Refine
Year of publication
Document Type
- Doctoral Thesis (17)
Language
- English (17) (remove)
Has Fulltext
- yes (17)
Is part of the Bibliography
- no (17)
Keywords
- Boltzmann-Gleichung (2)
- Delaunay-Triangulierung (2)
- Hadron (2)
- QCD (2)
- Quark <Physik> (2)
- Relativistische Hydrodynamik (2)
- ALICE experiment (1)
- Artificial Intelligence (1)
- CBM experiment (1)
- Cluster Hadronization (1)
Institute
This thesis presents a model for the dynamical description of deconfined quark matter created in ultra-relativistic heavy ion collisions, treating quarks and antiquarks as classical point particles subject to a colour-dependent, Cornell-type potential interaction. The model provides a dynamical handle for hadronization via the recombination of quarks and antiquarks in colour neutral clusters. Gluons are not included explicitly in the model,but are described in an effective manner by the means of the potential interaction. The model includes four different quark flavours (up, down, strange and charm) and uses current masses for the quarks. The dynamical evolution of a system of colour charges subject to the Hamiltonian equations of motion of the model yields the formation of colour neutral clusters of quarks and antiquarks, which are subject only to a small remaining interaction, the strong interquark potential notwithstanding. These clusters can be mapped onto hadrons and hadronic resonances. Thus, the model allows a dynamical description of quarks degrees of freedom in heavy ion collisions, including a recombination scheme for hadronization. The thermal properties of the model turn pout to be very satisfying. The model shows a transition from a confining phase to a deconfined phase with rising temperature, going hand in hand with a softest point in the equation of state and a rise of energy density and pressure to the Stefan-Boltzmann limit of a gas of quarks and antiquarks. Moreover, the potential interaction is screened in the deconfined phase. For the dynamical description of ultra-relativistic heavy ion collision, the qMD model is coupled to UrQMD as a generator for its initial conditions. In this way, a fully dynamical description of the expansion and hadronization of the fireball created in such collisions can be achieved. Non-equilibrium aspects of the expansion dynamics and hadronization by recombination of quarks and antiquarks are discussed in detail, and a comparison with experimental data of collisions at the CERN-SPS is presented. The big advantage of the qMD model is the possibility to study cluster formation, including exotic clusters, and fluctuations in a dynamical manner. As an example, event-by-event fluctuations in electric charge are studied. Such fluctuations have been proposed as a clear criterion to distinguish a deconfined system from a hadrons gas. However, experimental data show hadron gas fluctuation measures even at RHIC, where deconfinement is taken for granted. We will see how the dynamics of quark recombination washes out the quark-gluon plasma signal in the fluctuation criterion. Moreover, we will discuss briefly the problem of entropy at recombination. In a second application, the formation of exotic hadronic clusters, larger than usual mesons and baryons, is studied. Such clusters could provide new measures for the thermalization and homogenization of a deconfined gas of colour charges. Moreover, number estimates for exotic clusters from recombination are considerably lower than corresponding predictions from thermal models, providing a clear difference between statistical hadronization and hadronization via quark recombination. A detailed analysis is provided for pentaquark candidates such as the Theta-Plus. It turns out that the distribution of exotic states over strangeness, isospin, and spin could provide a sensitive measure for thermalization and decorrelation in the deconfined quark phase, if it could be measured.
