A simulação computacional de membranas biológicas, em particular membranas formadas por bicamadas lipídicas, é uma área de grande interesse na atualidade. Enquanto simulações moleculares são bastante populares, a simulação na escala de uma célula inteira requer modelos baseados na mecânica dos meios contínuos. Essas membranas apresentam um comportamento de fluido viscoso incompressível bidimensional. Além disso, as formas de equilíbrio são bem explicadas pela energia de Canham-Helfrich, que depende da curvatura da membrana. Neste trabalho, um novo método de simulação de membranas viscosas, baseado em elementos finitos, é apresentado. Ele se inspira no conceito de James Clerk Maxwell de elasticidade fugaz, o qual é usado para adaptar técnicas bem estabelecidas de simulação de membranas elásticas. Trata-se do primeiro método a levar em conta, de maneira rigorosa, o aspecto viscoso da membrana, que é dominante na escala de tamanho de uma célula biológica, além da sua característica de fluido incompressível / The computational simulation of biological membranes, in particular of those made of lipid bilayers, is currently an area of great interest. While molecular simulations are quite popular, the simulation on the scale of a whole cell requires models based on continuum mechanics. Those membranes behave like a bidimensional incompressible viscous fluid. Furthermore, the equilibrium shapes are well explained by means of the Canham-Helfrich energy, which depends on the curvature of the membrane. In this work, a novel finite element based method for the simulation of viscous membranes is presented. It is inspired by James Clerk Maxwells concept of fugitive elasticity, which is used to adapt well established simulation techniques for elastic membranes. This is the first method to take into account, in a rigorous fashion, the viscous aspect of the membrane, which is dominant at the length scale of a biological cell, in addition to its characteristics as an incompressible fluid

A simulação computacional de membranas biológicas, em particular membranas formadas por bicamadas lipídicas, é uma área de grande interesse na atualidade. Enquanto simulações moleculares são bastante populares, a simulação na escala de uma célula inteira requer modelos baseados na mecânica dos meios contínuos. Essas membranas apresentam um comportamento de fluido viscoso incompressível bidimensional. Além disso, as formas de equilíbrio são bem explicadas pela energia de Canham-Helfrich, que depende da curvatura da membrana. Neste trabalho, um novo método de simulação de membranas viscosas, baseado em elementos finitos, é apresentado. Ele se inspira no conceito de James Clerk Maxwell de elasticidade fugaz, o qual é usado para adaptar técnicas bem estabelecidas de simulação de membranas elásticas. Trata-se do primeiro método a levar em conta, de maneira rigorosa, o aspecto viscoso da membrana, que é dominante na escala de tamanho de uma célula biológica, além da sua característica de fluido incompressível / The computational simulation of biological membranes, in particular of those made of lipid bilayers, is currently an area of great interest. While molecular simulations are quite popular, the simulation on the scale of a whole cell requires models based on continuum mechanics. Those membranes behave like a bidimensional incompressible viscous fluid. Furthermore, the equilibrium shapes are well explained by means of the Canham-Helfrich energy, which depends on the curvature of the membrane. In this work, a novel finite element based method for the simulation of viscous membranes is presented. It is inspired by James Clerk Maxwells concept of fugitive elasticity, which is used to adapt well established simulation techniques for elastic membranes. This is the first method to take into account, in a rigorous fashion, the viscous aspect of the membrane, which is dominant at the length scale of a biological cell, in addition to its characteristics as an incompressible fluid

We present a new diffuse interface model for the dynamics of inextensible vesicles in a viscous fluid with inertial forces. A new feature of this work is the implementation of the local inextensibility condition in the diffuse interface context. Local inextensibility is enforced by using a local Lagrange multiplier, which provides the necessary tension force at the interface. We introduce a new equation for the local Lagrange multiplier whose solution essentially provides a harmonic extension of the multiplier off the interface while maintaining the local inextensibility constraint near the interface. We also develop a local relaxation scheme that dynamically corrects local stretching/compression errors thereby preventing their accumulation. Asymptotic analysis is presented that shows that our new system converges to a relaxed version of the inextensible sharp interface model. This is also verified numerically. To solve the equations, we use an adaptive finite element method with implicit coupling between the Navier-Stokes and the diffuse interface inextensibility equations. Numerical simulations of a single vesicle in a shear flow at different Reynolds numbers demonstrate that errors in enforcing local inextensibility may accumulate and lead to large differences in the dynamics in the tumbling regime and smaller differences in the inclination angle of vesicles in the tank-treading regime. The local relaxation algorithm is shown to prevent the accumulation of stretching and compression errors very effectively. Simulations of two vesicles in an extensional flow show that local inextensibility plays an important role when vesicles are in close proximity by inhibiting fluid drainage in the near contact region.

