Advertisement

Bayesian Heteroscedastic Regression for Diffusion Tensor Imaging

  • Bertil Wegmann
  • Anders Eklund
  • Mattias Villani
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

We propose a single-diffusion tensor model with heteroscedastic noise and a Bayesian approach via a highly efficient Markov Chain Monte Carlo ( MCMC) algorithm for inference. The model is very flexible since both the noise-free signal and the noise variance are functions of diffusion covariates, and the relevant covariates in the noise are automatically selected by Bayesian variable selection. We compare the estimated diffusion tensors from our model to a homoscedastic counterpart with no covariates in the noise, and to commonly used linear and nonlinear least squares methods. The estimated single-diffusion tensors within each voxel are compared with respect to fractional anisotropy (FA) and mean diffusivity (MD). Using data from the Human Connectome Project, our results show that the noise is clearly heteroscedastic, especially the posterior variance for MD is substantially underestimated by the homoscedastic model, and inferences from the homoscedastic model are on average spuriously precise. Inferences from commonly used ordinary and weighted least squares methods (OLS and WLS) show that it is not adequate to estimate the single-diffusion tensor from logarithmic measurements.

Notes

Acknowledgements

The authors would like to thank professor Hans Knutsson for his code for visualizing diffusion directions.

Anders Eklund was supported by the Information Technology for European Advancement (ITEA) 3 Project BENEFIT (better effectiveness and efficiency by measuring and modelling of interventional therapy) and by the Swedish research council (grant 2015-05356, “Learning of sets of diffusion MRI sequences for optimal imaging of micro structures“). Anders Eklund and Bertil Wegmann were supported by the Swedish research council (grant 2013-5229, “Statistical analysis of fMRI data”).

Data collection and sharing for this project was provided by the Human Connectome Project (HCP; Principal Investigators: Bruce Rosen, M.D., Ph.D., Arthur W. Toga, Ph.D., Van J. Weeden, MD). HCP funding was provided by the National Institute of Dental and Craniofacial Research (NIDCR), the National Institute of Mental Health (NIMH), and the National Institute of Neurological Disorders and Stroke (NINDS). HCP data are disseminated by the Laboratory of Neuro Imaging at the University of Southern California.

