# Molecular Diffusion in the Dynamics Brain Extracellular Space

8. In the thesis , we present a universal model of brain tissue microstructure that dynamically links osmosis and diffusion with geometrical parameters of brain extracellular space (ECS) . In the first part , we investigate the biological aspects of the model , and in the second , we analysis the model in the mathematical framework . The first part : Our model robustly describes and predicts the nonlinear time dependency of tortuosity ($\lambda = \sqrt{D/{D^{*}}}$) changes with very high precision in various media with uniform and nonuniform osmolarity distribution , as demonstrated by previously published experimental data ($D$ = free diffusion coefficient , $D^{*}$ = effective diffusion coefficient) . To construct this model , we first developed a multiscale technique for computationally effective modeling of osmolarity in the brain tissue . Osmolarity differences across cell membranes lead to changes in the ECS dynamics . The evolution of the underlying dynamics is then captured by a level set method . Subsequently , using a homogenization technique , we derived a coarse-grained model with parameters that are explicitly related to the geometry of cells and their associated ECS . Our modeling results in very accurate analytical approximation of tortuosity based on time , space , osmolarity differences across cell membranes , and water permeability of cell membranes . Our model provides a unique platform for studying ECS dynamics not only in physiologic conditions such as sleep-wake cycles and aging but also in pathologic conditions such as stroke , seizure , and neoplasia , as well as in predictive pharmacokinetic modeling such as predicting medication biodistribution and efficacy and novel biomolecule development and testing . The second part: We consider homogenization problem for a model in an evolving locally periodic perforated domain over time with the Robin boundary condition being posed on the boundary of the holes. Our main aim is to justify rigorously the homogenization limit for the upscaled system derived by means of asymptotic expansion technique. In other words, we obtain the so-called corrector homogenization estimate that specifies the convergence rate. The major challenge is the evolving of the heterogeneous media over time that constructs a moving boundary problem in microstructure . Also, we show existence and uniqueness results for the microscopic problem in with moving boundary