Chalmers Conferences, 9th European Conference on Mathematical and Theoretical Biology

Numerical treatment of chemotaxis related heterogeneous dynamics arising in cancer invasion simulations
N. Kolbe

Last modified: 2014-06-09


Chemotaxis plays an important role in a plethora of biomedical processes. In particular, in the 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-diffusion 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 fine discretization grids.

In this talk, we present our  findings from [3, 4]. In more details, we first 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 fine computational grids by employing mesh adaptation/refinement 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 refinement technique that we employ, c) we elaborate on the criteria that drive the refinement of the grid, d) we compare our results with the ones obtained without mesh refinement, 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-diffusion systems that involves 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.



[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-Universit at Mainz, 2013.


[4] N. Kolbe, J. Kat'uchova, N. Sfakianakis, and M. Luk   a cov   a-Medvid'ov   a , Heterogeneous complex dynamics in cancer invasion simulations under the prism of high resolution mesh refinement technicues. In preparation.


[5] A. Kurganov and M. Luka cov   a-Medvidov   a , Numerical study of two-species chemotaxis models, (2012). Submitted.