Ce travail, à l'interface entre les mathématiques appliquées et la physique, s'articule autour de la modélisation numérique des vésicules biologiques, un modéle pour les globules rouges du sang. Pour cela, le modéle de Canham et Helfrich est adopté pour décrire le comportement des vésicules. La modélisation numérique utilise la méthode des fonctions de niveau dans un cadre éléments finis. Un nouvel algorithme de résolution numérique combinant une technique de multiplicateurs de Lagrange avec une adaptation automatique de maillages garantit la conservation exacte des volumes et des surfaces. Cet algorithme permet donc de dépasser une limitation cruciale actuelle de la méthode des fonctions de niveau, à savoir les pertes de masse couramment observées dans ce type de problémes. De plus, les propriétés de convergence de la méthode des fonctions de niveau se trouvent ainsi grandement améliorées, comme l'indiquent de nombreux tests numériques. Ces tests comprennent notamment des problémes d'advection élémentaires, des mouvements par courbure moyenne ainsi que des mouvements par diffusion de surface. Concernant l'équilibre statique des vésicules, une condition générale d'équilibre d'Euler-Lagrange est obtenue à l'aide d'outils de dérivation de forme. En dynamique, le mouvement d'une vésicule sous l'action d'un écoulement de cisaillement est étudié dans le cadre des nombres de Reynolds élevés. L'effet du confinement est considéré, et les régimes classiques de chenille de char et de basculement sont retrouvés. Finalement, pour la premiére fois, l'effet des termes inertiels est étudié et on montre qu'au delà d'une valeur critique du nombre de Reynolds, la vésicule passe d'un mouvement de basculement à un mouvement de chenille de char. / This work, at the interface between the Applied Mathematics and Physics is connected about the numerical modelisation of biological vesicles, a pattern for the red blood cells. For this reason, the pattern of Canham and Helfrich is adopted to describe the behaviour of the vesicles. The numerical modelisation uses the Level Set method in finite element framework. A new algorithm of numerical resolution combining one technique of Lagrange multipliers with an automatic mesh adaptation ensures the accurate conservation of volumes and surfaces. Thus this algorithm enables to exceed an existing crucial restriction of the Level Set method, that's to say, the wastes of mass usually noticed in this kind of problems. Moreover, the proprieties of convergence of the Level Set method are thus much more improved, as shown in many numerical tests. Those tests chiefly include elementary problems of advection, motions by mean curvature just as motions by spread of surface. Concerning the static equilibrum of the vesicles, a mechanical equilibrum equation (Euler-Lagrange equation) of a vesicle membrane under a generalized elastic bending energy is obtained and the approach is based on shape optimization tools. In dynamics, the motion of a vesicle under the effect of a shear flow is elaborated in the frames of reference of high Reynolds numbers. The effect of confinement is respected, and the standard regimes of tank-treading and of tumbling motion are found again. Finally, for the first time, the effect of the inertia terms is elaborated and we show that beyond a critical value of the number of Reynolds the vesicle passes from a tumbling motion to a tank-treading motion.

