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Page 1: Cellular automaton models for tumor invasionrcshatzikir/invasion_review_final-june-16.pdf · Cellular automaton models for tumor invasion 5 † F is a deterministic or probabilistic

Cellular automaton models for tumor invasion

H. Hatzikirou, G. Breier and A. Deutsch

De�nition of the subject and its importance: Cancer cells display characteristictraits acquired in a step-wise manner during carcinogenesis. Some of these traitsare autonomous growth, induction of angiogenesis, invasion and metastasis. In thischapter, the focus is on one of the late stages of tumor progression, tumor inva-sion. Tumor invasion has been recognized as a complex system, since its behavioremerges from the combined effect of tumor cell-cell and cell-microenvironment in-teractions. Cellular automata (CA) provide simple models of self-organizing com-plex systems in which collective behavior can emerge out of an ensemble of manyinteracting �simple� components. Recently, cellular automata have been used to gaina deeper insight in tumor invasion dynamics. In this chapter, we brie�y introducecellular automata as models of tumor invasion and we critically review the mostprominent CA models of tumor invasion.

1 Introduction

Cancer describes a group of genetic and epigenetic diseases, characterized by un-controlled growth of cells, leading to a variety of pathological consequences andfrequently death. Cancer has long been recognized as an evolutionary disease [1].Cancer progression can be depicted as a sequence of traits or phenotypes that cells

H. HatzikirouCenter for Information Services and High Performance Computing, Technische Universitat Dres-den, Nothnitzerstr. 46, 01069 Dresden, Germany, e-mail: [email protected]. BreierInstitute of Pathology, Medical Faculty Carl Gustav Carus, Technische Universitat Dresden, Schu-bertstrasse 15, 01307 Dresden, Germany e-mail: [email protected]. DeutschCenter for Information Services and High Performance Computing, Technische Universitat Dres-den, Nothnitzerstr. 46, 01069 Dresden, Germany, e-mail: [email protected]

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2 H. Hatzikirou, G. Breier and A. Deutsch

have to acquire if a neoplasm (benign tumor) is to become an invasive and ma-lignant cancer. A phenotype refers to any kind of observed morphology, functionor behavior of a living cell. Hanahan and Weinberg [2] have identi�ed six cancercell phenotypes: unlimited proliferative potential, environmental independence forgrowth, evasion of apoptosis, angiogenesis, invasion and metastasis.

In this chapter, we concentrate on the invasive phase of tumor growth. Invasionis the main feature that allows a tumor to be characterized as malignant. The pro-gression of a benign tumor and delimited growth to a tumor that is invasive andpotentially metastatic is the major cause of poor clinical outcome in cancer patients,in terms of therapy and prognosis. Understanding tumor invasion could potentiallylead to the design of novel therapeutical strategies. However, despite the immenseamounts of funds invested in cancer research, the intracellular and extracellular dy-namics that govern tumor invasiveness in vivo remain poorly understood.

Biomedically, invasion involves the following tumor cell processes:• tumor cell migration, which is a result of down-regulation of cadherins, that is

loss of cell-cell adhesion,• tumor cell-extracellular matrix (ECM) interactions, such as cell-ECM adhesion,

and ECM degradation/remodeling, by means of proteolysis. These processes al-low for the penetration of the migrating tumor cells into host tissue barriers, suchas basement and interstitial stroma [3], and

• tumor cell proliferation.Tumor invasion facilitates the emergence of metastases, i.e. the spread of cancercells to another part of the body and the formation of secondary tumors. It is obvi-ous that tumor invasion comprises a central aspect in cancer progression. However,invasive phenomena occur not only in pathological cases of malignant tumors butalso during normal morphogenesis and wound healing.

Cancer research has been directed towards the understanding of tumor invasiondynamics and its implications in treatment design. In particular, research concen-trates along the following problems:

• Invasive tumor morphology: A wealth of empirical evidence links disease pro-gression with tumor morphology [4]. The tumor morphology can indicate thedegree of a tumor's malignancy. In particular, it is experimentally and clinicallyobserved that a morphological instability is related to invasive solid tumors, pro-ducing �nger-like spatial patterns. The question is which molecular and cellularmechanisms are responsible for this spatial pattern formation.

• Cell migration and in�uence of the ECM: Important aspects of invading tumorsare cell motion and the effect of the surrounding environment, especially theECM [3].