References

  1. 1.
    Andersson, J.: Maximum a posteriori estimation of diffusion tensor parameters using a Rician noise model: Why, how and but. NeuroImage 42, 1340–1356 (2008)CrossRefGoogle Scholar
  2. 2.
    Andersson, J.L., Sotiropoulos, S.N.: An integrated approach to correction for off-resonance effects and subject movement in diffusion MR imaging. NeuroImage 125, 1063–1078 (2016)CrossRefGoogle Scholar
  3. 3.
    Basser, P., Pierpaoli, C.: A simplified method to measure the diffusion tensor from seven MR images. Magn. Reson. Med. 39, 928–934 (1998)CrossRefGoogle Scholar
  4. 4.
    Basser, P., LeBihan, D., Mattiello, J.: Estimation of the effective self-diffusion tensor from the NMR spin echo. J. Magn. Reson. 103, 247–254 (1994)CrossRefGoogle Scholar
  5. 5.
    Beckmann, C.F., Jenkinson, M., Smith, S.M.: General multilevel linear modeling for group analysis in FMRI. NeuroImage 20(2), 1052–1063 (2003)CrossRefGoogle Scholar
  6. 6.
    Behrens, T., Woolrich, M., Jenkinson, M., Johansen-Berg, H., Nunes, R., Clare, S., Matthews, P., Brady, J., Smith, S.: Characterization and propagation of uncertainty in diffusion-weighted MR imaging. Magn. Reson. Med. 50, 1077–1088 (2003)CrossRefGoogle Scholar
  7. 7.
    Behrens, T., Berg, H.J., Jbabdi, S., Rushworth, M., Woolrich, M.: Probabilistic diffusion tractography with multiple fibre orientations: what can we gain? NeuroImage 34, 144–155 (2007)CrossRefGoogle Scholar
  8. 8.
    Chen, G., Saad, Z.S., Nath, A.R., Beauchamp, M.S., Cox, R.W.: FMRI group analysis combining effect estimates and their variances. NeuroImage 60(1), 747–765 (2012)CrossRefGoogle Scholar
  9. 9.
    Eierud, C., Craddock, R.C., Fletcher, S., Aulakh, M., King-Casas, B., Kuehl, D., LaConte, S.M.: Neuroimaging after mild traumatic brain injury: review and meta-analysis. NeuroImage: Clin. 4, 283–294 (2014)CrossRefGoogle Scholar
  10. 10.
    Eklund, A., Dufort, P., Forsberg, D., LaConte, S.M.: Medical image processing on the GPU - past, present and future. Med. Image Anal. 17(8), 1073–1094 (2013)CrossRefGoogle Scholar
  11. 11.
    Eklund, A., Lindquist, M., Villani, M.: A Bayesian Heteroscedastic GLM with Application to fMRI Data with Motion Spikes. arXiv:1612.00690 (2016)Google Scholar
  12. 12.
    Elhabian, S., Gur, Y., Vachet, C., Piven, J., Styner, M., Leppert, I.R., Pike, G.B., Gerig, G.: Subject-motion correction in HARDI acquisitions: choices and consequences. Front. Neurol. 5, 240 (2014)CrossRefGoogle Scholar
  13. 13.
    Essen, D.C.V., Smith, S.M., Barch, D.M., Behrens, T.E., Yacoub, E., Ugurbil, K.: The WU-Minn human connectome project: an overview. NeuroImage 80, 62–79 (2013)CrossRefGoogle Scholar
  14. 14.
    Glasser, M.F., Sotiropoulos, S.N., Wilson, J.A., Coalson, T.S., Fischl, B., Andersson, J.L., Xu, J., Jbabdi, S., Webster, M., Polimeni, J.R., Essen, D.C.V., Jenkinson, M.: The minimal preprocessing pipelines for the human connectome project. NeuroImage 80, 105–124 (2013)CrossRefGoogle Scholar
  15. 15.
    Gudbjartsson, H., Patz, S.: The Rician distribution of noisy MRI data. Magn. Reson. Med. 34(6), 910–914 (1995)CrossRefGoogle Scholar
  16. 16.
    Guo, G.: Parallel statistical computing for statistical inference. J. Stat. Theory Pract. 6(3), 536–565 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jones, D.: Diffusion MRI: Theory, Methods and Applications. Oxford University Press, Oxford (2011)Google Scholar
  18. 18.
    Koay, C.G.: Least squares approaches to diffusion tensor estimation. In: Jones, D. (ed.) Diffusion MRI: Theory, Methods and Applications, pp. 272–284. Oxford University Press, Oxford (2011)Google Scholar
  19. 19.
    Koay, C., Chang, L., Pierpaoli, C., Basser, P.: Error propagation framework for diffusion tensor imaging via diffusion tensor representations. IEEE Trans. Med. Imaging 26, 1017–1034 (2007)CrossRefGoogle Scholar
  20. 20.
