Analysis and control of complex
growth phenomena in physics and
biology
von Diplom-Physiker Diplom-Ingenieur (BA)
Michael Block
aus Waren/Müritz
Von der Fakultät II – Mathematik und Naturwissenschaften
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften,
Doctor rerum naturalium
genehmigte Dissertation
Promotionsausschuß:
Vorsitzender: Prof. Dr. E. Sedlmayr
Berichter/Gutachter: Prof. Dr. E. Schöll, PhD.
Berichter/Gutachter: Prof. Dr. S. Hess
zusätzlicher Gutachter: Dr. D. Drasdo
Tag der wissenschaftlichen Aussprache: 5. März 2007
Berlin 2007
D 83
Abstract
Pattern formation and the coarsening of growing surfaces have attracted wide interest
in scientific research during the last few decades.
Current fields of interest include not only the development of applications in nan-
otechnology combined with the fabrication of the corresponding microscopic struc-
tures but also the explanation of a wide range of biological growth processes.
In the area of nanotechnology there has, in the last decade, been particular inter-
est in the fabrication of quantum dots, because of the unique electronic and optical
properties of these "zero-dimensional" objects. The concept of self-organization holds
the key to the effective and cheap fabrication of such structures. Obviously the fabri-
cation of devices on an atomic scale requires rigorous theoretical observations of the
underlying processes.
Other fields in which self-organized growth is of great interest are biology and
medicine, where the interdisciplinary findings of both physicists and mathematicians
are increasingly providing detailed explanations of biomedical processes at a micro-
scopic level. During the last few years in particular, the use of theoretical models to
observe the development of cell tissues is becoming more and more important for the
development of effective therapies in the treatment of cancer. The aim of the present
work is to make a contribution to understanding self-organized growth and to provide
the basis for a possible method of control.
We use two well established models.
First we describe epitaxial growth by means of stochastic differential equations in
order to manipulate the crystal growth process. To do this we solve various growth
equations and combine them with existing methods from control theory to provide a
time-delayed feedback. This leads to the theoretical description of in situ influences
on the evolution of roughness, where we focus in particular on the experimentally
important early phase.
In the second part of the work we use a kinetic Monte Carlo method to describe
the formation of cell tissues in in-vitro mono-layers. Using the findings of an off-
lattice model and the experimental observations of tumor cells, a simulation tool is
generated which enables one to observe the dynamics and morphology of real size cell
populations. This tool makes possible the detailed analysis of biologically relevant
processes and their impact on growth.
Zusammenfassung
Die wissenschaftliche Untersuchung der Strukturbildung durch Wachstumsprozesse
ist seit Jahrzehnten von immenser Bedeutung.
Sowohl die Entwicklung von Anwendungen in der Nanotechnologie verbunden
mit der Herstellung entsprechender kleinster Strukturen, als auch die Erklärung von
Wachstum in seinen verschiedenen Variationen in der Biologie sind aktuelle For-
schungsgebiete.
Im Bereich der Nanotechnologien hat sich innerhalb des letzten Jahrzehnts unter
anderem die Fabrikation von Quantenpunkten als eine führende Forschungsrichtung
etabliert, nicht zuletzt durch die sehr speziellen elektronischen und optischen Eigen-
schaften dieser null-dimensionalen Objekte". Die Herstellung von Strukturen auf der
atomaren Längenskala erfordert dabei entsprechendes theoretisches Verständnis der
grundlegenden Prozesse. Als sehr vielversprechender Ansatz für eine effektive und
kostengünstige Herstellung entprechender Halbleiterstrukturen hat sich das Ausnut-
zen von selbstorganisiertem Wachstum herausgestellt.
Ein weiterer Bereich, in dem selbstorganisiertesWachstum eine grosse Rolle spielt,
ist die Biologie und Medizin, wobei zunehmend Kenntnisse aus der Physik und Ma-
thematik interdisziplinär kombiniert werden,um biologisch-medizinischeProzesse de-
tailliert zu beschreiben. Insbesondere das Verständnis der Entstehung von Zellgewebe
gewann in denletzten Jahren immer grössere Bedeutung für die Entwicklung effektiver
Therapien in der Krebsforschung.
Ziel der vorliegenden Arbeit ist es, einen Beitrag zum Verständnis von selbstorga-
nisierten Wachstumsprozessen zu leisten und einen Ansatz für eine mögliche Kontrolle
dieser zu erarbeiten.
Dazu werden in den Untersuchungen zwei etablierte Modelle genutzt. Zum einen
wird das epitaktische Wachstum mit Hilfe stochastischer Differentialgleichungen be-
schrieben, um anschliessend eine Anwendung zur gezielten Beeinflussung von Kri-
stallwachstum theoretisch herzuleiten. Dazu werden verschiedene bekannte Wachs-
tumsgleichungen numerisch gelöst und anschliessend die aus der Kontrolltheorie be-
kannte Methode der zeitverzögerten Rückkopplung in die Gleichungen eingeführt.
Dies führt zu einer theoretischen Beschreibung einer ’in situ’ Einflussnahme auf die
Rauigkeitsentwicklung, wobei besonderes Augenmerk auf die für Experimente wich-
tige Anfangsphase gelegt wurde.
Im zweiten Teil der Arbeit verwenden wir eine kinetische Monte-Carlo-Methode,
um die Bildung von Zellpopulationen in in-vitro Monolayern zu beschreiben. Auf der
Basis eines off-lattice Modells und von experimentellen Untersuchungen zu Tumor-
zellpopulationen wurde eine Simulation erstellt, mit der sich realistische Populations-
grössen hinsichtlich der Dynamik und der resultierenden Morphologie beschreiben
lassen. Dabei können im Modell gezielt verschiedene biologisch relevante Prozesse in
ihrem Einfluss untersucht werden.
Acknowledgments
First of all, it is a pleasure to thank Prof. Eckehard Schöll for his valuable advice and
support and for providing the challenging, interesting, and exciting topic of this thesis.
Cooperation plays a vital role in scientific research.
The part of this work concerning the model for cell population evolution was de-
voloped within a strong cooperation. For that I would like to thank Dr. Dirk Drasdo
from the Interdisciplinary Centre for Bioinformatics (IZBI) Leipzig.
For very intensive and helpful discussions concerning the part of the time-delayed
feedback control of stochastic differential equations I would like to thank Prof. Beate
Schmittmann from the Virginia Polytechnic Institute and State University Blacksburg
(USA).
Very fruitful and stimulating discussions with the members of the theoretical de-
partment of the Institute of Crystal Growth (IKZ) are acknowledged. In particular I
would like to thank Dr. Torsten Boeck, Dr. Thomas Teubner, and Dr. Wolfram Miller.
Some very helpful discussions with friends are acknowledged. Where Dr. Igor
Bjelakovic, Maika Felten, Dr. Gabriel Range, and Dr. Dirk Woywodpartly contributed
to the physics part, I thank Prof. Jochen Hühn from the German Rheumatism Research
Centre (DRFZ) in Berlin for very helpful discussions about the biological part of this
work. Where I had to enlarge my knowledgeabout cell biology, he had to act somehow
as a personal biologist.
For the nice working atmosphere in the group of Prof. Dr. E. Schöll I would
like to thank all the members, in particular Dr. Frank Elsholz, Pillip Hövel, Johanne
Hizanidis, Dr. Gerold Kiesslich, Dr. Roland Kunert, and Dr. Kathy Lüdge.
The Language support by Penny Salter is gratefully acknowledged.
Support in all situations outside this work and thus also support for the work was
given by my girlfriend Jana Zastrow. I am deeply grateful for that.
Last, but not least I would like to express warmest thanks to my mom Bürgny
Block.
List of publications
Parts of this work were already published as
•M. Block, R. Kunert, E. Schöll, T. Boeck, and Th. Teubner: “Kinetic Monte
Carlo simulation offormationof microstructuresin liquid droplet”. New Journal
of Physics 6, 166 (2004).
•M. Block and E. Schöll: “Adjusting surface roughness in growth processes by
time delayed feedback control”. In: Proceedings of the 28th International Con-
ference on the Physics of Semiconductors (ICPS-26), Vienna (2006).
•D. Drasdo, S. Hoehme and M. Block: “On the role of physics in the growth and
pattern formation of multi-cellular systems: What can we learn from individual-
cell based models?”. (2006) (accepted for publication in Journal of Statistical
Physics).
•M. Block and E. Schöll: “Time Delayed Feedback Control in growth phenom-
ena”. Journal of Crystal Growth (2006) doi:10.1016/j.jcrysgro.2006.10.254.
•M. Block, E. Schöll and D. Drasdo: “Classifying the expansion kinetics and
critical surface dynamics of growing cell populations”. (2006) (submitted to
Physical Review Letters).
Preface
If you asked fifty people of various ages what were the most important advances in
technology from the last few years, you would get many different answers. Some
would say computers, some the internet, photographers would say the digital camera,
business people might favour the mobile phone and children play stations.
But what most of the answers would undoubtedly have in common would be a
relationship to the miniaturization and optimization of electronic or optical devices.
Based on the answers one could say that nanotechnology is one of the most important
technological advance in recent years.
Since 2002 there has been a website for very smallscale images where the nanopic-
ture of the day is chosen 1. A lot of recent investigations into small scale science are
presented and there are also some futuristic speculations about the direction of nan-
otechnologies.
In Fig. 1 we can see one of the
Figure 1: Nanobot destroys a faulty red blood
cell [Mav03].
possible future applications of this
technology. The ’nanobot’ is to be
constructed to help doctors destroy
unwanted cells. As the authors say,
’this image portrays a tiny, nanome-
ter sized, fullyfunctional autonomous
robot helping to destroy a faulty red
blood cell.’ When we think about
the construction of such a nanobot,
we need to consider the problems
involved. One major goal is to solve
the problem of the materials needed
for the electronic devices in such a
small robot, where length scales are
of the order of atoms. Biological behaviour, on the other hand, is explained using
length scales of the order of biological cells. The solutionis going to involvenanoscale
work from a lot of different scientific fields. One could be forgiven for thinking that
such a robot is either impossible or will take the whole century to construct, but in fact
science is already beginning to solve the first part of the problem. One of the scientific
fields involved is the explanation of the properties of materials on an atomic scale and,
1http://www.nanopicoftheday.org
xii
of course, the development of the necessary experimental observational methods (see
Fig. 2).
In this thesis the reader will find in-
Figure 2: Chromosome image: scanning
force microscopy image [McM94].
vestigations into two specific systems
where the concept of self-organization
generates the kind of growth system we
want to explain by statistical methods.
First we focus on the formation of struc-
tures during the spatio-temporal evolu-
tion of the roughening surface. One of
the key questions in crystal growth to-
day is the problem of fabricating the
surface in a specified way, but cheaply.
Epitaxial growth is a well-established
method of preparing crystals where self-
organization plays a big role. The theo-
retical investigation of a possible in-situ influence on the growth process could there-
fore be very helpful. In the second part the reader will find a very different system, that
of tumor cell populations in in-vitro mono-layers. Although the length scale is totally
different in the two parts, the reader will find a lot of similarities. Both are growth
systems with their own self-organization and it was found that similar concepts can be
used to model the two systems. So, coming back to the nanobot, our work contributes
in a small way to the solution of the problem: both to the preparation of small scale ap-
plications and to the explanation of biological tissues, the aim being, of course, to find
effective methods of tackling tumor cells. Earlier we stated that work on a nanoscale
is a new field of interest for science, but nanotechnology was, of course, used by the
ancient Greeks, as Walter et al showed in their findings [Wal06]. A 2000-year-old
recipe for hair dye shows that they had a method of permanently colouring grey hair
black. Basically this method works by biologically inducing the growth of nanocrys-
tals. Presumably the ancient Greeks neither knew why their method worked nor could
explain the growth of nanocrystals. Nevertheless, these findings could lead to new
methods of growing nanocrystals, where the challenge will be much greater than that
of dying hair black.
Contents
Abstract iii
Zusammenfassung v
Acknowledgments vii
List of publications ix
Preface xi
1 Introduction 1
1.1 Crystal growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Tumor growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Structure of this work . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Crystal growth 7
2.1 Epitaxial Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Growth modes . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Processes in epitaxial growth . . . . . . . . . . . . . . . . . . 8
2.2 Methods in epitaxial growth . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Molecular Beam Epitaxy . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Metall Organic Chemical Vapour Deposition . . . . . . . . . 10
2.2.3 Liquid Phase Epitaxy . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Other methods in crystal growth . . . . . . . . . . . . . . . . . . . . 11
2.3.1 Czochralski growth . . . . . . . . . . . . . . . . . . . . . . . 11
3 The biology of tumor growth 13
3.1 The biology of the cell . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.1 The structure of an individual cell . . . . . . . . . . . . . . . 16
3.1.2 The cytoskeleton . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.3 The cell cycle . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.4 Cell types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1.5 Cell migration . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.6 Apoptosis and necrosis . . . . . . . . . . . . . . . . . . . . . 20
xiv Contents
3.2 Biology of cell populations . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 Extracellular matrix . . . . . . . . . . . . . . . . . . . . . . 21
3.2.2 Cell junctions . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.3 Cell adhesion . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Carcinogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Tumor cell types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 In vitro experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Modeling growth phenomena 25
4.1 Get the right view . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1.1 The microscopic view . . . . . . . . . . . . . . . . . . . . . 26
4.1.2 The macroscopic view . . . . . . . . . . . . . . . . . . . . . 26
4.1.3 The mesoscopic view . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Scaling theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2.1 Concept of self-similarity and self-affinity . . . . . . . . . . . 27
4.2.2 Roughening and scaling in growth systems . . . . . . . . . . 27
4.3 Lattice approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.4 General methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4.1 Monte Carlo approach . . . . . . . . . . . . . . . . . . . . . 32
4.4.2 Discrete models . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4.3 Stochastic SOS models . . . . . . . . . . . . . . . . . . . . . 36
4.4.4 Continuum Equations . . . . . . . . . . . . . . . . . . . . . 36
5 Stochastic Differential Equations 39
5.1 The Edwards-Wilkinson and the Kardar-Parisi-Zhang Equation . . . . 40
5.1.1 The Edwards-Wilkinson equation . . . . . . . . . . . . . . . 40
5.1.2 The Kardar-Parisi-Zhang Equation . . . . . . . . . . . . . . . 42
5.1.3 Relations beetween EW and KPZ equation . . . . . . . . . . 43
5.2 The Molecular Beam Epitaxy Equation . . . . . . . . . . . . . . . . 44
5.3 Crystal growth and stochastic differential equations . . . . . . . . . . 46
5.3.1 Observations by Raible et al . . . . . . . . . . . . . . . . . . 47
5.4 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.4.1 Numerical scheme of solving the growth equations . . . . . . 49
5.4.2 Discretization scheme . . . . . . . . . . . . . . . . . . . . . 50
5.4.3 Determination of the critical exponents . . . . . . . . . . . . 51
6 Control of stochastic differential equations 55
6.1 Control theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.1.1 Classical control methoods . . . . . . . . . . . . . . . . . . . 57
6.1.2 Chaos control . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.2 Control in this work . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2.1 Control variable . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2.2 Time delay . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Contents xv
6.2.3 Scheme of control . . . . . . . . . . . . . . . . . . . . . . . 60
6.2.4 Relation to control methods . . . . . . . . . . . . . . . . . . 63
7 Simulating Stochastic Differential Equations 65
7.1 The Kardar-Parisi-Zhang Equation . . . . . . . . . . . . . . . . . . . 65
7.1.1 The uncontrolled equation in 1+1 dimensions . . . . . . . . . 65
7.1.2 Definition of parameters for the control . . . . . . . . . . . . 68
7.1.3 Control of the KPZ equation in 1+1 dimensions . . . . . . . . 71
7.1.4 The uncontrolled equation in 2+1 dimensions . . . . . . . . . 86
7.1.5 With control in 2+1 dimensions . . . . . . . . . . . . . . . . 88
7.2 The MBE Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.2.1 Without control in 1+1 dimensions . . . . . . . . . . . . . . . 94
7.2.2 With control in 1+1 dimensions . . . . . . . . . . . . . . . . 97
7.2.3 Without control in 2+1 dimensions . . . . . . . . . . . . . . . 100
7.2.4 With control in 2+1 dimensions . . . . . . . . . . . . . . . . 104
7.3 Summary for the control of the growth equations . . . . . . . . . . . 111
7.3.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.3.2 Other control schemes . . . . . . . . . . . . . . . . . . . . . 112
7.3.3 Other equations . . . . . . . . . . . . . . . . . . . . . . . . . 113
8 The Model for the evolution of cell populations 115
8.1 Experiment and Off-lattice model . . . . . . . . . . . . . . . . . . . 115
8.1.1 Experiments by Bru et al . . . . . . . . . . . . . . . . . . . . 115
8.1.2 The off lattice model . . . . . . . . . . . . . . . . . . . . . . 116
8.2 The Dirichlet lattice construction . . . . . . . . . . . . . . . . . . . . 116
8.2.1 Voronoi diagrams and Delauney triangulation . . . . . . . . . 117
8.2.2 The construction in our model . . . . . . . . . . . . . . . . . 118
8.3 Modeling the basic processes . . . . . . . . . . . . . . . . . . . . . . 119
8.3.1 Cell division . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.3.2 Cell migration . . . . . . . . . . . . . . . . . . . . . . . . . 122
8.3.3 Apoptosis of cells . . . . . . . . . . . . . . . . . . . . . . . . 123
8.3.4 Necrosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.3.5 Mutations and fluctuations . . . . . . . . . . . . . . . . . . . 124
8.4 The Kinetic Monte Carlo method . . . . . . . . . . . . . . . . . . . . 125
8.5 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
9 Simulations of the evolution of cell populations 129
9.1 Lattice artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
9.2 Cell area distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.3 Proof of cell cycle time distributions . . . . . . . . . . . . . . . . . . 134
9.4 Expansion kinetics of cell populations . . . . . . . . . . . . . . . . . 135
9.4.1 General expansion . . . . . . . . . . . . . . . . . . . . . . . 135
9.4.2 Influence of the proliferating rim . . . . . . . . . . . . . . . . 136
xvi Contents
9.4.3 Influence of free migration . . . . . . . . . . . . . . . . . . . 136
9.4.4 Systematic parameter variation . . . . . . . . . . . . . . . . . 137
9.4.5 Proliferating rim . . . . . . . . . . . . . . . . . . . . . . . . 139
9.5 Comparison with experiments . . . . . . . . . . . . . . . . . . . . . 140
9.6 Cell density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
9.7 Surface dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.8 Apoptosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.8.1 Apoptosis with constant probability . . . . . . . . . . . . . . 147
9.8.2 Apoptosis with mutations . . . . . . . . . . . . . . . . . . . 148
9.9 Mutations of the cell cycle . . . . . . . . . . . . . . . . . . . . . . . 150
9.9.1 Global fluctuations . . . . . . . . . . . . . . . . . . . . . . . 151
9.10 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 152
9.10.1 Limited mutation of the cell cycle . . . . . . . . . . . . . . . 154
9.10.2 Correlated global fluctuations . . . . . . . . . . . . . . . . . 155
9.10.3 Different rules for division . . . . . . . . . . . . . . . . . . . 155
9.10.4 Different rules for migration . . . . . . . . . . . . . . . . . . 157
10 Conclusions and Outlook 159
A Simulations of stochastic growth equations 163
A.1 Additional simulations KPZ 1+1 . . . . . . . . . . . . . . . . . . . . 163
A.2 Additional simulations KPZ 2+1 . . . . . . . . . . . . . . . . . . . . 170
A.3 Additional simulations MBE 1+1 . . . . . . . . . . . . . . . . . . . . 171
A.4 Noisy Kuramoto-Sivashinsky equation . . . . . . . . . . . . . . . . . 173
B Deposition models 175
B.1 Ballistic deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
B.2 Random deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
C Simulation tool for the tumor model 177
C.1 Short manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
List of Figures 179
Bibliography 183
Chapter 1
Introduction
Key words like miniaturization,nano,lab on chip are connected to some of the major
challenges in science today, those involving the understanding of very small scale pro-
cesses down to the atomic scale and the development of applications which work on
that scale.
Obviously one important part in any kind of theoretical work concerning small
scale processes is that of understanding the formation of structures and the interplay
of the related particles. A lot of physical, chemical and biological processes can be
described as the spatio-temporal evolution of a system which we can explain as a kind
of a growth process. The wide range of growth phenomena exhibit very different
structures where we can see either very symmetric well-defined structures with for
instance circular symmetries like the snow flake crystals (Fig. 1.1), or totally different
structures like for example the fire front of a burning sheet of paper, where there is
a linear interface, or the growth of trees or the pattern formation on a snail’s shell
(Fig. 1.2 and Fig. 1.3).
Thus the consideration of growing systems in their spatio-temporal development is
an increasingly important field of interest in science, where the accurate description of
Figure 1.1: Snow crystals: capturing
snow flakes for observation with the low
temperature scanning electron microscope,
Wergin, W. P. and E. F. Erbe Electron
Microscopy Laboratory, Agricultural Re-
search Service, U.S. Department of Agri-
culture, Beltsville, MD 20705 USA 1994.
2 1.1. Crystal growth
Figure 1.2: Sample for growth: tree grown
on the island Hiddensee.
Figure 1.3: Sample for growth: snail
Conus marmoreus and behind a sim-
ulation result of the model describing
the pattern formation due to fronts of
pigment reactions. [Mei87](see also
[Mei03a] for further examples) .
the basic processes can give an explanation of the similarities and differences between
various growth phenomenaand thereby abetterunderstanding ofthebasicmechanisms
in general. The microscopic picture can then lead to the description of macroscopic
behaviour. We have concentrated here on two different kinds of growth, the roughening
of films in epitaxial crystal growth and the formation of cell populations in an in-vitro
environment. We shall now give short introductions to both.
1.1 Crystal growth
Current scientific work on crystal growth is focused on obtaining better electronic and
opto-electronic devices. Major tasks are the development of better memory chips and
effective solar cells together with the optimal miniaturization of these devices using
the newly discovered properties of materials. The fabrication of such devices was de-
veloped during the last decade on a truly atomic scale with nanocrystals, quantum dots
and quantum wells. One application was to lasers. Figure 1.4 shows examples of the
wide range of different crystal structures grown by epitaxial methods. It is obviously
not only essential to consider the properties of the devices prepared but also the related
growth processes needed for their fabrication. A wide branch of experimental meth-
Chapter 1. Introduction 3
Figure 1.4: Examples for crystal growth: (a) cross-section of an Indium droplet with
a Silicon nanocrystallite inside grown by Liquid Phase Epitaxy (LPE) [Boe99; Blo04],
(b) ’forest’ of ordered ’nanotrees’ grown by Metal Organic Vapor Phase Epitaxy
(MOVPE) [Dic04], (c) 14 ×14µm2AFM images of amorphous SiO2films after two
days deposition at a temperature of T= 611Kgrown by Chemical Vapor Deposition
(CVD) [Oje03] (d) silicon carbide nanobouquet grown by CVD [Ho04].
ods exist for preparing materials with a well defined structure related to the desired
and expected properties. There are a lot of experimental methods for growing well de-
fined crystal structures on an atomic scale in different ways for specific applications.
Sometimes this involves the growth of highly defined structures like single quantum
dots or single crystals, but we are going to concentrate here on roughening surfaces for
film growth in experiments. Usually the experimental setups have to be tuned to get
a specific structure in thin film growth. A helpful tool would be an ’in-situ’ control-
4 1.2. Tumor growth
Figure 1.5: Statistics of cancer diseases in Europe, estimated mortality from cancer in
Europe and the European Union 2004,* No data for Europe for all the individual sites
due to limitations of coding scheme employed. [Boy04].
lable setup to check the structure during growth and then retune the conditions to get
a more precise structure without having to start the experiment again. Controlling the
roughening process would give one the opportunity of growing a surface with a tuned
amplitude of roughness and a tuned correlation function within a well defined time.
We would like to present a first contribution to that control process.
1.2 Tumor growth
The most widespread disease in industrial countries today is cancer. As Fig. 1.5 shows,
there are a lot of very different types of cancers killing a lot of people every year.
Therapies developed during the last decades to tackle this ’scourge of mankind’ often
have very strong side effects on the human body and are not always effective. Indeed
for some types of cancer there are, as yet, no effective therapies. The development of
Chapter 1. Introduction 5
effective methods for destroying tumor cells without affecting the surrounding healthy
tissue is one of the major challenges to science today.
Scientific work in the field of understanding the mechanisms which lead to these
diseases has become more and more interdisciplinary during the last decade. Physical
and mathematical methods have been applied to biology and medicine. The develop-
ment of computers has lead to important advances in diagnostics using, for instance,
the new image-processing methods. Physics and mathematics have also helped explain
the behaviour of individual cells and their behaviour in a growing structure.
Models developed to explain behaviour on the scale of one cell and of many cells
have given us more knowledge about the behaviour of the cells in an organism and
helped in the development of effective therapies.
1.3 Structure of this work
In this work we have developed theoretical models for growth and we use two different
methods. Whereas stochastic differential equations using a continuous height function
are applied to problems in crystal growth, we have developed a kinetic Monte Carlo
algorithm as an individual cell based model to explain the growth of in-vitro tumor cell
monolayers.
So, this work can be seen both as a further development of the research on epitaxial
growth studied in our group during the last years and as a new field of research. Our
group has been using kinetic Monte-Carlo methods for about 12 years now as part of
an extensive study to explain the growth of semiconductor structures. A second field
of interest is the control theory used in part of this work. Both the explanation of
biological structures and the study of stochastic growth equations are new.
In this thesis the methods applied to crystal growth are different from those applied
to tumor growth. We shall give an introduction to the experimental setup and the
processes leading to growth in Ch. 2 and Ch. 3 and explain our modelling methods.
In order to take these different systems into account, we have to think both about
how to define the modelling conditions, and about what to take into account when
developing an effective and useful model to answer our specific questions.
In this context, Ch. 4 can be seen as a short guide to model growth from the micro-
scopic to the macroscopic range where we show how our findings fit into this general
overview.
After these more general aspects of the work we then go further to the first model
and growth system type, the control of the stochastic differential equations. We give a
detailed explanation for the stochastic differential equations used together with a de-
tailed description of the related processes and their correspondence to epitaxial crystal
growth (Ch. 5).
A summary of data analysis as an essential part is given in Sec. 5.4 together with
the time-delayed feedback control schemes we discuss in the next chapter (Ch. 6). The
combination of these first findings leads to the results for controlled and uncontrolled
6 1.3. Structure of this work
equations in 1+1 and 2+1 dimensions (Ch. 7). A detailed variation of the parame-
ters for different growth equations enables us to propose further possible experimental
setups.
Whereas up to now we have been working with a continuous description of a grow-
ing system, we now change both the growth system and the method in order to consider
the spatio-temporal evolution of a biological system. A kinetic Monte Carlo method is
used for observing the tumor growth of in-vitro cultures. The following chapter then
gives a detailed description of the individual cell based cellular automaton model (CA)
on the unstructured lattice used in this work. Ch. 8 explains our model and demon-
strates the ability of the simulation tool to study very different detailed mechanisms
and processes of biological interest. Thus, this chapter can be seen in part as a manual
for the use of the simulation tool in further investigations. We then, using extensive
simulations, demonstrate the behaviour of our model and the intrinsic properties of the
lattice whereby its advantages in comparison to common lattice types can be seen. We
then proceed to show in detail the mechanisms and influences of cell properties on the
critical surface dynamics. Comparison to experimental results are made and the result
of these findings demonstrates our expectations for realistic biological systems (Ch. 9).
Chapter 2
Crystal growth
In this chapter we want to give a short introduction to the most common methods
of growing crystals. We give a detailed description of epitaxial growth, on which
we have focused our model, and we look at Czochralski growth as one example of
a different method. More detailed overviews of possible experimental methods are
given by Scheel and Fukuda [Sch03a] and Byrappa [Byr03] in their books (see also
[Wil88; Pam75; Zan88]), where a more general overview of the theory can be found
[Mic04; Pim98; Bar95].
2.1 Epitaxial Growth
The theory of continuum stochastic differential equations together with epitaxial
growth can be seen as one single major task in the scientific investigation of crys-
tal growth. Epitaxial growth is the targeted deposition of one type of material on a
substrate of the same type of material (homoepitaxial growth) or on a different ma-
terial (heteroepitaxial growth). Heteroepitaxial systems exhibit different properties
because strain effects due to lattice mismatches become important.
The first findings of Volmer and Weber [Vol26] lead to the macroscopic description
by Becker and Doering [Bec35] which is still, with a few additions, the major theory
describing the formation of nuclei in crystal growth (see also [Sch03a; Wil88]). A lot
of recent studies deal with epitaxial methods for fabricating specific structures on an
atomic length scale, where for instance one major goal is the use of quantum dots.
Where some techniques fabricate the dots ’manually’ by putting the individual atoms
in the desired position [Eag90], a lot of observations show that self-organized growth
is a much more efficient and elegant method of growing such nanostructures. Probably
the first self-organized island formation in a semiconductor material system, namely
InAs/GaAs, was observed in 1985 by Goldstein et al [Gol85]. Self-organized growth
was then extensively studied and developed, starting with the first quantum dots [e.g.
Mo90; Eag90] and the first quantum dot lasers were developed experimentally in 1994
8 2.1. Epitaxial Growth
Figure 2.1: Growth modes of epitaxial growth, (a) Frank-van der Merwe growth, (b)
Volmer-Weber growth, (c) Stranski-Krastanov growth.
at the Technische Universität Berlin in collaboration with Ioffe Physico-Technical In-
stitute St. Petersburg [Led95].
The theory has become more and more important for applications to information
and communication technology. Quantum dot arrays and multilayer systems of quan-
tum dots are of very great interest [Bim96; Spr00; Wan04], and theoretical investiga-
tions have helped to explain the opto-electronic properties of these devices.
2.1.1 Growth modes
Epitaxial growth is normally divided into three different modes, where the interfa-
cial free energy and the lattice mismatch determine the growth mode [Bim99; Mar87;
Shc04a]. Fig. 2.1showsthese differentmodes. Frank-van derMerwe growthischarac-
terized by layer-by-layer growthor a tendency to fill the individualmonolayers [Fra49]
(Fig. 2.1 (a)). In contrast Volmer-Weber growth is characterized by the formation of is-
land structures [Vol26] (Fig. 2.1 (b)). The Stranski-Krastanov mode, where a phase of
building a wetting layer is followed by a nucleation of islands, is an intermediate mode
[Str39] (Fig. 2.1 (c)). In lattice matched systems only Frank-van der Merwe or Volmer-
Weber growth can occur, whereas in lattice mismatched material systems growth in the
Stranski-Krastanov mode is more favourable because of strain relaxations [Eag90, and
references therein].
2.1.2 Processes in epitaxial growth
The growth process can be explained by different individual atomic processes, namely
deposition or desorption processes and diffusion processes. Sometimes the nucleation
of islands is referred to as another process, where the nucleation can be seen as just a
product of diffusion at the surface together with binding energies, which lead to island
growth.
In Fig. 2.2 we can see a scheme of the possible processes on the surface (green
arrows show the direction of the events). Where deposition is not explicitly shown we
see desorption (Fig. 2.2 (a)) from the surface and desorption from an island (Fig. 2.2
(d)). Fig. 2.2 (b),(c),(e) refer to different diffusion processes, which can be explained
by a specific probability pto diffuse.
Chapter 2. Crystal growth 9
Figure 2.2: Processes at the surface in epitaxial growth: (a) desorption from the sur-
face, (b) diffusion along an island, (c) edge diffusion on an island, (d) desorption from
an island, (e) free diffusion.
If we assume that the atoms behave classically, the diffusion probability is expected
to follow Arrhenius law [Lai65]:
p=ν0exp(−E
kBT)(2.1)
where ν0is the so called attempt frequency, Eis the energy barrier for diffusion be-
tween the two states defined by the process, kBis Boltzmann’s constant and Tdenotes
the temperature.
Depending on the initial state and on the final state after diffusion we distinguish
here between free diffusion (Fig. 2.2 (e)), diffusion along an island (Fig. 2.2 (b)) and
edge diffusion (Fig. 2.2 (c)). In Eq. (2.1) these different types of diffusion refer to
different energy barriers E. We do not make use of this theory for the stochastic
differential equations but explain Arrhenius law in more detail for cell-cell adhesion in
the tumor growth model (Sec. 8.3.2).
10 2.2. Methods in epitaxial growth
2.2 Methods in epitaxial growth
A lot of different techniques exist for making semiconductor structures using epitaxial
methods.
2.2.1 Molecular Beam Epitaxy
As one of the leading techniques in the fabrication of crystals, Molecular Beam Epi-
taxy (MBE) offers the possibility of growing structures under well defined conditions
[Fra03; Shc04b]. This method deals with growth on a surface resulting from the con-
densation of single atoms or molecules out of the gas phase. The atomic source beams
come from the material, which is heated in evaporation cells. Mechanical shutters can
interrupt the atomic beam efficiently, so that it is possible to control the deposit of less
than one atomic layer. Ultra High Vacuum (UHV, ≈10−11 torr) conditions prevent
the incorporation of impurities and ensure that atoms and molecules follow a collision
free path towards the substrate. Most MBE systems are equipped with several in-situ
monitoring and analysis devices. These could be a mass analyzer, a Reflection High
Energy Electron Diffraction (RHEED), an Auger Electron Spectroscopy (AES) and/or
others. For detailed descriptions of MBE methods and instruments for the analysis of
systems grown by MBE, see the books by Parker [Par85] and Farrow [Far95].
2.2.2 Metall Organic Chemical Vapour Deposition
Chemical vapour deposition (CVD) is used for the deposition of thin films of various
materials. In a typical CVD process the substrate is exposed to one or more volatile
precursors, which react and/or decompose on the substrate surface to produce the de-
sired deposit. Volatile by-products are frequently produced too and are removed by gas
flow through the reaction chamber. CVD is used for a wide range of material systems,
for instance Si02,Ge/Si and TiN. The CVD method can be dividedinto a wide range
of slightly different methods. One kind of chemical vapour deposition is Metalorganic
Chemical Vapour Deposition (MOCVD). From the point of view of industrial prepa-
ration, MOCVD or Metalorganic Vapour Phase Epitaxy (MOVPE) has the advantage
that the source material can be provided continuously [Moo96]. The disadvantages, on
the other hand, are the complicated chemical processes and reactions that take place
before and during deposition in the gas phase. While UHV monitoring techniques can
not be applied because of the moderate pressure used in MOVPE systems, other in-situ
techniques, such as reflectance anisotropy spectroscopy or spectroscopic ellipsometry
[Ste96], are commonly used.
2.2.3 Liquid Phase Epitaxy
In contrast to the other methods, Liquid Phase Epitaxy (LPE) is a method of growing
semiconductor crystal layers from a melt on solid substrates. This happens at tempera-
Chapter 2. Crystal growth 11
tures well below the melting point of the deposited semiconductor. The semiconductor
is dissolved in the melt of another material. At conditions that are close to the equilib-
rium between dissolution and deposition the deposition of the semiconductor crystal
on the substrate is slow and uniform. The equilibrium conditions depend very much on
the temperature and on the concentration of the dissolved semiconductor in the melt.
The growth of the layer from the liquid phase can be controlled by a forced cooling of
the melt. Impurity introduction can be strongly reduced. Doping can be achieved by
the addition of dopants. For one special system, in which Liquid Phase Epitaxy is used
to fabricate silicon crystals inside indium droplets, see [Boe99; Blo04].
2.3 Other methods in crystal growth
Having explained the physical properties of epitaxial systems and the related experi-
mental methods, we would now like to refer briefly to another leading crystal growing
technique, Czochralski growth.
2.3.1 Czochralski growth
Czochralski growth is named after Jan Czochralski, who discovered the method in
1916. A seed crystal, mounted on a rod, is dipped into molten silicon. The seed crys-
tal’s rod is pulled upwards and rotated at the same time. By controlling the temperature
gradients, rate of pulling and speed of rotation precisely, it is possible to extract a large,
single-crystal, cylindrical ingot from the melt. This process is normally performed in
an inert atmosphere, such as argon, and in an inert chamber, such as quartz. While
the largest silicon ingots produced today are 400 mm in diameter and 1to 2meters in
length, 200 mm and 300 mm diameter crystals are the standard industrial size. Thin
silicon wafers cut from these ingots (typically about 0.75 mm thick) and polished to a
very high degree of flatness are used for creating integrated circuits. Other semicon-
ductors, such as gallium arsenide, can also be grown by this method, although in this
case lower defect densities are obtained. So, this method offers a precise fabrication
of semiconductor devices by a totally different method. For a detailed description of
some other methods we refer to the already mentioned books ([Sch03a; Byr03]).
12 2.3. Other methods in crystal growth
Chapter 3
The biology of tumor growth
The aim of cell biology is to understand the determining processes in nature in general
and to describe the mechanisms and actions at a cellular level in particular. Early
work on cell biology tried to observe the behaviour of cells as a kind of rough view
of phenomenological behaviour. However, as medicine and biology have developed,
scientific investigations have been going deeper and deeper into the detailed structure
of the human body and of course into the details of cell biology.
Exploring the details of cell structure and mechanisms requires a description of chem-
ical and physical actions on the cellular level. During the last decades the whole field
has become more and more interdisciplinary, and physical biology, mathematical biol-
ogy and bioinformatics are nowadays well established scientific fields.
Of course the development of better microscopes has opened up a new world of obser-
vations and helped us to understand what is happening on the microscopic level right
down to the molecular length scale. Increasing knowledge of the cellular structure has
also generated more interest in exploring the basic mechanisms of cell growth, the aim
being to help biologistsand doctors understand cell biologyin general and in particular
to find effective new therapies.
Until the mid-seventeenth century, scientists were unaware that cells even existed.
Probably the first observations of cell biology were made by Robert Hooke, which
he described in his ’Micrographia’ in 1665. Through his microscope he saw that plant
tissues were divided into tiny compartments. He termed them ’cellulae’, which is the
Latin word for the small rooms of monks. About 200 years later scientists really began
to understand the importance of these findings, when Jakob Schleiden and Theodor
Schwamm found similarities between animal and plant cells and deduced that all liv-
ing things are made up of cells.
