The distribution of axon diameters (ADD) is an important white matter feature: it influences action potential speed, it has been shown to evolve during development, and also to be affected during pathological processes. Diffusion Magnetic Resonance Imaging is a powerful non-invasive tool that is sensitive to the displacement of water molecules. In theory, as axon diameters are well bellow the imaging resolution, the ADD can be estimated indirectly by fitting a biophysical model to the measured signal. Such an approach is the basis of microstructure imaging. Microstructure imaging usually relies on compartment models, splitting the signal into the intra-axonal or the extra-axonal compartments. However, in practice, ADD reconstruction remains a challenging problem, mainly due to model degeneracy (different solutions can have similar signals), the diameter lower bound which dictates the smallest diameter visible by the scanner, and the difficulty in modeling properly the extra-axonal signal.

In this thesis, we addressed the challenges at stake by focusing on the intra-axonal and extra-axonal compartments separately. Model performance was evaluated by comparing estimates with the ground-truth provided by Monte Carlo simulations.

Model degeneracy was addressed by introducing Laplacian regularization, which was shown to provide better ADD estimates when considering the intra-axonal signal only. Sensitivity to small diameters was improved by using a richer diffusion protocol designed to maximize sensitivity to a set of diameters, which was shown to provide reconstruction of unimodal and bimodal distributions, with sensitivity to population specific changes.

Regarding the extra-axonal space, using a mixture of higher order tensors improved reconstruction of the signal compared to standard models. The proposed model is flexible enough to adapt to a variety of simulated signals, outperforming current models of hindered diffusion.

Combining the results for both compartments slightly improved estimates for some of the simulated signals. We highlighted the similarity between the intra-axonal and extra-axonal signals, which might be the main limiting factor when using single Pulsed Gradient Spin Echo sequences.