Diffusion Kurtosis Imaging (DKI) seeks to provide a better approximation of non-Gaussian spin displacements by expanding the expression to the fourth-order moment of the distribution, Kurtosis (odd orders are assumed to be zero in the absence of net coherent transport) [24]. This approach has the advantage that it makes no strong assumptions about what is causing the non-Gaussianity, but nonetheless quantifies it in a well-understood way. Indeed, non-zero excess Kurtosis is often described as being a signature of restriction [44] but is has also been shown to be special case of fractional diffusion models [20]

DKI can be derived from both physical and statistical arguments, but we will concentrate on the former. Here the Fickian flux from DTI is extended to include a fourth-order term which adds additional degrees of freedom. The fourth order flux is

where K is the fourth order Kurtosis tensor which modifies the Gaussian flux process. The factor of t is required to correct the dimensionality of the kurtosis term. Substituting this into Eq. (15) and solving yields the following solution

which is the 3D form of the usual diffusion kurtosis solution. As with DTI, this can be used to construct a linear system over several measurements that can be inverted to find the elements of the diffusion and kurtosis tensors.