Introduction

The MR-linac^[1], a combination of an MRI-scanner and linear accelerator, can enable real-time MR-guided radiotherapy (MRgRT) when given a continuous stream of high-quality and low-latency tumor motion estimates^[2]. Focusing on the latter, we previously introduced MR-MOTUS^[3]: a $$$k$$$-space domain framework which provides high-quality motion-fields with high temporal resolution (>10Hz)^[4]. However, MR-MOTUS has relatively long reconstruction times ($$$\sim$$$minutes) and is therefore unable to meet the low-latency requirement for real-time MRgRT.

In this work we propose low-rank MR-MOTUS, which does meet the low-latency requirement. Low-rank MR-MOTUS factorizes space-time motion-fields into fixed spatial components and dynamic temporal components. This allows to 1) exploit spatial and temporal correlations in motion, and 2) split the reconstruction into a large-scale off-line training phase, and a small-scale on-line inference phase. In the on-line phase the temporal components are estimated with a latency of 155ms, thereby paving the way for real-time MRgRT on the MR-linac.

Theory

MR-MOTUS
The MR-MOTUS forward model^[3] explicitly relates dynamic motion-fields $$$\mathbf{T}_t\in\mathbb{R}^{3N}$$$ and a reference image $$$\mathbf{q}_0\in\mathbb{C}^N$$$ to the dynamic $$$k$$$-space measurements $$$s_t$$$: $$s_t[\mathbf{k}]=F(\mathbf{T}_t|\mathbf{q}_0)[\mathbf{k}]=\int_\Omega q_0(\mathbf{r}_0)e^{-i2\pi\mathbf{k}\cdot\mathbf{T}_t(\mathbf{r}_0)}\ \text{d}\mathbf{r}_0.\quad\quad\quad(1)$$Space-time motion-fields can be reconstructed from minimal $$$k$$$-space snapshots $$$s_t$$$ by solving the inverse problem while exploiting the correlations of motion in both space and time, with appropriate regularization $$$\mathcal{R}$$$:$$\min_{\mathbf{T}} \ \frac{1}{M}\sum_{t=1}^M \left\lVert \mathbf{F}( \mathbf{T}_t ) - \mathbf{s}_t \right\rVert_2^2 + \lambda \mathcal{R}(\mathbf{T}).\quad\quad\quad(2)$$However, the number of parameters and reconstruction time of (2) scale with the required number of dynamics $$$M$$$ (~10²). Consequently, the reconstruction times are in the order of minutes which limits the applicability to low-latency processes such as real-time MRgRT.

Low-rank MR-MOTUS
We propose to factorize $$$\mathbf{T}_t$$$ in spatial ($$$\mathbf{\Phi}$$$) and temporal components ($$$\mathbf{\Psi}$$$):$$\mathbf{T}_t(\mathbf{r})=\sum_{q=1}^R \mathbf{\Phi}^q(\mathbf{r})\Psi^q(t).\quad\quad\quad(3)$$Here $$$\mathbf{\Phi}^q$$$ denote the voxel-by-voxel principal direction and magnitude, and $$$\Psi^q$$$ denote the temporal scaling (see Fig. 2a). Mathematically, (3) can be interpreted as the explicit constraint $$$\textrm{rank}(\mathbf{T})\le R$$$, which has appeared previously for image reconstruction^[5]. The factorization naturally leads to two phases in the motion-field reconstruction: 1) an off-line training phase which solves Equation (2) with $$$\textrm{rank}(\mathbf{T})\le R$$$, and 2) an on-line inference phase which reconstructs the time-dependent factor $$$\mathbf{\Psi}_t$$$:

