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INAUGURAL–DISSERTATION Zur Erlangung Der Doktorwürde Der Naturwissenschaftlich-Mathematischen Gesamtfakultät Der Ruprecht-Karls-Universität Heidelberg

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INAUGURAL–DISSERTATION Zur Erlangung Der Doktorwürde Der Naturwissenschaftlich-Mathematischen Gesamtfakultät Der Ruprecht-Karls-Universität Heidelberg INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der Naturwissenschaftlich-Mathematischen Gesamtfakultät der Ruprecht-Karls-Universität Heidelberg vorgelegt von Valeria Malieva geboren am 23. Mai 1986 in Lipezk, Russland Tag der mündlichen Prüfung: Mathematical Modelling and Simulations of Brain Cell Swelling Under Ischaemic Conditions Betreuer: Prof. Dr. Dr. h. c. mult. Willi Jäger Acknowledgements First and foremost I would like to express my sincere gratitude to my supervisors Prof. Willi Jäger and Prof. Peter Bastian. I am especially thankful to Prof. Willi Jäger for granting me the opportunity to work on this fascinating topic, for believing in me and for his scientific as well as moral guidance. From Prof. Willi Jäger I learned not only what it takes to be an accomplished researcher, but also how important it is to stay true to oneself. I am very thankful to Prof. Peter Bastian for supporting my research, for being understanding and for always finding time to discuss my work. For allowing me to pursue my scientific work in an academically excellent environment, I would like to thank the Faculty of Mathematics and Computer Science of Heidelberg University. I am very grateful to Dr. Felix Heimann for his contribution to the development of the simulation framework for my thesis and for advising me on the subject of physics. The discussions that I had with Dr. Felix Heimann and his enthusiastic participation played a great role in the development of this work. I would also like to acknowledge the contribution of Prof. Maria Neuss-Radu to the early development of the mathematical model. I am grateful for her interest in the project and for helping in its advancement. Being an active member of the Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences (HGS MathComp) was a great experience, and I am thankful to the Graduate School for providing me with opportunities to visit exciting conferences and workshops and for encouraging me to actively participate in the life of the School. I would like to especially thank the HGS MathComp and the German Research Foundation (DFG) for the funding of my doctoral studies. Additionally, I wish to thank the present and past members of the Scientific Computing Group and Applied Analysis Group for being genuinely friendly and helpful. I am infinitely thankful to my kind mother for her love and for always believing in me. And most importantly, I would like to thank my husband for his loving care, invaluable support and great patience. 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. Zusammenfassung Diese Arbeit ist ein Beitrag zur Forschung and ischämischen Schlaganfällen und widmet sich der Beschreibung osmotisch bedingter Schwellungen von Gehirnzellen durch Absorption extra- zellulärer Flüssigkeit. Ein physikalisch motiviertes mathematisches Modell zur Beschreibung der Interaktion zwischen einer einzelnen Zelle und der sie umgebenden extra-zellulären Flüssigkeit wird präsentiert. Deren dynamischen Beziehungen werden durch eine gekoppelte Form der Biot-Stokes Gleichungen auf zeitlich veränderlichen Geometrien beschrieben. Unter Betrachtung einzelner Zwischenresultate der Herleitung der Biot-Gleichungen durch Homogenisierung wird gezeigt, dass die zeitlichen Ableitungen des Drucks für den problem-relevanten Parameterraum vernachlässigbar klein sind. Filter-effekte der Zellmembran sowie osmotische Druck, werden in den Biot-Stokes Kopplungs- bedingungen abgebildet, welche den transmembranen Fluss mit dem Gesamtdruckunterschied an der Zellmembran in Beziehung setzen. Basierend auf einer Analyse relevanter Experimente zu dem betrachteten biologischen System wird eine Auswahl der wichtigsten physikalischen Prozesse getroffen, um daraus ein vereinfachtes (reduziertes) mathematisches Modell zu erhalten, welches einer numerischen Lösung zugänglich ist. Die vereinfachten gekoppelten Biot-Stokes Gleichungen werden mit finiten Elementen im Raum und mit impliziter Euler-Integration in der Zeit diskretisiert und schließlich unter Zuhilfenahme von Operator-Splitting gelöst. Die numerische Implementierung des (reinen) Biot-Systems wird durch Vergleich mit analytischen Lösungen in zwei und drei Dimensionen verifiziert. Die Ergeb- nisse der Simulation des vereinfachten mathematischen Modells, welches mit experimentell bes- timmten Parametern vervollständigt wurde, zeigen eine gute Übereinstimmung mit den exper- imentell zugänglichen Observablen des realen Systems. Die Sensitivität des Biot-Teilsystems auf einige Schlüsselparameter sowie der Einfluss der Kopplungseffekte der Lösung im Stokes- Teilsystems werden getestet und analysiert. Contents Introduction 1 Nomenclature 7 1 Description of the problem 11 1.1 Problem definition................................... 11 1.1.1 Extracellular space (ECS)........................... 12 1.1.2 Intracellular space (ICS)............................ 13 1.1.3 Cell membrane: mechanical properties.................... 14 1.2 Osmosis and osmotic pressure............................. 18 1.2.1 Membrane filtration.............................. 18 1.3 Effects of the surrounding media........................... 20 1.4 Summary: considered domains and effects...................... 22 1.4.1 List of assumptions............................... 24 2 Mathematical Model 25 2.1 ICS-ECS interaction problem: Biot-Stokes model.................. 25 2.1.1 Extracellular space: the Stokes equations.................. 25 2.1.2 Intracellular space: the Biot poroelasticity equations............ 26 2.1.3 Lagrangian and ALE formulations...................... 32 2.1.4 Initial and boundary conditions........................ 34 2.1.5 Biot-Stokes coupling: interface conditions.................. 35 2.2 Osmotic pressure.................................... 39 2.2.1 Osmotic pressure model............................ 39 2.2.2 Transport equations for the molarities.................... 40 2.2.3 Initial equilibrium relations.......................... 42 2.2.4 Fast diffusion model.............................. 43 2.2.5 Modified flux interface conditions....................... 45 2.3 Summary: general conceptual model......................... 46 3 Data and dimensional analysis 51 3.1 Data: parameters and characteristic values..................... 51 3.1.1 Material properties and parameters...................... 52 3.1.2 Biot-Stokes problem.............................. 55 3.1.3 Osmotic pressures and molar concentrations................. 63 3.1.4 Times and velocities.............................. 67 3.1.5 Molarity flux models.............................. 70 3.2 Dimensional equations................................. 74 3.2.1 ALE transformation terms........................... 76 3.2.2 Incompressible Navier-Stokes equations................... 76 3.2.3 Biot equations................................. 77 i 3.2.4 Convection–diffusion equations........................ 79 3.2.5 Interface conditions.............................. 80 3.3 Summary: data tables................................. 82 4 Numerics and simulations 87 4.1 Considered reduced problems............................. 87 4.1.1 Biot-Stokes problem.............................. 87 4.1.2 Pure Biot problem............................... 90 4.1.3 Variational formulations............................ 92 4.2 Discretisation...................................... 92 4.2.1 Spatial discretization.............................. 93 4.2.2 Temporal discretization............................ 99 4.3 Analytic solution of the Biot problem........................ 100 4.3.1 2D: polar coordinates............................
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