A framework for performing a multi-level sparse sampling time-resolved Alternating Direction Method of Multipliers (ADMM) reconstruction has been previously described.[1,2] Here "multi-level" refers to accelerating output volume updates via: (1) skipping phase encodes (SENSE [3]) in k-space, and (2) skipping phase encode updates (view sharing [4]) in time. We provide an updated ADMM framework where the age of each acquired view update is considered in the data fidelity penalty, and show the impact of this change with experimental data.

In this view-shared reconstruction, updates are performed such that for each new acquisition update -- consisting of a common low-pass region, and one of N differing high-pass regions -- freshly acquired views replace their respective encoding positions, and a complementary set of historical high-pass N-1 regions are used to fill out the remainder of the view-shared portion. This schedule provides minimal leading edge enhancement, or anticipation, of temporal change, but at the cost of some trailing-edge, or retention, effect. (A reversed schedule where the oldest center combines with newer high-pass regions will lead to increased anticipation artifact.) In contrast-enhanced studies, the leading edge (bolus inflow) will typically have a much higher rate of change, leading to a reduction of artifact with this, in contrast to the reversed, schedule.

While there are methods that perform filtering [5] or smoothing [6] of the acquired data, typically the shared data is weighted uniformly. As older data is less likely to represent the current state in a dynamic acquisition, reducing the importance of fidelity to older data is likely to improve fidelity to "ground truth." Iterative reconstruction frameworks such as ADMM provide an opportunity to assign a graduated weight or quality to sections of the source data; we leverage this flexibility to assign reduced importance to older data during the sparse reconstruction process.

A time-resolved 3D acquisition was used to evaluate the proposed technique. The experimental setup (Figure 1) consisted of dilute gadolinium filled "bolus" vials moved along tabletop tracks by a computer controlled motor at a known velocity.

We have introduced a scaled data fidelity weighting based upon the age of each view update relative to the most recent data. Explicitly, we linearly weight with respect to the total number of differing phases: with three view-shared phases, the data is weighted either with 1, 2/3, or 1/3, with decreasing weights for older data. We update the ADMM implementation starting with equation 3 from [1]:

$$\mathcal{J}(\mathbf{X}) \triangleq\lambda\sum_{b\in\Omega}\mathcal{P}(\mathbf{\Gamma_{b}M{X}}) +\sum_{t=0}^{T-1}\left\| \left( \mathbf{I}\otimes\mathbf{\Phi}_{t}\mathbf{F} \right) \mathbf{SMX}\delta_{t} - \mathbf{G}\delta_{t}\right\|_{\mathrm{2}}^{2}$$

We replace the time-dependent non-uniform binary sampler $$$\mathbf{\Phi}_t$$$ with a static sampler selecting the entire (still a subset of k-space) set of acquired views, $$$\mathbf{\Phi}$$$. To maintain the time dependence for each update, and impose time-varying data fidelity, the norm above is weighted with the fidelity term $$$\mathbf{\Lambda}_t$$$, which has the time-frame dependent weights on its diagonal:

$$\mathcal{J}(\mathbf{X})\triangleq\lambda\sum_{b\in\Omega}\mathcal{P}(\mathbf{\Gamma_{b}M{X}}) +\sum_{t=0}^{T-1}\left\| \left( \mathbf{I}\otimes\mathbf{\Phi}\mathbf{F} \right) \mathbf{SMX}\delta_{t} - \mathbf{G}\delta_{t}\right\|_{\mathbf{\Lambda}_{t}}^{2}$$

It follows that the data-fidelity subproblem in equation 7 from [1] is updated from:

$$\mathbf{W}_{i+1} = \sum_{t=0}^{T-1}\left( \mathbf{I}\otimes \left( \mathbf{\Phi}_{t}^{*}\mathbf{\Phi}_{t} + \mu_{1}\mathbf{I}\right)^{-1} \right)\mathbf{R}_{t}\delta_{t}^{*}\\\mbox{where }\mathbf{R}_{t} = \left(\mathbf{I}\otimes\mathbf{\Phi}_{t}^{*}\right)\mathbf{G}\delta_{t} + \mu_{1} ((\mathbf{I}\otimes\mathbf{F})\mathbf{SMX}_{i} + \eta_{1,i})\delta_{t}$$

to:

$$\mathbf{W}_{i+1} = \sum_{t=0}^{T-1}\left( \mathbf{I}\otimes \left( \mathbf{\Phi}^{*}\mathbf{\Lambda}_t\mathbf{\Phi} + \mu_{1}\mathbf{I}\right)^{-1} \right)\mathbf{R}_{t}\delta_{t}^{*}\\\mbox{where }\mathbf{R}_{t} = \left(\mathbf{I}\otimes\mathbf{\Phi}^{*}\mathbf{\Lambda}_{t}\right)\mathbf{G}\delta_{t} + \mu_{1} ((\mathbf{I}\otimes\mathbf{F})\mathbf{SMX}_{i} + \eta_{1,i})\delta_{t}.$$

The acquired 3D motion phantom data was reconstructed using the described ADMM reconstruction with and without data fidelity weighting. This data (Figure 1) contains a motion-controlled contrast-filled vial travelling through the FOV. Figure 2 focuses on the trailing edge of the upper vial, highlighting the space just vacated by the moving vial. The desired reduction in trailing edge artifacts is clear, as seen in Figure 2. As the high-pass regions contain the "historical" data, the ring-like artifact observed from the recently-departed cylindrical vial is as expected.As seen in Figure 2, the energy in the trailing edge / retention artifact from the phantom study had been reduced by ~7dB. This is without any change required to the acquisition technique. Future work will include considering alternate weighting profiles, as well as evaluating the impact of this change to DCE (Dynamic Contrast Enhanced) MR acquisitions, where changes to AIF response and enhancement fidelity will be of interest.