Nowadays cell biology makes use of modern microscopes to observe the molecular
structure of cells and a lot of mechanisms are now well understood. Since the 80’s the
major goal for cell biologyhas been to explain the developmentalprocesses, where cell
changes and grow. With new apparatuses and the development of computer science,
data analysis had a big role to play. A major step was announced in the November
6, 1998 in the Washington Post : "Scientists announced yesterday they had achieved
14 3.1. The biology of the cell
one of the most coveted goals in biology by isolating from human embryos and foe-
tuses a primitive kind of cell that can grow into every kind of human tissue, including
muscle, bone and brain." 1Gearhart [Gea98] and Thomson et al [Tho98] had isolated
embryonic stem cells. The major breakthrough of this work
Figure 3.1: Isolation of embryonic stem
cells [Tho98].
was the fact that one could now explain
how so many kinds of different cells can
develop from only a few cells to form an
individuum. A very important field of
research today is the explanation of the
uncontrolled growth of cells or their un-
controlled division, which is currently a
very important disease, cancer. Cancer
as a Latin word comes from the Greek
’karkinos’ which means a crayfish or a
crab, maybe because of the image of
a destructive crab in the human body.
’karkinos’ is also the origin of the word carcinoma, which means the cancer cells.
A similarly used word is tumor, meaning a medical excrescence that may be either
malignant or benign. The differences between the uses of these words will now be
explained. Tumor cells in general are so called because of two basic properties; their
uncontrolled reproduction and their invasion and colonization of territories reserved
for other cells. As long as the growing tumor or neoplasm is clustered in a single mass
the tumor is said to be benign. When the tumor cells become invasive and occupy
surrounding tissue or gain access to the blood stream to form secondary tumors, or
metastases, the tumor is malignant and in this case the tumors are also called cancer.
However these words are often used identically in the literature. In this chapter we are
going to give a short description of cell biology and tumor cells, very closely related to
our work. We are going to explain the main processes like the basic principles of cell
division and the structures inside and outside one individual cell in a cell population
but are not going to look into the cell on an atomic scale.
3.1 The biology of the cell
The cell is the structural and functional unit of all living organisms, and is therefore
also called the ’building block of life’. [Alb02]. Organisms are divided into unicel-
lular and multi-cellular types. Unicellular organisms consisting of a single cell are,
for instance, bacteria, whereas humans, with about 100 trillions of cells, obviously
belong to the multi-cellular group. A typical cell size is from 5to 30 µm in diame-
ter with typical masses around 1ng. Each cell is to some extent self-contained and
self-maintaining: it can take in nutrients, convert these nutrients into energy, carry out
specialized functions, and reproduce as necessary. Each cell stores its own set of in-
1http://www.washingtonpost.com/wp-srv/national/cell110698.htm
Chapter 3. The biology of tumor growth 15
Figure 3.2: View on length scales beetween living cells and atoms where each part
show an image magnified by a factor of ten from a thumb to a cluster of atoms part of
protein molecules, the scale, which our studies cover, are the image of 0.2mm and the
image of 20µm from cell population to an individual cell [Alb02].
structions for carrying out each of these activities. There are two basic kinds of cells,
prokaryotic and eukaryotic cells. Whereas eukaryotic cells keep their DNA in a dis-
tinct membrane-bounded intracellular compartment called the nucleus, the prokaryotes
have no such distinct nuclear compartment. Prokaryotes are normally small and often
live as unicellular organisms. According to one estimate, at least 99% of prokaryotic
species remain to be classified. A new classification of cells divides them into bac-
teria, achaea or archeabacteria and eukaryotes, where bacteria and archaea build the
prokaryote, but we don’t want to go into so much detail here (for more details see
[Alb02]). An individual cell is a very complex system and there is no place here to
describe all the details from the behaviour of the whole cell to the structure of DNA.
16 3.1. The biology of the cell
Figure 3.3: Schematic view of cells, left a typical eucaryotic cell, right a procaryotic
cell.
Fig. 3.2 shows the different lengthscales one can get by a look on cells. It is obviously
not possible to write an introduction to cell biology here.
We shall restrict ourselves here to some basic points related to the model we want
to construct later in this work. For very detailed descriptions we refer to the well
known and best compendium of molecular cell biology, ’The Cell’ by Alberts et al
[Alb02], where one can find not only an overview but a very detailed description of
everything related to an individual cell.
3.1.1 The structure of an individual cell
An individual cell consists of molecules from four major chemical families of organic
molecules, which are the important carbon based compounds, the sugars, the fatty
acids, the amino acids and the nucleotides. Linked into large macromolecules, these
compounds make up approximately 30% of the cell mass, where H2Ofills the re-
maining 70 %. Fig. 3.3 shows a schematic eukaryotic and a prokaryotic cell. Some
prokaryotic cells contain important internal membrane-bound compartments, but eu-
karyotic cells have a highly specialized endomembrane system characterized by reg-
ulated traffic and transport of vesicles. All cells, whether prokaryotic or eukaryotic,
have a membrane, which envelopes the cell, separates its interior from its environment,
regulates what moves in and out, and maintains the electric potential of the cell. Inside
the membrane, a salty cytoplasm takes up most of the cell volume. All cells possess
DNA, the hereditary material of genes, and RNA, containing the information needed
to build various proteins such as enzymes, the cell’s primary machinery.
3.1.2 The cytoskeleton
The cytoskeleton acts to organize and maintain the cell’s shape; it anchors organelles
in place, organizes the uptake of external materials by a cell, and cytokinesis, the
separation of daughter cells after cell division; and moves parts of the cell during
the processes of growth and mobility. The eukaryotic cytoskeleton is composed of
microfilaments, actin filaments and microtubules. There are a great number of proteins
associated with them, each controlling the cell’s structure by directing, bundling and
aligning filaments. Fig. 3.4 shows an experimental view of an eukaryotic cytoskeleton,
Chapter 3. The biology of tumor growth 17
Figure 3.4: Cytoskeleton: Actin filaments
are shown in red, microtubules in green,
and the nuclei are in blue.
where one can see the actin filaments (red), the microtubules (green) and the nuclei
(blue).
3.1.3 The cell cycle
’Where a cell arises, there must be a previous cell, just as animals can only arise from
animals and plants from plants’. Rudolf Virchow stated this ’cell doctrine’ in 1858.
Cell division is such that new cells can only come from existing cells. The ordered
sequence of such duplication and division is the cell cycle, the essential mechanism
for the reproduction of living cells. Cell division is the process by which hair, skin,
blood cells, and some internal organs are renewed. A specialized form of cell division
is responsible for cellular differentiation during embryogenesis and morphogenesis, as
well as for the maintenance of stem cells during adult life.
The cell cycle is specific to the cell type but there are some universalcharacteristics.
It consists of four distinct phases: G1 phase, S phase, G2 phase (collectivelyknown
as interphase) and M phase, which are schematically depicted in Fig. 3.5. The M phase
is itself composed of two tightly coupled processes: mitosis, in which the cell’s chro-
mosomes are divided between the two daughter cells (see Fig 3.6), and cytokinesis, in
which the cell’s cytoplasm divides physically. The S phase is characterized by DNA
duplication. The gap phases G1 and G2 are influenced by cell signallingand favourable
cell conditions, whereby the length can vary in a wide range for the cells. Cells that
have temporarily or reversibly stopped dividing are said to have entered a state of qui-
escence called G0 phase, while cells that have permanently stopped dividing due to
age or accumulated DNA damage are said to be senescent. In a typical human cell the
interphase I normally take 23 hours in a 24 hour cycle, whereas the M phase takes just
one hour. The molecular events that control the cell cycle are ordered and directional;
that is, each process occurs in a sequential fashion and it is impossible to ’reverse’ the
18 3.1. The biology of the cell
Figure 3.5: Scheme of the cell cycle: M
mitosis, G0, G1, G2 the gap phases, S the
synthesis phase, G0, G1, G2 and S build
the interphase.
Figure 3.6: Mitosis of a cell.
cycle. Regulatory molecules determine a cell’s progress through the cell cycle: cy-
clins and cyclin-dependent kinases. Leland H. Hartwell, R. Timothy Hunt, and Paul
M. Nurse won the 2001 Nobel Prize in Physiology or Medicine for their discovery of
these molecules which are central to the regulation of the cell cycle.
There has been a lot of work accorded to the cell cycle, but we don’t want to go
any further. As we will see in our model, the cell cycle is reduced to a one step event
in which the cell divides into two daughter cells. For a more detailed description of
the cell cycle see Alberts et al [Alb02] and references therein.
3.1.4 Cell types
The type determines the basic properties of a cell, so here we would like to give a short
description of how cells can be characterized as animal cells. There are basic types of
Chapter 3. The biology of tumor growth 19
tissue in the body of all animals and we are going to explain the most important types.
Epithelial tissue
Tissues composed of layers of cells that cover organ surfaces such as the surface of the
skin. The tissues serve for protection, secretion, and absorption.
Connective tissue
As the name suggests, connective tissue holds everything together. Blood is considered
to be a connective tissue. These tissues contain an extensive extra-cellular matrix.
Muscle tissue
Muscle cells contain contractile filaments that move past each other and change the
size of the cell. Muscle tissue also is separated into three distinct categories: visceral
or smooth muscle, which is found in the inner linings of organs; skeletal muscle, which
is found attached to bone in order for mobility to take place; and cardiac muscle which
is found in the heart.
Nervous tissue
Cells forming the brain, spinal cord and peripheral nervous system.
Areolar connective tissue
A pliable, mesh-like tissue with a fluid matrix whose function is to cushion and protect
body organs. There are also different types of tissues in plants.
3.1.5 Cell migration
Cell migration is the central process in the development and maintenance of multi-
cellular organisms. Tissue formation during embryonic development, wound healing
and immune responses all require the movement of specific cells in a particular di-
rection to a specific location. Errors during this process have serious consequences,
including mental retardation, vascular disease, rheumatoid arthritis, tumor formation
and metastasis. An understanding of the mechanisms by which cells migrate may lead
to the development of novel therapeutic strategies for controlling, for example, inva-
sive tumor cells. In animal tissues cells often migrate in response to, and towards,
specific external signals, a process called chemotaxis. For further information see
[Par99] [Lev06].
20 3.1. The biology of the cell
Figure 3.7: Scheme of programmed cell death (Apoptosis).
3.1.6 Apoptosis and necrosis
Apoptosis is also called programmed cell death. As such, it is the process of deliberate
life relinquishmentby a cell in a multi-cellularorganism. In contrast to necrosis, which
is a form of cell death that results from acute cellular injury, apoptosis is carried out in
an ordered process that generally confers advantages during an organism’s life cycle
[Ker72].
For example, the differentiation of human fingers in the developing embryo re-
quires the cells between the fingers to initiate apoptosis so that the fingers can separate.
Obviously such a mechanism must be well balanced, because too much apoptosis
causes cell-loss disorders, whereas too little results in uncontrolled cell proliferation,
namely cancerous tumors.
Apoptosis can occur, for instance, when a cell is damaged beyond repair, or in-
fected with a virus. The ’decision’ for apoptosis to occur can come from the cell itself,
from its surrounding tissue or from a cell that is part of the immune system. If a cell’s
capability for apoptosis is damaged (for example, by mutation), or if the initiation of
apoptosis is blocked (by a virus), a damaged cell can continue dividing without re-
strictions, developing into cancer. A cell undergoing apoptosis shows a characteristic
morphology which can be seen in Fig. 3.7.
The cell becomes circular. The chromatin then undergoes an initial degradation and
condensation. It then undergoes further condensation into compact patches against the
Chapter 3. The biology of tumor growth 21
nuclear envelope. At this stage, the double membrane that surrounds the nucleus still
appears complete. The nuclear envelope becomes discontinuous and the DNA inside it
fragments. The nucleus breaks into several discrete chromatin bodies or nucleosomal
units due to the degradation of DNA. The cell breaks apart into several vesicles called
apoptotic bodies, which are then phagocytosed.
3.2 Biology of cell populations
We now have the basic mechanisms of one individual cell. The next question is how
cells work together in a cell population. Cells are often motile and are normally de-
formable objects filled with some jelly-like medium, so some mechanisms must exist
to combine them to give them the strength of the human body. The mechanism is
similar to the game of breaking sticks one after the other or trying to break them all
together at the same time. Cells form cell-cell junctions, have cell-cell adhesions and
are connected by the extracellular matrix.
3.2.1 Extracellular matrix
A substantial part of cell tissues is normally extracellular space, which is largely filled
by a network of macromolecules constitutingthe extracellularmatrix. Produced by the
cells, this matrix composed of proteins and polysaccharides is organized in a mesh-
work in close association with the surface of the cell. The extracellular matrix in
connective tissues is extremely important for physical behaviour. It doesn’t determine
their behaviour but the properties of the epithelial cells depend on it.
3.2.2 Cell junctions
Cell junctions occur in all cell populations at the points of cell-cell or cell-matrix con-
tacts. They are normally classified into three groups. Occluding junctions seal cells
together in the epithelium in a way that prevents even small molecules from leaking
from one side of the epithelial sheet to the other. Anchoring junctions mechanically
attach cells and their cytoskeletons to their neighbours or to the extracellular matrix.
Communicating junctions mediate the passage of chemical or electrical signals from
one interacting cell to its partner.
3.2.3 Cell adhesion
The connection between junctions and adhesion is the fact that cells have to adhere
in order to build anchoring junctions. A bulky cytoskeletal apparatus must then be
assembled around the molecules that directly mediate the adhesion. This results in a
well-defined structure and different adhesions can be identified using the electron mi-
croscope. For example during the last decade there has been a lot of work on cadherins
22 3.3. Carcinogenesis
mediated Ca2+ -dependent cell-cell adhesions. The study of cell adhesion is part of
cell biology. Cells are often not found in isolation, but tend rather to stick to other
cells or to the non-cellular components of their environment. A fundamental question
is: what makes cells sticky? Cell adhesion generally involves protein molecules at the
surface of cells, so the study of cell adhesion involves cell adhesion proteins and the
molecules that they bind to.
3.3 Carcinogenesis
Cell division (proliferation) is a physiological process that occurs in almost all tissues
and under many circumstances. Normally homeostasis, the balance between prolifer-
ation and programmed cell death, usually in the form of apoptosis, is maintained by
the tight regulation of the processes. Carcinogenesis is caused by the mutation of the
genetic material of normal cells, which upsets the normal balance between prolifera-
tion and cell death. This results in uncontrolled cell division and tumor formation. The
uncontrolled and often rapid proliferation of cells can lead to benign tumors; some
types of these may turn into malignant tumors (cancer). More than one mutation is
necessary for carcinogenesis. In fact, a series of several mutations to certain classes of
genes is usually required before a normal cell is transformed into a cancer cell.
3.4 Tumor cell types
Cancers are generally classified according to the tissue and cell type from which they
arise. Tumor in medical language simply means swelling or lump, either neoplastic,
inflammatory or other. In common language, however, it is synonymous with ’neo-
plasm’, either benign or malignant. This is inaccurate, since some neoplasms do not
usually form tumors, for example leukaemia or carcinoma in situ.
Carcinoma
Tumor cells which arise from epithelial cells are carcinoma. Epithelial tissues are
well connected tissue divided into epithelial sheets. Cells are tightly bound and the
extracellular matrix consists of a thin mat called a basal lamina. So in carcinoma cells
are attached to each other by cell-cell adhesions.
Sarcoma
A sarcoma is a cancer of the connective or supportive tissue (bone, cartilage, fat, mus-
cle, blood vessels). The term comes from a Greek word meaning ’fleshy growth’.
Chapter 3. The biology of tumor growth 23
Figure 3.8: Scheme of an in vitro experiment with a petri dish where a solution of
nutrients (red) lead to a growth of a cell population monolayer.
Leukemias
Leukemia (or leukaemia; see spelling differences) is a cancer of the blood or of the
bone marrow characterized by an abnormal proliferation of blood cells, usually white
blood cells (leukocytes). It is one of the broad group of diseases called hematological
neoplasms.
3.5 In vitro experiments
All these cell types for tumor cell development have found a lot of interest in cell
biology, where the determination of possible explanations is feasible thanks to the
high resolution of modern microscopes which are able to distinguish the complexity
of the cells in a complex tissue. In vitro experiments have been established as a very
successful tool for studying the mechanisms of cells in a well defined environment
where the setup is such that unknown influences can, on the whole, be neglected. A
precise change of the properties of the cultured growth of cells therefore makes it
possible to study the basic mechanisms of cells in detail.
Fig. 3.8 gives an impression of a possible setup for in vitro experiments - of course
real apparatuses are much more complicated.
24 3.5. In vitro experiments
Chapter 4
Modeling growth phenomena
In the previous chapters we gave an overview of the systems we want to model. Ob-
viously it is impossible to include all the details of the real growth process in a useful
and effective model.
A model is, by definition, a simplification of reality, made in order to answer a
specific question about real behaviour. So of course the first very important part of the
work is to determine the limits and decide what assumptions have to be made, using
the questions we want to answer as guidelines. Basic tasks are the timescales, length
scales and the related methods.
In this chapter we follow this guideline in order to obtain the rules our representa-
tion have to fulfil. The method of building up the model may look obvious, but a closer
look at the systems we want to explain - and in our opinion all other growth systems -
shows that it is important and one of the first problems to solve. So this chapter may
be a help in the construction of growth models in general and we shall apply it later on
to crystal growth and cell population growth.
4.1 Get the right view
As shownin Fig. 4.1, different length scales explore very different views of the system,
where each is related to individual properties of the system. So posed in a slightly
different manner our question is, whether to look at the forest, at the individual tree or
at an individual leaf.
Here we want to describe many-particle systems in order to get results for the
dynamical behaviour of growth processes of as many particles as possible. On the
other hand we want to include details of the basic actions of the individual particles.
Thus, a well balanced description is required to ensure large scale simulations.
26 4.1. Get the right view
(a) (b) (c)
Figure 4.1: Zoom from macroscopic to microscopic view: Three different views on
the same problem, (a) forest, (b) tree (c) leaf.
4.1.1 The microscopic view
A first approach could be the explanation of the most detailed view. In the case of
particles such as atoms we go directly to the quantum mechanical potentials on the
surface. If we explain the cell as a complete system together with all the processes
inside one individualcell we rapidly come to the DNA structure and again to molecular
structures. All these processes are of great interest and importance, firstly for the
behaviour of individual particles and thereby also for collective behaviour.
However our questions are on a macroscopic scale. The microscopic view does
not help us with our problem if we do not want to derive a model that explains all the
processes of nature. But nevertheless findings from the microscopic view are essential
for our model in order for us to make suitable assumptions.
4.1.2 The macroscopic view
Another way of tackling the problem is to explain the system by a macroscopic view.
In case of atoms that view could be of the whole surface or in case of the cell the
grown cell population. But if we want to decide between processes which lead to this
behaviour and to model the growth itself this view seems too blunt.
4.1.3 The mesoscopic view
We need an approach that lies somewhere between the two previous approaches. This
is the mesoscopic view, where we don’t explain the structure of the particles but take
particles with known properties and see what happens when we model the processes.
So our scale for atomic behaviour is the lattice constant (∼0.5−1.0nm) and a
time scale of the order of milliseconds. For cells our scale starts with a cell size (∼
10 −40µm) and a time scale of hours to days (doubling time for cells 10 −30 hours).
Chapter 4. Modeling growth phenomena 27
4.2 Scaling theory
Having decided on our range for length scales, we are still left with the question of how
much information and how many assumptions to include. If one considers a surface
where in epitaxial growth some islands arise, an important question is whether we can
deduce the behaviour of the whole surface from looking at a small section of it. The
answer to that question is the concept of ’scaling’, where by measuring quantities on a
small section one can deduce values for bigger systems. A lot of growth systems show
such relations. In our case one can find scaling laws for the morphological structures
of the developing growth system.
4.2.1 Concept of self-similarity and self-affinity
As already explained, the basic concept in scaling approaches is the idea that one can
divide a big system into ’similar parts’. Similarity means that when one looks at two
maps with different magnifications and different measurements of a defined quantity,
they look similar. Mathematically that is either an isotropic transformation, in which
case the system is said to be self-similar, or anisotropic transformation, which defines a
self-affine system. We can then extrapolate from small parts of the system to behaviour
on a larger scale by using the scaling laws which we now want to introduce.
4.2.2 Roughening and scaling in growth systems
For surface growth the main quantities that describe the developmental processes are
the velocities and spatial dimensions of the system and its morphological structure.
Assuming either self-affinity or self-similarity we then measure the root mean square
(rms) surface roughness given by
wRMS(L, t) = v
u
u
t
1
L
L
X
i=1
[h(i, t)−h(t)]2(4.1)
where Lis the system-size, h(i, t)denotes the height function of the surface at the i-th
point at time t, and h(t)is its average.
It follows from this equation that the rms roughness describes the standard devia-
tion of the height function h(i, t). In Fig. 4.2 we can see an example of a rough surface.
Here the situations in 1+1 (1 spatial coord. + height) and in 2+1 (2 spatial coord. +
height) dimensions are depicted: In the middle a rough surface in 2+1 in terms of a
three dimensional height profile is shown, where the mean height is emphasized by the
green line. Under the profile a two dimensional projection is shown, where the height
increases from black to white colors, the plane on the left shows the same situation in
1+1 dimensions, where the height now depends on just one coordinate.
28 4.2. Scaling theory
Figure 4.2: Illustration of a rough surface: in the middle a 2+1 dimensional surface
with the height profile and the projected density plot of the height profile, on the left
the height function in 1+1 dimensions (red line), the mean height his shownby a green
line.
Figure 4.3: Typical temporal
evolutionof the root mean square
roughness wRMS (black line, ex-
ample taken form long-time sim-
ulations for the Molecular Beam
Epitaxy equation in Sec. 7.2),
blue dash-dotted line: the satura-
tion roughness, red dashed line:
the early phase with wRMS ∝tβ,
green dash-dotted line denotes
the crossover time tx(double-
logarithmic plot). 101102103104105
t (a.u.)
1
2
3
4
wRMS
tx
wsat
wRMS ~ tβ
If growth now starts from a flat surface with system size L, the system roughens.
In Fig. 4.3 we see a typical evolution divided into two phases, namely the roughening
and the saturation phase, divided by crossover time tx.
Chapter 4. Modeling growth phenomena 29
The early phase can be characterized by an exponent β, the so called growth expo-
nent, whereas the late phase can be explained by a roughness exponent α.
After a certain time, depending on the spatial dimension and system size, the
roughness saturates: The saturation roughness wsat has been reached (blue). The ex-
ponents are then defined by the following power laws:
wRMS(L, t)∝tβfor t≪tx(4.2)
wsat(L)∝Lαfor t≫tx(4.3)
where txis the crossover time between the two regimes of evolution (green). Fig. 4.3
shows this behaviour for the Molecular Beam Epitaxy equation in 2+1 dimensions.
We discuss this later in more detail (Sec. 7.2). Time tis in arbitrary units. We infer
the growth exponent β= 1/5from the simulated data, and then derive the roughness
exponent α= 2/3from the amplitudes of the saturation values. txand Lare linked by
a further power law.
tx∝Lz(4.4)
This third exponent z, the dynamic exponent, is not independent of αand β, as can
easily be checked using the Family-Vicsek scaling relation [Fam85],
wRMS(L, t)∝Lαft
Lz(4.5)
where fis the so-called scaling function and the exponents then obey the relation:
z=α
β(4.6)
The scaling exponents α,β(in ourcase theyare independent of one another) determine
the universality classes, which are then related to different kinds of growth. In general
these methods can be applied to a wide range of systems developing in time, wherever
one can define a height function and find self-similar growth in the system.
4.3 Lattice approaches
Once we knowthe relationship between the basic processes and the universality classes
we can describe the evolution of growth. For computer simulations of growth we obvi-
ously need a well defined underlying structure to work on. We now want to introduce
different approaches to defining it. In general there are two basic kinds, off-lattice
models and lattice models.
30 4.3. Lattice approaches
Off-lattice models
Off-lattice models are normally used to describe either the exact position of unstruc-
tured surfaces like glassy or amorphous materials in crystal growth or the changeable
or determined position of a cell in cell populations.
The question then arises as to whether the exact position is of crucial importance
in the model or whether one can use one’s a priori knowledge of an off-lattice model
to replace it by an effective model on a defined lattice. The choice here is between two
different explanatory systems. In the case of a crystal the best underlying structure is
given by the structure of the crystal itself, namely the discrete positions of the effective
atoms. The approximations for the processes are then in the choice of the method,
where one can either try to solve the many body quantum mechanical problems or
consider effective atoms and effective energies at the lattice points in order to simplify
the problem and move from the microscopic to the mesoscopic scale. The situation
changes if we then have an amorphous substrate where it is rather difficult to define
the lattice. For cell population growth the situation is totally different, because here the
possible positions are continuous, so a model which aims to reflect reality perfectly has
to be an off-lattice model. Here the lattice model is only the first approximation, which
not only fixes the position in some way but also restricts the overall area of the cells,
so that one can think about local changes. So these questions are again, as explained
in Sec. 4.1, the choice of including the microscopic view or staying on a mesoscopic
scale.
That question is obviously very important for computer experiments because work-
ing on an off-lattice model structure uses much more computer time. So, in the case
of both crystal growth and tumor growth off-lattice or detailed quantum mechanical
approaches are normally taken for small systems, whereas a coarse-grained approach
with lattices and without detailed solutions of the quantum mechanical wave functions
is successful for larger systems.
Lattice models
In most cases it is useful to take a well defined structure for large growth systems.
There are a lot of different very special lattice constructions and here we shall explain
the three most common.
First there is the square lattice or the cubic lattice which is also called the von
Neumann lattice, where every point is connected in two dimensions to the four neigh-
bors with equal xor yvalues. For a three dimensional structure every point then has
six neighbors (Fig. 4.4 (a)). The simplicity of this structure makes it easy to use in
computer experiments.
Depending on the structure, it can be useful for crystal growth to take a hexag-
onal lattice where every point has six neighbors in two dimensions and 12 in three
dimensions (Fig. 4.4 (b)).
Chapter 4. Modeling growth phenomena 31
Figure 4.4: Different lattice types: (a) von Neumann neighborhood, (b) hexagonal
lattice, (c) octogonal lattice Moore neighborhood with 8 neighbors, left the projections
to twodimensional systems with the neighbors, on the right are the neighbors in three
dimensions.
In a way similar to the cubic lattice one can also define the diagonals as neighbors
which leads to a so called Moore neighborhood with eight neighbors and 26 neighbors
in three dimensions (Fig. 4.4 (c)).
All of these structures are extensively used and implemented as models. The prob-
lem of lattice approaches is the reflection of the lattice structure in computer experi-
ments. These artefacts can cause mistakes if they don’t reflect realistic physical be-
haviour in the experiments. A new and different construction related to the special
conditions of growing cell populations will be introduced in Sec. 8.2. It has been
developed to avoid such artefacts.
32 4.4. General methods
4.4 General methods
Once we have made use of the scaling concepts and chosen a well defined structure to
workon, we have to choosethe method. We nowexplainsome of thecommon methods
and model types with their advantages, disadvantages and special constructions.
4.4.1 Monte Carlo approach
Monte Carlo simulations provide a very good tool for explaining growth. Monte Carlo
methods are numerical methods, where random numbers are used to describe statistical
quantities. Based on the early findings of Metropolis and Ulam [Met49], who named
the method after the famous city, a variety of different Monte Carlo techniques are
nowadays widely used to solve problems in statistical physics.
The name was chosen because of the relationship of the method to the huge random
number generators used in gambling. And in fact, Monte Carlo methods can still be
seen as a form of gambling, but just a little bit more advanced.
Nevertheless, the basic idea of such methods goes back to the 18th century to Buf-
fons’ famous needle problem to calculate the value of number π, which was solved in
1873 (A. Hall). These early experiments made use of known probabilities to solve in-
tegrals, and methods today still have the same rules. Monte Carlo simulations rely on
the assumption that the state of a system can be described by all its transition probabil-
ities to reach a different state. When one knows the transition rate of the incoming and
outgoing processes, one can then try to describe the global or macroscopic behaviour
of the system. In general there are two types of Monte Carlo methods, firstly time
independent methods which explain the equilibrium or local equilibrium behaviour of
the system and then time dependent methods which also try to give the development a
time scale.
Markov processes
Markov processes are stochastic processes which fulfil the Markov property. So by
definition all the possible states which can be reached from a given state depend solely
on the current state of the system and not on any past state.
A sequence of random variables X0,X1,...Xk−1,Xnthen is called a Markov chain,
if Xkjust depend on Xk−1.
Markov chains are said to be ergodic, if there is a nonzero probability of reaching
any possible state of the system from any other state.
So, if we have a system in the state iwith a transition probability pi→jof reaching
state jafter a certain timet, the probabilitiesfor all the transitions obey the relationship
pi→j>0(4.7)
Assuming a given ergodic Markov chain we can describe evolution in the state space
by a master equation:
Chapter 4. Modeling growth phenomena 33
∂tPi(t) = X
j
[Pj(t)rj→i−Pi(t)ri→j](4.8)
where Pj(t)rj→irepresents the processes that reach the state iand Pi(t)ri→jare the
processes which leave the initial state. Pi(t)are the probabilities of finding the system
at time tin state i, and ri→jis the rate of change to state j(transition probability per
unit time).
In general all simulations which are made by kinetic Monte Carlo techniques rep-
resent the solution of such a master equation.
When we want to explain systems which tend to an equilibrium state, the property
of detailed balance is required [Lan05].
Pipi→j=Pjpj→i(4.9)
Taking the required properties into account, one can now define the transition proba-
bilities which generate such Markov chains.
Classical Monte Carlo methods
The two most famous methods are
Metropolis algorithm
pi→j=(1if E(j)−E(i)<0
exp(−β(E(j)−E(i))) otherwise
Kawasaki algorithm
pi→j=1
1 + exp(β(E(j)−E(i))) ,
where β= 1/(kBT)where kBis Boltzmann’s constant and Tthe temperature.
In the simplest form of a Monte Carlo algorithm for simulating lattice dynamics, a
particle is chosen randomly and a jump direction is also chosen randomly. If the arrival
site is empty, the probability pi→jis computed and compared with a random number
0< rrand ≤1. If the final site is occupied or rrand > pi→jthe move is rejected. The
cycle now starts from the beginning again.
The essential drawback is clear. There are always a certain number of cycles which
do not produce new states since they are rejected, yet consume computing time. In low
temperature systems, where transition probabilities are low too, this effect becomes
dominant.
34 4.4. General methods
Continuous time Monte Carlo methods
To overcome this problem, each event needs to be chosen according to its a priori
probability,and every step needs to be accepted. Methods based on this idea are the so-
calledtime dependentMonte Carlo methods(themethodused in thiswork), sometimes
referred to as event based Monte Carlo,continuous time Monte Carlo,BKL algorithm
after Bortz, Kalos and Lebowitz [Bor75] or Gillespie algorithm after Gillespie [Gil76].
Consider a system with a total number of states Nin the state i. Labelling all states
jwhich may be reached from iwith k∈ {1,...,K}, the total transition rate is given
by
R(i) = X
j
ri→j=
K
X
k=1
r(i;k).
Here, the rate ri→jto the final state jbeing labeled by the number kis described by
r(i;k). The partial sums can be written as
R(i;k) =
k
X
l=1
r(i;l).
Now one specific event kcan be selected by a uniformly distributed random number
0<˜rrand ≤R(i), for which the condition
R(i; (k−1)) <˜rrand ≤R(i;k)(4.10)
must be met.
Under the constraint that time is incremented proportionally to the lifetime τ(i) =
τ0/R(i), a detailed balance is always ensured. One can therefore model the transition
probabilities with respect to the physical needs of the specific problem rather than
being restricted by the constraints mentioned above.
The time step ∆tin the event based Monte Carlo simulation is calculated as follows
[Fic91]:
∆t=−1
Pipi
ln(1 −ξ)(4.11)
where ξis a random number equidistributed in [0,1) and Pipiis the sum of all
possible events iwhich may occur at time t.
We now have a situation where, instead of wasting computation time on unneces-
sary rejections which do not contribute to a change of configuration, the main part of
the computing time is spent calculating total and partial transition rates. So by imple-
menting the time dependent Monte Carlo algorithm with care, the drawbacks on the
non-time dependent algorithm can be minimized and this method is much faster. Fi-
nally, as the last comment in this section, the difference between kinetic Monte Carlo
(KMC) and classical Monte Carlo should be emphasized. While the latter is used for
the calculation of a quantity in the thermodynamic equilibrium state of a system the
Chapter 4. Modeling growth phenomena 35
former describes the path of the system towards the equilibrium state. So, by using a
KMC algorithm, we ensure not only realistic equilibrium behaviour, but also realistic
kinetic behaviour.
4.4.2 Discrete models
Discrete models of crystal growth are closely related to kinetic Monte Carlo methods.
Here one defines the properties of the main processes and thereby gets different types
of models with well defined properties that can be identified by their critical exponents.
Because the stochastic differential equations and our tumor growth model work with
comparable quantities, we want to point out here the basic model types. In general they
differ in their definition of the deposition processes and in determining the diffusion, or
relaxation of particles on the surface, respectively. Where the discrete models and the
stochastic differential equations aimed to explain roughening by a height function, the
height here is discretized, normally corresponding to the actions of effective particles,
for instance atoms in a lattice.
Ballistic deposition models
In ballistic deposition models the particles which fall perpendicular onto the surface
stick to the first nearest neighbor (NN) they find, or to the next nearest neighbor (NNN)
(see [Mea93; Mea90; Bai88; Fam85]).
Solid on Solid models
The solid on solid approximation (SOS) is an idealization whereby neither bulk va-
cancy nor surface overhang is allowed to form during growth. One also normally
neglects desorption or evaporation processes from the front.
Random Deposition model
The easiest SOS model is the random deposition model, where we neglect diffusion
on the surface. The random deposition of particles at a position xon a given surface
at a deposition rate Fincreases the height function h(x, t)locally. Obviously, by
definition, in the random depositionmodel no correlations can occur without relaxation
processes.
Family model
Since most real growth systems show relaxations, a further developmentof the random
deposition model is the random deposition with surface relaxation [Fam86] sometimes
also referred to as the Family model. The deposited particles do not then stick irre-
versibly at the position, but can relax to a nearest neigbor with a lower height.
36 4.4. General methods
Wolf-Villain model
The Wolf-Villain model determines the relaxation after deposition by a move to the
neighboring site, when the particle is thereby able to increase the number of bonds
[Wol90].
Das Sarma-Tamborena model
This model is just a variation of the Wolf-Villain model, where in addition the particles
only relax if theydo not have any lateral neighbors, otherwisethey stay in their position
[Sar91].
A variety of other dynamic relaxation models exist but all these models have the
problem that the relaxation process is determined by the local environment at the po-
sition of the deposited particle.
The advantage of these models is their easy implementation in a computer simula-
tion with low computational demand.
4.4.3 Stochastic SOS models
The so called stochastic Solid on Solid models offer a more realistic modelling of
diffusion processes. The deposition of particles occurs in the same manner as in the
other models and the models are also only described by events to neighboring sites.
But in contrast to the other models, here any surface atom could be selected at any
time for a diffusion process, not only at deposition time. For instance a diffusion by
Arrhenius law can give the transition probabilities of such events.
In Sec. 2.1 we described the diffusion processes in epitaxial growth using Arrhe-
nius law, and we don’t want to go into the subject any further here but refer readers to
the publications for kinetic Monte Carlo simulations on SOS models extensively stud-
ied in our group during the last 12 years in the framework of Sfb 296 ([Sch98; Bos99a;
Bos99b; Bos00; Mei01c; Mei01a; Mei01b; Liu01; Mei02; Mei03c; Mei03b; Man03;
Els03; Wet04; Els04; Blo04; Els05b; Els05a; Kun06b; Kun06a; Kun06c].
4.4.4 Continuum Equations
A different method of describing growth or evolutionary processes is to use continuum
equations. Using scaling theory (see Sec. 4.2) there are different equations related
to different universality classes. To construct such continuum equations one has to
expand the so called generalized equation, which includes all the processes.
∂h(x, t)
∂t =G(h, x, t) + η(x, t)(4.12)
where Gis the generalized function depending on interface height, position and time
[Bar95]. If we now assume that the incoming flux of particles is not constant, then
Chapter 4. Modeling growth phenomena 37
we use the term η(x, t)to describe the random deposition. This means that random
fluctuations then have zero mean and normally the second moment is assumed to have
no correlations in space or time (Gaussian white noise).
hη(x, t)i= 0 (4.13)
hη(x, t)η(x′, t′)i= 2Dδd(x−x′)δ(t−t′)(4.14)
Whereas one can also introduce correlated fluctuations, in thiswork we use white noise
as defined in Eq. (4.13). Now the individual definition of the function Gcharacterizes
a specific growth process by a specific continuum equation. The general function can
be simplified by using the symmetry principles of roughening systems.
Time translation invariance
The growth equation does not depend on where we define the origin of time so
the invariance has to fulfil the relationship t→t+δt
Translation invariance along the growth direction
Growth has to be independent of the choice of h = 0 so the invariance has to
fulfil the relation h→h+δh
Translation invariance in perpendicular growth direction
The growth has to be invariant under translation perpendicular to growth x→
x+δx
Rotation and inversion symmetry about growth normal vector
Growth has to be invariant if we invert or rotate the height profile about the
growth normal n.
Up/down symmetry for h
One can include a symmetry which states that interface fluctuations are simi-
lar with respect to the mean height, but this property is only fulfilled by linear
equations.
Further reading about the symmetry principles can be found in text books [Bar95].
When we include the knowledge about growth gained from the symmetry princi-
ples, we first obtain an expansion of terms described as follows
∂h(x, t)
∂t = (∇2h)+(∇4h)+...+(∇2nh)+(∇2h)(∇h)2+...+(∇2kh)(∇h)2j+η(x, t)
(4.15)
were n,k,jcan take any positive value. For simplicity the coefficients in front have
not been written down explicitly. Neglecting the different coefficients in this expan-
sion now leads to different growth equations which are classified by different critical
exponents β,αand z.