• Metabolism and acidosis: The multi-step process of carcinogenesis is often de-scribed by somatic evolution, wherein phenotypic properties are retained or lostdepending on their contribution to the individual tumor cell survival and repro-ductive potential. One of the most prominent phenotypic changes involves theanaerobic glucose metabolism (glycolysis). A side-product of this metabolic ac-tivity is the production of H+ ions that increase the pH of the tumor's microenvi-

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Cellular automaton models for tumor invasion 3

ronment (acidosis). This gives rise to the questions: (i) Why does tumor evolutionlead to this kind of metabolism, which is energetically de�cient in comparisonwith the aerobic one? (ii) What are the advantages for the tumor? (iii) How doglycolytic tumor cells in�uence tumor invasion.

• Emergence of invasion: Typically, tumor invasion appears during the late stagesof carcinogenesis. Of ultimate importance is the question what are the mecha-nisms and the environmental conditions that trigger the progression from benignneoplasms to malignant invasive tumors.

• Robustness: There are several questions concerning the stability and the resis-tance of tumor invasion such as: (i) Why are malignant tumor so robust (resis-tant) against perturbations (i.e. therapies)? (ii) Is it possible to design intelligenttherapies (at the cellular level) that disturb the tumor's robustness? (iii) How canwe investigate the tumor's robustness?

Mathematical modeling and analysis provide invaluable tools towards answeringthe above questions. Tumor invasion involves processes, which occur at differentspatio-temporal scales, including processes at the subcellular, cellular and tissuelevels. Mathematical models allow description and linking of these levels. One candistinguish molecular, cellular and tissue scales, respectively [5, 6]:• The molecular scale refers to phenomena at the sub-cellular level and concen-

trates on molecular interactions and resulting phenomena, such as alterations ofsignaling cascades and cell cycle control, gene mutations, etc. In tumor invasion,the down-regulation of cadherins provides an example of a molecular process.

• The cellular scale refers to cellular interactions and therefore to the most promi-nent dynamics of cell populations, e.g. adhesion, contact inhibition, chemotaxisetc.

• The tissue scale focuses on tissue level processes taking into account macro-scopic quantities, such as volumes, �ows etc. Continuum phenomena include cellconvection and diffusion of nutrients and chemical factors, mechanical stress andthe diffusion of metastases.

For example, genetic alterations may lead to invasive cells (molecular scale) that areable to migrate (cellular scale) and interact with diffusible or non-diffusible signals(tissue scale). Models that deal with phenomena at multiple scales are called multi-scaled.

Recently, a variety of mathematical models have been proposed to analyze dif-ferent aspects of tumor invasion. Deterministic macroscopic models are used tomodel the spatio-temporal growth of tumors, usually assuming that tumor invasionis a wave propagation phenomenon [7, 8, 9, 10, 11]. Computational investigationsof the invasiveness of glioma tumors illustrate that the ratio of tumor growth andspatial anisotropy in cell motility can quantify the degree of tumor invasiveness[12, 13]. Whilst these models are able to capture the tumor structure at the tis-sue level they fail to describe the tumor at the cellular and the sub-cellular levels.Lately, multi-scale approaches attempt to describe and predict invasive tumor mor-phologies, growth and phenotypical heterogeneity [14, 15].

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4 H. Hatzikirou, G. Breier and A. Deutsch

Cellular automata (CA), and more generally cell-based models, provide an alter-native modeling approach, where a micro-scale investigation is allowed through astochastic description of the dynamics at the cellular level [16]. In particular, CAde�ne an appropriate modeling framework for tumor invasion since they allow for:• CA rules can mimic the processes at the cellular level. This fact allows for the

modeling of an abundance of experimental data that refer to cellular and sub-cellular processes related to tumor invasion.

• The discrete nature of CA can be exploited for investigations of the boundarylayer of a tumor. Bru et al. [17] have analyzed the fractal properties of tumor sur-faces (calculated by means of fractal scaling analysis) which can be comparedwith corresponding CA simulations to gain a better understanding of the tumorphenomemon. In addition, the discrete structure of CA facilitates the implemen-tation of complicated environments without any of the computational problemscharacterizing the simulation of continuous models.

• Motion of tumor cells through heterogeneous media (e.g ECM) involves phe-nomena at various spatial and temporal scales [18]. These cannot be capturedin a purely macroscopic modeling approach. On the other hand, discrete micro-scopic models, such as CA, can incorporate different spatio-temporal scales andthey are well-suited for simulating such phenomena.

