Abstract
The design of an optimal gradient encoding scheme (GES) is a fundamental problem in diffusion MRI. It is well studied for the case of second-order tensor imaging (Gaussian diffusion). However, it has not been investigated for the wide range of non-Gaussian diffusion models. The optimal GES is the one that minimizes the variance of the estimated parameters. Such a GES can be realized by minimizing the condition number of the design matrix (K-optimal design). In this paper, we propose a new approach to solve the K-optimal GES design problem for fourth-order tensor-based diffusion profile imaging. The problem is a nonconvex experiment design problem. Using convex relaxation, we reformulate it as a tractable semidefinite programming problem. Solving this problem leads to several theoretical properties of K-optimal design: (i) the odd moments of the K-optimal design must be zero; (ii) the even moments of the K-optimal design are proportional to the total number of measurements; (iii) the K-optimal design is not unique, in general; and (iv) the proposed method can be used to compute the K-optimal design for an arbitrary number of measurements. Our Monte Carlo simulations support the theoretical results and show that, in comparison with existing designs, the K-optimal design leads to the minimum signal deviation.
Highlights
Diffusion-weighted MRI is a noninvasive imaging technique to probe microstructures in living tissues, for example, the human brain
The only study on gradient encoding scheme (GES) design for high order tensors (HOTs) [16] is limited to comparison of existing GESs mainly devised for second-order tensor imaging, for example, the minimum condition number (MCN) scheme [12]
In this paper we solve the problem of K-optimal GES design for HOT-based apparent diffusion coefficient (ADC) profile imaging as follows
Summary
Diffusion-weighted MRI is a noninvasive imaging technique to probe microstructures in living tissues, for example, the human brain. The only study on GES design for HOTs [16] is limited to comparison of existing GESs mainly devised for second-order tensor imaging, for example, the minimum condition number (MCN) scheme [12]. A caveat here is BioMed Research International that the condition number is computed from the design matrix associated with the linear least square estimation of parameters of interest. In this paper we solve the problem of K-optimal GES design for HOT-based ADC profile imaging as follows. We reformulate it as a nonconvex experiment design problem. The former is used in the optimization context while the latter is used in the diffusion MRI (dMRI) community
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