A unifying theoretical and algorithmic framework for least squares methods of estimation in diffusion tensor imaging

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Abstract

A unifying theoretical and algorithmic framework for diffusion tensor estimation is presented. Theoretical connections among the least squares (LS) methods, (linear least squares (LLS), weighted linear least squares (WLLS), nonlinear least squares (NLS) and their constrained counterparts), are established through their respective objective functions, and higher order derivatives of these objective functions, i.e., Hessian matrices. These theoretical connections provide new insights in designing efficient algorithms for NLS and constrained NLS (CNLS) estimation. Here, we propose novel algorithms of full Newton-type for the NLS and CNLS estimations, which are evaluated with Monte Carlo simulations and compared with the commonly used Levenberg–Marquardt method. The proposed methods have a lower percent of relative error in estimating the trace and lower reduced χ2 value than those of the Levenberg–Marquardt method. These results also demonstrate that the accuracy of an estimate, particularly in a nonlinear estimation problem, is greatly affected by the Hessian matrix. In other words, the accuracy of a nonlinear estimation is algorithm-dependent. Further, this study shows that the noise variance in diffusion weighted signals is orientation dependent when signal-to-noise ratio (SNR) is low (⩽5). A new experimental design is, therefore, proposed to properly account for the directional dependence in diffusion weighted signal variance.

Introduction

Diffusion tensor imaging (DTI) is a novel noninvasive technique capable of providing important information about biological structures in the brain [1], [2], [3], [4]. This technique depends upon accurate and precise estimation of the diffusion tensor. The mathematical framework for diffusion tensor estimation is both elegant and simple [1], [4]. Its simplicity is due in part to the fact that the model is transformably linear [5]. However, the diffusion tensor in its original form as derived from first principles is a nonlinear model. Recent DTI studies have used several different models—from linear to nonlinear, and from unconstrained to constrained [4], [6], [7], [8], [9], [10], [11], [12].

In general, the methods of estimation in DTI can be classified as linear least squares (LLS), weighted linear least squares (WLLS), nonlinear least squares (NLS) and their corresponding constrained counterparts, which will be denoted as CLLS, CWLLS and CNLS, respectively [4], [6], [7], [8], [9], [10], [11], [12]. The constraint employed in the CLLS, CWLLS, and CNLS estimations is generally the positive definite constraint [11], [12], i.e., the requirement that every eigenvalue of the diffusion tensor estimate be positive. The statistical comparison among different methods of diffusion tensor estimation, both unconstrained and constrained, has been studied in [12]. In the present study, we present a theoretical and algorithmic framework for methods of estimation in DTI by investigating the properties of various least squares objective functions.

There are several numerical methods for solving the NLS problem in DTI. Yet, the Levenberg–Marquardt’s (LM) approach has been the method of choice, perhaps, due to its simple implementation. This simplicity is due in part to its approximation to the Hessian matrix of the NLS objective function. Another approach is Newton’s method (or full Newton-type method) where the complete Hessian matrix is required in the estimation process. It is well known that Newton’s method is more robust than the LM method and can speed up convergence in NLS problems [13], [14], but the complete Hessian matrix is often not available or known for a given problem. Fortunately, a previous account has shown that this is not the case in DTI [15]. In this study, we will show that the Hessian matrices for various methods of estimation in DTI have simple and compact forms.

We first review the basic estimation problem in DTI and discuss various least squares approaches for solving the problem. We then establish theoretical connections among the LLS, WLLS and NLS methods and among their constrained counterparts. We also derive all the Hessian matrices for the methods of estimation discussed in this paper. We propose an efficient strategy, which will be called Modified Full Newton’s method (MFN), for solving both the NLS and CNLS problem. This strategy entails using the WLLS solution as the initial guess, adjusting the LM parameter, and incorporating the full Hessian matrix of the NLS objective function. A similar strategy is also adapted for solving the CNLS problem in DTI.

The performance of the proposed method is compared with the LM method using Monte Carlo simulations. The robustness and accuracy of the MFN method is assessed with respect to the LM method in terms of percent relative error in the estimated trace and reduced χ2 value. The simulations are also used to assess the validity of the assumption of constant noise variance in a single voxel. The analysis and the results of this study provide new insights in constructing more appropriate experimental designs in which the direction-dependent noise variance is taken into account in the diffusion tensor estimation.

Section snippets

Review of DTI estimation

In a DT-MRI experiment, the measured signal in a single voxel has the following form [1], [4], [16]:s=S0exp(-bgTDg),where measured signal, s, depends on the diffusion encoding gradient vector, g, of unit length, the diffusion weight, b, the reference signal, S0, and the diffusion tensor, D. The symbol “T” denotes the matrix or vector transpose. Given m ⩾ 7 sampled signals based on at least six noncollinear gradient directions and at least one sampled reference signal, the diffusion tensor

Results and discussion

The results on the distributional properties of the χν2 estimate and of the trace estimate are summarized in Fig. 2, Fig. 3. The results on the average value of the relative error in estimating the trace in the simulated human brain map are shown in Fig. 4. Fig. 5 shows the difference in these average values among various methods considered in this paper. The results of Fig. 2, Fig. 3 are computed from a collection of 50,000 simulated tensors. In Fig. 4, Fig. 5, the results on each pixel are

Conclusion

The Hessian matrices for various least squares problems are explicitly derived. Simulation results indicate that the accuracy of a diffusion tensor estimate can be substantially improved by explicitly including the Hessian matrix in the least squares estimation algorithm. The proposed constrained nonlinear least squares estimation based on the modified full Newton’s method has lower relative error in estimating the trace than other methods discussed in this paper. The proposed method not only

Acknowledgments

C.G.K. thanks Dr. M. Elizabeth Meyerand and Dr. Andrew L. Alexander for the initial encouragement on this work. The authors thank Dr. Andrew L. Alexander for critically reading the early draft of this paper and Dr. Stefano Marenco for acquiring the human brain data set. We gratefully acknowledge Liz Salak for editing this paper. This research was supported by the Intramural Research Program of the National Institute of Child Health and Human Development (NICHD), National Institutes of Health,

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