• CA are paradigms of parallelizable algorithms. This fact makes them computa-tionally ef�cient.In the following section, we provide a de�nition of CA. In section 3, we review

the existing CA models for central processes of tumor invasion. Finally, in section4, we critically discuss the use of CAs in tumor invasion modeling and we identifyfuture research questions related to tumor invasion.

2 Cellular automata

The notion of a cellular automaton originated in the works of John von Neumann(1903-1957) and Stanislaw Ulam (1909-1984). Cellular automata may be viewed assimple models of self-organizing complex systems in which collective behavior canemerge out of an ensemble of many interacting �simple� components. In complexsystems, even if the basic and local interactions are perfectly known, it is possiblethat the global behavior obeys new laws that can not be obviously extrapolated fromthe individual properties, as if the whole is more than the sum of all the parts. Thisproperty makes cellular automata a very interesting approach to model complexsystems in physics, chemistry and biology (examples are introduced in [16, 19]). ACA can be de�ned as a 4-tuple (L ,S ,N ,F ), where:• L is an in�nite regular lattice of cells/nodes (discrete space),• S is a �nite set of states (discrete states); each cell i ∈ L is assigned a state

s ∈S ,• N is a �nite set of neighbors,

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Cellular automaton models for tumor invasion 5

• F is a deterministic or probabilistic map

F : S |N | → S (1){si}i∈N 7→ s, (2)

which assigns a new state to a cell depending on the state of all its neighborsindicated by N (local rule).

The evolution of a CA is de�ned by applying the function F synchronously to allcells of the lattice L (homogeneity in space and time).

The above features can be extended, giving rise to several variants of the classicalCA notion [20]. Some of these are:

Asynchronous CA: in such CA, the restriction of simultaneous update of all thenodes is revoked, allowing for asynchronous update.

Non-homogeneous CA: this variation allows the transition rules to vary depend-ing on node position. Agent-based models are �relatives� of CA that lost thehomogeneity property, i.e. each individual-particle may have its own set of rules.

Coupled-map lattices: in this case the constraint of discrete state space is with-drawn, i.e. the state variables are assumed to be continuous. An important typeof coupled-map lattices are the so-called Lattice Boltzmann models [21].

Structurally dynamic CA: in these systems, the underlying lattice is no longer apassive static object but becomes a dynamic component. Therefore, the latticestructure evolves depending on the values of the nodes state variables.

3 Models of tumor invasion

This section reviews the existing literature on cellular automata models of tumorinvasion. Categorizing these models is a non-trivial task. Moreover, existing CAmodels are describing tumor invasion at more than one scale (sub-cellular, cellularand tissue). In this review, we distinguish models that analyze: (i) the invasive mor-phology, (ii) tumor cell migration and the in�uence of the ECM, (iii) metabolismand acidosis, and (iv) the emergence of tumor invasion.

3.1 Invasive tumor morphology

The tumor morphology arising from the spatial pattern formation of the tumor cellpopulation, has been recognized as a very important aspect of tumor growth. Severalresearchers attempted to reveal the mechanisms of spatial pattern formation of in-vasive tumors. Here, we present the most representative CA models for the invasivetumor morphology.

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6 H. Hatzikirou, G. Breier and A. Deutsch

3.1.1 Effects of directed cell motion

Sander et al. [22] developed a CA model to investigate the branching morphologyof invasive brain tumors. In the model tumor cell motion is in�uenced by two keyprocesses: (i) chemotaxis, and (ii) �slime trail following�. A typical example of a�slime trail following� mechanism is found in the motion of certain myxobacteria[23].

The authors show that the branching morphology of tumors can be explained asa result of chemotaxis and slime trail following. In particular, simulations reproducethe branching pattern formation observed in in vitro cultures of glioma cells. How-ever, the assumption of �slime trail following� has not been proven biologically,yet.

3.1.2 Spatial structure of invasive tumors

Anderson [14, 24] proposed a model to examine the effects of the tumor cell het-erogeneity (at the genetic level) on the spatial morphology and to analyze the im-portance of cell-cell and cell-ECM adhesion. The model assumes a non-diffusible,�xed con�guration of ECM. The extracellular matrix can be degraded by diffusibleenzymes, such as metallo-proteinases, produced by tumor cells. Moreover, cells areallowed to mutate and evolve their phenotype from proliferative to invasive. Finally,an oxygen concentration �eld plays the role of nutrients in the model.

Simulations of the model show that: (i) the ECM heterogeneity is mainly respon-sible for the tumor branching morphology (�g. 2), (ii) cell-cell adhesion plays animportant role only in the early stages of tumor development, (iii) invasive tumorcells are located at the boundary of the tumor, and (iv) the tumor is a phenotypicallyheterogeneous object.

