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We develop a computational algorithm based on a diffuse interface approach to study the design of bioartificial organ scaffold architectures. These scaffolds, composed of poroelastic hydrogels housing transplanted cells, are linked to the patient’s blood circulation via an anastomosis graft. Before entering the scaffold, the blood flow passes through a filter, and the resulting filtered blood plasma transports oxygen and nutrients to sustain the viability of transplanted cells over the long term. A key issue in maintaining cell viability is the design of ultrafiltrate channels within the hydrogel scaffold to facilitate advection-enhanced oxygen supply ensuring oxygen levels remain above a critical threshold to prevent hypoxia. In this manuscript, we develop a computational algorithm to analyze the plasma flow and oxygen concentration within hydrogels featuring various channel geometries. Our objective is to identify the optimal hydrogel channel architecture that sustains oxygen concentration throughout the scaffold above the critical hypoxic threshold. The computational algorithm we introduce here employs a diffuse interface approach to solve a multi-physics problem. The corresponding model couples the time-dependent Stokes equations, governing blood plasma flow through the channel network, with the time-dependent Biot equations, characterizing Darcy velocity, pressure, and displacement within the poroelastic hydrogel containing the transplanted cells. Subsequently, the calculated plasma velocity is utilized to determine oxygen concentration within the scaffold using a diffuse interface advection-reaction-diffusion model. Our investigation yields a scaffold architecture featuring a hexagonal network geometry that meets the desired oxygen concentration criteria. Unlike classical sharp interface approaches, the diffuse interface approach we employ is particularly adept at addressing problems with intricate interface geometries, such as those encountered in bioartificial organ scaffold design. This study is significant because recent developments in hydrogel fabrication make it now possible to control hydrogel rheology and utilize computational results to generate optimized scaffold architectures. 

In this paper we introduce a constructive approach to study well-posedness of solutions to stochastic uid-structure interaction with stochastic noise. We focus on a benchmark problem in stochastic uidstructure interaction, and prove the existence of a unique weak solution in the probabilistically strong sense. The benchmark problem consists of the 2D time-dependent Stokes equations describing the ow of an incompressible, viscous uid interacting with a linearly elastic membrane modeled by the 1D linear wave equation. The membrane is stochastically forced by the time-dependent white noise. The uid and the structure are linearly coupled. The constructive existence proof is based on a time-discretization via an operator splitting approach. This introduces a sequence of approximate solutions, which are random variables. We show the existence of a subsequence of approximate solutions which converges, almost surely, to a weak solution in the probabilistically strong sense. The proof is based on uniform energy estimates in terms of the expectation of the energy norms, which are the backbone for a weak compactness argument giving rise to a weakly convergent subsequence of probability measures associated with the approximate solutions. Probabilistic techniques based on the Skorohod representation theorem and the Gyongy-Krylov lemma are then employed to obtain almost sure convergence of a subsequence of the random approximate solutions to a weak solution in the probabilistically strong sense. The result shows that the deterministic benchmark FSI model is robust to stochastic noise, even in the presence of rough white noise in time. To the best of our knowledge, this is the  rst well-posedness result for stochastic uid-structure interaction.