A Geometric Framework For Kurtosis Anisotropy

Diffusion kurtosis imaging (DKI) extends diffusion tensor imaging (DTI) by offering enhanced sensitivity to microstructural complexity. At the core of these metrics is the kurtosis part, a fourth-order tensor that quantifies the non-Gaussian behavior of water molecule diffusion in the tissue. Unlike the diffusion tensor in Diffusion tensor imaging (DTI), which is characterized by a relatively small set of independent parameters, the kurtosis tensor requires a much larger set for estimation, providing greater sensitivity to tissue microstructure but also increasing susceptibility to noise. To address these limitations, instead of working directly with the kurtosis tensor, we propose using mapped quadratic forms. These forms preserve positive definiteness and provide a robust geometric interpretation. We introduce a descriptor, geometric kurtosis fractional anisotropy (gKFA). Performance was systematically assessed through simulations, phantom studies, and in vivo datasets, including human and rodent brains. It showed higher sensitivity to angular variations in kurtosis, better responsiveness to microstructural heterogeneity, and robustness under low signal-to-noise conditions. In both phantom and biological datasets, it consistently performed better than KFA, particularly in capturing thalamic changes and differentiating deep brain regions. Statistical analyses confirmed significant regional differences detected by the descriptor.