3.2 Tumor cell migration and the in�uence of the extracellularmatrix

Cell migration and cell-ECM interactions are two of the most crucial invasion-related processes. Cellular automata provide an appropriate framework to modeland analyze the effect of cell motility and cell-environment interactions of tumorcell migration.

3.2.1 The role of cell-cell and cell-ECM adhesion

Turner and Sheratt [25] proposed a cellular Potts model [26] to investigate how cell-cell and cell-ECM adhesion in�uence the tumor invasion depth and tumor morphol-ogy. A cellular Potts model can be viewed as an extension of the CA idea allowing

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Cellular automaton models for tumor invasion 7

to analyze phenomena that take into account speci�c cell shapes. Cells are assumedto move according to intercellular adhesive interactions and haptotactical gradients.Moreover, cells are allowed to proliferate, while mitotic probabilities depend on thestrength of the adhesive interaction. Finally, cells are assumed to secrete proteolyticenzymes that degrade the ECM.

The authors show that adhesive dynamics can explain the ��ngering� patternsobserved in their simulations. Moreover, the authors demonstrate that the width ofthe invasion zone depends less on cell-cell adhesion and more on cell-ECM adhesionfacilitated by haptotaxis and proteolysis.

3.2.2 Cellular mechanisms of glioma cell migration

In the work of Aubert et al. [27], a CA model is introduced that allows for the inves-tigation of tumor cell migration, based on experimentally observed density pro�lesof glioma cell cultures. The goal is to identify the mechanisms of tumor (glioma)cell motion, which play a crucial role in tumor invasion. The authors do not con-sider proliferation of tumor cells. Only the in�uence of tumor cell migration andintercellular interactions are studied. The authors introduce and test two distinct cellmechanisms: (i) cell-cell adhesion, and (ii) a kind of �inertia� in cell motion, i.e. thecells tend to maintain the direction of their motion.

The authors carefully scale the model according to the experimental setup andcalibrate the corresponding model parameters. The simulation results indicate thatcell-cell adhesion can explain the experimental results. It is concluded that cell-celladhesion is an important process in glioma cell migration.

3.2.3 Effects of �ber tracts on glioma invasion

Wurzel et al. [28] model glioma tumor invasion with a lattice-gas cellular automa-ton (LGCA) [16]. The authors address the question of how �ber tracts, found inthe brain's white matter, in�uence the spatio-temporal evolution and the invadingfront morphology of glioma tumors. Cells are assumed to move, proliferate and un-dergo apoptosis according to corresponding stochastic processes. Fiber tracts arerepresented as a local gradient �eld that enhances cell motion in a speci�c direction.

The authors develop and analyze different scenarios of �ber tract in�uence. Agradient �eld may increase the speed of the invading tumor front. For high �eldintensities the model predicts the formation of cancer islets at distances away fromthe main tumor bulk. The simulated invasion patterns qualitatively resemble clinicalobservations.

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8 H. Hatzikirou, G. Breier and A. Deutsch

3.2.4 Effect of heterogeneous environments on tumor cell migration

Hatzikirou et al [29] developed a LGCA model to investigate the in�uence of het-erogeneous environments on tumor cell dispersal. This model is a simpli�ed ver-sion of [28] which facilitates the mathematical analysis. In this study no prolifera-tion or death of cells is considered. The authors distinguish two kinds of cell-ECMinteractions (i) cell-ECM adhesion leading to haptotactical motion along integrinconcentration gradients (environment with directional information) and (ii) contactguidance that promotes the alignment along ECM pores or �bres as seen in �g. 3(environment with orientational information).

The authors mathematically analyze tumor cell motion in dependence of bothkinds of cell-ECM interactions. In particular, macroscopic dispersal measures (likemean cell �ux) depending on cellular and environmental parameters are calculated.Accordingly, the models allow for prediction of cell motion in different environ-ments.

3.3 Metabolism and acidosis

In the course of cancer progression tumor cells undergo several phenotypic changes,in terms of motility, metabolism and proliferative rates. In particular, it is importantto analyze the effect of the anaerobic metabolism of tumor cells and the acidi�cationof the environment (as a side-product of glycolysis) on tumor invasion (�g. 4).

