Focus Article

Hybrid models of tumor growthKatarzyna A. Rejniak1,2∗ and Alexander R. A. Anderson1

Cancer is a complex, multiscale process in which genetic mutations occurringat a subcellular level manifest themselves as functional changes at the cellularand tissue scale. The multiscale nature of cancer requires mathematical modelingapproaches that can handle multiple intracellular and extracellular factors actingon different time and space scales. Hybrid models provide a way to integrate bothdiscrete and continuous variables that are used to represent individual cells andconcentration or density fields, respectively. Each discrete cell can also be equippedwith submodels that drive cell behavior in response to microenvironmental cues.Moreover, the individual cells can interact with one another to form and act asan integrated tissue. Hybrid models form part of a larger class of individual-based models that can naturally connect with tumor cell biology and allow forthe integration of multiple interacting variables both intrinsically and extrin-sically and are therefore perfectly suited to a systems biology approach to tumorgrowth. 2010 John Wiley & Sons, Inc. WIREs Syst Biol Med 2011 3 115–125 DOI: 10.1002/wsbm.102

INTRODUCTION

Mathematical and computational modeling oftumor growth is not new—in fact it goes back

over 50 years. However, to some extent it has largelybeen ignored by the biological and medical communi-ties. There are multiple reasons for this but two of themost significant revolve around the reductionist focusof biology and the lack of directly testable hypothesesfrom the models. By necessity, much of the models ofcancer were general, phenomenological, and not spe-cific to a type of cancer and therefore were plagued bya lack of experimental data to both parameterize andvalidate. That is not to say they were not useful. Attheir heart most mathematical models are mechanis-tic—focusing on the core processes that drive tumorgrowth and integrating them leading to predictionsthat are holistic by definition. This further contributedto the lack of biological interest in combining labo-ratory experiments with computational simulations.Most of the experimental biologists working in thisfield were more focused on the reductionist routerevolving around specific genetic mutations or sig-naling pathways that were found to be important incancer development. This led to the data explosion

∗Correspondence to: Kasia.rejniak@moffitt.org1Integrated Mathematical Oncology Department, H. Lee MoffittCancer Center & Research Institute, Tampa, FL, USA2Department of Oncologic Sciences, University of South FloridaCollege of Medicine, Tampa, FL, USA

DOI: 10.1002/wsbm.102

that motivated the advent of early systems biologyand the development of bioinformatics. Mathematicalbiology and the mechanistic cancer models it producedwere somewhat left behind, but little by little they havematured moving from simple nonspatial growth laws(gompertz) all the way to hybrid multiscale modelsdiscussed in this review. See also a list of previouslypublished reviews in the Future Reading section. Inthe last few years, mathematical and computationalmodels of cancer have become more accepted bythe biological community both as means to motivateexperimentation but also as a route to integrate mul-tiple experimental measurements to generate testablepredictions. This shift has been partly driven not onlyby the emergence of new modeling approaches (suchas hybrid models) but also by the refocusing of thebiological community on cancer as a system. Math-ematical and computational models of cancer havealmost always viewed cancer as a system of multi-ple interacting variables and processes and thereforeshould really be considered part of systems biology.

In this review, we will focus on the recentdevelopment of hybrid models of tumor growth.While not an exhaustive review we have triedto incorporate all of the most up-to-date models,constraining our search to key references withinthe last 5 years. Hybrid models integrate bothcontinuous and discrete variables and are ableto incorporate biological phenomena on varioustemporal and spatial scales. These models representcells as individual discrete entities and often use

Volume 3, January/February 2011 2010 John Wiley & Sons, Inc. 115

Focus Article www.wiley.com/wires/sysbio

continuous concentration or density fields to modelcell intracellular and extracellular environments. Bytheir very nature, hybrid models are ideal forexamining direct interactions between individual cellsand between the cells and their microenvironment, butthey also allow us to analyze the emergent propertiesof complex multicellular systems (such as cancer).It is worth noting that as these interactions takeplace on the intracellular and intercellular levels, butare manifested by changes on the tissue level, theemergent behavior of growing multiclonal tumors arealmost impossible to infer intuitively. Hybrid modelscan facilitate our understanding of the underlyingbiophysical processes in tumor growth. For example,by using high-throughput simulation techniques, wecan examine the impact that changes in specificcell interactions (or their microenvironment) haveon tumor growth and treatment. Hybrid models areoften multiscale by definition integrating processeson different temporal and spatial scales, such asgene expression, intracellular pathways, intercellularsignaling, cell growth, or migration. There are twogeneral classes of hybrid models, those that are definedupon a lattice and those that are off-lattice. Thestructure of this review will be to view these two broadclasses in terms of increasing cellular complexity. Wewill then revisit these models in terms of the levelof biological detail of the tumor growth process theyrecapitulate. Finally, we will discuss the critical rolethat integration needs to play if we want to makea direct impact on cancer research and treatmentboth from the perspective of integrating modelswith experiments but also from the perspective ofintegrating multiple modeling approaches.

HYBRID MODELS COMPLEXITYHybrid models can be divided into two classesthat depend reciprocally on the number of cellsthese models can handle and the included detailsof each individual cell structure, i.e., models dealingwith large cell populations but with simplified cellgeometry, and those that model small colonies offully deformable cells (Figure 1). Simplified geometrymodels are capable of handling large number ofcells (thousands to millions) and still treat them asindividual entities that can both act independentlyof other cells (individual cell cycle, cell mutations,cell phenotype) and interact with their immediateneighbors (cooperate or compete). With these kind ofmodels, one can simulate tumor growth up to clinicallyrelevant sizes, thereby allowing for incorporation ofdifferent kinds of tumor treatments, and enabling us totest in silico new and preexisting treatment protocols.

