Orientation-Dispersed Apparent Axon Diameter via Multi-Stage Spherical Mean Optimization

  • Marco PizzolatoEmail author
  • Demian Wassermann
  • Rachid Deriche
  • Jean-Philippe Thiran
  • Rutger Fick
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


The estimation of the apparent axon diameter (AAD) via diffusion MRI is affected by the incoherent alignment of single axons around its axon bundle direction, also known as orientational dispersion. The simultaneous estimation of AAD and dispersion is challenging and requires the optimization of many parameters at the same time. We propose to reduce the complexity of the estimation with an multi-stage approach, inspired to alternate convex search, that separates the estimation problem into simpler ones, thus avoiding the estimation of all the relevant model parameters at once. The method is composed of three optimization stages that are iterated, where we separately estimate the volume fractions, diffusivities, dispersion, and mean AAD, using a Cylinder and Zeppelin model. First, we use multi-shell data to estimate the undispersed axon micro-environment’s signal fractions and diffusivities using the spherical mean technique; then, to account for dispersion, we use the obtained micro-environment parameters to estimate a Watson axon orientation distribution; finally, we use data acquired perpendicularly to the axon bundle direction to estimate the mean AAD and updated signal fractions, while fixing the previously estimated diffusivity and dispersion parameters. We use the estimated mean AAD to initiate the following iteration. We show that our approach converges to good estimates while being more efficient than optimizing all model parameters at once. We apply our method to ex-vivo spinal cord data, showing that including dispersion effects results in mean apparent axon diameter estimates that are closer to their measured histological values.


Diffusion MRI Axon diameter Dispersion Spherical mean 



This work is supported by the Swiss National Science Foundation under grant number CRSII5\(\_\)170873 (Sinergia project) and by the ERC Advanced Grant agreement No. 694665 (CoBCoM).


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Marco Pizzolato
    • 1
    Email author
  • Demian Wassermann
    • 2
  • Rachid Deriche
    • 3
  • Jean-Philippe Thiran
    • 1
    • 4
  • Rutger Fick
    • 5
  1. 1.Signal Processing Lab (LTS5)École Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.Parietal, Inria, CEA, Université Paris-SaclayPalaiseauFrance
  3. 3.Athena, Inria, Université Côte d’AzurSophia AntipolisFrance
  4. 4.University Hospital Center (CHUV) and University of Lausanne (UNIL)LausanneSwitzerland
  5. 5.TheraPanaceaParisFrance

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