European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
08:30   MS33: Mathematical modelling and numerical simulation of tissue evolution
Chair: MaríaTeresa Sánchez
08:30
25 mins
A poroelastic approach of cell migration
María Teresa Sánchez, José Manuel García-Aznar
Abstract: Cell migration in 3D is a main research topic during the last years, and in many biological processes cells are confined in three-dimensional environments. In order to improve the understanding of this migration, the use of microfluidics devices to determine the mechanical properties of cells is widely extended. In this type of devices, cells can be confined by the walls in such a way that the mechanisms for their migration are mainly regulated by the role of the hydraulic pressure: using channels with different bifurcations and lengths, cells are able to identify the shorter path in absence of chemical stimuli. This phenomenon is called barotaxis, that is, the cell movement due to pressure variations. The aim of this work is the mathematical modelling and numerical simulation of the migration process when the cell is confined in microfluidic devices. We will propose a multiphysics model that couples the mechanical response of the cell and the dynamical behaviour of the actin and myosin molecules inside the cytoplasm. To do that, the cell will be assumed as a poroelastic material considering that the cytoplasm is a biphasic material which consists of an elastic phase (cytoskeleton) and a interstitial fluid phase (cytosol). Furthermore, the cell nucleus will be considered as an elastic material with higher rigidity than the cytoskeleton. Finally, we will propose a system of coupled reaction-diffusion equations to simulate the actin and myosin concentrations in order to model the actin polymerization and the myosin contractile effect, responsible of the polarization process of the cell.
08:55
25 mins
A mechanobiochemical model for 3D cell migration
Anotida Madzvamuse, Laura Murphy
Abstract: In this talk, I will present a mechanobiochemical model for 3D cell migration that couples a force balance equation describing the evolution of the displacement to a system of reaction-diffusion equations describing the spatiotemporal dynamics of actin and myosin. The mechanobiochemical model considers actin filament network as a viscoelastic and contractile gel. Within this approach the pressure and concentration forces are in the force balance equation are driven by actin and myosin. We employ linear stability analysis to determine parameter regimes that will give rise to cell movement close to bifurcation points. To demonstrate the applicability of the modelling framework, a novel moving grid finite element method is used to provide approximate numerical solutions. Numerical experiments exhibit various cell migration profiles reminiscent to those observed in experiments such as cell expansion, protrusion, contraction and translation. Cell migration is critical in multicellular organisms and play an important role in embryogenesis, wound healing, immune response, cancer metastasis, tumour invasion, and other processes.
09:20
25 mins
A multi-scale computational tool to assist wound care
Fred Vermolen
Abstract: Introduction: Wound contraction and closure are crucially important processes in which the first problem can lead to dramatic problems for patients. For the sake of prediction of the evolution of post-trauma skin, mathematical frameworks are indispensable. Material and Methods / Case Description: The mathematical frameworks that we consider are agent-based and continuum models, which respectively, are used on a micro-scale and on a macro-scale. Stochastic finite-element techniques are used to assess the large amount of uncertainty in the (patient) data. Results: We have established a good correspondence between the several modelling strategies, as well as good agreements with clinical experiments. Discussion: Mathematical models always suffer from uncertainties since input parameters have only been scarcely measured and most input data is patient dependent. Therefore, it is highly desirable to accommodate the modelling frameworks for uncertainties in the data. Furthermore, it is noted that this matter becomes more significant as the modelling framework becomes more complicated. Therefore, mathematical frameworks should focus on the quantification of uncertainty.
09:45
25 mins
A poroelastic model of avascular tumor growth
E. Javierre, F.J. Gaspar, C. Rodrigo
Abstract: The growth of a tumor in a confining environment creates physical forces and stresses that on one hand reduce the proliferation rate of tumor cells, but on the other hand increase their metastatic potential [1]. The compressive stresses on the tumor also act on the vascular and lymphatic systems, affecting negatively to the delivery of nutrients and drugs. Therefore, the development of predictive mathematical models are of great importance to help in the understanding of the intricate mechanochemical coupling regulating tumor cell function and (avascular) tumor growth. In this work, we incorporate into Biot’s poroelasticity theory an isotropic growth contribution due to the proliferation of tumor cells, in line with [2]. The proposed mechanochemical coupling is based on the following ideas: (i) the net production of the solid phase (cells and extracellular matrix) is enhanced by the presence of nutrients and inhibited by compressive stresses; (ii) the net production of the fluid phase (interstitial fluid) is formulated in terms of the fluid exchange between the vascular and lymphatic systems, in line with [3]. We take into account the spherical symmetry of the growing tumor, and propose a solution algorithm based on the finite volume discretization of the problem on a staggered grid. Numerical simulations of the reference problem will be presented and discussed in terms of their clinical implications. References [1] G. Helmlinger, P.A. Netti, H.C. Lichtenbeld, R.J. Melder, R.K. Jain, Nature biotech- nology 15, pp. 778-783, 1997. [2] T. Roose, P.A. Netti, L.L. Munn, Y. Boucher, R.K. Jain, Microvascular Research 66, pp. 204-212, 2003. [3] C. Voutouri T. Stylianopoulos, Journal of Biomechanics 47, pp. 3441-3447, 2014.