HOMOGENIZATION-LIMIT OF A MODEL SYSTEM FOR INTERACTION OF FLOW, CHEMICAL REACTIONS AND MECHANICS IN CELL TISSUES WILLI J ¨ AGER * , ANDRO MIKELI ´ C † , AND MARIA NEUSS-RADU ‡ Abstract. In this article we obtain rigorously the homogenization limit for a fluid-structure- reactive flow system. It consists of cell tissue and intercellular liquid, transporting solutes. The cell tissue is supposed linearly elastic and deforming with a viscous non-stationary flow. The elastic moduli of the tissue change with cumulative concentration value. In the limit, when the scale parameter goes to zero, we obtain the quasi-static Biot system, coupled with the upscaled reactive flow. Effective Biot’s coefficients depend on the reactant concentration. Additionally to the weak two-scale convergence results, we prove convergence of the elastic and viscous energies. This then implies a strong two-scale convergence result for the fluid-structure variables. Next we establish the regularity of the solutions for the upscaled equations. In our knowledge, it is the only known study of the regularity of solutions to the quasi-static Biot system. The regularity is used to prove the uniqueness for the upscaled model. Key words. reactive transport, fluid-structure interaction, homogenization, biological-tissue, generalized quasi-static Biot system. AMS subject classifications. 93A30, 35Q30, 47F10, 35B27, 92C50 1. Introduction. Information on biophysical and biochemical processes on all scales has enormously increased due to a revolution in experimental concepts and technologies. Consequently, quantitative methods, based on mathematical modeling and simulations are becoming more and more important in analyzing experimental data and designing theories based on mathematical concepts. One of the numerous challenges is modeling processes in tissues, including the molecular information on micro-scale. In this paper, the following processes in cell tissues are included: 1. fluid flow in the extracellular space, diffusion, transport and reactions of substances in the fluid, 2. exchange of substances at the cell membranes, 3. diffusion of substances and chemical reactions inside the cells, 4. changes of the mechanical properties of the cells due to the influence of chem- ical substances, small deformations of the structure. The corresponding microscopic system was formulated and analyzed by the authors in [10], where the existence and uniqueness of solutions was proven. Also in [10], the characteristic microscopic scale ε of this system was identified, depending on the application in cell biology and the available real data. In this paper, the scale limit ε → 0 is performed. For simplicity, the structure of the tissue is assumed to be periodic, that means generated by translations of a properly scaled geometric unit cell containing a biological cell connected with its adjacent neighbor cells. Here, fluid flow is restricted to the intercellular region Ω ε f (fluid region) and interacting with the cell region Ω ε s (solid region). A chemical substance is diffusing * IWR, University of Heidelberg, INF 368, 69120 Heidelberg, Germany Phone: +49 6221 548235 (jaeger@iwr.uni-heidelberg.de). † Universit´ e de Lyon, F-69003 France, Universit´ e Lyon 1, Institut Camille Jordan,Site de Gerland, Bˆ at. A, 50, avenue Tony Garnier, 69367 Lyon Cedex 07, France Phone: (Andro.Mikelic@univ- lyon1.fr). ‡ IWR, University of Heidelberg, INF 368, 69120 Heidelberg, Germany, Phone: +49 6221 546187 (maria.neuss-radu@iwr.uni-heidelberg.de). 1