38 4.4. General methods
If the growth is self-similar and fulfils the properties of scaling theory, then the
different growth equations extracted from the generalized equation lead to the different
universality classes described in Sec. 4.2 which can be classified using the related
scaling exponents.
Chapter 5
Stochastic Differential Equations
In the last chapter we gave a general explanation of methods of modelling growth
phenomena and stated that stochastic differential equations have been established as
one of the leading methods of modelling growth. We now want to describe this kind
of modelling in more detail, and will use it later on to control the roughening process.
The theory of stochastic differential equations for growth, also referred to as
stochastic growth equations, is based on Langevin equations.
Whereas Langevin equations were widely used earlier, Edwards and Wilkinson
first used stochastic differential equations for the roughening process in the early eight-
ies [Edw82].
With the observations of Kardar, Parisi and Zhang [Kar86] the theory of stochas-
tic differential growth equations became a well established tool to explain growing
systems.
A lot of different equations have been proposed during the last 20 years to describe
different universal classes of growth, but there are still a lot of unsolved problems.
Some of the questions arise because of the nonlinear form of some of the equations
and the impossibility of solving them analytically. The equations can be used as ide-
alized versions of realistic growth properties, but the relation of realistic growth to its
corresponding universality class is not always obvious. In Sec. 4.4.4 we developed the
equations as the result of an expansion with the addition of certain symmetry princi-
ples and we now want to describe the specific equations that are most frequently used.
We shall explain the terms related to the different processes together with their specific
physical meaning.
40 5.1. The Edwards-Wilkinson and the Kardar-Parisi-Zhang Equation
-3 -2 -1 0 1 2 3
x
-0,5
0
0,5
1
h(x,t)
h(x,t) = e-x2
2h(x,t) = 4x2e-x2 - 2e-x2
h(x,t + δt)
Figure 5.1: Behaviour of the Edwards-Wilkinsonterm (red dashed line) on an artificial
height profile h(x, t) = exp(−x2)(black line) leads to small variation in the resulting
profile (blue dash-dotted line).
5.1 The Edwards-Wilkinson and the Kardar-Parisi-
Zhang Equation
For an explanation of the relevant terms in this work we shall now discuss the hypo-
thetic generalized function G(h, x, t)in Eq. (5.1).
∂h(x, t)
∂t =G(h, x, t) + η(x, t)(5.1)
The basic question arising from this growth equation is the question as to which real-
istic processes dominate the roughening.
5.1.1 The Edwards-Wilkinson equation
The easiest generalized function to think about is the Edwards-Wilkinson (EW) equa-
tion [Edw82],
∂th(x, t) = ν∇2h(x, t) + η(x, t)(5.2)
where we take only the linear second order term from the expansion (Sec. 4.4.4). Orig-
inally developed to describe an Ising spin system, it also exhibits some properties rel-
evant to growth phenomena.
In Fig. 5.1 we see how this term acts on a given surface profile. We take a sim-
ple Gaussian height profile h(x, t) = exp(−x2)and calculate the second derivative
∇2h(x, t)for a pre-factor ν= 0.1in one spatial dimension.
Chapter 5. Stochastic Differential Equations 41
If we now add a small variation to the height function corresponding to a small
change of the function in time, we see that the Edwards-Wilkinson term acts as a
smoothing term on the height profile and as a conservative relaxation. Surface tension
behaves similarly, which is why this term is normally called the surface tension term.
The Edwards-Wilkinson equation is valid in the small gradient approximation, i.e.
in the limit |∇h| ≪ 1.
It corresponds to the well known discrete random deposition model with surface
relaxation [Fam86] (see also Sec. 4.4.2).
The main difference from a random deposition model without relaxation is the
presence of correlations.
There are different ways of solving the Edwards-Wilkinson equation and calcu-
lating the scaling exponents. Both an approach using scaling and an exact solution
[Nat92] are possible. The EW equation is one of the rare solvable equations and we
shall now show the solution. For the solution by scaling we only require a self-affine
interface with a height function h(x, t). As explained in Sec. 4.2, rescaling in the
horizontal and vertical directions produces interfaces which are statistically indistin-
guishable from the original one.
x→x′≡bx(5.3)
h→h′≡bαh
When we measure the height function at different times, the two interfaces are also
rescalable in time.
t→t′≡bzt(5.4)
Due to the fact that the rescaled quantities obey these relations, by substitution in d
dimensions we get
∂h′(x′, t′)
∂t′=ν∇2h′(x′, t′) + η(x′, t′)(5.5)
bα−z∂h(x, t)
∂t =νbα−2∇2h(x, t) + b−d
2−z
2η(x, t)(5.6)
∂h(x, t)
∂t =νbz−2∇2h(x, t) + b−d
2+z
2−αη(x, t)(5.7)
and by multiplying the term bα−zon both sides we come to the rescaled height function
which is invariant under the transformation and therefore fulfils the following relations
(independent of b, comparison of coefficients)
z−2 = 0 (5.8)
−d
2+z
2−α= 0 (5.9)
which leads to the exponents of the EW universality class,
α=2−d
2, β =2−d
4, z = 2 (5.10)
42 5.1. The Edwards-Wilkinson and the Kardar-Parisi-Zhang Equation
Figure 5.2: Scheme of the lateral growth,
height function h(x)(blue), lateral growth
on the surface (red dashed lines).
vδt
δh
h(x)
x
where dis the spatial dimension.
So in 1+1 dimensions the exponents are α= 0.5,β= 0.25, whereas in 2+1
dimensions the exponents for the above equations are α= 0.0,β= 0.0and z= 2.
This means that scaling is logarithmic in 2+1 dimensions for the Edwards-Wilkinson
equation.
Whereas we solved the equations before by scalingarguments ina phenomenologi-
cal way, this equation can be solvedexactly, as shown by the findings of Nattermann et
al [Nat92]. The Edwards-Wilkinson equation fulfils up/down symmetry (Sec. 4.4.4).
If the growth is non-linear, the scaling has to change and this property is no longer
fulfilled. We now come to an equation where processes related to nonlinear terms play
an essential role.
5.1.2 The Kardar-Parisi-Zhang Equation
Once again we first think of the easiest nonlinear term possible, which is the (∇h)2
term. The simplest such equation is the Kardar-Parisi-Zhang (KPZ) equation [Kar86]
which describes the growth of a surface in the absence of any conservation laws.
∂th(x, t) = ν∇2h(x, t) + λ
2(∇h(x, t))2+η(x, t)(5.11)
We have already explained the surface tension term ν. The nonlinear term determines
the strength and direction of both lateral growth and growth normal to the interface.
The origin of the nonlinear term can be seen in Fig. 5.2. Lateral growth normal to the
interface can be described locally by a term related to the Pythagorean theorem
δh2= (vδt)2+ (vδt∇h)2(5.12)
where δh is the small difference in the height function in the general growth direction
and (vδt)the lateral growth normal to the interface. We are using the small gradient
approximation, so one can easily see that an expansion of δh leads to
∂h(x, t)
∂t =v+v
2(∇h)2+... (5.13)
Chapter 5. Stochastic Differential Equations 43
-3 -2 -1 0 1 2 3
x
0
0,5
1
h(x,t)
h(x,t) = e-x2
( h)2 = 4x2 (e-x2)2
h(x,t+δt)
Figure 5.3: Behaviour of the nonlinear KPZ term (red dashed line) on an artificial
height profile h(x, t) = exp(−x2)(black line) leads to small variation in the resulting
profile (blue dash-dotted line).
and thereby exhibits the nonlinear term. The velocity is nothing but an included term
in the mean average height development of the flux Fto the surface. In Fig. 5.3 we
show the behaviour of the KPZ term on the roughening surface in the same way as we
showed it for the EW equation. Two main features are to be seen. Growth is related
to the normal of the interface and its strength to the local gradient. What one can also
observe is a lost mean height, so growth is not conserved relative to the mean height
development. So, whereas we can write a continuity equation for the total number of
particles for the Edwards-Wilkinson equation where jis the particle flux.
∂h
∂t =−∇j(x, t)(5.14)
this relation is not fulfilled by the Kardar-Parisi-Zhang equation. The consequence
is that, although we can generally describe a growth process with a constant flux to
surface which can be neglected for the continuity equation and conserved for growth
related to the mean height, this growth process can not bedescribed by a non-conserved
equation like the KPZ equation. The KPZ equation cannot be solved analytically be-
cause of its nonlinear character [Mic04; Bar95]. Nevertheless there are some proposed
scaling exponents for the equation.
5.1.3 Relations beetween EW and KPZ equation
The relationship between the Edwards-Wilkinson equation and the Kardar-Paris-
Zhang equation lies in the strength of the nonlinearity. Moser et al demonstrated
44 5.2. The Molecular Beam Epitaxy Equation
that fact by using an effective coupling constant gdefined by the parameters of the
KPZ equation [Mos91].
g=λ2D
ν3(5.15)
The coupling constant is related to the fixed point of a renormalization group theory
approach not discussed in this work. They describe the changed roughening for non-
zero nonlinearities due to a change in this coupling constant g. So, to ensure that the
behaviour we describe here is similar, we choose numerical parameters for our ’strong
coupling’ behaviour that ensure a coupling constant in the same range as that in this
paper. The critical exponents for the KPZ equation are well known in 1+1 dimensions
and are given by
α= 0.5, β =1
3, z =3
2(5.16)
In higher dimensions, where the renormalization group analysis fails, there exist two
different competing results from numerical simulations
α=1
d+ 1, β =1
2d+ 1, z =2d+ 1
d+ 1 (5.17)
and
α=2
d+ 3, β =1
d+ 2, z = 2d+ 2
d+ 3 (5.18)
which are both compatible with Eq. (5.16) for d= 1. The numerical observations of
Wolf and Kertész [Wol87] (Eq. (5.17)) and Kim and Kosterlitz [Kim89] (Eq. (5.18))
lead to the same exponents in 1+1 dimensions as given by Eq. (5.16) but they differ
for higher dimensions.
A lot of further calculations where made to determine these exponents. The values
of calculated growth exponents vary widely in a range from the models below, with
β= 0.20 from Wolf and Kertész to values close to and in between the two predictions,
where the exact value is still an open question (for numerical results see also [Mos91;
Ama90; Cha89; Guo90].
More recent findings by Lässig [Läs98] and Chin and den Nijs [Chi99] show the
values of the Kim-Kosterlitz model (Eq. (5.18)). A summary of the latter findings
together with some new numerical findings can be found in [Gha06]. We will see later
whether our findings without control fall within this range.
5.2 The Molecular Beam Epitaxy Equation
Molecular beam epitaxy (MBE) is a major technique in crystal growth of thin films.
Growth takes place in vacuum conditions under which particles from a molecular beam
are deposited on the surface (see also Sec. 2.2.1).
Chapter 5. Stochastic Differential Equations 45
Because of the growth temperature, desorption processes do not play an important
role in comparison with the diffusion processes on the surface. So models which aim
to describe a MBE process normally neglect desorption processes. Once one neglects
them one has to take surface diffusion asthe determining process. If one nowdescribes
the surface current jby the local chemical potential µ(x, t), it is driven by the gradient
j(x, t)∝ −∇µ(x, t)(5.19)
If one explains the movement of particles as a process depending on the number of
bonds, then this number increases with the local curvature. The chemical potential
then depends on −1/R and thereby on ∇2h(x, t), which gives us a relation
µ(x, t)∝ −∇2h(x, t)(5.20)
Combining that with the continuity equation (Eq. (5.14)) our height function is
∂h
∂t =−K∇4h(5.21)
where Kis the strength of this diffusion term.
So we now have a growth equation describing relaxation by diffusion just as we
have in epitaxial growth for Molecular Beam Epitaxy. The equation is also sometimes
referred to as the Mullins or Herring-Mullins equation, because the first findings came
from the observations of Herring [Her50] and Mullins [Mul57]. To avoid confusion
with the notation of the control strength, later on we use ν1instead of Kas the strength
of the diffusion. Equation 5.21 is deterministic. It was introduced for MBE growth by
Wolf and Villain [Wol90]. With some additional changes it becomes the normal type
of ’MBE growth equation’ we shall discuss later. Calculating the critical exponents
we arrive at [Sar91; Bar95]
α=4−d
2, β =4−d
8, z = 4 (5.22)
So in 1+1 dimensions the exponents are α= 1.5,β= 0.375 where in 2+1 dimensions
the exponents related to the above equations would be α= 1,β= 0.25 and z= 4.
The MBE growth equation that is normally used was described by Lai and das Sarma
[Lai91]. Also known as the conserved KPZ equation [Mic04], this equation takes an
additional term into account.
∂h(x, t)
∂t =−ν1∇4h+λ1∇2(∇h)2+η(x, t)(5.23)
The origin of the additional term is described as arising from the situation where ’par-
ticles landing at high steps (large derivatives)relax to lower steps (smaller derivatives)’
[Lai91]. The authors believe that it corresponds to ’high temperature’ regimes, where
the atoms at kink sites can break bonds and hop to steps with a smaller height and
a higher probability, so they propose the above equation as the ideal MBE growth
46 5.3. Crystal growth and stochastic differential equations
equation for intermediate to high temperatures. Obviously the explanation includes
a variation of the nonlinear term with temperature which is an essential factor in our
further findings. The change in this term also is quite similar to the situation explained
for the coupling in the KPZ equation, so here we have either strong or weak coupling
according to the different temperatures. Whereas these equations explain MBE growth
in an idealized way, the question arises as to what happens when the physical process
involves both surface relaxation related to deposition or desorption processes and a
diffusing term like in the following equation.
∂h
∂t =ν∇2h−ν1∇4h+η(x, t)(5.24)
The long term behaviour is obviously the behaviour of the EW equation, because
for large length scales the Laplacian either governs the equation or is its leading term.
The terms generate a characteristic length scale which determines whether the diffu-
sion term is still the leading term or whether the length is so large that the EW term is
the relevant one for the exponents.
That fact can be explained if we rescale the terms using the known exponents. We
then get νbα−2∇2hand ν1bα−4∇4h. Thus for b→0the diffusion term dominates and
for b→ ∞we get Edwards-Wilkinson scaling. In terms of length scales the term
L1=ν1
ν2(5.25)
describes the behaviour (L≫L1→MBE-like, L≪L1→EW-like). So which
length scale we choose depends on the growth conditions, but for realistic MBE con-
ditions one can normally neglect the EW term in comparison with diffusion, and the
length scale L1is so large that one can see the MBE exponents. There are a lot of
different models and equations related to Molecular Beam Epitaxy, a good overview
of the discrete models and their relations to the equations is given in [Sar96].
5.3 Crystal growth and stochastic differential equa-
tions
We have already discussed the behaviour of the most referenced and used equations
for growth. We are now going to have a short look at the different types. There are
a few articles which try to describe experiments using such growth equations but it is
still a developing field. It is quite difficult to find crystal growth experiments where
the growth conditions are as ideal as assumed in the generic equations. The measured
exponents vary over a wide range for the same system and it is not easy just to measure
the roughness and then write down one of the equations.
Chapter 5. Stochastic Differential Equations 47
5.3.1 Observations by Raible et al
The observations of Raible et al [Rai00a; Rai00b; Rai01] provide an example of a
more complicated growth equation applied to amorphous metallic thin films. Here a
specific equation is solved by numerical integration. The equation
∂h(x, t)
∂t =a1∇2h+a2∇4h+a3∇2(∇h)2+a4(∇h)2+η(x, t)(5.26)
is proposed to describe growth. One can see immediately that it is a combination of
the terms of the KPZ equation (Eq. (5.11)) and the conserved KPZ or MBE equa-
tion (Eq. (5.23)). Guided by the measured experimental roughness evolution of
Zr65Al7.5Cu27.5the equation was solved by a numerical scheme and fitted to the
experimental findings. The parameters were identified as
F= 0.79 nm/s D = 0.0174nm4/s (5.27)
a1=−0.0826nm2/s a2=−0.319nm4/s (5.28)
a3=−0.1nm3/s a4= 0.055nm/s (5.29)
The parameter a1, normally identified with surface tension in the equations, is negative.
The authors explain that irritating fact by growth instabilities "due to the deflection of
the initially perpendicular incident particles caused by the inter-atomic forces between
the surface atoms and the incident particles". The instability referred to is explained by
Villain as due to an instability on terraces in growth with a diffusion bias at a crystal-
lite layer [Vil91]. A test with our numerical simulation scheme shows agreement with
the findings but also shows that the numerical solution with exactly the same param-
eters is extremely sensitive to very small changes in the fitted parameters. Although
one can easily show that small differences in the pre-factors lead to a non-convergent
growth equation, small differences between experiments cannot be explained by this
model. Fig. 5.4 (a) shows that, with exactly the same parameters as used in the numer-
ical scheme, the evolution of roughness can be reproduced exactly by our numerical
scheme.
The authors do not explicitly report the roughness exponents. They showed the
height-difference correlation, where one can see from the plots, that αshould be close
to a value α∈[0.85,0.9]. We explicitely determined this quantity using the height-
height correlation (Fig. 5.4 (b)) and the height-difference correlation function (Fig. 5.4
(c)) and got exponents α= 0.88 and α= 0.87, so our numerical solution and the data
analysis are consistent with the findings of Raible (see next Section for the method).
Whereas the determination of the correlations once again reproduces the behaviour
found by Raible et al, one can easily check that the exponents are not related to any
one of the explained equations. Obviously a mixture of different terms can lead to
much more complicated behaviour in roughening. To summarize, it is quite difficult
to simulate very complex behaviour with the growth equation and to unambiguously
48 5.4. Data analysis
Figure 5.4: Verification of the Raible model for thin film growth: (a) height profile for
a400 ×400 l.s. simulation at time t= 1000, (b) the height-height correlation with a
fit function, (c) the height-difference correlation function with a fit function.
identify the basic processes which lead to the experimental behaviour. For amorphous
substrates discrete modelling with, for instance, KMC simulations can help to solve
these problems [Els05b; Els05a]. In general onehas to be very careful when explaining
the different processes.
Nevertheless, observations using the easier equations can lead to a better knowl-
edge of complicated roughening systems.
5.4 Data analysis
For computer experiments in general, and in our work too, it is fundamentally impor-
tant to calculate the quantities in a proper way and to construct the computer codes
Chapter 5. Stochastic Differential Equations 49
and the numerical scheme in a way most closely related to reality, in order to avoid
discretization artefacts as much as possible. It is also extremely important to include
the parameters in a simulation in a way that makes sense. The competing difficulty is
then nevertheless to ensure a numerical solution in an appropriate real time and not to
tune the computational expense to infinity. The first step in a data analysis is to think
about the discretization scheme to be applied to the equations, and only after that can
one think about how to calculate the observed quantities.
5.4.1 Numerical scheme of solving the growth equations
The numerical solutions of the stochastic growth equations are normally based on lat-
tice or on discretized points, for which the height function has to be solved. If we focus
on crystal growth, the natural approach is to identify the different points with atoms in
a lattice where the spatial position on the interface is discrete and the height function is
then quasi-continuous. The height function h(x, t)depends on discrete points, so when
we speak about the continuum height function for the growth equation our simulations
must approximate most closely not a continuum but a discrete version of it. The spatial
discretization ∆xand the discretization of the time steps dt are the discrete quantities
that determine the distance to a real growth process. In the ideal case dt goes to zero,
and the minimal discretization reflects a lattice constant a. In the case of an amorphous
substrate the normal approach is also an effective lattice constant [Els05b; Els05a].
Discrete growth equation →Continuum growth equation (5.30)
dt →0(5.31)
∆x→a(5.32)
In our case we want to investigate the behaviour of those equations where scaling
is dependent on the system size L. If we had, for instance, a system of real size
64×64 nm with a lattice constantaof approximately 0.5nm, we would have 128×128
discrete points on an atomic scale. If we now simulated a system with 256 ×256
discrete points, it would refer to a system of four times the area and would be related
to 128 ×128 nm, or with a lattice discretiation of ∆x= 1 nm it would refer to
256 ×256 nm. The scaling laws are not affected. But, for good results for different
system sizes, one needs to take the same underlying spatial discretization ∆xas we do
in this work. This shows that the choice of spatial discretization is not important for
the simulation itself but becomes important with its interpretation for real sizes.
So a rough view of a 128 ×128 nm with describing every second atom by a lattice
point of the system is an even rougher view of a 64 ×64 nm where all atoms are
represented by one point, but from the point of view of the simulation they are the
same and in the case of scaling laws they ought to be the same. These statements are
obviously a direct outcome of fulfilling the scaling laws.
The discretization of time dt is much more difficult. If the discretization is too
rough, the fluctuations that naturally appear in numerical schemes lead the growth
50 5.4. Data analysis
and the results don’t explain the equations. There is no general law to determine the
discretization that will ensure realistic behaviour. In general it has to be small in com-
parison with the timescale we want to analyze.
The lower cut-off is computationally demanding, so we have to find a rule to check
our simulations. If fluctuations dominate growth, in the worst case the height function
goes to infinity at a certain point. We then know that we have chosen the wrong
timescale for the simulation. That can easily be demonstrated using the algorithm. A
more difficult case occurs when the discrete version is not close enough to the realistic
equation. We can check this by using a smaller time discretization, and then, if the
behaviour does not change, we know that a suitable discretization has been chosen.
5.4.2 Discretization scheme
We have now explained how we prepare the discretization of the lattice and of time in
our numerical scheme. We now come to our discretization of the equation by terms.
The first observations by Moser and Kertész used a normal forward-backward differ-
ences scheme on a cubic grid and integrated it using an Euler algorithm [Mos91].
hn(t+ ∆t) = h(t) + ∆t
∆x2
d
X
i−1
(ν[hn+ei(t)−2hn(t) + hn−ei(t)] (5.33)
+1
8λ[hn+ei(t)−hn−ei(t)]2) + δ√12∆tRn(t)
While some of the newer investigations still make use of this simple discretization
scheme, Lam et al demonstrated that in 1+1 dimensions it produces some mistakes
in transitions from zero nonlinear terms to nonzero. They showed that the results of
individual roughness evolution produce the right exponents, but when one want to get,
for instance, the transition from the behaviour of the Edwards-Wilkinson equation to
the KPZ equation, some numerical mistakes occur [Lam98a; Lam98b].
Lam et al showed in their findings that in the transition from Edwards-Wilkinson
(λ= 0) to KPZ behaviour (λ > 0) a shift of the amplitude A for the saturation function
appears ([Kru92]).
wsat =A
121/2
Lα(5.34)
They conclude that this conventional discretization is not a genuine approximation to
the continuum KPZ equation. They propose a new discretization in 1+1 dimensions
and show in detail that their scheme is a solution of the continuum equation that does
not produce these instabilities.
A more generalized study of the problem provides a scheme which solves these
problems for more than just the 1+1 dimensional case [Buc05b].
Chapter 5. Stochastic Differential Equations 51
In order to ensure that the proposed numerical scheme really avoids unwanted be-
haviour we want to use here the scheme from Lam et al instead of the older scheme of
Moser and Kertesz [Lam98b; Mos91].
So as to avoid such mistakes we use the discretization scheme of Lam et al, while
ensuring that the exponents are not affected.
hn+1
i,j =hn
i,j +∆tn
(∆x)2[wn
i+1,j +wn
i−1,j +wn
i,j−1−4wn
i,j](5.35)
+a4
3(∆x)2[(hn
i+1,j −hn
i,j)2+ (hn
i+1,j −hn
i,j)(hn
i,j −hn
i−1,j)
+(hn
i,j −hn
i−1,j)2+ (hn
i,j+1 −hn
i,j)2
+(hn
i,j+1 −hn
i,j)(hn
i,j −hn
i,j−1) + (hn
i,j −hn
i,j−1)2]
+s24D∆tn
(∆x)2rn
i,j
wn
i,j =a1hn
i,j +a2
(∆x)2[hn
i+1,j +hn
i−1,j +hn
i,j+1 +hn
i,j−1−4hn
i,j](5.36)
+a3
3(∆x)2[(hn
i+1,j −hn
i,j)2+ (hn
i+1,j −hn
i,j)(hn
i,j −hn
i−1,j)
+(hn
i,j −hn
i−1,j)2+ (hn
i,j+1 −hn
i,j)2
+(hn
i,j+1 −hn
i,j)(hn
i,j −hn
i,j−1) + (hn
i,j −hn
i,j−1)2]
Here the hn
i,j is the discretized height function depending on xi,yjand time tn.rn
i,j is
a random number taken from a uniform distribution [−0.5,0.5).
From this general discretization we arrive at the specific equations by setting a2=
0and a3= 0 for the KPZ equation (a1=ν,a4=λ) or by setting a1= 0 and a4= 0
for the MBE equation (K=ν1=a2,a3=λ1).
5.4.3 Determination of the critical exponents
In our work the basic quantities calculated from simulations are the critical exponents
which determine the universality classes. We are going to use a lot of different methods
so we shall explain them here. The basic measured quantities in our work are the rms
roughness evolution in time wRMS(t)and the related exponents α β and z. We now
present the different methods of calculating of the exponents, and we shall use almost
all of them to obtain the resulting structures.
Calculation of the growth exponent β
There are different methods of determining the growth exponent β, direct and methods
using the exponents αand z. A direct measurement of βcan be made by tracking
52 5.4. Data analysis
the temporal evolution of roughness and then taking the slope of the double logarith-
mic plot which reflects the law explained in Eq. (4.2). In this case the roughness is
not saturated. We have already explained the roughening phase in Sec. 4.2.2 so will
not explain this direct method again. The second method of measuring the growth
exponents is by determining the other exponents, using Eq. (4.6).
Calculation of the roughness exponent α
First we have to measure the roughness exponents. Direct measurement of the rough-
ness exponent αis possible if we can reach the saturation point of the surface for
different system sizes L. Then the scaling law for saturation roughness (Eq. (4.3)) can
be mapped to the curves to get the related exponent. That may be the easiest method,
but it is not the best way of obtaining the roughness exponent, as we shall see later for
the tumor growth model (Sec. 9.7).
This method obviously fails for most crystal growth systems where saturation is
not always reached during growth. A different method assumes that, even if we do not
reach the saturation value, it is still possible to determine the roughness exponent from
locally saturated regions of the surface. The so called local width method then takes
the dependence of the locally saturated roughness (width)
wL(l, t) = p<[h(x, t)−hl(x, t)]2>x(5.37)
where hlis the mean height of the local window of size l. The scaling of local rough-
ness for small lis the same as for the whole system, so for small lwe obtain the
roughness exponent using the relation
wL(l, t)∝lα(5.38)
This method works very well for saturated systems but for unsaturated surfaces one
has to verify that l≪ξ||.
Another method is to determine the height-difference correlation function
H(r) = p[<(h(x)−h(x′))2>x] (|x−x′|=r)(5.39)
If we again assume a self-similar roughening system with an arbitrary factor bthen
h(x)→b−αh(bx)(5.40)
(see also Sec. 5.1), and invariance implies the relation
H(r) = b−2αH(br)(5.41)
By setting b= 1/|r|it follows that
H(r)∝ |r|2α(5.42)
Chapter 5. Stochastic Differential Equations 53
With this relationship it is possible to determine the roughness exponent from the
height-difference correlations.
A more general correlation is the height-height-correlation function
C′(r) = <(h(x+r)−h)(h(x)−h)>x(5.43)
C(r) = < C′(r)>|r|=r(5.44)
From the radius averaged correlation C(r)we can then calculate the roughness expo-
nent using the relation:
C(r) = C0exp(−(r
ξa
)2α)(5.45)
where ξais the so called self-affine correlation length. The functions H(r)and C(r)
are related to the correlation length by
H(r)→2w2
RMS for r≫ξa(5.46)
C(r)→0for r≫ξa(5.47)
The structure function S(k, t)makes use of the power spectrum of the interface. De-
fined by
S(k, t) =< h(k, t)h(−k, t)>(5.48)
with
h(k, t) = 1
Ld/2X
x
[h(x, t)−h]eik·x(5.49)
the Fourier transform of the height function h(x, t), the scaling concepts lead to the
relation
S(k, t) = k−d−2αg(t/k−z)(5.50)
with the Fourier space scaling function g(u)which fulfils the relations
g(u)∝u(2α+d)/z for u≪1(5.51)
g(u) = const. for u≫1(5.52)
and is quite similar to the scaling function in normal space (Eq. (4.5)). It allows one to
determine two of the three exponents directly.
If we now measure the slope of the log-log plot of the structure function we can
measure−(2α+d)directly. By then rescaling the function with themeasured exponent
we get the scaling function u.
Rescaling again with the expected value of zfor the structure function for different
times twe end up with a data collapse where the curves match one another, provided
we choose the right value of z.
This method also indirectly measures the growth exponent β. It is used for the
stochastic differential equations of the tumor growth model, as we shall see later.
Although there are a lot of other methods of calculating the exponents, in this work
we shall restrict ourselves to those we have already explained (see [Bar95]).
54 5.4. Data analysis
Chapter 6
Control of stochastic differential
equations
In the last chapter we described the equations whose universal exponents determine the
classes of the different growth phenomena. We discussed the properties and influence
of the terms corresponding to the physical mechanisms of realistic roughening; we
now want to answer the question of how to influence and control growth.
In crystal growth normally the first step, when growing defined structures, is to
calibrate the system. Although for a lot of systems it is rather difficult to measure
local conditions on the substrate, this procedure can be combined with an extensive
number of repeated tests, until one reaches the conditions under which growth shows
the required behaviour.
A more elegant and, of course, cheaper way is to tune the conditions during growth.
So an in-situ setup that adjusts the surface roughness is a very helpful tool.
When we use stochastic differential equations in the theoretical approach, the basic
question is how to implement a useful control in the equation in order to tune the
roughening process.
In this chapter we want to explain how we control the roughening surface using
the stochastic differential equations we described previously. We introduce the basic
concepts and then proceed to our method [Blo06b; Blo06a].
6.1 Control theory
Controllers are an essential part of daily life. Although one might first think of appli-
cations to engineering like the ’anti-skid system’ in a car or the automatically tuned
temperature of rooms, one of the most complex systems involving controllers is the
human body itself.
An exampleof this is the’erect posture’. We first use our tactile and visual sensesto
summarize the information from the environment. The brain then acts as the complex
56 6.1. Control theory
Figure 6.1: General scheme of a control system with the basic actions, measuring,
comparing, tuning.
controller which send thesignals tothe musclesto act in the rightway. If this controller
fails for any reason, the corresponding actions fail to occur.
Although the design of this controller is very complicated, it shares certain univer-
sal properties with other systems.
The design of a controller like that in Fig. 6.1 can be described by the process of
adjusting a specific quantity, which we measure in the system. The required value
of this quantity then gives, by comparison, the direction for tuning. So the cycle of
measuring, comparing and tuning is the basic concept and the controller determines
the changes needed to reach the desired value.
The properties of the system then decide the specific design.
There are two general types of controllers, the feedback method and the non-
feedback method (often called feed-forward). In this work we only use the feedback
method.
Figure 6.2: Mathematical scheme for a control of a system with ydthe desired value
of y,ethe difference from the measured value to ydand y(t)the output of the system
acting together with the measuring section, the controller and the plant.
Chapter 6. Control of stochastic differential equations 57
In Fig. 6.2 we see how feedback control works. The control is designed to tune
quantity yto the desired value yd. During the development of the system, the time
dependent quantity y(t)is measured in the measuring section.
It is compared to the required value and the difference eis given to the controller.
The controller then uses the information and responds by tuning the so-called plant,
which is the system to be controlled.
By constantly following the defined feedback loop, a properly designed controller
will reach yd. In the ideal case egoes to zero. The control has to be reset if some
disturbance zoccurs in the system.
The properties of the system we want to tune help us to decide between the various
methods of control theory. We call the adjustment ofthe value of a quantity in a system
which doesn’t show any chaotic behaviour "classical control". However, during the
last decade, methods of controlling systems with a huge number of unstable periodic
orbits have been developed in the field of nonlinear dynamics. When such systems
show chaotic behaviour we have to decide between "classical" and "chaos control".
Although the basic concepts like the choice of feedback or feed-forward methods
are similar, the systems exhibit different behaviour under control.
6.1.1 Classical control methoods
The most important class of mathematical approaches in classical control methods
is the so called "Proportional Integral Differential" (PID) controller. As the name
indicates, the controller is made up of three different terms, some of which can be
neglected, depending on the specific problem.
These three parts determine the behaviour of the control. The P-part works as an
amplifier of the difference e, the I-part sums up the measured values of eand thereby
memorizes the development of the quantity and the D-part measures the gradient of
the difference.
In general these parts of a controller are well defined by the answer from the step
response. The equation ufor a PID control follows directly from the transfer function
f.
f(s) = KP+KI
s+KDs(6.1)
u(t) = KPe+KIZe dt +KD
de
dt (6.2)
The weights of thecontroller parts are the pre-factors K, and sdenotesthe timeinterval
in which we measure the differences e.u(t)is then given by the controller to the plant
of the system.
More complicated kinds of P-parts or time delay parts can be included. These
make the controller much more complicated. For a detailed overview of the concepts
of PID we refer the reader to the book by Åstrom and Hägglund [Åst95].
58 6.1. Control theory
Fuzzy control is a different approach to controlling a system. The basic concepts
were developed from fuzzy logic theory by Lotfi Asker Zadeh in 1965. Fuzzy logic
includes not only the set ’true’ or ’false’ but also logical states in between. Fuzzy
control then answers the question of how close the measurement is to the correct value.
The detection of edges on a poor grey colored picture is an example. If we define
white to be the edge and black to be not at the edge, then most of the points are in
between. For further reading we suggest the book by Passino and Yurkovich [Pas98].
6.1.2 Chaos control
Where normally, for classical methods, the aim is to tune a specific quantity to a certain
target value, for fluctuating systems the aim is sometimes to stabilize or destabilize
certain chaotic attractors. Control then usually means adjusting the essential oscillation
properties of the system by imposing a small perturbation.
Control of complex irregular motion is one of the central problems in nonlinear
dynamics [Sch01; Sch04; Sch89; Sch99; Sch07].
The phase space of such systems contains a large number of unstable periodic
orbits embedded in a chaotic attractor. Therefore a small change in initial conditions
can lead to a completely different evolution.
There are several different methods, which once again can be divided into feedback
and feed-forward methods.
An important class is called non-invasive control. Here only weak external forces
are coupled into the system. They do not change the dynamics of the system com-
pletely, but stabilize an already existing orbit embedded in the chaotic attractor.
The most importantnon-invasivemethods are theOtt-Grebogi-Yorke(OGY) method
[Ott90], and the Pyragas control scheme [Pyr92], which is also known as time-delayed
auto-synchronization (TDAS). The TDAS scheme uses the time-delayed feedback of
a system variable, which is coupled back into the system. It can easily be applied
to a great number of systems and has proved to be successful in real experiments.
See [Ben02; Boc00; Jus03a; Sch06a], or for various classes of theoretical models
[Bab02; Bec02; Fra99; Höv04; Höv03; Jus03b; Bal05; Höv05; Yan06; Sch06b],
and for models of semiconductor nanostructures [Sch93; Ama03; Ama02; Sch03b;
Unk03]. A wide range of applications of this method have been tested and there
have been a great many theoretical investigations. A further development of control
was then proposed as extended time-delayed auto-synchronization (ETDAS) [Soc94].
Time-delayed feedback control has also been applied to noise-induced oscillations
[Jan04; Bal04; Sch05; Pom05; Ste06; Hiz06; Hau06; Bal06].
We have briefly discussed chaos control, but for more detailed explanations of the
methods and a state of the art review of theoretical methods and their experimental
applications see the book by [Sch07].
Chapter 6. Control of stochastic differential equations 59
6.2 Control in this work
In this work we use the observed methods explained above to control the stochastic
differential equations and thereby the behaviour of the roughening interface by means
of a time-delayed feedback control method.
6.2.1 Control variable
The main quantity for the stochastic differential equations is the rms-roughness. The
development of roughness is correlated to three exponents, which determine the uni-
versality class of growth and thereby determine the growth process. Any possible
control has to influence this evolution and so our control variables are restricted to the
growth exponent β, the roughness exponent αand the dynamic exponent z.
In order to be included in a setup, the variables have to be directly measurable. We
have seen in Sec. 4.2, that, in self-affine systems, the exponents are not independent.
While it is rather difficult to determine the dynamic exponent zdirectly by measuring
the crossover time txor by using the structure function, we can in general control both
other exponents.
For αwe have to calculate the correlation functions (see Sec. 5.4) or the structure
function during control. If we want our tool to be related to realistic growth, the direct
method is very complicated, because it requires the comparison of different system-
sizes. And as already pointed out (see Sec. 5.4), a lot of systems do not reach saturation
during growth. So both for the theoretical model and for an experiment the natural
choice of control is the growth exponent β.
Our control variable in the roughness evolution represents the early phase, so for a
lot of epitaxial systems it can be measured directly from development.
6.2.2 Time delay
The determination of βrequires the roughness evolution to be tracked during devel-
opment. According to the definition of the growth exponent β, our algorithm has to
calculate βin-situ by taking the slope of the roughness w(t)on a logarithmic scale and
therefore requires the previous roughness values to be memorized.
For a measurement in a numerical scheme it is obviously important not to measure
the growth exponent at every single time step dt in order to avoid large effects of
discretization. As control theory is widely used for a lot experimental setups, we give
the system a time delay τbefore it reacts to control tuning. So our scheme calculates
the roughness for a time interval [t−τ, t]from the actual and the memorized value of
time t−τand therefore is a time-delayed feedback control method.
60 6.2. Control in this work
6.2.3 Scheme of control
We have now explained the basic quantities and our control scheme follows as a direct
result.
Figure 6.3: Control of the growth exponent β.
As shown in Fig. 6.3, measurements from the stochastic differential equations we
have solved give the time dependent roughness, which determines the behaviour. The
algorithm calculates theactual growthexponent at time tfrom the roughness evolution,
compares it with the desired value of βand then changes the behaviour using a well
defined strategy.
