Respiratory motion remains a problem in free-breathing abdominal MRI, introducing artefacts in the reconstructed images. Self-gating techniques have been proposed to estimate respiratory motion and reconstruct data acquired within a respiratory gating window. Most of these techniques estimate simple 1D respiratory motion from the repeatedly acquired k-space centre [1]. However, respiratory motion is only approximately periodic, so the use of such simple signals for self-gating may only partially correct for respiratory motion. In this work we propose a novel method based on manifold alignment (MA) which permits richer data to be used to identify k-space data acquired at similar respiratory positions. This enables reconstruction of motion-free abdominal images throughout the respiratory cycle to better capture respiratory intra- and inter-cycle variations.

The intuition behind our technique is that respiratory motion correspondences can be established in a reduced-dimensional space (or manifold space (MS)) [2] of the acquired k-space data using MA [3]. This is achieved by dividing the data into groups. Each group contains data that is directly comparable, but acquired at different respiratory positions. A common MS is formed in which data from different groups that were acquired at similar respiratory positions are close together. This alignment of the groups is performed using our recently proposed MA scheme [3], which is based on locally linear embeddings (LLE), and involves no inter-group comparisons of data.

An overview of the proposed technique is shown in Fig. 1. Acquisition is performed using a golden-angle radial trajectory. All k-space radial profiles are evenly divided into $$$L$$$ groups according to their profile angles (Fig. 1, left), where each group contains $$$N$$$ profiles ($$$N>L$$$). For forming the MS only, each profile is Gaussian-weighted ($$$\sigma_1$$$) based on its distance to the centre of the profile. The MA scheme is used to embed all $$$L$$$ profile datasets into the common MS in which data acquired at similar respiratory positions are close together (Fig 1, middle), by minimising a joint cost function [3]:

$$argmin_{\space Y_1...Y_L}(\sum_{l=1}^{L}\phi_{l}(Y_{l})+\mu\sum_{n=1,m=1,m\neq n}^L\sum_{i,j}^{N}U_{ij}^{(nm)}\parallel y_{i}^{(n)}-y_{j}^{(m)}\parallel ^2) $$

where $$$\phi_{l} $$$ is the LLE cost function [4], $$$y^{(n)}_i$$$ represents the MS coordinates of profile $$$i$$$ in group $$$n$$$, $$$Y_l$$$ is a matrix containing the $$$y$$$ for group $$$l$$$, $$$U^{(nm)} $$$ is an inter-group similarity kernel [3], and $$$\mu$$$ is a weighting parameter that balances the intra-group/inter-group terms.

Image reconstruction (at as many respiratory positions as k-space profiles acquired) is performed by selecting the $$$P$$$ highest weighted profiles from each of the $$$L$$$ groups, followed by non-uniform inverse FFT. The weight is a product of two terms: Gaussian-weighted profile distances in MS ($$$\sigma_2$$$) and time ($$$\sigma_3$$$) respectively.

Our method was evaluated on simulated and in-vivo datasets. For the simulated dataset, 100 2D sagittal dynamic liver images were acquired. A smooth synthetic dynamic sequence of 10000 images was simulated from this acquisition using B-spline interpolation. One k-space radial profile per image was simulated to mimic a realistic acquisition process. Parameter settings for reconstruction are reported in the caption of Fig. 2.

A 2D radial golden-angle acquisition was performed under free-breathing for ~45s on a healthy subject, resulting in 15000 k-space profiles. Data was acquired on a Philips 1.5T scanner using a 28 channel-coil (b-SSFP, 2x2x8mm resolution, flip angle 70$$$^o$$$, TR/TE=3.08/1.54).

A virtual navigator (VN) was used to measure the superior-inferior translation of the liver-lung boundary on the simulations. When visualising the MS, each profile was colour-coded using the VN value. The resulting MS (Fig. 2) shows that profiles with similar VN values (colours) are embedded close to each other. The VN values for the ground-truth (red) and reconstructed images (blue) of the entire simulated sequence are shown in Fig. 3 (Pearson’s r=0.9504).

In-vivo data was reconstructed with the proposed method. Example reconstructions at 4 different respiratory positions are shown in Fig. 4. A k-space centre self-gating reconstruction was also performed for comparison purposes at mid-inhale (Fig. 5). Less blurring can be observed with the proposed method, especially at small structures.

We have presented a novel technique based on MA, which enables reconstruction of motion-free abdominal images throughout the respiratory cycle. The method allows reconstruction of images at as many respiratory positions as k-space profiles acquired and also provides information for retrospective motion correction. The proposed MA-based scheme enables much richer profile data to be used for self-gating, resulting in less blurring when compared to the central k-space gating method.