Off-line training:
$$\mathbf{\Phi}^*,\mathbf{\Psi}^*=\text{argmin}_{\mathbf{\Phi},\mathbf{\Psi}}\quad\frac{1}{M_\text{train}}\sum_{t=1}^{M_{\text{train}}} \left\lVert \mathbf{F}\left( \mathbf{\Phi}\mathbf{\Psi}^T_t \right) - \mathbf{s}_t \right\rVert_2^2 + \lambda \mathcal{R}(\mathbf{\Phi},\mathbf{\Psi}).\quad\quad\quad(4)$$

On-line inference:
$${\mathbf{\Psi}_t} = \text{argmin}_{\mathbf{\Psi}_t}\label{eq:onlineinference} \ \left\lVert \mathbf{F}\left( \mathbf{\Phi}^*\mathbf{\Psi}^T_t \right) - \mathbf{s}_t \right\rVert_2^2, \qquad t>M_\text{train}.\quad\quad\quad(5)$$
In addition to the explicity low-rank constraint, (4) was regularized by enforcing orthonormal and orthogonal columns in $$$\mathbf{\Phi}$$$ and $$$\mathbf{\Psi}$$$, respectively. Here $$$\mathbf{\Psi}_t\in\mathbb{R}^R$$$, where typically $$$R\le3$$$ suffices. Hence, (5) is very small-scale and can be solved in under 130ms for $$$R=1$$$.

Methods

Low-rank motion analysis on 2D data
Two-dimensional $$$k$$$-space data were acquired in a single volunteer using a continuous 2D sagittal golden-angle radial readout and a spoiled gradient echo sequence. The volunteer was instructed to start with 10 second breath-hold and then continue breathing. From the breath-hold data we reconstructed a reference image $$$\vec{q}_0$$$ and motion-fields were reconstructed off-line from all free-breathing data using 5 spokes/dynamic. The motion-field reconstructions were performed with different ranks ($$$R=1,\dots,4)$$$ and compared with ground-truth images reconstructed with 25 spokes/dynamic using compressed sensing^[6].

Real-time 3D respiratory motion-field reconstructions
3D respiratory motion-fields were reconstructed from prospectively undersampled non-Cartesian $$$k$$$-space data with a rank-1 motion model. Data was continuously acquired during breath-hold and free-breathing using a spoiled gradient echo with a 3D golden mean cone trajectory^[7]. From $$$\sim$$$30s breath-hold a reference image was reconstructed. First, the off-line large-scale problem (4) was solved using 96 dynamics and 40 cones/dynamic. Subsequently, the on-line small-scale problem (5) was solved in 130ms/dynamic, using a total of only 6 $$$k$$$-space points/dynamic extracted from 3 different cones. See Fig. 2b for a complete overview of the low-rank reconstructions.

Results

The low-rank analysis for $$$R=1,\dots,4$$$ is shown in Fig. 3. All reconstructions show good correspondence to the ground-truth compressed sensing reconstruction and minimal differences can be observed between the different ranks.

Fig. 4 shows 3D rank-1 respiratory motion-fields that were reconstructed in 130ms/dynamic in the on-line training phase, from data acquired in 24ms/dynamic. The temporal resolution was decreased for visualization purposes.

Discussion & Conclusion

We have enhanced MR-MOTUS with an explicit low-rank representation to reconstruct time-dependent motion-fields. Results confirm the validity of the low-rank representation. Reconstructions in the on-line phase show promising results with reconstruction times of about 130ms/dynamic. The required data was acquired in just 24ms/dynamic, yielding a total reconstruction latency of 154ms.

In practice we envision the following workflow. A reference image could be obtained with a respiratory-sorted 3D golden-mean-cone acquisition of 2-3 minutes. This same data can be used to train the low-rank model. Finally, motion-fields can be inferred from continuously acquired data during free-breathing with a latency of 154ms and a temporal resolution of 40Hz. Thus, low-rank MR-MOTUS can thereby open the window for real-time MRgRT on the MR-linac.

Acknowledgements

This research is funded by the Netherlands Organisation for Scientific Research, domain Applied and Engineering Sciences, Grant number 15115.