Last modified: 2014-06-09

#### Abstract

case of cancer it is threefold: a) it promotes the vascularization of the cancer tumour, b) it is used to

inactivate/evade the immune system, and c) it directs the migration of the cancer metastating cells

into the circulatory system; leading hence to possible metastasis to another location of the organism.

Accordingly, the deterministic mathematical modelling of cancer dynamics takes chemotaxis into

account, see e.g. [2, 1].

Our work is inspired by a deterministic model, proposed in [2], that describes the cancer cell invasion

of the surrounding tissue by including interactions between the cancer cells, the extracellular matrix,

and several proteases.

From a numerical point of view, the derived system of reaction-advection-di_usion equations is

challenging mostly due to the chemotaxis driven appearance of merging and emerging concentrations

that occur whenever relevant parameters are chosen, see [1]. Classical numerical methods, either fail

completely in resolving the heterogeneous dynamics in a quantitatively consistent way, or necessitate

very _ne discretization grids.

In this talk, we present our _ndings from [3, 4]. In more details, we _rst present an extended

numerical study of the cancer invasion model of [2] using classical and non-classical numerical methods.

We analyse the reasons these methods either fail or crave for large numbers of grid cells, in order to

consistently resolve the dynamics.

We alleviate the need for very _ne computational grids by employing mesh adaptation/re_nement

techniques. More precisely, a) we propose a higher order stable and consistent numerical method,

that follows the guidelines of [5] and which we adapt to both uniform and non-uniform grids, b) we

present the mesh re_nement technique that we employ, c) we elaborate on the criteria that drive the

re_nement of the grid, d) we compare our results with the ones obtained without mesh re_nement,

and the ones found in the literature.

We further demonstrate, by a properly adjusted two equation subsystem of the model, that similar

heterogeneous dynamics can be obtained by other reaction-advection-di_usion systems that involve

chemotaxis. We close this presentation by presenting the wellness of the overall proposed method

-numerical scheme and mesh adaptation- on a more involved -biologically and mathematically- model

of tissue invasion by cancer cells that we have developed.

References

[1] V. Andasari, A. Gerisch, G. Lolas, A. P. South, and M. A. J. Chaplain, Mathematical modeling of cancer

cell invasion of tissue: biological insight from mathematical analysis and computational simulation, Journal of mathematical

biology, 63 (2011), pp. 141{171.

[2] M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase

plasminogen activation system, Mathematical Models and Methods in Applied Sciences, 15 (2005), pp. 1685{1734.

[3] N. Kolbe, Mathematical Modeling and Numerical Simulations of Cancer Invasion, Master's thesis, Johannes

Gutenberg-Universitat Mainz, 2013.