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

Abstract

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.

A Multi‐scale and Multi‐phase Model for the Description of Toxicity caused by Paracetamol in Biological Tissue using the Example of the Human Liver

“…Figure 1 shows the division of the liver into smaller sub structures, the liver lobules and the liver cells, also called hepatocytes. Ricken et al [2] introduced a multi-scale approach focusing on the organ-, lobule-and cell-scale. To describe the complex biological tissue on the lobulescale we use the Theory of porous media (TPM), that was developed by Ehlers [4] and de Boer [5], and embed it into the Finite Element Method (FEM).…”

“…The external nutrients glucose, lactate, free fatty acid, oxygen and paracetamol are transported through the liver via the blood flow, whereas the internal substances glycogen and glutathione are stored in the liver tissue. Furthermore, we use an anisotropic perfusion model to describe the blood flow on the lobulescale, see Ricken [2]. Using a thermodynamically consistent model we gain the weak equations with the unknown quantities {u s , p To calculate the metabolism processes on the cell-scale and describe the production, utilization and storage of the metabolites, we use a set of coupled ordinary differential equations (ODE).…”

“…Using a thermodynamically consistent model we gain the weak equations with the unknown quantities {u s , p To calculate the metabolism processes on the cell-scale and describe the production, utilization and storage of the metabolites, we use a set of coupled ordinary differential equations (ODE). The 0-D model contains the glucose metabolism, the fat metabolism [see [2], [3]] and is extended by the paracetamol metabolism. Since the depletion of paracetamol is dependent on the concentration of glutathione in the liver tissue, we develop pharmacokinetic equations to calculate the paracetamol and the glutathione concentrations, using the approach of Geenen [6] .…”

Model Prediction and Validation of an Order Mechanism Controlling the Spatiotemporal Phenotype of Early Hepatocellular Carcinoma

“…Fundamentally, the computational study of many processes in liver lobules may also be amenable to sophisticated continuum models (Ricken et al 2010(Ricken et al , 2014, whereby cells are not tracked individually, but local averages are performed usually approximating tissue pieces as a continuous body. However, if the tissue piece is only one cell thick, as is the case in the hepatic lobule for the hepatocyte sheets located between neighboring sinusoids, and at the same time cell growth, division and death of cells occur, then a continuous approach mimicking realistic growth or remodeling processes in liver lobules might be difficult to justify.…”

On the Influence of Growth in Perfusion Dependent Biological Systems – at the Example of the Human Liver

“…For the microscopic cell level use has been made of an embedded set of ordinary differential equations (ODE) that mimics a simplified metabolism, e.g. glucose utilization and production; [2]. In case of growth of the fat phase we follow a phenomenological approach.…”