In detail, the scheme is as follows. First we choose the desired value of the growth
exponent, β0, and select an appropriate time delay τ. Generating sufficiently many
samples of h(x, t), we record w(t−τ)and w(t)(the argument Lwill be omitted from
now on). The local exponent βlocal at time tis defined as
βlocal(t)≡log w(t)−log w(t−τ)
log t−log(t−τ)(6.3)
Depending on the sign and value of βlocal(t)−β0, we adjust the nonlinear coupling, λ,
of the KPZ equation, as follows. First we introduce a control function F(t).
For digital control, we define
F(t)≡(a, if βlocal ≤β0
−a, if βlocal > β0
(6.4)
where the parameter adefines the control ’bit’, i.e. the amount by which λchanges at
each control step.
Alternatively, we also investigate a differential method for which
Chapter 6. Control of stochastic differential equations 61
F(t)≡K(β0−βlocal)(6.5)
and Ksets the amplitude of the control strength. Given one of the two choices of F(t),
the control scheme kicks in at time t0and acts on the nonlinear terms of the different
equations, as we will explain in more detail for each specific equation.
Our scheme is successful if βlocal(t)approaches β0and then settles at the desired
value within a reasonable period of time after the control has been activated.
Control of the KPZ equation
We have already explained in Sec. 5.1 the relationship between the KPZ equation
(Eq. (5.11)) and the EW equation (Eq. (5.2)) when we have zero nonlinearity. The
best method of controlling the exponents is to control the leading term, which is the
nonlinearity λ. The value of the nonlinear term is then no longer constant in time but
changed by the control force F(t)in the following way
λ(t) =
λ0, if t < t0
λ(t−τ) + F(t), if t =tn
λ(tn), if tn< t < tn+1
(6.6)
The control scheme starts at time t0. From then on, the nonlinearity λis updated at
times tn≡t0+nτ,n= 1,2, ..., starting from an initial value λ0. As we know,
zero nonlinearity leads to EW like behaviour of the growth exponent β, where the
value βEW = 0.25 (for 1+1 dimensions) is smaller than that for the KPZ equation
βKP Z = 1/3.
The algorithm has to include that fact. Therefore the control force is added to the
nonlinearity if the local exponent is smaller than that desired (Eq. (6.6)). This assump-
tion is valid if we look at positive nonlinearities λ > 0. For negative nonlinearities
the situation is just the opposite. We have to subtract if the difference βlocal −β0is
negative, showing that the KPZ equation is symmetric in that sense.
Negative lateral growth corresponds to a negative nonlinear term. Such a process
seems to be unusual in crystal growth. But there are quite similar systems with corro-
sive behaviour at the interface which exhibit negative lateral growth.
In Table 6.1 we see a typical setup for our simulations with the initial parameters
ν= 0.1and D= 0.5kept constant for all simulations. The parameters of the control
β0,τ,λ0and the control strength aand Kdetermine the control force F(t).
Control of the MBE equation
The situation for controlling the MBE equation (Eq.(5.23)) is in some ways different
from the control of the KPZ equation. The MBE equation with just the fourth order
term has a proposed exponent of β= 0.375 in 1+1 dimensions and the exponent
62 6.2. Control in this work
EW term (surface tension) ν= 0.1nm2/s
Strength of the Gaussian white noise D= 0.5nm4/s
Initial nonlinear term λ0= 0.00 ...0.40 nm/s
Strength of the digital control a= 0.001 ...0.100 nm/s
time discretization dt = 0.001 ...0.1s
System size L= 256 ...16384 lattice sites (l.s.)
desired value of the growth exponent β0= 0.25 ...0.33
time delay for the feedback τ= 0.1...10 s
Table 6.1: Typical set of parameters for the control of the KPZ equation.
Surface diffusion term ν1= 0.1nm4/s
Strength of the Gaussian white noise D= 0.5nm4/s
Initial nonlinear term λ1,0= 0.00 ...0.10 nm3/s
Strength of the digital control a= 0.001 ...0.100 nm/s
Strength of the differential control K= 0.001 ...0.100
time discretization dt = 0.001 ...0.1s
System size L= 256 ...8192 lattice sites (l.s.)
desired value of the growth exponent β0= 0.33 ...0.375
time delay for the feedback τ= 0.1...10 s
Table 6.2: Typical set of parameters for the control of the MBE equation.
decreases with a nonzero additional term of the nonlinearity λ1. For two dimensions
the same situation occurs when the exponent for a zero nonlinearity λ1is higher.
That is essential for the control scheme. If we want to tune an effective exponent
β0by our time-delayed feedback control scheme, we have to increase the nonlinearity
λ1to get smaller values of the exponents by increasing the function λ1(t). So our
control force F(t)has to work in opposition to the force from the KPZ equation.
For simplicity we change the sign in the control scheme, but one could alternatively
redefine the control forces with a change of sign in Eq. (6.4) (6.5). If we control the
exponents using λ1, our control scheme has to be
λ1(t) =
λ1,0, if t < t0
λ1(t−τ)−F(t), if t =tn
λ1(tn), if tn< t < tn+1
(6.7)
where the procedure for the time delay is the same as for the KPZ equation. In Ta-
ble 6.2 we see a setup of the parameters for this type of equation.
Chapter 6. Control of stochastic differential equations 63
We have now defined a control procedure for the growth equations which will be
applied to control the growth exponent. So the question arises as to whether it is
possible to tune βto all the desired values if the choice is restricted to universality
classes. As one would expect, the answer is no.
But if we conceive of the exponents as defined quantities for the long term and long
range behaviour of the roughening surface, we can control the early states with a well
defined system size Lby effective exponents, so the behaviour can be different.
A second question concerns the behaviour of the roughness exponent αduring
the control. αdoes not differ for the Edwards-Wilkinson and the KPZ equation. So
a useful and successful control maintains these values during the control, otherwise
we would leave the universality classes and the above equation would not explain a
adjustment of local growth exponents within these classes.
For the MBE equation the situation changes, the roughness exponents are not the
same for the two extreme cases, but one would expect αto lie somewhere in a range
between the universality classes for the controlled equation. Table 6.3 sums up the
exponents of the different regimes for the equations we want to control by the meth-
ods explained above. The different exponents for the KPZ equation in 2+1 dimen-
sions denote the different values proposed by Wolf and Kertész [Wol87] and Kim and
Kosterlitz [Kim89].
Kardar-Parisi-Zhang equation β α
1 + 1 dimensions λ= 0 0.25 0.50
1 + 1 dimensions λ > 0 0.33 0.50
2 + 1 dimensions λ= 0 0.00 0.00
2 + 1 dimensions λ > 0 0.20 0.33 Wolf-Kertesz
2 + 1 dimensions λ > 0 0.25 0.40 Kim-Kosterlitz
MBE equation
1 + 1 dimensions λ1= 0 0.375 1.50
1 + 1 dimensions λ1>0 0.33 1.00
2 + 1 dimensions λ1= 0 0.25 1.00
2 + 1 dimensions λ1>0 0.20 0.66
Table 6.3: The critical exponents for the growth equations.
6.2.4 Relation to control methods
A short look at our control scheme shows its relationship to the general methods ex-
plained above. First, of course, we have a time-delayed feedback method as for the
classical control and for chaos control.
Obviously we do not control a chaotic system. On the other hand we have a system
driven by stochastic noise as noise induced roughening. The main difference from a lot
64 6.2. Control in this work
of chaotic systems is the absence of a chaotic attractor and in this sense it is closer to
the classical methods of tuning a developing system. So it is rather difficult to include
classical or chaos control.
The method contains not only the properties of PID controllers but also those of
the TDAS scheme. Our differential control scheme acts in a similar way to a propor-
tional controller, in that it amplifies the difference of the desired exponent. The digital
scheme answers with a step on a step function.
So our scheme combines some properties and findings of both the classical and
chaos control approaches.
Chapter 7
Simulating Stochastic Differential
Equations
In the previous chapters we showed how to control surface roughness by adjusting
the growth exponent βin the early phase of the roughening. We now apply the time-
delayed feedback control method defined in Ch. 6.2 to the equations for crystal growth.
7.1 The Kardar-Parisi-Zhang Equation
In case of the KPZ equation we want to tune the growth exponent βby means of the
nonlinear term λ. We have to ensure that the related exponents are valid for 1+1 dimen-
sions and have to check if we get similar results in comparison to previous findings for
2+1 dimensions or to the proposed exponents, respectively (see table 6.3). So we first
check our numerical scheme for this equation without any control and then continue
with the feedback scheme.
7.1.1 The uncontrolled equation in 1+1 dimensions
For a satisfactory check of the exponents without control the best method is to look
at the long time behaviour of scaling. In Fig. 7.1 we provide a data collapse for the
equation in 1+1 dimensions, where we used two different system sizes L= 1024 and
L= 4096 for the numerical scheme.
Simulations were made for long times. ’Long time’ here means that the roughness
is in the saturation phase as shown in Fig. 7.1. The data collapse was made using the
Family-Vicsek relation (Eq. (4.5)).
First the roughness w(t)was rescaled by w→w/Lα, then the timescale was
rescaled by t→t/Lz. If we have chosen the right values for αand z, the curves
collapse into single curves, as can be seen for the three initial setups.
In all three cases α= 0.50 measured by the height-difference correlation H(r)
(see Sec. 5.4.3 for the method) and is the right value for both the EW universality class
66 7.1. The Kardar-Parisi-Zhang Equation
Figure 7.1: Data collapse for the KPZ equation in 1+1 dimensions w/Lαvs t/Lz
plotted on a logarithmic scale for three different initial setups for the nonlinearity λfor
systemsizes L= 1024 (red lines) and L= 4096 (black lines) (i) λ= 0 shifted by a
factor of 4(ii) λ= 0.1shifted by a factor of 2(iii) λ= 0.25, insets show the height
difference correlation function H(r)measured for r∈[0, L], the broken green lines
are guides to the eyes for the extracted exponents αand β, respectively.
and the KPZ universality class. The dynamic exponent zdiffers and therefore so does
the growth exponent β.
Three different initial nonlinearities λwere used. For zero nonlinearity we see that
the Edwards-Wilkinson equation (β= 0.25,α= 0.5and z= 2) behaves as expected.
The other extreme case has a sufficiently strong nonlinearity (λ= 0.25) to provide a
case of KPZ behaviour (β= 1/3). Here the other exponents α= 0.5and z= 3/2stay
at the KPZ values.
The third case (λ= 0.1) is surprising. We find a local exponent βlocal ∼0.30
which is neither the EW value nor the KPZ value but is nevertheless constant over a
range of more than two decades.
Chapter 7. Simulating Stochastic Differential Equations 67
There are several possible explanations for this behaviour.
•It is possible that saturation sets in before the exponent reaches the KPZ value.
Alternatively the opposing processes, the roughening phase and early saturation,
cancel one another out to give an effective exponent.
•It is also possible, though extremely unlikely, that we have found a totally new
universality class.
•It is also possible that there are some difficulties with the numerical scheme. If
this is true then the numerical scheme needs to be changed.
To get a more detailed view, we made simulations of the early phase for a wide range
of different nonlinearities λ. As simulations of long time behaviour are extremely
computationally demanding, we used shorter simulations and took ensemble averages.
In Fig. 7.2 we have changed the nonlinear term in the range λ∈[0.0,0.8]. The
dashed lines help one to see the limiting exponents, which are the EW (β= 1/4) and
the KPZ value (β= 1/3).
For small nonlinearities we get clear effective exponents, which seem to increase
monotonically with the value of λ. If λis not too large we see the behaviour we expect
1 10 100 1000
t (a.u.)
1
w(t)
λ = 0.00
λ = 0.05
λ = 0.10
λ = 0.15
λ = 0.20
λ = 0.25
λ = 0.40
λ = 0.80
wrms = 0.50 t0.33
wrms = 0.28 t0.25
Figure 7.2: Early roughness evolution of the KPZ equation w(t)vs time twith λ∈
[0.0,0.8] for L= 4096 and time t= 1000 with a time discretization dt = 0.02,
dashed lines denote the limits of the growth exponents for βEW = 0.25 (orange) and
βKP Z = 1/3(green) as guides to the eye.
68 7.1. The Kardar-Parisi-Zhang Equation
and have observed before in Fig. 7.1. For a large enough nonlinearity λ= 0.80 we see
a local exponent which increases and then saturates at the KPZ value β= 1/3. This
indicates that we have not found a new universality class for the equation but rather
a local regime of early development in which the growth exponent βis tunable to a
certain value.
Quite similar behaviour has been found by other authors [Mos91; Gha06]. So we
are not proposing a new universality class, and the limits of the numerical solutions
are still within the expected range. The universality class explains long time and,
more importantly, large scale behaviour, so our results explain behaviour during early
development. But this behaviour could be relevant to realistic setups when we have a
defined limited scale and obviously also a limited time scale.
We have determined the limitsand range of the KPZ equation in 1+1 dimensions where
control can adjust the growth exponent β.
We now take a more detailed look at the calculation of the roughness exponents for
long time behaviour. As one can see in Fig. 7.3, the methodswe have already described
(see Sec. 5.4.3) do actually work. We test the calculation using the height-difference
correlation H(l)(Fig. 7.3 (a)), the height-height correlation C(l)(Fig. 7.3 (b)) and the
structure function S(k, t)(Fig. 7.3 (c)). As expected, the exponent α= 0.5is the same
in all three cases, with a zero nonlinearity λ= 0, a strong nonlinearity λ= 0.25 and
an intermediate value of λ= 0.10. In Fig. 7.3 (b) the simulation plots (solid lines) are
fitted (dashed lines) to the function C(l) = C0exp(−(l/ξ)2α)(see Sec. 5.4.3). For the
structure function S(k, t)we see that all the curves match a single curve with a slope
corresponding to α= 0.5. Here kis scaled so that k= 1 corresponds to l=Lin
phase space and we shall use this scaling in further calculations.
7.1.2 Definition of parameters for the control
The first thing to determine for control is the range of time in which we want to apply
the control. As we can see, for our simulations in arbitrary units the range of clear
effective exponents is t∈[10,1000], where saturation normally sets in after t= 1000.
So, in order to avoid effects produced by the saturation process, control should not be
applied for too long a period. We have therefore restricted our control to this range for
all our simulations.
The next step is to define the time delay τand the strength of the control forces by
means of the parameters aand Kfrom Eq. (6.4), (6.5). Because our simulations are
highly computationally demanding, we aim as far as possible to restrict the range and
use the parameters as a control before starting any simulation. This avoids both long
parameter changes and wasting too much time reaching the right parameters.
Obviously, τ,aand Kare not independent of one another during control. When
we have a very small time delay, we do not take too large a control parameter in order
to avoid numerical instabilities in the scheme. If we take a larger time delay, we have
to ensure that the control force can tune the exponent to the desired value in time, or
else we have to choose parameters aand Kthat are not too small. There are limits
Chapter 7. Simulating Stochastic Differential Equations 69
to the parameters defined by the equation and the control range determined above.
If we assume a limit for our control function λ(t)∈[0,0.25], which is a reasonable
assumption looking at the changeable exponents in Fig. 7.2, then we have to determine
our control using that range. To get a more generalized view we now define a control
factor for digital control.
Ca=a
τ(7.1)
From this factor we can easily calculate the maximum of our range for λ(t)by
∆λ(t) = Ca(tc,end −tc,0)(7.2)
where tc,0is the time of the beginning of control and tc,0is the end. So for a time delay
τ= 1 and a control step a= 0.005 the range is ∆λ∼0.5which is twice the range of
λ(t)and therefore a good choice. Similar possible choices would then be τ= 10 and
10 100 1000
l (l.s.)
10
100
1000
H(l)
λ = 0
λ = 0.1
λ = 0.25
H(l) ~ l2α with α = 0.5
(a)
0 1000
l (l.s.)
0
50
100
C(l)
λ = 0
λ = 0.1
λ = 0.25
fit with C(l)=C(0)exp(-(l/ξ)2α) α ~ 0.51
fit with C(l)=C(0)exp(-(l/ξ)2α) α ~ 0.52
fit with C(l)=C(0)exp(-(l/ξ)2α) α ~ 0.48
(b)
10-4 10-2 100
k
10-2
100
102
104
106
S(k)
λ = 0
λ = 0.1
λ = 0.25
S(k) ~ k-2α+1, α = 0.5
(c)
Figure 7.3: Calculation of the roughness exponent αfor the KPZ equation in 1+1
dimensions in the long time behaviour: (a) the height-difference correlation func-
tion H(l), (b) the height-height correlation function C(l), (c) the dynamic structure
function S(k, t)for three different nonlinearities λ= 0.0(black), λ= 0.1(red) and
λ= 0.25 (blue) with L= 4096,t= 106a.u. and dt = 0.01, the dashed lines show the
fit functions for calculation of the roughness exponents.
70 7.1. The Kardar-Parisi-Zhang Equation
a= 0.05 or τ= 0.1and a= 0.0005 which give the same maximum ∆λ. So the factor
Cagives general predictions as to how to set the initial parameters. The best choice
then depends on the specific development of roughness. A ’coarse control’ or a control
which only changes a few times in the control range of the function λ(t)defines the
upper limit of time delay. A very small time delay is more sensitive to the fluctuations
from the numerical scheme which appear in the numerical integration. Of course one
also has to ensure that the time delay is large enough in comparison with the time step
dt.For differential control the control factor has a similar definition
CK=K
τ(7.3)
which is obviously impossibleto calculate without some ’test’ simulations of the initial
conditions for Kand τ. This is because it depends directly on the difference β0−βlocal.
Digital control only reacts to the sign of this difference. In this sense differential
control is more difficult to apply but, on the other hand, is probably a much faster
control method.
We now need to define either the range in which we want our control to influence
the roughening phase or the times t0for the onset of control and for the end of control.
We therefore clarify the restrictions in our numerical scheme. In Fig. 7.4 we see the
development up to a time t= 10000 for a setup with L= 4096 and a nonlinearity
value of λ= 0.25. In the left panel we show the linear plot and the insets show that
the power law is relatively stable up to t= 1000, with fluctuations appearing in the
range from t= 1000 to t= 2000 and becoming very obvious at t= 10000, whereas
this does not show up so clearly in the logarithmic plot.
0 2000 4000 6000 8000 10000
t (a.u.)
0
1
2
3
4
5
6
7
w(t)
L = 4096 l.s., ν = 0, λ = 0.25, dt = 0.01
200 400 600 800 1000
1,5
2
2,5
3
1500
1000 2000
3
3.5
(a)
100101102103104
t (a.u.)
10-1
100
101
w(t)
L = 4096 l.s., ν = 0, λ = 0.25, dt = 0.01
(b)
Figure 7.4: Roughening of the early time KPZ equation with L= 4096,w(t)vs
t: (a) linear plot, (b) logarithmic plot with a time discretization of dt = 0.01 and a
nonlinearity of λ= 0.25
The origin of these fluctuations can be explained both by the start of a change
in roughening before the saturation phase and by the strong influences of numerical
Chapter 7. Simulating Stochastic Differential Equations 71
System size L= 4096
Time of onset of control t0= 10
End of control te= 1000
Initial nonlinear term λ0∈[0,0.25]
Time delay τ∈(0,1000]
Strength of the digital control a∈(0,0.05]
Strength of the differential control K∈(0,0.10]
desired value of the growth exponent β0∈[1/4,1/3]
time discretization dt ∈(0,0.05]
Averages 25 realizations
Table 7.1: Parameter ranges for the control of the KPZ equation in 1+1 dimensions.
fluctuations. When the evolution obeys a power law, small fluctuations in the numbers
lead to bigger changes in the local exponent for the later times. This is because the
absolute values of the differences between the values of the roughness decrease due to
the logarithmic scale.
Of course, we also get these fluctuations for smaller values, but ours seems to be a
suitable choicefor controllingroughening up tot= 1000. Up to t= 10 theroughening
depends on the initial flat surface, so we set our time t0= 10.
So we now have determined our basic parameters for the control of the KPZ-
equation and also the range within which control of the local exponent is possible.
Table 7.1 lists a summary of these parameters.
7.1.3 Control of the KPZ equation in 1+1 dimensions
We now test our control scheme for the KPZ equation in 1+1 dimensions with these
restriction on the parameters. We check how control works and to what extent the
scheme depends on the basic parameters for certain setups.
Influence of τon control
First we want to test the reaction on different time delays τ. We set an initial nonlin-
earity λ0= 0 and take a control strength with constant values a= 0.01 and K= 0.01.
The desired growth exponent is set to be β0= 0.29.
We now test this setup for three time delays τ∈ {0.01,0.1,1}. In order to make
the influence of the time discretization dt as negligible as possible, we set it to dt =
0.0005. This increases the simulation time but we get clear results that only depend
on τ. In Fig. 7.5 we see the results of the control for a variation of the time delay
τ. In Fig. 7.5 (a) we see that digital control works for the time delay τ= 1 (blue)
and for a value of τ= 5 (orange), where the control adjusts the exponent a little
bit later in the second case. For the smaller values control fails (black, red). In the
72 7.1. The Kardar-Parisi-Zhang Equation
10 100 1000
t (a.u.)
1
2
3
w(t)
a = 0.01, τ = 0.01
a = 0.01, τ = 0.1
a = 0.01, τ = 1
a = 0.01, τ = 5
fit function f(x) = 0.27 x0.293
(a)
10 100 1000
t (a.u)
1
2
3
4
w(t)
K = 0.1, τ = 0.01
K = 0.1, τ = 0.1
K = 0.1, τ = 1
K = 0.1, τ = 5
0.265586 x0.290
(b)
0 200 400 600 800 1000
t (a.u.)
0
0,1
0,2
0,3
0,4
0,5
0,6
λ(t)
a = 0.01, τ = 0.01
a = 0.01, τ = 0.1
a = 0.01, τ = 1
a = 0.01, τ = 5
(c)
0 200 400 600 800 1000
t (a.u.)
0
1
2
3
4
5
λ(t)
K = 0.01, τ = 0.01
K = 0.01, τ = 0.1
K = 0.01, τ = 1
K = 0.01, τ = 5
(d)
Figure 7.5: Influence of time delay on control for digital and differential control, (a) w
vs tfor the digital control with a= 0.01, (b) for the differential control with K= 0.01,
(c) λ(t)for the digital control, (d) λ(t)for the differential control, time delay vary in
τ∈ {0.01,0.1,1,5},dt = 0.0005,λ0= 0.00,β0= 0.29 and 25 averages for all
simulations.
corresponding control functions (Fig. 7.5 (c)) we see the reason for this behaviour.
Whereas for the successful control the function first increases and then stays nearly
constant, it fluctuates widely for smaller τ. This is obviously a reaction to the much
faster control with the smaller time delays. So differences from the ideal case of the
power law of roughening here lead to over-controlled behaviour and thereby to a larger
effective exponent β.
For differential control the situation is much more extreme. In general we see
similar behaviour: the control works for τ= 1 and in the case of τ= 5 does not reach
the value of β0, but stays close to β= 0.25. Control fails for the smaller time delays.
Because of the direct dependence of the control strength on the absolute value of
the difference from the desired value of β0, the fluctuations are much stronger here.
In conclusion, we have found a possible control but anticipate better tuning of
the control strength for other cases of digital control using different time delays. So
although differential control reacts faster, the digital scheme of changing the time delay
Chapter 7. Simulating Stochastic Differential Equations 73
under constant conditions offers a wider range of possibilities.
Influence of control strength on control efficiency
We now take a closer look at how controllers react to a change in the strength of the
control parameters aand K. We again take setups with time delays of dt = 0.0005,
β0= 0.29,λ0= 0 and set the time delay to a constant value τ= 1 for both types
of control. In Fig. 7.6 (a) we see that the control works for a digital parameter of
a= 0.005 (red), but fails for a∈ {0.001,0.02,0.05}. The dashed line here is a fit to
the working control parameter which shows only a slight difference from the desired
value of the effective exponent. Obviously, too small an aleads to a control function
which does not adjust the exponent in the given range of time. This is because of the
absolute added value of the parameter. For parameters that are too large, the changes
are too large for a given difference, so the control functions λ(t)fluctuate more and
the required exponent cannot be reached: the control is too fast for the system to react
normally. That can be seen from looking at the functions λ(t)in Fig. 7.6 (c). For
differential control the behaviour is very similar: the lowest value K= 0.001 gives a
smaller effective exponent and the control strengths K= 0.02 and K= 0.05 produce
larger effectiveexponentsthan desiredand alsocause large fluctuations inthe functions
λ(t)(Fig. 7.6 (d)). The adjustment β0= 0.29 only works for K= 0.006 (red).
So a change in the strength of control using parameters aand Kleads to quite
similar behaviour in both types of control.
If we look at the introduced control factors Ca(Eq. (7.1)) and CK(Eq. (7.3)),
control works here for values of Ca= 0.005 and CK= 0.006.
Simulations with constant Caand CK
We now want to take a look at these artificial parameters.
We again take our setups with time discretizations of dt = 0.0005,β0= 0.29,
λ0= 0.00 and now set the factors at Ca= 0.01 and CK= 0.01, close to the values
of our previous working control. Then we change both the control parameters and the
time delay in simulations and ensure these factors stay constant. In Fig. 7.7 show the
results for constant factors. For digital control, the roughness evolution is adjusted
perfectly for two setups, a= 0.005 with τ= 0.5(orange) and a= 0.002 with τ= 0.2
(blue). For the other setups control fails. The absolute changeable range during control
using the factor Cais constant (here ∆λmax = 0.99, (see Eq. (7.1)), the reason being
slow reaction to changes in the local effective exponent.
For the differential control method this is not the case and therefore all setups
show very similar behaviour. Due to the direct amplifying nature of CKthis leads to a
working control in all cases. The inset in Fig. 7.7 (b) shows that roughness increases
slightly for higher τand K.
If we take a look at the control functions for digital control we see that values
increase for increases in τand a. In the early stage the function increases fast with
74 7.1. The Kardar-Parisi-Zhang Equation
10 100 1000
t (a.u.)
1
2
w(t)
τ = 1, a = 0.001
τ = 1, a = 0.005
τ = 1, a = 0.02
τ = 1, a = 0.05
fit function f(x) = 0.27 x0.288
(a)
10 100 1000
t (a.u)
1
2
w(t)
τ = 1, K = 0.001
τ = 1, K = 0.006
τ = 1, K = 0.02
τ = 1, K = 0.05
fit function f(t) = 0.26 x0.289
(b)
0 200 400 600 800 1000
t (a.u.)
0
0,1
0,2
0,3
0,4
λ(t)
τ = 1, a = 0.001
τ = 1, a = 0.005
τ = 1, a = 0.02
τ = 1, a = 0.05
(c)
0 200 400 600 800 1000
t (a.u.)
0
0,1
0,2
0,3
λ(t)
τ = 1, K = 0.001
τ = 1, K = 0.006
τ = 1, K = 0.02
τ = 1, K = 0.05
(d)
Figure 7.6: Influence of control strength on control for digital and differential control,
(a) wvs tfor the digital control with a∈ {0.001,0.005,0.02,0.05}, (b) for the dif-
ferential control with K∈ {0.001,0.006,0.02,0.05}, (c) λ(t)for the digital control,
(d) λ(t)for the differential control, time delay in τ= 1,dt = 0.0005,λ0= 0.00,
β0= 0.29 and 25 averages for all simulations.
high values, whereas later, because of fluctuations, it cannot decrease fast enough to
give the right exponent.
To conclude: the fast reacting differential control has the advantage of being in-
dependent of τand Kfor constant values of CK, thereby reducing the degrees of
freedom.
Simulations with nonconstant Caand CK
Fig. 7.8 gives a summary of a wide range of possible variations for τ,aand Kfor
the setup we used before. We have classified the results using a color code: green
squares for a very good adjustment in the range ∆β < 0.005 around β0, blue squares
for a functional but imperfect control at ∆β < 0.01 and red squares denote a non-
functional control for ∆β > 0.01. For digital control we see in Fig. 7.8 (a) that the
possible control works around values of τ= 1 and a= 0.01 for small changes. In
Chapter 7. Simulating Stochastic Differential Equations 75
10 100 1000
t (a.u.)
1
w(t)
a = 0.0001, τ = 0.01
a = 0.0005, τ = 0.05
a = 0.002, τ = 0.2
a = 0.005, τ = 0.5
a = 0.05, τ = 5
fit function f(t) = 0.27 x0.287
(a)
10 100 1000
t (a.u.)
1
2
λ(t)
K = 0.0001, τ = 0.01
K = 0.0005, τ = 0.05
K = 0.005, τ = 0.5
K = 0.05, τ = 5
f(t) ~ x0.29
600 1000
t (a.u.)
2
1.8
w(t)
(b)
200 400 600 800 1000
t (a.u.)
0,05
0,1
0,15
0,2
0,25
λ(t)
a = 0.0001, τ = 0.01
a = 0.0005, τ = 0.05
a = 0.002, τ = 0.2
a = 0.005, τ = 0.5
a = 0.05, τ = 5
(c)
0 200 400 600 800 1000
t (a.u.)
0,1
λ(t)
K = 0.0001, τ = 0.01
K = 0.0005, τ = 0.05
K = 0.005, τ = 0.5
K = 0.05, τ = 5
(d)
Figure 7.7: Influence of constant factors Ca=a/τ and CK=K/τ on control, (a)
roughness w(t)vs time tfor different parameters aand τand Ca= 0.01 by digital
control, (b) roughness w(t)vs time tfor different parameters Kand τand CK= 0.01
by differential control, insets show the curve in smaller range to see the differences,
(c) and (d) the corresponding control functions λ(t)
comparison with the differential control in Fig. 7.8 (b), the range is larger but generally
more limited by an upper and lower bound to both τand a. The differential control for
constant K only shows control for a smaller range of τbut does not seem to be limited
by choice of K. So for all Ka corresponding τcan be found.
Nevertheless there are limits due to the fact that when small τare of the same order
of magnitude as the time discretization , τand Kdo not lead to useful control if the
function λ(t)reacts strongly to differences.
So, as explained above, in the case of differential control we can reduce the pa-
rameters over a wide range to the factor CKwhich determines the efficiency of con-
trol. In the case of the initial setup of β0= 0.29 and λ0= 0 this control works for
CK∈[0.005,0.01].
76 7.1. The Kardar-Parisi-Zhang Equation
10-2 10-1 100101
time delay τ (a.u.)
10-4
10-3
10-2
10-1
control strength a
β = 0.29, ∆β < 0.005
β = 0.29, ∆β < 0.01
β = 0.29, ∆β > 0.01
(a)
10-2 10-1 100101
time delay τ (a.u.)
10-4
10-3
10-2
control strength K
β = 0.29, ∆β < 0.005
β = 0.29, ∆β < 0.01
β = 0.29, ∆β > 0.01
(b)
Figure 7.8: Influence of the constants Caand CKdelay on control, digital and dif-
ferential, (a) digital control for different setups of τand a, (b) differential control
for different setups of τand a, categorization in both cases by green squares (good
working control), blue quares (working control) and red squares (no working control),
parameters in all cases λ0= 0.0,β0= 0.29,dt = 0.0005 for 25 averages.
Other control setups
We have made a detailed investigation of one specific initial setup for the time delayed
feedback and know how the method works and what influences determine and restrict
the ranges. It is now possible to tune the control parameters more efficiently for other
setups.
As we saw for the uncontrolled case, we can generate the full range of exponents
between the EW and the KPZ universality classes by changing λwithin a range of
λ∈[0,0.25].
There are three different setups in which our control works for extreme cases.
These cases are:
•an initially zero λ0= 0 corresponding to the KPZ universality class (β= 1/4)
to be controlled to a desired β0= 1/3corresponding to the KPZ universality
class
•an initially strong λ0= 0.25 corresponding to the EW universality class (β=
1/3) to be controlled to a desired β0= 1/4corresponding to the KPZ universal-
ity class
•different initial λ0which stabilize the effective exponents in a range of β0∈
(1/4,1/3)
Chapter 7. Simulating Stochastic Differential Equations 77
By testing these setups we showed that all other possible relevant setups with initial
partial nonlinearities in the range between them also work.
We have already partly shown the third case of analysis of the parameters; we
now check the whole range for an initial nonlinearity of λ= 0 and desired values of
β0∈ {0.25; 0.27; 0.29; 0.31; 0.33}. From the simulations for β0we know that τ∼1
seems to be the best choice for optimal control in both digital and differential control.
We therefore normally restrict simulations to τ= 1, although we have also partly
tested setups with other time delays.
Forthecontrolstrengthswe usesetups ofa∈ {0.005,0.01}and K∈ {0.005; 0.01}
and partly test other setups for the differential control to reproduce the behaviour ex-
plained above, where control seems to depend only on CK. Because simulations with
the prior time discretization dt = 0.0005 are too computationally demanding (a few
hours for single simulation), we reduce the time discretization by a factor of 10 to
dt = 0.005. We thereby reduce the whole simulation time from days to hours, which
suggests that the precision of control is slightly affected.
To analyze the results we took the roughness evolution and measured the change
in the control function λ(t)in situ starting at time t0with λ0. The insets in the upper
left of the diagram show the development of this function during control. We have
already seen slight changes in the late phase of the control time range due to the more
important numerical fluctuations in this range and we shall now take a closer look at the
late phase of all the simulations, as shownin the lower right of the diagram (also double
logarithmic plot). The figures show the fitted effective exponent in the time range after
the control has tuned it to a nearly constant value. In Fig. 7.9 we show the results for
digital control with λ0= 0,a= 0.005 and τ= 1. In general there is the possibility of
control in all cases. Whereas there is nearly perfect control for the required exponent
β0>0.25 (Fig. 7.9 (b - e)), the case of β0= 0.25 is more problematic. Normally
one would expect that, when the initial nonlinearity λ= 0 corresponds to this required
value, it would be easily adjustable, as the scheme just has to stay at a zero value. In
fact we see the effects of numerical solutions, where small changes in the roughness
evolution activate a change of λ. So, in all cases of nonzero λ, which we always get
in the case of partly measured values βeffective <0.25, the tendency is to produce
β > 0.25. Summing gives an exponent of β > 0.25. Changing the condition that
λ≥0does not change the problem, because negative λalso leads to bigger growth
exponents due to the symmetric nature of the equation (see [Mos91; Bar95]).
As already explained, we are going neglect that case, because in experimental se-
tups it would be difficult to change the sign of the nonlinear term corresponding to
a real physical quantity. Nevertheless we also tested the control without any restric-
tions on the sign of λ, but did not find noticeable differences, so here we only show
results that neglect such schemes. In the control functions we see that a small increase
in nonlinearity leads to the control behaviour, which then stabilizes for higher values
of β0. For the higher values of β0the control is perfectly stable, and in the case of
β0= 0.27 too, as can be seen in the inset, small fluctuations lead to a bigger local
exponent, which is then compensated for by the control.
78 7.1. The Kardar-Parisi-Zhang Equation
In the differential scheme we see exactly the same behaviour, except in the prob-
lematic case of β0= 0.25 (Fig. 7.10). Here we get better control behaviour with a
faster control, which gives a stable exponent of β= 0.254. Nevertheless fluctuations,
which are then controlled by the scheme, can also be seen in this case (inset).
If there is a problem adjusting the exponents to the EW universality class from a
zero nonlinearity due to the numerical behaviour described above, then other setups
usually fail to stabilize β= 0.25. A control that works for β0>0.25 can be seen in
Fig. 7.11. The question is whether the good working control in the case of β0is only
an effect of the numerics or if it is relevant for experimental setups. We do not want
numerical fluctuations in experimental setups, so the control of a setup which normally
tends to have EW-behaviour should tune the nonlinearity to zero.
A good indication that there is a numerical reason for the behaviour is that the
initial conditions are chosen so that the value of λstays at zero.
For the reasons already explained for the symmetric border at λ= 0, this behaviour
is not seen in other setups, not even in the opposite case of a strong nonlinearity (λ0=
0.25), which can be controlled to a KPZ exponent (β0= 0.33).
We now check other setups for the control with different initial nonlinearities. We
restrict ourselves here to setups with λ0= 0.10 and λ0= 0.25, which mark the
important changes in the initial nonlinearity. For further information about additional
simulations see the Appendix.
Now we look at an initial nonlinearity of λ= 0.10 for both the digital and the
differential control. In Figs. 7.12 and 7.13 we can see the setups for λ0= 0.10.
As already described, control fails for β0= 0.25, but the other cases show stable
behaviour and differential control seems to be nearly perfect in all cases.
The range in which control changes the nonlinearity is much smaller than for the
case λ= 0.00. That is obviously the case for these setups, because the initial condi-
tions are closer to those required. So, as can be seen in the scheme without control
(Fig. 7.4), this initial setup without control produces an exponent between the EW and
the KPZ class. So here it is much easier to tune the function λ(t)to the correct value.
That is why, in the case of β0= 0.29, the range for both control types fluctuates be-
tween λ(t)∈[0.1,0.12], and increases in the case of higher exponents to a maximum
of λ(t)∼0.16 for β0= 0.33.
These values give also an indication of how the system tends to behave in the KPZ
class. It complies with the proposed value λ= 0.25 as a "strong coupling" value.
For the setup of this strong nonlinearity we now look again at the results. In
Fig. 7.14 we see that the control works very well for higher β0and higher control
strengths also lead to control behaviour (see appendix). It is not surprising that when
the control works for small initial nonlinearities, it also works for larger ones. We can
see that the function decreases slightly and then stabilizes, with more fluctuations in
the late phase, but with a clearly stabilized growth exponent.
Chapter 7. Simulating Stochastic Differential Equations 79
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.25, a = 0.005
β = 0.259
600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.08
0.04
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.27, a = 0.005
β = 0.268
600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.08
0.04
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.29, a = 0.005
β = 0.290
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
0.1
λ(t)
(c)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.31, a = 0.005
β = 0.308
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.08
0.06
0.04
0.12
λ(t)
(d)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.33, a = 0.005
β = 0.326
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.08
0.06
0.04
0.1
0.14
λ(t)
(e)
Figure 7.9: Digital control for the KPZ equation in 1+1 dimensions with a control
setup: λ0= 0.00 and a= 0.005 for five different desired control values of: (a) β0=
0.25, (b) β0= 0.27, (c) β0= 0.29, (d) β0= 0.31, (e) β0= 0.33, time discretization
dt = 0.005, upper left insets show the functions λ(t), lower right insets show the
roughness in the late phase in double logarithmic plot.
