Minisymposium “Multiscale Problems in the Life Sciences”
Organizers: PD Dr. Maria Neuss-Radu and Dr. Nadja Ray
(Wednesday, 04.03.20, 10:20-12:20, lecture hall H12)
Prof. Dr. Willi Jäger (University of Heidelberg):
“Dynamics of cells, cell layers and tissue under the influence of biophysical, biochemical and biomechanical processes, fluid flow and growth – Modelling, analysis and simulation of subprocesses in inflammation”
This lecture discusses basic physiological processes that take place on different scales and are relevant for the issues like inflammation, the response of the immune system to an infection, an injury or disfunctions like hypoxia. Biological research provides increasingly detailed information about cellular processes that needs to be integrated into the macroscopically described processes in cell assemblies and tissues. The problems at hand require the coupling of processes of multiple type, in order to understand the dynamics of the system.
As a first example we discuss the swelling of a single cell, surrounded by water in which the osmotic pressure increases due to a change to anaerobic respiration caused by hypoxia. This leads to the influx of fluid into the cell and to the swelling of the cell. The properties of the cell membrane play a central role, also in the mathematical modelling of the system. A free boundary value problem arises, with a model equation for the fluid and diffusion reaction equations in a poro-elastic medium coupled to the fluid via the membrane. The model system is calibrated and simulated. As a second example we consider the start of an inflammation in an artery. We present a model system and simulations, investigating the dynamics of the barrier function of the endothelial cell layer that separates the intima from the lumen of the artery, the transport between the lumen and the intima and the structural changes induced in the vessel wall. We consider processes only in three compartments: the lumen, the endothelial layer and the intima. The wall is assumed to be a poro-elastic medium modelled by a Biot system and growing in volume due to the actions of monocytes, causing a swelling of the layer. A fluid structure interaction problem arises, which is far more complex than those treated before. A computational tool is provided to study the dynamics in dependence of the system parameters.
This lecture is based on joint research with Maria Neuss-Radu, Valeria Heimann, Adelia Sequeira, Telma Silva-Fortes, Yifan Yang, Marek Netrušil, Jerimi Mizerski and Thomas Richter.
Prof. Dr. Thomas Richter (University of Magdeburg):
“The Candywrapper Effect – Predicting the long term evolution of complex temporal multiscale problems”
We present a computational method for accelerating the simulation of temporal multiscale problems that appear in the mechanical material weathering of structures but also when modeling growth processes in biological tissue. In such problems one is often interested in quantities that evolve very slowly in time, like the grade of a stenosis in blood vessels that however are influenced by components with a rapid temporal development, like a pulsating blood flow that acts on the boundaries of the vessel. Given that the time horizon of interest is in the range of months and the fast scale asks for a resolution of less than a second, resolving all scales in one coupled simulation is out of bounds.
In this talk we describe a numerical approach that is based on averaging the dynamical system to derive a decoupled equation for the long term dynamics. To incorporate the impact of the fast scale, localized solutions which are periodic in time enter the long term equation. The focus of the talk is on handling problems with detailed modeling that are usually three dimensional and that involve fluid-structure interactions. We use this approach to model the “candy wrapper problem” a special kind of re-stenosis that sometimes affects coronary arteries after treatment with drug-eluting stents and we demonstrate the computational efficiency of the temporal multiscale scheme. In simplified problems, where it is possible to run fully resolved simulations, we can show tremendous savings in computational time at similar accuracy.
This is joint work with S. Frei (University of Konstanz) and J. Mizerski (University of Magdeburg).
Timo Koch (University of Stuttgart):
“Multiscale subvoxel models for tissue perfusion, contrast agent transport, and perfusion MRI in brain tissue”
The development of so-called mixed-dimension embedded models allows to efficiently model fluid and drug transport on the capillary scale. We present the model concept and give an overview over different variants of the method. We show how such models can be applied to simulate contrast agent transport as it occurs during perfusion MRI acquisition in the brain. The fluid-mechanical model is coupled with a multiscale MRI model. The simulation MR results are compared to clinical data of multiple sclerosis patients. Using numerical Bayesian inference for the model parameters under consideration of the data, posterior probability distributions for the parameters are obtained. The parameter uncertainty is quantified and the parameter values are physically interpreted.
PD Dr. Maria Neuss-Radu (FAU Erlangen-Nürnberg):
“Mathematical modeling and multiscale analysis of transport processes through membranes”
In this presentation, we develop multiscale methods for the derivation and analysis of effective models in environments containing membranes. At the microscopic level, where membranes are modeled as thin heterogeneous layers, the model consists of nonlinear reaction-diffusion equations within each subdomain. At the macroscopic level, membranes are reduced to interfaces, and effective transmission conditions and/or effective equations at these interfaces are derived. It turns out that the form of the effective laws at the interface depends on the scaling of the microscopic system as well as of the type of microscopic transmission conditions imposed at the bulk-layer interface. By adding corrector terms to the macroscopic solutions, we obtain higher order approximations. To validate these approximations, we prove error estimates with respect to the scaling parameter ε.
This is joint work with Markus Gahn (University of Heidelberg) and Willi Jäger (University of Heidelberg).