## Abstract

Diffusion-weighted magnetic resonance imaging (MRI) is sensitive to ensemble-averaged molecular displacements, which provide valuable information on e.g. structural anisotropy in brain tissue. However, a concrete interpretation of diffusion-weighted MRI data in terms of physiological or structural parameters turns out to be extremely challenging. One of the main reasons for this is the multi-scale nature of the diffusion-weighted signal, as it is sensitive to the microscopic motion of particles averaged over macroscopic volumes. In order to analyze the geometrical patterns that occur in (diffusion-weighted measurements of) biological tissue and many other structures, we may invoke tools from the field of stochastic geometry. Stochastic geometry describes statistical methods and models that apply to random geometrical patterns of which we may only know the distribution. Despite its many uses in geology, astronomy, telecommunications, etc., its application in diffusion-weighted MRI has so far remained limited. In this work we review some fundamental results in the field of diffusion-weighted MRI from a stochastic geometrical perspective, and discuss briefly for which other questions stochastic geometry may prove useful. The observations presented in this paper are partly inspired by the Workshop on Diffusion MRI and Stochastic Geometry held at Sandbjerg Estate (Denmark) in 2019, which aimed to foster communication and collaboration between the two fields of research.

Original language | English |
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Title of host publication | Anisotropy Across Fields and Scales |

Editors | Evren Özarslan, Thomas Schultz, Eugene Zhang, Andrea Fuster |

Publisher | Springer |

Publication date | 2021 |

Pages | 193-202 |

ISBN (Print) | 978-3-030-56214-4 |

DOIs | |

Publication status | Published - 2021 |

Event | 2018 Workshop on Visualization and Processing of Anisotropy in Imaging, Geometry, and Astronomy - Dagstuhl, Germany Duration: 28 Oct 2018 → 2 Nov 2018 |

### Conference

Conference | 2018 Workshop on Visualization and Processing of Anisotropy in Imaging, Geometry, and Astronomy |
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Country/Territory | Germany |

City | Dagstuhl |

Period | 28/10/2018 → 02/11/2018 |

Series | Mathematics and Visualization |
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ISSN | 1612-3786 |

### Bibliographical note

Funding Information:Acknowledgments The authors thank the Villum Foundation for funding this work through a block stipendium and a Villum Experiment grant, as well as for funding the Centre for Stochastic Geometry and Advanced Bioimaging (CSGB) which fostered 10 years of interactions between the authors and experts in stochastic geometry, including the 2019 Workshop on Diffusion MRI and Stochastic Geometry. In particular, the authors wish to thank Eva B. Vedel Jensen and Markus Kiderlen, for their engagement, input, and encouragement in our ideas for collaboration, as well as their kind patience with us when beginning to engage with stochastic geometry. We are also indebted to Andrea Fuster and Luc Florack for their contributions to the PhD project of Tom Dela Haije, which included the introduction of the barrier ODF. The authors would like to acknowledge Schloss Dagstuhl and the organizers and participants of Dagstuhl seminars 16142 and 18442 for facilitating discussions that further supported developing the ideas discussed in this work. Finally, the authors wish to thank all the participants in the 2019 Workshop on Diffusion MRI and Stochastic Geometry.

Publisher Copyright:

© 2021, The Author(s).