    Koay, C., Nevo, U., Chang, L., Pierpaoli, C., Basser, P.: The elliptical cone of uncertainty and its normalized measures in diffusion tensor imaging. IEEE Trans. Med. Imaging 27, 834–846 (2008)CrossRefGoogle Scholar
  21. 21.
    Kubicki, M., Westin, C.F., Maier, S.E., Mamata, H., Frumin, M., Ersner-Hershfield, H., Kikinis, R., Jolesz, F.A., McCarley, R., Shenton, M.E.: Diffusion tensor imaging and its application to neuropsychiatric disorders. Harv. Rev. Psychiatry 10(6), 324–336 (2002)CrossRefGoogle Scholar
  22. 22.
    Power, J.D., Mitra, A., Laumann, T.O., Snyder, A.Z., Schlaggar, B.L., Petersen, S.E.: Methods to detect, characterize, and remove motion artifact in resting state fMRI. NeuroImage 84, 320–341 (2014)CrossRefGoogle Scholar
  23. 23.
    Setsompop, K., Kimmlingen, R., Eberlein, E., Witzel, T., Cohen-Adad, J., McNab, J., Keil, B., Tisdall, M., Hoecht, P., Dietz, P., Cauley, S., Tountcheva, V., Matschl, V., Lenz, V., Heberlein, K., Potthast, A., Thein, H., Horn, J.V., Toga, A., Schmitt, F., Lehne, D., Rosen, B., Wedeen, V., Wald, L.: Pushing the limits of in vivo diffusion MRI for the human connectome project. NeuroImage 80, 220–233 (2013)CrossRefGoogle Scholar
  24. 24.
    Shenton, M.E., Hamoda, H.M., Schneiderman, J.S., Bouix, S., Pasternak, O., Rathi, Y., Vu, M.A., Purohit, M.P., Helmer, K., Koerte, I., Lin, A.P., Westin, C.F., Kikinis, R., Kubicki, M., Stern, R.A., Zafonte, R.: A review of magnetic resonance imaging and diffusion tensor imaging findings in mild traumatic brain injury. Brain Imaging Behav. 6(2), 137–192 (2012)CrossRefGoogle Scholar
  25. 25.
    Siegel, J.S., Power, J.D., Dubis, J.W., Vogel, A.C., Church, J.A., Schlaggar, B.L., Petersen, S.E.: Statistical improvements in functional magnetic resonance imaging analyses produced by censoring high-motion data points. Hum. Brain Mapp. 35(5), 1981–1996 (2014)CrossRefGoogle Scholar
  26. 26.
    Smith, S.M., Jenkinson, M., Johansen-Berg, H., Rueckert, D., Nichols, T.E., Mackay, C.E., Watkins, K.E., Ciccarelli, O., Cader, M.Z., Matthews, P.M., Behrens, T.E.: Tract-based spatial statistics: Voxelwise analysis of multi-subject diffusion data. NeuroImage 31(4), 1487–1505 (2006)CrossRefGoogle Scholar
  27. 27.
    Veraart, J., Sijbers, J., Sunaert, S., Leemans, A., Jeurissen, B.: Weighted linear least squares estimation of diffusion MRI parameters: strengths, limitations, and pitfalls. NeuroImage 81, 335–346 (2013)CrossRefGoogle Scholar
  28. 28.
    Villani, M., Kohn, R., Giordani, P.: Regression density estimation using smooth adaptive gaussian mixtures. J. Econom. 153(2), 155–173 (2009)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Villani, M., Kohn, R., Nott, D.: Generalized smooth finite mixtures. J. Econom. 171(2), 121–133 (2012)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Walker, L., Chang, L.C., Koay, C.G., Sharma, N., Cohen, L., Verma, R., Pierpaoli, C.: Effects of physiological noise in population analysis of diffusion tensor MRI data. NeuroImage 54(2), 1168–1177 (2011)CrossRefGoogle Scholar
  31. 31.
    Wegmann, B., Eklund, A., Villani, M.: Non-central chi regression for neuroimaging. arXiv:1612.07034 (2016)Google Scholar
  32. 32.
    Westin, C.F., Knutsson, H., Pasternak, O., Szczepankiewicz, F., Ozarslan, E., van Westen, D., Mattisson, C., Bogren, M., O’Donnell, L.J., Kubicki, M., Topgaard, D., Nilsson, M.: Q-space trajectory imaging for multidimensional diffusion MRI of the human brain. NeuroImage 135, 345–362 (2016)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Bertil Wegmann
    • 1
  • Anders Eklund
    • 1
    • 2
    • 3
  • Mattias Villani
    • 1
  1. 1.Division of Statistics and Machine Learning, Department of Computer and Information ScienceLinköping UniversityLinkopingSweden
  2. 2.Division of Medical Informatics, Department of Biomedical EngineeringLinköping UniversityLinkopingSweden
  3. 3.Center for Medical Image Science and Visualization (CMIV)Linköping UniversityLinkopingSweden

Personalised recommendations