Burhaneddin Yaman^{1,2}, Sebastian Weingärtner^{1,2,3}, Steen Moeller^{2}, Nikolaos Kargas^{1}, Nicholas Sidiropoulos^{4}, and Mehmet Akçakaya^{1,2}

^{1}Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN, United States, ^{2}Center for Magnetic Resonance Research, University of Minnesota, Minneapolis, MN, United States, ^{3}Computer Assisted Clinical Medicine, University Medical Center, Heidelberg University, Mannheim, Germany, ^{4}University of Virginia, Charlottesville, VA, United States

### Synopsis

Recently a new method called TOPAZ was developed
for cardiac phase-resolved myocardial T_{1} mapping, which is performed in
breath-hold duration that subsequently limits its spatial resolution. These datasets
are multidimensional which makes tensor regularization a natural fit for
regularization. In this work, we sought to compare different tensor
regularization techniques to enable high-resolution TOPAZ acquisitions.

### INTRODUCTION

Myocardial T_{1} mapping is typically performed using single-shot imaging with 150-200 ms temporal resolution. However, for subjects with high heart rates, improved temporal resolution may be needed, especially for quantifying mobile structures, like papillary muscles. Recently a new technique, called TOPAZ [1], which is performed in a breath-hold, was proposed to acquire T_{1} maps in a cardiac phase-resolved manner at improved temporal resolutions. However, spatial resolution of TOPAZ is limited due to breath-hold durations. Thus, regularization may be desirable for high-resolution imaging in order to alleviate the noise amplification. TOPAZ acquisitions are multi-dimensional consisting of two spatial, a cardiac phase and a contrast dimension. Low-rank matrix regularization has been studied in this context [2]. However, low-rank tensor regularization (LRTR) [3] may be a better fit to mitigate the noise amplification for such datasets since low-rank matrix regularization is limited to pairwise interactions, which hinders capturing interaction between multiple dimensions. In this work, we sought to compare different LRTR techniques based on PARAFAC [4] and Tucker [5,6] decompositions for high-resolution TOPAZ acquisitions.### METHODS

Imaging was performed on five healthy subjects at 3T, using a 30-channel receiver coil array with resolution=1.3×1.3mm^{2}, FOV=300×225 mm^{2}, temporal resolution=60ms, partial Fourier=6/8, in-plane acceleration=3 and ACS lines=24. Datasets used in this study *$$$ X (x,y,t,c) \in {\mathbb C}^{M\times N \times T \times C} $$$ * were 4-dimensional where $$$(x,y)$$$ were the spatial dimensions, $$$t$$$ was the cardiac phase and $$$c$$$ represented different T_{1} contrasts. This led to $$$T$$$ cardiac phases and $$$C$$$ different T_{1}-weightings per phase for a total of $$$T*C$$$ images in each dataset. Subsampled k-space data were first reconstructed using GRAPPA [7] and combined using SENSE-1 coil combination. Different tensor decompositions (**Fig.1**) were applied to these image series to reduce noise amplification arising from GRAPPA. PARAFAC decomposition represents a tensor as a sum of rank one tensors, while Tucker decomposition factorizes a tensor into a core tensor multiplied with factor matrices along each mode. For both techniques, global processing was applied by processing the whole dataset, which contains all tissue types and contrasts, and local processing was performed by processing smaller patches in x-y plane, which is likely to have similar tissue properties. The patches were of size $$$8\times 8 \times T \times C$$$, and the processed patches were combined via averaging. Rank for PARAFAC decomposition was selected empirically [4]. Since Tucker decomposition has multilinear rank, the high degrees of freedom hinders empirical rank selection. Thus, we proposed a rank selection approach for Tucker method using a principal component analysis based approach, which applies proportion of variance (PoV) to each factor matrix in the tensor. In PoV approach, eigenvalues with negligible contribution to variance and the largest eigenvalue value [8] are discarded and a proportion of the eigenvalues is used to choose the rank for the factor matrix. Hence, multilinear Tucker decomposition was formed using the ranks of factor matrices. For each method, the T_{1} values and spatial variability are measured as the mean and standard deviation myocardial T_{1} for each cardiac phase. These are reported as averages across all cardiac phases.### RESULTS

Representative quantitative TOPAZ T_{1} maps are shown in **Fig. 2**. For both tensor decompositions, local LRTR substantially alleviated the noise artifacts compared to global LRTR and GRAPPA, although some residual blurring is apparent in the local Tucker LRTR approach. Across all subjects, for both tensor decomposition approaches, the absolute differences in estimated mean T_{1} values were <1.5% compared to parallel imaging for all subjects. The spatial variability of myocardial T1 times across all subjects were 352±33ms for GRAPPA, 213±23ms for global PARAFAC and 138±22ms for local PARAFAC, 190±27ms for global Tucker and 149±22ms for local Tucker methods. For both tensor decompositions, local LRTR substantially alleviated the noise artifacts compared to global LRTR.** Fig. 3** depicts the T_{1} times through cardiac phases across a cross-section of the heart. Both local LRTR methods significantly eliminate noise artifacts present in GRAPPA, while not hindering identification of tissue borders. ### DISCUSSION

In this study, LRTR was applied to an image series from TOPAZ T_{1} mapping acquisition in order to mitigate noise amplification. Local processing of small patches in both PARAFAC and Tucker decompositions outperforms global processing of the whole volume data. Additionally, local LRTR has a computational advantage over global LRTR since each patch can be processed independently on GPU or multi-cores. ### CONCLUSION

Local low-rank tensor regularization techniques improve
the spatial resolution of TOPAZ T_{1} mapping to 1.3 mm in-plane resolution despite scan-time limitations imposed by the breath-hold duration.

### Acknowledgements

Grant support: NIH R00HL111410, NIH P41EB015894, NSF CCF-1651825
and NSF IIS-1704074### References

[1] Weingärtner, MRM, 2018; [2] Zhang, MRM,
2015; [3] Trzasko, ISMRM 2015; [4] Yaman, IEEE CAMSAP, 2017; [5] He, IEEE Trans
Med Imaging, 2016; [6] Christodoulou, Nat Biomed Eng, 2018 [7] Griswold, MRM,
2002; [8] Baik, Ann. Probab, 2005.