Gegenstand dieser Arbeit sind Eigenschaften angeregter hadronischer Materie sowie physikalische Systeme, in denen diese Materie auftritt bzw. produziert wird. Die Beschreibung der stark wechselwirkenden Materie erfolgt in einem hadronischen, chiral-symmetrischen SU(3)L x SU(3)R Modell, welches die Saturierungseigenschaften von Kernmaterie und die Eigenschaften von Atomkernen reproduziert. Die Untersuchung heißer und dichter unendlicher hadronischor Materie zeigt, dass das vom Modell vorhergesagte Phasendiagramm stark von den Kopplungen der Baryonenresonanzen abhängt. Für kalte hadronische Materie ergibt die Einbeziehung des Baryonendekupletts und die Freiheit in deren Vektorkopplungen eine sehr große Bandbreite an verschiedenen Zustandsgleichungen. Für heiße hadronische Materie mit verschwindendem baryochemischen Potential zeigt sich ebenfalls eine starke Abhängigkeit der Eigenschaften hadronischer Materie von der Ankopplung der baryonischen Resonanzen. Es werden drei verschiedene Parametrisierungen betrachtet. Das resultierende Phasenübergangsverhalten variiert von einem "Crossover" über einen schwachen, zu einem doppelten Phasenübergang erster Ordnung. Es zeigt sich jedoch, dass die beobachteten Eigenschaften von Neutronensternen die Unbestimmtheit bzgl. der Vektorkopplung dieser Freiheitsgrade und damit der Zustandsgleichung deutlich verringern. Das Raum-Zeit Verhalten relativistischer Schwerionenkollisionen bei SPS- und RHIC-Energien wird mittels einer hydrodynamischen Simulation unter Benutzung der chiralen Zustandsgleichungen untersucht. Dabei spiegelt sich das unterschiedliche Phasenübergangsverhalten deutlich im Ausfrierverhalten der hadronischen Materie wider. Die im chiralen Modell berechneten Teilchenzahlverhältnisse werden mit den aus Schwerionenkollisionen von AGS- bis RHIC-Energien erhaltenen experimentellen Daten verglichen. Dabei zeigt sich, dass die verschiedenen Parametersätze des chiralen Modells und die Rechnungen für ein nichtwechselwirkendes, ideales Hadronengas eine ähnlich gute Beschreibung der gemessenen Weite liefern. Die deduzierten Ausfrierwerte für die Temperatur sind sensitiv auf das Phasenübergangsverhalten und liegen unterhalb der jeweiligen kritischen Temperatur. Die vorhergesagten Ausfriermassen sind in allen Parametrisierungen sehr ähnlich mit Abweichungen bis zu 15% von den entsprechenden Vakuumwerten. Die Untersuchung der Eigenschaften von Vektormesonen in dichter Materie erfolgt in der Mittleren-Feld- und in der HartreeNäherung. Hierbei zeigt sich eine signifikante Reduzierung der Teilchenmassen durch Vakuumpolarisationseffekte.
Nonequilibrium phase transitions in chiral fluid dynamics including dissipation and fluctuation
(2011)
Chiral fluid dynamics combines the fluid dynamic expansion of a hot and dense plasma created in a heavy-ion collision with the explicit propagation of fluctuations at the chiral phase transition of quantum chromodynamics. From systems in equilibrium long-range fluctuations are expected at a conjectured critical point. Heavy-ion collisions are, however, finite in size and time and very dynamic. It is thus likely that nonequilibrium effects diminish the signal of a critical point. They can, however, stimulate phenomena at a first order phase transitions, like nucleation and spinodal decomposition. Both of phase transition scenarios are investigated in this work. Based on the linear sigma model with constituent quarks a consistent quantum field theoretical approach using the two-particle irreducible effective action is developed to derive both, the local equilibrium properties of the expanding quark fluid and the damping and noise terms in the Langevin equation of the order parameter of the phase transition, the sigma field. Within this formalism it is possible to obtain a conserved energy-momentum tensor of the coupled system. It describes the energy dissipation from the sigma field to the heat bath during relaxation. Within this model we investigate nonequilibrium phenomena in a scenario with a critical point and a first order phase transition. We observe long relaxation times at the phase transition, phase coexistence at the first order phase transition and critical slowing down at the critical point. We find a substantial supercooling in a first order phase transition in our model and due to the energy-momentum exchange also reheating is present. While at the critical point the correlation length increases slightly we find an enhanced intensity of nonequilibrium fluctuations at the first order phase transition, which leads to an increased production of sigma mesons.
