BioMed Research International

Volume 2015 (2015), Article ID 760230, 10 pages

http://dx.doi.org/10.1155/2015/760230

*K*-Optimal Gradient Encoding Scheme for Fourth-Order Tensor-Based Diffusion Profile Imaging

^{1}Department of Signals and Systems, Chalmers University of Technology, 41296 Gothenburg, Sweden^{2}Centre for Microscopy, Characterisation and Analysis, The University of Western Australia, Perth, WA 6009, Australia^{3}Department of Radiology, Sahlgrenska University Hospital, Gothenburg University, 41345 Gothenburg, Sweden^{4}Department of Radiation Physics, Institute of Clinical Sciences, University of Gothenburg, 41345 Gothenburg, Sweden^{5}Department of Medical Physics and Biomedical Engineering, Sahlgrenska University Hospital, 41345 Gothenburg, Sweden

Received 22 June 2015; Revised 26 August 2015; Accepted 27 August 2015

Academic Editor: Sergio Murgia

Copyright © 2015 Mohammad Alipoor et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 (-optimal design). In this paper, we propose a new approach to solve the -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 -optimal design: (i) the odd moments of the -optimal design must be zero; (ii) the even moments of the -optimal design are proportional to the total number of measurements; (iii) the -optimal design is not unique, in general; and (iv) the proposed method can be used to compute the -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 -optimal design leads to the minimum signal deviation.

#### 1. Introduction

Diffusion-weighted MRI is a noninvasive imaging technique to probe microstructures in living tissues, for example, the human brain. It involves acquiring a series of diffusion-weighted images, each corresponding to diffusion sensitization along a particular gradient direction. Non-Gaussian diffusion models have gained wide attention among researchers because of their potential ability to resolve complex multifiber microstructures. Özarslan and Mareci [1] introduced high order tensors (HOTs) as an alternative to conventional second-order tensor model. In regions with complex microstructures, HOTs can model the apparent diffusion coefficient (ADC) with higher accuracy than the conventional second-order model [2]. Several aspects of HOT-based ADC profile estimation have been addressed in the literature [3–5]. HOTs have also been used to represent orientation distribution functions that are required for tractography [6, 7].

The need for robust estimation of diffusion parameters in a limited acquisition time has given rise to many studies on optimal gradient encoding scheme (GES) design. In the case of the classical second-order model they include [8–14]. However, there are few studies tackling the problem of optimal GES design for non-Gaussian diffusion models [15, 16]. 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 that the condition number is computed from the design matrix associated with the linear least square estimation of parameters of interest. Thus, by definition, it is model-dependent. This implies that the minimum condition number GES for second-order tensor estimation is not an optimal GES for fourth-order tensor estimation. An experiment design that minimizes the condition number of the design matrix is called -optimal design. In this paper we solve the problem of -optimal GES design for HOT-based ADC profile imaging as follows. First, we reformulate it as a nonconvex experiment design problem. Then, by convex relaxation we obtain a tractable semidefinite programming (SDP) problem. The last step is to extract design points (the gradient encoding directions) from the optimal design matrix. Finally, to show the relevance of the proposed design approach, we evaluate our solutions using the rotational variance test and Monte Carlo simulations. Throughout the paper “*experiment design*” and “*gradient encoding scheme (GES)*” are used interchangeably. The former is used in the optimization context while the latter is used in the diffusion MRI (dMRI) community.

#### 2. Problem Statement

This section briefly reviews the basics of HOT-based ADC profile estimation both for the sake of completeness and to define notation. The reader is referred to [4, 5] for more details. For definitions of symmetry, positive semidefiniteness and eigendecomposition of high order tensors () see [5, 17]. The Stejskal-Tanner equation for dMRI signal attenuation is [18]where is the diffusivity function, is the measured signal when the diffusion sensitizing gradient is applied in the direction , is the observed signal in the absence of such a gradient, and is the diffusion weighting taken to be constant over all measurements. The diffusivity function is modeled using even order symmetric positive semidefinite tensors as follows:where contains distinct entries of the th-order tensor. Here we focus on the case of , where . It is worth mentioning that both vectors and are vectors in and is used for simplification. Given measurements in different directions , the least squares estimator (LSE) of the HOT is obtained as follows:where is an * design matrix* defined as and . The closed-form solution is .