On-lattice

Cellular Potts

Hexagonal CA

Square-lattice CA

Multi-compartmentCA

Sphericalcell-centered

Spa

tial s

cale

Cellular detail

Ellipsoidcell-centered

Cell-centered withVoronoi tessellation

Off-lattice

IBCell

FIGURE 1 | Reciprocal relation between the number of cells handledby the models and the level of included cellular details. In each class(on-lattice and off-lattice), the models complexity rises from cellsrepresented by single points to fully deformable bodies.

Models with deformable cells allow us to investigatethe intimate interactions between individual cellsand between cells and their environment. Variouscellular processes can be represented in these modelsin a more realistic way, by incorporating, e.g., thetime- and space-dependent enlargement of growingcells, the orientation of cell division, the elongationduring cell migration. Both classes, however, canbe coupled with additional equations, such asordinary differential (ODE), partial differential (PDE),and/or stochastic equations, to describe signalingor metabolic pathways, as well as mechanical ormolecular details of cell life processes.

Technically, hybrid models can also be dividedinto two classes, on- and off-lattice (Figure 1),however, this term actually refers only to the imposedpositions of the cells [a square, hexagonal, or cubiclattice versus unconstrained locations in the two- (2D)or three-dimensional (3D) space], but the underlyingchemical or physical fields are typically defined onregular grids in both kinds of models (as the simplestway to solve standard reaction-diffusion equations).We elaborate on both classes of model below,

116 2010 John Wiley & Sons, Inc. Volume 3, January/February 2011

WIREs Systems Biology and Medicine Hybrid models of tumor growth

(a) (c)(b)

(d) (e) (f)

(g)

(j) (k) (l)

(i)(h)

400500600300400500

54.8

= 0.25

FIGURE 2 | Snapshots from simulations of various hybrid models of tumor growth. (a) Three-dimensional (3D) tumor spheroid simulated by ahybrid cellular automaton (Reprinted with permission from Ref 12. Copyright 2007 Birkhauser-Verlag). (b) Tumor invasion in prostate ducts simulatedby a hybrid cellular automaton (Reprinted with permission from Ref 25. Copyright 2009 American Association of Cancer Research). (c) 3D tumorspheroid simulated by an agent-based on-lattice model (Reprinted with permission from Ref 14. Copyright 2009 Springer). (d) 3D tumorself-metastatic spheroids simulated by a hybrid cellular automaton (Reprinted with permission from Ref 5. Copyright 2009 Nature Publishing Group).(e) 3D model of ductal carcinoma in situ simulated by a square-grid cellular automaton (Reprinted with permission from Ref 29. Copyright 2007Elsevier). (f) Two-dimensional (2D) tumor spheroids simulated by a hexagonal cellular automaton (Reprinted with permission from Ref 34. Copyright2008 BioMed Central, the Open Access Publisher). (g) 3D vascularized tumor spheroid simulated by Potts model (Reprinted with permission fromRef 46. Copyright 2009 Public Library of Science, open access article). (h) 2D tumor spheroid in a heterogeneous environment composed onextracellular matrix (ECM) fibers simulated by Potts model (Reprinted with permission from Ref 42. Copyright 2008 Elsevier). (i) 2D model ofcolorectal tumor simulated using the particle model with Voronoi triangulation (Reprinted with permission from Ref 47. Copyright 2009 John Wileyand Sons, Inc.). (j) 2D tumor spheroid modeled using the cell-centered off-lattice model (Reprinted with permission from Ref 48. Copyright 2009Springer). (k) 2D hybrid model of tumor growth simulated by particle center-based ellipsoid model (Reprinted with permission from Ref 49. Copyright2007 World Scientific). (l) 2D multiclonal tumor growth simulated by a model of deformable fluid-based cells (Reprinted with permission from Ref 50.Copyright 2007 Birkhauser-Verlag).

Volume 3, January/February 2011 2010 John Wiley & Sons, Inc. 117

Focus Article www.wiley.com/wires/sysbio

discussing in briefly the different models that fit ineach class and how they have been applied to tumorgrowth.

On-Lattice ModelsOn-lattice models are constrained by a latticestructure (square, hexagonal, cubical) that definesthe locations of cells and cell–cell interactionneighborhoods. Technically, they may seem to bemore straightforward to program than the off-lattice models, since usually the underlying grid iscommon for modeling both cellular locations andthe chemical/metabolic fields. Also, the algorithms forcellular neighborhood and cellular microenvironmentare easier to handle in computational implementationsince cell neighborhood relation is defined by thefixed number of surrounding grid sites, and thereforethe search for cell neighbors (e.g., to determinecell–cell adhesion or communication) is simplified.However, the common underlying grid implies thatthe changes in the chemical fields are modeled on thecell scale (unless a multigrid approach is used), andtherefore the ‘jumps’ in these values may not reflectthe smooth changes in chemical gradients. Anotherdisadvantage of the lattice-based models is a limitednumber of directions in which the cells can move andcommunicate with their neighbors. Several examplesof on-lattice models are shown in Figure 2(a)–(h).

