Many algorithms were proposed to provide useful images from undersampled MRI data to facilitate higher temporal resolutions in dynamic MRI. One family of such techniques exploits spatio-temporal correlations in MR image series by describing temporal progression of each pixel by a low-rank model. Such models are typically constructed from several temporal basis functions either pre-estimated from low-resolution images¹ or estimated during image reconstruction under sparsity constraints^2,3. While based on a powerful concept, this approach is limited by the fundamental inability of low-rank models to adequately describe the multitude of temporal dynamics due to physiological and patient motion typically present in dynamic datasets. Recently, Model Consistency COndition (MOCCO) reconstruction was proposed in an attempt to overcome the problems of low-rank techniques⁴. MOCCO still employs low-rank models pre-estimated from data as a source of prior knowledge. However, unlike its predecessors, MOCCO yields realistic full-rank solutions and alleviates well-known problems of model order selection, accuracy and precision of solution. Here, we describe next generation of MOCCO which makes use of new Weighted low-rank models and attains high image quality through Iterative model adaptation to complexity of the local temporal dynamics (WI-MOCCO).

The original MOCCO estimates dynamics series $$${S}=\left(s(x_\rho,t_k)\right)\in {C}^{n_p\times n_t}$$$ solving$$\min_{\bf{S}}\left(\|vect(\bf{ES})-\bf{d}\|_2+\lambda\|\bf{S}(\bf{U}_K\bf{U}_K^*-\bf{I}_t)\|_X\right)\qquad[1]$$where E is the joint encoding matrix for all time frames, d is a stacked vector of k-space data, $$$I_t$$$ is the $$$n_t\times n_t$$$ identity matrix, U_k is a low-rank model formed by the first K pre-estimated principal components (PCs). In general, higher K favors more accurate and lower K more precise solutions. Sparsity promoting norm X=L1 enables higher rank solutions than implied by the model order K. We note that MOCCO projection may be written as $$$\bf{U}_K\bf{U}_K^*=\bf{U}\bf{H}_K\bf{U}^*$$$ where $$$\bf H_K$$$ is a diagonal filter matrix,$$\left({H}_K\right)_{n,n}=1\ \rm{ if }\ n\le K;\ 0\ \rm{ otherwise},\ $$selecting the first K components from all PCs U. Here, we propose to use a more general weighting matrix W(K) with a smoothly decaying function of K on the main diagonal for a smooth transition from lower to higher order PCs (Fig. 1). Second, we note that using a temporal model of the same rank for all pixels is hardly optimal, as pixels representing different anatomy require different order models. Therefore, we propose to adapt the model order on the per-pixel basis to achieve overall lower average model rank by solving$$\min_{\bf{S,K}}\left(\|vect(\bf{ES})-\bf{d} \|_2+\lambda\sum_\rho\|\bf{s}_\rho(\bf{UW}(K_\rho)\bf{U}^*-\bf{I}_t\|_X+\beta\sum_\rho\|\bf{W}(K_\rho)\|_1\right)\qquad[2]$$Here, $$$s_\rho$$$ is $$$\rho_{th}$$$ row of S. Starting with $$$K_\rho=1$$$, the model adaptation stage involves solution of Eq. [2] by alternating between two minimization problems:$$\bf{S}^{(n+1)}=\min_{\bf{S}}\left(\| vect(\bf{ES})-\bf{d}\|_2+\lambda\sum_\rho\|\bf{s}_\rho\bf{UW}^{(n)}\left(K_\rho^{(n)}\right)\bf{U}^*-\bf{I}_t)\|_X \right)\qquad[3]$$$$K_\rho^{(n+1)}=\min_{K_\rho}\left(\|\bf{s}^{(n+1)}_\rho(\bf{UW}(K_\rho)\bf{U}^*-\bf{I}_t \|_2\right),\forall\rho\qquad[4]$$The first problem is solved with X=L2 using conjugate gradients. The second problem is solved using an incremental fitting similar to orthogonal marching pursuit⁵, which makes a decision on increasing local model order in each iteration by testing objective function value in Eq. [4] for predefined tolerance. On the final stage, the adapted model is used to solve Eq. [3] with X=L1.

The dynamic contrast-enhanced myocardial perfusion data were acquired on a 1.5T Siemens Sonata MR system with saturation recovery TrueFISP protocol (TR/TI=2.3/90ms, FA=50°, R=2, 40 frames) using an 8-element coil and combined to generate a single channel 2D VD Cartesian k-space data. The MOCCO filter was constructed using Hermitian splines (Fig. 1). The methods were compared to adaptive blind compressed sensing (Blind CS) technique2. Parameters of all reconstructions were optimized to minimize corresponding root-mean-squared errors (RMSE). Figure 2 shows RMSE dependance on the model order K. Standard low-rank model produced by the step function shows typical diverging behavior as K increases, which is reversed by the proposed smooth filter. The proposed adaptation of weighted models results in minimized error compared to all reconstructions including Blind CS. Basic trends observed in Figure 2 persist in image errors (Fig. 3) exhibiting increased noise/structured artifacts in standard MOCCO and their reduction in weighted and weighted+adaptive MOCCO. The latter also shows significantly less pronounced loss of structural details than Blind CS reconstruction which comes at expense of slighly increased noise observed in the MOCCO images.The weighted models proposed in the paper demonstrated superior performance over the standard (non-weighted) PC models. Weighted models de-emphasize contributions from higher order singular vectors reducing the sensitivity of reconstruction to noise while preserving the span of the model to accurately approximate a variety of temporal dynamics. Further, proposed pixel-adaptive selection of weighted model orders results in maximized combination of accuracy (preservation of structural details) and precision (noise level) in the reconstructed images. The combination of these features resulted in superior performance of WI-MOCCO without extensive computational load (3 min vs. 20 min for WI-MOCCO and Blind CS, respectively).