gKFA: a geometric framework for kurtosis fractional anisotropy

Diffusion kurtosis imaging (DKI) extends diffusion tensor imaging (DTI) by incorporating non-Gaussian diffusion effects through a fourth-order kurtosis tensor. In this work, we investigate the three-dimensional geometric structure of the kurtosis tensor and introduce a geometry-inspired anisotropy descriptor based on mapped symmetric second-order quadratic forms. This representation enables kurtosis anisotropy to be studied through lower-order tensors whose geometry is easier to analyze and whose positive definiteness can be assessed more conveniently. Based on this framework, we define geometric kurtosis fractional anisotropy (gKFA) as the geometric mean of the fractional anisotropies computed from the mapped quadratic forms. The proposed descriptor was evaluated using simulations, physical phantoms, and in vivo human and rodent brain data. In simulations and phantom experiments, gKFA showed sensitivity to directional heterogeneity and tissue-complexity changes that were only weakly reflected by conventional KFA. In the human brain dataset, additional analyses compared gKFA with conventional kurtosis fractional anisotropy (KFA), mean kurtosis (MK), axial kurtosis (AK), radial kurtosis (RK), and anisotropy measures derived from the mapped quadratic forms. These results indicate that gKFA behaves as a kurtosis-anisotropy descriptor that is closely related, but not identical, to KFA, and distinct from standard scalar kurtosis measures such as MK, AK, and RK. In both human and rodent data, gKFA showed better tissue contrast and complementary behavior relative to KFA, including sensitivity to regional differences in the rodent brain.