Due to an ageing society and unhealthy living conditions, liver diseases like non-alcoholic fatty liver disease (NAFLD) or liver cancer will account for an increasing proportion of deaths in the coming years. Using a mathematical model, the underlying function-perfusion processes of both diseases are investigated. We developed a multiscale and multiphase model for the simulation of hepatic processes on the lobular and cell scale. The lobular scale is described with partial differential equations (PDEs) based on the Theory of Porous Media (TPM), whereas on the cellular scale the metabolic processes are calculated using ordinary differential equations (ODEs). Since NAFLD and the development of a liver tumor lead to tissue growth as well as changes in the blood perfusion, growth and remodelling processes in the human liver are evaluated.

Malignant neoplasms are one of the most dangerous diseases. Within the framework of the well-established Theory of Porous Media (TPM), a multi-constituent model is derived. The model is mathematically formulated by a set of coupled partial differential equations which are solved within the well-known framework of the finite-element method. The general TPM model is applied to basal-cell carcinoma in the avascular skin and representative numerical examples show the capabilities of the model.

The medical relevance of brain tumours is characterised by its locally invasive and destructive growth. With a high mortality rate combined with a short remaining life expectancy, brain tumours are identified as highly malignant. A continuummechanical model for the description of the governing processes of growth and regression is derived in the framework of the Theory of Porous Media (TPM). The model is based on medical multi-modal magnetic resonance imaging (MRI) scans, which represent the gold standard in diagnosis. The multi-phase model is described mathematically via strongly coupled partial differential equations. This set of governing equations is transformed into their weak formulation and is solved with the software package FEniCS. A proof-of-concept simulation based on one patient geometry and tumour pathology shows the relevant processes of tumour growth and the results are discussed.

Brain tumours are among the most serious diseases of our time. A continuum‐mechanical model is proposed to represent the basic processes of growth and regression. The physical multi‐constituent approach is derived in the framework of the Theory of Porous Media (TPM). This modelling approach can be expressed mathematically via strongly coupled partial differential equations (PDEs), that are solved using the well‐known Finite Element Method with the software toolkit FEniCS. A realistic initial‐boundary‐value problem is used to demonstrate the workflow with the used software and the capabilities of the model.