80 7.1. The Kardar-Parisi-Zhang Equation
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.25, K = 0.005
β = 0.254
400 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.09
0.06
0.04
0.02
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.27, K = 0.005
β = 0.268
600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.08
0.06
0.04
0.12
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.29, K = 0.005
β = 0.289
600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.08
0.06
0.04
0.12
λ(t)
(c)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.31, K = 0.005
β = 0.312
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.08
0.06
0.04
0.12
λ(t)
(d)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.33, K = 0.005
β = 0.329
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.08
0.06
0.04
0.1
0.14
λ(t)
(e)
Figure 7.10: Differential control for the KPZ equation in 1+1 dimensions with λ0=
0.00 and K= 0.005 for five different desired control values of: (a) β0= 0.25, (b)
β0= 0.27, (c) β0= 0.29, (d) β0= 0.31, (e) β0= 0.33, time discretization dt = 0.005,
upper left insets show the functions λ(t), lower right insets show the roughness in the
late phase in double logarithmic plot.
Chapter 7. Simulating Stochastic Differential Equations 81
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.27, a = 0.01
β = 0.279
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.08
0.06
0.04
0.12
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.27, K= 0.01
β = 0.268
400 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.08
0.06
0.04
0.1
0.14
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.29, a = 0.01
β = 0.296
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.08
0.06
0.04
0.1
0.14
λ(t)
(c)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.29, K = 0.01
β = 0.294
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.08
0.06
0.04
0.1
0.16
λ(t)
(d)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.31, a = 0.01
β = 0.314
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.08
0.06
0.04
0.1
0.14
λ(t)
(e)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.31, K = 0.01
β = 0.308
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.08
0.06
0.04
0.1
0.16
λ(t)
(f)
Figure 7.11: Digital and differential control for the KPZ equation in 1+1 dimensions
with λ0= 0.00,a= 0.01 respectively K= 0.01 for three different desired control
values of: (a,b) β0= 0.27, (c,d) β0= 0.29, (e,f) β0= 0.31, time discretization
dt = 0.005, upper left insets show the functions λ(t), lower right insets show the
roughness in the late phase in double logarithmic plot.
82 7.1. The Kardar-Parisi-Zhang Equation
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.25, a = 0.005
β = 0.265
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.08
0.06
0.04
0.1
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.27, a = 0.005
β = 0.275
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.08
0.1
0.12
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.29, a = 0.005
β = 0.296
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.08
0.1
0.12
0.14
λ(t)
(c)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.31, a = 0.005
β = 0.314
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.1
0.12
0.16
0.14
λ(t)
(d)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.33, a = 0.005
β = 0.333
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.12
0.18
0.16
0.14
λ(t)
(e)
Figure 7.12: Digital control for the KPZ equation in 1+1 dimensions with a control
setup: λ0= 0.10 and a= 0.005 for five different desired control values of: (a) β0=
0.25, (b) β0= 0.27, (c) β0= 0.29, (d) β0= 0.31, (e) β0= 0.33, time discretization
dt = 0.005, upper left insets show the functions λ(t), lower right insets show the
roughness in the late phase in double logarithmic plot.
Chapter 7. Simulating Stochastic Differential Equations 83
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.25, K = 0.005
β = 0.264
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.08
0.06
0.04
0.1
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.27, K = 0.005
β = 0.274
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.08
0.06
0.1
0.12
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.29, K = 0.005
β = 0.290
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.08
0.1
0.12
0.14
λ(t)
(c)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.31, K = 0.005
β = 0.310
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.1
0.12
0.14
λ(t)
(d)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.33, K = 0.005
β = 0.328
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.1
0.12
0.16
0.14
λ(t)
(e)
Figure 7.13: Differential control for the KPZ equation in 1+1 dimensions with λ0=
0.10 and K= 0.005 for five different desired control values of: (a) β0= 0.25, (b)
β0= 0.27, (c) β0= 0.29, (d) β0= 0.31, (e) β0= 0.33, time discretization dt = 0.005,
upper left insets show the functions λ(t), lower right insets show the roughness in the
late phase in double logarithmic plot.
84 7.1. The Kardar-Parisi-Zhang Equation
100101102103
t (a.u.)
1
w(t)
λ0 = 0.25, β0 = 0.25, a = 0.005
β = 0.266
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.24
0.1
0.18
0.14
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.25, β0 = 0.31, a = 0.005
β = 0.31
600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.24
0.18
0.14
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.25, β0 = 0.33, a = 0.005
β = 0.331
600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.22
0.26
0.18
0.14
λ(t)
(c)
Figure 7.14: Digital control for the KPZ equation in 1+1 dimensions with a control
setup: λ0= 0.25 and a= 0.005 for three different desired control values of: (a)
β0= 0.25, (b) β0= 0.31, (c) β0= 0.33, time discretization dt = 0.005, upper left
insets show the functions λ(t), lower right insets show the roughness in the late phase
in double logarithmic plot.
Chapter 7. Simulating Stochastic Differential Equations 85
100101102103
t (a.u.)
1
w(t)
λ0 = 0.25, β0 = 0.25, K = 0.005
β = 0.266
600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.24
0.1
0.18
0.14
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.25, β0 = 0.27, K = 0.005
β = 0.273
600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.24
0.18
0.14
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.25, β0 = 0.29, K = 0.005
β = 0.295
600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.24
0.18
0.14
λ(t)
(c)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.25, β0 = 0.31, K = 0.005
β = 0.312
600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.24
0.18
0.14
λ(t)
(d)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.25, β0 = 0.33, K = 0.005
β = 0.332
600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.22
0.2
0.24
λ(t)
(e)
Figure 7.15: Differential control for the KPZ equation in 1+1 dimensions with λ0=
0.25 and K= 0.005 for five different desired control values of: (a) β0= 0.25, (b)
β0= 0.27, (c) β0= 0.29, (d) β0= 0.31, (e) β0= 0.33, time discretization dt = 0.005,
upper left insets show the functions λ(t), lower right insets show the roughness in the
late phase in double logarithmic plot.
86 7.1. The Kardar-Parisi-Zhang Equation
7.1.4 The uncontrolled equation in 2+1 dimensions
We now want to look at the behaviour of the KPZ equation in 2+1 dimensions. As
depicted in Fig. 7.16, the situation here is much more complicated. In the 1+1 dimen-
sional case we got clear effective exponents in the early phase but they do not appear
here. We used setups for the nonlinear terms in the equation in the range λ∈[0,0.1]
1 10 100 1000 10000
t (a.u.)
1
2
3
w(t)
λ = 0.00
λ = 0.02
λ = 0.05
λ = 0.06
λ = 0.08
λ = 0.10
w(t)~ t0.09
w(t) ~ t0.31
(a)
1 10 100 1000 10000
t (a.u.)
1
2
3
w(t)
λ = 0.00
λ = 0.02
λ = 0.04
λ = 0.05
λ = 0.07
λ = 0.09
λ = 0.10
w(t) ~ t0.07
w(t) ~ t0.33
(b)
Figure 7.16: Longtime roughness evolution of the KPZ equation in 2+1 dimensions:
wvs time tfor different values of λ(λ∈[0.0,0.10]) for (a) L= 32 ×32 and (b)
L= 64×64 and time t= 1000, dashed lines denote the limits of the growth exponents
for (a) β∼0.09 (b) β∼0.07 (green) and (a) β∼0.31 and (b) β∼0.33.
with system sizes L= 32 ×32 l.s. (Fig. 7.16 (a)) and L= 64 ×64 l.s. (Fig. 7.16 (b)).
Although we can not determine the exponents exactly by a direct method, we can
nevertheless get an impression of the range in which βis valid for our scheme.
We can thereby check if our numerical scheme works. The bounds of possible ex-
ponents in the early phase are denoted by the dashed lines. We do not see a wide range
of exponents over more than one decade, but on the other hand the dashed lines indi-
cate that the exponents could be in the range β∈[∼0.09; ∼0.3] for both system sizes.
This agrees with recent studies of the values of the KPZ-equation in 2+1 dimensions
[Mos91]. Additionally, as in our findings for 1+1 dimensions, the local exponents and
also the roughness w(t) increase with the value of λ.
This behaviour is not surprising since, as we have already pointed out in Sec. 5.1.2,
the determination of the critical exponents of the KPZ equation in 2+1 dimensions is
still an open problem. It would be much more surprising if we could determine them
by a simple direct method.
Most recent studies have tried to tackle this problem using stochastic models and
Monte Carlo or Kinetic Monte Carlo methods. The renormalization method fails in
this case [Bar95; Mic04]. Models (see also Sec. 4.4.2) proposed for KPZ behaviour
are expected to be in the class of ballistic deposition models (see Sec. 4.4.2). A variety
of these models have appeared during the last decade, some using deposition to ex-
plain crystal growth ([Sar96; Osk06; Chi99]), others describing the two type particle
Chapter 7. Simulating Stochastic Differential Equations 87
system ([Kol06]) and treating a lot of very different problems concerning growth and
fluctuation phenomena.
All these slightly different approaches make the assumption that ballistic deposi-
tion models can explain phenomena related to the KPZ equation described above. The
extracted exponents vary between β∈[0.1,0.25] and are close to the expected ex-
ponent β= 0.25. But the exact values remain unknown (newer findings for specific
problems can be found in [Hor06; Rei06; Gha06; Fog06]). A very short analysis of
simple ballistic deposition where we implemented the simple version introduced by
Meakin et al [Mea86] and later further explained by Baiod et al [Bai88] can be seen in
the App. B.1. Our results agree with their findings and the exponents are in the range
expected for the KPZ universality class.
If our numerical solution of the equation does not give the growth exponent di-
rectly, nevertheless the control can give some indication of its value.
Because of the behaviour shown in Fig. 7.16 we can not ensure that our numerical
scheme will work, so we have to strengthen the approach by looking at the roughness
exponents.
For this calculation we make use of both the height-height correlation and the
height-difference correlation function. In Fig. 7.17 we see the behaviour of the correla-
tion functions for one setup with three different initializations of the random generator.
We took a nonzero nonlinearity λ= 0.05 for a now larger system of L= 128×128 l.s.
and we use this for the control too. Although the extracted exponents αfor the height-
1510 50 100
l (l.s.)
4
6
8
10
H(l)
fit with H(l) ~ l2α, α = 0.37
fit with H(l) ~ l2α, α = 0.42
fit with H(l) ~ l2α, α = 0.39
(a)
0510 15 20 25 30
l (l.s.)
0
1
2
3
4
5
6
C(l)
fit with C(l) = C(0)exp(-(l/ξ)2α), α = 0.33
fit with C(l) = C(0)exp(-(l/ξ)2α), α = 0.37
fit with C(l) = C(0)exp(-(l/ξ)2α), α = 0.36
(b)
Figure 7.17: Determination of the roughness exponent for different samples for L=
128 ×128,λ= 0.05 and time t= 1000, (a) height-difference correlation, dashed lines
are fits for the small length behaviour to calculate α(b) height-height correlation, three
initializations of the random generator were used, dashed lines are fits with C(l) =
C0exp(−(l/ξ)2α).
height correlation give smaller values (α∼0.35) than for the height-difference method
(α∼0.40) both values are in the range proposed for the KPZ model by numeri-
88 7.1. The Kardar-Parisi-Zhang Equation
System size L= 128 ×128
Time of onset of control t0= 10
End of control te= 1000
Initial nonlinear term λ0∈[0,0.1]
Time delay τ= 1
Strength of the digital control a∈(0.005 : 0.01]
Strength of the differential control K∈(0,0.10]
desired value of the growth exponent β0∈[0,1/4]
time discretization dt = 0.005
Averages 10 realizations
Table 7.2: Parameter ranges for the control of the KPZ equation in 2+1 dimensions.
cal solutions of the equation [Mos91] or using different ballistic deposition models
[Bai88; Bar95; Mic04; Gha06; Sar96].
We do not see clear growth exponents. That may be due to the fact that saturation
sets in when the nonlinear term in the equation becomes responsible for roughening.
Alternatively we see a short very early phase, also called random growth ([Rei06]),
which then reaches the saturation phase very fast. The phase in between, called the
correlated growth phase by Reis (it is responsible for the growth exponent), then can
become very small (see [Rei06]).
In this case, control can be applied to the equation to stabilize it in a given range.
7.1.5 With control in 2+1 dimensions
We now determine the range for control. We again use a range t∈[10,1000]. Once
again the control sets in at time t0= 10 in order to exclude effects occurring during
the very early phase (Fig. 7.16).
The question is, if it is useful to apply our scheme here. Further investigations will
have to clarify what control can tell us about the behaviour of the continuum function,
but we nevertheless tried control and got surprising results.
At first setups for L= 128 ×128 l.s. with a strong initial nonlinearity λ= 0.10
were investigated, which should give larger βcorresponding to the KPZ class (consis-
tent with Moser et al [Mos91]).
As we can see in Fig. 7.18, the equation for the digital scheme shows control
behaviour in 2+1 dimensions, too. We tried to adjust the exponents between those
expected from the EW class (β= 0) and the KPZ class (β∼0.25). Control for
β > 0.30 failed in all cases, but we got local control for β0≤0.25. The scheme
adjusts the lower value of β0for only a very small time range, but seems to work very
well for the desired exponents β0∈ {0.20; 0.25}.
Chapter 7. Simulating Stochastic Differential Equations 89
Looking at the insets, it can be seen that fluctuations arise in the cases of β0= 0.2
and β0= 0.25, where the roughness increases briefly and is then restabilized by the
scheme to the desired exponent.
For values of βcloser to the EW class the local increase in roughness is not resta-
bilized. So if the aim is to adjust and then stabilize the values using the time delayed
feedback, then control obviously fails in this case.
The function λ(t)shows similar behaviour as in the 1+1 dimensional case: it first
decreases and then stabilizes at a certain value. When the setups fail we observe first
an initial decrease and then a monotonic increase in the fluctuations of λ. We had
similar problems controlling the EW value in the case of the 1+1 dimensional equa-
tion, but here this deviation is much more relevant and, in contrast to the case in 1+1
dimensions, can not be controlled. Different factors could give rise to this behaviour:
•the problem of a zero nonlinearity λwhich acts as a border, where all other λ
lead to higher exponents (symmetry of the equation)
•too large fluctuations in the late time range
•too small system sizes, which encourage fluctuations
•the EW class exponent is generally not adjustable
The first point is partly responsible but, as we saw for 1+1 dimensions, its influence
can be decreased by decreasing the time discretization dt. We proved that point, but
got no noticeable differences.
If those fluctuations which can notbe compensated for fast enough play an essential
role, then differential control should be more stabilizing as a fast reacting control (see
the results for KPZ equation in 1+1 dimensions). And in fact, if we look at the results
for the same initial setups with the differential scheme, the control is also better for
smaller exponents (see Fig. 7.19).
But there are stilleffects on the evolution of roughness. The tendency tolate rough-
ening against the control is still present.
The fourth point we can simply not prove here. If we see an improvement when we
change from digital time delay to differential time-delayed feedback, it might indicate
that control is also possible for small values of β0.
However the EW class with β= 0 is a special case. Here we can not see a really
stable exponent in the roughness evolution (see Fig. 7.16). So β= 0 just means that
the roughness scales logarithmically with t.
Although we have not entirely solved this problem, we strongly suggest that tests
be made with ballistic deposition models and control to reproduce the behaviour and
give further information.
90 7.1. The Kardar-Parisi-Zhang Equation
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.25, a = 0.01
β = 0.03
100 200 600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.05, a = 0.01
β = 0.055
100 200 600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.10, a = 0.01
β = 0.104
200 600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.04
λ(t)
(c)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.15, a = 0.01
β = 0.157
200 600 1000
t (a.u.)
3
3.5
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.04
λ(t)
(d)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.20, a = 0.005
β = 0.208
600 1000
t (a.u.)
4.5
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.04
λ(t)
(e)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.25, a = 0.005
β = 0.255
400 1000
t (a.u.)
5
4.5
5.5
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.08
0.06
0.04
0.1
λ(t)
(f)
Figure 7.18: Digital control for the KPZ equation in 2+1 dimensions with λ0= 0.10,
β0∈ {0.00; 0.05; 0.10; 0.15; 0.20; 0.25}, time delay τ= 1 and control strengths of
a= 0.01 ((a), (b), (c), (d)) and a= 0.005 ((e), (f)), upper left insets show the functions
λ(t), lower right insets show the roughness in the late phase in a logarithmic plot.
Chapter 7. Simulating Stochastic Differential Equations 91
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.00, K = 0.01
β = 0.008
100 400 600
t (a.u.)
2.2
2.4
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.08
0.06
0.04
0.02
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.05, K = 0.01
β = 0.053
100 400 800
t (a.u.)
2.4
2.6
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.08
0.06
0.04
0.02
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.10, K = 0.01
β = 0.107
100 400 800
t (a.u.)
2.1
2.4
2.7
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.08
0.06
0.04
0.02
λ(t)
(c)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.15, K = 0.01
β = 0.151
200 600 1000
t (a.u.)
4
3.5
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.08
0.06
0.04
0.1
λ(t)
(d)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.20, K = 0.01
β = 0.203
600 800 1000
t (a.u.)
5
4.5
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.08
0.06
0.1
λ(t)
(e)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.1, β0 = 0.25, K = 0.01
β = 0.251
200 600 1000
t (a.u.)
4
5
4.5
5.5
6
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.08
0.06
0.04
0.1
λ(t)
(f)
Figure 7.19: Differential control for the KPZ equation in 2+1 dimensions with λ0=
0.10,β0∈ {0.00; 0.05; 0.10; 0.15; 0.20; 0.25}, time delay τ= 1 and a control strength
of K= 0.01, upper left insets show the functions λ(t), lower right insets show the
roughness in the late phase in a logarithmic plot.
92 7.1. The Kardar-Parisi-Zhang Equation
We get the best results for the desired exponents around the values expected for
KPZ-behaviour (β0∈ {0.2; 0.25; 0.3). Here we see really stable exponents in the
region where the control acts on the equation. This is surprising, because we do not
get such clear behaviour without control. There could be various reasons for this.
Where the nonlinearity is the leading term in the equation and depends strongly on
local gradients of the height function, then, by definition, the control acts not only to
control the equation but also to control the unwanted numerical fluctuations.
The nonlinearitydecreases briefly in all cases, and then stabilizes at different values
for the desired exponents. In the late phase, where numerical fluctuations are much
more in evidence, small β0can not be controllled. So roughness increases again for
small β0, whereas it does not increase for the nearby KPZ exponents.
This may indicate that in this case we can control roughness and also get some
information about a realistic KPZ exponent in 2+1 dimensions.
We now also show digital control and differential control setups for λ0= 0.00 in
Fig. 7.20,
Whereas the control adjusts the exponents in the expected way for the higher ex-
ponents, it fails for small β0(not shown). In the setups the control functions λ(t)show
behaviour similar to that observed for 1+1 dimensions. For small initial nonlinearities
and higher exponents the functions increase to a certain value.
So also in this case, the growth exponent seems to be only adjustable around the
value β0= 1/4, where it is very well tunable.
Nevertheless, there should be some comparable results from, for instance, ballistic
deposition models related to KPZ-behaviour to ensure that the above explanations are
indeed responsible for the behaviour.
For initial nonlinearities in between the presented values we see very similar be-
haviour, some of the additional simulations can be seen in the Appendix A.2.
Chapter 7. Simulating Stochastic Differential Equations 93
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.20, a = 0.01
β = 0.21
600 1000
t (a.u.)
3.5
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.04
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.25, a = 0.01
β = 0.255
600 1000
t (a.u.)
4
4.5
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.08
0.04
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.20, K = 0.01
β = 0.203
600 1000
t (a.u.)
3
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.04
λ(t)
(c)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.0, β0 = 0.25, K = 0.01
β = 0.251
600 1000
t (a.u.)
3
3.5
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.04
λ(t)
(d)
Figure 7.20: Digital and differential control for the KPZ equation in 2+1 dimen-
sions with λ0= 0.0,β0∈ {0.20; 0.25}, time delay τ= 1 and a control strength
of a= 0.01,K= 0.01, respectively. Upper figures: digital contro, lower figures: the
corresponding differential cases. Upper left insets show the functions λ(t), lower right
insets show the roughness in the late phase in a logarithmic plot.
94 7.2. The MBE Equation
7.2 The MBE Equation
We have already shown that our control scheme works for the KPZ equation in 1+1
dimensions and have the impression that it could also partly work in 2+1 dimensions.
We now want to apply our scheme to the growth equation proposed to explain Molecu-
lar Beam Epitaxy. In general control should also be applicable to the growth exponent
in this equation. For the MBE equation with λ1= 0 (Eq.(5.21)) and the equation with
λ1>0(Eq.(5.23)) the roughness exponents αare not the same, as has already been
explained in Sec. 5.2.
The proposed exponents are:
β= 3/8α= 3/2z= 4 for λ1= 0
β= 1/3α= 1 z= 3 for λ1>0(7.4)
We follow the same steps as for the KPZ equation. First we apply our scheme to 1+1
dimensions.
7.2.1 Without control in 1+1 dimensions
In the case of 1+1 dimensions the fourth order term makes it rather difficult to see any
saturation in the roughness. Therefore we first use very small systems L= 32 l.s.
and L= 64 l.s. which obviously are not useful for control. Nevertheless they should
show saturation and thereby help test the numerical scheme and the determination of
the exponents without control by rescaling.
In Fig. 7.21 we see the data collapse from the rescaled functions and can determine
the exponents. As before, we have used three different setups, λ1∈ {0,0.05; 0.10}for
a time t= 100000. We see that the data collapse into single curves for all setups and
we get slightly different values for the growth exponent consistent with the proposed
exponents.
We now look more closely at the early behaviour of a larger system L= 8192,
which appears reasonable for the control scheme. As we can see in Fig. 7.22, we get
differences that look very small on the logarithmic scale. Obviously the differences
appear more clearly in the late phase between t∼500 and t= 10000.
The zero nonlinear term gives an exponent of β= 0.374 close to β= 3/8and the
nonlinear term λ1= 0.08 gives a value β= 0.336 close to the value β= 1/3.
Of course, if our solution reproduces the scaling of the MBE class, the saturation
for this system size can not be reached in a computationally useful time. If we assume
an exponent of α= 1 for nonzero nonlinearity and get a time txfrom the above
simulation, the saturation then sets in at t∼500000 and can be clearly seen for t >
10000000 with one decade saturation. For zero nonlinearities the case would be much
more extreme. As we see in Fig. 7.21, saturation sets in later for tx∼5000 and with an
expected exponent of α= 1.5we would see saturation at approximately t∼20 ·106.
Chapter 7. Simulating Stochastic Differential Equations 95
0,0001 1
t/Lz
0,001
0,01
0,1
w/Lα
λ1 = 0.10
λ1 = 0.05
λ1 = 0
α = 1, z = 3
α ~ 1.3, z ~ 3.6
α = 1.5, z = 4
Figure 7.21: Roughening of the MBE equation in 1+1 dimensions for setups with L=
32 and L= 64,t= 100000 and three different initial nonlinearities λ∈ {0; 0.05; 0.10}
with a time step dt = 0.05, setup for λ= 0.1shifted by a factor 4in y-axis, setup for
λ= 0.05 shifted by a factor 2in y-axis, 20 averages for both systems.
So it is obviously impossible to reach saturation with a simulation. The time scale
is different for the MBE equation in 1+1 dimensions so we have to change the time
range in which we apply the scheme of control. We can use the same onset of control
time t0= 10 and, if early roughening from the flat surface does not influence the
behaviour of the equation, we expand the control to te= 10000. This is much more
computationally demanding but promises clear results. In the case of the KPZ equation
we saw that distinguishing the five different desired growth exponents β0is quite easy,
but here we restrict ourselves to three different values: the limiting exponents β0= 3/8
and β0= 1/3and one exponent in between, β0= 0.35. The control results would
otherwise be speculative.
As already explained, the critical exponents depend on long time scaling behaviour.
To ensure scaling for the early development of the interface as well we have to check
the roughness exponents.
Therefore we calculate the structure function. In Fig. 7.23 we see the results of
rescaling, once again using the Family-Vicsek relation. We used very different setups
with different system sizes L∈ {256; 1024; 4096; 8192}, a nonlinearity λ1= 0 and
different times. In this case not all interfaces reach saturation. In the left hand panel
we see the unrescaled functions, which in the case of scaling have to match the others
in the descending part of the curve. After rescaling by k→kt1/z and S(k, t)→
S(k, t)k−(2α+1) they have to collapse into one single curve as we can see in the right
96 7.2. The MBE Equation
100 1000 10000
t (a.u.)
2
3
4
5
6
7
w (t)
λ1 = 0, β = 0.374
λ1 = 0.04, β= 0.354
λ1 = 0.08, β= 0.336
Figure 7.22: Early roughening of the MBE equation in 1+1 dimensions for a setup
with L= 8192,t= 10000 and five different initial nonlinearities λ∈ {0; 0.04; 0.08}
with a time step dt = 0.01.
hand panel. This rescaling only works if we use the correct exponents, so we used
α= 3/2and z= 4, giving a growth exponent β= 3/8(β=α/z). Having described
10-4 10-3 10-2 10-1 100
k
10-2
100
102
104
S(k,t)
L = 4096, λ1 = 0.00, t = 1000
L = 4096, λ1 = 0.00, t = 10000
L = 8192, λ1 = 0.00, t = 10000
L = 256, λ1 = 0.00, t = 100000
L = 1024, λ1 = 0.00, t = 100000
fit with k-(2α+1), α = 1.5
(a)
100
10-2
10-3
k t1/z
10-12
10-4
S(k,t) k2a+1
L = 4096, λ1 = 0.00, t = 1000
L = 4096, λ1 = 0.00, t = 10000
L = 8192, λ1 = 0.00, t = 10000
L = 256, λ1 = 0.00, t = 100000
L = 1024, λ1 = 0.00, t = 100000
(b)
Figure 7.23: Data collapse by structure function S(k, t)for the MBE equation in
1+1 dimensions for five different setups (legends): (a) the structure function, (b) the
rescaled function by the Family-Vicsek relation.
the scheme without control, we now take a look at behaviour for "small" times. In
case of the MBE equation for 1+1 dimensions we get relatively clear-cut behaviour
for the growth exponent over a wide range. In comparison to the KPZ equation, the
difference between zero and nonzero nonlinearity λ1appears at a much later phase
of the roughness evolution for similar values of the simulation parameters, namely
Chapter 7. Simulating Stochastic Differential Equations 97
System size L= 8192 l.s.
Time of onset of control t0= 10
End of control te= 10000
Initial nonlinear term λ1,0∈[0,0.1]
Time delay τ= 1
Strength of the digital control a∈(0.0005,0.002]
Strength of the differential control K∈(0.0005,0.002]
desired value of the growth exponent β0∈[1/3,3/8]
time discretization dt = 0.01
Averages 10 realizations
Table 7.3: Parameter ranges for the control of the MBE equation in 1+1 dimensions
L= 4096l.s.,D= 0.50 and ν1= 0.10.
7.2.2 With control in 1+1 dimensions
By tests for control with that parameters, our control seemed to work, but did not show
really clear exponents without large fluctuations (not shown).
So we have to enlarge the systemsize and the control time.
In the simulations here shown we used L= 8192 l.s. (see appendix for further
simulations) and a larger time te= 10000. Time discretization is set to dt = 0.01 in
all setups. Table 7.3 lists the used parameters for the shown results.
In Fig. 7.24 and Fig. 7.25 we show the results for an initial nonlinearity of λ1,0=
0.0.In contrast to the KPZ equation, this initialization corresponds to a higher value of
the growth exponent as explained in detail in Sec. 6.2. A working control also has to
react in a contrasting way. This behaviour can be seen for the setups in both digital
and differential control. In order to adjust the desired exponents, the control functions
now increase to tune the lower β0, as predicted by the theory.
As expected we get a control behaviour for the MBE equation in 1+1 dimensions.
Due to the required long time simulations to clarify the difference between the two
universality classes we here restrict to the setups shown, where the uncontrolled results
and the results with digital and differential control for λ1show, that also for other
setups one can expect working adjustment of the desired exponents (see Appendix for
further simulations).
98 7.2. The MBE Equation
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.0, β0 = 0.33, a = 0.0005
β = 0.338
1000 t (a.u.)
3
4
5
6
w(t)
simulation curve
fitted curve
ideal β0 curve
0 3000 10000
t (a.u.)
0.02
0.06
0.04
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.0, β0 = 0.35, a = 0.0015
β = 0.353
10001000 10000
t (a.u.)
3
4
5
6
w(t)
simulation curve
fitted curve
ideal β0 curve
0 3000 10000
t (a.u.)
0.02
0.06
0.04
0.1
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.0, β0 = 0.375, a = 0.0015
β = 0.373
10001000 10000
t (a.u.)
3
4
5
6
w(t)
simulation curve
fitted curve
ideal β0 curve
0 3000 10000
t (a.u.)
0.02
0.06
0.04
λ(t)
(c)
Figure 7.24: Digital control for the MBE equation in 1+1 dimensions with a control
setup: λ1,0= 0.00 and a= 0.005 for three different desired control values of: (a)
β0= 0.33, (b) β0= 0.35, (c) β0= 0.375, time discretization dt = 0.01, upper left
insets show the functions λ(t), lower right insets show the roughness in the late phase
in a logarithmic plot.
Chapter 7. Simulating Stochastic Differential Equations 99
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.0, β0 = 0.33, K = 0.0005
β = 0.337
10001000 10000
t (a.u.)
3
4
5
6
w(t)
simulation curve
fitted curve
ideal β0 curve
0 3000 10000
t (a.u.)
0.02
0.06
0.1
0.16
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.0, β0 = 0.35, K = 0.0005
β = 0.352
10001000 10000
t (a.u.)
3
4
5
6
w(t)
simulation curve
fitted curve
ideal β0 curve
0 3000 10000
t (a.u.)
0.02
0.04
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.0, β0 = 0.375, aK = 0.0005
β = 0.372
10001000 10000
t (a.u.)
3
4
5
6
w(t)
simulation curve
fitted curve
ideal β0 curve
0 3000 10000
t (a.u.)
0.02
0.06
0.04
λ(t)
(c)
Figure 7.25: Differential control for the MBE equation in 1+1 dimensions with λ1,0=
0.00 and K= 0.005 for three different desired control values of: (a) β0= 0.33, (b)
β0= 0.35, (c) β0= 0.375, time discretization dt = 0.01, upper left insets show the
functions λ(t), lower right insets show the roughness in the late phase in a logarithmic
plot.
100 7.2. The MBE Equation
7.2.3 Without control in 2+1 dimensions
We now come to the 2+1 dimensional case. For the KPZ equation in 2+1 dimensions
we obtained a control that was very difficult to apply. The control behaviour was very
difficult to interpret as we could not find a clear exponent for the roughening phase.
Obviously here the situation seems to be much easier, as can be seen from Fig. 7.26.
We used setups for L= 32 ×32 l.s. and L= 64 ×64 l.s. for the data collapse. The
0.0001 1
t/Lz
0.1
1
β = 0.200
β = 0.225
β = 0.250 z = 4
z = 11/3
z = 10/3
λ1 = 0.0
λ1 = 0.1
λ1 = 0.2
w/Lα
Figure 7.26: Data collapse for the MBE equation in 2+1 dimensions with two system-
sizes L×L= 32 ×32 (black) L×L= 64 ×64 (red) and with three different initial
λ10,ν1= 0.1and D= 0.5kept constant for all simulations.
parameters for the uncontrolled case were λ1∈ {0; 0.1; 0.2}with times t= 10000000
for the smaller system and t= 1000000 for the second system size. All setups give
a data collapse for rescaling with t→t/Lzand w→w/Lα. For zero nonlinearity
λ1= 0 we see that the exponents for rescaling α= 1 and z= 4 agree with the
expected exponents for the equation. The strongly nonlinear term of λ1= 0.2also
produces the expected exponents α= 2/3and z= 10/3. As for the other equations,
the third case shows clear behaviour, where exponents of α∼0.82 and z∼3.7lead
to a data collapse.
So the requirements for possible control are fulfilled. We have a clear behaviour at
the limiting borders determined by the zero nonlinearity and a strong enough nonlinear
term. Additionally we see an effective exponent in between the limiting borders.
Chapter 7. Simulating Stochastic Differential Equations 101
We now look at the short time behaviour of larger systems L= 128 ×128 l.s..In
Fig. 7.27 we see that we can measure the different values of βvery well in this case.
The borders are given by the critical exponents of the universality classes. We now
1 10 100 1000 10000
t (a.u)
1
2
3
4
5
6
w(t)
λ1 = 0.00
λ1 = 0.05
λ1 = 0.10
λ1 = 0.15
λ1 = 0.20
w ~ t0.25
w ~ t0.2
Figure 7.27: Early roughness evolution of the MBE equation in 2+1 dimensions w(t)
vs time twith λ1∈[0.0,0.2] for L= 128 ×128 and time t= 1000 with a time
discretization dt = 0.01, dashed lines denote the limits of the growth exponents for
β1= 0.25 (violet) and β2= 0.20 (green) as guides to the eye.
want to look briefly at roughening by means of the surface structure. In Fig. 7.28 we
compared the surface structure after t= 10 (upper panels) and after t= 10000 (lower
panels) for λ1= 0 (left) and λ1= 0.20. Whereas after t= 10 we see very similar
results with a rough surface, the structure formation differs at t= 10000, as can be
seen in the lower panels. For λ1= 0.20 we can see small clear structures overlaying
a local rough surface. These structures do not arise in the left hand surface for λ1=
0. Although this analysis is just visual, a more precise analysis is possible using the
height-height correlations in Fig. 7.29. The results of the extracted exponents for λ1=
0,α∼0.93 and for λ1= 0.2α∼0.67 are close to those expected. In addition we
can see greater roughness for the nonzero nonlinearity and a more pronounced first
maximum in the correlation, which indicates the mean distance between the structures
that occur (note that C(0) = w2). Although it is quite difficult to depict this behaviour
in 1+1 dimensions, we have the impression that it could be changed experimentally by
controlling the roughening.
102 7.2. The MBE Equation
Figure 7.28: Roughening in the MBE equation in 2+1 dimensions, system size L=
256 ×256 l.s.,t= 10000 for two nonlinear terms λ1= 0 and λ1= 0.2, (a),(c) show
for zero λ1the surface after t= 10,t= 10000 respectively, (b), (d) show the surface
for λ1= 0.2after t= 10,t= 10000 respectively, the images are scaled from lowest
to highest value of the height function, the roughness for t= 10000 are both given in
Fig. 7.29 by C(0).
Chapter 7. Simulating Stochastic Differential Equations 103
050 100
l (l.s.)
0
10
20
30
40
C(l)
L = 256x256, λ1 = 0.00
L = 256x256, λ1 = 0.20
fit with C(l)=C(0)exp(-(l/ξ)2α), α = 0.93
fit with C(l)=C(0)exp(-(l/ξ)2α), α = 0.67
Figure 7.29: Correlations of roughening surfaces for the MBE equation in 2+1 dimen-
sions, here we used the same setups as for Fig 7.28.
104 7.2. The MBE Equation
System size L= 256 ×256 l.s.
Time of insetting control t0= 10
End of control te= 1000
Initial nonlinear term λ1,0∈[0 : 0.2]
Time delay τ= 1
Strength of the digital control a∈(0.005 : 0.01]
Strength of the differential control K∈(0.005 : 0.02]
desired value of the growth exponent β0∈[1/5 : 1/4]
time discretization dt = 0.04
Averages 10 realizations
Table 7.4: Parameter ranges for the control of the MBE equation in 2+1 dimensions.
7.2.4 With control in 2+1 dimensions
We now go on to the control of the MBE equation in 2+1 dimensions. Here we use
a system L= 256 ×256 l.s., time discretization dt = 0.04 and set the time range to
t0= 10 and te= 1000. Where the system size is chosen as large as possible, the other
parameters are again guided by the detailed investigations for the KPZ equation and
partly tested for some setups before generally applied.
In Fig. 7.30 and Fig. 7.31 we showed solutions of the equation for initial λ1,0= 0.0
with control strengths of a= 0.005 and K= 0.005.
We get really clear control behaviour with both types of control for all setups .
So from a zero initial nonlinearity, the MBE equation is adjustable to any desired
growth exponent between the universality classes β0∈[0.2,0.25]. The behaviour
of the control functions, well known from the other equations, is also present in the
solution. As can be seen in the other setups with λ1,0= 0.1and λ1,0= 0.2, the
function at first either increases or decreases, depending on the desired exponent, until
it reaches the exponent, and then stabilizes at the corresponding value. For λ0= 0
and β0= 0.2we get a strong increasing function λ(t)which then stabilize at a value
λ∼0.08. for the other cases the stabilization values are lower as expected, where the
desried value of βincreases.
Chapter 7. Simulating Stochastic Differential Equations 105
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.0, β0 = 0.20, a = 0.005
β = 0.20
100 200 600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
0.1
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.0, β0 = 0.225, a = 0.006
β = 0.218
100 200 600 1000
t (a.u.)
2
3
w(t)
ation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.08
0.04
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.0, β0 = 0.25, a = 0.006
β = 0.229
200 600 1000
t (a.u.)
2
3
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.08
0.04
λ(t)
(c)
Figure 7.30: Digital control for the MBE equation in 2+1 dimensions λ1,0= 0.0,
L= 256 ×256 l.s.,dt = 0.04, (a) β0= 0.2, (b) β0= 0.225, (c) β0= 0.25, upper left
insets show the functions λ(t), lower right insets show the roughness in the late phase
in a logarithmic plot.
106 7.2. The MBE Equation
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.0, β0 = 0.20, K = 0.005
β = 0.202
100 200 600 1000
t (a.u.)
2
3
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
λ(t)
(a)
101102103
t (a.u.)
1
w(t)
λ1,0 = 0.0, β0 = 0.225, K = 0.005
β = 0.225
200 600 1000
t (a.u.)
