Tess E Wallace^{1,2}, Simon K Warfield^{1,2}, and Onur Afacan^{1,2}

Radial MRI is intrinsically more robust to motion than Cartesian sampling; however, if large rotational motion occurs, the uniform sampling of conventional 3D radial acquisitions is disrupted and is difficult to recover retrospectively. The golden angle ratio has been used to generate a quasi-isotropic distribution of spokes over time in 2D, but is limited to fully correct for motion, which occurs in three dimensions. Extending the flexibility of golden-ratio spoke ordering to 3D radial sampling, combined with rigid-body motion tracking using electromagnetic sensors, enables robust retrospective correction by maintaining relatively uniform sampling, even in the presence of large-amplitude rotational motion.

We
modified a 3D radial pulse sequence to acquire *k*-space data with a golden-ratio sampling scheme. The two-dimensional
golden ratios $$$\phi_1=0.4656$$$ and $$$\phi_2=0.6823$$$ were used to increment the azimuthal and polar
angles of each 3D radial projection, generating a set of spatially and
temporally uniformly distributed spokes (Fig. 1).

*Phantom experiment. *An ACR phantom was scanned at
3T (Siemens, Erlangen, Germany) using a 3D GRE sequence with the following scan
parameters: TR/TE = 7.4/2.4
ms, $$$\alpha$$$ = 6°, RBW = 400 Hz/pix, FOV = 220 mm, 1 mm^{3}
isotropic resolution, 48,000 spokes for radial acquisitions, acquisition time ~ 6 min. Five scans were
acquired for each trajectory; the phantom was translated and rotated during
scans two and four, respectively. Rigid-body motion measurements from four electromagnetic
(EM) sensors (Robin
Medical, Baltimore, MD) placed on
the surface of the phantom were combined using singular value decomposition^{5} and used to retrospectively
correct the *k*-space lines from each
scan. Radial reconstruction was performed in Matlab (R2016b; MathWorks) by applying a weighting
function computed using an iterative numerical approach^{6} and regridding the data using
the NUFFT toolbox.^{7}

*Volunteer experiment.* Three volunteers were scanned
at 3T after obtaining informed consent. An MPRAGE sequence was used to generate
a T_{1}-weighted image with the following scan parameters: TR/TE/TI = 1540/2.77/800 ms, $$$\alpha$$$ = 5°, RBW = 300 Hz/pix, FOV = 256 mm, 1 mm^{3}
isotropic resolution, 48,000 spokes, acquisition time ~ 6.5 min. Three scans were acquired for each
sampling trajectory with: 1) no motion; 2) six abrupt head movements; 3) slow
continuous nodding. Reconstruction was performed as described above using EM
tracking motion measurements to retrospectively correct the *k*-space data. The normalized
root-mean-square error (NRMSE) and structural similarity index (SSIM)^{8} were computed relative to the
‘no motion’ scan. Mean motion scores^{9} were also estimated for each
scan to ensure head movements were comparable for each sampling trajectory.

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