Quantum chromodynamics predicts the existence of a phase transition from hadronic to quark-gluon matter when temperature and pressure are sufficiently high. Colliding heavy nuclei at ultra-relativistic speeds allows to deposit large amounts of energy in a small volume of space, and is the only available experimental mean to produce the extreme conditions necessary to obtain the deconfined state. Numerous models and ideas were developed in the last decades to study heavy ion physics and understand the properties of extremely heated and compressed nuclear matter. With the ever increasing energy available in the center of mass frame (and thus number of particles produced) and the development of large acceptance detectors, it has become possible to study the fluctuations of physical quantities on an event-by-event basis, and access thermodynamical properties not present in particle spectra. The characteristics of the highly excited matter produced, e.g. thermalization, effect of resonance decay. . . can be investigated by fluctuation analyses. In fact, fluctuations are good indicators for a phase transition and a plethora of fluctuation probes have been proposed to pin down the existence and the properties of the QGP. We study various fluctuation quantities within the Ultra-relativistic Quantum Molecular Dynamics UrQMD and the quantum Molecular Dynamics qMD models. UrQMD is based on hadron and string degrees of freedom and allows to disentangle purely hadronic effects. In contrast, the qMD model includes an explicit transition from quark to hadronic matter and can serve to test adequate probes of the initial QGP state. We show that the qMD model can reasonably reproduce various experimental particles rapidity distributions and transverse mass spectra in wide energy range. Within the frame of the dynamical recombination procedure used in qMD, we study the enhancement of protons over pions (p/π) ratio in the intermediate pt range (1.5 < pt < 2.5). We show that qMD can reproduce the large p/π ≈ 1 observed experimentally at RHIC energies at hadronization. However, the subsequent decay of resonances makes the ratio fall to values incompatible with experimental data. We thus conclude that resonance decay might have a drastic influence on this observable in the quark recombination picture. Charged particles multiplicity fluctuations measured at SPS by the NA49 collaboration are enhanced in midperipheral events for Pb+Pb collisions at Elab = 160 AGeV. This feature is not reproduce by hadron-string transport approaches, which show a flat centrality dependence, within the proper experimental acceptance and with the proper centrality selection procedure. However, we show that the behavior of multiplicity fluctuations in transport codes is similar to the experimental result in full 4π acceptance. We identify the centrality selection procedure as the reason for the enhanced particle multiplicity fluctuations in midperipheral reactions and argue that it can be used to distinguish between different scenarios of particle productions. We show that experimental data might indicate a strong mixing of projectile and target related production sources. Strangeness over entropy K/π and baryon number over entropy p/π ratio fluctuations have been measured by the NA49 experiment in the SPS energy range, from Elab = 20 AGeV up to Elab = 160 AGeV. We investigate the sensitivity of this observable to kinematical cuts and discuss the influence of resonance decay. We find the dynamical p/π ratio fluctuations to increase with beam energy, in agreement with the measured data points. On the contrary, the dynamical K/π ratio fluctuations are essential flat as a function of centrality and depend only weakly on the kinematical cuts applied. Our results are in line with the simulations performed earlier by the NA49 collaboration in their detector acceptance filter. Finally, we focus on the correlations and fluctuations of conserved charges. It was proposed that these fluctuations are sensitive to the fractional charge carried by the quarks in the initial QGP stage and survive the whole course of heavy ion reactions. A crucial point is the influence of hadronization that may relax the initial QGP fluctuation/correlation signals to their hadronic values. We use the quark Molecular Dynamics qMD model to disentangle the effect of recombination-hadronization on charged particles ratio fluctuations, charge transfer fluctuations, baryon number-strangeness correlation coefficient and various ratios of susceptibilities (i.e. correlations over fluctuations). We find that the dynamical recombination procedure implemented in the qMD model destroys all studied initial QGP fluctuations and correlations and might ex- plain why no signal of a phase transition based on event-by-event fluctuations was found in the experimental data until now.
