Maria Neuss-Radu

Mathematical modeling and simulation of the evolution of plaques in blood vessels.

In this paper, a model is developed for the evolution of plaques in arteries, which is one of the main causes for the blockage of blood flow. Plaque rupture and spread of torn-off material may cause closures in the down-stream vessel system and lead to ischemic brain or myocardial infarctions. The model covers the flow of blood and its interaction with the vessel wall. It is based on the assumption that the penetration of monocytes from the blood flow into the vessel wall, and the accumulation of foam cells increasing the volume, are main factors for the growth of plaques. The dynamics of the vessel wall is governed by a deformation gradient, which is given as composition of a purely elastic tensor, and a tensor modeling the biologically caused volume growth. An equation for the evolution of the metric is derived quantifying the changing geometry of the vessel wall. To calculate numerically the solutions of the arising free boundary problem, the model system of partial differential equations is transformed to an ALE (Arbitrary Lagrangian-Eulerian) formulation, where all equations are given in fixed domains. The numerical calculations are using newly developed algorithms for ALE systems. The results of the simulations, obtained for realistic system parameters, are in good qualitative agreement with observations. They demonstrate that the basic modeling assumption can be justified. The increase of stresses in the vessel wall can be computed. Medical treatment tries to prevent critical stress values, which may cause plaque rupture and its consequences.

Homogenization of Reaction--Diffusion Processes in a Two-Component Porous Medium with Nonlinear Flux Conditions at the Interface

In this paper, we are dealing with the mathematical modeling and homogenization of nonlinear reaction-diffusion processes in a porous medium that consists of two components separated by an interface. One of the components is connected, and the other one is disconnected and consists of periodically distributed inclusions. At the interface, the fluxes are given by nonlinear functions of the concentrations on both sides of the interface. Thus, the concentrations may be discontinuous across the interface. For the derivation of the effective (homogenized) model, we use the method of two-scale convergence. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we give a new approach which involves the boundary unfolding operator and a compactness result for Banach-space-valued functions. The model is motivated by metabolic and regulatory processes in cells, where biochemical species are exchanged between organelles and cytoplasm through the organellar membranes. In this context the nonlinearities are given by kinetics corresponding to multispecies enzyme catalyzed reactions, which are generalizations of the classical Michaelis--Menten kinetics to multispecies reactions.

Model-Based Design of Biochemical Microreactors

Mathematical modeling of biochemical pathways is an important resource in Synthetic Biology, as the predictive power of simulating synthetic pathways represents an important step in the design of synthetic metabolons. In this paper, we are concerned with the mathematical modeling, simulation, and optimization of metabolic processes in biochemical microreactors able to carry out enzymatic reactions and to exchange metabolites with their surrounding medium. The results of the reported modeling approach are incorporated in the design of the first microreactor prototypes that are under construction. These microreactors consist of compartments separated by membranes carrying specific transporters for the input of substrates and export of products. Inside the compartments of the reactor multienzyme complexes assembled on nano-beads by peptide adapters are used to carry out metabolic reactions. The spatially resolved mathematical model describing the ongoing processes consists of a system of diffusion equations together with boundary and initial conditions. The boundary conditions model the exchange of metabolites with the neighboring compartments and the reactions at the surface of the nano-beads carrying the multienzyme complexes. Efficient and accurate approaches for numerical simulation of the mathematical model and for optimal design of the microreactor are developed. As a proof-of-concept scenario, a synthetic pathway for the conversion of sucrose to glucose-6-phosphate (G6P) was chosen. In this context, the mathematical model is employed to compute the spatio-temporal distributions of the metabolite concentrations, as well as application relevant quantities like the outflow rate of G6P. These computations are performed for different scenarios, where the number of beads as well as their loading capacity are varied. The computed metabolite distributions show spatial patterns, which differ for different experimental arrangements. Furthermore, the total output of G6P increases for scenarios where microcompartimentation of enzymes occurs. These results show that spatially resolved models are needed in the description of the conversion processes. Finally, the enzyme stoichiometry on the nano-beads is determined, which maximizes the production of glucose-6-phosphate.

Homing to the Niche: A Mathematical Model Describing the Chemotactic Migration of Hematopoietic Stem Cells

It has been shown that hematopoietic stem cells migrate in vitro and in vivo following the gradient of a chemotactic factor produced by stroma cells. In this contribution, a quantitative model for this process is presented. The model consists of chemotaxis equations coupled with an ordinary differential equation on the boundary of the domain and subjected to nonlinear boundary conditions. The existence and uniqueness of a local solution is proved and the model is simulated numerically. It turns out that for adequate parameter ranges, the qualitative behavior of the stem cells observed in the experiment is in good agreement with the numerical results. Our investigations represent a first step in the process of elucidating the mechanism underlying the homing of hematopoietic stem cells quantitatively.