Elsevier

NeuroImage

Volume 81, 1 November 2013, Pages 335-346
NeuroImage

Weighted linear least squares estimation of diffusion MRI parameters: Strengths, limitations, and pitfalls

https://doi.org/10.1016/j.neuroimage.2013.05.028Get rights and content

Highlights

  • Linear least squares estimators are widely used in diffusion MRI.

  • Weighting of the linear least squares estimator is needed to improve the precision.

  • Different weighting strategies are routinely used.

  • The actual accuracy of linear estimators strongly depends on the weight definition.

  • The squares of the noisy diffusion-weighted signals should not be used as weights.

Abstract

Purpose

Linear least squares estimators are widely used in diffusion MRI for the estimation of diffusion parameters. Although adding proper weights is necessary to increase the precision of these linear estimators, there is no consensus on how to practically define them. In this study, the impact of the commonly used weighting strategies on the accuracy and precision of linear diffusion parameter estimators is evaluated and compared with the nonlinear least squares estimation approach.

Methods

Simulation and real data experiments were done to study the performance of the weighted linear least squares estimators with weights defined by (a) the squares of the respective noisy diffusion-weighted signals; and (b) the squares of the predicted signals, which are reconstructed from a previous estimate of the diffusion model parameters.

Results

The negative effect of weighting strategy (a) on the accuracy of the estimator was surprisingly high. Multi-step weighting strategies yield better performance and, in some cases, even outperformed the nonlinear least squares estimator.

Conclusion

If proper weighting strategies are applied, the weighted linear least squares approach shows high performance characteristics in terms of accuracy/precision and may even be preferred over nonlinear estimation methods.

Introduction

Diffusion magnetic resonance imaging (dMRI) is currently the only method for the in vivo and non-invasive quantification of water diffusion in biological tissue (Le Bihan and Johansen-Berg, 2012). Several diffusion models have been proposed to obtain quantitative diffusion measures, which could provide novel information on the structural and organizational features of biological tissues, the brain white matter (WM) in particular. Typical examples of such diffusion models are diffusion tensor imaging (DTI; Basser et al., 1994a) and diffusion kurtosis imaging (DKI; Jensen et al., 2005, Veraart et al., 2011a). Both diffusion models have in common that they can be linearized by the natural log-transformation for computing the model parameters. Despite the overwhelming literature on advanced diffusion (kurtosis) tensor estimation (e.g., Andersson, 2008, Jones and Basser, 2004, Koay et al., 2009, Kristoffersen, 2007, Kristoffersen, 2011, Landman et al., 2007a, Veraart et al., 2011b, Veraart et al., 2012), the class of linear least squares (LLS) estimators is still widely used in diffusion MRI due to its low computational cost and ease of use.

It is well recognized that the variance of a log-transformed diffusion-weighted (DW) signal depends on the signal itself (Basser et al., 1994b, Koay et al., 2006, Salvador et al., 2005). Given the signal-dependency of their variances, a set of log-transformed DW signals is heteroscedastic. Consequently, a weighted linear least squares (WLLS) approach with well-defined weights, i.e. the inverse of the log-transformed signals' variances, is expected to provide more precise diffusion parameter estimates, at least, compared to its unweighted linear alternative. Indeed, Optimal performance of the linear estimators can only be achieved if the variance of the log-transformed MR signals is known. Salvador et al. (2005) showed that this variance is a function of the noise-free signal if the magnitude MR data is Rice distributed. Obviously, the noise-free signals are not known and, as such, the weight terms need to be estimated. Different ways to approximate the theoretically optimal weights have been suggested (Koay et al., 2006, Basser et al., 1994b, Salvador et al., 2005) and adopted by the community. A consensus on how to use the WLLS in practice has thus not been reached yet. Moreover, information on the weighting is often omitted in scientific reports or software documentation. In this study, we will evaluate the impact of the different weighting strategies on the performance of the linear diffusion parameter estimators. Additionally, we will compare the linear estimators to their nonlinear alternative (Koay et al., 2006) in terms of accuracy and precision. By doing so, we aim to obtain more insight in the strengths, limitations, and potential pitfalls of the simple, yet elegant, class of linear estimators.

Section snippets

Diffusion models

The natural logarithm of the DW MR signal, S, can be expanded in powers of the wave number q in a zero-centered neighborhood (Kiselev, 2010). This expansion is known as the cumulant expansion, with DTI and DKI being the three-dimensional generalization of the second and fourth order cumulant expansion, respectively (Basser et al., 1994a, Lu et al., 2006). The expansion is generally expressed in terms of the diffusion sensitizing gradient strength (b) and its direction (g):lnSbg=lnS(0)bi,j=13gi

Simulation experiment 1

In Fig. 2, the different estimators are compared in terms of accuracy, precision, and MSE as a function of SNR 1. The FA, MD, and MK values – calculated from the average model parameters to exclude nonlinear effects such as eigenvalue repulsion (Pierpaoli and Basser, 1996) – strongly vary across the different estimators (see Figs. 2(a–c)). Generally, the NLS estimator shows a large difference to the reference values, compared to the unweighted and weighted linear approaches. However, note that

Discussion

The DTI and DKI models have in common that they can be structured into a linear regression form depending on the natural logarithm of the DW MR signals. The unknown model parameters can be estimated with a LLS estimator, or its weighted variants. Those estimators are widely used in DTI and DKI studies, because they come with several strengths. First, the (weighted) LLS estimators have a closed-form solution. Therefore, unlike iterative nonlinear strategies, the linear estimators are

Conclusion

Within the wealth of different diffusion parameter estimators, the WLLS estimator stands out by its simplicity and elegance. To date, no consensus has been reached on how to use the estimator in practice. In this work, it has been shown that the accuracy of the WLLS estimator strongly depends on the selected weight terms. A comparison of the most common weighting strategies indicated that the squares of the noisy DW signals should not be used if one is interested in designing a linear estimator

Acknowledgment

This work was supported by the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen), the Fund for Scientific Research-Flanders (FWO), and by the Interuniversity Attraction Poles Program (P7/11) initiated by the Belgian Science Policy Office.

Conflicts of Interest

The authors declare no conflicts of interest.

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