Patel et al. [30] proposed a model of tumor growth to examine the roles of nativetissue vascularity and anaerobic metabolism on the growth and invasion ef�cacyof tumors. The model assumes a vascularized host tissue. Anaerobic metabolisminvolves the consumption of glucose and the production of H+ ions, leading to theacidi�cation of the local tissue. The vascular network allows for the absorption ofH+ ions. Cells are assumed to be proliferative and non-motile. The pH level, i.e.is the H+ concentration, and the glucose concentration determine the survival anddeath of the cells.

Simulations of the model show: (i) high tumor H+ ion production favors tu-mor invasion by the acidi�cation of the neighboring host tissue, and (ii) there is anoptimal density of microvessels that maximizes tumor growth and invasion, by min-imizing the acidi�cation effects on tumor cell proliferation (absorption of H+ ions)and maximizing the negative effect of H+ ions on the neighboring tissue.

3.4 Emergence of tumor invasion

Recently, several model have been suggested that concentrate on the evolutionarydynamics of tumors (�g. 5). The main goal of these models is to understand under

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Cellular automaton models for tumor invasion 9

which environmental conditions a certain phenotype appears. Here, we review thosemodels that focus on the mechanisms that allow the emergence of invasive behavior.

3.4.1 In�uence of metabolic changes

Smallbone et al. [31] developed an evolutionary CA model to investigate the cell-microenvironmental interactions that mediate somatic evolution of cancer cells. Inparticular, the authors investigate the sequence of tumor phenotypes that ultimatelyleads to invasive behavior. The model considers three phenotypes, (i) the hyper-plastic phenotype that allows growth away from the basement membrane, (ii) theglycolytic phenotype that allows anaerobic metabolism (the �fuel� is glucose), and(iii) the acid-resistant phenotype that enables the cell to survive in low pH. Cells areallowed to proliferate, die or adapt, i.e. change their phenotype. No cell motion isexplicitly considered.

The model predicts three phases of somatic evolution: (i) Initially, cell survivaland proliferation are dependent on the oxygen concentration. (ii) When the oxygenbecomes scarce, the glycolytic phenotype confers a signi�cant proliferative advan-tage. (iii) The side-products of glycolysis, e.g. galactic acid, increase the microenvi-ronmental pH and promote the selection of acid-resistant phenotypes. The latter celltype is able to invade the neighboring tissue since it takes advantage of the death ofhost cells, due to acidi�cation, and proliferates using the available free space.

3.4.2 The game of invasion

Recently, Basanta et al. [32] have developed a game theory inspired CA that ad-dresses the question of how invasive behavior emerges during tumor progression(see also [33]). The authors study the circumstances under which mutations thatconfer increased motility to cells can spread through a tumor composed of rapidlyproliferating cells. The model assumes the existence of only two phenotypes: �pro-liferative� (high division rate and no motility) and �migratory� (low division rateand high motility). Mutations allow the random change of phenotypes. Nutrientsare assumed to be uniformly distributed over the lattice.

Simulations show that low nutrient conditions confer a reproductive advantageto motile cells over the proliferative ones. The model suggests novel ideas for thera-peutic strategies, e.g. by increasing the oxygen supply around the tumor to favor thereproduction of proliferative cells over the migrating ones. This is not necessarilya therapy since there are benign tumors that are life threatening even if they do notbecome invasive. Despite that, in most cases a growing but non aggressive tumorwill have a much better prognosis than a smaller but invasive one.

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10 H. Hatzikirou, G. Breier and A. Deutsch

4 Discussion

In this review, we have focused on one of the most important aspects of cancerprogression: tumor invasion. The main processes involved in tumor invasion arerelated to tumor cell migration, cell-ECM interactions, especially ECM degrada-tion/remodeling and tumor cell proliferation. These processes are evolving at dif-ferent scales, e.g. cell-ECM adhesion is the response of tumor cells to ECM in-tegrins (molecular level) leading to a haptotactical cell motion (cellular level) andin�uencing the tumor morphology (macroscopic level). Therefore, in order to un-derstand tumor invasion dynamics, it is important to use mathematical tools thatallow for modeling sub-cellular or cellular processes and to analyze the emergentmacroscopic behavior. Individual-based models, and especially CA, are well-suitedfor this task. Moreover, some types of CA models, such as lattice-gas cellular au-tomata [16, 29], facilitate analytical investigations allowing for deeper insight intothe modeled phenomena.

In this chapter, we reviewed the existing CA models of tumor invasion. The pre-sented models explore central aspects of tumor invasion. Some of the models are ingood agreement with biomedical observations for in-vitro and in-vivo tumors. In thefollowing, we list the most interesting biological insights that can be gained fromthe reviewed models:• The signi�cance of hypoxia in the process of tumor progression: Activation of

glycolysis and acidi�cation of the host tissue facilitate tumor invasion. Low nu-trient conditions, such as hypoxia, may trigger invasive behaviors.

