It is important to assess liver stiffness and monitor the progression of fibrosis in patients with liver diseases using non-invasive imaging. Exploiting the fact that the motion of the heart during cardiac cycle is an intrinsic driving source to deform the liver, dynamic tagged MRI can be employed to measure the cardiac motion induced displacements using harmonic phase images and thereby evaluate the corresponding strain maps in the liver.¹ In this work, we propose a novel framework of compressed sensing (CS) for dynamic tagged liver MRI, in which: 1) data is acquired using tag-constrained, incoherent undersampling in $$$k$$$-$$$t$$$ space, 2) a time series of morphologies is decomposed into transient tag-only images and stationary tag-free liver images, 3) both morphological components are then reconstructed directly from the tag-constrained, undersampled $$$k$$$-$$$t$$$ space, and 4) the transient tag-only images are employed to estimate time-varying displacements and the corresponding strain maps while the stationary tag-free liver images are used as a structural roadmap.

Tag-Constrained Data Sampling: Tags in MR images are represented as periodic spectral peaks in the Fourier domain (Fig.1a-b). Since a time series of the spectral peaks contains information on transient liver motion, it is important to sample them during data acquisition such that tags are not lost during image reconstruction. In this work, we propose tag-constrained, incoherent variable density (VD) undersampling in $$$k$$$-$$$t$$$ space where data is densely acquired in the central region of $$$k$$$-space as well as in the region of spectral peaks closest to the center of $$$k$$$-space. Figs.1c-1e represent the proposed, tag-constrained VD undersampling in the in-plane $$$k$$$-space, probability density function of sampling in the $$$k_y$$$ direction, and the resulting sampling pattern in $$$k$$$-$$$t$$$ space, respectively.

Separation of Transient Tag and Stationary Liver Components: Dynamic tagged MR images are composed of the following two morphological components: tag (texture) and liver (cartoon): $$$X = L + T + N$$$ where $$$X$$$ is the Casorati matrix ($$$x$$$-$$$t$$$ space) that contains a time series of tagged MR images, $$$L$$$ is the liver image, $$$T$$$ is the tag image, and $$$N$$$ is the additive noise matrix. However, it is noted that the proposed signal model doubles the number of unknowns and becomes highly underdetermined with increasing reduction factors. Exploiting that tags vary slowly with changing time, $$$T$$$ in the Casorati martrix form is highly correlated and thus has low rank properties. Reconstructing $$$L$$$ is interpreted as an interpolation problem for image inpainting and thus can be formulated as low rank matrix completion problem. Since a tag-free liver image is used only as a structural roadmap, the thresholding value of the matrix completion problem, $$$|| H(L)||_∗$$$, was selected nearly rank 1 for the image to be stationary. Given the considerations above, image reconstruction from the tag-constrained undersampled $$$k$$$-$$$t$$$ space is performed by solving the following optimization problem: $$L,T=\underset{L,T}{\mathrm{min}}|| d-F_u (L+T)||_2^2+λ_L || H(L)||_∗+λ_T || Γ(T)||_∗ $$

where $$$d$$$ is the measured data in $$$k$$$-$$$t$$$ space, $$$F_u$$$ is the undersampled Fourier transform operator, $$$H$$$ is the Hankel matrix operator, $$$Γ$$$ is the Casorati matrix operator, $$$λ_L$$$ and $$$λ_T$$$ are the thresholding value parameters, $$$||·||_2$$$ is the Euclidian norm, and $$$||·||_*$$$ is the nuclear norm.

Assessment of Liver Stiffness: The transient tag-only images are used to calculate both displacement and strain maps using conventional HARP method,² while the stationary tag-free liver image is used as a structural roadmap.

In vivo liver data were acquired in a healthy volunteer and a patient (suspicious of liver disease) on 3T (Siemens Trio) using ECG-triggered, breath-hold cardiac cine imaging with spatial modulation of magnetization (FOV = 300mm×300mm, acquisition matrix = 256×174 , slice thickness = 6mm, TE = 3.8ms, TR = 47.6ms, flip angle=11°, six segments per cardiac cycle, tag spacing=7mm, tag orientation = 45°, total acquisition time per slice = 15-20ms). Fig.2 shows a representative, reconstructed k-space and its corresponding decomposed images, $$$X$$$ (Fig.2a), $$$T$$$ (Fig.2b), and $$$L$$$ (Fig.2c), at different cardiac phases. Fig.3 demonstrates that $$$T$$$ and $$$L$$$ are well separated with increasing reduction factors up to 5. Fig.4 compares displacement, P1 strain, and P2 strain maps in between a healthy subject (Figs.4a-c) and a liver patient (Figs.4d-f). Displacement and strains in a patient appear much lower than those in a healthy subject.We successfully demonstrated the feasibility of the proposed, novel CS framework for dynamic tagged liver MRI that employs the tag-constrained sampling and separation to accurately assess liver stiffness. It is expected that the proposed method widens the clinical utility of dynamic tagged liver MRI for diagnosing liver diseases.