European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
15:45   MS12: Modeling and Simulation of temporal multiscale problems
Chair: Thomas Richter
25 mins
Mathematical and numerical models of atherosclerotic plaque progression in carotid arteries
Silvia Pozzi, Christian Vergara
Abstract: We propose a mathematical model for the description of plaque progression in carotid arteries. This is based on the coupling of a fluid-structure interaction problem, arising between blood and vessel wall, and differential problems for the cellular evolution. A numerical model is also proposed. This is based on the splitting of the coupled problem based on a suitable strategy to manage the multiscale-in-time nature of the problem. We present some preliminary numerical results both in ideal and real scenarios.
25 mins
A temporal multi-scale approach for flow problems and its application to atherosclerotic plaque growth
Stefan Frei, Thomas Richter
Abstract: We present an efficient approach for the long-term simulation of flow problems that carry a multiscale character in time. Our developments are motivated by the process of atherosclerotic plaque formation, where fluid forces acting on a scale of milliseconds to seconds influence plaque growth within the arterial wall, which typically takes place over several months. A major issue within the design of temporal multiscale algorithms is the unknown initial data for the fast scale problem. For flow problems a possible remedy is to approximate the problems locally by a periodic solution in time. We derive an averaging scheme that is of first order with respect to the ratio of time scales. Further, we construct second-order accurate time discretisation schemes on both scales and analyse the interplay between modelling and discretisation errors. Finally, we present numerical examples for both a Navier-Stokes problem with a slowly moving boundary and the full mechano-chemical fluid-structure interaction problem corresponding to atherosclerotic plaque growth.
25 mins
Multiscale Modelling of Cancer Invasion Process: macrophages influence on cancer cell motility within fibrous environment
Dumitru Trucu
Abstract: Recognised as one of the hallmarks of cancer, cancer cell invasion into tissue is a complex process that plays a key role in the growth and spread of cancer, culminating in metastatic spread (secondary cancers). One common aspect of all cancer progression is the secretion of matrix degrading enzymes (MDEs) by the cancer cells that modify or destroy the surrounding tissue or various components of extracellular matrix (ECM) and support local cancer cell invasion. In conjunction with MDE activities, increased cancer cell motility due to changes in cell-adhesion properties further exacerbates the invasion. Transmembrane calcium-dependent adhesion molecules (cadherins) interact with intra-cellular proteins, such as β-catenin and give rise to adhesion junctions. Of particular importance in cancer invasion are the dynamics between the calcium-sensing receptor distribution and the calcium ions (Ca^(2+)) from the ECM. In addition to cell-cell adhesion, the binding of various ECM ligands to cell-surface receptors (integrins) enables cell-matrix adhesion. Thus, processes occurring at a molecular (micro) scale give rise to processes occurring at the tissue (macro) scale, via processes taking place at the cellular (meso) scale. Despite recent mathematical modelling advances, the understanding of the biologically multiscale process of cancer invasion remains an open question. In this work we introduce a novel multiscale moving boundary approach for cancer invasion that accounts for the influence of the macrophages on cell-adhesion in the context of the multiphase nature of the ECM dynamics. Distinguishing here between the fibres components and the rest of the ECM components and incorporating their multiscale dynamics within the new modelling approach, this framework connects the tissue-scale macro-dynamics of the heterotypic tumour cell population with both the proteolytic cell-scale dynamics occurring at the tumour invasive edge and the micro-scale ECM fibres dynamic degradation and realignment occurring inside the tumour domain. The new multiscale modelling framework will be accompanied by details of the computational approach and a discussion of the numerical simulation results.
25 mins
Numerical Continuation of Stochastic Problems
Christian Kuehn
Abstract: In this talk, I shall explain a method how to analyze certain aspects of stochastic dynamical systems from a purely deterministic and computational perspective. In particular, we study fluctuations around steady states in stochastic differential equations using ellipsoids calculated via Lyapunov matrix equations. The method will be embedded in a numerical continuation framework to effectively study parametrized problems. The method will then be applied to singular limits for several temporal multiscale problems for SODEs and SPDEs.