• Cell-cell adhesion: Intercellular adhesion has been shown to play an importantrole only in the early stages of tumor progression. It is evident that intercellularadhesion has a great impact in the early stages of tumor growth. However, intumor invasion the role of cell-cell adhesion is minor, since mainly the cell-ECMinteractions appear to dictate the tumor cell behavior.

• Cell-ECM adhesion: This is understood as an important process for tumor inva-sion. In particular, the heterogeneous structure of the ECM strongly in�uencesthe spatial morphology of invasive tumors.Mathematical modeling offers potentially signi�cant insight into tumor invasion.

Several crucial questions have not been adequately addressed so far by modelingefforts:• Branching morphology: In the literature several mechanisms have been proposed

that lead to branching patterns, e.g. diffusion-limited aggregation, the interplay ofcell-cell and -ECM adhesion, as well as chemotaxis or �slime following� motion.However, biologists and modelers have not yet identi�ed a unique mechanismthat drives the branching morphology of invasive tumors.

• Go or grow: The mechanisms of invasive tumor cell migration are still not un-derstood. Recently, Fedotov et al. [34] have analyzed the effect of a postulatedmigration/proliferation dichotomy on cell migration.

• Emergence: Concerning the emergence of invasion in tumor progression little isknown. Mechanisms related to tumor cell motion and other cell processes, such

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Cellular automaton models for tumor invasion 11

as proliferation (migration/proliferation dichotomy), may play an important rolefor the dominance of invasive phenotypes [32, 35].

• Angiogenesis: Another open issue is the in�uence of angiogenesis and vasculo-genesis on tumor invasion. Despite signi�cant efforts to describe the mechanismsof angio- and vasculogenesis, little is known about the effect of these processeson tumor invasion [30].

• Robustness: The identi�cation of cellular mechanisms that are responsible fortumor robustness is a great challenge.Finally, for clinical purposes, future models should be able to provide accurate

and quantitative predictions. Simpli�ed models considering only the essential in-gredients for tumor growth, and especially tumor invasion, but validated with actualclinical data may be helpful in this regard. We sincerely hope that a more profoundknowledge of important tumor characteristics, such as tumor invasion, will eventu-ally lead to the design of more effective therapeutic strategies.

Acknowledgments

We are grateful to D. Basanta, L. Brusch, A. Chauviere, E. Flach and F. Peruani for the commentsand the fruitful discussions. We acknowledge support from the systems biology network HepatoSysof the German Ministry for Education and Research through grant 0313082J. Andreas Deutsch is amember of the DFG Research Center for Regenerative Therapies Dresden - Cluster of Excellence- and gratefully acknowledges support by the Center. The research was supported in part by fundsfrom the EU Marie Curie Network �Modeling, Mathematical Methods and Computer Simulationof Tumour Growth and Therapy� (EU-RTD IST-2001-38923). Finally, the authors would like tothank for the �nancial support of Gottfried Daimler- and Karl Benz foundation throurgh the project�Biologistics: From bio-inspired engineering of complex logistical systems until nanologistics�(25-02/07).

Glossary

• Somatic evolution: Darwinian-type evolution that occurs on soma (as opposedto germ) cells and characterizes cancer progression [36].

• Extracellular matrix: Components that are extracellular and composed of se-creted �brous proteins (e.g. collagen) and gel-like polysaccharides (e.g. gly-cosaminoglycans) binding cells and tissues together.

• Cadherins: Important class of transmembrane proteins. They play a signi�cantrole in cell-cell adhesion, ensuring that cells within tissues are bound together.

• Chemotaxis: Motion towards high concentrations of a diffusive chemical sub-stance.

• �Slime trail motion�: Cells secrete a non-diffusive substance; concentration gra-dients of the substance allow the cells to migrate towards already explored paths.

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12 H. Hatzikirou, G. Breier and A. Deutsch

• Haptotaxis: Directed motion of cells along adhesion gradients of �xed substratesin the ECM, such as integrins.

• Fiber tracts: Bundle of nerve �bers having a common origin, termination, andfunction and especially one within the spinal cord or brain.

References

1. Nowell, P. C.: The clonal evolution of tumor cell populations. Science, 4260, 194: 2328 (1976)2. Hanahan, D., Weinberg, R.: The hallmarks of cancer. Cell 100: 57-70 (2000)3. Friedl, P.: Prespeci�cation and plasticity: shifting mechanisms of cell migration. Curr. Opin.

