Homogenized modeling for vascularized poroelastic materials

Abstract

A new mathematical model for the macroscopic behavior of a material composed of a poroelastic solid embedding a Newtonian fluid network phase (also referred to as vascularized poroelastic material ), with fluid transport between them, is derived via asymptotic homogenization. The typical distance between the vessels/channels (microscale) is much smaller than the average size of a whole domain (macroscale). The homogeneous and isotropic Biot's equation (in the quasistatic case and in absence of volume forces) for the poroelastic phase and the Stokes' problem for the fluid network are coupled through a fluid-structure interaction problem which accounts for fluid transport between the two phases; the latter is driven by the pressure difference between the two compartments. The averaging process results in a new system of partial differential equations (PDEs) that formally reads as a double poroelastic, globally mass conserving, model, together with a new constitutive relationship for the whole material which encodes the role of both pore and the fluid network pressures. The mathematical model describes the mutual interplay among fluid filling the pores, flow in the network, transport between compartments, and linear elastic deformation of the (potentially compressible) elastic matrix comprising the poroelastic phase. Assuming periodicity at the microscale level, the model is computationally feasible, as it holds on the macroscale only (where the microstructure is smoothed out), and encodes geometrical information on the microvessels in its coefficients, which are to be computed solving classical periodic cell problems. Recently developed double porosity models are recovered when deformations of the elastic matrix are neglected. The new model is relevant to a wide range of applications, such as fluid in porous, fractured rocks, blood transport in vascularized, deformable tumors, and interactions across different hierarchical levels of porosity in the bone.

The influence of anisotropic growth and geometry on the stress of solid tumors

“…Morover, in [23] and [19], the authors solve the homogenized fluid and drug transport models developed in [34] and [24] for vascularized tumors, respectively. Their analysis, which was extended to mechanic deformations in [26], supports the argument that geometric regularization of the microvasculature improves transport of blood and advected drugs transported into the tumor mass. Cancer evolution is extremely complex and cannot be reduced only to its mechanical stress response; however some features in the tumor progression can be associated with the generation and accumulation of mechanical stresses.…”

Effective balance equations for elastic composites subject to inhomogeneous potentials

“…Enforcing the multiplicative decompositions (100) and (101) in the cell problem for the first order displacements (42)(43)(44)(45), the local force f y that appears on the righ hand sides rewrites as…”

“…We then apply the asymptotic homogenization technique via formal expansions and by adopting a strong form formulation approach (see [43,44]), which has been recently enforced also to investigate various problems of practical interest such as bone modeling [46], filter efficiency [14], cardiac electrical activity [48], transport, growth, and heat exchange in tumors ( [27,40,41,42,45,47,54]). We admit both macroscale and microscale variations of the elastic coefficients of the individual phases of the composite, as well as possible discontinuities of the elastic moduli across the interface between two different phases.…”

Homogenized out-of-plane shear response of three-scale fiber-reinforced composites

“…Multiscale composites organized across two or more length scales are often encountered in nature, as well as in artificial materials designed to optimize specific properties (see, e.g., [10]). Relevant applications involving hierarchical systems include, but are not limited to, rocks and fracture [11], biomechanics and nanomedicine [28], poroelasticity [21], and the bone tissue [26].…”