3
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.04
λ(t)
(b)
101102103
t (a.u.)
1
w(t)
λ1,0 = 0.0, β0 = 0.25, K = 0.005
β = 0.245
200 600 1000
t (a.u.)
3
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
λ(t)
(c)
Figure 7.31: Differential control for the MBE equation in 2+1 dimensions λ1,0= 0.0,
L= 256 ×256 l.s.,dt = 0.04, (a) β0= 0.2, (b) β0= 0.225, (c) β0= 0.25, upper left
insets show the functions λ(t), lower right insets show the roughness in the late phase
in a logarithmic plot.
Chapter 7. Simulating Stochastic Differential Equations 107
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.10, β0 = 0.20, a = 0.005
β = 0.20
100 200 600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.1
0.12
0.16
0.14
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.10, β0 = 0.225, a = 0.005
β = 0.223
100 200 600 1000
t (a.u.)
2
3
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
0.1
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.10, β0 = 0.25, a = 0.005
β = 0.23
100 200 600 1000
t (a.u.)
2
3
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
λ(t)
(c)
Figure 7.32: Digital control for the MBE equation in 2+1 dimensions λ1,0= 0.1,
L= 256 ×256 l.s.,dt = 0.04, (a) β0= 0.2, (b) β0= 0.225, (c) β0= 0.25, upper left
insets show the functions λ(t), lower right insets show the roughness in the late phase
in a logarithmic plot.
108 7.2. The MBE Equation
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.10, β0 = 0.20, K = 0.02
β = 0.20
100 200 600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.1
0.12
0.16
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.10, β0 = 0.225, K = 0.01
β = 0.224
100 200 600 1000
t (a.u.)
2
3
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
0.1
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.10, β0 = 0.25, K = 0.02
β = 0.242
100 200 600 1000
t (a.u.)
2
3
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
0.1
λ(t)
(c)
Figure 7.33: Differential control for the MBE equation in 2+1 dimensions λ1,0= 0.1,
L= 256 ×256 l.s.,dt = 0.04, (a) β0= 0.2, (b) β0= 0.225, (c) β0= 0.25, upper left
insets show the functions λ(t), lower right insets show the roughness in the late phase
in a logarithmic plot.
Chapter 7. Simulating Stochastic Differential Equations 109
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.20, β0 = 0.20, a = 0.006
β = 0.196
100 200 600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.1
0.12
0.16
0.14
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.20, β0 = 0.225, a = 0.006
β = 0.223
100 200 600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.1
0.12
0.16
0.14
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.20, β0 = 0.25, a = 0.006
β = 0.245
100 200 600 1000
t (a.u.)
2
3
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.1
0.20
0.14
λ(t)
(c)
Figure 7.34: Digital control for the MBE equation in 2+1 dimensions λ1,0= 0.2,
L= 256 ×256 l.s.,dt = 0.04, (a) β0= 0.2, (b) β0= 0.225, (c) β0= 0.25, upper left
insets show the functions λ(t), lower right insets show the roughness in the late phase
in a logarithmic plot.
110 7.2. The MBE Equation
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.20, β0 = 0.20, K = 0.02
β = 0.199
100 200 600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.20
0.1
0.12
0.16
0.14
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.20, β0 = 0.225, K = 0.02
β = 0.224
100 200 600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.1
0.20
0.14
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.20, β0 = 0.25, K = 0.015
β = 0.247
100 200 600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.1
0.20
0.14
λ(t)
(c)
Figure 7.35: Differential control for the MBE equation in 2+1 dimensions λ1,0= 0.2,
L= 256 ×256 l.s.,dt = 0.04, (a) β0= 0.2, (b) β0= 0.225, (c) β0= 0.25, upper left
insets show the functions λ(t), lower right insets show the roughness in the late phase
in a logarithmic plot.
Chapter 7. Simulating Stochastic Differential Equations 111
7.3 Summary for the control of the growth equations
To conclude, the stochastic differential equations, described here for modelling growth
phenomena, are adjustable, within a certain range, to values of the effective growth
exponent β0for different universality classes.
We explained in great detail the mechanisms and considerations for the KPZ equa-
tion in 1+1 dimensions, showing the restrictions on the possible ranges and the borders
of the possible control setups. In the case of the EW exponent β= 1/4it is difficult to
tune the control strength due to numerical fluctuations, but for all other setups we get
clear control behaviour.
We applied the knowledge acquired from this equation to other equations. In case
of the KPZ equation in 2+1 dimensions there are indications that the control might
also work. For the proposed KPZ value β= 1/4we can stabilize by control the
effective exponent very well, where a control for small values is rather difficult. In
these cases without more detailed investigations we can not be sure that the behaviour
as explained is responsible for a realistic control. This is because of the numerical
fluctuations, which can not be determined without controlling the exponents. This is
not surprising as the exact values are still unknown. Our control here could establish a
different method to determine the right exponents, as by definition a working control is
possible in the range between the universality classes and a nonworkingcontrol defines
the border and therefore the realistic exponents.
But it would be helpful to get such behaviour with a different type of model, such as
a Kinetic Monte Carlo method with a stochastic model, currently under consideration
[Wün07].
For the MBE equation the control works in both 1+1 dimensions and 2+1 dimen-
sions.
We have demonstrated control for 1+1 dimensions for different setups with ex-
tensive numerical simulations restricted to a smaller parameter space than for other
equations due to the computational expense.
For 2+1 dimensions the exponents are also adjustable between the two values
which determine the universality classes. For these equations, control of the growth
exponents βleads to an automatic change of the roughness exponent αrelating to the
same universality class as the desired growth exponent.
The MBE equation is proposed to explain the corresponding experimental setup
where also KPZ-behaviour was found in crystal growth. The difference between
the equations is the absence of relevant lateral growth for the MBE equation. KPZ-
behaviour is found for low temperature systems, where under high temperatures, dif-
fusion dominates the growth and the MBE equations are corresponding.
So, where simulations by means of a Kinetic Monte Carlo method can help to
reproduce the findings of the time-delayed feedback control the aim then has to be a
test with experimental setups.
112 7.3. Summary for the control of the growth equations
7.3.1 Experiments
By our theoretical investigations we showed in detail that in case of the stochastic dif-
ferential equations a time-delayed feedback control scheme can lead to an adjustment
of the growth exponent and thereby to a deliberate tuning of the surface roughness.
Now we want to discuss some hyptheses, how those schemes could be applied to ex-
periments.
While in the literature there is still a lack of comparison of scaling theory with
roughening of crystal growth systems, the work by Ojeda et al [Oje00; Oje03] can
be seen as a guide and a proof that such schemes are relevant, where the hypothetic
specific application depend on the experiment.
If we want to predict, how one can tune the roughness, we first have to look at the
parameters of the equation, in case of KPZ namely νand λor ν1and λ1, respectively,
for the MBE equation.
The question for real-world systems then is, what is a corresponding tunable quan-
tity. In case of crystal growth that could be first the temperature and then pressure
as influences from the experimental conditions and second, all material parameters as
intrinsic conditions.
What can we influence by temperature? Of course, the surface tension is coupled
to the temperature, diffusion of particles strongly depends on the temperature and the
deposition for almost all crystal growth experiments is related to temperature.
Obviously we just repeated all terms we used in the equations, thus the next ques-
tion has to be: Can we tune these parameters independent from each other by temper-
ature?
Ojeda et al showed in great detail in their findings that in case of a chemical vapor
deposition of silicon films, where they find exponents corresponding to KPZ class, it is
first possible to change the nonlinear term experimentally bychange of the temperature
and second, this procedure does not affect in their setup the smoothing term νof the
equation. For different temperatures they showed thereby, that different exponents
appear in response to that change.
That is exactly what we use in our setups, an independently changable λ.
Our control is related to constant νand ν1, so λand λ1seem to be needed indepen-
dent from the other parameters. But as we already have shown in Sec. 5.1.3 the KPZ
equation can be also characterized by a factor gdepending on the equation parameters.
So in case νand λare not independent changable by temperature, a study concerning
this factor ghave to be the method, to characterize a control scheme of the equation.
7.3.2 Other control schemes
We have applied a control scheme which is similar toclassical Proportional-controllers
and the Time-delayed autosynchronization method in chaos control.
Chapter 7. Simulating Stochastic Differential Equations 113
If we think of Proportional-Integral-Differential control (PID), we could possibly
improve the working adjustment by including integral parts or differential parts as
explained.
An integral control would average memorized values of the local growth exponents
within a pre-defined time interval and then the control force would depend on these
values.
β= 1/L
L
X
i=1
βi(7.5)
F(t) = K(β−β0)(7.6)
where tidenote the times where βihas to be measured. The range from t1to tLwould
define an additional time delay. We tested such a scheme, but did not find noteworthy
differences to the presented setups. Some selected results are shown in the appendix.
A additional part of the controller could also react on the changes of the differences
to the desired value and thereby establish a D-part.
We did not test such a scheme here.
We explained the difficulties to get good results for large time control due to numer-
ical fluctuations. The absolute differences of the roughness between to points, where
the control acts on the development, decreases due to the power law behaviour. So,
a possibly better control setup up could be a changed strength of the control for late
phase, especially for the differential control. We have seen, that for the MBE equa-
tion in 1+1 dimensions the enlarged time range lead to smaller control strengths for
a working control, that may be an indication for a changed control strength for later
times. So this might neglect strong fluctuations in the functions λ(t).
A further development of our findings could also be the test of a control of the
roughness exponents α. While for our equations such a scheme would be only mean-
ingful for the MBE equation, also in other equations it could be interesting to apply
methods for the other exponents.
7.3.3 Other equations
While we explained and reproduced by our numerical scheme the observations by
Raible et al, a time-delayed feedback could also be applied to such a complexequation.
As already explained, KPZ behaviour is proposed for low to medium temperature
behaviour, whereas the MBE universality should be obtained in high temperature sys-
tems. So, a generalized equation, containing both situations and the transition between
the classes would be an equation, where all terms here explained in two different equa-
tions are included. A change of the prefactors ν,ν1,λand λ1then would correspond
to a change of the temperature.
114 7.3. Summary for the control of the growth equations
We madefirsts steps towardsthe control of theso-called noisy Kuramoto-Sivashinsky
equation (KS), where we just solved the equation without control.
∂th(x, t) = ν∇2h(x, t) + λ
2(∇h(x, t))2−ν1∇4h+η(x, t)(7.7)
As can be seen in Eq.7.7 the KS equation combines the terms of both, the MBE and
the KPZ equation. While the long time behaviour thus must be KPZ-like, for early
times the exponents should depend on the strengths of the terms, so this equation
could explain the transition from low-temperature (KPZ) to high-temperature (MBE)
behaviour, where a control possibly could act to tune the universality class.
The problem of more complex equations is the fact that we normally can not see
a clear scaling in the early roughening due to the different terms responsible for the
behaviour. So, control of the growth exponents here ismuch more difficult and requires
extensive precending investigations of the uncontrolled equations (see appendix for
results of the uncontrolled KS equation).
Chapter 8
The Model for the evolution of cell
populations
A lot of models describing the development of cell populations have been used during
the last few years [Mor02; Dra05a; Dra05b; And05]. In this chapter we introduce the
individual cell based model we used to observe the growth of tumor cells in an in-vitro
environment.
Following the basic steps for modelling growth phenomena explained in Ch. 4, we
first want to look at the system and define the underlying structure (lattice) on which
our simulation has to work.
We want to describe the dynamics and surface morphologyof large cell populations
and to include the most relevant biological properties about the cells themselves and
their interactions at a multi-cellular level. We consider the system at an individual cell
length scale that does not explicitly explain the sub-cellular structure and is therefore
a kind of mesoscopic view.
We have used the experiments of Bru et al [Brú03] as guidelines for our model. We
have also used detailed information about the off-lattice model introduced by Drasdo
and Hoehme [Dra05b]. Here extensive simulations were used to explain cell structure
and then give information about the multi-cellular structure.
8.1 Experiment and Off-lattice model
We now want briefly to introduce the main results from the observations of Bru et al
and the off-lattice model of Drasdo and Hoehme before we proceed to our model.
8.1.1 Experiments by Bru et al
In these experiments [Brú03] colonies of 15 in vitro cell lines and 16 types of in vivo
cultures were extensively studied to explain the growth dynamics and to study the
morphological structure of the tumor border. The cell lines were grown in Petri dishes
116 8.2. The Dirichlet lattice construction
of diameter 5cm under specific conditions and analyzed by taking photographs at
24 hour intervals. Previous results needed scaling analysis methods to explain the
border structure for one cell line [Brú98], but here [Brú03] they analyzed the critical
exponents for all the cell lines studied.
For the growth dynamics they found an initial exponential growth law followed by
a regime, in which the cell population radius grows linearly in time. They concluded
that a proliferating zone restricted to a rim at the tumor border is responsible for such
a behaviour.
For the scaling analysis they took photographs of the tumor border and analyzed
the structure. Due to their interpretation, the critical exponents correspond to the MBE
universality class (see Sec. 5.2) and explained this behaviour by a migration at the
tumor border which depends either on the coordination number of the cell, or on the
number of neighboring cells.
They found these critical exponents for all cell lines and suggested a general MBE
like critical surface dynamic for tumor cell lines.
The mathematical treatment of the universality class was critically discussed by
Buceta and Galeano [Buc05a], who concluded that the analysis was incorrect. They
suggested that the critical exponents could belong to other universality classes, for
instance to the KPZ universality class. We aim to clarify this discrepancy and also to
explain the dynamics found in the studies.
8.1.2 The off lattice model
For our model we use the results of an off lattice model. Drasdo and Höhme [Dra05b]
developed a model based on individual cells. Cells are described as sticky, elastic
particles of limited compressibility and deformability. Cell division is modelled by
the spherical shape of the cell after division, though it deforms during mitosis into a
dumb-bell.
Cell adhesion is defined by adhesive bonds which are affected by the distance be-
tween cell centers. When cell pressure and nutrient supply are taken into account, the
results for the dynamics are in good agreement with the findings of Bru.
So our lattice model, which also aims to explain the dynamics of Bru, uses some
of the results from observations of that model.
8.2 The Dirichlet lattice construction
We consider a model on a lattice. We want to combine the advantage of lattice struc-
tures without the artifacts often produced by such models. We are now going to in-
troduce, as an alternative construction to the common lattice types, the construction of
the cell structure on an irregular lattice by a Delauney triangulation.
Chapter 8. The Model for the evolution of cell populations 117
8.2.1 Voronoi diagrams and Delauney triangulation
We construct a lattice based onconcepts of Dirichlet, Voronoi and Delauney 1. Descartes
first used Voronoi like diagramsin 1644. In the nineteenthcentury Dirichlet (1850)used
Voronoi diagrams in theoretical studies (Dirchlet) and Snow used them in a study of
the Soho cholera epidemic of 1854 (John Snow) [Oka00]. He showed that the people
who died lived closer to the infected pump than to any other water pump. This also
illustrates one of the fields where Voronoi or Dirichlet tessellations are most often
applied, because of the properties of a Voronoi cell.
Figure 8.1: An individual Voronoi cell: seven points distributed on a twodimensional
surface define the set M, the white point denote xiand the grey area defines the
Voronoi cell (black polygon), where all interior points have xias the closest point
of set M.
The diagram in Fig. 8.1 demonstrates the properties of a Voronoi cell. If we have a
set Mof points in a space, then the set of all points closer to a point xithan to any other
point of set Mdefines the ’Voronoi cell’ or ’Dirichlet domain’ (black polygon). If we
optimize the distributed points in an area and then define the nearest point for cells, the
solution is a Voronoi diagram. Taking post offices as points this optimization problem
is very famous as the "Post Office" problem. The tessellation of polytypes then defines
the Voronoi diagram, named after Georgy Voronoi [Vor08]. The dual graph of the
Voronoi diagram is the Delauney triangulation. Delauney triangulations are a well-
covered topic; for an overview of the possible applications for these concepts see e.g.
[Oka00; Ber97].
1Delauney is the french pronouncation of the Sovjet mathematician Boris Nikolajevitch Delone
118 8.2. The Dirichlet lattice construction
8.2.2 The construction in our model
In our model we apply these concepts to construct a lattice that is unstructured but has
a well defined distribution of the cell area. The algorithm is depicted in Fig. 8.2. Our
Figure 8.2: Construction of the Dirichlet lattice in four steps: (a) distribution of
Voronoi points (black) in the square lattice, (b) Delauney triangulation (red), (c) De-
launey triangulation and the corresponding dual graph, the Voronoi tesselation (black),
(d) the Dirichlet lattice corresponding to cells.
construction is divided into the following steps:
We take a simple square lattice of size L=l×lpoints with a lattice constant a. All
cells here then have a cell area A=a2, where the overall area is ((l−1)a)2(denoting
that a lattice with lpoints has l−1divisions).
The second step is to distribute points randomly in every square. Later we briefly
discuss different ways of doing this, but for these simulations we insert exactly one
point into each square. So we now have (l−1) ×(l−1) newly constructed points as
the points of our lattice (Fig. 8.2 (a)).
We now define the neighborhood of all points by a Delauney triangulation. As
the name says, we construct triangles using the selection rule, which says that if, by
connecting the points, we produce a square, we divide the square into two triangles by
the shorter connecting line (Fig. 8.2 (b)). In the case where two connections have the
same length, the choice is random [Oka00].
Chapter 8. The Model for the evolution of cell populations 119
So our lattice fulfils the basic properties of an unstructured lattice with a well de-
fined neighborhood.
The cell structure corresponding to our construction is given by the Voronoi graph,
which is the dual graph of the Delauney triangulation (Fig. 8.2 (c)). If we take the
perpendicular bisector of the connecting lines from the Delauney triangulation the in-
tersections determine our cell structure (Fig. 8.2 (d)).
The lattice given by this algorithm has the following properties.
•lattice of (l−1)×(l−1) points with a well defined neighborhood of on average
six neighbors as a result of the Delaunay triangulation.
•a pre-described average cell area of A=a2with a well defined sharply peaked
distribution around the average (determined by the choice of one point in each
square)
•a well-defined correspondence of the lattice points to the cell structure on the
dual graph
In order to be able to compare simulations using our lattice with other types of lattice
simulations, we included the possibility of loading the lattice types as explained in
Sec. 4.3, namely the square lattice, the hexagonal lattice and the octagonal lattice.
To ensure a direct comparison, we consider types with the same cell area as those in
our Voronoi tessellation, where in the regular case their size is exactly the cell area
(A=A) which in our case is the mean cell area.
With this lattice construction we now proceed to explain the model for our growth
simulation.
8.3 Modeling the basic processes
The basic processes in a cell population growth model are obviously the division and
migration of cells. Additionally we here also include other relevant processes like
apoptosis, mutations and fluctuations which could be responsible for a change in the
developmental behaviour.
8.3.1 Cell division
The lattice structure in our model does not determine anything about the structure of
the cell, so cell division is reduced to modelling the cell cycle time and its distribution.
As already explained, the cell cycle consists of distinct phases, namely the mitosis
phase (M-phase), the DNA duplication phase (S-phase) and Gap phases, in which cell
signalling and individual cell conditions determine the time.
120 8.3. Modeling the basic processes
Because the cell cycle is controlled by cell cycle check points [Alb02] and exper-
iments indicate a Γ-like distribution, we here model the cell cycle time τusing the
discrete analogue to the Γdistribution, the Erlang distribution in Eq. (8.1).
f(τ′) = λm
(λmτ′)m−1
(m−1)! exp{−λmτ′}(8.1)
Here λm=msuch that hτ′i ≡ τ= 1.
As can be easy seen from the equation, m= 0 corresponds to a Poisson distribu-
tion. So the parameter mensures a realistic distribution of the cell cycle time.
In our model cell division is the same as the occupation of a new lattice point. We
describe the biological process of one mother cell dividing into two daughter cells by
choosing one cell to divide and then setting the new cell at a neighboring site on the
lattice, adjacent to the mother cell. Volume exclusion (one cell on one point) then
determines the possible choices of newly occupied cell as shown in Fig. 8.3. Although
Figure 8.3: Division in the model: a) a dividing cell with two possible choices to
divide to a lattice point, b) randomly chosen point of possible choices is occupied.
we normally choose the point for division randomly, we also include different rules for
division in the model. As already explained, the cell is able to sense its environment.
If one considers a choice of position which promises the best environment for the cell,
for instance maximum nutrients or maximum free volume, then the rules have to be
changed. We discuss these different choices later, but the first approach has to be
random choice.
Proliferating rim
By experimental observations of many tumor cell lines Bru et al found a dynamics
that shows an exponential growth in the early phase of the development of the cell
diameter that then changes to a linear growth rate, so correspondingly a proliferating
rim has to be included in the model. The experimental growth velocities do not agree
with a proliferating rim ∆L= 1. Thus, ∆L > 1is needed. Cells are able to divide
Chapter 8. The Model for the evolution of cell populations 121
inside this rim. We model this by a environment of size ∆Lfor each lattice point. A
cell then is able to divide if a free lattice point is available within a circle radius ∆L.
If there is no free point in the direct neighbourhood, we give the cell the ability to
push aside other cells in its neighborhood. This algorithm allows a cell to divide if, and
only if, there is at least one free neighboring site within a circle of radius ∆Laround
the dividing cell. We see this environment for one individual cell in Fig. 8.4. A sample
simulation for 641 cells shows both the cell and the lattice structure with an enlarged
section showing the movement of cells (red cells) inside the rim along a line (green).
One interpretation is that a dividing cell is able to exert a sufficiently large force to
Figure 8.4: Cell and Point structure in the simulation, (a) the cell structure by the
Voronoi tesselation, (b) the corresponding Delauney triangulation with the lattice
points and the connections to the neighboring points, light blue cells are quiescent
and the dark blue proliferating, insets show a sample for pushing cells inside the pro-
liferating rim along the greeen line.
push at most ∆L/l cells aside in a certain direction in order to obtain free space for its
division. Another interpretation of this rule is that only a limited number of cells can
be stimulated to migrate away and leave free space for a dividing cell. It is noteworthy
that as ∆L→ ∞ lattice asymmetries in the growth patterns disappear from a regular
(square) lattice; usually ∆L/l ∼2−3already gives reasonable results [Dra05a].
To determine the growth sites we draw a circle of radius ∆L/l around the dividing
cell and shift the neighboring cells towards the closest free site within this circle (shifts
by more that ∆L/l lattice positions are prohibited). If division is permitted, we place
one of the daughter cells on the site of the mother cell, and the other daughter cell on
the neighboring site that has become free as a consequence of the previous cell shift. A
biological interpretation of the assumption of limited shifts is that a cycling cell stops
in one of the cell cycle check points if the division would require a shift of surrounding
cells over a distance of more than ∆L/l cell diameters. As a consequence, the size of
the proliferating rim within the expanding monolayer cannot exceed ∆Lif the cells are
dense (as they are here), which is why we call ∆Lthe proliferation depth. In the lattice
model ∆Lis a free parameter, while in the off-lattice model ∆Lis a consequence of
122 8.3. Modeling the basic processes
the biomechanical and migrational properties of the cells and may be influenced by,
for example, the cell stiffness and motility [Dra05b].
8.3.2 Cell migration
We want to describe the dynamics and surface morphology of large cell clusters. The
migration of cells is responsible for changes in the general behaviour. For tumor cell
populations in general, processes related to migration play a crucial role. If a mutation
causes a cell to lose its ability to adhere to other cells, it becomes invasive. Migration
can then cause these cells to invade other parts of the human body and form new tu-
mors. This metastatic process is one of the most important processes in tumor growth.
Figure 8.5: Migration in the model: a) a migrating cell with two possible choices to
migrate to a lattice point, b) by type of migration chosen point of the two possible
points is occupied, where the old position is now free again.
Although it is not our aim to model the metastatic processes of invasive cells ex-
plicitly, we consider cell migration (Fig. 8.5). Bru et al explained the behaviour of cell
population growth relying on the Molecular Beam Epitaxy universality class. We use
our model to explain how different migration rules change the growth behaviour. We
consider the following types of cell migration.
Free migration
A cell moves with rate φto an unoccupied neighboring site, irrespectively of the num-
ber of neighboring cells before and after its move. This rule corresponds to the case of
no cell-cell adhesion.
Border migration
Cells move with rate φif by this move the cell is not isolated. This may be seen as the
easiest way to model cell-cell adhesion.
Chapter 8. The Model for the evolution of cell populations 123
Cell-Cell adhesion
The most complex behaviour to model is cell-cell adhesion by the kinds of bonds
between the cells. Cells move with a rate φexp{−∆E/FT}with ∆E=E(t+ ∆t)−
E(t), where ∆tis the time step, E(t)is the total interaction energy of the multi-cellular
configuration, FT∼10−16Jis a "metabolic" energy [Bey00], ∆E/FT∼ O(1) −
O(10) [Dra05b]. This induces migration towards locations with a larger number of
neighboring cells. After considering the basic properties of particle diffusion in other
systems with energies corresponding to neighboring sites, we define our energy by
E=Es+n·EB(8.2)
where Esis an energy normally related to the bonds to the substrate, here it may
correspond to the bond to the extra-cellular matrix, nis the number of occupied neigh-
boring cells and EBdenote the bond energy stored in each cell-cell contact. Whereas
in simulations for crystal growth such definitions are widely used in the application of
Arrhenius law (see Sec. 2.1), here the pre-factor φcorresponds to the frequency with
which a cell is able to perform a hopping trial.
While the findings of Bru et al suggest a migration related to Molecular Beam
Epitaxy, which is a diffusion-dominated type of growth (see Sec.5.2), the above as-
sumptions should explain the migration of cells, as also shown in the off-lattice model
[Dra05b].
8.3.3 Apoptosis of cells
Simple apoptosis
We partly include apoptosis (programmed cell death) in our model in order to obtain
the specific dynamics for a change from no apoptotic cells to a situation where cells are
undergoing development apoptosis. The simple way is to include the rate γat which
the cells undergo apoptosis.
Complex apoptosis
We have defined a constant rate which defines the ability of the cell to undergo apopto-
sis, however other rates might be needed to describe realistic behaviour. Carcinogene-
sis is the process whereby cells mutate into tumor cells and it is often partly associated
with a change in the rate at which cells undergo apoptosis. A combination of knocked-
outs of tumor suppressor genes and the suppression of apoptosis are processes which
can lead to the uncontrolled growth of the cell population. We therefore add to the rate
of apoptosis a probability for the suppression of apoptosis.
In detail: if we have a general rate of apoptosis in a cell population giving a 5%
rate of cells undergoing apoptosis, this rate is decreased by a mutation rate γko.
124 8.3. Modeling the basic processes
8.3.4 Necrosis
Whereas cells dying of apoptosis die without damaging their neighbors, those dying
of necrosis normally die as a result of acute injury, causing a potentially damaging
inflammatory response. In in-vitro monolayers of cultured cells, destruction is an un-
wanted process and does not have to be taken into account. But in general necrosis
plays a role in cell populations, so we include a necrosis rate in our model.
8.3.5 Mutations and fluctuations
Tumor growth in general is a result either of changes in cell cycle behaviour or of a
change in the suppression and promotion of cell conditions, and is therefore a kind of
mutated condition.
Mutation in the cell cycle
Of course, during the development of a cell population consisting of cells mutated and
thereby supporting the uncontrolled growth, additional mutations can occur. We want
to denote a mutation of the cell cycle by a change the rate to divide within a certain
range around the original cell division rate.
So we randomly mutate the rate τby ∆τwith
τnew =τold + ∆τ(8.3)
where ∆τis a random number ∆τ∈[−∆τmax/2,∆τmax/2].
So our mutation does not show a preference for faster or slower divisions of the
individual cells, but a change in the cell population caused by such a mutation can of
course effect the dynamics.
Mutation of apoptosis rate
Whereas a mutation of the cell cycle probably has more effects on the dynamics, the
properties of cells undergoing apoptosis can also be changed and could have effects
on the behaviour of the cell population. We include a change of the apoptosis rate γ
similar to the mutation of the cell cycle time τ.
γnew =γold + ∆γ(8.4)
Here the cell cycle mutation ∆γis a randomly selected number in the range
[−∆γmax/2; ∆γmax/2].
Fluctuations of the environment
We have already included parameters for change which take into consideration intrin-
sic cell conditions such as mutation and apoptosis, but the cell cycle is also influenced
Chapter 8. The Model for the evolution of cell populations 125
by external properties like, for instance, the accessibility of nutrients. So fluctuations
of the external conditions may affect the cell cycle. In our model such fluctuations
are shown either by the underlying structure or by the lattice. So we have built into
our model the possibility of defining the lattice with random fluctuations related to the
lattice sites. In our structure this leads to a local change of the cell cycle time τ. We
would like to emphasize that this is a fluctuation of the environment in which the cells
grow as opposed to the mutation of a single cell, where change is an intrinsic property
of the cell.
8.4 The Kinetic Monte Carlo method
We have already defined our underlying structure, namely the Delauney triangulation,
and we have described the possible processes and parameters in the model. We are
now going to describe our method of observing cell population growth.
In contrast to the simulations for the crystal growth equations, we here use the Ki-
netic Monte Carlo method. This method has been described in Sec. 4.4.1 and we now
describe the specific conditions for our simulation. We have already defined the pro-
cesses we included by mean cell cycle times and mean migration and we now include
probabilities.
The rules given in this chapter can be formalized by the master equation
∂tp(Z, t) = X
Z′→Z
WZ′→Zp(Z′, t)−X
Z→Z′
WZ→Z′p(Z, t).(8.5)
Here p(Z, t)denotes the multivariate probability of finding the cells in configuration
Zand WZ→Z′denotes the transition rate from configuration Zto configuration Z′. A
configuration Z={..., xi−1, xi, xi+1, ...}consists of local variables xi={0,1}with
xi= 0 if lattice site iis empty, and xi= 1 if it is occupied by a cell.
The kinetic Monte Carlo method or event-based Monte Carlo then makes use of all
possible events in the system at time t[Bor75; Gil76; Fic91]. According to the specific
probability of the event, we then step by step choose an event to happen and increase
the time by the well known time step
∆t=−1
Rln(1 −ξ)(8.6)
Here, ξis a random number uniformly distributed in [0,1), and R=Pipiis the sum
of all transition probabilities piof possible events which may occur at time t.
We have now included all the parameters and can analyze tumor growth using our
simulation tool. For a detailed description of the options for the simulation tool see
Appendix C.
126 8.5. Data analysis
8.5 Data analysis
We want to explain the development of the cell population and the critical surface
dynamics, so our main quantities are the cell diameter of the population and the border
cells.
Gyration radius
It is obviouslyimportant to have a measurement of the size of the cell population which
is independent of the morphology. Although we can also analyze growth kinetics using
the cell population size N(t), in this case we take the gyration radius defined by
Rgyr =v
u
u
t
1
N
N
X
i=1
(ri−R0)2(8.7)
Here R0=1
NPN
i=1 riis the position of the center of mass. For a compact
circular cell aggregate (in d= 2 dimensions), Rgyr is related to the mean radius
R(t) = 1
2πR2π
0R(ϕ, t)dϕ (polar angle ϕ) of the aggregate by R=Rgyr√2.
Structure function
To determine the cell population border we use the structure function we have already
described above. For completeness we would like to point out the special conditions
needed for this approach, and for the scaling theory and data analysis for the critical
exponents we refer the reader to sections 4.2 5.4.
Here we want to explain a roughening process that is different in the sense that we
have a circular environment. So the border is not as easy to determine as in the case of
the development of a single line by roughening. Additionally we have an unstructured
lattice.
There are two basic differences from the scaling in 1+1 dimensions we described
before: the circular environment and the size of the developing system. Curvature may
have an effect on the structure function and we make the assumption that the tumor
border is large enough to avoid artifacts.
The developing system should reflect the basic properties of scaling. Although
this effect makes it difficult to observe a stable growth exponent, the assumption that
the tumor border shows scaling requires the observation of αand zby the structure
function.
S(k, t) = hR(k, t)R(−k, t)i(8.8)
where R(k, t)is the Fourier transform of the local radius R(s, t)and h...idenotes the
average of the growth process over different realizations (e.g. [Ram00]).
Chapter 8. The Model for the evolution of cell populations 127
Figure 8.6: Structure extracted from simulations, red cells here denote the tumor bor-
der, where the light blue cells are quiescent and the dark blue proliferating.
In order to determine the tumor border using an algorithm, we can either use all
the cells at the border and follow the individual points along the border by arc length,
or use a discretization by angles ∆φfrom the center of mass. When we include both
types, we normally make use of the first method guided by the algorithm described in
the findings of Bru et al [Brú03].
In Fig. 8.7 we show the scheme of the simulation tool. A detailed description of
the options can be found in App.C.
128 8.5. Data analysis
Selection of one event
from probability list according
to its probability
Initialization of lattice type
Initialization of
simulation parameters
Initialization of lattice
with specifi c parameters
Output of measured quantities
for single simulation
Update of the affected
processes in the environment
of the selected cell
Vor onoi structure
Square lattice
Hexagonal lattice
Octagonal lattice
proliferating rim
fi r st cell
division probability
division
migration
t = t + ∆t
apoptosis
execute the selected process
and update the time
migration probability
division distribution m
type of division
type of migration
Calculation and output of
measured quantities for
time evolution
Dynamic structure function
Gyrationradius
Statistics of processes
Graphics output
type of migration
Start of an average simulation
Output of measured quantities
for average of simulations
Dynamic structure function
Gyrationradius
migration statistics
Graphics output
division statistics
apoptosis statistics
population border statistics
Figure 8.7: Scheme of the Kinetic Monte Carlo simulation.
Chapter 9
Simulations of the evolution of cell
populations
We now apply different parameters to our Kinetic Monte Carlo simulation tool to in-
vestigate how expansion kinetics and critical surface dynamics depend on the various
properties and mechanisms. In the first part we want to test the properties of our model
and show how its behaviour differs from that of other types of model.
For all simulations we use reference time scales and length scales, more specifi-
cally the mean cell cycle time τand the mean cell area Aor, for linear quantities in
space, √A. So all quantities are multiples of these scaling factors; for instance, the
gyration radius Rgyr is described by the mean average cell diameter A1/2=l. For
direct comparison with experiments the quantities are then rescaled, and we use these
scales to investigate generic growth behaviour in our model.
9.1 Lattice artifacts
First we consider our lattice. We need to show that our construction avoids lattice arti-
facts. Such lattice-induced asymmetries could significantly disturb the analysis of the
surface growth dynamics in circular geometries. If the lattice type is chosen properly
for crystal growth, it reflects the actual lattice and therefore the actual properties of
the crystal, but here realistic behaviour is not directly coupled to a regular morphology
because of the absence of a regular underlying lattice structure.
Our simulationtool can decide between four different types of the underlying struc-
ture (see Sec. 8.2.2) and we now use this to compare our lattice to the other lattice
types, namely the square lattice (von Neumann neighborhood), the hexagonal neigh-
borhood and the Moore neighborhood with eight neighbors.
In order to check for any possible lattice artifacts we let the parameter τcorre-
sponding to the sharpness of the cell cycle time increase m→ ∞. This corresponds to
aδ-function for the distribution where all cells divide after exactly τ, and reduces the
effects of the random nature of realistic cell cycle times.
130 9.1. Lattice artifacts
At large m the tumor border then becomes smoother and the tumor shape reflects
the symmetryof the underlying lattice. This effect is known as noisereduction [Bat91].
Figure 9.1: Lattice artifacts: (a) von Neumann neighborhood, (b) hexagonal neigh-
borhood with six neighbors, (c) Moore neighborhood with eight neighbors, (d) the
Dirichlet lattice construction with an average of six neighbors, (e) for comparison a
simulation with the off-lattice model [Dra05a], all lattice simulations with no migra-
tion ∆L= 0 and m= 10000 for time t= 120.
In Fig. 9.1 we show the resulting morphology of the noise reduced simulations. For
the setups we used m= 10000 for all lattices and let all the cell populations expand
for t= 120. There was no migration and we took a proliferating rim ∆L= 0 to avoid
the effects of parameters other than m.
As anticipated, the three regularlattice types show the underlying structure (Fig. 9.1
(a - c), whereas our lattice type (Fig. 9.1 (d)) and, of course, the off-lattice model
(Fig. 9.1 (e)) do not show any regular structure other than the circular shape of the
cluster.
It can be seen that, in this case, the lattice construction of our model produces a
simulation free from lattice artifacts. So the construction our model is an advance on
the regular structure based models.
There is one additional property of the regular lattices. For the same simulation
times the number of cells increase with the number of neighbors in the regular lat-
tice, whereas our lattice shows similar values of cell divisions in comparison with the
hexagonal structure. This agrees with the fact that our points have, on average, six
neighbors due to the triangulation procedure.
Chapter 9. Simulations of the evolution of cell populations 131
The underlying lattice structure does not only appear as a result of noise reduction
in the cell cycle. Similar behaviour can be seen in simulations for crystal growth if we
have a large rate of diffusion at the surface and a comparatively low deposition rate
[Blo04].
In this case the islands that are grown normally tend to form squares of a specific
size on a cubic lattice. The diffusion in such simulations is defined by an Arrhenius
law with the energy difference depending on the number of neighbors (coordination
number) as explained in Sec. 2.1.2 and as also defined in our tool (Sec. 8.3.2).
Because Bru et al [Brú03] suggested exactly this type of migration, we test our
lattice type again to compare it with the others. If our lattice type has no artifacts asso-
ciated with this type of noise reduction, then it should not show any regular structure.
Figure 9.2: Lattice artifacts: (a) von Neumann neighborhood (square lattice), (b) the
Dirichlet lattice construction with an average of six neighbors, all lattice simulations
with migration φ= 100,∆L= 0 and Es= 1 EB= 6 and divided cells N= 10000
before proliferation stops.
In Fig. 9.2 we compare the square lattice (Fig. 9.2 (a)) with our structure (Fig. 9.2
(b)). We let 10000 cells divide, then stop the proliferation and let the cells migrate
(φ= 100) only by means of a coordination number depending on diffusion, as defined
in Sec. 2.1.2. This procedure corresponds to the fast equilibration of a conserved
model. Conservation here means that the number of particles is constant due to the
elimination of the division process after 10000 divisions. If the migration is chosen
in such a way that the cells tend to adhere to a maximum number of other cells, then
migration could cause this tendency of equilibrium behaviour to reflect the underlying
structures.
