Estimating Shared Relaxation and Susceptibility Tensor Eigenvectors Enhances STI Tractography in the Heart, Kidney, and Brain
Russell Dibb1,2, Luke Xie1,3, and Chunlei Liu4,5

1Center for In Vivo Microscopy, Duke University, Durham, NC, United States, 2Biomedical Engineering, Duke University, Durham, NC, United States, 3Utah Center for Advanced Imaging Research, University of Utah, Salt Lake City, UT, United States, 4Brain Imaging & Analysis Center, Duke University, Durham, NC, United States, 5Radiology, Duke University, Durham, NC, United States

### Synopsis

Conjoint relaxation and susceptibility tensor imaging (CRSTI) uses both magnitude- and phase-derived tensor data to compute susceptibility-based tractography in magnetically anisotropic tissues. CRSTI reduces image artifacts that appear in traditional susceptibility tensor imaging by taking advantage of mutual eigenvector data in relaxation and susceptibility tensors. We present an efficient conjoint tensor estimation algorithm and demonstrate improved susceptibility-based tractography in myofibers, renal tubules, and axon fiber bundles. As susceptibility imaging is sensitive to both microstructure and cellular content, CRSTI is a potential tool for studying disease in tissues throughout the body.

### Purpose

Susceptibility tensor imaging (STI) can probe tissue microstructure and cellular content1,2, but is limited by incomplete or incorrect phase information. As a result, STI noticeably differs from DTI when used to map fiber orientations in the brain3 and heart4. By including R2* information, one study showed that these differences can be mitigated to improve susceptibility-based fiber tractography in the heart using conjoint relaxation and susceptibility tensor imaging (CRSTI)4. This work presents a fast and effective CRSTI algorithm that improves STI tractography in not only the heart, but also the kidney and brain.

### Methods

Organ specimens were excised from adult, male C57BL/6 mice. One mouse was perfusion fixed with 50 mM Gd-HP-DO3A in 10% buffered formalin4. The chambers of the heart were then perfused with agarose gel that solidified before the organ was excised. A second mouse was perfusion fixed with 10% buffered formalin5. The kidney was excised and immersed in formalin overnight. A third mouse was perfusion fixed with 50 mM of Gd-HP-DO3A in 10% buffered formalin following a procedure described by Johnson et al.6 The head was removed with the brain intact and immersed in formalin overnight. All three specimens were stored in a solution of 2.5 mM Gd-HP-DO3A in 10 mM phosphate-buffered saline prior to scanning. MRI data was acquired from each organ using the protocols in Table 1.

Normalized phase data were calculated from the multi-orientation GRE image data using iHARPERELLA7. Susceptibility and relaxation tensors were calculated using MATLAB’s LSQR algorithm with mean-value regularization8 and constraints on high spatial frequencies3. The susceptibility tensor (${\bfχ}$) was reconstructed by inverting the susceptibility-phase relationship2. A second-order relaxation tensor (${\bf R}$) was fit to the apparent R2* observed in each subject orientation4. R2* is smallest in the direction parallel to the long axis of axons9, myofibers4, and renal tubules, so the orientation of these structures is indicated by the minor eigenvector of ${\bf R}$. In ${\bfχ}$, orientation is indicated by the major eigenvector in myofibers and axon fiber bundles2,10, and by the minor eigenvector in the renal tubules5. To estimate a set of shared eigenvectors (${\bf Q}$), a CRSTI algorithm was implemented to eigendecompose a weighted combination of ${\bfχ}$ and ${\bf R}$: $${\ttν{\bfχ}–{\bf R}={\bf Q}(ν{\bfΛ_χ}–{\bfΛ_R}){\bf Q}^T}$$ Here, ν (Hz) attempts to assign equal weight to the susceptibility and relaxation tensors, and ${\bfΛ_χ}$ and ${\bfΛ_R}$ are diagonal matrices of the eigenvalues of ${\bfχ}$ and ${\bf R}$, respectively. Relaxation is defined as –${\bf R}$, and the sign of ν is selected to ensure that the major eigenvector of ${\bf Q}$ aligns with the tissue structure orientation. Tractography was then performed on the STI, CRSTI, and DTI data using Diffusion Toolkit and TrackVis11.

