Robbert Leonard Harms^{1} and Alard Roebroeck^{1}

Diffusion MRI microstructure approaches use point estimates ignoring the uncertainty in these estimates. In this work, we evaluate two general methods to quantify uncertainty and generate uncertainty maps for any microstructure model. We find that the Fisher Information Matrix method based in nonlinear optimization is fast and accurate for models with few parameters. The Markov Chain Monte Carlo (MCMC) based method takes more time, but provides robust uncertainty estimates even for sophisticated models with more parameters. Uncertainty estimates of microstructure measures can help power evaluations for group/population studies and assist in data quality control and analysis of microstructure model fit.

1. Assaf Y, Basser PJ. Composite hindered and restricted model of diffusion (CHARMED) MR imaging of the human brain. Neuroimage. 2005;27(1):48-58. doi:10.1016/j.neuroimage.2005.03.042.

2. Zhang H, Schneider T, Wheeler-Kingshott CA, Alexander DC. NODDI: Practical in vivo neurite orientation dispersion and density imaging of the human brain. Neuroimage. 2012;61(4):1000-1016. doi:10.1016/j.neuroimage.2012.03.072.

3. Assaf Y, Blumenfeld-Katzir T, Yovel Y, Basser PJ. AxCaliber: A method for measuring axon diameter distribution from diffusion MRI. Magn Reson Med. 2008;59(6):1347-1354. doi:10.1002/mrm.21577.

4. Alexander DC, Hubbard PL, Hall MG, et al. Orientationally invariant indices of axon diameter and density from diffusion MRI. Neuroimage. 2010;52(4):1374-1389. doi:10.1016/j.neuroimage.2010.05.043.

5. Jones DK. Determining and visualizing uncertainty in estimates of fiber orientation from diffusion tensor MRI. Magn Reson Med. 2003;49(1):7-12. doi:10.1002/mrm.10331.

6. Jones DK, Basser PJ. “Squashing peanuts and smashing pumpkins”: How noise distorts diffusion-weighted MR data. Magn Reson Med. 2004;52(5):979-993. doi:10.1002/mrm.20283.

7. Behrens TEJ, Woolrich MW, Jenkinson M, et al. Characterization and Propagation of Uncertainty in Diffusion-Weighted MR Imaging. Magn Reson Med. 2003;50(5):1077-1088. doi:10.1002/mrm.10609.

8. Jbabdi S, Woolrich MW, Andersson JLR, Behrens TEJ. A Bayesian framework for global tractography. Neuroimage. 2007;37(1):116-129. doi:10.1016/j.neuroimage.2007.04.039.

9. Sotiropoulos SN, Behrens TEJ, Jbabdi S. Ball and rackets: Inferring fiber fanning from diffusion-weighted MRI. Neuroimage. 2012;60(2):1412-1425. doi:10.1016/j.neuroimage.2012.01.056.

Estimation pipelines for the Fractional Anisotropy of the
Diffusion Tensor model using optimization (top, red) and sampling
(bottom, blue). The optimization pipeline uses the inverse of the
Hessian at the optimal point for computing the uncertainties of the
model parameters. These uncertainties are then propagated to the
derived measures (FA in this case). In MCMC sampling, derived
measures can be calculated per sample, after which a standard
deviation can be obtained by fitting a Gaussian to these derived
samples. Both these pipelines and methodologies extend naturally to
any model derived measure.

A) comparison maps between optimization and sampling of Tensor
Fractional Anisotropy (FA) values and corresponding standard
deviations (std), with for a highlighted voxel the computed values.
B) Information about the highlighted point in figure A, with in gray
the sampled FA histogram and in red and blue the predicted
distributions of, respectively, the optimization and sampling
methods. The red and blue colors correspond to figure 1.

Comparison between the point estimates and corresponding
uncertainties using optimization and sampling for CHARMED_r1 and
NODDI. The maps represent, from left to right, the point estimate of
optimization, the corresponding standard deviation (std.) calculated
using the inverse of the Hessian, the mean of the marginal likelihood
distribution obtained from MCMC sampling and last, the std. of those
samples. The histograms in gray are calculated from the MCMC samples
of the voxel highlighted in the maps, with the predicted
distributions of optimization and sampling plotted on top.