Section: New Results

Tissue growth, regeneration and cell movements

At the cell scale, one uses kinetic equations for which numerical simulations have been performed. Behaviour of solutions can be understood by performing a hydrodynamical limit of the kinetic equation. It leads to aggregation type equations for which finite time blow up is observed [42] . Then measure solutions for this system should be considered. A theoretical framework for the existence of weak solutions has then been developed [17] , [34] where duality solutions for such system has been investigated which are equivalent to gradient flow solutions [33] .

Our understanding of traveling waves has progressed considerably in three directions: fitting continuous models and IBMs [21] , fitting precisely models with experiments based on known biological values of parameters, and opening new paradigms: traveling waves can connect a dynamically unstable state to a Turing unstable state, certainly the stable wave connects the unstable state to a pulsating state.

The recent biomechanical theory of cancer growth considers solid tumours as liquid-like materials comprising elastic components. In this fluid mechanical view, the expansion ability of a solid tumour into a host tissue is mainly driven by either diffusion of cells (emerging on the mesoscopic scale by coarse graining from the cell micro-motility) or by cell division depending either on the local cell density (contact inhibition), on mechanical stress in the tumour, or both. For the two by two degenerate parabolic/elliptic reaction-diffusion system that results from this modelling, we prove there are always travelling waves above a minimal speed and we analyse their shapes. They appear to be complex with composite shapes and discontinuities. Several small parameters allow for analytical solutions; in particular the incompressible cells limit is very singular and related to the Hele-Shaw equation. These singular travelling waves are recovered numerically. See [21] . Besides this work, a direct comparison with agent-based and continuum models has been performed, showing very good agreement over a large parameter range.

1. Ammonia metabolism in healthy and damaged liver The model on ammonia detoxification in liver, integrating a compartment model for the glutamine synthetase-active peri-central and the glutamine-inactive peri-portal liver lobule compartment (see Bang report 2012) with the spatial - temporal model of liver regeneration after drug-induced peri-central damage [41] has been extended to include the mass balance of other body compartments. The analysis shows that some body compartments that in the healthy liver produce ammonia, in the damaged liver detoxify blood from ammonia. The detoxification model of liver in combination with the body ammonia balance can be found in ref. (Schliess et. al., Hepatology, accepted [20] ).

2. Drug metabolism in hepatocytes Since the begining of 2013 animal experiments for testing of cosmetics are forbidden within the EU. This has triggered initiatives towards how modeling may help to investigate drug toxicity, circumventing animal testing. The basic conceptual idea is to test drugs (cosmetics, perspectively also other drugs) in in-vitro systems such as monolayers, sandwich cultures, or multi-cellular spheroids, and use the emerging data to infer the expected toxicity in-vivo using novel experimental and computational approaches [16] . We have integrated an intracellular mathematical model of paracetamol drug metabolism in a mathematical agent-based cell model for monolayer and multi-cellular spheroids and compared simulation results with experimental findings in the same systems. We find that cell-to-cell variabilty can largely explain the experimentally observed cell population survival fractions. The mathematical model is now refined based on measurements of intermediate drug metabolites.

3. Cell mechanics and its impact on cell proliferation A novel numerical methodology has been developed to simulate the mechanics of cells and tissues using a continuum approach. Analogously to the Center Based Models, particles are used to represent (parts of) the cells but rather than discrete interactions they represent a continuum. This approach can be used for tissue mechanics simulations in where the individual cell-cell interactions are discarded but instead a constitutive law is proffered [23] .

   Moreover, a new model in where cell adhesion dynamics is addressed. The cell model is constructed by a triangulated surface and a coarse-grained internal scaffolding structure. A model cell can adapt to realistic cell shapes, and is able to interact with a substrate or other cells. The parameters in this model can be determined by canonical experiments performed on cells informing about cell deformation, compression and cell-cell adhesion [19] .

   A computational model for the confined growth of cells in a capsule has been developed. This model represents a realistic simulation tool for a novel experimental system (Institut Curie, Prof P. Nassoy) in where cells are grown in an elastic environment to mimic the effects of mechanical stress on cells and while monitoring their fate. Model parameter calibration is now ongoing to reproduce the correct quantitative behavior of the cells in order to unravel the relationship between cell mechanical stress and cell behavior.

4. Playing the game of life with yeast cells Within a collaboration with a synthetic biology lab at MIT, multicellular modelling of engineered yeast cell populations is performed. Those cells secrete a messenger molecule (IP) which diffuse in the medium, bind to other cells, and trigger a signalling cascade, which finally induces expression of lethal genes. A model has been established based on our single-cell-based model framework associated with PDE simulations, and it is currently used to explain and guide experiments conducted at the MIT. In 2013, the project has achieved significant progress on several aspects. First, we were able to quantitatively reproduce newly produced, rich data on the signaling cascade behavior with a kinetic model describing signaling reactions. Second, comparison between simulations and data allowed to identify key characteristics of the death module, which is positioned downstream of the signaling cascade: there is a rapid and stochastic commitment to death, followed by a deterministic and long delay (2-4 cell generations) needed before cells actually die. Finally, data production and analysis iterations with our collaborators alloweded to optimize the procedures for experimental measurements and the quantitative analysis of data in a synergistic manner.

5. Other projects in short Further progress have been achieved on the reconstruction of lung cancer micro-architecture from bright field micrographs. In partial hepatectomy (PHx), pig data on the changes of microarchitecture during regeneration after PHx have been generated and stained now being processed. The image processing chain for liver architecture reconstruction has been refined and extensive analysis has been performed on the architecture of the bile canaliculi network in healthy liver and in disease states of liver. Moreover, non-small-cell lung cancer cell invasion pattern have been analyzed leading to interesting observations now being studied by modelling.

   For multi-scale modeling of liver regeneration after drug-induced pericentral damage, integration of a molecular model of hepatocyte growth factor signalling with an agent-based model of liver regeneration has been extended to include blood flow in the lobule, as well as the contributions of the body compartment to the degradation and production of hepatocyte growth factor (HGF).

1. Flow and perfusion scenarios in cancer. We started reconstruction of the blood vessel system of lung cancers removed by surgery. For this purpose, patients underwent DCE-MRI prior to surgery. Part of the tumors after surgery was sliced and stained for nuclei, proliferation and endothelial cells. The slice data were recorded (Mevis, Luebeck) to allow identification of the position of the indiviudal structures in 3D space. The structures were then segmented. The work turned out to be particularly challenging because of staining artifacts for which image algorithms had to correct for. Nevertheless, last results look promising so that at least the network formed by larger vessels can be segmented and reconstructed in 3D. The so emerging data will be used for modeling of blood flow using the models developed in 2012.

2. Flow in liver lobules. We integrated blood flow in the new software CellSys (see above under software) and refined the algorithms. Moreover, we increased the resolution of the capillaries by triangulating them from high resolution confocal scanning micrographs.