A mecânica celular jaz nas propriedades materiais da membrana plasmática, fundamentalmente uma bicamada fosfolipídica com espessura de dimensões moleculares. Além de forças elásticas, tal material bidimensional também experimenta tensões viscosas devido ao seu comportamento fluido (presumivelmente newtoniano) na direção tangencial. A despeito da notável concordância entre teoria e experimentos biofísicos sobre a geometria de membranas celulares, ainda não se faz presente um método computacional para simulação de sua (real) dinâmica viscosa governada pela lei de Boussinesq-Scriven. Assim sendo, introduzimos uma formulação variacional mista de três campos para escoamentos viscosos de superfícies fechadas curvas. Nela, o fluido circundante é levado em conta considerando-se uma restrição de volume interior, ao passo que uma restrição de área corresponde à inextensibilidade. As incógnitas são a velocidade, o vetor curvatura e a pressão superficial, todas estas interpoladas com elementos finitos lineares contínuos via estabilização baseada na projeção do gradiente de pressão. O método é semi-implícito e requer a solução de apenas um único sistema linear por passo de tempo. Outro ingrediente numérico proposto é uma força que mimetiza a ação de uma pinça óptica, permitindo interação virtual com a membrana, onde a qualidade e o refinamento de malha são mantidos por remalhagem adaptativa automática. Extensivos experimentos numéricos de dinâmica de relaxação são apresentados e comparados com soluções quasi-analíticas. É observada estabilidade temporal condicional com uma restrição de passo de tempo que escala como o quadrado do tamanho de malha. Reportamos a convergência e os limites de estabilidade de nosso método e sua habilidade em predizer corretamente o equilíbrio dinâmico de compridas e finas elongações cilíndricas (tethers) que surgem a partir de pinçamentos membranais. A dependência de forma membranal com respeito a uma velocidade imposta de pinçamento também é discutida, sendo que há um valor limiar de velocidade abaixo do qual um tether não se forma de início. Testes adicionais ilustram a robustez do método e a relevância dos efeitos viscosos membranais quando sob a ação de forças externas. Sem dúvida, ainda há um longo caminho a ser trilhado para o entendimento completo da mecânica celular (há de serem consideradas outras estruturas tais como citoesqueleto, canais iônicos, proteínas transmembranares, etc). O primeiro passo, porém, foi dado: a construção de um esquema numérico variacional capaz de simular a intrigante dinâmica das membranas celulares. / Cell mechanics lies on the material properties of the plasmatic membrane, fundamentally a two-molecule-thick phospholipid bilayer. Other than bending elastic forces, such a two-dimensional interfacial material also experiences viscous stresses due to its (presumably Newtonian) surface fluid tangential behaviour. Despite the remarkable agreement on membrane shapes between theory and biophysical experiments, there is no computational method for simulating its (actual) viscous dynamics governed by the Boussinesq- Scriven law. Accordingly, we introduce a mixed three-field variational formulation for viscous flows of closed curved surfaces. In it, the bulk fluid is taken into account by means of an enclosed-volume constraint, whereas an area constraint stands for the membranes inextensible character. The unknowns are the velocity, vector curvature and surface pressure fields, all of which are interpolated with linear continuous finite elements by means of a pressure-gradient-projection scheme. The method is semi-implicit and it requires the solution of a single linear system per time step. Another proposed ingredient is a numerical force that emulates the action of an optical tweezer, allowing for virtual interaction with the membrane, where mesh quality and refinement are maintained by adaptively remeshing. Extensive relaxation experiments are reported and compared with quasi-analytical solutions. Conditional time stability is observed, with a time step restriction that scales as the square of the mesh size. We discuss both convergence and the stability limits of our method, its ability to correctly predict the dynamical equilibrium of the tether due to tweezing. The dependence of the membrane shape on imposed tweezing velocities is also addressed. Basically, there is a threshold velocity value below which the tethers shape does not arise at first. Further tests illustrate the robustness of the method and show the significance of viscous effects on membranes deformation under external forces. Undoubtedly, there is still a long way to track toward the understanding of celullar mechanics (one still has to account other structures such as cytoskeleton, ion channels, transmembrane proteins, etc). The first step has given, though: the design of a numerically robust variational scheme capable of simulating the engrossing dynamics of fluid cell membranes.

