POTENTIAL BIOMARKERS FROM POSITIVE DEFINITE 4TH ORDER TENSORS IN HARDI Sumit Kaushik, Jan Kybic Faculty of Electrical Engineering Czech Technical University in Prague Czech Republic Avinash Bansal, Temesgen Bihonegn, Jan Slovak Department of Mathematics and Statistics Masaryk University Brno, Czech Republic ABSTRACT In this paper, we provide a framework to evaluate new scalar quantities for higher order tensors (HOT) appearing in high angular resolution diffusion imaging (HARDI). These can po- tentially serve as biomarkers. It involves flattening of HOTs and extraction of the diagonal D-components. Experiments performed in the 4th order case reveal that D-components en- code geometric information unlike the isometric 6D 2nd order Voigt form. The existing invariants obtained from the Voigt form are considered for comparison. We also notice that D- components can be useful in segmentation of white matter structures in crossing regions and classification. Results on phantom and the synthetic dataset support the conclusions. Index Terms— HARDI, biomarker, 4th order tensor, D- component 1. INTRODUCTION Diffusion MRI (dMRI) is a powerful tool in the study of microstructures in human brain non-invasively. Based upon dMRI principle, diffusion tensor imaging (DTI) model was introduced by Basser and others [1], [2] and its importance as a clinical standard grows. The eigenvalues of the 2nd order DTI tensors remain invariant under 3D rotations. Some of the scalars obtained as from function of these eigenvalues are: mean diffusivity (MD), fractional anisotropy (FA), radial diffusivity (RD) etc. These scalars serve as the biomark- ers which are helpful in discerning unhealthy tissues from healthy ones. This change occurs due to disorganization of tissue structure due to present anomaly. The DTI modality looses its efficiency if the diffusion of water molecules is not Gaussian, which happens e.g. for voxels including cross- ing or merging fibers. For such situations, we may increase the number of the measured diffusion gradient directions [3] and approximate the diffusion by higher order tensors. This acquisition protocol is known as high angular diffusion imaging (HARDI). There are some works which considered rotational invariance property to obtain scalars in higher order tensors (HOT). These scalars are shown to serve as potential biomarkers and are considered better than those derived from 2nd order tensors. Ghosh et al proposed such invariants of 4th order tensor [4]. They are obtained from the characteristic polynomial of the Voigt form, a 6x6 isometric representation of the 4th order tensors. Their eigenvalues are also known as Kelvin eigenvalues and they are invariant under 6D rotation group SO(6) instead of the real 3D rotations of interest. Fur- ther works deal with the complete set of invariants under 3D rotations [5]. The biological and clinical significance of these HOT ro- tation invariant scalars is still largely a matter of research. In this paper, we present an approach which extracts scalar mea- sures of positive definite higher order tensors using the diag- onal component (D) approach. It is observed that the three 3x3 (blocks) diagonal components in case of 3D flattened 4th order tensor carry useful geometric information. The scalar obtained by combining these components is shown to be ro- bust under rotations than those aforementioned. The princi- pal eigenvalues of the these components reflect the number of fibers at a voxel level. These scalars can serve as poten- tial biomarkers. We have shown they are also effective in segmenting white matter fibers in heterogeneous region and classification of tissues with respect to underlying number of fibers. The approach is extendable to HOT of any order. We discuss experiments on phantom and synthetic dataset. 2. THEORY 2.1. Diffusion Model The generalized Stejskal-Tanner equation which is a mono exponential model of the diffusion of water molecules in bi- ological tissues. The attenuated signal corresponding to a gradient pulse, with the diffusion weighting coefficient b, is S(g)= S 0 exp(-bD(g)), where D(g)= 3  j1=1 3  j2=1 ··· 3  jn=1 D j1j2...jn g j1 g j2 ...g jn (1) and g k is the kth component of the magnetic gradient vector with |g| =1. The number of independent coefficients for ℓth order symmetric tensors is N ℓ = 1 2 (ℓ + 1)(ℓ + 2). Thus, for 4th order symmetric tensors, 3 4 = 81 coefficients of general tensors reduce to 15. 2021 IEEE 18th International Symposium on Biomedical Imaging (ISBI) April 13-16, 2021, Nice, France 978-1-6654-1246-9/21/$31.00 ©2021 IEEE 1003 2021 IEEE 18th International Symposium on Biomedical Imaging (ISBI) | 978-1-6654-1246-9/20/$31.00 ©2021 IEEE | DOI: 10.1109/ISBI48211.2021.9434144 Authorized licensed use limited to: CZECH TECHNICAL UNIVERSITY. Downloaded on October 12,2021 at 20:30:15 UTC from IEEE Xplore. Restrictions apply.