  Effective characterisation of the brain grey matter cytoarchitecture with
quantitative sensitivity to soma density and volume remains an unsolved
challenge in diffusion MRI (dMRI). Solving the problem of relating the dMRI
signal with cytoarchitectural characteristics calls for the definition of a
mathematical model that describes brain tissue via a handful of
physiologically-relevant parameters and an algorithm for inverting the model.
To address this issue, we propose a new forward model, specifically a new
system of equations, requiring a few relatively sparse b-shells. We then apply
modern tools from Bayesian analysis known as likelihood-free inference (LFI) to
invert our proposed model. As opposed to other approaches from the literature,
our algorithm yields not only an estimation of the parameter vector $\theta$
that best describes a given observed data point $x_0$, but also a full
posterior distribution $p(\theta|x_0)$ over the parameter space. This enables a
richer description of the model inversion, providing indicators such as
credible intervals for the estimated parameters and a complete characterization
of the parameter regions where the model may present indeterminacies. We
approximate the posterior distribution using deep neural density estimators,
known as normalizing flows, and fit them using a set of repeated simulations
from the forward model. We validate our approach on simulations using dmipy and
then apply the whole pipeline on two publicly available datasets.
