Julia Velikina^{1} and Alexey Samsonov^{2}

^{1}Medical Physics, University of Wisconsin, Madison, WI, United States, ^{2}Radiology, University of Wisconsin, Madison, WI, United States

### Synopsis

**We describe next generation of Model Consistency COndition (MOCCO) reconstruction 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). ****Purpose**

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 images

^{1} 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 techniques

^{4}. 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

**W**eighted low-rank models and attains
high image quality through

**I**terative model adaptation to complexity of the
local temporal dynamics (WI-MOCCO).

** Theory**

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 pursuit^{5}, 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.

**Methods and Results**

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.

** Discussion**

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).

### Acknowledgements

The
work was supported by NIH (R21EB018483).### References

1. Liang ZP. Spatiotemporal imaging with
partially separable functions. In: Proc of ISBI; 2007; Washington, DC, USA. p
988-991.

2. Lingala
SG, Jacob M. Blind compressive sensing dynamic MRI. IEEE Trans Med Imaging
2013;32(6):1132-1145.

3. Zhao
B, Haldar JP, Brinegar C, Liang ZP. Low rank matrix recovery for real-time
cardiac MRI. In: Proc of ISBI; 2010; Rotterdam, The Netherlands. p 996-999.

4. Velikina
JV, Samsonov AA. Reconstruction of dynamic image series from undersampled MRI
data using data-driven model consistency condition (MOCCO). Magn Reson Med
2014.

5. Davis
G, Mallat S, Avellaneda M. Adaptive greedy approximations. Constr Approx
1997;13(1):57-98.