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