1. Multicompartment cellular automata (CA) maycontain more than one cell in each grid site;however, they have a predefined grid capacityand upon reaching this limit some cells need tobe replaced to another grid site. The efficientalgorithms for cell migration between neighbor-ing grid sites need to be defined. In the contextof tumor development, the multicompartmentCA were used to model multicellular spheroidsgrowth1 and the emergence of invasive tumors.2

2. Square-lattice CA assume the mutually exclusivespace management, i.e., each grid site canaccommodate at most one cell. Typically, eachcell has either four or eight potential neigh-bors (von Neumann neighborhood or Mooreneighborhood, respectively). Cells move to theone of the unoccupied lattice sites (eitherrandomly or via the directional stimulus, suchas chemoaxis or haptotaxis). This kind of modelhas been widely used to simulate avascular3–9

and vascularized tumor growth,10,11 tumorcell invasion,12–16 and tumor interactionswith various environmental factors, including

oxygen, glucose, growth factors,17–23 stromalcomposition,24–28 and tissue architecture.29–32

3. Hexagonal-lattice CA use a hexagonal grid todefine cell positions and cell neighborhood con-sisting of six symmetrically located cells sur-rounding the host cell. This kind of modelhas been used to simulate multicellular tumorgrowth,33–35 to investigate different arrange-ments of growing ductal carcinoma in situ,36

and to reproduce patterns of migrating gliomacells.37,38

4. Potts models or Glazier-Granier-Hogeweg(GGH) models extend the square CA by allow-ing individual cells to occupy several 2D squareor 3D cubical lattice sites that together define cellvolume and cell surface area. Cell shape defor-mation and direct cell–cell interactions are basedon the concept of Monte Carlo simulations andenergy minimalization. This model has beenused to model tumor growth and invasion,39–43

and angiogenic vascularization.44–46

Off-Lattice ModelsOff-lattice models, in comparison to on-lattice models,have a more realistic representation of cell spatiallocations in the sense that they do not need tobe uniformly spaced on a fixed grid. However, todetermine cell placement of daughter cells during celldivision some additional steps need to be undertakento account for mutually exclusive cell areas orvolumes (i.e., nonoverlapping). Moreover, the mainadvantage of the lattice-free models, i.e., the freedomto move each cell in any direction, brings a riskthat an inappropriately chosen discrete-time step willresult in cell collisions (especially in densely packedpopulations). The main disadvantage of this kind ofmodel is that one needs to design special algorithms toefficiently handle cell–cell neighborhoods, and someinterpolation techniques need to be applied to transfervalues between the cellular off-lattice individuals andthe chemical fields that are usually computed onregular grids. Several examples of off-lattice modelsare shown in Figure 2(i)–(l).

1. Spherical cell-centered models represent the cellsas single points with either the springs orenergy potentials defined between the neigh-boring cells to keep them within a minimumpredefined distance (circular or spherical geom-etry). These have been used to simulate tumorcell colonies growing in the form of monolayersand spheroids,51–54 various patterns of ductal

118 2010 John Wiley & Sons, Inc. Volume 3, January/February 2011

WIREs Systems Biology and Medicine Hybrid models of tumor growth

carcinoma in situ,55 tumor invasion,48,56,57 andblood vessel intravasation.58

2. Ellipsoidal cell-centered models are similar inconcept to the spherical cell-centered models;however, each cell in this model is defined by twoaxes with different lengths to form an ellipticalshape. Thus allow for a more intuitive definitionof cell orientation and polarization.49,59

3. Voronoi tessellation technique can be used forthe cell-centered models to overlay polygonalshapes around cell nuclei, which in turn leadsto a variable number of cell neighbors, butalso defines cell–cell interactions based onvariable contact between neighboring cells,e.g., based on the length of adherent cell sides.This model has been used to simulate thegrowth of multicellular spheroids60,61 and thedevelopment of colorectal tumors.47,62

4. Fluid-based elastic cell models allow formodeling cell plasticity, geometrical adaptabil-ity, and cell deformation during migration,polarization, and differentiation. Moreover,since cell elastic boundaries are discretized,all boundary points may be treated as cellmembrane receptors/sensors and various cell–cell and cell–extracellular matrix (ECM)interactions can be defined based on cellmembrane receptor dynamics. These kind ofmodels have been used to simulate the growthof multicellular spheroids,63,64 various cellularpatterns in developing ductal carcinoma insitu,50,65 invasive tumors66,67 as well as tomodel normal development of epithelial ductalmonolayers and their various mutants.68,69

BIOLOGICAL COMPLEXITY

Cancer development is a complex multiscale processthat depends on both the intrinsic factors (such asgenetic mutation, gene expression, cell adaptability,robustness, and phenotypic evolution) and on extrin-sic cues sensed from the cell microenvironment (suchas multiple metabolite and nutrient gradients, differ-ent densities and alignments of ECM fibers, or diversetissue architectures). Experimentally, cancer evolutionand development are generally only considered at thegene or protein scale; however, recently there has beena great deal of interest in the impact of this evolutionat the cellular scale. After all, selection occurs uponthe cellular phenotype even if mutations take place inthe genotype. This selection pressure is often driven bythe changes in the tumor microenvironment. Hybridmodels seem particularly well suited to investigate

the evolutionary aspects of cancer and variousstrategies have been developed to model evolution ofboth cell phenotypes and genotypes, as well as thecomplex interactions between cancer cells and theirsurrounding microenvironment. Evolution of cellphenotypes is often modeled using deterministic flowcharts in which a decision to enter the specific cellularprocess (such as cell growth, division, death, ormovement) is determined sequentially by comparingcell status (e.g., cell age, nutrients level, the number ofcell neighbors, or the configuration of cell membranereceptors) to predetermined thresholds. 5,15,25,29,68

Another approach involves the introduction ofrandom mutations that determine the evolutionof a given cellular phenotype (e.g., doubling time,death rate, or sensitivity to contact inhibition) or cellinteractions with external factors (such as concentra-tion of metabolites or ECM degradation).15,30 Suchinteractions can be also modeled using the neuralnetworks18,21 or systems of ODEs defining certainsignaling pathways or protein networks.14,25,40,47

