Understanding the interaction between a viscous, incompressible fluid and a multi-layered poroelastic structure modeling bioartificial pancreas

I will present the modeling and a well-posedness result for a fluid-structure interaction problem describing the flow of an incompressible, viscous fluid interacting with a multi-layered poroelastic structure describing a bioartificial pancreas.  The poroelastic structure consists of two layers: a thin poroelastic plate and a thick poroelastic medium modeled by the (nonlinear) Biot equations. The thin poroelastic plate serves as an interface between the Biot poroelastic medium, and the free fluid flow, which is modeled by the time-dependent Stokes equations. This mathematical problem was motivated by the design of a first implantable bioartificial pancreas without the need for immunosuppressant therapy. The design is based on transplanting the healthy (donor) pancreatic cells into a poroelastic medium and encapsulating the cell-containing medium between two semi-permeable poroelastic plates. The encapsulated poroelastic medium containing the pancreatic cells is connected to the blood flow via a tube (anastomosis graft). The blood flow in the tube (free fluid flow) brings oxygen and nutrients to the transplanted pancreatic cells. The semi-permeable poroelastic plate allows the passage of oxygen and nutrients from the free fluid (blood) flow to the transplanted cells, while blocking the patient’s own immune cells from attacking the transplant.  We obtain existence of a weak solution to this fluid-structure interaction problem. The solution is unique under additional regularity assumptions (weak-strong uniqueness). The modeling introduces interesting coupling (boundary) conditions at the interface, while the constructive existence proof introduces a novel proof methodology that is particularly suitable for this class of problems. I will show in the end how we use numerical simulations to optimize the pancreas design in order to maximize oxygen supply to the transplanted cells.