In the framework described above, the precision of the estimation problem is dependent on the experiment designs , . For independent and zero mean measurement noise with constant variance the LSE is unbiased and has the following covariance matrix [19]:where and is usually called the “*information matrix*.” Optimal experiment design entails making the covariance matrix* small* in some sense. It is usual to minimize a scalar function of the covariance matrix. One design approach is to minimize the condition number of the information matrix (-optimal design) [11, 12, 19]. In this paper, we solve the -optimal experiment design problem for HOT-based ADC profile imaging.

*Remark 1. *For isotropic diffusion, it has been shown that (4) holds [9, 20]. We investigate the significance of the noise assumptions in the case of anisotropic diffusion, later in Section 4. Therein we present Monte Carlo simulations for a more realistic case (with anisotropic tensor and Rician distributed noise on ).

#### 3. Proposed GES Design Approach

In Section 3.1 we present mathematical formulations for the -optimal GES design problem. The solutions are given in Section 3.2. Section 3.3 then considers the problem of extracting the design points from the optimal information matrix. Finally, the properties of the obtained solutions and some theoretical results are discussed in the last subsection.

##### 3.1. Mathematical Formulations of the -Optimal Design Problem

The condition number measures the sensitivity of the solution to changes in measurements [11]. Hence, it is desirable to minimize the condition number of (denoted by ) or equivalently to minimize . The -optimal design in HOT-based ADC profile imaging can be performed with respect to either the design matrix or information matrix because [11]where and are the maximum and minimum eigenvalues of , respectively. The -optimal experiment design problem in HOT estimation can be written as follows:This problem, in its current form, is not convex. Our aim here is to reformulate this problem as an SDP problem that can be efficiently solved. Before describing the approach, it is worth mentioning that conventional experiment design problems (as in [21]) seek to minimize the objective function over a finite and thus countable set , that is, . In the present case, however, is not a countable set but includes the whole set of feasible solutions. Note that the degree of freedom in this design problem is 45. In other words, can be parameterized in 45 independent variables. For example, , , . To reformulate the problem, we first parameterize in 45 distinct variables assuch that we obtain the affine mapping (its range is the set of symmetric positive semidefinite matrices of size fifteen). This can equivalently be expressed aswhere is a symmetric matrix and . To clarify how is decomposed into s, consider the following example:where is the element of placed in the th row and th column. Carefully note the relationship between and the original design variables (s) because this is used in Section 3.3. For example, and .

It is possible to relax the constraints , , and solve the problem by the algorithm given in [22] to obtain a lower bound on the optimal value of problem (6). However, we instead convert the constraints in (6) to a convex constraint as follows:where has only fifteen nonzero elements. We then have the following relaxed problem:Given that the conversion in (10) is not reversible, the optimal value of the problem in (11) is a lower bound on the optimal value of the problem in (6). The objective function is a quasiconvex function [22]. Thus, an approximate solution of (11) may be obtained by solving a sequence of feasibility problems [21, 22]. Alternatively, this problem can be formulated as an SDP problem:where is the identity matrix, is the condition number, and equals . This is a bilinear matrix inequality problem that can be solved by the line search method. For a constant , it becomes a tractable linear matrix inequality (LMI) problem. The optimal value of (12) can be obtained by performing a line search on . Let the optimal value of the following problem be , where is a real nonnegative constant:Then we have . The problem in (13) can be efficiently solved by LMI solvers.

##### 3.2. Solutions to the -Optimal Design Problem

The -optimal design problem in (13) can be solved for different values of , using the YALMIP [23] and SDPT3 solvers [24]. By close inspection of the results for different values of , one can conclude the following about -optimal solutions:(i)If is a solution to (13) with , then is a solution to (13) with for any real positive . Thus, the optimal solution () is proportional to .(ii)The minimum condition number is independent of and is given by .(iii)The -optimal solution is(iv)And (where ).

##### 3.3. Extracting Design Points

The task of extracting the design points (, ) from the optimal information matrix is straight forward, as outlined in [25]. By expressing the optimal in terms of the original decision variables, one obtains 45 equations as listed in (15) and (16). Furthermore, equations of the form can be added to guarantee that the resulting solutions belong to the feasible set of the original problem. Thus, one obtains a nonlinear system of equations in unknowns. Given that is required, the system is usually underdetermined. By numerically solving the nonlinear system, one can extract the design points. The odd moments of the optimal design must be zero (). We refer to this fact as* symmetry* of the optimal design. This property means that the following holds true:The even moments of the optimal design must satisfy the following conditions ():As an example, Table 1 lists the -optimal design points for derived using our proposed method. We solved the above-mentioned nonlinear system of equations using the fsolve command in MATLAB. For a discussion on the uniqueness of this solution, see the next subsection where we explain some properties of the -optimal design.