Modeling function-perfusion behavior in liver lobules including tissue, blood, glucose, lactate and glycogen by use of a coupled two-scale PDE-ODE approach

This study focuses on a two-scale, continuum multicomponent model for the description of blood perfusion and cell metabolism in the liver. The model accounts for a spatial and time depending hydro-diffusion-advection-reaction description. We consider a solid-phase (tissue) containing glycogen and a fluid-phase (blood) containing glucose as well as lactate. The five-component model is enhanced by a two-scale approach including a macroscale (sinusoidal level) and a microscale (cell level). The perfusion on the macroscale within the lobules is described by a homogenized multiphasic approach based on the theory of porous media (mixture theory combined with the concept of volume fraction). On macro level, we recall the basic mixture model, the governing equations as well as the constitutive framework including the solid (tissue) stress, blood pressure and solutes chemical potential. In view of the transport phenomena, we discuss the blood flow including transverse isotropic permeability, as well as the transport of solute concentrations including diffusion and advection. The continuum multicomponent model on the macroscale finally leads to a coupled system of partial differential equations (PDE). In contrast, the hepatic metabolism on the microscale (cell level) was modeled via a coupled system of ordinary differential equations (ODE). Again, we recall the constitutive relations for cell metabolism level. A finite element implementation of this framework is used to provide an illustrative example, describing the spatial and time-depending perfusion-metabolism processes in liver lobules that integrates perfusion and metabolism of the liver.