Dynamic contrast-enhanced magnetic resonance angiography(DCE-MRA) has been widely used for diagnostic assessment in clinical practices^1,2. To enhance the conspicuity of arteries relative to unwanted background tissues, subtraction between the pre-contrast and the post-contrast images was typically performed displaying maximum intensity projection images(MIP)^3-5. Nevertheless, if there exists non-stationary signal transition due to time-drifting field inhomogeneity, and subject motion etc, the subtraction leads to incomplete background suppression, impairing the detectability of arteries as well as small vessel particularly at high reduction factors^6,7. In this work, we propose a novel DCE-MRA method exploiting motion subspace learning and sparsity priors for robust angiogram separation, in which the motion subspace is learned using partial Casorati matrix without any motion information while image reconstruction with sparsity priors is performed to jointly estimate motion-induced artifacts and DCE angiograms of interest under the framework of the decomposition. Simulation and experimental studies show that the proposed method is highly competitive with the competing methods including subtraction and fast reconstruction techniques.

1) DCE Signal Model: we model DCE signals in x-t space in the presence of noises $$$\bf{N}$$$ by linear combination of: stationary background signals $$$\bf L$$$, 2) motion-induced signals $$$\bf{S_M}$$$, and 3) DCE angiograms of interest $$$\bf{S_D}$$$: $$\bf{X =L+S_M+S_D+N}$$ Defining the stationary background image $$$\bf L$$$ as the mean image of the partial Casorati matrix $$$\bf{X_L =\left[ x_0 \ x_1 \ \cdot\cdot\cdot x_{T_{p}-1} \right]}$$$ constructed by pre-contrast images column by column , the remainder of the DCE signal model can be sparsely represented using $$$\bf{S_M}$$$ and $$$\bf{S_D}$$$.

2) Motion Subspace Learning: To avoid signal contamination among $$$\bf{S_M}$$$ and $$$\bf S_D$$$, we employ subspace projection(SP)-based separation for motion-induced artifacts and DCE angiograms. Assuming that the neighboring pixels share similar distribution across time frames while showing strong correlations inversely proportional to the distances between pixels, we emulate dynamic information to generate pixel statistics by including all possible variations of the background motions where: 1) pre-contrast images are translated or rotated and 2) each pixel of the pre-contrast images is replaced by the value randomly taken within a specific radius. To enhance the representation of artifacts, the synthesized images are subtracted from the background mean image, and the motion subspace is learned by using singular value decomposition and taking the left singular vectors from the residual images $$$\bf X_R $$$:

$$\bf{X_R = \left[ x_{r,0}- \boldsymbol{\mu} \ \ \ x_{r,1}- \boldsymbol{\mu} \ \ \cdot\cdot\cdot \ x_{r,T_{p}-1} - \boldsymbol{\mu} \right] \overset{SVD}{\rightarrow} P_M \Sigma_M V_M^H} $$

where $$$ \boldsymbol{\mu} =\bf{\left (x_0 + x_1 + \cdots +x_{T_p-1} \right )/T_p} $$$ is the background mean vector. The SP-based separation is performed by projecting sparse signal model ($$$\bf{S=S_M + S_D)}$$$ onto the subspace spanned by $$$\bf{P_M}$$$:

$$ \bf S_M=P_M P_M^H S$$

The DCE signal model is modified:

$$ \bf X = L+P_M P_M^H S+S_D +N \\ \ = L+ AS_M + BS_D +N $$

where $$$ \bf A=P_M P_M^H$$$, $$$\bf B=P_M P_M^H+I$$$ , and $$$\bf I $$$ is the identity matrix.

3) Problem Formulation: Exploiting SP-based signal model, the proposed image reconstruction is formulated as:

$$ \bf \underset{S_M,S_D}{min} \lambda_M \left \| \varphi \left ( S_M \right ) \right \|_1 + \left \| \varphi \left ( S_D \right ) \right \|_1 \\ s.t. \ \ \ r=F_u \left ( AS_M+BS_D \right ) $$

where $$$ \lambda_M$$$ is the balancing parameter between $$$\bf S_M$$$ and $$$\bf S_D$$$, $$$ \bf r=y-F_u \left ( L \right ) $$$ is the subtracted k-space between measured and background mean signals, and $$$ \varphi \left ( \cdot \right ) $$$ is a sparsifying transform function. To solve the objective function, we employ an alternating minimization with respect to $$$\bf S_M$$$ and $$$\bf S_D$$$.To validate the proposed method, 3D DCE-MR images (320×240×144×28) were acquired on a 3T clinical scanner. Fig. 1 shows the effective decomposition of the sparse components. Fig. 2 compares the proposed method with subtraction. The subtraction yields motion-induced artifacts in the region of jaw and eyes while the proposed method clearly depicting vascular structures with robust suppression of background signals. Fig. 3 compares the performance of the accelerated reconstruction against existing methods(R=40): low-rank completion(LRC), dynamic compressed-sensing(DCS), and k-t robust-principle-component-analysis(k-t RPCA). The proposed method leads to the most superior quality of DCE angiogram in delineating vessel morphology and suppressing motion artifacts over the existing methods.We successfully demonstrated a novel DCE-MRA method with sparsity priors under the framework of decomposition. It is expected that the proposed method would enable a rapid, robust DCE-MRA in the presence of motion without additional registration algorithms, thus widening its applications in a clinical routine.