Robust Estimation of Kurtosis and Diffusion Tensors in Diffusional Kurtosis Imaging

Diffusion of water molecules in biological tissues is conventionally quantified via diffusion tensor imaging (DTI) [1]. DTI enables a Gaussian approximation to the probability distribution governing the random displacement of water molecules. In some circumstances of great interest, however, the displacement probability distribution can deviate considerably from a Gaussian form. Diffusional kurtosis imaging (DKI) allows for characterization of this deviation via inclusion of a kurtosis term in the expression for the displacement distribution [2]. The DKI model is parameterized by the diffusion and kurtosis tensors from which scalar measures such as mean diffusivity, fractional anisotropy, and axial, radial, and mean kurtoses are derived. Obtaining physically and biologically plausible estimates of these tensors in a computationally tractable framework is crucial to reliable estimation of the scalar kurtosis and diffusion metrics. In previous work concerning the estimation of generalized DTI tensors, unconstrained linear [3] and nonlinear least squares (ULLS and UNLS) [2] algorithms have been utilized. Unfortunately, in the presence of noise, unconstrained schemes do not necessarily produce plausible tensor estimates. To address this drawback, a parameterization has been proposed in a UNLS framework to guarantee a positive diffusivity function in a fourth-order tensor-only model of diffusion [4]. We present a tractable computational framework to obtain plausible estimates of kurtosis and diffusion tensors in DKI. The estimation problem is formulated as linearly constrained linear least squares (CLLS). Mean kurtosis (MK) maps for a human subject obtained using this formulation are compared to those estimated using ULLS.