A multiscale theoretical model for fluid flow in cellular biological media

An integrated methodology is developed for the theoretical analysis of momentum transfer in cellular biological media, such as biofilms and tissues. First, the method of local spatial averaging via a weight function is used to establish the equations that describe momentum transfer at the cellular biological medium scale, starting with a continuum-based formulation of the process at finer spatial scales. The constitutive behavior of each constituent phase is postulated at the polymer- or cell-scale and, through the averaging procedure, appropriate constitutive relations are developed for the upscaled stress tensors and the fluid–structure interaction forces. Further, closure problems are developed for the theoretical calculation of the effective material properties that appear in the constitutive relations. The developed closure problem for the static hydraulic permeability tensor is solved using a finite element method in the context of a periodic spherocylinder-in-cell model, which accounts for salient geometric features of microbial aggregates and biofilms at the cell-scale. The degree of structural anisotropy resulting from the shape, orientation, and spatial arrangement of biological cells (from stack formation to nematic alignment), is examined and shown to affect strongly the permeability tensor. Very good agreement is observed with results from previous theoretical studies for sphere packings and experimental data for the hydraulic permeability of mycelial cakes.