19th European Conference on Mathematics for Industry: book of Abstracts, June, 13-17, 2016, Santiago de Compostela (Spain)

In the last decades, the biomedical relevance of mathematical models has been demonstrated and comparison of experiments against computer simulations has been encouraged. Blood circulation in the human liver and in particular perfusion, the process of delivering blood to the capillary bed, is an open problem and inherently multiscale in nature. Models currently available in the literature [2, 1] either present a macroscale approach in which liver is assumed as a homogeneous anisotropic porous medium and therefore flow within it is simulated using Darcy’s equation, or they work at the microscale where the vascular and extravascular domains need to be treated differently solving Stokes’ equation in the former and Darcy’s equation in the latter and applying suitable coupling condition at the interface. In this communication, instead, we present an approach where the Darcy-Stokes-Brinkmann [3] equation is used on the entire computational domain, different areas of the tissue being represented by a (possibly discontinuous) friction coefficient. This approach allows to run simulations at the capillary scale on real-life geometries deduced from medical images avoiding complex and costly preprocessing such as edge detection, and mesh generation. The peculiar properties of IsoGeometric discretization methods [4, 5] such as stability and ability to provide exactly divergence free velocity are exploited in the simulation. After validating the numerical method on 2D and 3D test cases based on syntetic images, we apply it to actual micro-CT images of the liver and perform an upscaling procedure to determine the macroscale parameters of the tissue such as the local permeability tensor.