We propose a new method to solve the inverse problem of relating the diffusion MRI signal with cytoarchitectural characteristics in brain gray matter. Specifically, our method has quantitative sensitivity to soma density and volume. Our solution is twofold. First, we propose a new forward model that relates summary statistics of the dMRI signal with tissue parameters, relying on six b-shells only. We then apply a likelihood-free inference based algorithm to invert the proposed model, which not only estimates the tissue parameters that best describe the acquired diffusion signal, but also a full posterior distribution over the parameter space.