Evolution of cell phenotypes depends not onlyon cell genotype but also on cues sensed by thecells from their neighborhood. Moreover, the evolv-ing cells modify also their immediate vicinity, andthese mutual interactions may lead to the emer-gence of certain microenvironments promoting tumordevelopment. The establishment of a three-layeredstructure (consisting of a proliferating rim, a ring ofquiescent cells, and a necrotic core) that arises intumor spheroids as a consequence of nutrient deple-tion has been reproduced by virtually every kind ofmodeling approach, and has become a test prob-lem for every newly developed mathematical modelof solid tumor growth.3,6,15,34,40,42,46,47,49,50,53,56,67,70

Gradients of nutrients, such as oxygen or glucose,are not the only chemical species present in thestroma surrounding normal and tumor tissues. In fact,tumor cells are exposed to various enzymes [morethan 20 kinds of matrix metalloproteinases (MMPs)and tissue inhibitors of matrix metalloproteinases(TIMPs)], a multitude of growth factors and a range ofchemokines. Mathematical models were used exten-sively to investigate the relations between gradientsof various metabolites and the emerging morpholo-gies of developing tumors. 7,15,19,21,22,25,34,40,43,47,49,67

In addition to responding to various chemical factors,tumor cells can mechanically interact with other tumorcells as well as with various other stromal cells, such asfibroblasts, macrophages, and immune cells. Tumorcell behavior depends also on the interactions withits physical environment, e.g., variable densities andalignment of different ECM fibers (such as collagen,laminin, elastin, or fibronectin). The intimate adhesive

Volume 3, January/February 2011 2010 John Wiley & Sons, Inc. 119

Focus Article www.wiley.com/wires/sysbio

relations between neighboring tumor cells, cells andthe ECM, and the interactions between tumor cellsand other stromal cells have been addressed by mul-tiple investigators.11,15,25–27,38,42,45,52,56,57 The initia-tion and progression of most tumors depends stronglyon the architecture of the host tissue. Various compu-tational models have addressed the issues arising fromconfined microenvironments such as the structure ofepithelial ducts29,35,47,63,65 or brain geometry.30,71

BRIDGING SCALES AND MODELSIn principle, it is possible to build a model thatwill span multiple scales from the genotype andvarious biochemical reactions to the details on cellmorphology, and the collective behavior of millionsof individual cells forming the whole tumor tissue.However, such a model may acquire structural com-plexity that is comparable with biological cells andfar less effective computationally than the real liv-ing organism. It is therefore more desirable to findways to bridge independent models rather than builda single ‘mega-model’ that encompasses all the com-plexity of tumor development. This bridging maybe in terms of separate models that consider dis-tinct parts of the cancer process or the same processbut on different scales. Our group has undertakensuch an approach to address genetic, mechanistic,and evolutionary mechanisms of disruption of tis-sue homeostasis and initiation of tumor growth,31 aswell as to investigate how the local tumor microen-vironment can select for cells with an invasiveadvantage.13,66 Similarly, the questions of vascularendothelial growth factor (VEGF) transport in healthyand cancerous vascular systems were investigatedby Popel and collaborators using a multicompart-ment model.72,73 The emergence of glycolytic phe-notype in carcinogenesis was addressed by Gatenbyand colleagues using a combination of approaches

including CA, evolutionary dynamics, information, orcompetition theories.74–77 The advantage of applyingseveral distinct models in answering the same scientificquestion is manifold. If these models produce similar(or comparable) outcomes, the common assumptionsunderlying the investigated phenomena can be identi-fied, and used to infer underlying mechanisms that canthen be further investigated experimentally. If thesemodels result in different outcomes, further investiga-tion can be carried to determine which features specificto each model have influenced the contrasting resultsand how this relates to the underlying biology. Again,this may lead to further experimentation to confirmor rule out the contrasting results.

CONCLUSION

As we hinted at in the opening section of this article,computational models developed and implementedwithout real experimental data to neither parameterizenor validate their predictions was one of the majorlimitations in them gaining biological acceptance.What has recently become clear is that there is notonly a need for greater integration between models andexperimentation but also a requirement.28,23,78,79 Thisdialogue must go both ways—experiments shoulddrive models and models should drive experiments.Models can utilize experimental data and producenovel hypotheses but without the experimental testingto validate or negate such hypotheses, it becomes avery limited academic exercise. Although to be fair, itcan be very difficult to find appropriate collaboratorsmotivated to provide such experimental support.

Models need to drive experimentation and tosome extent this requires an understanding of theexperimental systems that are currently being usedby the cancer research community. The schematicpresented in Figure 3 highlights the multiple scales

Molecules

Kinetics modelsNeural/genetic networks

Biochemical reaction networks

Subcellular Cellular

Hybrid models

Tissues Clinicaloutcome

Continuous modelsHomogenization

Game theory

GenesProteinsSignaling

OrgansSystems

Organisms Populations

Reductionism

High-throughput screening

FIGURE 3 | A schematic of modeling scales and techniques. Multiple biological scales can be bridged by various types of mathematical models.