A mecânica celular jaz nas propriedades materiais da membrana plasmática, fundamentalmente uma bicamada fosfolipídica com espessura de dimensões moleculares. Além de forças elásticas, tal material bidimensional também experimenta tensões viscosas devido ao seu comportamento fluido (presumivelmente newtoniano) na direção tangencial. A despeito da notável concordância entre teoria e experimentos biofísicos sobre a geometria de membranas celulares, ainda não se faz presente um método computacional para simulação de sua (real) dinâmica viscosa governada pela lei de Boussinesq-Scriven. Assim sendo, introduzimos uma formulação variacional mista de três campos para escoamentos viscosos de superfícies fechadas curvas. Nela, o fluido circundante é levado em conta considerando-se uma restrição de volume interior, ao passo que uma restrição de área corresponde à inextensibilidade. As incógnitas são a velocidade, o vetor curvatura e a pressão superficial, todas estas interpoladas com elementos finitos lineares contínuos via estabilização baseada na projeção do gradiente de pressão. O método é semi-implícito e requer a solução de apenas um único sistema linear por passo de tempo. Outro ingrediente numérico proposto é uma força que mimetiza a ação de uma pinça óptica, permitindo interação virtual com a membrana, onde a qualidade e o refinamento de malha são mantidos por remalhagem adaptativa automática. Extensivos experimentos numéricos de dinâmica de relaxação são apresentados e comparados com soluções quasi-analíticas. É observada estabilidade temporal condicional com uma restrição de passo de tempo que escala como o quadrado do tamanho de malha. Reportamos a convergência e os limites de estabilidade de nosso método e sua habilidade em predizer corretamente o equilíbrio dinâmico de compridas e finas elongações cilíndricas (tethers) que surgem a partir de pinçamentos membranais. A dependência de forma membranal com respeito a uma velocidade imposta de pinçamento também é discutida, sendo que há um valor limiar de velocidade abaixo do qual um tether não se forma de início. Testes adicionais ilustram a robustez do método e a relevância dos efeitos viscosos membranais quando sob a ação de forças externas. Sem dúvida, ainda há um longo caminho a ser trilhado para o entendimento completo da mecânica celular (há de serem consideradas outras estruturas tais como citoesqueleto, canais iônicos, proteínas transmembranares, etc). O primeiro passo, porém, foi dado: a construção de um esquema numérico variacional capaz de simular a intrigante dinâmica das membranas celulares. / Cell mechanics lies on the material properties of the plasmatic membrane, fundamentally a two-molecule-thick phospholipid bilayer. Other than bending elastic forces, such a two-dimensional interfacial material also experiences viscous stresses due to its (presumably Newtonian) surface fluid tangential behaviour. Despite the remarkable agreement on membrane shapes between theory and biophysical experiments, there is no computational method for simulating its (actual) viscous dynamics governed by the Boussinesq- Scriven law. Accordingly, we introduce a mixed three-field variational formulation for viscous flows of closed curved surfaces. In it, the bulk fluid is taken into account by means of an enclosed-volume constraint, whereas an area constraint stands for the membranes inextensible character. The unknowns are the velocity, vector curvature and surface pressure fields, all of which are interpolated with linear continuous finite elements by means of a pressure-gradient-projection scheme. The method is semi-implicit and it requires the solution of a single linear system per time step. Another proposed ingredient is a numerical force that emulates the action of an optical tweezer, allowing for virtual interaction with the membrane, where mesh quality and refinement are maintained by adaptively remeshing. Extensive relaxation experiments are reported and compared with quasi-analytical solutions. Conditional time stability is observed, with a time step restriction that scales as the square of the mesh size. We discuss both convergence and the stability limits of our method, its ability to correctly predict the dynamical equilibrium of the tether due to tweezing. The dependence of the membrane shape on imposed tweezing velocities is also addressed. Basically, there is a threshold velocity value below which the tethers shape does not arise at first. Further tests illustrate the robustness of the method and show the significance of viscous effects on membranes deformation under external forces. Undoubtedly, there is still a long way to track toward the understanding of celullar mechanics (one still has to account other structures such as cytoskeleton, ion channels, transmembrane proteins, etc). The first step has given, though: the design of a numerically robust variational scheme capable of simulating the engrossing dynamics of fluid cell membranes.

