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Effective characterisation of the brain grey matter cytoarchitecture withquantitative sensitivity to soma density and volume remains an unsolvedchallenge in diffusion MRI (dMRI). Solving the problem of relating the dMRIsignal with cytoarchitectural characteristics calls for the definition of amathematical model that describes brain tissue via a handful ofphysiologically-relevant parameters and an algorithm for inverting the model.To address this issue, we propose a new forward model, specifically a newsystem of equations, requiring a few relatively sparse b-shells. We then applymodern tools from Bayesian analysis known as likelihood-free inference (LFI) toinvert 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 fullposterior distribution $p(\theta|x_0)$ over the parameter space. This enables aricher description of the model inversion, providing indicators such ascredible intervals for the estimated parameters and a complete characterizationof the parameter regions where the model may present indeterminacies. Weapproximate the posterior distribution using deep neural density estimators,known as normalizing flows, and fit them using a set of repeated simulationsfrom the forward model. We validate our approach on simulations using dmipy andthen apply the whole pipeline on two publicly available datasets.

Inverting brain grey matter models with likelihood-free inference: a tool for trustable cytoarchitecture measurements