=2 results. The NUFFT, CG-SENSE, and SMS HAPI
reconstructions from 34 spokes are shown here. The CG-SENSE algorithm fails at
such high total acceleration factors while HAPI is able to largely remove
streaking artifacts without the use of spatial regularization.