Mathematical framework for biological tissue perfusion modeling and simulation (Mathieu Barré, INRIA Saclay)Séminaire « Analyse numérique et équations aux dérivées partielles »
Many biological tissues can be modeled as porous media, namely continuous media composed of a solid skeleton filled by a fluid. In such applications, the fluid is incompressible and the porous medium itself can be considered as nearly-incompressible. In this talk, we will analyze a recent partial differential equation poromechanics model adapted to this biomedical context. In this model, the solid and fluid equations show a hyperbolic – parabolic behavior and are in addition coupled through the interstitial pressure associated with the incompressibility divergence constraint. We will show the existence and uniqueness of strong and weak solutions in the nearly-incompressible and incompressible cases by combining semigroup theory, energy estimates and T-coercivity. T-coercivity theory, originally developed for unconstrained problems, is extended here to treat general saddle-point and perturbed saddle-point problems. Then, we will present two schemes to discretize the model: a monolithic one and a fractional-step method. For both schemes, spatial and temporal convergence analysis are performed, leading to robust error estimates with respect to incompressibility, porosity and permeability. Finally, the relevance of the model to biomedical applications is illustrated by comparing microvessels-on-chip simulations with experimental data.