Tumour Invasion

Responsible: Dirk Schumann, Jens Lang

Literature: D. Schumann, Eine anwendungsbezogene Einführung in die adaptive Finite Elemente Methode,  Diplomarbeit, Tech. Fachhochschule Berlin, 2001.

A.R.A. Anderson, M.A.J. Chaplain, E.L. Newman, R.J.C. Steele, and A.M. Thompson, Mathematical modelling of tumor invasion and metastasis, J. of Theor. Medicine, 2000.

A. Gerisch, J.G. Verwer, Operator splitting and approximate factorization for taxis-diffusion-reaction models,  to appear in Applied Numerical Mathematics (2001).

A tumour arises from a single cell which is genetically disturbed. A local tumour is growing but it doesn't grow arbitrarily. In fact, we get a balance between new and dying cells because the tumour cannot be sufficiently supplied with oxygen and other nutrients. This equilibrium can take months or years. However, some tumours are able to produce proteins initializing the growth of blood vessels. If these proteins come close to existing blood vessels then new blood vessels are generated growing in direction of the tumour and penetrating it. This improves the supplement of the tumour with oxygen and nutrients. The tumour strengthens its growth.
The tumour starts to produce metastasis when meeting some blood vessels. We distinguish the following steps:
1. Tumour cells are seperated from the original tumour.
2. The seperated cells penetrate the neighbouring tissue and enter the circulation of blood and lymph being transported to other locations in the body.
3. Somewhere the tumour cells leave the circulation and penetrate healthy tissue. There they generate new tumours called metastasis.
Each of these steps is influenced by different factors, e.g., the presence of special proteins.
We consider a taxis-diffusion-reaction model of tumour cell invasion Described in [2] and used in [3]. It describes the cell propagation and the process of penetration the neighbouring tissue. The interaction between extracellular matrix (ECM) and matrix degradative enzymes (MDE) is responsible for that. ECM integrates regular cells into tissue. ECM is reduced by MDE when healing a wound or developing an embryo. The increase of metastasis is also determined by the reduction of ECM. The MDE necessary for that can be produced by the tumour itself.
The model describes the behaviour of three components: the density of the tumour cells, the density of ECM, and the concentration of MDE.
In particular, we can simulate the fragmentation of an initial cell mass.

Fragmentation of initial cell mass and corresponding FEM mesh

Last update: July 2007
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