  With the recent introduction of the MR-LINAC, an MR-scanner combined with a
radiotherapy LINAC, MR-based motion estimation has become of increasing
interest to (retrospectively) characterize tumor and organs-at-risk motion
during radiotherapy. To this extent, we introduce low-rank MR-MOTUS, a
framework to retrospectively reconstruct time-resolved non-rigid 3D+t
motion-fields from a single low-resolution reference image and prospectively
undersampled k-space data acquired during motion. Low-rank MR-MOTUS exploits
spatio-temporal correlations in internal body motion with a low-rank motion
model, and inverts a signal model that relates motion-fields directly to a
reference image and k-space data. The low-rank model reduces the
degrees-of-freedom, memory consumption and reconstruction times by assuming a
factorization of space-time motion-fields in spatial and temporal components.
Low-rank MR-MOTUS was employed to estimate motion in 2D/3D abdominothoracic
scans and 3D head scans. Data were acquired using golden-ratio radial readouts.
Reconstructed 2D and 3D respiratory motion-fields were respectively validated
against time-resolved and respiratory-resolved image reconstructions, and the
head motion against static image reconstructions from fully-sampled data
acquired right before and right after the motion. Results show that 2D+t
respiratory motion can be estimated retrospectively at 40.8
motion-fields-per-second, 3D+t respiratory motion at 7.6
motion-fields-per-second and 3D+t head-neck motion at 9.3
motion-fields-per-second. The validations show good consistency with image
reconstructions. The proposed framework can estimate time-resolved non-rigid 3D
motion-fields, which allows to characterize drifts and intra and inter-cycle
patterns in breathing motion during radiotherapy, and could form the basis for
real-time MR-guided radiotherapy.
