Diffusion kurtosis imaging with free water elimination: A bayesian estimation approach

Abstract

Purpose: Diffusion kurtosis imaging (DKI) is an advanced magnetic resonance imaging modality that is known to be sensitive to changes in the underlying microstructure of the brain. Image voxels in diffusion weighted images, however, are typically relatively large making them susceptible to partial volume effects, especially when part of the voxel contains cerebrospinal fluid. In this work, we introduce the "Diffusion Kurtosis Imaging with Free Water Elimination" (DKI-FWE) model that separates the signal contributions of free water and tissue, where the latter is modeled using DKI.

Theory and methods: A theoretical study of the DKI-FWE model, including an optimal experiment design and an evaluation of the relative goodness of fit, is carried out. To stabilize the ill-conditioned estimation process, a Bayesian approach with a shrinkage prior (BSP) is proposed. In subsequent steps, the DKI-FWE model and the BSP estimation approach are evaluated in terms of estimation error, both in simulation and real data experiments.

Results: Although it is shown that the DKI-FWE model parameter estimation problem is ill-conditioned, DKI-FWE was found to describe the data significantly better compared to the standard DKI model for a large range of free water fractions. The acquisition protocol was optimized in terms of the maximally attainable precision of the DKI-FWE model parameters. The BSP estimator is shown to provide reliable DKI-FWE model parameter estimates.

Conclusion: The combination of the DKI-FWE model with BSP is shown to be a feasible approach to estimate DKI parameters, while simultaneously eliminating free water partial volume effects. Magn Reson Med 80:802-813, 2018. © 2018 The Authors Magnetic Resonance in Medicine published by Wiley Periodicals, Inc. on behalf of International Society for Magnetic Resonance in Medicine. This is an open access article under the terms of the Creative Commons Attribution NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.

Keywords: Bayesian estimation; diffusion kurtosis imaging; free water elimination; partial volume effects; shrinkage prior.

Figure 1

Condition numbers in a real data slice for (a) the Diffusion Kurtosis Imaging (DKI) model and (b) the “Diffusion Kurtosis Imaging with Free Water Elimination” (DKI‐FWE) model. Panel (c) shows the histograms of the condition numbers of both the DKI and DKI‐FWE fit. Condition numbers larger than 15e4 have been set to 15e4 for visual clarity.

Figure 2

a: The maximum likelihood estimate of the probability of success P_s where the DKI‐FWE model is significantly better at describing the data compared to the standard DKI model for simulated data in function of the true free water signal fraction f. Colored bands indicate the 95% confidence interval (b) B ₀ image overlayed with the voxels where the DKI‐FWE model describes the data significantly better compared to the standard DKI model (red). Cerebrospinal fluid (CSF) regions are indicated with a green contour plot.

Figure 3

Error distributions of f (a), fractional anisotropy (FA) (b), mean diffusivity (MD) (c), and mean kurtosis (MK) (d), for 3 different diffusion model and estimation technique combinations: BSP with the DKI‐FWE model (solid blue), ML with the DKI‐FWE model (dashed red) and BSP with the standard DKI model (dotted yellow).

Figure 4

Real data maps of DKI‐FWE model metrics (f, FA, MD, and MK) using the BSP estimator.

Figure 5

Real data difference maps of DKI‐FWE model metrics (f, FA, MD, and MK) between the BSP estimator and the ML estimator.

Figure 6

Distribution of BSP estimator (blue) and ML estimator (orange) parameter estimates of a selection of four parameters from the DKI‐FWE model in a real human data set. CSF was masked out to make parameters representative of brain tissue.

Figure 7

Real data maps of DKI model metrics (FA, MD, and MK) using the BSP estimator.

Figure 8

Real data difference maps between the DKI‐FWE model and the DKI model in terms of their common model metrics (FA, MD, and MK). Both model parameters are estimated using the BSP estimator.