An optimization based 3D-1D coupling strategy for tissue perfusion and chemical transport during tumor-induced angiogenesis

The application of an optimization based 3D-1D coupling strategy is proposed, for the first time, for the simulation of fluid and chemical exchanges between a growing capillary network and the surrounding tissue, in the context of tumor-induced angiogenesis. A well posed mathematical model is worked out, based on the coupling between a three-dimensional and a one-dimensional equation (3D-1D coupled problem). The problems are then solved in a PDE-constrained optimization framework, under which no mesh conformity is required. This makes the method particularly suitable for this kind of application, since no remeshing is required as the capillary network grows. In order to handle both the evolution of the quantities of interest and the changes in the geometry, a discrete-hybrid strategy is adopted, combining a continuous modeling of the tissue and of the chemicals with a discrete tip-tracking model to account for the vascular network growth. The tip-tracking strategy, together with some proper rules for branching and anastomosis, is able to provide a realistic representation of the capillary network