120 2010 John Wiley & Sons, Inc. Volume 3, January/February 2011

WIREs Systems Biology and Medicine Hybrid models of tumor growth

that are experimentally studied in cancer research bymeans of the experimental systems that are utilized.If we truly want to build integrated models, then weneed to think of what sort of experiments will beneeded to drive our models and validate them. Fromour personal experience, this leads to a significantshift in thinking in relation to which components areincorporated into a model and which are not. It alsodictates what type of model should be utilized andthis review would not be about hybrid models if wedid not believe that hybrid approaches are perfectlysuited to facilitate such integration. Owing to theircell-centric nature, hybrid models naturally connectwith cell biology and readily incorporate microenvi-ronmental components. The interface between tumorcells and their microenvironment being one of thecritical drivers of cancer progression, the other beingthe intracellular changes that result from mutations,altered intracellular and intercellular signaling or pro-tein trafficking, which can also be captured usinghybrid models.14,18,40,57,68

It is worth restating that cancer is a multi-scale process, whereby mutations at the molecular

scale effect protein formation that effects signalingpathways, which modulate cell behavior that trans-forms the tissue leading to damaged organs andpotentially death. This complex multiscale processcan be broken down into smaller units that aremore amenable to both experimental and theoreticalapproaches. This again brings into focus the bridg-ing nature of mathematical models that are criticalfor understanding how the different biological scalesof cancer impact upon one another. The models wehave focused on this review bridge several scales bothabove and below the fundamental unit of the cell(Figure 3), however, they cannot bridge all—this mostcertainly will require different modeling approachessuch as continuous80 or statistical models.81 In addi-tion, there is an unspoken void between in vitro andin vivo models and between in vivo and the clinic. Insilico models have the power to link these approachesand in doing so can give some insight into the pro-cesses that translate well between them and those thatdo not. This is a severely understudied area for mod-eling in cancer research and should be a ripe focus forfuture work.

ACKNOWLEDGEMENTS

Both authors were partially supported by the National Cancer Institute Integrative Cancer Biology Program(U54 CA113007) and by the NCI Physical Sciences-Oncology Centers Program (U54 CA143970).

REFERENCES

1. Piotrowska MJ, Angus SD. A quantitative cellularautomaton model of in vitro multicellular spheroidtumour growth. J Theor Biol 2009, 258:165–178.

2. Basanta D, Hatzikirou H, Deutsch A. Studying theemergence of invasiveness in tumours using the gametheory. Eur Phys J B 2008, 63:393–397.

3. Dormann S, Deutsch A. Modeling of self-organizedavascular tumor growth with a hybrid cellular automa-ton. In Silico Biol 2002, 2:393–406.

4. Spencer SL, Gerety RA, Pienta KJ, Forrest S. Modelingsomatic evolution in tumorigenesis. PLoS Comput Biol2006, 2:e108.

5. Enderling H, Hlatky L, Hahnfeldt P. Migration rules:tumours are conglomerates of self-metastases. Br JCancer 2009, 100:1917–1925.

6. Kansal AR, Torquato S, Chiocca EA, DeisboeckTS. Emergence of a subpopulation in a computa-tional model of tumor growth. J Theor Biol 2000,207:431–441.

7. Zhang L, Strouthos CG, Wang Z, Deisboeck TS.Simulating brain tumor heterogeneity with multiscaleagent-based model: linking molecular signatures, phe-notypes and expansion rate. Math Comp Model 2009,49:307–319.

8. Wcislo R, Dzwinel W, Yuen DA, Dudek AZ. A3-D model of tumor progression based on complexautomata driven by particle dynamics. J Mol Model2009, 15:1517–1539.

9. Enderling H, Anderson ARA, Chaplain MAJ, BeheshtiA, Hlatky LR, Hahnfeldt PJ. Paradoxical dependen-cies of tumor dormancy and progression on basic cellkinetics. Cancer Res 2009, 69:8814–8821.

10. Stephanou A, McDougall SR, Anderson ARA, Chap-lain MAJ. Mathematical modelling of flow in 2D and3D vascular networks: applications to anti-cancer andchemotherapeutic drug strategies. Math Comp Model2005, 41:1137–1156.

Volume 3, January/February 2011 2010 John Wiley & Sons, Inc. 121

Focus Article www.wiley.com/wires/sysbio

11. Owen MR, Alarcon T, Maini PK, Byrne HM. Angio-genesis and vascular remodelling in normal and cancer-ous tissues. J Math Biol 2009, 58:689–721.

12. Anderson ARA. A hybrid multiscale model of tumourinvasion: evolution and the microenvironment. In:Anderson A, Chaplain M, Rejniak KA, eds. Single-Cell-Based Models in Biology and Medicine, Chapter I.1.Basel, Switzerland: Birkhauser-Verlag; 2007.

13. Anderson ARA, Rejniak KA, Gerlee P, Quaranta V.Microenvironment driven invasion: a multiscale modelinvestigation. J Math Biol 2009, 58:579–624.

14. Zhang L, Wang Z, Sagotsky JA, Deisboeck TS. Multi-scale agent-based cancer modeling. J Math Biol 2009,58:545–559.

15. Anderson ARA. A hybrid mathematical model of solidtumour invasion: the importance of cell adhesion. MathMed Biol 2005, 22:163–186.

16. Sottoriva A, Verhoeff JJC, Borovski T, McWeeney SK,Naumov L, Medema JP, Sloot PMA, Vermeulen L. Can-cer stem cell tumor model reveals invasive morphologyand increased phenotypical heterogeneity. Cancer Res2010, 70:46–56.

17. Gatenby RA, Smallbone K, Maini PK, Rose F, AverillJ, Nagle RB, Worrall L, Gillies RJ. Cellular adaptationsto hypoxia and acidosis during somatic evolution ofbreast cancer. Br J Cancer 2007, 97:646–653.

18. Gerlee P, Anderson ARA. Evolution of cell motility inan individual-based model of tumour growth. J TheorBiol 2009, 259:67–83.

19. Silva AS, Yunes JA, Gilles RJ, Gatenby RA. Thepotential of systemic buffers in reducing intratumoralextracellular pH and acid-mediated invasion. CancerRes 2009, 69:2677–2684.

