Fig. 2.

A hierarchy of effective theories relevant for tissue microstructure (bottom to top): From the atomic scale (many-body Schrödinger’s equation governing the dynamics of atoms and of their electronic orbitals); to a simple diffusion equation within any small domain of cytoplasm or extra-cellular space, with classical particle density ψ emerging after coarse-graining ∣Ψ∣² and integrating out the environment coordinates (Feynman and Vernon, 1963; Caldeira and Leggett, 1981; Kamenev, 2011); to the diffusion equation with heterogeneous local diffusivity D(x) describing the microstructural complexity at the cellular level; to the dynamics of the diffusion-weighted signal, Eq. (2), acquired over an MRI voxel, with ${\partial }_x^2\to -q^2$, ${\partial }_x^4\to q^4$, …, corresponding to the cumulant expansion in the powers of diffusion wave vector q after the Fourier transform. The effective-medium retarded response functions $\mathcal{D}(t)$, W(t), … give rise to the time-dependent diffusivity D(t), time-dependent kurtosis K(t), and so on (Novikov and Kiselev, 2010), (Novikov et al., 2019, Sec. 2).