The parameters for the adhesion energies in our case are Es= 1 and EB= 6 for
both lattice types.
132 9.1. Lattice artifacts
Once again we can see the underlying structure of the regular lattice, which does
not appear in our construction. Obviously the underlying symmetry is not as clear
as for noise reduction by cell division, but whereas in this case the cells can reach
local equilibrium, the probability of reaching global equilibrium is not so large. One
therefore has to run the simulation for very long times to see the perfect square lattice
structure, whereas we can already see the underlying structure locally.
So for both types of noise reduction we have shown that our lattice type, as opposed
to the regular type, seems to be free of any lattice artifacts.
Of course we have shown extreme cases, but whereas the artifacts may not be ob-
vious in other simulations, they affect the results to a greater or lesser degree and so
normally require explanations for a recalculation of the measured quantities, if possi-
ble.
Chapter 9. Simulations of the evolution of cell populations 133
9.2 Cell area distribution
The second major difference between our construction and regular lattice types is that
it produces a realistic cell area distribution. We demonstrate the distribution our con-
struction produces using the steps that have already been explained. Therefore we
make the Voronoi tessellation inside the simulation and calculate the cell area by
Herons formula (see appendix for details).
Obviously for regular lattice types, like the three included in the tool, the distribu-
tion is sharp where biologically it should vary slightly around the average area.
0 1 2 3 4
Cell area A [a.u.]
0
2000
4000
6000
8000
10000
f(A) one point to 1x1 square
four points to a 2x2 square
16 points to a 4x4 square
total random
Figure 9.3: Cell area distribution for different lattices: Lattice points: 1000 ×1000
for the basic lattice construction, four different distributions to the squares, random
distribution of 1000 ×1000 points to the square lattice (brown), 16 points to a 4×4
squares (blue), 4 points to a 2×2squares (red) and 1 point to each square (black)
To emphasize the nature of our algorithm for construction, we have chosen differ-
ent methods of distributing the Voronoi points to the square lattice.
In Fig. 9.3 we see the dispersion of the cell area for our construction with one point
in each square, giving a totally random distribution on the predefined lattice.
So as expected, our distributiongives a pre-described average with a sharpdistribu-
tion. An upper border of the sharp distribution is of course given by A= 4a2because
of the maximum distance between two points.
If we now distribute more points randomly to a larger area by definition of our
construction this maximum increases and the distribution disperses.
If we expect the cell area to be sharply peaked around an average, our best choice
seems to be the first (black line), whereas for very flexible and fast growing cell lines
other choices might be better.
134 9.3. Proof of cell cycle time distributions
9.3 Proof of cell cycle time distributions
We now proceed to prove the cell cycle time τusing the Erlang distribution. We tested
the distribution of the cell cycle for different setups with different m.
0 1 2
τ
0
1000
2000
3000
4000
f(τ)M = 0
M = 2
M = 4
M = 6
M = 8
Erlang distr. M = 8
Figure 9.4: Cell cycle time distribution for different parameters m:300000 cells were
grown for the different setups, no migration and ∆L= 0, the dash-dotted line denote
for M= 8 the corresponding Erlang distribution.
We can see that for m= 0 the cell cycle has a Poisson distribution and for larger
mthe cell cycle becomes sharper around the average of τ= 1 for all distributions. We
show the "ideal" Erlang distribution (dash-dotted black line) for comparison for the
setup with m= 8.
Chapter 9. Simulations of the evolution of cell populations 135
9.4 Expansion kinetics of cell populations
We now want to investigate the expansion kinetics of cell populations with specific
properties. We therefore vary the basic parameters ,i.e., migration and the proliferating
rim, for the division of cells overa wide range. Additionallywe look at the dependence
of the cell cycle time distribution on the mean velocity of developing cell populations.
9.4.1 General expansion
We first want to focus on the general growth behaviour we see in all simulations with-
out mutations.
We took the simplest case, namely a simulation with ∆L= 0 m= 0 and φ= 0.
In Fig. 9.5 (a) we see the development of the gyration radius Rgyr vs time tand in
Fig. 9.5 (b) the morphology of the developed cluster. The general growth behaviour
can be seen. After an early exponential phase the gyration radius enters a linear phase
where the velocity stay at a nearly constant value (inset). vgyr denote the velocity of
the gyration radius Rgyr.
0 20 40 60 80 100
t (a.u.)
0
20
40
60
80
Rgyr(t)
∆L = 0, t = 100
0 20 40 60 80 100
t (a.u.)
0,3
0,4
0,5
0,6
0,7
0,8
0,9
vgyr(t)
(a)
Figure 9.5: General dynamics of cell populations, parameters: proliferating rim ∆L=
0, rate for proliferation 1/τ = 1, rate of diffusion φ= 0, (a) Gyration radius Rgyr vs
time scale t/τ, inset show the velocity for the Gyration radius vgyr =˙
Rgyr, points
denote the steps, where the clusters are depicted in (b); (b) shows the development of
the morphology of the cell cluster, dark blue: the proliferating cells at the border, light
blue: quiescent cells in the interior.
The behaviour corresponds to the expansion observed by Bru et al and also to
the findings for the off-lattice model [Dra05b]. As we shall see for other setups, this
general behaviour is found for all simulations without mutations.
136 9.4. Expansion kinetics of cell populations
9.4.2 Influence of the proliferating rim
We now proceed to observe the influence of the proliferating rim, one of the basic
parameters in our model. In experiments the proliferating rim is responsible for linear
expansion in the late phase of development.
0 10 20 30 40 50 60
t/τ
0
50
100
200
250
Rgyr(t)
∆L = 6, t = 60
∆L = 0
(a)
Figure 9.6: General dynamics of cell populations, parameters: proliferating rim ∆L=
6, rate for proliferation 1/τ = 1, rate of diffusion φ= 0, (a) Gyration radius Rgyr vs
time scale t/τ, dashed black line show the setup from Fig. 9.5, points denote the steps,
where the clusters are depicted in (b); (b) shows the development of the morphology
of the cell cluster, dark blue: the proliferating cells at the border, light blue: quiescent
cells in the interior.
We tested a change of expansion using a setup with ∆L= 6 and otherwise the
same conditions as before. In Fig. 9.6 we see the general influence of the proliferating
rim on both the velocity and the morphology. The linear phase can also be seen in this
case: under constant conditions the velocity increases with ∆Land the tumor border
smoothes out, as can be seen in Fig. 9.6 (b). A larger number of proliferating cells
obviously leads to an increase in the expansion velocity as the dividing cells push their
neighbors in a direction corresponding to the local radius of ∆Land this then leads to
the smoothening effect. We discuss the role of the proliferating rim in detail later.
9.4.3 Influence of free migration
The first main process in our simulation is the division of cells, but we now look at
the second main process, migration. We include here the free migration of cells or the
absence of any cell-cell adhesion, respectiveley.
When we look at the behaviour under conditions of free migration, we see that
the gyration radius again increases in comparison with the first setup, but that the
behaviour of the developing cluster is slightly different. Initially the morphological
Chapter 9. Simulations of the evolution of cell populations 137
0510 15
t/τ
0
20
40
60
80
Rgyr(t)
N = 100000 cells, φ = 50
φ = 0
(a)
Figure 9.7: General dynamics of cell populations, parameters: proliferating rim ∆L=
0, rate for proliferation 1/τ = 1, rate of diffusion φ= 50, (a) Gyration radius Rgyr vs
time scale t/τ, dashed black line show the setup from Fig. 9.5, points denote the steps,
where the clusters are depicted in (b); (b) shows the development of the morphology
of the cell cluster, dark blue: the proliferating cells at the border, light blue: quiescent
cells in the interior.
structure is not a compact cluster (Fig. 9.7 (b)), but later, as more cells divide, the tumor
population becomes denser. So although the linear phase is similar to the increases of
the proliferating rim, the way it is reached is very different. The early development
can be described by a square root function of the gyration radius corresponding to a
free migration of particles.
9.4.4 Systematic parameter variation
We have shown the influence of the basic parameters by looking at the morphology
and the general expansion using the gyration radius Rgyr, and we now proceed to a
more systematic study using the parameters ∆L,φ,m.
Fig. 9.8 shows a systematic study of growth kinetics for free migration.
Initially, the cell population size grows exponentially fast with
N(t) = N(0)exp(t/τeff )(9.1)
where the relationship
τ−1
eff = (21/m −1)mτ−1(9.2)
is fulfilled [Dra05a].
The duration of the initial phase increases with ∆Land φ. The growth law for the
diameter depends on φ. If φ= 0, the initial expansion of the diameter is exponentially
fast. If φ > 0, cells initially detach from the main cluster and the diameter grows
diffusively, with
138 9.4. Expansion kinetics of cell populations
L≡2√2Rgyr ∝pA(φ+ 1/τeff)t(9.3)
where A≈1.2is a lattice-dependent fit constant (Fig. 9.8(a)).
0 2 4 6 8
t/τ
0
2
4
6
8
Yφ = 2000
φ = 1000
φ = 500
φ = 100
0 50 100
(∆L/l)2
0
50
100
150
v2
0 5 10 15 20
φ
0
50
100
v2
0 10 20 30 40 50 60
m
0,8
1
1,2
1,4
v
a) b)
c) d)
Figure 9.8: Dynamics of tumor cell populations: (a) Y=R2
gyr/(φ+ 1/τeff)vs. t/τ
for m= 0,∆L= 1 and different values for φ. (b-d): Growth in the linear expansion
regime (N∼105). (b) Square of expansion velocity, v2, vs. square of the proliferation
zone, ∆L2(triangles: φ= 0, circles: φ= 10, squares: φ= 20;m= 0). (c) v2vs. φ
(triangles: ∆L= 1, circles: ∆L= 3, squares: ∆L= 6, stars: ∆L= 10;m= 0). (d)
vvs. m(∆L= 1,φ= 0). The lines are fits using eqn. (9.4).
For t/τ ≤2,Rgyr ∝t(Fig. 9.8(a)). This regime disappears for N(0) ≫1(see
[Dra05a]). As soon as cells in the interior of the aggregate are incapable of further
division the exponential growth crosses over to a linear expansion phase.
Fig. 9.8 shows v2vs. (b) (∆L)2, (c) φ, and (d) mfor large N(N∼105cells).
The model can explain theexperimentallyobservedvelocity-rangein Ref. [Brú03].
As t→ ∞,L=v(m, φ, ∆L)twith
v2≈B2([∆L′(∆L)]2/τ2
eff +φ/τeff),(9.4)
Chapter 9. Simulations of the evolution of cell populations 139
B≈1.4(lines in Fig. 9.8b-c). ∆L′(∆L)(≈1 + 0.685(∆L−1)) results from the
average over all permutations to pick boundary cells within a layer of thickness ∆L.
For∆L/τeff ≪pφ/τeff eqn. (9.4) hasthe same form as forthe Fisher-Kolmogorov-
Petrovskii-Piskounov (FKPP) equation. (e.g. [Mor01][Mur02]). This equation is fre-
quently used to model tumor growth phenomena by continuum models [Swa00]. Here
the FKPP equation is used to predict the distribution of tumor cells for high-grade
glioma in regions which are below the detection threshold of medical image tech-
niques. Where we can get the same velocities for expansions depending on different
proliferating rims and migration and as we will see, for different apoptotic behaviour,
we believe, that these predictions require additionally measurement to decide the dif-
ferent biologically parameters which can lead (as we show) to the same expansion
velocity.
9.4.5 Proliferating rim
Where the role of division and migration is clear, we want to explain the role of ∆L
here in more detail.
The size ∆Lof the proliferating rim controls the growth velocity in both, the
off-lattice and the cellular automaton model. In the simulations we found that v≈
∆L′/τeff with τeff =τ/ω being the cell cycle time
Here ∆L′/l ≈[1 + (∆L/l −1)0.685] and ω= (21/m −1)m(and thereby the ex-
pansion velocity) depends on the dispersion of cycle time distribution. The parameter
m∈[0,1,2, ...)controls the shape of the cycle time distribution f(τ′).
Hence the larger the dispersion of the cycle time distribution (by choosing mto
be smaller) the smaller is ω, and the larger are τeff and consequently the expansion
velocity vof the monolayer. At no dispersion the expansion velocity is the smallest.
The factor 0.685 results from the order in which the cell divisions take place. Al-
though our simulations are in two dimensions, the occurrence of this factor can best be
understood if one considers a one-dimensional segment of a two-dimensional growing
cell population, ideally a one-cell-thick column ranging from the center of mass of the
monolayer until its surface.
If only the outermost cell is able to divide (∆L/l = 1), the increment within τis
∆L. However, if the proliferation depth is ∆L≫lthen the order of divisions deter-
mines whether a cell is able to divide or not. To see this assume an almost precise cell
cycle length (i.e., a cycle time distributionsharply peaked at τ=hτiwhich is obtained
for m≫1). Then, if it is the innermost cell that divides first then all cells closer to the
border are still able to divide while, if it is not the innermost cell that divides first, then
the innermost cell cannot divide anymore since this would require to shift more than
∆L/l cells. So even if f(τ′)→∼ δ(τ′−τ)the order at which the cells divide matters
since for ∆L > l the cell divisions are not completely parallel. The factor ∼0.685 can
be calculated from investigating the expected growth increment from all permutations
of choosing the cells in the proliferative rim for division. Note that the factor ∼0.685
marks the difference between an asynchronous and a parallel update. To understand
140 9.5. Comparison with experiments
this first note, that since we start each simulation with a single cell, a precise length of
the cycle time would mean that all cells divide at the same point of time. The factor
∼0.685 results from the asynchrony as argued above. For a parallel update this factor
would not be expected; the expansion velocity should instead be v≈∆L/τ. (Note
that in a circular geometry the expansion velocity may slightly deviate from this value
due to the boundary curvature which decreases with increasing monolayer size as 1/R
with Rbeing the monolayer radius.)
Note also, however, that the factor ∼0.685 may disappear also in asynchronous up-
dates if the choice of how cells are divided is slightly changed. If one would assume
that a cell that once has passed the restriction point divides with probability one that is,
if one assumes the decision on whether a cell divides or not is made immediately after
its birth and not when it is chosen for division, then the dependency of the velocity
upon the order at which the cell divisions in the proliferating rim are performed would
no longer be expected.
9.5 Comparison with experiments
Now we want to compare our model directly with the experimental data.
Findings from the off-lattice model [Dra05b] were able to explain the growth ve-
locity found by Bru et al [Brú03] for the developing population, and in our simulation
we use parameters that are consistent with these findings, namely a proliferating rim
of ∆L= 9 and a parameter for cell cycle time distribution m= 60.
0 10 20 30
t[days]
0
1000
2000
R[µm] exp. Bru
CA lattice
Off-lattice
11.50.5 τ’/τ
0
1
2
3
4
fCA lattice
Off-lattice
Erlang distr.
a) b)
Figure 9.9: Dynamics of experiments:(a) Mean radius Rof the cell aggregate vs. time
t. Full circles: experimental findings for C6 rat astrocyte glioma cells ([Brú03]). (b)
Cell cycle time distribution f(τ′)for the off-lattice model and the CA growth model
in comparison with the Erlang distribution (m= 60,∆L= 9,φ= 0).
After the simulations we rescaled the resulting expansion parameters using the real
size of the cell diameter as also used by [Dra05b] (cell size l= 10µm cell cycle time
Chapter 9. Simulations of the evolution of cell populations 141
τ= 19h). As can be seen in Fig. 9.9 our simulation is consistent with both, the
experimental data and the off-lattice model.
142 9.6. Cell density
9.6 Cell density
We have already explained how different parameters and therefore different biophysi-
cal properties can lead to the same velocities in the linear phase. We now consider the
properties that determine them. A variety of mechanisms can give the same velocity,
one being the cell density at the tumor border. If, for example, we have the same ve-
locity but a different migration strength at the border, and cells are also able partly to
migrate away from the cluster, this can be determined by measuring the cell density.
Cell density here means the mean volume filled within a given radius.
0510 15 20 25 30
t/τ
0
50
100
150
200
Rgyr(t)
∆L = 10.5
N = 100000 cells, φ = 50
0510 15 20
t/τ
0
2
4
6
8
vgyr
(a)
120 160
140
R
0
0,5
1
f(R)
φ = 0
φ = 50
average radius (Rgyr = 100)
(b)
Figure 9.10: Comparison of cell density at the tumor border for simulations with
m= 0 and two different setups: proliferating rim ∆L= 10.5and migration rate
φ= 0 and ∆L= 0 and φ= 50,Rgyr = 100 for both simulations, profiles are rescaled
to normal radius (factor √2) and shifted to recent region.
We have used simulations with the same velocity and the setup of a proliferating
rim ∆L= 10.5and zero migration (black) compared to ∆L= 0 and φ= 50 (red). In
both simulations m= 0. To make the simulations comparable, we stop the simulations
at a gyration radius Rgyr = 100.
At first we see the same expansion velocities (inset of Fig. 9.10 (a)), but the initial
phase is different. The velocity measurement alone obviously does not give us enough
information to decide between the two setups, but when clusters without migration are
denser, large migration rates lead to more active cells at the border and additionally
to unoccupied points, so the density decreases more slowly at the border as shown in
Fig. 9.10 (b).
This setup shows that further measurement of either the initial phase or the cell
density is required for the model in order to decide between expansions with the same
velocity. So the relationship to the FKPP equation can not determine all the relevant
parameters.
Chapter 9. Simulations of the evolution of cell populations 143
9.7 Surface dynamics
We now go further to look at the behaviour of the tumor border in terms of the structure
function. As already explained, different suggestions have been made for the critical
surface dynamics of the tumor cell lines. Whereas Bru et al suggest an MBE like
behaviour, the critical comments by Buceta and Galeano suggest a KPZ like behaviour.
First we want to look at the behaviour of the case with no migration and ∆L= 0
for different times t. In Fig. 9.11 we see the structure functions for different times
10-4 10-3 10-2 10-1 100
k
100
102
104
104
106
S(k,t)
t = 60, s = 865
t = 100, s = 1478
t = 150, s = 2265
~k-2(α+1)
(a)
10-2 10-1 100101
kt1/z
100
102
S(k,t)k2α+1
(b)
0 100 200 300 400 500
x
0
100
200
300
400
500
y
t = 60
t = 100
t = 150
(c)
Figure 9.11: Dynamic structure function for S(k, t)vs. kfor different times t,∆L=
0,φ= 0 and m= 0, (b) rescaled structure function S(k, t)k2α+1 vs. kt1/z by α= 0.5
and z= 3/2, (c) surface border for the different times.
t= 60,100,150. The slope suggests a roughness exponent α= 0.5. Rescaling
using the Family-Vicsek relation (see Sec. 4.2.2) we get data collapse for the function
(Fig. 9.11). When we use z= 3/2the data collapse into a single curve, giving us clear
exponents corresponding to the KPZ universality class.
144 9.7. Surface dynamics
Obviously this setup leads to very different scaling to that suggested by Bru et al.
We now proceed to vary the other parameters. In Fig. 9.12 (a) we see the behaviour
for ∆L= 6 under otherwise constant conditions. Here we have simulations where we
calculate the structure function S(k, t)for four different times and we can see that all
simulations show similar scaling behaviour.
In Fig. 9.12 (b) we see Arrhenius law migration with parameters ν= 2,Es= 1
and EB= 2 which require large migration rates and define the migration according to
the explanation of Bru et al.
In Fig. 9.12 (e) we than take the same type of migration with realistic ’slower’ rates
for the parameters derived from the off-lattice model.
We also varied the sharpness of the cell cycle by m= 5 (Fig. 9.12 (c)). We used a
setup with free migration φ= 100 (Fig. 9.12 (d)) For both we did not see any MBE-
like behaviour.
We have included the migration explained by Bru which should be responsible for
the behaviour of MBE like growth. We also tested a lot of different setups for the
binding energies but we did not find any MBE-like behaviour in the structure function,
but in all simulations values for the roughness exponent close to the KPZ universality
class value α= 0.5. So we need to explain why we did not find MBE behaviour but
rather exponents related to KPZ-behaviour. First we want to remind ourselves about
the behaviour for 1+1 dimensions on a single line.
Here MBE-like behaviour corresponds to a system where particles are deposited at
a constant rate and then relax due to diffusion on the surface [Bar95; Sar96; Mic04].
MBE describes conserved growth, so, after subtracting the mean deposition, the evolv-
ing height function has the same mean average height as when it just roughens.
Physical properties eventually require some of the terms ∇2h(smoothing surface
tension) or (∇h)2(lateral growth) but the critical surface dynamics can not rely on
the MBE universality class for long term behaviour [Sar96] due to the non-dominant
fourth order term (see Sec. 4.4.4Sec. 5.2).
The universality class then is either EW or KPZ.
In MBE modelling the particles fall onto the surface and then relax due to diffu-
sion. Here we have a different case. The particles form the interior of the surface.
This behaviour is similar to the deposition of particles and locally the cells can grow
laterally. If we take a specific radius vector from the center of mass, we find that the
cells can grow in a direction perpendicular to this line.
This behaviour corresponds to lateral growth or, in terms ofdeposition, to a ballistic
deposition model (see [Bar95; Mic04] and references therein) where before relaxation
particles stick to the nearest neighbor thereby producing voids and overhangs.
Both explanations lead to KPZ-like behaviour, and we have already pointed out in
Sec.7.1.4 that ballistic deposition models belong to the KPZ universality class.
These overhangs can be seen in our model and also in the observations of Bru
[Brú03], so it can be seen that we have included such mechanisms in the growth.
Consequently, the behaviour in our model belongs to the KPZ class. If we include
the precise mechanisms explained by Bru, then either the calculations of Bru et al are
Chapter 9. Simulations of the evolution of cell populations 145
wrong or different mechanisms are causing the observed behaviour.
Our results therefore agree with the critical comments by Buceta and Galeano
[Buc05a].
146 9.7. Surface dynamics
10-4 10-3 10-2 10-1 100
k
100
102
104
106
108
S(k,t)
t = 20, s = 681
t = 30, s = 1189
t = 40, s = 1647
t = 48, s = 2020
~k-(2α+1)
(a)
10-3 10-2 10-1 100
k
100
102
104
106
S(k,t)
t = 40, s = 434.5
t = 60, s = 679
t = 80, s = 922.8
~k-(2α+1)
(b)
10-4 10-3 10-2 10-1 100
k
100
102
104
106
108
S(k,t)
t = 100, s = 874
t = 150, s = 1320
~k-(2α+1)
(c)
10-3 10-2 10-1 100
k
100
102
104
106
108
S(k,t)
pdiff = 100, t = 20, s = 447
~k-(2α+1)
(d)
10-3 10-2 10-1 100
k
100
102
104
106
108
S(k,t)
ν = 20, E0 = 3, EB = 10, s = 606
ν = 20, E0 = 5, EB = 10, s = 627
ν = 20, E0 = 10, EB = 10, s = 637
~k-(2α+1)
(e)
Figure 9.12: Dynamic structure function for different parameters, (a) ∆L= 6,m= 0,
φ= 0 for different times, (b) ∆L= 0,m= 0, migration depending on the coordi-
nation number (Arrhenius law) with ν= 2,Es= 1 ,EB= 2 for different times, (c)
m= 5,∆L= 0,φ= 0 (d) φ= 100 with border migration (e) ∆L= 0,m= 0, mi-
gration depending on the coordination number (Arrhenius law) with ν= 20, varying
Es,EB= 10 corresponding to the energy derived from the off-lattice model, for the
same number of cells, in all figures sdenote the arclength of the border in average cell
sizes, clusters contain a cell number N∼3·104...3·105cells (Bru et al ∼105), s
denotes the arclength in units of average cell sizes (see Fig.8.6).
Chapter 9. Simulations of the evolution of cell populations 147
9.8 Apoptosis
In normal cell populations proliferation is balanced by apoptosis. In tumor cells this
balance is destroyed, so although apoptosis can still occur, the cells do not stop their
uncontrolled proliferation and the population size increases. We now want to look at
the two types of apoptosis we included in the simulation and their influence on the
expansion of the monolayer.
9.8.1 Apoptosis with constant probability
0 10 20 30 40 50 60
t (a.u.)
0
10
20
30
40
Rgyr(t)
γ = 0.4
(a)
Figure 9.13: Apoptosis with constant rate. Parameters: φ= 0,m= 0 and rate for
apoptosis γ= 0.4, (a) Rgyr vs t/τ, (b) shows the development of the morphology of
the cell cluster, dark blue: the proliferating cells at the border, light blue: quiescent
cells in the interior. φ= 0,m= 0 and γ= 0.4.
As we can see in Fig. 9.13, apoptosis, as expected, changes the velocity of popu-
lation growth. It only affects the border at extremely large rates; otherwise it leads to
smaller expansion velocities. The linear phase is reached later, at a stage where the
rate at which cells undergo apoptosis and proliferate determines the velocity. A larger
proliferating rim would compensate for this effect, but one could still not determine
the expansions.
We now want to retest those setups where we expect to see the same expansion
velocities. In Fig. 9.14 we have chosen setups for very different conditions both with
and without migration, neglecting apoptosis and with varying proliferating rims.
In Fig. 9.14 we see that three setups show exactly the same expansion velocity for
different mechanisms. So we once again see behaviour that supports the assumption
that the velocitycan not be the only parameter which determines the growth conditions.
148 9.8. Apoptosis
0 10 20 30
t/τ
0
50
100
Rgyr
∆L=9,γ=0,φ=0
∆L=12,γ=0,φ=0
∆L=12,γ=0.22,φ=0
∆L=9,γ=0.22,φ=0
∆L=1,γ=0,φ=25
(a)
0510 15 20 25 30
t/τ
0
1
2
3
4
5
6
vgyr
∆L=9,γ=0,φ=0
∆L=12,γ=0,φ=0
∆L,γ=0.22,φ=0
∆L=9,γ=0.22,φ=0
∆L=1,γ=0,φ=25
(b)
Figure 9.14: Setups with the same velocity for five setups with and without migration,
different proliferating rims and different apoptosis rates: (a) Rgyr(t), (b) the velocities
of the same setups.
9.8.2 Apoptosis with mutations
For apoptosis with constant probability we now let the rate γmutate with a variation of
∆γas explained in Sec. 8.3.5. In Fig. 9.15 (a) we see the expansion of the monolayers
0 20 40 60 80 100
t/τ
0
20
40
60
80
Rgyr
γ = 0.4, ∆γ = 0
γ = 0.4, ∆γ = 0.2
γ = 0.4, ∆γ = 0.4
(a)
0 0,2 0,4
pγ
0
100
200
300
500
600
700
f(pγ)
γ = 0.4, ∆γ = 0
γ = 0.4, ∆γ = 0.2
γ = 0.4, ∆γ = 0.4
(b)
Figure 9.15: Mutation of apoptosis rate. Parameters: γ= 0.4,t= 100,m= 0 and
∆L= 0, (a) Gyration radius Rgyr vs time scale t/τ for different mutations of the
apoptosis ∆γ(b) distribution f(pγ)of probability to undergo apoptosis.
for three different rates of apoptosis mutation ∆γ∈ {0; 0.2; 0.4}with a constant initial
rate of apoptosis γ= 0.4. In Fig. 9.15 (b) we have depicted the histogramof successful
apoptotic processes and the corresponding rates of the individual cells, and we can see
in both the expansion and the histogram that the velocity increases with the mutation
of the apoptosis rate, where, not surprisingly, the monolayer with more cells to divide
expands faster.
Chapter 9. Simulations of the evolution of cell populations 149
Here we see a kind of competition between the cells where the mutation gives
higher apoptotic rates and those with lower rates. Obviously the cells that win have a
lower probability of dying, as can be seen in the distribution.
150 9.9. Mutations of the cell cycle
9.9 Mutations of the cell cycle
Tumor cells are characterized by uncontrolled proliferation and one basic mechanism
which leads to this is cell mutation. Defects in tumor suppressor genes are one reason
for the behaviour. During uncontrolled proliferation changes in the cell cycle can also
appear. If the cell cycle decreases, the DNA replication phase can also be shortened,
which makes it more difficult for the cell to repair defects and this again leads to
mutations. We now look at simple kinds of mutations in the cell cycle.
In Fig. 9.16 we can see that both cell dynamics and expansions significantly change
under mutation and that the resulting morphology reflects the mutation. Whereas in
the early phase no significant differences can be seen, in the late phase the mutations
lead to totally different behaviour. The expansion velocity increases rapidly and the
Figure 9.16: Comparison of mutated and unmutated cell morphology: φ= 0,∆L=
0, and mutation of the cell cycle time ∆τ= 10%.
nearly round shape of the cell cluster is destroyed. We start with a mutation equally
distributed around the average cell cycle time and by definition no side is preferred.
But as we can see, faster cells are in the lead in the expanding tumor monolayer.
Chapter 9. Simulations of the evolution of cell populations 151
That is not really surprising, since, when faster cells lead, new cells also divide fast
and thereby overgrow the slow cells, which are then not equally distributed over the
monolayer and so don not dominate the growth conditions.
So we have a kind of competition between the initially equally distributed fast and
slow cells. The fast cells win the competition and are responsible for the behaviour
of the monolayer.. In Fig. 9.17 we see that, for different strengths of the mutation,
0 10 20 30 40 50 60
t/τ
0
50
100
150
200
Rgyr
∆τ = 25%
∆τ = 0%
∆τ = 10%
∆τ = 10%, γ = 0.4
k=9,M=60
Figure 9.17: Mutation of the cell cycle and apoptosis for four different setups, all
simulations with ∆L= 0, no migration and m= 0.
velocity increases with strength and apoptosis causes a strong increase in the gyration
radius in the expansion that sets in later due to the mutation.
We have now tested one specific setup, where we only varied the apoptosis rate.
We took a setup with ∆L= 9,m= 60 and φ= 0, zero apoptosis and γ= 0.4.
Expansion is, as expected, initially faster in the setup without apoptosis (Fig. 9.18).
But, surprisingly, the expansion velocity of the setup with apoptosis increases faster
and reaches the velocity of the non-apoptotic case at the intersection point. The reason
is that, in the apoptotic case with a constant rate, the mutated fast and slow dividing
cells undergo apoptosis. When the fast dividing cells dominate growth, the slow are
destroyed by apoptosis faster than in the non-apoptotic case.
So under apoptosis the contest between fast dividing and slower dividing cells is
lost earlier. Until then, velocity increases more strongly than in the non-apoptotic case
and reaches it at the intersection point shown in the figure.
9.9.1 Global fluctuations
We now want to see how growth is affected by fluctuations which are not intrinsic to
the individual cells but to the underlying structure. So we take a pre-described random
152 9.10. Summary and outlook
050 100 150
Rgyr
0
5
10
v
∆L=9,∆τ=0.1,γ=0
∆L=9,∆τ=0.1,γ=0.4
intersection
Figure 9.18: Mutation and mutation with apoptosis, parameters the same as in 9.9,
additionally we include mutation ∆τ= 10% and for the second setup additionally
apoptosis γ= 0.4.
distribution of the cell cycle time around the average τ∈[τ−∆τ/2, τ +∆τ/2]. In our
simulation we change the corresponding probabilities for a cell to divide at that point.
We here test different setups with and without apoptosis and with different fluctuations.
We also compare our non-intrinsic mutation with the mutation where the change of the
cell cycle is coupled to the cell.
As we can see in Fig. 9.19 (a), our change does not seems to affect the dynamics
for all setups. We take the setups with γ= 0.0and γ= 0.1as references and see
that the setups with additional fluctuations (∆τ= 40%,∆τ= 5%) show the same
behaviour. A look at the velocities in Fig. 9.19 (b) confirms this behaviour. Although
we do not see differences in the general behaviour, a closer look at the setups with
γ= 0.1and zero fluctuation and ∆τ= 40% shows that there are larger fluctuations in
the velocity. We conclude that a random fluctuating underlying cell cycle distribution
has no effects on the general dynamics, as opposed to the case explained before, where
an intrinsic cell cycle mutation leads to extreme changes in the dynamics with the
faster cells dominating growth (shown for comparison in Fig. 9.19 (a)).
9.10 Summary and outlook
We explained in this chapter the development of tumor cell in-vitro monolayers under
specific growth conditions. By means of a Kinetic Monte Carlo method we observed
the expansion kinetics depending on the basic processes, namely division and migra-
tion of cells.
Chapter 9. Simulations of the evolution of cell populations 153
050 100 150 200
t/τ
0
50
100
150
Rgyr
τ = 1, γ = 0
τ = 1, γ = 0.1
τ = 1, ∆τ = 5%, γ = 0
τ = 1, ∆τ = 5%, γ = 0.1
τ = 1, ∆τ = 40%, γ = 0
τ = 1, ∆τ = 40%, γ = 0.1
τ = 1, γ = 0, mutation rate 5%
(a)
050 100 150
t/τ
0,5
1
1,5
v
τ = 1, γ = 0
τ = 1, γ = 0.1
τ = 1, ∆τ = 5%, γ = 0
τ = 1, ∆τ = 5%, γ = 0.1
τ = 1, ∆τ = 40%, γ = 0
τ = 1, ∆τ = 40%, γ = 0.1
(b)
100 150
t/τ
1,2
1,5
vτ = 1, γ = 0.1
τ = 1, ∆τ = 40%, γ = 0.1
avaraged v = 1.364
averaged v = 1.360
(c)
Figure 9.19: Fluctuations of cell cycle depending on the individual lattice site.
We introduced a new type of lattice, which under different kinds of noise reduction
opposite to a regular structure does not show any lattice artifacts.
Guided by an off-lattice model the simulation can explain the kinetics observed in
experiments.
A detailed analysis of the additionally determining parameters ∆L,φand the pa-
rameter mwhich ensure a realistic cell cycle time distribution we observed an asymp-
totic expansion velocity that is reminiscent of the front velocity of the FKPP equation.
We have shown by variation of parameters that different biologically relevant
mechanisms can lead to the same velocities in the development and concluded, that
the velocity in the linear phase can not be the only parameter which determine this
quantity.
Additionally one has to explain the proliferating rim ∆and the migration rules and
the cell density at the tumor border to get indications, which mechanisms lead to the
expansion.
We then included different kinds of apoptosis as a relevant parameter and again
showed setups, where very different mechansims lead to the same velocityin the linear
154 9.10. Summary and outlook
expansion. Here, a detailed view in experiments to the early is required to decide
between the different mechanisms.
Guided by the experiments of Bru et al and additionally motivated by the critical
comments of Buceta and Galeano, we explained in detail the critical surface dynamics
of the tumor border. By use of the scaling theory for self-affine types of growth we
calculated the three critical exponents α,βand z. Therefore we varied the growth con-
ditions in a wide range. In particular we also introduced the migration rules proposed
by Bru et al to be responsible for the tumor growth of different cell lines.
While Bru et al claimed a MBE-like critical surface dynamics by these migration
rules, we did not find by parameter variation any MBE-like behaviour but, opposite to
their findings, a KPZ-like behaviour for all setups.
Thus, our observations assert the critical comments of Buceta and Galeano.
We then additionally implemented different kinds of mutations and fluctuations of
the cell cycle and explained how mutated cells affect the kinetics and the morphology.
We found that randomly distributed non-intrinsic fluctuations (fluctuations of the cell
cycle time due to conditions depending on the underlying structure) don’ t lead to
significant changes, but just to a more strongly fluctuating velocity.
We have shown that a special type of the underlying structure leads to an absence
of lattice artifacts, which in comparison can be clearly seen for regular lattice types.
We included a realistic cell cycle time distribution by the Erlang distribution. So
our cell cycle has a predefined distribution around the mean cell cycle time.
Guided by the experiments and by use of results from an off-lattice model we could
reproduce the dynamics for tumor cells observed in experiments.
Our model can explain and distinguish a variety of biologically relevant actions for
the developing system and give the ability to observe the behaviour without unknown
influences.
We explained the expansion kinetics and the dependence of it on the determin-
ing parameter proliferating rim ∆L, the strength of migration φand the parameter m
related to the sharpness of the cell cycle distribution.
We now want to briefly explain some other possible further observations, which
could be made by use of the developed simulation tool.
9.10.1 Limited mutation of the cell cycle
We described the mutation of the cell cycle as a variation of the probability for division
equally distributed corresponding to the variation of the cell cycle time (see Sec. 9.9).
This mutation generally include the possibility of the cells to mutate to a regime, where
the cells divide very fast. If we take into regard that the mitosis phase (1∼2h) in
comparison to the whole cell cycle (∼24h, in experiments for the expansion 19h) is
very small, then this approach appropiate for a model. A more detailed assumption
would be the inclusion of a lower border for the cells to divide.
Chapter 9. Simulations of the evolution of cell populations 155
Mutations to lower cell cycle times lead to the reduction of the interphase, so there
is less time for the cell to activate their repair mechanisms. Nevertheless there is a
minimum time, which the cell need to duplicate.
In Fig. 9.20 we show a setup with such a minimum time for the cells to divide.
00,5 11,5 2
τ
0
500
1000
1500
2000
f(τ)without mutation
∆ τ = 10%, no limiting τ
∆ τ = 10%, limiting τmin = 2/3 τ
(a)
0 20 40 60 80
t/τ
0
20
60
80
Rgyr(t)
without mutation
∆ τ = 10%, no limiting τ
∆ τ = 10%, limiting τmin = 2/3 τ
0 20 60 80
t (a.u.)
0
1
2
vgyr(t)
(b)
Figure 9.20: Mutation with limiting lower border τmin setups used for 100000 cells,
without mutations, with cell cycle mutation ∆τ= 10%
If we consider also the upper border, such an assumption is not so evident and may
be not realistic, since the cells can enlarge their gap phases in a wide range [Alb02].
The nondominating nature of the cells with larger cell cycle times in our model we
have already shown for the mutations without borders, where the fast dividing cells
dominate the expansion kinetics and the slower cells do not affect the growth.
