arXiv subjects

Maria Lukacova-Medvidova

Publications and source records attributed to Maria Lukacova-Medvidova.

40 records · Page 3Linked to original sources

arXiv2016

Numerical simulation of a contractivity based multiscale cancer invasion model

We present a problem-suited numerical method for a particularly challenging cancer invasion model. This model is a multiscale haptotaxis advection-reaction-diffusion system that describes the macroscopic dynamics of two types of cancer cells coupled with microscopic dynamics of the cells adhesion on the extracellular matrix. The difficulties to overcome arises from the non-constant advection and diffusion coefficients, a time delay term, as well as stiff reaction terms. Our numerical method is a second order finite volume implicit-explicit scheme adjusted to include a) non-constant diffusion coefficients in the implicit part, b) an interpolation technique for the time delay, and c) a restriction on the time increment for the stiff reaction terms.

math.NA↗

arXiv2016

A multiscale approach to the migration of cancer stem cells: mathematical modelling and simulations

We propose a multiscale model of the invasion of the extracellular matrix by two types of cancer cells, the differentiated cancer cells and the cancer stem cells. We assume that the epithelial mesenchymal-like transition between them is driven primarily by the epidermal growth factors. We moreover take into account the transidifferentiation program of the cancer stem cells and the cancer associated fibroblast cells as well as the fibroblast-driven remodelling of the extracellular matrix. The proposed haptotaxis model combines the macroscopic phenomenon of the invasion of the extracellular matrix with the microscopic dynamics of the epidermal growth factors. We analyse our model in a component-wise manner and compare our findings with the literature. We investigate pathological situations regarding the epidermal growth factors and accordingly propose "mathematical-treatment" scenarios to control the aggressiveness of the tumour.

q-bio.CB↗

arXiv2014

Numerical study of cancer cell invasion dynamics using adaptive mesh refinement: the urokinase model

In the present work we investigate the chemotactically and proteolytically driven tissue invasion by cancer cells. The model employed is a system of advection-reaction-diffusion equations that features the role of the serine protease urokinase-type plasminogen activator. The analytical and numerical study of this system constitutes a challenge due to the merging, emerging, and travelling concentrations that the solutions exhibit. Classical numerical methods applied to this system necessitate very fine discretization grids to resolve these dynamics in an accurate way. To reduce the computational cost without sacrificing the accuracy of the solution, we apply adaptive mesh refinement techniques, in particular h-refinement. Extended numerical experiments exhibit that this approach provides with a higher order, stable, and robust numerical method for this system. We elaborate on several mesh refinement criteria and compare the results with the ones in the literature. We prove, for a simpler version of this model, $L^\infty$ bounds for the solutions, we study the stability of its conditional steady states, and conclude that it can serve as a test case for further development of mesh refinement techniques for cancer invasion simulations.

math.NA↗

arXiv2012

Entropy dissipation of moving mesh adaptation

Non-uniform grids and mesh adaptation have been a growing part of numerical simulation over the past years. It has been experimentally noted that mesh adaptation leads not only to locally improved solution but also to numerical stability of the underlying method. There have been though only few results on the mathematical analysis of these schemes due to the lack of proper tools that incorporate both the time evolution and the mesh adaptation step of the overall algorithm. In this paper we provide a method to perform the analysis of the mesh adaptation method, including both the mesh reconstruction and evolution of the solution. We moreover employ this method to extract sufficient conditions -on the adaptation of the mesh- that stabilize a numerical scheme in the sense of the entropy dissipation.

math.NA↗