A coupled approach for fluid saturated poroelastic media and immersed solids for modeling cell‐tissue interactions

In this paper, we propose a finite element–based immersed method to treat the mechanical coupling between a deformable porous medium model (PM) and an immersed solid model (ISM). The PM is formulated as a homogenized, volume‐coupled two‐field model, comprising a nearly incompressible solid phase that interacts with an incompressible Darcy‐Brinkman flow. The fluid phase is formulated with respect to the Lagrangian finite element mesh, following the solid phase deformation. The ISM is discretized with an independent Lagrangian mesh and may behave arbitrarily complex (it may, eg, be compressible, grow, and perform active deformations). We model two distinct types of interactions, namely, (1) the immersed fluid‐structure interaction (FSI) between the ISM and the fluid phase in the PM and (2) the immersed structure‐structure interaction (SSI) between the ISM and the solid phase in the PM. Within each time step, we solve both FSI and SSI, employing strongly coupled partitioned schemes. This novel finite element method establishes a main building block of an evolving computational framework for modeling and simulating complex biomechanical problems, with focus on key phenomena during cell migration. Cell movement is strongly influenced by mechanical interactions between the cell body and the surrounding tissue, ie, the extracellular matrix (ECM). In this context, the PM represents the ECM, ie, a fibrous scaffold of structural proteins interacting with interstitial flow, and the ISM represents the cell body. The FSI models the influence of fluid drag, and the SSI models the force transmission between cell and ECM at adhesions sites. We propose a comprehensive, versatile computational framework for modeling and simulation of an individual cell interacting with its surrounding tissue. The cell is modeled as a visco‐hyperelastic solid interacting with a biphasic poroelastic medium model representing the extracellular matrix coupled to interstitial flow. A novel immersed finite element approach and a combination of monolithic and strongly coupled partitioned solution schemes is utilized to discretize and solve the arising three field multiphysics problem in a robust manner.