Optimal Gradient Encoding Schemes for Diffusion Tensor and Kurtosis Imaging

Diffusion-derived parameters find application in characterizing pathological and developmental changes in living tissues. Robust estimation of these parameters is important because they are used for medical diagnosis. An optimal gradient encoding scheme (GES) is one that minimizes the variance of the estimated diffusion parameters. This paper proposes a method for optimal GES design for two diffusion models: high-order diffusion tensor (HODT) imaging and diffusion kurtosis imaging (DKI). In both cases, the optimal GES design problem is formulated as a D-optimal (minimum determinant) experiment design problem. Then, using convex relaxation, it is reformulated as a semidefinite programming problem. Solving these problems we show that: 1) there exists a D-optimal solution for DKI that is simultaneously D-optimal for second- and fourth-order diffusion tensor imaging (DTI); 2) the traditionally used icosahedral scheme is approximately D-optimal for DTI and DKI; 3) the proposed D-optimal design is rotation invariant; 4) the proposed method can be used to compute the optimal design ( $b$ -values and directions) for an arbitrary number of measurements and shells; and 5) using the proposed method one can obtain uniform distribution of gradient encoding directions for a typical number of measurements. Importantly, these theoretical findings provide the first mathematical proof of the optimality of uniformly distributed GESs for DKI and HODT imaging. The utility of the proposed method is further supported by the evaluation results and comparisons with with existing methods.