20. Zhang L, Chen LL, Deisboeck TS. Multi-scale, multi-resolution brain cancer modeling. Math Comput Simul2009, 79:2021–2035.

21. Gerlee P, Anderson ARA. A hybrid cellular automa-tion model of clonal evolution in cancer: the emer-gence of the glycolytic phenotype. J Theor Biol 2008,250:705–722.

22. Smallbone K, Gatenby RA, Gillies RJ, Maini PK, Gav-aghan DJ. Metabolic changes during carcinogenesis:potential impact on invasiveness. J Theor Biol 2007,244:703–713.

23. Anderson ARA, Hassanein M, Branch KM, Lu J,Lobdell NA, Maier J, Basanta D, Weidow B, NarasannaA, Arteaga CL, et al. Microenvironmental indepen-dence associated with tumor progression. Cancer Res2009, 69:8797–8806.

24. Anderson ARA, Weaver AM, Cummings PT, QuarantaV. Tumor morphology and phenotypic evolution drivenby selective pressure from the microenvironment. Cell2006, 127:905–915.

25. Basanta D, Strand DW, Lukner RB, Franco OE, Clif-fel DE, Ayala GE, Hayward SW, Anderson ARA. The

role of transforming growth factor-B–mediated tumor-stroma interactions in prostate cancer progression:an integrative approach. Cancer Res 2009,69:7111–7120.

26. De Pillis LG, Mallet DG, Radunskaya AE. Spatialtumor-immune modeling. Comput Math Methods Med2006, 7:159–176.

27. Enderling H, Alexander NR, Clark ES, Branch KM,Estrada L, Crooke C, Jourquin J, Lobdell N, ZamanMH, Guelcher SA, et al. Dependence of invadopodiafunction on collagen fiber spacing and cross-linking:computational modeling and experimental evidence.Biophys J 2008, 95:2203–2218.

28. Anderson ARA, Quaranta V. Integrative mathematicaloncology. Nat Rev Cancer 2008, 8:227–234.

29. Bankhead A, Magnuson N, Heckendorn R. Cellularautomaton simulation examining progenitor hierarchystructure effects on mammary ductal carcinoma in situ.J Theor Biol 2007, 246:491–498.

30. Gevertz J, Torquato S. Growing heterogeneous tumorsin silico. Phys Rev E 2009, 80:051919.

31. Anderson ARA, Basanta D, Gerlee P, Rejniak KA. Evo-lution, regulation and disruption of homeostasis and itsrole in carcinogenesis. In: Deisboeck TS, Stamatakos G,eds. Multiscale Cancer Modeling. New York: Chapman& Hall; 2010.

32. Silva AS, Gatenby RA, Gillies RJ, Tunes JA. A quan-titative theoretical model for development of malig-nancy in ductal carcinoma in situ. J Theor Biol 2010,262:601–613.

33. Deutsch A. Lattice-gas cellular automata modeling ofdeveloping cell systems. In: Anderson A, Chaplain M,Rejniak KA, eds. Single-Cell-Based Models in Biol-ogy and Medicine, Chapter I.2. Basel, Switzerland:Birkhauser-Verlag; 2007.

34. Engelberg JA, Ropella GEP, Hunt CA. Essential oper-ating principles for tumor spheroid growth. BMCSystBiol 2008, 2:110.

35. Kim SHJ, Debnath J, Mostov K, Park S, Hunt CA. Acomputational approach to resolve cell level contribu-tions to early glandular epithelial cancer progression.BMC SystBiol 2009, 3:122.

36. Shumate SD, El-Shenawee M. Computational model ofductal carcinoma in-situ: the effects of contact inhibi-tion on pattern formation. IEEE Trans Biomed Eng2009, 5:1341–1347.

37. Aubert M, Badoual M, Christov C, Grammaticos B.A model for glioma cell migration on collagen andastrocytes. J R Soc Interface 2008, 5:75–83.

38. Aubert M, Badoual M, Grammaticos B. A modelfor short- and long-range interactions of migrating.Tumour Cell Acta Biotheor 2008, 56:297–314.

39. Turner S, Sherratt JA, Cameron D. Tamoxifen treat-ment failure in cancer and the nonlinear dynamics ofTGF-beta. J Theor Biol 2004, 229:101–111.

122 2010 John Wiley & Sons, Inc. Volume 3, January/February 2011

WIREs Systems Biology and Medicine Hybrid models of tumor growth

40. Jiang Y, Pjesivac-Grbovic J, Cantrell C, Freyer JP. Amultiscale model for avascular tumor growth. BiophysJ 2005, 89:3884–3894.

41. Maree AFM, Grieneisen VA, Hogweg P. The cellularPotts model and biophysical properties of cells, tissuesand morphogenesis. In: Anderson ARA, Chaplain MAJ,Rejniak KAR, eds. Single Cell Based Models in Biologyand Medicine. Basel, Switzerland: Birkhauser; 2007.

42. Rubenstein BM, Kaufman LJ. The role of extracellu-lar matrix in glioma invasion: a cellular Potts modelapproach. Biophys J 2008, 95:5661–5680.

43. Poplawski NJ, Agero U, Gens JS, Swat M, GlazierJA, Anderson ARA. Front instabilities and invasivenessof simulated avascular tumors. Bull Math Biol 2009,71:1189–1227.

44. Merks RMH, Perryn ED, Shirinifard A, Glazier JA.Contact-inhibited chemotaxis in de novo and sprout-ing blood-vessel growth. PLoS Comput Biol 2008,4:e1000163.

