A Conditional Autoregressive Model for Estimating Slow and Fast Diffusion from Magnetic Resonance Images

  • Ettore LanzaroneEmail author
  • Elisa Scalco
  • Alfonso Mastropietro
  • Simona Marzi
  • Giovanna Rizzo
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 296)


The Intra-Voxel Incoherent Motion (IVIM) model is largely adopted to estimate slow and fast diffusion parameters of water molecules in biological tissues, which are used as biomarkers for different diseases. However, the standard approach to obtain the maps of these parameters is based on a voxel-by-voxel estimation and neglects the spatial correlations, thus resulting in noisy maps. To get better maps, we propose a Bayesian approach that exploits a Conditional Autoregressive (CAR) prior density. We consider a pure CAR model and a mixture CAR model, and we compare the outcomes with two benchmark approaches. Results show better maps under the CAR models.


Conditional autoregressive model Diffusion parameters Intra-voxel incoherent motion Magnetic resonance imaging Spatial correlation 


  1. 1.
    Alston, C.L., Mengersen, K.L., Thompson, J.M., Littlefield, P.J., Perry, D., Ball, A.J.: Extending the Bayesian mixture model to incorporate spatial information in analysing sheep CAT scan images. Aust. J. Agr. Res. 56, 373–388 (2005)CrossRefGoogle Scholar
  2. 2.
    Barbieri, S., Donati, O.F., Froehlich, J.M., Thoeny, H.C.: Impact of the calculation algorithm on biexponential fitting of diffusion-weighted MRI in upper abdominal organs. Magn. Reson. Med. 75, 2175–2184 (2016)CrossRefGoogle Scholar
  3. 3.
    Dyvorne, H.A., Galea, N., Nevers, T., Fiel, M.I., Carpenter, D., Wong, E., Orton, M., de Oliveira, A., Feiweier, T., Vachon, M.L., Babb, J.S., Taouli, B.: Diffusion-weighted imaging of the liver with multiple b values: effect of diffusion gradient polarity and breathing acquisition on image quality and intravoxel incoherent motion parameters - a pilot study. Radiology 266, 920–929 (2013)CrossRefGoogle Scholar
  4. 4.
    Freiman, M., Perez-Rossello, J.M., Callahan, M.J., Voss, S.D., Ecklund, K., Mulkern, R.V., Warfield, S.K.: Reliable estimation of incoherent motion parametric maps from diffusion-weighted MRI using fusion bootstrap moves. Med. Image Anal. 17, 325–336 (2013)CrossRefGoogle Scholar
  5. 5.
    Feng, D., Tierney, L., Magnotta, V.: MRI tissue classification using high-resolution Bayesian hidden Markov normal mixture models. J. Am. Stat. Assoc. 107, 102–119 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gudbjartsson, H., Patz, S.: The Rician distribution of noisy MRI data. Magn. Reson. Med. 34, 910–914 (1995)CrossRefGoogle Scholar
  7. 7.
    Gustafsson, O., Montelius, M., Starck, G., Ljungberg, M.: Impact of prior distributions and central tendency measures on Bayesian intravoxel incoherent motion model fitting. Magn. Reson. Med. 79, 674–1683 (2018)CrossRefGoogle Scholar
  8. 8.
    Jeong, J., Vannucci, M., Ko, K.: A wavelet-based Bayesian approach to regression models with long memory errors and its application to fMRI data. Biometrics 69, 184–196 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kurugol, S., Freiman, M., Afacan, O., Perez-Rossello, J.M., Callahan, M.J., Warfield, S.K.: Spatially-constrained probability distribution model of incoherent motion (SPIM) for abdominal diffusion-weighted MRI. Med. Image Anal. 32, 173–183 (2016)CrossRefGoogle Scholar
  10. 10.
    Le Bihan, D.: Intravoxel incoherent motion imaging using steady-state free precession. Magn. Reson. Med. 7, 346–351 (1988)CrossRefGoogle Scholar
  11. 11.
    Leroux, B.G., Lei, X., Breslow, N.: Estimation of disease rates in small areas: a new mixed model for spatial dependence. In: Halloran, M.E., Berry, D. (eds); Statistical Models in Epidemiology, the Environment, and Clinical Trials; the IMA Volumes in Mathematics and its Applications 116, 179–191 (2000)Google Scholar
  12. 12.
    Neil, J.J., Bretthorst, G.L.: On the use of Bayesian probability theory for analysis of exponential decay date: an example taken from intravoxel incoherent motion experiments. Magn. Reson. Med. 29, 642–647 (1993)CrossRefGoogle Scholar
  13. 13.
    Orton, M.R., Collins, D.J., Koh, D.M., Leach, M.O.: Improved intravoxel incoherent motion analysis of diffusion weighted imaging by data driven Bayesian modeling. Magn. Reson. Med. 71, 411–420 (2014)CrossRefGoogle Scholar
  14. 14.
    Spinner, G.R., von Deuster, C., Tezcan, K.C., Stoeck, C.T., Kozerke, S.: Bayesian intravoxel incoherent motion parameter mapping in the human heart. J. Cardiovasc. Magn. Reson. 19, 85 (2017)CrossRefGoogle Scholar
  15. 15.
    Stan Development Team: Stan modeling language users guide and reference manual (version 2.9.0), (2015)
  16. 16.
    Suo, S., Lin, N., Wang, H., Zhang, L., Wang, R., Zhang, S., Hua, J., Xu, J.: Intravoxel incoherent motion diffusion-weighted MR imaging of breast cancer at 3.0 Tesla: comparison of different curve-fitting methods. J. Magn. Reson. Imaging 42, 362–370 (2015)Google Scholar
  17. 17.
    While, P.T.: A comparative simulation study of Bayesian fitting approaches to intravoxel incoherent motion modeling in diffusion-weighted MRI. Magn. Reson. Med. 78, 2373–2387 (2017)CrossRefGoogle Scholar
  18. 18.
    Zhang, L., Guindani, M., Versace, F., Vannucci, M.: A spatio-temporal nonparametric Bayesian variable selection model of fMRI data for clustering correlated time courses. Neuroimage 95, 162–175 (2014)CrossRefGoogle Scholar
  19. 19.
    Zhang, L., Guindani, M., Vannucci, M.: Bayesian models for functional magnetic resonance imaging data analysis. WIREs Comp. Stat. 7, 21–41 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Zhang, L., Guindani, M., Versace, F., Vannucci, M.: A spatiotemporal nonparametric Bayesian model of multi-subject fMRI data. Ann. Appl. Stat. 10, 638–666 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ettore Lanzarone
    • 1
    Email author
  • Elisa Scalco
    • 2
  • Alfonso Mastropietro
    • 2
  • Simona Marzi
    • 3
  • Giovanna Rizzo
    • 2
  1. 1.Institute for Applied Mathematics and Information Technologies (IMATI), CNRMilanItaly
  2. 2.Institute of Molecular Bioimaging and Physiology (IBFM), CNRLeccoItaly
  3. 3.Medical Physics Laboratory, Regina Elena National Cancer InstituteRomeItaly

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