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Mathematical Modelling and Simulations of Brain Cell Swelling Under Ischaemic Conditions

Malieva, Valeria

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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 Biot-Stokes 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 Biot-Stokes 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 Biot-Stokes coupled problem is discretized using FEMs (in space) and the implicit Euler scheme (in time), and solved following an operator-splitting 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 Biot-Stokes 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
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