45. Bauer AL, Jackson TL, Jiang Y. Topography of extra-cellular matrix mediates vascular morphogenesis andmigration speed in angiogenesis. PLoS Comput Boil2009, 5:e1000445.

46. Shirinifard A, Gens JS, Zaitlen BL, Poplawski NJ,Swat M, Glazier JA. 3D Multi-cell simulation of tumorgrowth and angiogenesis. PLoS ONE 2009, 4:e7190.

47. van Leeuwen IMM, Mirams GR, Walter A, Fletcher A,Murray P, Osborne J, Varma S, Young SJ, Cooper J,Doyle B, et al.An integrative computational model forintestinal tissue renewal. Cell Prolif 2009, 42:617–636.

48. Galle J, Hoffmann M, Aust G. From single cells to tissuearchitecture—a bottom-up approach to modelling thespatio-temporal organisation of complex multi-cellularsystems. J Math Biol 2009, 58:261–283.

49. Kim Y, Stolarska MA, Othmer HG. A hybrid modelfor tumor spheroid growth in vitro, I: theoretical devel-opment and early results. Math Models Methods ApplSci 2007, 17:1773–1798.

50. Rejniak KA. Modelling the development of complextissues using individual viscoelastic cells. In: AndersonA, Chaplain M, Rejniak KA, eds. Single-Cell-BasedModels in Biology and Medicine. Basel, Switzerland:Birkhauser-Verlag; 2007.

51. Drasdo D, Hoehme S. A single-cell-based model oftumor growth in vitro: monolayers and spheroids. PhysBiol 2005, 2:133–147.

52. Galle J, Sittig D, Hanisch I, Wobus M, Wandel E, Loef-fler M, Aust G. Individual cell-based models of tumor-environment interactions. Multiple effects of CD97 ontumor invasion. Am J Pathol 2006, 169:1802–1811.

53. Galle J, Loeffler M, Drasdo D. Modeling the effects ofderegulated proliferation and apoptosis on the growthdynamics of epithelial cell populations in vitro. BiophysJ 2005, 88:62–75.

54. Jeon J, Quaranta V, Cumming PT. An off-lattice hybriddiscrete-continuum model of tumor growth and inva-sion. Biophys J 2010, 98:37–47.

55. Norton K-A, Wininger M, Bhanot G, Ganesan S,Barnard N, Shinbrot T. A 2D mechanistic model ofbreast ductal carcinoma in situ (DCIS) morphologyand progression. J Theor Biol 2010, 263:393–406.

56. Drasdo D, Hoehme S, Block M. On the role of physicsin the growth and pattern formation of multi-cellularsystems: what can we learn from individual—cell basedmodels? J Stat Phys 2007, 128:287–345.

57. Ramis-Conde I, Drasdo D, Anderson ARA, ChaplainMAJ. Modeling the influence of the cadherin-b-cateninpathway in cancer cell invasion: a multiscale approach.Biophys J 2008, 95:155–165.

58. Ramis-Conde I, Chaplain MAJ, Anderson ARA,Drasdo D. Multi-scale modelling of cancer cell intrava-sation: the role of cadherins in metastasis. Phys Biol2009, 6:016008.

59. Stolarska MA, Kim Y, Othmer HG. Multi-scale modelsof cell and tissue dynamics. Phil Trans R Soc A 2009,367:3525–3553.

60. Schaller G, Meyer-Hermann M. Multicellular tumorspheroid in an off-lattice Voronoi-Delaunay cell model.Phys Rev 2005, 71:051910.

61. Beyer T, Meyer-Hermann M. Multiscale modeling ofcell mechanics and tissue organization. IEEE Eng MedBiol 2009, 28:38–45.

62. Meineke FA, Potten CS, Loeffler M. Cell migration andorganization in the intestinal crypt using a lattice-freemodel. Cell Prolif 2001, 34:253.

63. Rejniak KA. An immersed boundary framework formodelling the growth of individual cells: an applicationto the early tumour development. J Theor Biol 2007,247:186–204.

64. Dillon RH, Owen M, Painter K. A single-cell-based model of multicellular growth using theimmersed boundary method. AMS Contemp Math2008, 466:1–15.

65. Rejniak KA, Dillon RH. A single cell based model ofthe ductal tumour microarchitecture. Comput MathMethods Med 2007, 8:51–69.

66. Quaranta V, Rejniak KA, Gerlee P, Anderson ARA.Invasion emerges from cancer cell adaptation to com-petitive microenvironments: quantitative predictionsfrom multiscale mathematical models. Semin CancerBiol 2008, 18:338–348.

67. Rejniak KA. A single-cell approach in modeling thedynamics of tumor microregions. Math Biosci Eng2005, 2:643–655.

68. Rejniak KA, Anderson ARA. A computational studyof the development of epithelial acini. I. Sufficient con-ditions for the formation of a hollow structure. BullMath Biol 2008, 70:677–712.

Volume 3, January/February 2011 2010 John Wiley & Sons, Inc. 123

Focus Article www.wiley.com/wires/sysbio

69. Rejniak KA, Anderson ARA. A computational studyof the development of epithelial acini. II. Necessaryconditions for structure and lumen stability. Bull MathBiol 2008, 70:1450–1479.

70. Macklin P, McDougall S, Anderson ARA, ChaplainMAJ, Cristini V, Lowengrub J. Multiscale modellingand nonlinear simulation of vascular tumour growth.J Math Biol 2009, 58:765–798.

