Computational Diffusion MRI: Optimal Gradient Encoding Schemes
Abstract: Diffusion-weighted magnetic resonance imaging (dMRI) is a non-invasive structural imaging technique that provides information about tissue microstructures. Quantitative measures derived from dMRI reflect pathological and developmental changes in living tissues such as human brain. Such parameters are increasingly used in diagnostic and prognostic procedures and this has motivated several studies to investigate their estimation accuracy and precision. The precision of an estimated parameter is dependent on the applied gradient encoding scheme (GES). An optimal GES is one that minimizes the variance of the estimated parameter(s). This thesis focuses on optimal GES design for the following dMRI models: second and fourth-order diffusion tensor imaging (DTI), ADC imaging and diffusion kurtosis imaging (DKI). A unified framework is developed that comprises three steps. In the first step, the original problem is formulated as an optimal experiment design problem. The optimal experiment design is the one that minimizes the condition number (K-optimal) or the determinant (D-optimal) of the covariance matrix of the estimated parameters. This yields a non-convex optimization problem. In the second step, the problem is re-formulated as a semi-definite programming (SDP) problem by introducing new decision variables and convex relaxation. In the final step, the SDP problem is solved and the original decision variables are recovered. The proposed framework is comprehensive; it can be applied to DTI, DKI, K-optimal design, D-optimal design, single-shell and multi-shell acquisitions and to optimizing directions and b-values. The main contributions of this thesis include: (i) proof that by uniformly distributing gradient encoding directions one obtains a D-optimal design both for DKI and DTI; (ii) proof that the traditionally used icosahedral GES is D-optimal for DTI; (iii) proof that there exist rotation-invariant GESs that are not uniformly distributed; and (iv) proof that there exist GESs that are D-optimal for DTI and DKI simultaneously. A simple algorithm is presnted that can compute uniformly distributed GESs. In contrast to previous methods, the proposed solution is strictly rotation-invariant. The practical impact/utility of the proposed method is demonstrated using Monte Carlo simulations. The results show that the precision of parameters estimated using the proposed approach can be as much as 25% better than that estimated by state-of-the-art methods. Validation of these findings using real data and extension to non-linear estimators/diffusion models provide scope for future work.
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