*Published Paper*

**Inserted:** 4 mar 2012

**Last Updated:** 26 jul 2012

**Journal:** Comm. Pure Appl. Analysis

**Volume:** 11

**Number:** 5

**Pages:** 2157-2177

**Year:** 2012

**Abstract:**

Diffusion Magnetic Resonance Imaging (MRI) is used to (noninvasively)study neuronal fibers in the brain white matter. Reconstructing fiber paths from such data (tractography problem) is relevant in particular to study the connectivity between two given cerebral regions. Fiber-tracking models rely on how water molecules diffusion is represented in each MRI voxel. The Diffusion Spectrum Imaging (DSI) technique represents the diffusion as a probability density function (DDF) defined on a set of predefined directions inside each voxel. DSI is able to describe complex tissue configurations (compared e.g. with Diffusion Tensor Imaging), but ignores the actual density of fibers forming bundle trajectories among adjacent voxels, preventing any evaluation of the real physical dimension of these fiber bundles. By considering the fiber paths between two given areas as geodesics of a suitable well-posed optimal control problem (related to optimal mass transportation) which takes into account the whole information given by the DDF, we are able to provide a quantitative criterion to estimate the connectivity between two given cerebral regions, and to recover in principle the actual distribution of neuronal fibers between them.

**Keywords:**
Variational methods, imaging, tractography, optimal mass transportation

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