Constrained maximum likelihood estimation of the diffusion kurtosis tensor using a Rician noise model

A computational framework to obtain an accurate quantification of the Gaussian and non-Gaussian component of water molecules' diffusion through brain tissues with diffusion kurtosis imaging, is presented. The diffusion kurtosis imaging model quantifies the kurtosis, the degree of non-Gaussianity, on a direction dependent basis, constituting a higher order diffusion kurtosis tensor, which is estimated in addition to the well-known diffusion tensor. To reconcile with the physical phenomenon of molecular diffusion, both tensor estimates should lie within a physically acceptable range. Otherwise, clinically and artificially significant changes in diffusion (kurtosis) parameters might be confounded. To guarantee physical relevance, we here suggest to estimate both diffusional tensors by maximizing the joint likelihood function of all Rician distributed diffusion weighted images given the diffusion kurtosis imaging model while imposing a set of nonlinear constraints. As shown in this study, correctly accounting for the Rician noise structure is necessary to avoid significant overestimation of the kurtosis values. The performance of the constrained estimator was evaluated and compared to more commonly used strategies during simulations. Human brain data were used to emphasize the need for constrained estimators as not imposing the constraints give rise to constraint violations in about 70% of the brain voxels.