  Machine learning is a powerful approach for fitting microstructural models to
diffusion MRI data. Early machine learning microstructure imaging
implementations trained regressors to estimate model parameters in a supervised
way, using synthetic training data with known ground truth. However, a drawback
of this approach is that the choice of training data impacts fitted parameter
values. Self-supervised learning is emerging as an attractive alternative to
supervised learning in this context. Thus far, both supervised and
self-supervised learning have typically been applied to isotropic models, such
as intravoxel incoherent motion (IVIM), as opposed to models where the
directionality of anisotropic structures is also estimated. In this paper, we
demonstrate self-supervised machine learning model fitting for a directional
microstructural model. In particular, we fit a combined T1-ball-stick model to
the multidimensional diffusion (MUDI) challenge diffusion-relaxation dataset.
Our self-supervised approach shows clear improvements in parameter estimation
and computational time, for both simulated and in-vivo brain data, compared to
standard non-linear least squares fitting. Code for the artificial neural net
constructed for this study is available for public use from the following
GitHub repository: https://github.com/jplte/deep-T1-ball-stick