### Results

The proposed CRSTI algorithm improved susceptibility-based tractography in each organ. As similarly demonstrated in an earlier study4, the mouse heart tractography data (Fig. 1) show that CRSTI fiber orientations are more consistent with DTI compared to traditional STI. In the kidney tractography data (Fig. 2), CRSTI yielded improvements over STI that were most obvious in the medullary regions, where susceptibility and diffusion anisotropy were greatest. STI tractography exhibited the greatest differences from DTI in the mouse brain (Fig. 3). CRSTI corrected the orientation and improved the continuity of axon fiber tracts throughout the white matter regions of the brain, particularly in the corpus callosum, cerebellum, and anterior commissure.

### Discussion and Conclusion

CRSTI is able to compensate for inadequate phase information by including eigenvector data from ${\bf R}$4. This is possible because R2* in a set of parallel cylinders is positively correlated with the magnetic susceptibility difference between the objects and the surrounding medium12. Hence, the anisotropy of ${\bfχ}$ engenders anisotropy in ${\bf R}$13, and the two tensors share an eigenvector corresponding to the orientation of the cylindrical tissue structure. Though not necessary for CRSTI, contrast agents can also be used to enhance the inherently anisotropic component of ${\bf R}$ in organs with capillaries that align with the tissue structure14, such as the heart15 and kidney16.

Susceptibility-based tensor mapping is useful for studying the organized molecular sources of susceptibility contrast and anisotropy in the heart, kidney, and brain. This technique is made more robust by conjoint tensor estimation, which requires no additional scan time since phase and relaxation data can be acquired simultaneously. By simplifying the shared eigenvector optimization problem4, this particular CRSTI algorithm requires very little computational overhead. The efficiency and chemical sensitivity of this technique make CRSTI a potential tool for studying disease in a variety of magnetically anisotropic tissues throughout the body.

### Acknowledgements

The authors wish to thank G. Allan Johnson, PhD, Gary Cofer, MS, and Yi Qi, MD, for their assistance in this study. All imaging was performed at the Center for In Vivo Microscopy of Duke University This work was supported in part by the National Institutes of Health through NIBIB P41 EB015897, T32 EB001040, NIMH R01 MH096979, Office of the Director 1S10ODO10683-01, and NHLBI R21 HL122759, and by the National Multiple Sclerosis Society through grant RG4723.

### References

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4. Dibb R, Liu C. Whole-heart myofiber tractography derived from conjoint relaxation and susceptibility tensor imaging. 2015; Proceedings of the 23nd Annual ISMRM, Toronto, Canada. p 287.

5. Xie L, Dibb R, Cofer GP, Li W, Nicholls PJ, Johnson GA, Liu C. Susceptibility tensor imaging of the kidney and its microstructural underpinnings. Magn Reson Med 2014;73(3):1270-1281.

6. Johnson GA, Cofer GP, Gewalt SL, Hedlund LW. Morphologic phenotyping with MR microscopy: the visible mouse. Radiology 2002;222(3):789-793.

7. Li W, Wu B, Liu C. iHARPERELLA: and improved method for integrated 3D phase unwrapping and background phase removal. 2015; Proceedings of the 23nd Annual ISMRM, Toronto, Canada. p 3313.

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10. Dibb R, Xie L, Liu C. Magnetic susceptibility anisotropy of the myocardium. 2014; Proceedings of the 22nd Annual ISMRM, Milan, Italy. p 627.

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16. Hook JB. Toxicology of the kidney: Raven Press; 1981.

### Figures

Table 1. MRI protocols for STI, CRSTI, and DTI data acquisitions.

Fig. 1. STI, CRSTI, and DTI tractography in a mouse heart section. STI exhibits spurious, incoherent fibers, whereas CRSTI tractography produces more continuous tracts that better resemble those of DTI (yellow arrows). Unlike STI, CRSTI shows the trilayer structure of the entire left ventricle wall (white arrows).

Fig. 2. STI, CRSTI, and DTI tractography in the medullary regions of the mouse kidney. CRSTI improved the continuity (orange arrows) and orientation (yellow arrows) of tracts relative to STI. The short, incoherent tracts in the cortex of the kidney are not shown for clarity.

Fig. 3. Mouse brain tractography. CRSTI improved the continuity and orientation of the tracts in the corpus callosum (white arrows), cerebellum (yellow arrows), anterior commissure and optic chiasm (orange arrows). CRSTI also better delineates the genu of the corpus callosum (white double arrowheads) and the cingulum (yellow double arrowheads).

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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