Diffusion of water molecules in biological tissues is conventionally quantified via diffusion tensor imaging (DTI) (1
). DTI is a valuable tool for noninvasive characterization of tissue microstructural properties (2
). The DTI model uses a Gaussian approximation to the probability distribution governing the random displacement of water molecules. In many biological tissues, however, the displacement probability distribution can deviate considerably from a Gaussian form. Several techniques have been proposed to characterize non-Gaussian diffusion, with diffusion spectrum imaging (DSI) (4
) being arguably the most comprehensive. Although it is being used in human research, the application of DSI as a routine clinical protocol has been limited by its extended acquisition time and hardware requirements.
Diffusional kurtosis imaging (DKI) is a clinically feasible extension of DTI which enables the characterization of non-Gaussian diffusion by estimating the kurtosis of the displacement distribution (5
), in addition to the estimation of the standard DTI-derived parameters. DKI has shown promising results in studies of human brain aging (9
), tumor characterization in gliomas (10
) and head and neck cancers (11
), and rodent brain maturation (12
). Moreover, the additional information provided by DKI has been exploited to resolve intravoxel fiber crossings (13
), which could help to improve upon DTI-based fiber tracking methods.
The DKI model is parameterized by the diffusion tensor (DT) and kurtosis tensor (KT) from which several rotationally-invariant scalar measures are extracted. The most common DT-derived measures are mean, axial, and radial diffusivity (2
), as well as fractional anisotropy (FA) (2
); and the KT-derived measures are axial, radial, and mean kurtoses (6
). The interpretability of these metrics is influenced by the estimation accuracy of the tensors. Noise, motion, and imaging artifacts can introduce errors into the estimated tensors. Sufficiently large errors can cause the tensor estimates to be physically and/or biologically implausible. For instance, the directional diffusivities may become negative, that is, the DT may become non-positive definite (NPD). A well-known consequence of NPD DT estimates is that FA values, which in theory should range between 0 and 1, may exceed 1, particularly in high FA regions of the brain such as the corpus callosum (16
). Inaccuracies in the estimated tensors may also drive the directional kurtoses outside of an acceptable range. Empirical evidence in the brain as well as idealized multi-compartment diffusion models suggest that directional kurtoses should typically be positive and should not exceed a certain level depending on tissue complexity (6
). The maximum allowable kurtosis is also influenced by the maximum b
-value used in image acquisition (8
In our previous work, the DT and KT were estimated using unconstrained nonlinear least squares (UNLS) (7
). In a related work, higher-order DTs were estimated using unconstrained linear least squares (ULLS) (17
). These unconstrained schemes do not guarantee acceptable tensor estimates. To address this drawback in the context of DTI, the Cholesky decomposition has been utilized to impose the non-negative definiteness constraint on the DT (16
) using either the ULLS or UNLS algorithms. Moreover, a parameterization has been proposed in a UNLS framework to guarantee a positive diffusivity function in a fourth-order tensor-only model of the diffusion signal (19
). While these methods outperform other algorithms for imposing the positive diffusivity function in the context of a second order-only and a fourth order-only diffusion signal model, their generalization to impose the constraints on the DKI signal model is not straightforward.
We propose that the tensor estimation problem be cast as linear least squares (LS) subject to linear constraints. The constraints ensure that the directional diffusivities and kurtoses along the imaged gradient directions remain within a physically and biologically plausible range. The proposed constrained linear LS (CLLS) formulation yields a convex objective function that permits efficient solutions via convex quadratic programming or a fast heuristic algorithm. The two algorithms proposed for solving the CLLS problem strike different tradeoffs between the speed and exactness of the solution as well as algorithm flexibility. The quadratic programming-based (CLLS-QP) algorithm exactly satisfies the constraints and can also handle an arbitrary number of diffusion weightings and different gradient sets for each diffusion weighting, but it does so at a moderately high computational cost. On the contrary, the heuristic (CLLS-H) algorithm produces an approximation to the optimal solution, but it accomplishes this at almost no computational overhead compared to the ULLS. Both constrained algorithms substantially improve the estimation fidelity of the tensors.
Once the tensors are estimated, they are utilized to determine the scalar diffusion and kurtosis measures. In our previous work, mean kurtosis (MK) was being approximated as the average of directional kurtoses (5
). Here, we present an exact analytical formula for the MK, expressing it as an explicit function of the DT and KT. We also present an exact analytical expression for the radial kurtosis, defining it as the diffusional kurtosis averaged over all directions perpendicular to the DT eigenvector with the largest eigenvalue. Our definition is consistent with the definition of radial diffusivity in DTI and the definitions given in (8
), but is different from that of (15
). The analytic formulae for the MK and radial kurtosis were previously quoted in (8
) but here for the first time we present their derivations and discuss important practical details of their implementation. The application of these formulae improves both the accuracy and efficiency of DKI parameter estimation.