Dependence of the extra-cellular diffusion coefficient on the fractions of neurites and cell bodies in gray matter
Purpose: The dependence of the long-time (tortuosity) limit of the extra-cellular diffusivity on the intra-cellular volume fraction is of fundamental importance for microstructure modeling. While such dependencies have been explored for the white matter, the tortuosity limit in gray matter is unknown due to complex cell composition and geometry. Here we rationalize and validate numerically the analytical relation between the extra-cellular diffusivity and intra-cellular fractions of cell bodies (somas) and neurites. Methods: The tortuosity relation for extra-cellular diffusivity qualitatively follows from effective medium theory, coarse-grained by diffusion outside somas (spheres) and neurites (cylinders), respectively. This problem is equivalent to finding the overall conductivity in a medium of grains in a matrix, with methodology dating back to the 19th century. We extend the effective medium methodology to populations of impermeable spheres and randomly oriented cylinders with various volume fractions, yielding closed-form expressions corroborated by Monte Carlo simulations. Results: We establish the power-law scaling of the extra-cellular diffusivity with the volume fractions of the extra-soma and extra-neurite spaces. We further evaluate the proposed framework using simulations in realistic tissue geometries. Conclusion: Theory and simulations relate extra-cellular tortuosity to soma and neurite fractions, potentially offering a diffusion MRI protocol design optimized for in vivo assessment of soma size and soma/neurite fractions within clinical scan times. Such in vivo measurements can be used to study development, aging and neurodegenerative disorders.