The MR-Linac is a combination of an MR-scanner and radiotherapy linear accelerator (Linac) which holds the promise to increase the precision of radiotherapy treatments with MR-guided radiotherapy by monitoring motion during radiotherapy with MRI, and adjusting the radiotherapy plan accordingly. Optimal MR-guidance for respiratory motion during radiotherapy requires MR-based 3D motion estimation with a latency of 200-500 ms. Currently this is still challenging since typical methods rely on MR-images, and are therefore limited by the 3D MR-imaging latency. In this work, we present a method to perform non-rigid 3D respiratory motion estimation with 170 ms latency, including both acquisition and reconstruction. The proposed method called real-time low-rank MR-MOTUS reconstructs motion-fields directly from k-space data, and leverages an explicit low-rank decomposition of motion-fields to split the large scale 3D+t motion-field reconstruction problem posed in our previous work into two parts: (I) a medium-scale offline preparation phase and (II) a small-scale online inference phase which exploits the results of the offline phase for real-time computations. The method was validated on free-breathing data of five volunteers, acquired with a 1.5T Elekta Unity MR-Linac. Results show that the reconstructed 3D motion-field are anatomically plausible, highly correlated with a self-navigation motion surrogate (R = 0.975 +/- 0.0110), and can be reconstructed with a total latency of 170 ms that is sufficient for real-time MR-guided abdominal radiotherapy.
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