In this thesis, we study mathematical models that describe the morphology of a generalized biological cell in equilibrium or under the influence of external forces. Within these models, the cell is considered as a thermodynamic system, where streaming effects in the cell bulk and the surrounding are coupled with a Helfrich-type model for the cell membrane. The governing evolution equations for the cell given in a continuum formulation are derived using an energy variation approach. Such two-phase flow problems that combine streaming effects with a free boundary problem that accounts for bending and surface tension can be described effectively by a diffuse interface approach. An advantage of the diffuse interface approach is that models for e.g. different biophysical processes can easily be combined. That makes this method suitable to describe complex phenomena such as cell motility and multi-cell dynamics. Within the first model for cell motility, we combine a biological network for GTPases with the hydrodynamic Helfrich-type model. This model allows to account for cell motility driven by membrane protrusion as a result of actin polymerization. Within the second model, we moreover extend the Helfrich-type model by an active gel theory to account for the actin filaments in the cell bulk. Caused by contractile stress within the actin-myosin solution, a spontaneous symmetry breaking event occurs that lead to cell motility. In this thesis, we further study the dynamics of multiple cells which is of wide interest since it reveals rich non-linear behavior. To apply the diffuse interface framework, we introduce several phase field variables to account for several cells that are coupled by a local interaction potential. In a first application, we study white blood cell margination, a biological phenomenon that results from the complex relation between collisions, different mechanical properties and lift forces of red blood cells and white blood cells within the vascular system. Here, it is shown that inertial effects, which can become of relevance in various parts of the cardiovascular system, lead to a decreasing tendency for margination with increasing Reynolds number. Finally, we combine the active polar gel theory and the multi-cell approach that is capable of studying collective migration of cells. This hydrodynamic approach predicts that collective migration emerges spontaneously forming coherently-moving clusters as a result of the mutual alignment of the velocity vectors during inelastic collisions. We further observe that hydrodynamics heavily influence those systems. However, a complete suppression of the onset of collective migration cannot be confirmed. Moreover, we give a brief insight how such highly coupled systems can be treated numerically using finite elements and how the numerical costs can be limited using operator splitting approaches and problem parallelization with OPENMP. / Diese Dissertation beschäftigt sich mit mathematischen Modellen zur Beschreibung von Gleichgewichts- und dynamischen Zuständen von verallgemeinerten biologischen Zellen. Die Zellen werden dabei als thermodynamisches System aufgefasst, bei dem Strömungseffekte innerhalb und außerhalb der Zelle zusammen mit einem Helfrich-Modell für Zellmembranen kombiniert werden. Schließlich werden durch einen Energie-Variations-Ansatz die Evolutionsgleichungen für die Zelle hergeleitet. Es ergeben sie dabei Mehrphasen-Systeme, die Strömungseffekte mit einem freien Randwertproblem, das zusätzlich physikalischen Einflüssen wie Biegung und Oberflächenspannung unterliegt, vereinen. Um solche Probleme effizient zu lösen, wird in dieser Arbeit die Diffuse-Interface-Methode verwendet. Ein Vorteil dieser Methode ist, dass es sehr einfach möglich ist, Modelle, die verschiedenste Prozesse beschreiben, miteinander zu vereinen. Dies erlaubt es, komplexe biologische Phänomene, wie zum Beispiel Zellmotilität oder auch die kollektive Bewegung von Zellen, zu beschreiben. In den Modellen für Zellmotilität wird ein biologisches Netzwerk-Modell für GTPasen oder auch ein Active-Polar-Gel-Modell, das die Aktinfilamente im Inneren der Zellen als Flüssigkristall auffasst, mit dem