71. Szeto MD, Chakraborty G, Hadley J, Rockne R, MuziM, Alvord EC, Krohn KA, Spence AM, Swanson KR.Quantitative metrics of net proliferation and invasionlink biological aggressiveness assessed by MRI withhypoxia assessed by FMISO-PET in newly diagnosedglioblastomas. Cancer Res 2009, 69:4502–4509.

72. Stefanini MO, Wu FTH, Mac Gabhann F, Popel AS.A compartment model of VEGF distribution in blood,healthy and diseased tissues. BMC Syst Biol 2008, 2:77.

73. Wu FTH, Stefanini MO, Gabhann FM, Popel AS. Acompartment model of VEGF distribution in humansin the presence of soluble VEGF receptor-1 acting as aligand trap. PLoS ONE 2009, 4:e5108.

74. Fang JS, Gillies RD, Gatenby RA. Adaptation tohypoxia and acidosis in carcinogenesis and tumor pro-gression. Semin Cancer Biol 2008, 18:330–337.

75. Vincent TL, Gatenby RA. An evolutionary model forinitiation, promotion, and progression in carcinogene-sis. Int J Oncol 2008, 32:729–737.

76. Gatenby RA, Frieden BR. Application of informationtheory and extreme physical information to carcino-genesis. Cancer Res 2002, 62:3675–3684.

77. Gatenby RA. Application of competition theory totumour growth: implications for tumour biology andtreatment. Eur J Cancer 1996, 32A:722–726.

78. Wolkenhauer O, Auffray Ch, Baltrusch S, Bluthgen N,Byrne H, Cascante M, Ciliberto A, Dale T, Drasdo D,Fell D, et al. Systems biologists seek fuller integrationof systems biology approaches in new cancer researchprograms. Cancer Res 2010, 70:12–13.

79. Wolkenhauer O, Fell D, De Meyts P, Bluthgen N,Herzel H, Le Novere N, Hofer T, Schurrle K,van Leeuwen I. SysBioMed report: advancing systemsbiology for medical applications. IET Syst Biol 2009,3:131–136.

80. Hinow P, Gerlee P, McCawley LJ, Quaranta V,Ciobanu M, Wang SZ, Graham JM, Ayati BP, ClaridgeJ, Swanson KR, et al. A spatial model of tumor-hostinteraction: application of chemotherapy. Math BiosciEng 2009, 6:521–546.

81. Komarova NL, Sadovsky AV, Wan FYM. Selectivepressures for and against genetic instability in cancer:a time-dependent problem. J R Soc Interface 2008,5:105–121.

FURTHER READING

Reviews: Cancer GrowthAnderson ARA, Chaplain MAJ, Rejniak KA, eds. Single-Cell-Based Models in Biology and Medicine. Basel, Switzerland:Birkhauser-Verlag; 2007.

Araujo RP, McElwain DLS. A history of the study of solid tumour growth: the contribution of mathematical modeling. BullMath Biol 2004, 66:1039–1091.

Bellomo N, Chaplain M, de Angelis E, eds. Selected Topics in Cancer Modeling: Genesis, Evolution, Immune Competitionand Therapy. Basel, Switzerland: Birkhauser; 2008.

Byrne HM, Alarcon T, Owen MR, Webb SD, Maini PK. Modelling aspects of cancer dynamics: a review. Phil Trans R SocA 2006, 364:1563–1578.

Byrne H, Drasdo D. Individual-based and continuum models of growing cell populations: a comparison. J Math Biol 2009,58:657–687.

Byrne HM. Dissecting cancer through mathematics: from the cell to the animal model. Nat Rev Cancer 2010, 10:221–230.

Edelman LB, Eddy JA, Price ND. In silico models of cancer. WIREs Syst Biol Med 2010, 2:438–459.

Lowengrub JS, Frieboes HB, Jin F, Chuang Y-L, Li X, Macklin P, Wise SM, Cristini V. Nonlinear modelling of cancer:bridging the gap between cells and tumours. Nonlinearity 2010, 23:R1–R91.

Martins ML, Ferreira SC, Vilela MJ. Multiscale models fro the growth of avascular tumors. Phys Life Rev 2007, 4:128–156.

Preziosi L, ed. Cancer Modeling and Simulation. New York: Champan & Hall; 2003.

Rejniak KA, McCawley LJ. Current trends in mathematical modeling of tumor-microenvironment interactions: a survey oftools and applications. Exp Biol Med 2010, 235:411–423.

124 2010 John Wiley & Sons, Inc. Volume 3, January/February 2011

WIREs Systems Biology and Medicine Hybrid models of tumor growth

Sanga S, Frieboes HB, Zheng X, Gatenby R, Bearer EL, Cristini V. Predictive oncology: a review of multidisciplinary,multiscale in silico modeling linking phenotype, morphology and growth. Neuroimage 2007, 27:S120–S134.

Tracqui P. Biophysical models of tumour growth. Rep Prog Phys 2009, 72:056701.

Wang Z, Deisboeck TS. Computational modeling of brain tumors: discrete, continuum or hybrid? Sci Model Simul 2008,15:381–393.

Reviews: Integration of Biological and Mathematical ModelingHunt CA, Ropella GEP, Lam TN, Tang J, Kim SHJ, Engelberg JA, Sheikh-Bahaei S. At the biological modeling andsimulation frontier. Pharm Res 2009, 26:2369–2400.

Meier-Schellersheim M, Fraser IDC, Klauschen F. Multiscale modeling for biologists. WIREs Syst Biol Med 2009, 1:4–14.

Strand DW, Franco OE, Basanta D, Anderson ARA, Hayward SW. Perspectives on tissue interactions in development anddisease. Curr Mol Med 2010, 10:95–112.

Volume 3, January/February 2011 2010 John Wiley & Sons, Inc. 125