In this thesis the first fully integrated Boltzmann+hydrodynamics approach to relativistic heavy ion reactions has been developed. After a short introduction that motivates the study of heavy ion reactions as the tool to get insights about the QCD phase diagram, the most important theoretical approaches to describe the system are reviewed. To model the dynamical evolution of the collective system assuming local thermal equilibrium ideal hydrodynamics seems to be a good tool. Nowadays, the development of either viscous hydrodynamic codes or hybrid approaches is favoured. For the microscopic description of the hadronic as well as the partonic stage of the evolution transport approaches have beeen successfully applied, since they generate the full phse-space dynamics of all the particles. The hadron-string transport approach that this work is based on is the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) approach. It constitutes an effective solution of the relativistic Boltzmann equation and is restricted to binary collisions of the propagated hadrons. Therefore, the Boltzmann equation and the basic assumptions of this model are introduced. Furthermore, predictions for the charged particle multiplicities at LHC energies are made. The next step is the development of a new framework to calculate the baryon number density in a transport approach. Time evolutions of the net baryon number and the quark density have been calculated at AGS, SPS and RHIC energies and the new approach leads to reasonable results over the whole energy range. Studies of phase diagram trajectories using hydrodynamics are performed as a first move into the direction of the development of the hybrid approach. The hybrid approach that has been developed as the main part of this thesis is based on the UrQMD transport approach with an intermediate hydrodynamical evolution for the hot and dense stage of the collision. The initial energy and baryon number density distributions are not smooth and not symmetric in any direction and the initial velocity profiles are non-trivial since they are generated by the non-equilibrium transport approach. The fulll (3+1) dimensional ideal relativistic one fluid dynamics evolution is solved using the SHASTA algorithm. For the present work, three different equations of state have been used, namely a hadron gas equation of state without a QGP phase transition, a chiral EoS and a bag model EoS including a strong first order phase transition. For the freeze-out transition from hydrodynamics to the cascade calculation two different set-ups are employed. Either an in the computational frame isochronous freeze-out or an gradual freeze-out that mimics an iso-eigentime criterion. The particle vectors are generated by Monte Carlo methods according to the Cooper-Frye formula and UrQMD takes care of the final decoupling procedure of the particles. The parameter dependences of the model are investigated and the time evolution of different quantities is explored. The final pion and proton multiplicities are lower in the hybrid model calculation due to the isentropic hydrodynamic expansion while the yields for strange particles are enhanced due to the local equilibrium in the hydrodynamic evolution. The elliptic flow values at SPS energies are shown to be in line with an ideal hydrodynamic evolution if a proper initial state is used and the final freeze-out proceeds gradually. The hybrid model calculation is able to reproduce the experimentally measured integrated as well as transverse momentum dependent $v_2$ values for charged particles. The multiplicity and mean transverse mass excitation function is calculated for pions, protons and kaons in the energy range from $E_{\rm lab}=2-160A~$GeV. It is observed that the different freeze-out procedures have almost as much influence on the mean transverse mass excitation function as the equation of state. The experimentally observed step-like behaviour of the mean transverse mass excitation function is only reproduced, if a first order phase transition with a large latent heat is applied or the EoS is effectively softened due to non-equilibrium effects in the hadronic transport calculation. The HBT correlation of the negatively charged pion source created in central Pb+Pb collisions at SPS energies are investigated with the hybrid model. It has been found that the latent heat influences the emission of particles visibly and hence the HBT radii of the pion source. The final hadronic interactions after the hydrodynamic freeze-out are very important for the HBT correlation since a large amount of collisions and decays still takes place during this period.
