Abstract
Purpose It has been shown previously that for the conventional Stejskal-Tanner pulsed gradient, or linear tensor encoding (LTE), as well as planar tensor encoding (PTE) and in tissue in which diffusion exhibits a ‘stick-like’ geometry, the diffusion-weighted MRI signal at extremely high b-values follows a power-law. Specifically, the signal decays as a 
in LTE and 1/b in PTE. Here, the direction-averaged signal for arbitrary diffusion encoding waveforms is considered to establish whether power-law behaviors occur with other encoding wave-forms and for other (non-stick-like) diffusion geometries.
Methods We consider the signal decay for high b-values for encoding geometries ranging from 2-dimensional planar tensor encoding (PTE), through isotropic or spherical tensor encoding (STE) to linear tensor encoding. When a power-law behavior was suggested, this was tested using in-silico simulations and in-vivo using an ultra-strong gradient (300 mT/m) Connectom scanner.
Results The results show that using an axisymmetric b-tensor a power-law only exists for two scenarios: For stick-like geometries, (i) the already-discovered LTE case; and (ii) for pure planar encoding. In this latter case, to first order, the signal decays as 1/b. Our in-silico and in-vivo experiments confirm this 1/b relationship.
Conclusion A complete analysis of the power-law dependencies of the diffusion-weighted signal at high b-values has been performed. Only two forms of encoding result in a power-law dependency, pure linear and pure planar tensor encoding and when the diffusion geometry is ‘stick-like’. The different exponents of these encodings could be used to provide independent validation of the presence of stick-like geometries in-vivo.