Multi-Phasen-Modell kombiniert. Beide Modelle erlauben es, komplexe Vorgänge bei der selbst hervorgerufenen Bewegung von Zellen, wie das Vorantreiben der Zellmembran durch Aktinpolymerisierung oder auch die Kontraktionsbewegung des Zellkörpers durch kontraktile Spannungen innerhalb des Zytoskelets der Zelle, zu verstehen. Weiterhin ist die kollektive Bewegung von vielen Zellen von großem Interesse, da sich hier viele nichtlineare Phänomene zeigen. Um das Diffuse-Interface-Modell für eine Zelle auf die Beschreibung mehrerer Zellen zu übertragen, werden mehrere Phasenfelder eingeführt, die die Zellen jeweils kennzeichnen. Schließlich werden die Zellen durch ein lokales Abstoßungspotential gekoppelt. Das Modell wird angewendet, um White blood cell margination, das die Annäherung von Leukozyten an die Blutgefäßwand bezeichnet, zu verstehen. Dieser Prozess wird dabei bestimmt durch den komplexen Zusammenhang zwischen Kollisionen, den jeweiligen mechanischen Eigenschaften der Zellen, sowie deren Auftriebskraft innerhalb der Adern. Die Simulationen zeigen, dass diese Annäherung sich in bestimmten Gebieten des kardiovaskulären Systems stark vermindert, in denen die Blutströmung das Stokes-Regime verlässt. Schließlich wird das Active-Polar-Gel-Modell mit dem Modell für die kollektive Bewegung vom Zellen kombiniert. Dies macht es möglich, die kollektive Bewegung der Zellen und den Einfluss von Hydrodynamik auf diese Bewegung zu untersuchen. Es zeigt sich dabei, dass der Zustand der kollektiven gerichteten Bewegung sich spontan aus der Neuausrichtung der jeweiligen Zellen durch inelastische Kollisionen ergibt. Obwohl die Hydrodynamik einen großen Einfluss auf solche Systeme hat, deuten die Simulationen nicht daraufhin, dass Hydrodynamik die kollektive Bewegung vollständig unterdrückt. Weiterhin wird in dieser Arbeit gezeigt, wie die stark gekoppelten Systeme numerisch gelöst werden können mit Hilfe der Finiten-Elemente-Methode und wie die Effizienz der Methode gesteigert werden kann durch die Anwendung von Operator-Splitting-Techniken und Problemparallelisierung mittels OPENMP.

In this thesis, we study mathematical models that describe the morphology of a generalized biological cell in equilibrium or under the influence of external forces. Within these models, the cell is considered as a thermodynamic system, where streaming effects in the cell bulk and the surrounding are coupled with a Helfrich-type model for the cell membrane. The governing evolution equations for the cell given in a continuum formulation are derived using an energy variation approach. Such two-phase flow problems that combine streaming effects with a free boundary problem that accounts for bending and surface tension can be described effectively by a diffuse interface approach. An advantage of the diffuse interface approach is that models for e.g. different biophysical processes can easily be combined. That makes this method suitable to describe complex phenomena such as cell motility and multi-cell dynamics. Within the first model for cell motility, we combine a biological network for GTPases with the hydrodynamic Helfrich-type model. This model allows to account for cell motility driven by membrane protrusion as a result of actin polymerization. Within the second model, we moreover extend the Helfrich-type model by an active gel theory to account for the actin filaments in the cell bulk. Caused by contractile stress within the actin-myosin solution, a spontaneous symmetry breaking event occurs that lead to cell motility. In this thesis, we further study the dynamics of multiple cells which is of wide interest since it reveals rich non-linear behavior. To apply the diffuse interface framework, we introduce several phase field variables to account for several cells that are coupled by a local interaction potential. In a first application, we study white blood cell margination, a biological phenomenon that results from the complex relation between collisions, different mechanical properties and lift forces of red blood cells and white blood cells within the vascular system. Here, it is shown that inertial effects, which can become of relevance in various parts of the