A mathematical model for neuronal fibers

Abstract

Diffusion Magnetic Resonance Imaging (MRI) is a powerful non-invasively method producing images of biological tissues exploiting the water molecules diffusion into the living tissues under a magnetic field. This technique enhances the highly non--homogenous character of the diffusion medium, revealing underlying microstructure. Recently, this method has been widely applied to the study of the neuronal fibers in the brain white matter, and several methods have been proposed to reconstruct the fiber paths from such data (tractography). Results rely on the model chosen to represent water molecules diffusion into an MRI voxel: among all the proposed model, we recall the Diffusion Spectrum Imaging (DSI) model, which describes the diffusion inside each voxel as a probability density function defined on a set of predefined directions inside the voxel. DSI is able to successfully describe more complex tissue configurations than other models as, for example, Diffusion Tensor Imaging (DTI), but lacks to consider the density of fibers going to make up a bundle trajectory among adjacent voxels, preventing any evaluation of the real physical dimension of neuronal fiber bundles. We describe here a new approach, based on ideas from mass transportation theory, that takes into account the whole information given by DSI in order to reconstruct the underlying water diffusion process, and recover the actual distribution of neuronal fibers.