The aim of this work is to develop an effective equation of state for QCD, having the correct asymptotic degrees of freedom, to be used as input for dynamical studies of heavy ion collisions. We present an approach for modeling an EoS that respects the symmetries underlying QCD, and includes the correct asymptotic degrees of freedom, i.e. quarks and gluons at high temperature and hadrons in the low-temperature limit. We achieve this by including quarks degrees of freedom and the thermal contribution of the Polyakov loop in a hadronic chiral sigma-omega model. The hadronic part of the model is a nonlinear realization of an sigma-omega model. As the fundamental symmetries of QCD should also be present in its hadronic states such an approach is widely used to describe hadron properties below and around Tc. The quarks are introduced as thermal quasi particles, coupling to the Polyakov loop, while the dynamics of the Polyakov loop are controlled by a potential term which is fitted to reproduce pure gauge lattice data. In this model the sigma field serves a the order parameter for chiral restoration and the Polyakov loop as order parameter for deconfinement. The hadrons are suppressed at high densities by excluded volume corrections. As a next step, we introduce our new HQ model equation of state in a microscopic+macroscopic hybrid approach to heavy ion collisions. This hybrid approach is based on the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) transport approach with an intermediate hydrodynamical evolution for the hot and dense stage of the collision. The present implementation allows to compare pure microscopic transport calculations with hydrodynamic calculations using exactly the same initial conditions and freeze-out procedure. The effects of the change in the underlying dynamics - ideal fluid dynamics vs. non-equilibrium transport theory - are explored. The final pion and proton multiplicities are lower in the hybrid model calculation due to the isentropic hydrodynamic expansion while the yields for strange particles are enhanced due to the local equilibrium in the hydrodynamic evolution. The elliptic and directed flow are shown to be not sensitive to changes in the EoS while the smaller mean free path in the hydrodynamic evolution reflects directly in higher flow results which are consistent with the experimental data. This finding indicates qualitatively that physical mechanisms like viscosity and other non equilibrium effects play an essentially more important role than the EoS when bulk observables like flow are investigated. In the last chapter, results for the thermal production of MEMOs in nucleus-nucleus collisions from a combined micro+macro approach are presented. Multiplicities, rapidity and transverse momentum spectra are predicted for Pb+Pb interaction at different beam energies. The presented excitation functions for various MEMO multiplicities show a clear maximum at the upper FAIR energy regime making this facility the ideal place to study the production of these exotic forms of multistrange objects.
This dissertation is devoted to the study of thermodynamics for quantum gauge theories.The poor convergence of quantum field theory at finite temperature has been the main obstacle in the practical applications of thermal QCD for decades. In this dissertation I apply hard-thermal-loop perturbation theory, which is a gauge-invariant reorganization of the conventional perturbative expansion for quantum gauge theories to the thermodynamics of QED and Yang-Mills theory to three-loop order. For the Abelian case, I present a calculation of the free energy of a hot gas of electrons and photons by expanding in a power series in mD/T, mf /T and e2, where mD and mf are the photon and electron thermal masses, respectively, and e is the coupling constant.I demonstrate that the hard-thermal-loop perturbation reorganization improves the convergence of the successive approximations to the QED free energy at large coupling, e ~ 2. For the non-Abelian case, I present a calculation of the free energy of a hot gas of gluons by expanding in a power series in mD/T and g2, where mD is the gluon thermal mass and g is the coupling constant. I show that at three-loop order hard-thermal-loop perturbation theory is compatible with lattice results for the pressure, energy density, and entropy down to temperatures T ~ 2 - 3 Tc. The results suggest that HTLpt provides a systematic framework that can be used to calculate static and dynamic quantities for temperatures relevant at LHC.
Ziel dieser Arbeit war, die Reaktion von biologischen Gewebeproben auf dünn- und dicht-ionisierende Strahlung zu evaluieren. Dafür wurden die Gewebeproben konventioneller Röntgenstrahlung sowie einem ausgedehnten 12C-Ionen Bragg-Peak ausgesetzt. Zur Bestrahlung der biologischen Proben mit 12C wurde mit dem GSI-eigenen Simulationsprogramm TRiP98 ein Tiefendosisprofil eines ausgedehnten Bragg-peaks erstellt. Ein weiteres Ziel dieser Arbeit war, dieses Tiefendosisprofil mit drei anderen Simulationsprogrammen (ATIMA, MCHIT, TRIM) zu reproduzieren und zu vergleichen.
ATIMA und TRIM sind allgemeine Programme für den Energieverlust von Ionen in Materie. Sie können das von TRiP98 berechnetet Tiefendosisprofil nur ungenügend reproduzieren, da sie aufgrund fehlender Fragmentierung ein linear ansteigendes Tiefendosisprofil berechnen. Das Monte Carlo-Programm MCHIT, welches speziell für die Wechselwirkung von Ionen mit Materie in medizinischer Anwendung entwickelt wurde, zeigt die beste Übereinstimmung mit der TRiP98-Referenzkurve. Bis auf eine leicht höhere Durchschnittsdosis um 0.1 Gy konnte das Tiefendosisprofil nahezu exakt reproduziert werden.