cardiovascular system, lead to a decreasing tendency for margination with increasing Reynolds number. Finally, we combine the active polar gel theory and the multi-cell approach that is capable of studying collective migration of cells. This hydrodynamic approach predicts that collective migration emerges spontaneously forming coherently-moving clusters as a result of the mutual alignment of the velocity vectors during inelastic collisions. We further observe that hydrodynamics heavily influence those systems. However, a complete suppression of the onset of collective migration cannot be confirmed. Moreover, we give a brief insight how such highly coupled systems can be treated numerically using finite elements and how the numerical costs can be limited using operator splitting approaches and problem parallelization with OPENMP. / Diese Dissertation beschäftigt sich mit mathematischen Modellen zur Beschreibung von Gleichgewichts- und dynamischen Zuständen von verallgemeinerten biologischen Zellen. Die Zellen werden dabei als thermodynamisches System aufgefasst, bei dem Strömungseffekte innerhalb und außerhalb der Zelle zusammen mit einem Helfrich-Modell für Zellmembranen kombiniert werden. Schließlich werden durch einen Energie-Variations-Ansatz die Evolutionsgleichungen für die Zelle hergeleitet. Es ergeben sie dabei Mehrphasen-Systeme, die Strömungseffekte mit einem freien Randwertproblem, das zusätzlich physikalischen Einflüssen wie Biegung und Oberflächenspannung unterliegt, vereinen. Um solche Probleme effizient zu lösen, wird in dieser Arbeit die Diffuse-Interface-Methode verwendet. Ein Vorteil dieser Methode ist, dass es sehr einfach möglich ist, Modelle, die verschiedenste Prozesse beschreiben, miteinander zu vereinen. Dies erlaubt es, komplexe biologische Phänomene, wie zum Beispiel Zellmotilität oder auch die kollektive Bewegung von Zellen, zu beschreiben. In den Modellen für Zellmotilität wird ein biologisches Netzwerk-Modell für GTPasen oder auch ein Active-Polar-Gel-Modell, das die Aktinfilamente im Inneren der Zellen als Flüssigkristall auffasst, mit dem Multi-Phasen-Modell kombiniert. Beide Modelle erlauben es, komplexe Vorgänge bei der selbst hervorgerufenen Bewegung von Zellen, wie das Vorantreiben der Zellmembran durch Aktinpolymerisierung oder auch die Kontraktionsbewegung des Zellkörpers durch kontraktile Spannungen innerhalb des Zytoskelets der Zelle, zu verstehen. Weiterhin ist die kollektive Bewegung von vielen Zellen von großem Interesse, da sich hier viele nichtlineare Phänomene zeigen. Um das Diffuse-Interface-Modell für eine Zelle auf die Beschreibung mehrerer Zellen zu übertragen, werden mehrere Phasenfelder eingeführt, die die Zellen jeweils kennzeichnen. Schließlich werden die Zellen durch ein lokales Abstoßungspotential gekoppelt. Das Modell wird angewendet, um White blood cell margination, das die Annäherung von Leukozyten an die Blutgefäßwand bezeichnet, zu verstehen. Dieser Prozess wird dabei bestimmt durch den komplexen Zusammenhang zwischen Kollisionen, den jeweiligen mechanischen Eigenschaften der Zellen, sowie deren Auftriebskraft innerhalb der Adern. Die Simulationen zeigen, dass diese Annäherung sich in bestimmten Gebieten des kardiovaskulären Systems stark vermindert, in denen die Blutströmung das Stokes-Regime verlässt. Schließlich wird das Active-Polar-Gel-Modell mit dem Modell für die kollektive Bewegung vom Zellen kombiniert. Dies macht es möglich, die kollektive Bewegung der Zellen und den Einfluss von Hydrodynamik auf diese Bewegung zu untersuchen. Es zeigt sich dabei, dass der Zustand der kollektiven gerichteten Bewegung sich spontan aus der Neuausrichtung der jeweiligen Zellen durch inelastische Kollisionen ergibt. Obwohl die Hydrodynamik einen großen Einfluss auf solche Systeme hat, deuten die Simulationen nicht daraufhin, dass Hydrodynamik die kollektive Bewegung vollständig unterdrückt. Weiterhin wird in dieser Arbeit gezeigt, wie die stark gekoppelten Systeme numerisch gelöst werden können mit Hilfe der Finiten-Elemente-Methode und wie die Effizienz der Methode gesteigert werden kann durch die Anwendung von Operator-Splitting-Techniken und Problemparallelisierung mittels OPENMP.