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Abstract
Motivated by the ischaemic brain stroke research, this work is devoted to the description of the osmotic swelling of a brain cell due to the absorption of the extracellular fluid during the formation of the cytotoxic (cellular) oedema. A physically motivated mathematical model describing the interaction between a single swelling cell and the extracellular fluid surrounding it is developed. In particular, the dynamics of the interaction is approximated by the coupled BiotStokes equations, resulting in a free boundary interaction problem. The Biot equations derived using homogenization techniques are considered and it is shown, that for the relevant data range, the temporal pressure derivative term of the Biot equations is negligible. Filtering effects of the cell membrane and the driving force of the transmembrane osmotic pressure difference are reflected in the BiotStokes coupling condition relating the normal fluid flux to the total pressure difference across the membrane. The analysis of the relevant experimentally obtained data for the considered biological system suggests that certain effects and processes included into the developed general coupled model can be neglected. As a result, a simpler (reduced) mathematical model is obtained and numerically implemented. The reduced BiotStokes coupled problem is discretized using FEMs (in space) and the implicit Euler scheme (in time), and solved following an operatorsplitting approach. The numerical implementations of the (pure) Biot problem are verified by comparing the analytic and numerical solutions, and are available for two and three dimensions. The simulation results for the reduced mathematical model parametrized with the estimated experimental parameters showed good agreement with the experimental observations. The sensitivity of the Biot problem solution to the variations of the key parameters and domain geometry, as well as the overall effect of the Stokes domain solution on the solution of the coupled BiotStokes problem are tested and analysed.
Item Type:  Dissertation 

Supervisor:  Jäger, Prof. Dr. Dr. h. c. mult. Willi 
Date of thesis defense:  22 June 2015 
Date Deposited:  01 Jul 2015 09:45 
Date:  2015 
Faculties / Institutes:  The Faculty of Mathematics and Computer Science > Department of Mathematics The Faculty of Mathematics and Computer Science > Department of Applied Mathematics The Faculty of Mathematics and Computer Science > Department of Computer Science 
Subjects:  000 Generalities, Science 500 Natural sciences and mathematics 510 Mathematics 530 Physics 570 Life sciences 610 Medical sciences Medicine 
Controlled Keywords:  Modellierung, Strömungsmechanik, Osmose, Elastizitätstheorie, Modellierung, Numerische Mathematik, Cytologie 
Uncontrolled Keywords:  Biot equations, Stokes equations, mathematical modelling, ischaemic stroke, cytotoxic swelling, cell swelling, cell mechanics, osmotic pressure, free boundary interaction problem, finite elements, DUNE 