Cell. Biol. 16(1): 14-23 (2004)4. Sanga, S., Frieboes, H., Zheng, X., Gatenby, R., Bearer, E. and Cristini, V. Predictive on-

cology: multidisciplinary, multi-scale in-silico modeling linking phenotype, morphology andgrowth. Neuroim. 37(1): 120134 (2007)

5. Hatzikirou, H., Deutsch, A., Schaller, C., Simon, M. and Swanson, K.: Mathematical mod-elling of glioblastoma tumour development: a review. Math. Mod. Meth. Appl. Sc., 15(11):1779-1794 (2005)

6. L. Preziozi (ed.), Cancer modelling and simulation. Chapman & Hall/CRC Mathematical Bi-ology & Medicine (2003)

7. Marchant, B. P., Norbury, J. and Perumpanani, A. J.: Traveling shock waves arising in a modelof malignant invasion. SIAM. J. Appl. Math. 60(2): 263�276 (2000)

8. Perumpanani, A. J., Sherratt, J. A., Norbury, J. and Byrne, H. M.: Biological inferences froma mathematical model of malignant invasion. Invas. Metast. 16: 209�221 (1996)

9. Perumpanani, A. J., Sherratt, J. A., Norbury, J. and Byrne, H. M.: A two parameter familyof travelling waves with a singular barrier arising from the modelling of extracellular matrixmediated cellular invasion. Phys. D 126: 145�159 (1999)

10. Sherratt, J. A. and Nowak, M. A.: Oncogenes, anti-oncogenes and the immune response tocancer: a mathematical model. Proc. Roy. Soc. Lond. B 248: 261�271 (1992)

11. Sherratt, J. A. and Chaplain, M. A. J.: A new mathematical model for avascular tumour growth.J. Math. Biol. 43: 291�312 (2001)

12. Swanson, K. R., Alvord, E. C. and Murray, J.: Quantifying ef�cacy of chemotherapy of braintumors (gliomas) with homogeneous and heterogeneous drug delivery. Acta Biotheor. 50:223�237 (2002)

13. Jbabdi, S., Mandonnet, E., Duffau, H., Capelle, L., Swanson, K., Pelegrini-Issac, M.,Guillevin, R. and Benali, H.: Simulation of anisotropic growth of low-grade gliomas usingdiffusion tensor imaging. Magn. Res. Med. 54: 616624 (2005)

14. A. Anderson, A. Weaver, P. Cummings and V. Quaranta. Tumor morphology and phenotypicsevolution driven by selective pressure from the microenvironment. Cell 127: 905-915 (2006)

15. Frieboes, H., Lowengrub J., Wise S., Zheng X., Macklin P., Bearer E., Cristini V.: Computersimulation of glioma growth and morphology. Neuroim. 37 (Suppl 1): 59-70 (2007)

16. Deutsch, A., Dormann, S.: Cellular Automaton Modeling of Biological Pattern Formation,Birkhauser (2005)

17. Bru, A., Albertos, S., Subiza, J. L., Lopez Garcia-Asenjo, J. and Bru, I.: The universal dynam-ics of tumor growth, Bioph. J. 85: 2948-2961 (2003)

18. A. Lesne. Discrete vs continuous controversy in physics. Math. Struct. Comp. Sc., 17: 185223(2007)

19. Chopard, B., Dupuis, A., Masselot, A. and Luthi, P.: Cellular automata and lattice Boltzmanntechniques: an approach to model and simulate complex systems. Adv. Compl. Syst. 5(2): 103- 246 (2002)

20. Moreira, J., Deutsch, A.: Cellular automaton models of tumour development: a critical review.Adv. Compl. Syst. 5: 1-21 (2002)

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21. Succi, S.: The lattice Boltzmann equation: for �uid dynamics and beyond. Series NumericalMathematics and Scienti�c Computation. Oxford New York: Oxford University Press (2001)

22. Sander, L. M. and Deisboeck, T. S.: Growth patterns of microscopic brain tumours. Phys. Rev.E, 66: 051901 (2002)

23. Wolgemuth, C. W., Hoiczyk, E., Kaiser, D. and Oster, G. F. , How myxobacteria glide. Curr.Biol., 12 (5): 369-377 (2002)

24. Anderson A. R. A.: A hybrid model of solid tumour invasion: the importance of cell adhesion.Math. Med. Biol. 22: 163186 (2005)

