The incessant, random motion of water molecules within biological tissues reveals unique information about the tissues’ structural and functional characteristics. Diffusion magnetic resonance imaging is sensitive to this random motion, and since the mid-1990s it has been extensively employed for studying the human brain. Most notably, measurements of water diffusion allow for the early detection of ischaemic stroke and for the unveiling of the brain’s wiring via reconstruction of the neuronal connections. Ultimately, the goal is to employ this imaging technique to perform non-invasive, in vivo virtual histology to directly characterise both healthy and diseased tissue. 

Recent developments in the field have introduced new ways to measure the diffusion process in clinically feasible settings. These new measurements, performed by employing generalised magnetic field gradient waveforms, grant access to specific features of the cellular composition and structural organisation of the tissue. Methods based on them have already proven beneficial for the assessment of different brain diseases, sparking interest in translating such techniques into clinical practice. This thesis focuses on improving the methods currently employed for the analysis of such diffusion MRI data, with the aim of facilitating their clinical adoption. 

The first two publications introduce constrained frameworks for the estimation of parameters from diffusion MRI data acquired with generalised gradient waveforms. The constraints are dictated by mathematical and physical properties of a multi-compartment model used to represent the brain tissue, and can be efficiently enforced by employing a relatively new optimisation scheme called semidefinite programming. The developed routines are demonstrated to improve robustness to noise and imperfect data collection. Moreover, constraining the fit is shown to relax the requirements on the number of points needed for the estimation, thus allowing for faster data acquisition. 

In the third paper, the developed frameworks are employed to study the brain’s white matter in patients previously hospitalised for COVID-19 and who still suffer from neurological symptoms months after discharge. The results show widespread alterations to the structural integrity of their brain, with the metrics available through the advanced diffusion measurements providing new insights into the damage to the white matter. 

The fourth paper revisits the modelling paradigm currently adopted for the analysis of diffusion MRI data acquired with generalised gradient waveforms. Hitherto, the assumption of free diffusion has been employed to represent each domain in a multi-compartmental picture of the brain tissue. In this work, a model for restricted diffusion is considered instead to alleviate the paradoxical assumption of free but compartmentalised diffusion. The model is shown to perfectly capture restricted diffusion as measured with the generalised diffusion gradient waveforms, thus endorsing its use for representing each domain in the multi-compartmental model of the tissue.