Die biologischen Proben bestanden aus Schnittkulturen gesunder Maus-Lebern und Explantatkulturen gesunder Maus-Pankreata, um Nebenwirkungen ionisierender Strahlen abzuschätzen. Zusätzlich wurde die Reaktion auf 12C-Bestrahlung in neoplastischem Lebergewebe transgener c-myc/TGF-α Mäuse mit induzierbarem Lebertumor bestimmt. Um eine mögliche Tageszeitabhängigkeit der Gewebereaktion auf die Bestrahlung zu untersuchen, wurden die Schnitt- und Explantatkulturen zu zwei unterschiedlichen Tageszeiten präpariert: zur Mitte des subjektiven Tages und zur Mitte der subjektiven Nacht.
Die Präparate wurden für mehrere Tage auf einer Membran an einer Grenzschicht von Flüssigkeit und Luft kultiviert. Leber- und Pankreaskulturen gesunder C3H wildtyp Mäuse wurden mit einer Dosis von 2 Gy, 5 Gy oder 10 Gy Röntgenstrahlen bestrahlt. Leber- und Pankreaskulturen transgener Mäuse wurden mit ausgedehnten C-Ionen Bragg Peaks gleicher Dosen bestrahlt. Als Kontrolle dienten unbestrahlte Proben. Alle Proben wurden 1 h bzw. 24 h nach der Bestrahlung fixiert und immunhistochemisch auf Marker für Proliferation (Ki67), Apoptose (Caspase3) und DNA- Doppelstrangbrüche (γH2AX) untersucht.
Während die Pankreas-Präparate im Hinblick auf die untersuchten Parameter leider keine auswertbaren Ergebnisse ergaben, zeigten die untersuchten Parameter im gesunden Lebergewebe deutliche Tag-Nacht Unterschiede: die Proliferationsrate war zur Mitte des subjektiven Tages signifikant höher als zur Mitte der subjektiven Nacht. Umgekehrt waren die Raten für DNA-Doppelstrangbrüche zur Mitte der subjektiven Nacht signifikant erhöht. Diese Tag-Nacht Unterschiede ließen sich in neoplastischem Lebergewebe nicht nachweisen. Unabhängig von der Art und Dosis, hatte die Bestrahlung im gesunden Lebergewebe keinen Einfluss auf die untersuchten Parameter. In neoplastischem Lebergewebe hingegen wird die Rate an DNA-Doppelstrangbrüchen durch eine Bestrahlung dosisabhängig erhöht.
Die Auswirkungen ionisierender Strahlen auf das circadiane Uhrwerk wurden in Gewebeproben transgener Per2luc-Mäuse überprüft. Per2luc-Mäuse exprimieren das Enzym Luziferase unter der Kontrolle des Promoters von Per2, einem wichtigen Bestandteil des circadianen Uhrwerks. Daher erlaubt die Analyse dieser Tiere, den circadianen Rhythmus des molekularen Uhrwerks in Leber und anderen Geweben durch Messung der Luziferase-Aktivität in Echtzeit aufzuzeichnen. Wie in Leber- und Nebennierenkulturen dieser Tiere gezeigt werden konnte, führten ioniserende Strahlen dosisabhängig zu einem Phasenvorsprung des circadianen Uhrwerks.
Die Ergebnisse erlauben die Schlussfolgerung, dass ionisierende Strahlen das circadiane Uhrwerk verstellen, Proliferation und Apoptose in gesundem Lebergewebe jedoch kaum beeinflussen.