25. Turner, S. and Sherratt, J. A.: Intercellular adhesion and cancer invasion: A discrete simulationusing the extended Potts model, J. Theor. Biol. 216: 85-100 (2002)

26. Graner, F. and Glazier, J.: Simulation of biological cell sorting using a two-dimensional ex-tended Potts Model. Phys. Rev. Lett. 69: 2013-2016 (1992)

27. Aubert, M., Badoual, M., Freol, S., Christov, C.and Grammaticos, B: A cellular automatonmodel for the migration of glioma cells. Phys. Biol. 3: 93-100 (2006)

28. M. Wurzel, C. Schaller, M. Simon, A. Deutsch , Cancer cell invasion of normal brain tissue:Guided by Prepattern? J. Theor. Med. 6(1): 21-31 (2005)

29. Hatzikirou, H., Deutsch, A.: Cellular automata as microscopic models of cell migration inheterogeneous environments. Curr. Top. Dev. Biol. 81: 401-434 (2008)

30. Patel A., Gawlinski E., Lemieux S., Gatenby R.: Cellular automaton model of early tumorgrowth and invasion: the effects of native tissue vascularity and increased anaerobic tumormetabolism. J. Theor. Biol., 213: 315-331 (2001)

31. Smallbone, K., Gatenby, R., Gillies, R., Maini, P., Gavaghan, D.: Metabolic changes duringcarcinogenesis: Potential impact on invasiveness. J. Theor. Biol. 244: 703713 (2007)

32. Basanta, D., Hatzikirou, H. and Deutsch, A.: The emergence of invasiveness in tumours: agame theoretic approach, Eur. Phys. Journal B (to appear)

33. Basanta, D., Simon, M., Hatzikirou, H. and Deutsch, A.: An evolutionary game theory per-spective elucidates the role of glycolysis in tumour invasion. Cell Prolif. (to appear)

34. Fedotov, S. and Iomin, A.: Migration and proliferation dichotomy in tumor-cell invasion. Phys.Rev. Let. 98: 118101-4 (2007)

35. Hatzikirou, H, Basanta, B., Simon, M., Schaller, C. and Deutsch, A.: �Go or Grow�: the keyto the emergence of invasion in tumor progression? (under submission)

36. Bodmer, W.: Somatic evolution of cancer cells. J. R. Coll. Physicians Lond. 31(1): 82-89(1997)

37. Habib, S., Molina-Paris, C., Deisboeck, T. S.: Complex dynamics of tumors: modeling anemerging brain tumor system with coupled reaction-diffusion equations. Phys. A 327: 501-524 (2003)

38. Gillies, R. J. and Gatenby, R. A.: Hypoxia and adaptive landscapes in the evolution of carcino-genesis. Canc. Metast. Rev. 26: 311317 (2007)

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14 H. Hatzikirou, G. Breier and A. Deutsch

Figures

Fig. 1 Hanahan and Weinberg have identi�ed six possible types of cancer cell phenotypes: unlim-ited proliferative potential, environmental independence for growth, evasion of apoptosis, angio-genesis, invasion and metastasis (Reprinted from [2], with permission from the authors.)

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Fig. 2 Left: Microscopy image of a multicellular tumor spheroid, exhibiting an extensive branch-ing system that rapidly expands into the surrounding extracellular matrix gel. These branches con-sist of multiple invasive cells. (Reprinted from Habib et al. [37] with permission). Right: Simula-tion of Anderson's model [14] reproducing the experimentally observed morphology of invasivetumors.

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Fig. 3 The effect of the brain's �ber tract on glioma growth. Top: in the left �gure a simulationis shown without taking into account the in�uence of �ber tracts. In the top right �gure the �bertracts in the brain strongly drive the evolution of the tumor growth. Bottom: The left (right) �guredisplays a zooming of the tumor area of the top left (right) simulations (Reprinted from Hatzikirouet al. [29]).

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Fig. 4 Typically, tumors exhibit abnormal levels of glucose metabolism. Positron emission imag-ing (PET) techniques localize the regions of abnormal glycolytic activity and identify the tumorlocus.

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18 H. Hatzikirou, G. Breier and A. Deutsch

Fig. 5 The evolving microenvironment of breast cancer. The multiple stages of breast carcinogene-sis are shown progressing from left to right, along with histological representations of these stages.As indicated the pre-invasive stages occur in an avascular environment, whereas cancer cells havedirect access to vasculature following invasion. (Reprinted from Gilles et al. [38])