The microstructure of the brain white matter is not visible to the naked eye but would be of invaluable help to the clinician in the diagnosis and treatment of many brain pathologies. Diffusion MRI is an in-vivo non invasive imaging technique that probes the cyto-architecture of the white matter through the diffusion of water. However, diffusion MRI is limited in resolution, which makes forward models of the diffusion at the voxel level rather complex. In this paper, we provide a statistical framework for recovering the maximum-likelihood estimators of the parameters of mixture models of the diffusion. We calibrate different methods on simulated data to guarantee convergence to the maximum likelihood and show that profile likelihood maximization using variable projection together with a Levenberg-Marquardt algorithm with analytic Jacobian is the most efficient method to obtain the MLE.