The goal of this project is to develop a framework for a cell that takes in consideration its internal structure, using an agent-based approach. In this framework, a cell was simulated as many sub-particles interacting to each other. This sub-particles can, in principle, represent any internal structure from the cell (organelles, etc). In the model discussed here, two types of sub-particles were used: membrane sub-particles and cytosolic elements. A kinetic and dynamic Delaunay triangulation was used in order to define the neighborhood relations between the sub-particles. However, it was soon noted that the relations defined by the Delaunay triangulation were not suitable to define the interactions between membrane sub-particles. The cell membrane is a lipid bilayer, and does not present any long range interactions between their sub-particles. This means that the membrane particles should not be able to interact in a long range. Instead, their interactions should be confined to the two-dimensional surface supposedly formed by the membrane. A method to select, from the original three-dimensional triangulations, connections restricted to the two-dimensional surface formed by the cell membrane was then developed. The algorithm uses as starting point the three-dimensional Delaunay triangulation involving both internal and membrane sub-particles. From this triangulation, only the subset of connections between membrane sub-particles was considered. Since the cell is full of internal particles, the collection of the membrane particles' connections will resemble the surface to be obtained, even though it will still have many connections that do not belong to the restricted triangulation on the surface. This "thick surface" was called a quasi-surface. The following step was to refine the quasi-surface, cutting out some of the connections so that the ones left made a proper surface triangulation with the membrane points. For that, the quasi-surface was separated in clusters. Clusters are defined as areas on the quasi-surface that are not yet properly triangulated on a two-dimensional surface. Each of the clusters was then re-triangulated independently, using re-triangulation methods also developed during this work. The interactions between cytosolic elements was given by a Lennard-Jones potential, as well as the interactions between cytosolic elements and membrane particles. Between only membrane particles, the interactions were given by an elastic interaction. For each particle, the equation of motion was written. The algorithm chosen to solve the equations of motion was the Verlet algorithm. Since the cytosol can be approximated as a gel, it is reasonable to suppose that the sub-cellular particles are moving in an overdamped environment. Therefore, an overdamped approximation was used for all interactions. Additionally, an adaptive algorithm was used in order to define the size of the time step used in each interaction. After the method to re-triangulate the membrane points was implemented, the time needed to re-triangulate a single cluster was studied, followed by an analysis on how the time needed to re-triangulate each point in a cluster varied with the cluster size. The frequency of appearance for each cluster size was also compared, as this information is necessary to guarantee that the total time needed by to re-triangulate a cell is convergent. At last, the total time spent re-triangulating a surface was plotted, as well as a scaling for the total re-triangulation time with the variation. Even though there is still a lot to be done, the work presented here is an important step on the way to the main goal of this project: to create an agent-based framework that not only allows the simulation of any sub-cellular structure of interest but also provides meaningful interaction relations to particles belonging to the cell membrane.
Different numerical approaches and algorithms arising in the context of modelling of cellular tissue evolution are discussed in this thesis. Being suited in particular to off-lattice agent-based models, the numerical tool of three-dimensional weighted kinetic and dynamic Delaunay triangulations is introduced and discussed for its applicability to adjacency detection. As there exists no implementation of a code that incorporates all necessary features for tissue modelling, algorithms for incremental insertion or deletion of points in Delaunay triangulations and the restoration of the Delaunay property for triangulations of moving point sets are introduced. In addition, the numerical solution of reaction-diffusion equations and their connection to agent-based cell tissue simulations is discussed. In order to demonstrate the applicability of the numerical algorithms, biological problems are studied for different model systems: For multicellular tumour spheroids, the weighted Delaunay triangulation provides a great advantage for adjacency detection, but due to the large cell numbers the model used for the cell-cell interaction has to be simplified to allow for a numerical solution. The agent-based model reproduces macroscopic experimental signatures, but some parameters cannot be fixed with the data available. A much simpler, but in key properties analogous, continuum model based on reaction-diffusion equations is likewise capable of reproducing the experimental data. Both modelling approaches make differing predictions on non-quantified experimental signatures. In the case of the epidermis, a smaller system is considered which enables a more complete treatment of the equations of motion. In particular, a control mechanism of cell proliferation is analysed. Simple assumptions suffice to explain the flow equilibrium observed in the epidermis. In addition, the effect of adhesion on the survival chances of cancerous cells is studied. For some regions in parameter space, stochastic effects may completely alter the outcome. The findings stress the need of establishing a defined experimental model to fix the unknown model parameters and to rule out further models.