Numerical analysis of viscous flow through fibrous media: a model for glomerular basement membrane permeability

Viscous flow through fibrous media is characterized macroscopically by the Darcy permeability (KD). The relationship between KD and the microscopic structure of the medium has been the subject of experimental and theoretical investigations. Calculations of KD based on the solution of the hydrodynamic flow at fiber scale exist in literature only for two-dimensional arrays of parallel fibers. We considered a fiber matrix consisting of a three-dimensional periodic array of cylindrical fibers with uniform radius (r) and length connected in a tetrahedral structure. According to recent ultrastructural studies, this array of fibers can represent a model for the glomerular basement membrane (GBM). The Stokes flow through the periodic array was simulated using a Galerkin finite element method. The dimensionless ratio K* = KD/r2 was determined for values of the fractional solid volume (phi) in the range 0.005 < or = phi < or = 0.7. We compared our numerical results, summarized by an interpolating formula relating K* to phi, with previous theoretical determinations of hydraulic permeability in fibrous media. We found a good agreement with the Carman-Kozeny equation only for phi > 0.4. Among the other theoretical analysis considered, only that of Spielman and Goren (Environ. Sci. Technol. 2: 279-287, 1968) gives satisfactory agreement in the whole range of phi considered. These results can be useful to model combined transport of water and macromolecules through the GBM for the estimation of the radius and length of extracellular protein fibrils.