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 ( K D). The relationship between K D and the microscopic structure of the medium has been the subject of experimental and theoretical investigations. Calculations of K D 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* = K D/ r 2 was determined for values of the fractional solid volume (φ) in the range 0.005 ≤ φ ≤ 0.7. We compared our numerical results, summarized by an interpolating formula relating K* to φ, with previous theoretical determinations of hydraulic permeability in fibrous media. We found a good agreement with the Carman-Kozeny equation only for φ > 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 φ 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.