The Jensen Gaps of Directional Diffusion: Model-Free Microstructural White Matter Markers of Aging
Object: To develop the logarithmic Jensen gap as a model-free measure of directional diffusion nonuniformity, computable from any multi-direction acquisition without tensor fitting, in both the diffusivity and signal domains. Materials and Methods: The diffusivity-domain gap is the log-ratio of the arithmetic and geometric means of the per-direction diffusivities. The signal-domain gap is the b-weighted difference between the mean per-direction apparent diffusion coefficient (ADC) and the ADC of the directionally averaged signal. Nonnegativity and the Hölder ladder were derived. Both gaps were computed voxelwise in 1,379 Human Connectome Project–Aging participants at b = 1500 and 3000 s/mm², and correlated with age. Results: The gaps vanish only when the directional diffusivities are uniform. Although the ADC maps appear identical, the gap between them exposes directional structure. In the corona radiata, the diffusivity and signal gaps dissociated in sign with age (r = −0.47 and r = +0.30, both p < 10⁻²⁵), reproduced at b = 3000 s/mm² and within subjects. After the full tensor was regressed out, both retained broad age effects. Discussion: The gap yields two scalar maps computable retrospectively at negligible cost. Because it is b-invariant under Gaussian diffusion, its cross-shell change is a fit-free non-Gaussianity signature, complementing tensor- and kurtosis-based methods.