The same behaviour we get by a limiting border. In Fig. 9.20 (a) we see, that the
cell cycle distribution has changed to faster dividing cells also for the setup with a
limiting minimum cell cycle time of τmin = 2/3τ(Fig. 9.20 (b)). Then the expansion
velocity increase in comparison to the unmutated case, but has a lower velocity than
the case of mutation without limiting borders (Fig. 9.20 (b)).
9.10.2 Correlated global fluctuations
We defined before the influence of the nonintrinsic fluctuations totally randomly on the
lattice and see as a result no general changes in the expansion kinetics but a stronger
fluctuation in the velocity as expected. A further development of thes concept would
be the inclusion of nonrandom fluctuations, but defined pattern, by which the cell cycle
change due to fluctuations of the environment which could be explained by differently
distributed nutrient supply.
9.10.3 Different rules for division
The most important process is obviously the division of the cells. We used a division
which includes a random selection of the new place in the environment for one of
156 9.10. Summary and outlook
the daugther cells, where the mother cell stay on the old position. For a cell in the
proliferating rim, the cell select the shortest way to push the cells in this direction.
This leads to a shift along this cell pushing path for the cells.
Where cells are able to sense their environment, this rule for division could differ.
Possible non-random divisions could be the selection of the longest distance motivated
by the aim to get as much volume for the cell as possible. Another way to reach this
aim is to make the algorithm able to count the coordination number and to select the
position which as less as possible neighbors.
By these different division rules we can define different model types which could
lead to very different expansion kinetics. In particular we included 5 different divi-
sions.
•0random selection of the new cell, shift to the shortest distance inside ∆L
•1random selection of the new cell, shift to direction of a random cell inside ∆L
•2selection of the new cell by the longest distance, shift inside ∆Lto the longest
distance
•3selection of the daughter cell by the minimum coordination number, shift to
the cell inside ∆Lwith the lowest coordination number
•4selection of the daughter cell by the maximum coordination number, shift to
the cell inside ∆Lwith the highest coordination number
050 100 150
t/τ
0
25
50
100
125
Rgyr(t)
Type of division 0
Type of division 1
Type of division 2
Type of division 3
Type of division 4
(a)
0 10 20 30 40
t/τ
0
25
50
100
125
Rgyr(t)
Type of division 0
Type of division 1
Type of division 2
Type of division 3
Type of division 4
(b)
Figure 9.21: Expansion kinetics for different division rules for 100000 cells, (a) ∆L=
0in all cases, (b) ∆L= 6 in all cases, simulations without migration.
In Fig. 9.21 wesee that the expansion kinetics differ depending on thedivisionrule. We
do not see different velocities for the first three types and a proliferating rim ∆L= 1.
However the rule depending on the coordination number changes here the kinetics.
That is not surprising, since the the rules here just affect the cells, which are inside the
Chapter 9. Simulations of the evolution of cell populations 157
proliferating rim and not at the border. For a larger proliferating rim all expansion ki-
netics differ, where not only the linear phase is changed, but also the initial expansion.
So defining these rules, we can investigate by the simlation tool different model types
for the division guided by the assumption, that cells could sense their environment.
9.10.4 Different rules for migration
Before, we used different migration rules, where we included free migration, free mi-
gration at the tumor border and a migration depending on the coordination number by
an Arrhenius law. In Fig. 9.22 we show that by all of these different migrationsdefining
different types of model one can reach the same expansion velocities as for the exper-
iments of Bru [Brú98]. Here the velocity (Fig. 9.22 (b)) is in µm/days corresponding
to the shown development of the radius in Fig. 9.22 (a). The velocity is consistent
with the experimentally observed velocity v= 2.9µm/h for C6 rat astrocyte glioma
[Brú98].
0 10 20 30
t (days)
0
500
1000
2000
R[µm]
exp. Bru
CA lattice
Off-lattice
φ = 25, ∆L = 1, free migration
φ = 25, ∆L = 8, border migration
φ = 25, ∆L = 9, E0 = 2, EB = 10
0 10 20 30
t (days)
0
50
100
v (µm/day)
exp. Bru
CA lattice
Off-lattice
φ = 25, ∆L = 1, free migration
φ = 25, ∆L = 8, border migration
φ = 25, ∆L = 9, E0 = 2, EB = 10
(b)
Figure 9.22: Expansion kinetics for different migration rules for (a) Mean radius R
of the cell aggregate vs. time t. Full black circles experimental findings for C6 rat
astrocyteglioma cells ([Brú03]), three different migration rules, free migration (green),
to border restricted free migration (violet) and Arrhenius law migration (light blue) (b)
expansion velocity for the same setups.
In case of the coordination number dependent rule, the choice of different setups
is thereby possible by definition of the ’binding energies’ which define the ∆Ein the
Arrhenius law. For particles in crystal growth, namely effective atoms, the effective
binding isalwayspositive. We havenormallya surface bindingand a neighbor binding.
In case of cells, which we inlcude as points, the behaviour may vary. If we assume a
cell-cell adhesion to the tissue, the cells could nevertheless by sensing tend to migrate
to position with more free volume. This could be included by including the cells to
migrate preferently to positions with less neighbors. In our algorithm, that is just a
setting of the different sign of the neighbor binding.
158 9.10. Summary and outlook
In conclusion here we also have shown some possibilities which the simulation
tool additionally offer for further investigations.
Chapter 10
Conclusions and Outlook
In this work the self-organized growth was extensively studied for two different types
of systems.
First we modeled epitaxial crystal growth by use of the well-established stochastic
differential equations. Additionally we applied the theory of time-delayed feedback
methods to develop a tool to study the control of the roughening phase of surfaces by
time-delayed feedback control. For different growth equations we showed, how the
corresponding growth exponent βcould be adjusted by such a scheme.
In the second part a powerful model for simulation of cell population growth by
means of a Kinetic Monte Carlo method was developed. Aimed to model the growth
of tumor cells in an in-vitro monolayer, the tool includes a large variety of properties
of biological relevance. By extensive simulations we have investigated the generic
kinetical behaviour and have shown that our single cell based cellular automaton model
reproduces the kinetics of experimental studies and can explain the critical surface
dynamics of the tumor borders.
In both parts we made use of the well-established scaling theory, which gives for
self-affine types of growth phenomena the ability to determine the surface roughness
evolution by means of three exponents, namely the growth exponent β, the roughness
exponent αand the dynamic exponent z, where only two of those are independent.
For the crystal growth we additionally established a new type of control method
to adjust the growth exponent. For the tumor growth we developed a simulation tool
which combines advantages of lattice models and off-lattice models by definition of
an irregular lattice free of artefacts.
In particular by numerical schemes we solved the stochastic growth equations, namely
the Kardar-Parisi-Zhang and the Molecular Beam Epitaxy equation in 1+1 and 2+1
dimensions. Detailed analysis lead to observations of the three critical exponents β,α
and zwhich determine the universality classes for the growth.
160
We could exactly reproduce by our scheme the exponents for the MBE equation in
both dimensions, but for the KPZ equation we get stable values only for 1+1 dimen-
sions and some indications during control for the 2+1 dimensional case.
We then defined a time-delayed feedback method to control the early roughness
evolution by adjusting the growth exponent βduring the roughening process.
Our method in particular includes two different schemes, the digital control, which
acts by a control step aon the sign of the difference to the desired exponent and a
differential control which contains an amplification factor K, which determines the
control force F.
We explained in detail, how one can define, restrict and calculate parameters which
could be useful for control.
The control after that gave precise results for two types of control with predictions
for possible experiments. Indications for possible setups were explained by compari-
son with recent experiments [Oje00; Oje03]. Here, for a specific system, the relation
between the nonlinear term λfrom the KPZ equation to the temperature is explained
in detail and it is shown that one can tune it by changing the temperature.
A lot of additional observations identify the KPZ equation as relevant for low tem-
perature behaviour in experiments due to the nonlinear term which is related to lateral
growth. In high temperature systems, diffusion processes dominate the growth process,
so the MBE equation then is responsible for the universality of the growth.
For both types of behaviour, the tuning of temperature can change the behaviour
and a relation to the theory could be given by experiments where the exponents depen-
dent on temperature have to be measured.
While further explanations by experimental setups have to reproduce the theoreti-
cal investigations, the method could then give predictions how the roughness develop-
ment can be tuned by time-delayed feedback.
We have explained in detail limits of control for both the digital and the differential
scheme. These findings should also be reproduced by different types of methods,
namely a Kinetic Monte Carlo method for a solid-on-solid approximation of crystal
growth.
For the single cell based tumor growth model we explained in detail the dynamics and
the surface morphology depending on different parameters.
We have defined a new lattice type consisting of Voronoi cells related to the bio-
logical cells. A construction by a Delauney triangulation gives a well defined average
cell size with a well defined sharp distribution around the mean area.
The relation of the cell cycle to an Erlang distributionincluded in the model ensures
realistic cell cycle time distributions.
By extensive simulations we observed the expansion kinetics of tumor cell in-vitro
monolayers.
By the special lattice construction we ensured that our model is free from any
lattice artefacts. So the model establishes a tool where one combines the advantage of
Chapter 10. Conclusions and Outlook 161
off-lattice models which are independent from any underlying lattice structure and the
advantage of well-defined neighborhood which leads to a faster simulation.
We have shown that the expansion kinetics covers the findings observed in experi-
ments and the observations made by an off-lattice model. It was in detail explained that
very different biological actions included in our model can lead to the same expansion
velocities in growth. Recently, mathematical models based on the Fisher-Kolmogorov-
Petrovskii-Piskounov (FKPP) equation were used to predict the distribution of tumor
cells for high-grade glioma in regions which are below the detection threshold of med-
ical image techniques [Swa00]. We found that the asymptotic expansion velocity has
a form that is reminiscent of the front velocity of the FKPP equation, nevertheless the
same expansion velocity can be obtained for different combinations of the migration
and division activities of the cell and of the cycle time distribution.
So in conclusion we believe such predictions must fail since the FKPP equation
lacks some important parameters such as the proliferation depth which is why it is not
sensitive to relative contributions of the proliferation depth and free migration.
We observed in our simulations that these relative contributions in fact determine
the cell density profile at the tumor-medium interface: the larger the fraction of free
migration is, the wider is the front profile even if the average expansion velocity is
constant.
We additionally included apoptosis with different rules consistent with biological
interpretations of that process and again determined the expansion kinetics, where we
showed in detail that a large variety of different mechanims leads to the same velocities
in the linear regime of the expansion.
We found the determining processes and thus can give suggestions for possible
experiments to decide these different cell actions, for instance the measurement of the
cell density at the tumor border or the migration activity or the early phase to observe
large apoptosis rates.
By additional inclusion of various intrinsic mutations of the cell cycle and nonin-
trinisic fluctuations of the underlying structure we then showed scenarios which could
determine the kinetics in cell lines under strong mutational behaviour.
For these observations by construction we don not prefer mutations to fast or slow
dividing cells, nevertheless we see a strong regime, in which after a certain time range
the faster cells always dominate the growth and thus determine the expansion.
Bru et al propose the cell lines, they investigated to show universal scaling related
to the MBE universality class, we included a calculation of the corresponding critical
exponents. For a wide range of different setups under inclusion of the migration pro-
posed by Bru et al to be responsible for this type of universality class, we did not find
any MBE-like behaviour, but strong KPZ-like critical behaviour. Our findings thereby
comply with the critical comment of Buceta and Galeano.
We here stronlgy suggest further experimental investigations.
162
So in conclusion we investigated two systems related to complex growth phenomena,
where in both parts scaling theory played an essential rule. For stochastic differential
equations applied to epitaxial growth we established a new method of a time-delayed
feedback control and gave predictions, how possible experimental setups have to act
to tune the roughness evolution ’in situ’.
In addition, these findings could in general be applied to any system, which be-
long to the explained equations, where one then has to define the relation between the
equation parameters and the growth phenomena.
For the second system, the tumor growth of an in-vitro monolayer, we explained in
detail how the biological actions on the scale of an individual cell determine both the
expansion kinetics and the critical surface dynamics.
We could reproduce the kinetics in consitstency with an off-lattice model and with
experiments. However, our investigations for the universality class of tumor growth
don not comply with previous interpretations of the experiments and require new ex-
perimental investigations.
Thus we investigatedproblems on the nanometer scale for materials grown by epitaxial
methods and cell behaviour from the length scale of an individual cell to large cell
populations and hopefully contributed in some way to the problem of the ’nanobot’
outlined in the preface.
Appendix A
Simulations of stochastic growth
equations
A.1 Additional simulations KPZ 1+1
In Fig. A.1 and Fig. A.2 we show the control for a larger system size L= 32768. In
Fig. A.1 the results for three initial setups λ0= 0 and β0= 0.33 (black), λ0= 0.1and
β0= 0.29 (red), and λ0= 0.25 and β0= 0.25 (blue) are shown, the digital (Fig. A.1
(a)) and the differential (Fig. A.1 (b)). The roughness evolution shows, that all setups
can be controlled and the evolution of the nonlinearity λ(t)show the general properties
of the control method, increase of the function for the first setup (black), nearly stable
function for the second setup (red) and a decrease for the third setup (blue).
In Fig. A.2 the results for three initial setups λ0= 0 and β0= 0.29 (black),
λ0= 0.1and β0= 0.29 (red), and λ0= 0.25 and β0= 0.29 (blue) are shown,
the digital (Fig. A.1 (a)) and the differential (Fig. A.2 (b)). The roughness evolution
shows, that all setups can be adjusted to the same desired exponent β0= 0.29 (guide
to the eyes: green).
In Fig. A.3 - A.6 we show the results for the KPZ equation in 1+1 dimensions for
digital and differential control with initial nonlienarities λ0= 0.05 and λ0= 0.15.
As for the results shown in Sec. 7.1.3 (Fig. 7.9 - 7.15), five setups with different β0
for each control type and nonlinearity are chosen. The results show again the general
behaviour of the control methods.
164 A.1. Additional simulations KPZ 1+1
1 10 100 1000
t(a.u.)
1
w(t)
λ0 = 0, β0 = 0.33
λ0 = 0.1, β0 = 0.29
λ0 = 0.25, β0 = 0.25
10 100 1000
t (a.u.)
0
0.05
0.1
0.15
0.25
λ(t)
(a)
1 10 100 1000
t(a.u.)
1
w(t)
λ0 = 0, β0 = 0.33
λ0 = 0.1, β0 = 0.29
λ0 = 0.25, β0 = 0.25
10 100 1000
t (a.u.)
0
0.05
0.1
0.15
0.25
λ(t)
(b)
Figure A.1: Control for the KPZ equation in 1+1 dimensions with L = 32768: Three
setups for the digital and the differential control λ0= 0 and β0= 0.33 (black), λ0=
0.1and β0= 0.29 (red), and λ0= 0.25 and β0= 0.25 (blue). (a) digital control with
a= 0.01, (b) differential control with K= 0.02, time discretization dt = 0.01 for all
setups, upper left insets show the functions λ(t)..
Appendix A. Simulations of stochastic growth equations 165
1 10 100 1000
t(a.u.)
1
w(t)
λ0 = 0
λ0 = 0.1
λ0 = 0.25
w ~ t0.29
10 100 1000
t (a.u.)
0
0.05
0.1
0.15
0.25
λ(t)
(a)
1 10 100 1000
t(a.u.)
1
w(t)
λ0 = 0
λ0 = 0.1
λ0 = 0.25
w ~ t0.29
10 100 1000
t (a.u.)
0
0.05
0.1
0.15
0.25
λ(t)
(b)
Figure A.2: Control for the KPZ equation in 1+1 dimensions with L = 32768: Three
setups for the digital and the differential control with constant β0= 0.29,λ0= 0
(black), λ0= 0.1(red), and λ0= 0.25 (blue). (a) digital control with a= 0.01, (b)
differential control with K= 0.02, time discretization dt = 0.01 for all setups, upper
left insets show the functions λ(t).
166 A.1. Additional simulations KPZ 1+1
100101102103
t (a.u.)
1
w(t)
λ0 = 0.05, β0 = 0.25, a = 0.005
β = 0.258
600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.05, β0 = 0.27, a = 0.005
β = 0.271
600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
0.1
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.05, β0 = 0.29, a = 0.005
β = 0.290
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
0.1
λ(t)
(c)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.05, β0 = 0.31, a = 0.005
β = 0.312
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.1
0.14
λ(t)
(d)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.05, β0 = 0.33, a = 0.005
β = 0.332
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.1
0.18
0.14
λ(t)
(e)
Figure A.3: Digital control for the KPZ equation in 1+1 dimensions with a control
setup: λ0= 0.05 and a= 0.005 for five different desired control values of: (a) β0=
0.25, (b) β0= 0.27, (c) β0= 0.29, (d) β0= 0.31, (e) β0= 0.33, time discretization
dt = 0.005, upper left insets show the functions λ(t), lower right insets show the
roughness in the late phase in double logarithmic plot.
Appendix A. Simulations of stochastic growth equations 167
100101102103
t (a.u.)
1
w(t)
λ0 = 0.05, β0 = 0.25, K = 0.005
β = 0.259
600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
0.1
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.05, β0 = 0.27, K = 0.005
β = 0.27
600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.04
0.1
0.14
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.05, β0 = 0.29, K = 0.005
β = 0.289
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.1
0.14
λ(t)
(c)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.05, β0 = 0.31, K = 0.005
β = 0.309
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.1
0.14
λ(t)
(d)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.05, β0 = 0.33, K = 0.005
β = 0.325
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.1
0.14
λ(t)
(e)
Figure A.4: Digital control for the KPZ equation in 1+1 dimensions with a control
setup: λ0= 0.05 and K= 0.005 for five different desired control values of: (a) β0=
0.25, (b) β0= 0.27, (c) β0= 0.29, (d) β0= 0.31, (e) β0= 0.33, time discretization
dt = 0.005, upper left insets show the functions λ(t), lower right insets show the
roughness in the late phase in double logarithmic plot.
168 A.1. Additional simulations KPZ 1+1
100101102103
t (a.u.)
1
w(t)
λ0 = 0.15, β0 = 0.25, a = 0.005
β = 0.263
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.1
0.14
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.15, β0 = 0.27, a = 0.005
β = 0.278
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.1
0.16
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.15, β0 = 0.29, a = 0.005
β = 0.294
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.1
0.18
0.14
λ(t)
(c)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.15, β0 = 0.31, a = 0.005
β = 0.314
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.1
0.18
0.14
λ(t)
(d)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.15, β0 = 0.33, a = 0.005
β = 0.331
600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.2
0.1
0.14
λ(t)
(e)
Figure A.5: Digital control for the KPZ equation in 1+1 dimensions with a control
setup: λ0= 0.15 and a= 0.005 for five different desired control values of: (a) β0=
0.25, (b) β0= 0.27, (c) β0= 0.29, (d) β0= 0.31, (e) β0= 0.33, time discretization
dt = 0.005, upper left insets show the functions λ(t), lower right insets show the
roughness in the late phase in double logarithmic plot.
Appendix A. Simulations of stochastic growth equations 169
100101102103
t (a.u.)
1
w(t)
λ0 = 0.15, β0 = 0.25, K = 0.005
β = 0.262
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.1
0.18
0.14
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.15, β0 = 0.27, K = 0.005
β = 0.275
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.1
0.18
0.14
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.15, β0 = 0.29, K = 0.005
β = 0.294
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.1
0.18
0.14
λ(t)
(c)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.15, β0 = 0.31, K = 0.005
β = 0.312
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.1
0.18
0.14
λ(t)
(d)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.15, β0 = 0.3, K = 0.005
β = 0.329
600 1000
t (a.u.)
2
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.1
0.18
0.14
λ(t)
(e)
Figure A.6: Digital control for the KPZ equation in 1+1 dimensions with a control
setup: λ0= 0.15 and K= 0.005 for five different desired control values of: (a) β0=
0.25, (b) β0= 0.27, (c) β0= 0.29, (d) β0= 0.31, (e) β0= 0.33, time discretization
dt = 0.005, upper left insets show the functions λ(t), lower right insets show the
roughness in the late phase in double logarithmic plot.
170 A.2. Additional simulations KPZ 2+1
A.2 Additional simulations KPZ 2+1
In Fig. A.7 we show the control of the KPZ equation in 2+1 dimensions with an initial
nonlinearity λ0= 0.05. The left figures (Fig. A.7 (a,c,e)) show the digital control for
three values of the desired exponent β0with a= 0.005, the right figures (Fig. A.7
(b,d,f)) show the control with the same setups for K= 0.005. The results show the
same behaviour as for the control with other initial nonlinearities.
100101102103
t (a.u.)
1
w(t)
λ0 = 0.05, β0 = 0.15, a = 0.005
β = 0.17
600 1000
t (a.u.)
3
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
λ(t)
(a)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.05, β0 = 0.15, K = 0.005
β = 0.153
600 1000
t (a.u.)
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
λ(t)
(b)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.05, β0 = 0.2, a = 0.005
β = 0.212
600 1000
t (a.u.)
3.5
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.06
0.04
λ(t)
(c)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.05, β0 = 0.2, K = 0.005
β = 0.199
600 1000
t (a.u.)
3
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
λ(t)
(d)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.05, β0 = 0.25, a = 0.005
β = 0.255
600 1000
t (a.u.)
4
4.5
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
λ(t)
(e)
100101102103
t (a.u.)
1
w(t)
λ0 = 0.05, β0 = 0.25, K = 0.005
β = 0.25
600 1000
t (a.u.)
4
w(t)
simulation curve
fitted curve
ideal β0 curve
0 200 400 800 1000
t (a.u.)
0.02
0.06
0.04
λ(t)
(f)
Figure A.7: Control for the KPZ equation in 2+1 dimensions with a control setup:
λ0= 0.05,a= 0.005 for digital and K= 0.005 for differential control, three desired
control values of: (a,b) β0= 0.15, (c,d) β0= 0.20, (e,f) β0= 0.25, time discretization
dt = 0.005, upper left insets show the functions λ(t), lower right insets show the
roughness in the late phase in double logarithmic plot.
Appendix A. Simulations of stochastic growth equations 171
A.3 Additional simulations MBE 1+1
In Fig. A.8 we show the results for the control of the MBE equation in 1+1 dimensions
with an initial nonlinearity λ1,0= 0.05. The left figures (Fig. A.8 (a,c,e)) show the
digital control for three values of the desired exponent β0with a= 0.0005, the right
figures (Fig. A.8 (b,d,f)) show the control with the same setups for K= 0.0005.
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.05, β0 = 0.33, a = 0.0005
β = 0.335
1000 t (a.u.)
3
4
5
6
w(t)
simulation curve
fitted curve
ideal β0 curve
0 3000 10000
t (a.u.)
0.02
0.06
0.04
λ(t)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.05, β0 = 0.33, K = 0.0005
β = 0.334
1000 t (a.u.)
3
4
5
6
w(t)
simulation curve
fitted curve
ideal β0 curve
0 3000 10000
t (a.u.)
0.08
0.06
0.04
0.12
λ(t)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.05, β0 = 0.35, a = 0.0015
β = 0.356
1000 t (a.u.)
3
4
5
6
w(t)
simulation curve
fitted curve
ideal β0 curve
0 3000 10000
t (a.u.)
0.02
0.06
0.04
0.1
0.14
λ(t)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.05, β0 = 0.35, K = 0.0005
β = 0.354
1000 t (a.u.)
3
4
5
6
w(t)
simulation curve
fitted curve
ideal β0 curve
0 3000 10000
t (a.u.)
0.02
0.06
0.04
λ(t)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.05, β0 = 0.375, a = 0.0005
β = 0.371
1000 t (a.u.)
3
4
5
6
w(t)
simulation curve
fitted curve
ideal β0 curve
0 3000 10000
t (a.u.)
0.02
0.06
0.04
λ(t)
100101102103
t (a.u.)
1
w(t)
λ1,0 = 0.05, β0 = 0.375, K = 0.0005
β = 0.372
1000 t (a.u.)
3
4
5
6
w(t)
simulation curve
fitted curve
ideal β0 curve
0 3000 10000
t (a.u.)
0.02
0.06
0.04
λ(t)
Figure A.8: Digital control for the MBE equation in 1+1 dimensions with a control
setup: λ1,0= 0.05,a= 0.0005 for digital and K= 0.0005 for differential control,
three desired control values of: (a,b) β0= 0.33, (c,d) β0= 0.35, (e,f) β0= 0.375, time
discretization dt = 0.01, upper left insets show the functions λ(t), lower right insets
show the roughness in the late phase in double logarithmic plot.
172 A.3. Additional simulations MBE 1+1
The results show the same general properties as explained for λ1,0.
Appendix A. Simulations of stochastic growth equations 173
A.4 Noisy Kuramoto-Sivashinsky equation
With the time-delayed feedback method we investigated the KPZ and the MBE equa-
tion. An equation combining the terms of both and possibly also controllable is the
noisy Kuramoto-Sivashinsky equation. Fig. A.9 show for 1+1 and 2+1 dimensions
100101102103104105
t (a.u.)
1
10
w(t)
ν = -1, λ = 1, ν1 = 1
ν = 0, λ = 1, ν1 = 1
ν = 1, λ = 1, ν1 = 1 β = 0.25
β = 0.33
β = 0.39
β = 0.28
100101102103104105
t (a.u.)
1
w(t)
ν = -1, λ = 1, ν1 = 1
ν = 0, λ = 1, ν1 = 1
ν = 1, λ = 1, ν1 = 1
Figure A.9: Solutions of the noisy KS equation in 1+1 and 2+1 dimensions with three
different parameter setups: ν= 1,λ= 1 and ν1= 1 (blue), ν= 0,λ= 1 and ν1= 1
(red), and ν= 1,λ= 1 and ν1= 1 (black), (a) in 1+1 dimensions with local exponents
βas guides to the eye, (b) in 2+1 dimensions.
solutions for different initial terms ν(EW term). In 1+1 dimensions we show, that
different phases of roughening appear. Further investigations could make a control as
in this work explained possible also for this type of equation.
174 A.4. Noisy Kuramoto-Sivashinsky equation
Appendix B
Deposition models
B.1 Ballistic deposition
In Fig. B.1 we show the results of the simple ballistic deposition in 1+1 dimensions.
The rule for the deposited particles is to stick on the first nearest neighbor [Bar95].
The ballistic deposition is often used to get a relation form Solid-on-solid models to
the KPZ equation. In Fig. B.1 we show that the effective exponent β∼0.3is close to
the KPZ exponent (β= 1/3) as expected.
100101102103
t (a.u.)
1
5
10
20
w (t)
β = 0.3
Figure B.1: Roughness evolution in the simple ballistic deposition model with nearest
neighbor sticking rule [Bar95] for L= 131072 and t= 1000.
176 B.2. Random deposition
100101102103
t (a.u.)
1
5
10
20
30
40
w (t)
β = 0.5
(b)
Figure B.2: Roughness evolution in the random deposition model, (a) shows a density
plot of the height profile from lower values (blue) to higher values (green), (b) show
the roughness vs time tfor a system of 256 ×256l.s..
B.2 Random deposition
In Fig. B.2 we show the results for a random deposition on a 256×256l.s. system. We
get the well-known exponent β= 0.50 and do not see any correlations in the density
plot (B.2 (a)) as expected.
Appendix C
Simulation tool for the tumor model
C.1 Short manual
Table C.1 and table C.2 give short descriptions for the options of the simulation tool.
Option Description
-h show the help
-i load lattice file (see options z, w)
-v the probability for division (corresponding to the rate)
-f the probability for migration (corresponding to the rate)
-k the proliferating rim ∆L
-a the factor τ/value by which the cell cyle time mutates
-m the parameter for the Erlang distribution (sharpness)
of the cell cycle time
-c the probability for a cell to undergo apoptosis
-G the probability for a mutation of the cell cyle time
depending on the lattice point
-x the number of averages
-y time
-A animation flag for graphic output
Table C.1: Short manual for the usage of the tool for cell population evolution
178 C.1. Short manual
Option Description
-D Type of migration
0 free migration
1 free migration restricted to the border
2 Arrhenius law migration
3 Arrhenius law migration just depending on the migrating cell
4 Arrhenius law migration restricted to the border
-M Type of division
0 migration to randomly selected free points, shift
to shortest distance
1 migration and shift to randomly selected free points
2 migration and shift to the free points with the longest distance
3 migration and shift to the free point with the
lowest coordination number
4 migration and shift to the free point with the
highest coordination number
-N prefactor for Arrhenius migration
-E E0for Arrhenius migration
-B EBfor Arrhenius migration
-z size of lattice to create (100 for 100 ×100 lattice)
-w type of lattice to create
4 square lattice
6 hexagonal lattice
8 octagonal lattice
-C Maximum of cells
-U Maximum of Gyration radius
-Z the factor of mutation of the apoptosis probability
-T maximum of the probability to divide under mutation
(corresponds to an average minimum of the cell cycle time
-K probability to knock out apoptosis
-s seed for random number generator
-o Output rate
-n number of divisions
Table C.2: Short manual for the usage of the tool for cell population evolution
List of Figures
1 Nanobot destroys a red blood cell. . . . . . . . . . . . . . . . . . . . xi
2 Chromosome image. . . . . . . . . . . . . . . . . . . . . . . . . . . xii
1.1 Snow crystal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Sample for growth: tree. . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Sample for growth: snail. . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Examples for crystal growth. . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Statistics of cancer diseases in Europe. . . . . . . . . . . . . . . . . . 4
2.1 Growth modes of epitaxial growth. . . . . . . . . . . . . . . . . . . . 8
2.2 Processes at the surface in epitaxial growth. . . . . . . . . . . . . . . 9
3.1 Isolation of embryonic stem cells. . . . . . . . . . . . . . . . . . . . 14
3.2 View on length scales beetween living cells and atoms. . . . . . . . . 15
3.3 Schematic view of cells. . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 Cytoskeleton. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5 Scheme of the cell cycle. . . . . . . . . . . . . . . . . . . . . . . . . 18
3.6 Mitosis of a cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.7 Scheme of programmed cell death (Apoptosis). . . . . . . . . . . . . 20
3.8 Scheme of an in vitro experiment. . . . . . . . . . . . . . . . . . . . 23
4.1 Zoom from macroscopic to microscopic view. . . . . . . . . . . . . . 26
4.2 Illustration of a rough surface. . . . . . . . . . . . . . . . . . . . . . 28
4.3 Temporal evolution of the rms-roughness. . . . . . . . . . . . . . . . 28
4.4 Different lattice types. . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.1 Behaviour of the Edwards-Wilkinson term. . . . . . . . . . . . . . . 40
5.2 Scheme of the lateral growth. . . . . . . . . . . . . . . . . . . . . . . 42
5.3 Behaviour of the nonlinear KPZ term. . . . . . . . . . . . . . . . . . 43
5.4 Verification of the Raible model. . . . . . . . . . . . . . . . . . . . . 48
6.1 General Control of a system. . . . . . . . . . . . . . . . . . . . . . . 56
6.2 Mathematical scheme for a control of a system. . . . . . . . . . . . . 56
6.3 Control of a the growth exponent β. . . . . . . . . . . . . . . . . . . 60
180 List of Figures
7.1 Data collapse for the KPZ equation in 1+1 dimensions. . . . . . . . . 66
7.2 Early roughness evolution of the KPZ equation . . . . . . . . . . . . 67
7.3 Calculation of the roughness exponent αfor the KPZ equation in 1+1
dimensions in the long time behaviour . . . . . . . . . . . . . . . . . 69
7.4 Roughening of the early time KPZ equation . . . . . . . . . . . . . . 70
7.5 Influence of time delay on control . . . . . . . . . . . . . . . . . . . 72
7.6 Influence of control strength on control . . . . . . . . . . . . . . . . . 74
7.7 Influence of constant factors Caand Ckon control . . . . . . . . . . . 75
7.8 Influence of time delay on control . . . . . . . . . . . . . . . . . . . 76
7.9 Digital controlfor the KPZ equation in1+1 dimensionswith λ0= 0.00
and a= 0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.10 Differential control for the KPZ equation in 1+1 dimensions with λ0=
0.00 and K= 0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.11 Digital and differential control for the KPZ equation in 1+1 dimen-
sions with λ0= 0.00,a= 0.01 respectively K= 0.01 ......... 81
7.12 Digital controlfor the KPZ equationin 1+1 dimensionswith λ0= 0.10
and a= 0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.13 Differential control for the KPZ equation in 1+1 dimensions with λ0=
0.10 and K= 0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.14 Digital controlfor the KPZ equationin 1+1 dimensionswith λ0= 0.10
and a= 0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.15 Differential control for the KPZ equation in 1+1 dimensions with λ0=
0.10 and K= 0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.16 Longtime roughness evolution of the KPZ equation in 2+1 dimensions. 86
7.17 Determination of the roughness exponent. . . . . . . . . . . . . . . . 87
7.18 Digital control for the KPZ equation in 2+1 dimensions with λ0= 0.10. 90
7.19 Differential control for the KPZ equation in 2+1 dimensions with λ0=
0.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.20 Digital and differential control for the KPZ equation in 2+1 dimen-
sions with λ0= 0.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.21 Roughening of the MBE equation in 1+1 dimensions. . . . . . . . . . 95
7.22 Early roughening of the MBE equation in 1+1 dimensions. . . . . . . 96
7.23 Data collapse by structurefunction for the MBE equation in 1+1 di-
mensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.24 Digital control for the MBE equation in 1+1 dimensions with λ1,0=
0.00 and a= 0.0005,0.0015. . . . . . . . . . . . . . . . . . . . . . . 98
7.25 Differential control for the MBE equation in 1+1 dimensions with
λ1,0= 0.00 and K= 0.0005. . . . . . . . . . . . . . . . . . . . . . . 99
7.26 Data collapse for the MBE equation in 2+1 dimensions. . . . . . . . . 100
7.27 Early roughness evolution of the MBE equation in 2+1 dimensions. . 101
7.28 Roughening in the MBE equation in 2+1 dimensions. . . . . . . . . . 102
7.29 Correlations of roughening surfaces for the MBE equation in 2+1 di-
mensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
List of Figures 181
7.30 Digital control for the MBE equation in 2+1 dimensions λ1,0= 0.0. . 105
7.31 Differential control for the MBE equation in 2+1 dimensions λ1,0= 0.0106
7.32 Digital control for the MBE equation in 2+1 dimensions λ1,0= 0.1. . 107
7.33 Differential control for the MBE equation in 2+1 dimensions λ1,0= 0.1.108
7.34 Digital control for the MBE equation in 2+1 dimensions λ1,0= 0.2. . 109
7.35 Differential control for the MBE equation in 2+1 dimensions λ1,0= 0.2.110
8.1 An individual Voronoi cell. . . . . . . . . . . . . . . . . . . . . . . . 117
8.2 Construction of the Dirichlet lattice. . . . . . . . . . . . . . . . . . . 118
8.3 Division in the model. . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8.4 Cell and Point structure in the simulation. . . . . . . . . . . . . . . . 121
8.5 Migration in the model. . . . . . . . . . . . . . . . . . . . . . . . . . 122
8.6 Structure extracted from simulations. . . . . . . . . . . . . . . . . . . 127
8.7 Scheme of the Kinetic Monte Carlo simulation. . . . . . . . . . . . . 128
9.1 Lattice artifacts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
9.2 Lattice artifacts by large migration rates. . . . . . . . . . . . . . . . . 131
9.3 Cell area distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.4 Cell cycle time distribution. . . . . . . . . . . . . . . . . . . . . . . . 134
9.5 General dynamics of cell populations . . . . . . . . . . . . . . . . . . 135
9.6 General dynamics of cell populations with changed proliferating rim . 136
9.7 General dynamics of cell populations with free migration. . . . . . . . 137
9.8 Dynamics of tumor cell populations. . . . . . . . . . . . . . . . . . . 138
9.9 Dynamics of experiments. . . . . . . . . . . . . . . . . . . . . . . . 140
9.10 Comparison of cell density. . . . . . . . . . . . . . . . . . . . . . . . 142
9.11 Dynamic structure function for ∆L= 0. . . . . . . . . . . . . . . . . 143
9.12 Dynamic structure function for different parameters. . . . . . . . . . 146
9.13 Apoptosis with constant rates. . . . . . . . . . . . . . . . . . . . . . 147
9.14 Setups with the same velocity. . . . . . . . . . . . . . . . . . . . . . 148
9.15 Mutation of apoptosis rate. . . . . . . . . . . . . . . . . . . . . . . . 148
9.16 Comparison of mutated and unmutated cell morphology. . . . . . . . 150
9.17 Mutation of the cell cycle and apoptosis. . . . . . . . . . . . . . . . . 151
9.18 Mutation and Mutation with apoptosis. . . . . . . . . . . . . . . . . . 152
9.19 Fluctuations of cell cycle depending on the individual lattice site. . . . 153
9.20 Mutation with limiting lower border . . . . . . . . . . . . . . . . . . 155
9.21 Expansion kinetics for different division rules. . . . . . . . . . . . . . 156
9.22 Expansion kinetics for different migration rules. . . . . . . . . . . . . 157
A.1 Control for the KPZ equation in 1+1 dimensions with L = 32768 and
different β0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
A.2 Control for the KPZ equation in 1+1 dimensions with L = 32768 and
β0= 0.29 ................................ 165
182 List of Figures
A.3 Digital controlfor the KPZ equation in1+1 dimensionswith λ0= 0.05
and a= 0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
A.4 Differential control for the KPZ equation in 1+1 dimensions with λ0=
0.05 and K= 0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
A.5 Digital controlfor the KPZ equation in1+1 dimensionswith λ0= 0.15
and a= 0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
A.6 Differential control for the KPZ equation in 1+1 dimensions with λ0=
0.15 and K= 0.005 . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
A.7 Control for the KPZ equation in 2+1 dimensions with λ0= 0.05,a=
0.005 and K= 0.005. . . . . . . . . . . . . . . . . . . . . . . . . . . 170
A.8 Control for the MBE equation in 1+1 dimensions with λ1,0= 0.05,
a= 0.0005 and K= 0.0005. . . . . . . . . . . . . . . . . . . . . . . 171
A.9 Solutions of the noisy KS equation in 1+1 and 2+1 dimensions. . . . . 173
B.1 Roughness evolution in the simple ballistic deposition model. . . . . . 175
B.2 Roughness evolution in the random deposition model. . . . . . . . . . 176
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